. Since 1955 _ Strojniški vestnik Journal of Mechanical ^^mp' Engineering no. 6 year 2019 volume 65 Strojniški vestnik - Journal of Mechanical Engineering (SV-JME) Aim and Scope The international journal publishes original and (mini)review articles covering the concepts of materials science, mechanics, kinematics, thermodynamics, energy and environment, mechatronics and robotics, fluid mechanics, tribology, cybernetics, industrial engineering and structural analysis. The journal follows new trends and progress proven practice in the mechanical engineering and also in the closely related sciences as are electrical, civil and process engineering, medicine, microbiology, ecology, agriculture, transport systems, aviation, and others, thus creating a unique forum for interdisciplinary or multidisciplinary dialogue. The international conferences selected papers are welcome for publishing as a special issue of SV-JME with invited co-editor(s). 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Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6 Contents Contents Strojniški vestnik - Journal of Mechanical Engineering volume 65, (2019), number 6 Ljubljana, June 2019 ISSN 0039-2480 Published monthly Papers Lin Zhou, Guoqiang Wang, Kangkang Sun, Xin Li: Trajectory Tracking Study of Track Vehicles Based on Model Predictive Control 329 Ozgur Poyraz, Melih Cemal Ku§han: Residual Stress-induced Distortions in Laser Powder Bed Additive Manufacturing of Nickel-based Superalloys 343 Yu Zhang, Hongzhi Yan, Zhiyong Wang, Qing Zhao: The Influence Rules of Cutter Parameters on the Contact Characteristics of Tooth Surfaces by Five Cut Processes and the Duplex Helical Method 351 Spasoje Trifkovic, Nebojsa Zdravkovic, Milomir Gasic, Mile Savkovic, Goran Markovic: Analysis of the Influence Parameters on the Support Structure Stiffness of Large Radial-Axial Bearings 366 Tien Dung Hoang, Nhu-Tung Nguyen, Buc Quy Tran, Van Thien Nguyen: Cutting Forces and Surface Roughness in Face-Milling of SKD61 Hard Steel 375 Endashaw T. Woldemariam, Hirpa G. Lemu: Numerical Simulation-Based Effect Characterization and Design Optimization of a Micro Cross-Flow Turbine 386 Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, 329-342 © 2019 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2019.5980 Original Scientific Paper Received for review: 2019-01-04 Received revised form: 2019-04-14 Accepted for publication: 2019-05-07 Trajectory Tracking Study of Track Vehicles Based on Model Predictive Control Lin Zhou - Guoqiang Wang* - Kangkang Sun - Xin Li Jilin University, School of Mechanical and Aerospace Engineering, China This paper proposes a model predictive control (MPC) algorithm for trajectory tracking of vehicles. Using MPC can reduce tracking errors and random disturbances in complex environments in time. According to the linear kinematics model of the vehicle, a kinematics trajectory tracking controller and an electromechanical coupling dynamics trajectory tracking controller are designed. The drive system of the electrically driven tracked vehicle is non-linear, and an electromagnetic system and mechanical system interact with each other. Taking the electromechanical coupling characteristics into consideration can ensure the matching of the electromechanical performance and the stability of the system during the trajectory tracking control. To verify the algorithm, kinematic simulations and dynamic simulations are performed. The simulation results show that the algorithm has good tracking ability. In addition, a set of test devices is designed to confirm the performance of the trajectory-tracking control algorithm in a real environment. Vision recognition is used to obtain vehicle deviation, and the Kalman filter is used to reduce signal interference. The result shows that the algorithm can meet trajectory tracking requirements. Keywords: model predictive control, trajectory tracking, tracked vehicle, electromechanical coupling Highlights • A kinematics model, a dynamic model, and an electromechanical coupling model of the tracked vehicle are established. • The control system in trajectory tracking is designed based on model predictive control. • The electromechanical coupling characteristic is added to the trajectory tracking control. • The simulation and vehicle experiment results further validate the control algorithm. 0 INTRODUCTION A track vehicle has good passing ability, low grounding pressure, and high adhesion to the ground. It can work in harsh environments, such as soft land and wetlands. Especially in dangerous or narrow situations, a crawler travel mechanism is often used, such as mine-sweeping robots, fire-fighting robots, and deep-sea mining robots [1] and [2]. Automatic navigation or trajectory-tracking technology for track vehicles is necessary for these vehicles. [3] and [4] The automatic movement of the vehicle mainly refers to automatic control the vehicle to reach the designated target, according to the predetermined reference trajectory and the state of the vehicle and the environmental information [5]. Furthermore, trajectory tracking requires the controlled object to reach the specified reference point within a given time [6]. Vehicle mathematical model analysis plays a key role in solving the trajectory tracking control problem of unmanned track vehicles. However, track vehicles are complex nonlinear systems, and it is difficult to establish accurate mathematical models. At the same time, high-precision control problems need to consider many uncertain factors, which can lead to a high complexity of control problems. Relevant scholars have used different control strategies to explore the trajectory tracking of unmanned track vehicles. Yeu et al. use a modified proportional-differential controller for the path tracking of soft road track vehicles [7]. Zou et al. use a backstepping method based on a modified proportional-integral-derivative (PID) computed-torque control for unmanned track vehicle tracking on hard ground [8]. Huang et al. use a PI path-tracking controller based on look-ahead point information for a differential-drive crawler-type robot [9]. Chen et al. [10] and Li et al. [11] use adaptive sliding mode control theory to control wheeled vehicles. Other scholars also have used other methods, such as fuzzy logic-based controllers [12], adaptive neural-fuzzy inference controllers [13] and slip-compensating control strategies [14]. In the methods commonly used for trajectory tracking control, PID control has poor control effect on nonlinear systems and structurally uncertain systems due to fixed control parameters. Optimal control and pure tracking control depend on specific mathematical models. Fuzzy control does not need to establish an accurate mathematical model, so it is suitable for nonlinear time-varying systems, but the choice of fuzzy rules is not systematic, and it is difficult to adjust online. Model predictive control (MPC) is based on the system using rolling optimization techniques to solve constrained real-time problems. Predictive control has clear basic ideas and strong development *Corr. Author's Address: Jilin University, School of Mechanical and Aerospace Engineering, China, linzhou17@mails.jlu.edu.cn 329 Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 329-342 capabilities. It can be combined with various system models, control theory and optimization algorithms to form various control algorithms. MPC can overcome tracking errors and random disturbances in complex environments, and eliminate the uncertainty caused by model mismatch and external disturbances in time; it is also suitable for systems in which mathematical models are inaccurate and have constraints. Consequently, MPC is widely used in the motion control of smart cars, drones and mobile robots [15] and [16]. Shen et al. performed trajectory tracking control of autonomous underwater vehicle using Lyapunov-based model predictive control [17]. Falcone et al. applied MPC to an active front steering system in an autonomous vehicle [18]. Based on the vehicle dynamics model, Lee et al. [19] and Kim et al. [20] applied the MPC method to laterally control a vehicle and achieved good control results. Researchers studied the trajectory control method of a tractor-trailer system based on MPC, and the tests proved that the control method is effective [21] and [22]. Therefore, applying the model predictive control method to the electric drive track vehicles can achieve good control effects, but at this stage there is less research in this direction. The electromechanical coupling system of the electrically driven track vehicle consists of an alternating current motor and a working machine. The electromagnetic system and the mechanical system interact with each other and form a complex nonlinear system. When the electromechanical performance is not suitable, the induction motor is likely to be blocked and cannot drive the crawler device. In severe cases, it may even burn out the motor [23]. Therefore, it is necessary to perform electromechanical coupling dynamics analysis when analysing and controlling a track vehicle. In the current research, many scholars have established kinematic and dynamic models of track vehicles and applied them to motion analysis [24] and [25]. However, there are few studies combining vehicle trajectory tracking control with electromechanical coupling dynamic analysis. This paper establishes the kinematic model, the dynamic model, and the electromechanical coupling dynamic model of track vehicles. According to the kinematics model, it designs a trajectory tracking algorithm based on the model predictive control. In the simulation environment, the kinematics model and the electromechanical coupling dynamics model are tested under two conditions to verify the correctness of the control algorithm. To verify the effectiveness of the algorithm under real conditions, physical prototype tests of the track vehicle based on monocular vision and Kalman filter under different initial conditions are performed. 1 ELECTROMECHANICAL COUPLING MODEL OF A TRACK VEHICLE 1.1 Kinematics Model When the track is steering in a plane without any slippage, as shown in Fig. 1, according to the triangular relationship: (1) R ± B R + B' Fig. 1. Track steering schematic diagram According to Eq. (1), the steering radius, the centre-of-mass velocity, and the steering angular velocity of the track can be obtained: R = B. vR + vl 2 VR - Vl\ m = = -c R B (2) (3) (4) To obtain the plane motion equation of the track vehicle, the ground coordinate system OXY is established, as shown in Fig. 2. When vr or v{ is consistent with the vehicle's heading direction, it takes a positive sign; otherwise, it is negative. According to Fig. 2, the differential equation of motion of the track vehicle is as follows: x = vc • cosp = - -• cosp VR + VL y = vc • sin^ = —R-— • cosp. (5) p = rnc = - B v Vc = 2 2 330 Zhou, L. - Wang, G. - Sun, K. - H, X. Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 329-342 Fig. 2. Plane motion analysis Take x = (x,y, z)T as the state vector and u = (vR, vL)T as the input vector. Eq. (5) can be rewritten as a state equation in matrix form: cos^ sin^ 1 B cos^ sin^ _ 1 -B (6) 1.2 Dynamic Model When the track vehicle is moving at low speed in the plane, the external forces include running resistance, steering resistance, and driving force. Since the centrifugal force is negligible at low speed, the analysis of the available force on the track is shown in Fig. 3. Fig. 3. Force analysis According to the differential equation of rigid body plane motion, the dynamic equation of the track vehicle can be written: Smi-c = Fqr + F - F - F B B B (7) JA = 2(Fqr -Fql) -~(F„ -Frl) -2(Fsr -F„) The driving force on both sides can be obtained with Eq. (7): Fr = F = SmvC + JC®C 2 B 8mVC JC®C 2 B -F +F, -Fr! +Fsl (8) The load torque of the motor shafts on both sides can be written: T = F — -1Lr 1 qr . T = F _ 1Ll 1 ql . (9) Introduce the motor's general drag equation: J> = Tem - Tl - f>. (10) Joint Eqs. (7) to (10): m r (8mvc JA . JWr = TemR--I —C + ^ + Frr +Fsr \- fmWR in v 2 B t- T r I SmVc J ®c , JAL = emL --^ + Frl +F1 \ - (11) in V 2 B Bring Eqs. (3) and (4) into Eq. (11) and calculate: r ®R = -tfo,®R - + tTemR - t ~ Frr in - t — F„ + uTemL - u — Frl - u—Fsl in in in (12) ®L = -Uf«PR - tf^L + uTemL - U~ Frr in r r r - u-F— + tTemL - t — Fri - t — Fd in in in The components of t and u in Eq. (12) are expressed as follows: ( 1 t = -2 1 u = — 2 1 1 J + - 2i2 J + 2jy B 2i2 (13) J+ 2i2 J+ 2Jcr2 B 2i2 Take x = (mR ,mL, x, y,qf as the state vector, U = (TemRi / r, Frr, Fsr, T'„J / r, Fh , Fs,) as the input. Joint Eqs. (6) and (12) can obtain the complete dynamics equation, and the writing matrix form is as follows: x = Ax + B,u, Trajectory Tracking Study of Track Vehicles Based on Model Predictive Control 393 (14) 331 L + 2 mr 1 1 mr Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 329-342 where A = -tf -uf -uf -tf r —cosö 2i r — cosö 2i r — sinö 2i r — sinö 2i r r B B 4 = - in t -t -tu -u -u u -u -u t -t -t (15) 1.3 Electromechanical Coupling Model To establish the electromechanical coupling model of the track vehicle, a basic dynamic equation of the motor should first be obtained. Take the current, the rotor angle, and the rotor mechanical angular velocity as state variables, and taking the applied voltage of the winding and the load torque on the shaft as the input vector, which are x = (i,rnm,d)T and v = (u,TL) . The equation for the induction motor in the ABC coordinate system is: d = - L-1 dt r+dL e 30 i + L da dt de dt = JpLr dLi -faa - TL. (16) 2J de ' ~ ' f . -— ■ J m J = Po^m Since the Eq. (16) is a differential equation with time-varying coefficients, the solution is complicated. If a certain degree of decoupling is achieved with coordinate changes, the equation of state can be simplified. The equation of the motor in the aß - dq system can be expressed as: - = - L-1 (R + G0)i + L-1u At V / dt dm dt d'Q_ dt . = p0iTGi -i^m -II J = Pomrn J J (17) In Eq. (17), both L 1 and G are constant arrays, which is easy to solve. Combining dynamic Eq. (14) of the track vehicle with Eq. (17), the equation for the electromechanical coupling of the track vehicle can be obtained. Take x = (isaR,ispR,irdR,irqR,mR,0R , isaL , isfiL , irdL,irqL,mL,dL,x,y,(p)T as the state vector and take U = (UsaR , UspR, UrdR , UrqR , Frr , Fsr , UsaL ,Us$L , UrdL, UrqL , Frl, Fsl) as the input vector. The induction motor rotor is always sitort-circmte^ so urdR = urqR = urdL = urqL = 0 . Then the track vehicle electromechanical coupling equation is as follows: x = Ax + Bv, (18) where A = -L 1 (R + G8r ) 04x1 04x1 04x6 tPoiRG -tf. 0 uPoiLG -uf. 0 01x4 po 0 01x6 04x6 -L1 (R + G6l ) 04x1 04x1 WR -uf. 0M 0 tp0iTLG 01x4 -tf. P0 0 0 015x3 . (19) r 2i 2c* 03x4 2 S* r Bi 03x1 03x4 r -Bi 03x1 B = L-1 r r -t— -t — in in rr -u — -u — in in rr -u — -u — in in L-1 rr -t— -t — in in (20) (21) 2 MODEL PREDICTIVE CONTROL ALGORITHM The discrete-time state-space model of a linear system can be expressed as: fx(k +1) = Ax(k) + Bu(k) [y (k) = Cx(k) • If the state x is measurable and the system is undisturbed, set the prediction time domain to P and the control time domain to M (where P >M). At the sampling instant k, the system control input is u(k). Let the control variable changes only in the future M steps k, k+1, ..., k+M-1, and remain unchanged outside the control time domain, then the future P steps k+1, k+2, ..., k+P-1 are as follow: X(k + j) = A x x(k) + B x U(k), (22) where: 0 332 Zhou, L. - Wang, G. - Sun, K. - Li, X. Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 329-342 ' x(k +1) " " u(k) ■ u(k +1) X(k + j)= x(k + M ) , U(k) = u(k + 2) x(k + VP) _ u(k + M -1) A, = A AM , B, = AP B AM-1B Am-2B A B AP-B B ■ (23) Z AB For the track vehicle kinematics model Eq. (6), it can be written as a general form: x = f (x, u). (24) Using linear systems can make it easy to check local stability and accelerate the simulation. Since the kinematic model is simple in form, and the coefficient matrix elements are constants, the kinematic model should be linearized. Assuming that the target trajectory is the running trajectory of the reference vehicle, and the state quantity and the control amount of the reference vehicle at any time are given, then the target trajectory can be tracked by the deviation between the controlled vehicle and the reference vehicle. Every point on the target trajectory satisfies Eq. (6): X, = f (xr, u,) (25) In Eq. (25), xr = (xr, yr, zr )T is the reference state variable, and ur = (vrr, vlr )T is the reference manipulated variable. Expanding the above equation with a Taylor series at the reference track point and only retaining one degree term: df (x, u) x r = f (xr, ur )- df (x, u) dx dx (U - Ur ). (x - xr ) + (26) Shortened as: X r = f (xr, u r) + fx,r (x - xr) + fur (u - u r). (27) In Eq. (27), fx,r and fu,r are the Jacobian matrices of f relative to x and u, respectively. Subtracting Eq. (24) and Eq. (27), we obtain: x = A(t )x + B(t )ii. (28) In Eq. (28), x = x - xr and fi = u - ur are deviation vectors; A(t) = fxr , B(i) = fu,r. The above Eq. (28) is a continuous time system, which needs to be discretized to obtain the general predictive control form as Eq. (21). Discretize it with the forward difference: x (k +1) = A(k )x (k) + B(k )fi (k), (29) where, A(k) = I + TA(t) = 1 0 - 2(vR + vl )sin^'T 0 0 0 1 - 2(vr + vL )cosv-T (30) B(k) = TB(t) = T cos^ cos^ 2 2 sin^ sin^ 2 2 1 -1 B -B (31) Predictive control needs to solve an optimization problem in each sampling time, so a performance index function needs to be set in advance. By controlling the amount of optimization, we can obtain the maximum or minimum. The performance index function usually uses a quadratic function, and the normal control objective is to make the output predicted value as close as possible to the target value. The trajectory tracking control involved in this paper belongs to the output optimal control problem, and its performance function can be expressed as: J = XT (k + j )QX(k + j) + UT (k) RU(k). (32) State-weighting matrix Q is a diagonal matrix, Q = diag(qpq2,...,qp). Each element qt represents the importance of tracking errors at different times; control weighting matrix R is also generally taken as a diagonal matrix. It achieves a certain degree of suppression of the fluctuation of the control variable and improves the stability of the system to some extent. Since the performance index function is a quadratic function, it can be represented as a quadratic programming problem to solve. Write Eq. (32) as the standard form of quadratic programming: J = - Ur (k) H (k )U(k) + f r (k )U(k) + d (k), s.t. Umin(k) < U(k) < Umax(k), (33) =0 1 Trajectory Tracking Study of Track Vehicles Based on Model Predictive Control 333 Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 329-342 where: H (k) = 2(Bt (k )QB(k) + Rx), f (k) = 2Bt (k )QA(k )x (k|k), d (k) = x T (k|k )At (k )QA(k )x (k|k). (34) H is a Hessian matrix, symmetrical and positive definite, and describing the quadratic part of the objective function; vector f describes the linear part of the objective function; d is an unrelated item to U(k) and does not affect the optimization result. Therefore, the optimization control problem can be described as finding a control sequence U*(k), so that: U*(* )=argmin J. (35) U ( * ) After obtaining the optimal sequence, executing the first control amount: u(k) = (1,0,...,0)U*(k). (36) The optimization shown in Eq. (35) is an unconstrained optimization. The fluctuation of prediction output and the control input can be suppressed to some extent by the weighting matrix Q and R. However, the control variable cannot be precisely constrained. In the actual process, if the amplitude of the applied control signal or its increment too large, it can cause a significant impact on the system and even affect the stability of the system. Therefore, for Eq. (33), the boundary constraint of the control variable needs to be considered. Umn(k) = Umin - U, Umx(k) = Umax - U where Umm and Ur = [ur(k),...,ur(k + N-1)] control sequence. (37) and Umax are control variable constraints is the reference ^ Ending ^ Fig. 4. MPC trajectory tracking process Using the solution algorithm to solve Eq. (33) in each control cycle and obtain a control sequence U*(k)= {uk(k),uk(k +1),...,uk(k + N-1} . Applying the first control variable uk(k) of the control sequence to the system. Repeat this step to achieve track vehicle trajectory tracking. The process of MPC trajectory tracking control is shown in Fig. 4. 3 NUMERICAL SIMULATION To verify the performance of the control algorithm, the trajectory tracking ability simulation is carried out under the conditions of kinematics and dynamics. 3.1 Kinematic Trajectory Tracking Simulation To perform kinematics tracking simulation, this paper established a kinematics model of the vehicle in MATLAB and built a controller in Simulink. At present, the MPC controller is based on the present position and uses the model to simulate several kinds of outputs, and the optimizer finds the optimal one. After obtaining the optimal control sequence, the controller applies the first step to the vehicle. Then, it calculates the position at the next moment using Eq. (6) and uses it as the starting point for the new optimization. It repeats at this process until the control is completed. 3.1.1 Straight Linear Conditions When the track vehicle is in line tracking, the reference trajectory equation is set to: ' x(t) = 0.15t < y(t) = 1 . (38) angle between vc and the X axis, [-] b width of the track shoes, [m] l track grounding length, [m] m vehicle mass, [kg] Fqr, Fql driving forces on both sides, [N] Frr, Frl running resistance on both sides, [N] Fsr, Fsl steering resistance on both sides, [N] Jc moment of inertia, [kgm2] d rotational mass factor, [-] Jm total moment of inertia, [kgm2] m motor shaft speed, [r/s] Tem motor electromagnetic torque, [Nm] Tl motor load torque, [Nm] fm rotational drag coefficient, [kgm2/s] n transmission system efficiency, [-] J moment of inertia of the motor shaft, [kgm2] r radius of the drive wheel, [m] i transmission ratio, [-] L inductance matrix of the motor, [-] R resistance matrix of the motor, [-] p0 number of pole pairs, [-] G rotational inductance matrix, [-] i current vector, [A] u voltage vector, [V] x(k) state vector, [-] u(k) control input, [-] y(k) control output, [-] A state matrix, [-] B control matrix, [-] C output matrix, [-] 8 REFERENCES [1] Marquardt, J.G., Alvarez, J., Von Ellenrieder, K.D. 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D0l:10.5545/sv-jme.2019.6004 Original Scientific Paper Received for review: 2019-01-15 Received revised form: 2019-05-10 Accepted for publication: 2019-05-31 Residual Stress-induced Distortions in Laser Powder Bed Additive Manufacturing of Nickel-based Superalloys Ozgur Poyraz1* - Melih Cemal Ku§han2 1TEI, Tusa§ Engine Industries Inc., Turkey 2Eski§ehir Osmangazi University, Faculty of Engineering, Department of Mechanical Engineering, Turkey One of the significant process limits of laser powder bed additive manufacturing technologies is residual stresses and distortions induced by heating the fine metal powder to the melting point and sudden cooling to the initial temperature. While this phenomenon applies to most types of metal alloys, it becomes more important in nickel-based superalloys, which have low thermal conductivity and high-quality requirements for aerospace utilization. Together with the mechanical integrity and geometric problems in the final product, residual stress-induced distortions carry the potential risk of interrupting the job process above a certain distortion limit as a result of part-to-re-coater rubbing. Process simulations performed in a computer environment are of critical importance to predict and to prevent this risk. However, simulation models need to be verified through experiments, and they need to be cost-effective in terms of simulation times. This paper presents a comprehensive study to introduce a fast thermo-mechanical simulation model, together with experiments on nickel-based superalloy demonstrators. It also contributes to the scientific knowledge base with novel coupon design, manufacturing and evaluation methodology for distortion studies. Keywords: additive manufacturing, finite element analysis, laser powder bed fusion, thermo-mechanical simulations Highlights • A fast thermo-mechanical finite element model was introduced for the analysis of distortions and residual stresses in laser powder bed additive manufacturing. • Novel benchmark coupons were designed for the benchmarking of residual stress-induced distortions considering circumferential effects. • The thermo-mechanical finite element model was employed for the simulation of benchmark coupons. • Experiments were conducted for an Inconel 718 nickel-based superalloy using a laser powder bed fusion additive manufacturing system • Results were evaluated via a coordinate measuring machine equipped with a laser scanner head. • Successful convergence was achieved with the experimental results. 0 INTRODUCTION Laser powder bed fusion (LPBF) is a powder bed additive manufacturing (AM) process used for the production of three dimensional (3D) parts from a wide range of powder materials; it is known with different commercial names, such as selective laser melting (SLM), selective laser sintering (SLS) or direct metal laser sintering (DMLS) [1]. It expands its sectorial utilization by offering good mechanical properties, design freedom [2], easy transition from design to manufacturing, post-processing ability, and low material consumption [3]. Regardless of the various advantages offered by LPBF, remaining challenges such as surface quality [3], residual stresses, distortions [4], or the use of support structures prevent the wider acceptance of this process. LPBF AM uses a laser beam to selectively melt the metal powders by scanning cross sections on the surface of a powder bed layer by layer into an object that has a desired 3D shape based on a computer-aided design (CAD) model [5]. After each cross-section is scanned through a mirror system and solidifies with the help of blowing inert gas, the powder bed is lowered by one layer of thickness, a new layer of material is spread on top with a re-coater, and the process is repeated until the part is complete [4]. In this respect, heat transfer mechanisms include phase change, conduction, convection, and radiation (Fig. 1). As a result of the above-mentioned phenomenon, occurring thermal gradients cause cyclic thermal expansions, which exceed the elastic strain of the material and generate plastic strains, especially at high temperatures. These cumulated strains can generate internal stresses (also known as residual stresses) in the part [6] and cause distortions, which may lead failure by cracking [7] or layer delamination [8]. To eliminate the risk of cyclic thermal expansions and the resultant problems of residual stresses and distortions, several measures can be taken. In this regard, to check residual stresses, inspection techniques can be employed such as neutron diffractometry [9] or X-ray diffraction [10], and distortions with dimensional measurement systems [11] can be used. However, due to the rapid nature of additive manufacturing, these techniques are not *Corr. Author's Address: TEI, Tusa§ Engine Industries Inc., 26210 Eski§ehir, Turkey, ozgur.poyraz@tei.com.tr 343 Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, 343-350 available for in-process control but are commonly used for the final product. Most of the time, detected problems of residual stresses and distortions [12] are overcome on the final product using additional operations, such as stress relieving and/or machining adding extra costs on the product. Moreover, in certain cases, these additional operations may not be useful owing to high stresses or large distortions. On the other side, they are still insufficient to eliminate in-process risks, such as delamination, cracking, or job interruption, due to part-to-re-coater rubbing. Fig. 1. Heat transfer mechanisms occur as the result of the process [4] For these reasons, computer-based numerical simulation tools or methods are developed and used to predict residual stresses and distortions. Within the literature, studies on LPBF AM simulations mainly focus on numerical modelling to understand the process itself and the effect of process parameters on the quality of the produced parts. The state-of-the-art in this field includes different studies and applications on various problems. Accordingly, thermal models coupled with mechanical ones can be used to numerically simulate LPBF processes to understand the effect of different process parameters [13]. The results of these simulations may include the shape and size of the melt pool, temperature distribution, as well as residual stresses and deformations [14]. While this approach is useful for meso-scale process simulations, it may be computationally expensive for macro-scale ones [15]. To overcome this computational expense, several researchers have employed diverse approaches. In the beginning phases of additive manufacturing simulations, researchers focused on reducing the mesh count and thus the computation time by using the element birth-and-death technique provided by the general-purpose finite element method (FEM) software packages such as ABAQUS [16] and ANSYS [17]. However, due to the lack of a recordable time reduction in the simulation times, there was a tendency towards other methods in the following periods. The applied plastic strain method is one of the most commonly used methods, using highresolution models of the transient thermo-mechanical analysis, it calculates plastic strain tensor components with equivalent plastic strain and maps the obtained plastic strains on the complete part [18]. This study deals with this challenge by introducing a fast thermo-mechanical finite element model. Together with the proposed model, experimental studies were also conducted on complex parts made of nickel-based superalloys. In this regard, innovative circumferential coupon designs were employed to promote a comprehensive understanding of residual stress induced distortion phenomenon. Verification of the simulation model was accomplished using experimental values. The rest of the paper is organized as follows: In the next section, details of the thermo-mechanical model were presented among with the coupon geometry designs, necessary material properties and boundary conditions for finite element analysis. The following section explains the experimental procedure in terms of manufacturing and inspection steps. Findings are discussed in the subsequent section with the comparison of modelling results versus experiments. The last section summarizes the paper and emphasizes the potential work to be done. 1 METHODS 1.1 Coupon Designs Various types of coupon designs can be utilized to understand the residual stress-induced distortions as a result of LPBF AM. Due to the longer computation times, previous studies used simplistic coupon designs of prismatic shapes to have a constant cross-section along the part [19]. In a few studies, which employs complex geometry, it can be seen that unidirectional cantilever beams are used with arrayed combs to allow distortions [20]. Although these provide useful information about the distortions as a result of LPBF AM and process parameters, they are not sufficient to reflect different aspects such as the multi-directional differences of distortions due to re-coater movement and/or inert gas blowing. Multi-directional situations were also considered in other previous research, 344 Poyraz, 0. - Ku¡han, M.C. Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, 343-350 but they still lack the evaluation of asymmetry as a significant issue [21]. In the context of this research, innovative coupon geometries were designed with two configurations. These are circumferentially arrayed cantilever beams with 120° between each beam. The first configuration (C1) employs combs on all beams for investigating the symmetrical situation in terms of distortion results. The second configuration (C2) employs combs on two beams and one of the beams as a full solid without any comb. This is to see the effect of asymmetry on the geometry and also unbalanced material volumes on the parts to effect residual stress-induced distortions. The external dimensions of both coupons were kept equivalent for a fair benchmark, and these dimensions were also kept as low as possible for minimum material consumption. Each coupon has an outer diameter of 70 mm and each beam has outer dimensions of 35 mm x 10 mm x 7 mm. Combs have a thickness of 1 mm and a height of 4.5 mm. A 1 mm gap is present between each comb. Fig. 2 shows the coupons with external dimensions. Boeing 747, Boeing 767, and Airbus 300, is 39 %, and the total utilization rate of Inconel 718 in these nickel alloys is 57 %. In this study, EOS Nickel Alloy IN718 was considered for the part material, and all the necessary material properties (physical, thermal, mechanical) for modelling were calculated with JMatPro® software. According to the proposed methodology, which is explained in the following section, the fast thermo-mechanical model predicts the distortions in the solid phase and uses material properties below melting temperatures. The physical phenomena of the process were taken into account, and elasto-plastic behaviour was assumed for the material. Within the elastic region, linear elasticity calculations were conducted using the elastic modulus and Poisson ratio. For the plastic region, von Mises yield criteria were followed, and the temperature dependence of yield strength was adopted. Moreover, the bilinear hardening rule was included using the tangent modulus. Essential properties are given in Table 1 for three different temperature levels. The temperatures between this range and below melting point were extrapolated using polynomial curve fitting of second order. Table 1. Material properties of as-built IN718 Fig. 2. Circumferential coupons and dimensions 1.2 Materials and Properties Material property [°C] 20 400 600 Density [kg/m3] 8195 8100 8000 Thermal conductivity [W/(m-K)] 11.25 13.45 15.55 Specific heat capacity [J/(kgK)] 415 495 580 Thermal expansion x10-6 [1/K] 13.0 14.8 16.6 Elastic modulus [GPa] 172 156 143 Poisson ratio [-] 0.29 0.28 0.27 Yield strength [MPa] 707 512 260 Tangent modulus [MPa] 920 785 603 Among various LPBF processed superalloys, Inconel 718 was chosen for the material; it has major elements of nickel (52.5 %), chromium (19 %), iron (balance), and niobium (5.12 %). It possesses both oxidation resistance and high strength at elevated temperatures. Moreover, different forms of Inconel 718 are easily producible with conventional and non-conventional processes, such as machining, welding, and additive manufacturing [22]. Inconel 718, which is preferred in different sectors with its many advantages in terms of design and production, is used in the highest ratio for aircraft engines. An example of this is the PW4000 engine from Pratt & Whitney [22]. The total ratio of nickel alloys used in the example engine, which provides power to the commercial platforms such as 1.3 Modelling To overcome the challenge of computationally expensive LPBF AM modelling, a fast thermo-mechanical finite element model was introduced based on previous studies [13] and [15]. The model was developed using COMSOL Multiphysics FEM software as a result of a series of studies. In the scope of initial studies, a meso-scale model based on a moving heat source with Gaussian distribution was adopted just for thermal simulations [13] and for IN718 material was verified through microstructural investigations on the melt pool dimensions [15]. The convergence of less than 10 % was achieved with the meso-scale model (Fig. 3). 345 Residual Stress-induced Distortions in Laser Powder Bed Additive Manufacturing of Nickel-based Superalloys 407 Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, 343-350 .«.lit fKmn-?. Fig. 3. Results of previously developed meso-scale model [15] Later, comprehensive parametric analyses were conducted, and the model was accelerated to be used in macro-scale problems having real part geometries [15]. In this regard, the meso-scale model was utilized for various scanning strategies such as band, island, and spiral scanning. Average energy input per unit area and temperature distribution were extracted from the results of these studies. Equivalent energy input was applied to whole layers as a surface heat source simultaneously during the build process by multiplying the energy input per unit area value with the area value of each layer (Fig. 4). The time for the application of the energy input was calculated through dividing surface area value of each layer by the hatch distance multiplied with laser scanning speed (Fig. 4). Hacth distance x Scanning speed Fig. 4. Fast thermo-mechanical model and input time calculation Temperature rise due to high loading was reported to be between 5 °C to 30 °C in the literature [23]. Since there is low-level loading in AM due to low strain rates and the temperature levels in AM process are extremely high, such as over 1000 °C comparing to 5 °C to 30 °C, the effect of deformation on temperature change was neglected. A one-way or weakly coupled thermo-mechanical model was developed, and multiphysics simulations were conducted accordingly. In this type of modelling strategy, the temperature distribution as the result of energy input and heat transfer was mapped simultaneously and transiently to structural mechanics in the same FEM software. 1.4 Boundary Conditions The designed coupon geometries among with the calculated material properties were used for the modelling study, employing the developed fast thermo-mechanical model. For the thermal boundary conditions, the initial condition at time t = 0 was set to 20 °C, and the bottom surface temperature of the part geometry was set to 80 °C to simulate the actual condition of build platform pre-heating step. As indicated in the modelling section, equivalent energy input was applied to whole layers as a surface heat source and a convective heat flux condition with a heat transfer co-efficient of 0.0178 W/(m2 K) [24] was also applied to simulate the effect of argon flow inside the build chamber. The thermal insulation condition was applied for the side surfaces of the part geometry. The reason for this that the side faces of the part geometry are surrounded by powder material, and its thermal conductivity is extremely low [13] and can be neglected comparing it to solid material. A fixed constraint was assigned to bottom surfaces of the coupon parts. Layer addition was also taken into account with the moving mesh algorithm of the layer plane. For this purpose, the part geometry was modelled at full height, and the block envelope for the geometry was also generated. A predefined Boolean operation (subtract) of block envelope from part geometry was assigned, and the block envelope was transiently transformed with a step function to consider 0.04 mm layer thickness as a step movement interval and also the layer time as explained in the modelling section. In the final step of simulations, the fixed constraint was deactivated to reflect the real situation of part separation from the build platform. Mesh size was utilized as the result of mesh sensitivity analyses to allow proper heat dissipation through layer plane and through the side faces. In this 346 Poyraz, 0. - Ku¡han, M.C. Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, 343-350 regard, constant mesh distribution was maintained on the layer plane to facilitate a minimum of two mesh elements for related geometry width and a minimum of five mesh elements for geometry length. Mesh elements on side faces were maintained with a geometric sequence distribution function to re-mesh the part and update a fine mesh size beneath the layer plane and coarsen the elements close to the part bottom for each layer addition. With this aim, the layer plane surface of the part was meshed with fixed sized triangular elements, and the rest of the geometry was swept with tetrahedral elements. The size of the tetrahedral elements was limited to a minimum value of 0.25 mm and a maximum value of 5 mm. An element growth ratio of 1.25 was defined to regulate the mesh growth from the minimum to maximum value in each build stage (Fig. 5). The adaptive solver algorithm feature of the software package was used to transfer nodal history variables for each re-meshing step. For this case of a transient situation, the solution at a certain time interval (of a certain part layer) to the partial differential equation problem on the previous mesh was mapped to the new mesh of the following time interval step. This cycle continued automatically until the end of the simulations. EXAMPLE MESH AT HALF HEIGHT EXAMPLE MESH AT FULL HEIGHT Fig. 5. Mesh examples in different part heights 1.5 Computations The computational studies were conducted in a notebook computer with a central processing unit that had two physical cores and four logical cores with hyper-threading option by a clock speed of 2.40 GHz. Moreover, the computer featured 16 GB of random access memory to facilitate random storing of intermediate simulation data resulting because of the transient iterative solver. Benchmark studies were conducted by using the above-explained hardware. First, the full layer of the part geometry was simulated using the meso-scale model, and the computation time was 3 hours and 16 minutes. Later, the meso-scale model was used for energy input calibration on a reference unit geometry, ending in 21 minutes. Finally, the simulation for the whole part geometry was conducted in 1 hour 7 minutes. With the proposed fast modelling strategy, the overall computation time was the sum of the calibration, and the whole part simulation and the value was 1 hour 28 minutes. The whole part simulation with the meso-scale model was not conducted, since it could be seen from the one layer simulation that it would take hundreds of hours. From this benchmark, it is obvious that the introduced fast thermo-mechanical model represents considerable benefits in terms of computational expense. 2 EXPERIMENTAL 2.1 Part Manufacturing The designed coupons are manufactured in EOS M290 machine of EOS IN718 metal powder. To eliminate any collision or rubbing of coupon parts with the powder re-coater, one of the arms of the coupon was aligned with the X-axis and then rotated 5°. Laser scanning strategies were adjusted for a proper comparison between the modelling and experimental studies, considering the fact that simulation was conducted with an accelerated thermo-mechanical model that simultaneously applies equivalent energy input to whole layer surface. In this respect, parallel scanning was chosen as the scanning strategy, and the angles of the scanning paths were rotated by a certain degree in each layer so that the Table 2. Process parameters for IN718 Parameter Value Laser power 285 W Laser scanning speed 960 mm/s Hatch distance 0.11 mm Layer thickness 0.040 mm Stripe width 10.0 mm Stripe overlap 0.12 mm Hatching X & Y Strategy Alternating Angle 67° Residual Stress-induced Distortions in Laser Powder Bed Additive Manufacturing of Nickel-based Superalloys 347 Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, 343-350 heat distribution did not intensify anywhere through the building process. Essential LPBF processing parameters are given in Table 2. 2.2 Part Measurement Part measurements were conducted after the parts were removed from the building platform by cutting with an electric discharge machine. The distorted parts as a result of relieving residual stresses after removal from building platform were measured with a Nikon Altera coordinate measuring machine equipped with a non-contact laser scanner (Fig. 6). For a detailed evaluation, many points were obtained during measurement, and the resultant point cloud was exported in stereolithography (STL) file format. Fig. 6. Part measurement with a non-contact laser scanner 3 RESULTS AND DISCUSSIONS The results of the study were evaluated and discussed from two different perspectives. In the first perspective, modelling and experimental results were compared with each other to understand the convergence of the proposed modelling methodology. In the second perspective, experimental results were discussed for a better understanding of the process and the use of proposed coupon designs for residual stress induced distortion evaluation. Finite element analyses were performed for the proposed model following to assignment of part geometry, material properties and boundary conditions, and outputs were obtained as part deformation (Fig. 7). Deformed part geometry was also exported in STL file format compatible to measurement results. Both STL files (one from measurement results and one from modelling) were imported to single CAD software for comparison purposes, and both geometries were aligned from their top faces. The comparison was made using the height (Z value) references of edges at the end of each beam. Modelling and measurement results are presented in Table 3. In Table 3, the convergence ratio was calculated as the ratio of the measurement to the modelling difference divided by measurement value. Fig. 7. Example of the simulation results and reference edge numbers of coupons Table 3. Comparison of modelling and experimental results Reference edge Modelling [mm] Measurement [mm] Convergence ratio [%] C1-B1 8.02 8.34 3.8 C1-B2 8.07 8.31 2.8 C1-B3 7.99 8.27 3.3 C2-B1 7.55 8.06 6.3 C2-B2 8.01 8.38 4.4 C2-B3 8.09 8.41 3.9 As can be seen from Table 3, the convergence between the modelling and measurement results is better than 10 %, based on the overall height of the edges of both coupons. The success of the convergence was obtained as the result of the proposed modelling method, since the modelling method considers equivalent energy input similar to the real situation utilized by parallel scanning with alternating angles. The proposed modelling method can also be used for the processes utilizing island (chessboard) or spiral scanning strategies [3]. Nevertheless, convergence in fixed-angle parallel or zigzag scans is not expected 348 Poyraz, 0. - Ku¡han, M.C. Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, 343-350 to be as successful due to the remarkable thermal gradients between X and Y directions [19]. The experiments with the use of proposed coupon designs have also revealed noteworthy results. It can be observed from the results of coupon C1 that all deformation values in reference edges of B1, B2 and B3 are considerably close to each other, within a few microns. In contrast, it is evident from the results of coupon C2 that the distortion results of the reference edge B1 of the full beam is noticeably lower than B2 and B3, with 0.3 mm between each. According to the extracted deduction from this study, it can be said that the effect of variables such as gas blow or powder re-coating direction is low in terms of distortion values of the same geometries in different layout positions and/or angles. In contrast, the difference in distortion values of geometries with varying features is high. In light of this information, it is recommended to reinforce the specific geometrical features in cases in which the residual stress-induced distortion problems are experienced and cannot be solved by the different positioning of the part. 5 CONCLUSIONS One of the process limits of LPBF AM is residual stress induced distortions to expose mechanical integrity and geometric problems in the final product. Compared to other metallic materials, this problem becomes more important for nickel-based superalloys having low thermal conductivity and high quality requirements for aerospace utilization. The lack of in-process measurement techniques, difficulty in overcoming resultant problems and the risk of job failures show the importance of employing computer-based numerical simulation tools or methods to predict residual stresses and distortions. This study introduced a fast thermo-mechanical finite element model together with experiments conducted on complex parts made of nickel-based superalloys, in this case, Inconel 718. Unlike previous studies in the literature, innovative distortion coupon designs were also presented to facilitate a comprehensive understanding of circumferential effects during process. The introduced thermo-mechanical model provided a computationally feasible method and showed good convergence results, better than 10 % for the case of strategy to rotate scanning vectors with an alternating angle. Via the equivalent energy input principle, the proposed method can also be used for the processes utilizing island or spiral scanning strategies. The experimental results revealed that geometrical difference are mostly dominant in comparison to circumferential effects such as gas blow or powder re-coater on residual stresses and residual stress induced distortions. Future studies may include detailed measurement of residual stresses using x-ray diffraction techniques, research on other nickel-based superalloys, parametric simulations on representative part geometries and consideration of other scanning strategies, such fixed-angle parallel or zigzag scans, by improving the modelling method. 6 ACKNOWLEDGEMENTS The authors would like to thank TEI (Tusa§ Engine Industries Inc.) for supporting this research. 7 REFERENCES [1] Levy, G.N., Schindel, R., Kruth, J.P. (2003). Rapid manufacturing and rapid tooling with layer manufacturing (LM) technologies, state of the art and future perspective. CIRP Annals-Manufacturing Technology, vol. 52, no. 2, p. 589-609, DOI:10.1016/S0007-8506(07)60206-6. [2] Thompson, M.K., Moroni, G., Vaneker, T., Fadel, G., Campbell, R.I., Gibson, I., Bernard, A., Schulz, J., Graf, P., Ahija, B., Martina, F. (2016). Design for additive manufacturing: trends, opportunities, considerations, and constraints. 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D0l:10.5545/sv-jme.2018.5944 Original Scientific Paper Received for review: 2018-12-21 Received revised form: 2019-05-05 Accepted for publication: 2019-05-10 The Influence Rules of Cutter Parameters on the Contact Characteristics of Tooth Surfaces by Five Cut Processes and the Duplex Helical Method Yu Zhang1* - Hongzhi Yan2 - Zhiyong Wang1 - Qing Zhao1 1 Central South University of Forestry and Technology, Institute of Modern Mechanical Transmission Engineering Technology, China 2 Central South University, State Key Laboratory of High-Performance Complex Manufacturing, China The principles of cutter parameters, such as the cutter tilt angle, the blade angle, and the cutter radius, are investigated in the five-cut process and the duplex helical method. The duplex helical method is an advanced and primary manufacturing method of face-milled spiral-bevel and hypoid gears, so it is necessary and urgent to understand its generalized theory. First, the different working principles of the cutter tilt mechanism in the two processing methods are analysed in detail, and the principal idea of the generalized theory for the duplex helical method is fully explored. Next, the influences of the blade angle and the cutter radius on the tooth surface meshing characteristics in the two processing methods are compared and analysed with the numerical analysis method; the results show that the relationships between the cutter parameters and the meshing characteristics are more regular and that the tooth-bending strength is higher in the duplex helical method, which provides a theoretical basis for effectively obtaining the optimal machine settings of spiral-bevel and hypoid gears. Keywords: duplex helical method, spiral-bevel and hypoid gears, five-cut process, generalized theory, cutter parameter Highlights • The different working principles of the cutter tilt mechanism in the two processing methods are analysed. • The principal idea of the generalized theory for the duplex helical method is fully explored. • The influences of the blade angles and the cutter radius on the meshing characteristics in the two processing methods are compared and analysed. • The research work provides a theoretical basis for obtaining the optimal machine settings of spiral bevel and hypoid gears. • The bending strength of the duplex helical method is significantly higher than that of the five-cut process. 0 INTRODUCTION At present, the machining technology of face-milling, spiral-bevel and hypoid gears has two major methods. One method is the traditional face-milling method called the 'five-cut process' or the 'fixed setting process', which is still widely used in developing countries, such as China [1] to [3]. The other method is called the 'duplex helical method' or the 'completing process', which has many advantages, such as higher machining efficiency, lower production cost, higher tooth strength, and dry cutting. Therefore, the five-cut process is gradually being replaced by the duplex helical method, and the duplex helical method will become an advanced and primary manufacturing method for face-milling, spiral-bevel and hypoid gears in China. However, there are significant differences and correlations between the generalized theories of the two methods. The significant differences between the five-cut process and the duplex helical method are: (1) In the five-cut process, the convex and concave tooth surfaces of a pinion are separately cutting with two different head-cutters; the outside head-cutter processes the concave tooth surfaces, and the inside head-cutter cuts the convex tooth surfaces. The duplex helical method can cut both sides of the pinion or gear tooth slots completely, with a spread blade head-cutter, in one operation, from a solid blank. (2) There is a helical motion (does not exist in the five-cut process) in the duplex helical method to control bias on the pinion, which is a relative axial motion between the workpiece and the cutter, along the generating gear axis, during the generating roll. With either method, the cutter represents a tooth of an imaginary gear which rolls with the gear or the pinion being cut. Therefore, its shape and position determine the shape of the tooth surface, and the influences of its parameters on the tooth surface topography play an important role in revealing the generalized theories of the two methods. The generalized theory of spiral bevel and hypoid gears manufactured with the five-cut process have been comprehensively presented by several gear scientists [4] to [6]. Gleason's most representative gear scientists establish the spatial geometry of Gleason *Corr. Author's Address: Central South University of Forestry and Technology, Institute of Modern Mechanical Transmission Engineering Technology, Shaoshan South Road, No. 498, Changsha 410004, Hunan, China, zhyu@csuft.edu.cn 351 Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 351-365 spiral bevel and hypoid gears, analyse the calculation approach of the tooth surface curvature, and introduce the generalized theory of the five-cut process [7] to [9]. The generalized theory and calculation method for the five-cut process are developed by Litvin et al. [10], Fuentes et al. [11], and Litvin and Fuentes [12], which is different from Gleason's technology. To avoid the root profile interference and the tooth edge contact, the blade geometry consists of four sections (tip, toprem, profile, and flankrem), as presented by Fan [13]. The application of the parabolic and Top-Rem profile blade are proposed by Litvin et al.[14] and Fuentes et al. [15] to avoid or reduce the areas of severe contact stresses and increase the endurance of spiral bevel gears. An optimal tooth modification for spiral bevel gears is introduced into the pinion tooth-surface by using a head-cutter with the bicircular profile and the optimal diameter to improve the load distribution and reduce the maximum tooth contact pressure and transmission errors [16] and [17]. Ma et al. [18] investigate the influences of the cutter diameter on the meshing performance of spiral bevel gears, such as the contact area, the contact pressure, the bending stress, the torsional stiffness, and the transmission error, with the finite element analysis methods. Gleason invented the duplex helical method several decades earlier, but has not clearly published its generalized theory and only shows few formulas and calculating instructions [19]. To the best knowledge of the authors of this paper, little attention has been given to the generalized theory of the duplex helical method until now. The conversion between the specific machine-tool settings of a given hypoid generator and the neutral machine-tool settings of spiral bevel gears manufactured by the duplex helical method is investigated, and the parabolic blade profiles are applied to improve the tooth bearings described in [20]. A mathematical model of the general hypoid machine is proposed by Fong [21], this model includes several common modified motions, such as the modified generating roll ratio, the helical motion, and the cutter tilt, to simulate all face-milled spiral-bevel and hypoid gear-machining methods. A new methodology is presented to determine the basic machine settings of spiral bevel and hypoid gears manufactured by the duplex helical method in [22], and the effect of straight-lined and circular cutting-edge profiles on the meshing and contact of gear pair is analytically investigated to effectively improve meshing performance [23]. In summary, the differences and relations between the theories of the five-cut process and the duplex helical method, the primary idea and many other key scientific problems of the duplex helical method have not yet been solved. Therefore, on the basis of analysing the generalized theory of the five-cut process and the duplex helical method [22], this paper investigates the different working principles of the cutter tilt angle and the blade angle in the two methods and reveals the major idea of the duplex helical method. Furthermore, the influences of the cutter parameters on the contact characteristics of tooth surfaces in the two processing methods are studied, which provides a theoretical reference for the optimization of machining parameters. 1 DIFFERENT WORKING PRINCIPLES OF CUTTER TILT ANGLE In this paper, the following definitions are made: the convex side of the gear and the concave side of pinion are the drive sides; the other mating surfaces are the coast sides. In general, the meshing mode for all types of spiral-bevel and hypoid gears is local conjugate contact (or called 'point contact'). To receive the length crowning between the mating flanks of a bevel gearset, the length curvatures (the curvature is inverse to the radius) of the convex flanks have to be larger than the length curvature of the concave flanks, that is, a pair of tooth surfaces with meshing must have a curvature difference. In addition, the normal vector of the two mating flanks at the reference point should be the same, or the two mating flanks should have equal spiral angle and pressure angle at the reference point. The cutter point radius is a good measure for the tooth length curvature in the five-cut process. To obtain a suitable curvature for a pair of the mating tooth surfaces, an inside cutter point radius for the convex flank must be smaller than an outside cutter point radius for the concave flank. The blade angle in the five-cut process is very similar to the pressure angle of the tooth. If there are some small differences, then they are related to the basic setting adjustments which are done in order to fine-tune the rolling performance. With equal pressure angles, the inside blade (IB) radius at the blade reference point (not at the tip) has to be larger than the outside blade (OB) radius at the reference point, as shown in Fig. 1. In this case, the IB and OB for pinion obviously cannot be mounted on one spread blade head-cutter, which cannot be used in the duplex helical method. In the duplex helical method, the cutter physically requires a smaller inside point radius and a larger OB radius to be able to mount the IB and OB of the pinion on a head-cutter. Comparing the cutter radii in the two 352 Zhang, Y. - Yan, H. - Wang, Z. - Zhao, Q. Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 351-365 methods, it can be seen that this is actually equivalent to result in the correct tooth thickness and a very large amount of the length crowning on both sides of the pinion. When machining spiral bevel and hypoid gears, by changing the blade angle simultaneously with the cutter tilt, the large "natural" length crowning can be reduced to the desirable crowning. This is the primary idea of the generalized theory for the duplex helical method. That is, a suitable curvature radius (the curvature radius is the distance from the blade reference point to the cutter axis, perpendicular to the cutting edge) at the reference point can be changed by adjusting the blade angles and the cutter tilt angles, so that the proper length curvature can be obtained on both sides of the pinion. Fig. 1. Corresponding relationship among cutter radii of gear and pinion In the following, how the blade angle and the cutter tilt angle affect the length curvature in the five-cut process and the duplex helical method, respectively and how they meet the required meshing conditions will be analysed. In the five-cut process, the blade angles are very similar to the pressure angles of the gear tooth, especially when the gear is cut with the non-generated method (formate). The pressure angle of the gear at the reference point must be equal to the pressure angle of the mating pinion at the reference point to make the pair of tooth surfaces mesh correctly; the relative position of the gear and pinion cutters can be represented in Fig. 2 at the reference point. In the five-cut process, the gear cutter generally has the same IB and OB angles, or the OB angle is slightly larger than the IB angle, as shown in Fig. 2. However, in the five-cut process or the duplex helical method, the pinion cutter generally has a smaller OB angle and a larger IB angle. When the cutter tilt is used to process the pinion, the curvature radius of the pinion OB rn1 is increased by an amount Arn1 relative a) b) Fig. 2. Relative position of gear and pinion cutters in five-cut process; a) drive side and b) coast side to that of the gear IB Rn1 on the dive side, that is, rn1 = Rn1 + Arn1. The curvature radius of the pinion IB rn2 is reduced by an amount Arn2 relative to that of the gear OB Rn2 on the coast side, that is, rn2 = Rn2 + Arn2. The values of Arn1 and Arn2 are determined by the angle Aa caused by the tilt of the cutter. Therefore, the cutter tilt in the five-cut process meets precisely the condition that a pair of meshing flanks have a curvature difference. That is, at the reference point, the curvature radius of the concave side must be larger than that of the convex side. In addition, it should be noted that Fig. 2 is drawn under the same mean cutter radius of the gear and the pinion, and the length curvature crowning is performed in the five-cut process (it is equivalent to counter-crowning the cutter radius), which can further reduce the OB radius and increase the IB radius for the pinion, this will cause the curvature difference required for the correct engagement of the two mating flanks to increase further. The above situation is the reason why the OB radius is smaller than the IB radius of the pinion in the five-cut process. 2 The Influence Rules of Cutter Parameters on the Contact Characteristics of Tooth Surfaces by Five Cut Processes and the Duplex Helical Method 353 Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 351-365 a) n2 b) Fig. 3. Relative position of gear and pinion cutters in duplex helical method; a) drive side and b) coast side In the duplex helical method, the spread blade cutter with the smaller IB angle and the larger OB angle are generally chosen to generate the gear and the pinion, and the asymmetry degree between the IB and OB blade angles is larger (the so-called asymmetry degree is the extent to which the IB and OB angles deviate from the mean pressure angle). The reason for this is to control the tooth contact pattern length. A smaller OB angle and a larger IB angle will lengthen the contact pattern, a larger OB angle and a smaller IB angle will shorten the contact pattern. As shown in Fig. 3, when the cutter tilt is used to process the pinion, the curvature radius of the pinion OB rn1 is reduced by an amount Arn1 relative to that of the gear IB Rn1 on the dive side, that is, rn1 = Rn1 - Arn1. The curvature radius of the pinion IB rn2 is increased by an amount Arn2 relative to that of the gear OB Rn2 on the coast side, that is, rn2 = Rn2 + Arn2. This is equivalent to increasing the concave curvature and reducing the convex curvature for the pinon, and the purpose of effectively reducing the large "natural" length crowning on both sides of the pinion is achieved at the same time, which can satisfy the desirable curvature required for the correct engagement of a gearset. According to the above analysis, it can be seen that a desirable curvature radius for a pair of mating tooth surfaces can be obtained by the suitable tilt angles and blade angles, and then the different cutter radii required to meet the two processing methods are obtained, which is the essential difference between the five-cut process and the duplex helical method. 2 INFLUENCE RULES OF BLADE ANGLES ON MESHING PERFORMANCE The selection principle of the blade angles in the five-cut process, and the duplex helical method is significantly different, especially regarding the blade angles of the gear. The tooth contact characteristics are an important indicator of the gearset meshing performance. Therefore, a numerical example of the faced-milled hypoid gearset of the gear ratio 11*45 is considered to investigate the different influences of the blade angle and the cutter radius on meshing performance in the two processing methods. For comparison purposes, the basic design parameters of the hypoid gearset processed by the two methods are as identical as possible, and the HFT (one of the five-cut process, format gear and pinion finished with cutter tilt) and HFDH (one of the duplex helical method, format gear and pinion generated by the duplex helical method) method are used to perform the numerical example, the design parameters are given in Table 1. A computer program is developed to implement the formulation provided above. Table 1. Basic design parameters of hypoid gears Parameters Pinion Gear Number of teeth 11 45 Module [mm] - 6 Face width [mm] 44.35 40 Pinion offset [mm] 30 - Shaft angle [°] 90 - Mean pressure angle [°] 20 20 Mean spiral angle [°] 48 33.82 Hand of spiral left right Whole depth [mm] 11.89 11.73 Mean cone distance [mm] 125.33 120.98 Outer diameter [mm] 99.68 271.02 Face angle [°] 20.53 74.08 Pitch angle [°] 16.32 73.2 Root angle [°] 15.45 68.87 Based on keeping other machining parameters unchanged (for example, the cutter radius is 114.3 mm), the OB and IB angles of the pinion and gear cutter are changed to investigate the influences of the blade angle on meshing performance in the five-cut process. Fig. 4 shows some of the results of tooth 354 Zhang, Y. - Yan, H. - Wang, Z. - Zhao, Q. Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 351-365 Blade angle Contact bearing Function of transmission errors Ease-off topography [urn] Gear OB:13° Gear IB:27° Pinion OB:15° Pinion IB:25° Drive side Coast side Drive side Gear OB:16° Gear IB:24° Pinion OB:15° Pinion IB:25° Coast side Gear OB:22° Gear IB:18° Pinion OB:15' Pinion IB:25° Drive side Gear OB:25° Gear IB:15° Pinion OB:15° Pinion IB:25° Coast side Fig. 4. TCA results when changing gear blade angles for HFT method The Influence Rules of Cutter Parameters on the Contact Characteristics of Tooth Surfaces by Five Cut Processes and the Duplex Helical Method 355 Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 351-365 Blade angle Contact bearing Function of transmission errors Ease-off topography [urn] Gear rotation angle ] Drive side Gear 0B:20o Gear IB:20° Drive side Gear 0B:20° Gear IB:20° Pinion 0B:15° Pinion IB:25° Coast side Drive side Gear 0B:20° Gear IB:20° Pinion 0B:17° Pinion IB:23° Coast side Fig. 5. TCA results when changing pinion blade angles for HFT method surface contact analysis (TCA) when the pinion blade angles remain unchanged, and the gear blade angles change; Fig. 5 shows some of the results of the TCA when the gear blade angles remain unchanged, and the pinion blade angles change. Comparison of results of the TCA in Figs. 4 and 5 have confirmed the following: 356 (1) As the asymmetry degree of the gear blade angles increases, the length of the contact bearings and the inclination angle of the contact path become larger. The difference of the contact bearing length between the drive side and the coast side becomes significantly larger. (2) It can be seen from the comparison of different ease-off topographies that the amount of the tooth Zhang, Y. - Yan, H. - Wang, Z. - Zhao, Q. Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 351-365 surface mismatch increases with the growth of the asymmetry degree for the blade angles. (3) From the comparison of different transmission error functions, it can be shown that the maximum levels of the transmission error function gradually become larger with the growing of the asymmetry degree for the blade angles, and the maximum levels of the drive side are significantly larger than that of the coast side. As shown in Figs. 6 and 7, the relationship between the blade angles and the size of the contact bearing is further analysed, it is known that the change of the major axis radius of the contact ellipse has no apparent regularity and the variation tendency of the drive side, and the coast side is different, while the minor axis radius of the contact ellipse has almost no change with the variation of the OB angles for the gear and the unchanged pinion blade angles (OB: 15° and IB: 25°). When the gear blade angles (OB: 20° and IB: 20°) do not change, the major axis radius for the drive side increases, and the coast side decreases substantially with the OB angle of the pinion increasing gradually, while the minor axis radius of the contact ellipse remains approximately unchanged. The influences of the blade angle on meshing performance in the duplex helical method are investigated with the same analysis method mentioned above, Figs. 8 and 9 show some of the TCA results when the gear and pinion blade angles change respectively, and other head-cutter parameters remain unchanged. Figs. 10 and 11 show the relationship between the contact bearing size and the blade angles. The analysis results obtained are as follows: (1) Whether with the increase of the OB angles of the gear or the pinion, the length of the contact bearing becomes shorter, which is completely different from the five-cut process and very regular. The slopes of the curves between the drive side and the coast side change only when the OB and IB angles of the gear are 20° (mean pressure angle). (2) Similar to the five-cut process, the minor axis radius of the contact bearing changes very little, and the minor axis radius of the drive side is larger than that of the coast side. (3) Compared with the five-cut process, the relationships between the transmission error function, the ease-off topography and the blade angle are more regular. That is, as the OB angle of the gear increases, the maximum level of the transmission error function and the amount of the tooth surface mismatch increase gradually when aj b) Fig. 6. Relation between gear blade angles and size of contact ellipse for HFT method; aj major axis radius and bj minor axis radius a) bj Fig. 7. Relation between pinion blade angles and size of contact ellipse for HFT method; aj major axis radius and bj minor axis radius The Influence Rules of Cutter Parameters on the Contact Characteristics of Tooth Surfaces by Five Cut Processes and the Duplex Helical Method 357 Strajniski vestnik - Journal of Mechanical Engineering 65(2019)6, 351-365 Blade angle Contact bearing Function of transmission errors Ease-off topography [urn] Gear OB:13° Gear IB:27° Pinion OB:15° Pinion IB:25° Drive side Coast side Gear OB:16° Gear IB:24° Pinion OB:15° Pinion IB:25° Drive side Coast side Gear OB:19° Gear IB:21° Pinion OB:15° Pinion IB:25° Drive side Coast side Gear OB:22° Gear IB:18° Pinion OB:15° Pinion IB:25° Drive side Coast side Gear OB:25° Gear IB:15° Pinion OB:15° Pinion IB:25° Drive side Coast side 358 Fig. 8. TCA results when changing gear blade angles for HFDH method Zhang, Y. - Yan, H. - Wang, Z. - Zhao, Q. Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 351-365 Blade angle Contact bearing Function of transmission errors Ease-off topography [urn] Gear 0B:20o Gear IB:20° Pinion 0B:13° Pinion IB:27° Drive side Coast side Gear 0B:20° Gear IB:20° Pinion 0B:15° Pinion IB:25° Drive side Coast side Gear 0B:20° Gear IB:20° Pinion 0B:17° Pinion IB:23° Drive side Coast side Gear 0B:20° Gear IB:20° Pinion 0B:19° Pinion IB:21° Drive side Coast side Fig. 9. TCA results when changing pinion blade angles for HFDH method the blade angles of the pinion are constant. The variation trends are similar as above with the increase of the OB angle of the pinion when the blade angles of the gear are unchanged. Therefore, unlike the five-cut process, the blade angle is an important control parameter for adjusting the size of the contact bearing in the duplex helical method. In general, increasing the asymmetry degree of the IB and OB angles can effectively increase the area of the contact bearing. This is also the reason that the gear and the pinion often can be manufactured by one head-cutter with the same blade angles (for 359 The Influence Rules of Cutter Parameters on the Contact Characteristics of Tooth Surfaces by Five Cut Processes and the Duplex Helical Method 421 Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 351-365 a) Fig. 10. b) Relation between gear blade angles and size of contact ellipse for HFDH method; a) major axis radius and b) minor axis radius a) b) Fig. 11. Relation between pinion blade angles and size of contact ellipse for HFDH method; a) major axis radius and b) minor axis radius example, the OB and IB angle can be 15° and 25° in this numerical example, respectively) and the mean cutter radius. 3 INFLUENCE RULES OF CUTTER RADIUS ON PERFORMANCE The relationship between the cutter radius and the meshing performance is studied by changing the cutter radius under keeping the blade angles unchanged (Gear OB:20° and IB:20°, Pinion OB:15° and IB:25°) in the two processing methods, respectively. Figs. 12 and 13 show some of the TCA results when the mean cutter radius changes and other head-cutter parameters remain unchanged, Figs. 14 and 15 show the relationship between the contact bearing size and the cutter radius for the five-cut process and the duplex helical method, respectively. The analysis results obtained are as follows: (1) In the five-cut process, the size, position and direction of the contact bearing and the amount of the tooth surface mismatch have little influence on the cutter radius; only when the cutter radius is less than 114.3 mm does the major axis radius of the contact ellipse become longer with the increase of the cutter radius for the drive side; the coast side is just the opposite, but the minor axis 360 radius of the contact ellipse always changes very little on both surfaces. (2) In the duplex helical method, the size of the contact bearing increases with the increase of the cutter radius all the time, but the change rates of the drive side and the coast side are different. The amount of the tooth surface mismatch decreases with the increase of the cutter radius. (3) The influence rules between the maximum level of the transmission error function and the cutter radius are the same in the two processing methods, that is, the maximum level decreases with the increase of the cutter radius, while the coast side is exactly the opposite. The relationship between the cutter radius and the tooth bending strength is investigated, as shown in Fig. 16; the data in the figure are obtained according to the calculation method of ISO 10300 [24]; the comparison and analysis of the tooth bending strength evolution for the two processing methods are as follows: (1) Except for a few cutter radii, in general, the bending strength of the gear and the pinion decreases with the increase of the cutter radius. (2) For the gear, the bending strength in the duplex helical method is slightly higher than that in the five-cut process, but the difference between the two methods is very small, which is because the Zhang, Y. - Yan, H. - Wang, Z. - Zhao, Q. Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 351-365 Cutter radius [mm] Contact bearing Function of transmission errors Ease-off topography [urn] 95.25 Drive side Coast side 114.3 Drive side Coast side 133.35 Drive side Coast side 160 Drive side Coast side 177.8 Drive side Coast side Fig. 12. TCA results when changing cutter radius for HFT method The Influence Rules of Cutter Parameters on the Contact Characteristics of Tooth Surfaces by Five Cut Processes and the Duplex Helical Method 361 Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 351-365 Cutter radius [mm] Contact bearing Function of transmission errors Ease-off topography [urn] 95.25 Drive side Coast side 114.3 Drive side Coast side 133.35 Drive side Coast side 160 Drive side Coast side 177. Drive side 362 Fig. 13. TCA results when changing cutter radius for HFDF method Zhang, Y. - Yan, H. - Wang, Z. - Zhao, Q. Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 351-365 aj Fig. 14. Relation between cutter radius and size of contact ellipse for HFT method; a) major axis radius and b) minor axis radius. aj b) Fig. 15. Relation between cutter radius and size of contact ellipse for HFDH method; a major axis radius and bj minor axis radius. gears are all processed by the format in the two processing methods. (3) For the pinion, the bending strength of the duplex helical method is significantly higher than that of the five-cut process, which is because the slot width has a constant width along lengthwise when the pinion is manufactured with the duplex helical method; the characteristic of the equal slot width ensures that the maximum point width and the maximum cutter edge radius can be used during machining. Therefore, not only the tooth strength can be improved, but also the cutter life can be extended. In addition, both sides of the pinion and gear tooth slots are cut with a single cutter in the duplex helical method; the connection smoothness of the tooth root fillet and the tooth bottom surface is much better than that cut by the five-cut process in actual manufacturing, which also improves the tooth strength of the pinion machined with the duplex helical method. 4 CONCLUSIONS The following conclusions can be drawn from the performed investigation: (1) The major idea of the generalized theory for the duplex helical method is revealed by analysing the different working principles of the cutter tilt mechanism in the two processing methods, that is, by changing the OB and IB angles of a spread blade head-cutter simultaneously with the cutter 363 The Influence Rules of Cutter Parameters on the Contact Characteristics of Tooth Surfaces by Five Cut Processes and the Duplex Helical Method 425 Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 351-365 tilt, the large "natural" length crowning can be reduced, down to the desirable crowning. (2) Compared with the five-cut process, the influence of the blade angle and the cutter radius on the tooth surface contact characteristics is more regular in the duplex helical method. Therefore, they are often used as important parameters to optimize the meshing performance of spiral bevel and hypoid gears. (3) In the case of the same design parameters and cutter parameters, the gearset processed by the duplex helical method has higher tooth-bending strength than that processed by the five-cut process; the difference can be up to 2.8 %. (4) The relationships between the cutter parameters and the contact characteristics of the tooth surface by the five-cut process and the duplex helical method are established and analysed; the research results of this paper provide a theoretical basis for effectively obtaining the optimal machine settings of spiral bevel and hypoid gears. 5 ACKNOWLEDGEMENTS The authors express their deep gratitude to the National Natural Science Foundation of China (NSFC) (Grant No. 51805555), Hunan Provincial Natural Science Foundation of China (Grant No. 2018JJ2679), Introducing Talent Startup Fund of Central South University of Forestry and Technology of China (Grant No.2016YJ006) for the financial support of the research. 6 REFERENCES [1] Hu, Z.H., Ding, H., Peng, S.D., Tang, Y., Tang, J.Y. (2019). Numerical determination to loaded tooth contact performances in consideration of misalignment for the spiral bevel gears. International Journal of Mechanical Sciences, vol. 151, p. 343-355, DOI:10.1016/j.ijmecsci.2018.11.014. [2] Peng, S.D., Ding, H., Zhang, G., Tang, J.Y., Tang, Y. (2019). New determination to loaded transmission error of the spiral bevel gear considering multiple elastic deformation evaluations under different bearing supports. Mechanism and Machine Theory, vol. 137, p. 37-52, DOI:10.1016/j. mechmachtheory.2019.03.013. [3] Zheng, F.Y., Hua, L., Chen, D.F., and Han, X.H. (2016). Generation of noncircular spiral bevel gears by face-milling method. Journal of Manufacturing Science and Engineering, vol. 138, no.8, p. 081013, DOI:10.1115/1.4033045. [4] Simon, V.V. (2016). Micro tooth surface topography of face-milled hypoid gears. Mechanism and Machine Theory, vol. 104, p. 370-381, DOI:10.1016/j.mechmachtheory.2016.05.012. [5] Wang, J., Kong, L., Liu, B., Hu, X., Yu, X., Kong, W. (2014). The Mathematical Model of Spiral Bevel Gears - A Review. 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Design, manufacture, and evaluation of prototypes of low-noise high-endurance spiral bevel gear drives. Proceedings of ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, p. 585-595, DOI:10.1115/DETC2005-84013. [16] Simon, V. (2009). Head-cutter for optimal tooth modifications in spiral bevel gears. Mechanism and Machine Theory, vol. 44, no. 7, p. 1420-1435, DOI:10.1016/j. mechmachtheory.2008.11.007. [17] Simon, V.V. (2011). Optimal modifications of gear tooth surfaces. Gear Technology, no. March/April, p. 62-72. [18] Ma, Z.H., Han, X.H., Hua, L., Xiong, X.S., Zheng, F.Y. (2016). Influence of cutter diameter on meshing performance in spiral bevel gears. Journal of Central South University, vol. 23, no. 1, p. 102-111, DOI:10.1007/s11771-016-3053-9. [19] Gleason Works (1971). Calculation instructions generated hypoid gears duplex helical method. The Gleason Works, New York. [20] Gonzalez-Perez, I., Fuentes, A., Hayasaka, K. (2011). Computerized design and tooth contact analysis of spiral bevel gears generated by the duplex helical method. ASME International Design Engineering Technical Conferences and 364 Zhang, Y. - Yan, H. - Wang, Z. - Zhao, Q. Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 351-365 Computers and Information in Engineering Conference, p. 149-158, D0l:10.1115/DETC2011-47108. [21] Fong, Z.H. (2000). Mathematical model of universal hypoid generator with supplemental kinematic flank correction motions. ASME Journal of Mechanical Design, vol. 122, no. 1, p. 136-142, D0I:10.1115/1.533552. [22] Zhang, Y., Yan, H.Z., Tao, Z. (2015). Computerised design and simulation of meshing and contact of formate hypoid gears generated with a duplex helical method. Strojniski vestnik -Journal of Mechanical Engineering, vol. 61, no. 9, p. 523-532, D0I:10.5545/sv-jme.2015.2627. [23] Zhang, Y., Yan, H.Z., Zeng, T., Zeng, Y.Y. (2016). Tooth surface geometry optimization of spiral bevel and hypoid gears generated by duplex helical method with circular profile blade. Journal of Central South University, vol. 23, no. 3, p. 544-554, DOI:10.1007/s11771-016-3101-5. [24] ISO 10300-3:2014(E) (2014). Calculation of Load Capacity of Bevel Gears - Part 3: Calculation of Tooth Root Strength. International Organization of Standardization, Geneva, DOI:10.3403/02362607. The Influence Rules of Cutter Parameters on the Contact Characteristics of Tooth Surfaces by Five Cut Processes and the Duplex Helical Method 365 Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, 366-374 © 2019 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2019.6006 Original Scientific Paper Received for review: 2019-01-15 Received revised form: 2019-04-13 Accepted for publication: 2019-05-16 Analysis of the Influence Parameters on the Support Structure Stiffness of Large Radial-Axial Bearings Spasoje Trifkovic1 - Nebojsa Zdravkovic2 - Milomir Gasic2 - Mile Savkovic2 - Goran Markovic2* 1 University of East Sarajevo, Faculty of Mechanical Engineering, Bosnia and Herzegovina 2 University of Kragujevac, Faculty of Mechanical and Civil Engineering in Kraljevo, Serbia In certain types of crane and earth-moving machinery, such as portal cranes, loaders and excavators, the transfer of loads to crane tracks or the terrain is accomplished by means of undercarriage frames composed of box-like girders. The hypothesis that all four supports of the undercarriage frame do not lie in the horizontal plane is the basis for the formation of a calculation model. This paper analyses the influence of geometric parameters of box-like girders on the magnitude of additional forces at the supports of those frames when one of the supports is raised or lowered relative to the horizontal plane for the size A. Theoretical dependences between moments of inertia and stiffness under bending and torsion of those girders were thus established. Obtained relations leads to more concise forms of expressions for influential coefficients in Maxwell-Mohr integrals and simplifies optimization methods in the design of support structures. With experimental verification of the results, the influences of geometric parameters on the stiffness of the support structure are confirmed, and preconditions are created for further analysis of the connection made by large diameter bearings. Keywords: support structure, large diameter bearing, box-like girders, geometrical parameters, stiffness, experimental verification Highlights • A calculation method was created for determination of additional forces in the undercarriage frame supports in relation to its geometric parameters and initiative deflection in one support. • The dependence between the ratio of moments of inertia for bending and torsion and the height-width ratio of the box-like cross-section was established. • Dependences were confirmed using the measurement results on the models of undercarriage frames of different geometrical values. • The obtained results lead, to a great extent, to the simplification of optimization methods in the design of carrying structures with box-like girders, and they are the precondition for further analysis of proper functioning of the connection realized by large diameter bearings. 0 INTRODUCTION The connections between the undercarriage and upper structure in certain types of cranes and earth-moving machines are accomplished by large diameter bearings, considering the influence of various operational and structural requirements. The purpose of analysing the functioning of these connections in real conditions is to provide an adequate distribution of loads, as well as the reliable and long-lasting operation of large bearings. The main research topics about large slewing bearings are related to empirical investigations and computational analyses of the failure mechanisms and the determination of internal contact load distributions and load capacity [1]. Generally, authors use both the analytical and the numerical approach for solving these problems. The detailed review of the former analytical approaches is presented in [2] and [3], where both articles offer the calculation models for determination of the contact angle and the carrying capacity of a four contact-point ball bearing. Modern approaches for computing the load capacity of slewing bearings are based on the finite element method (FEM). These computational methods ([4] and [5]) include the most influential parameters, such as the raceway/ring deformations, non-parallel ring displacements, and bearing clearances. In general, it has been shown that all of these parameters have a significant role when determining the load capacity of large slewing bearings. Olave et al. [6] used two different ways for obtaining the force distribution in four contact-point slewing bearings (FEM analysis and new calculation procedure considering the effect of the structure's elasticity). This analysis shows that the flexibility of the structures must be taken into account during the calculation of load distribution. Authors in [7] outline a procedure for the determination of the interferences between balls and raceways in four contact-point slewing bearings due to the manufacturing errors. Therefore, an inadequate hardened raceway depth can cause raceway failure. In contrast, an excessively large hardened raceway depth can increase the overall vibration and production cost. 366 *Corr. Author's Address: University of Kragujevac, Faculty of Mechanical and Civil Engineering in Kraljevo, Serbia, markovic.g@mfkv.kg.ac.rs Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 366-374 Authors in [8] analysed a three-row roller slewing bearing with a hardened raceway, by using a nonlinear spring instead of a solid roller, to quickly obtain the maximum contact load. Another direction of research leads to the analysis of influences of the undercarriage frame on the slewing bearing operation. Many researchers have found the stiffness of the supporting structure to be a crucial constructional problem of large slewing mechanisms ([9] and [10]). The supporting structure and the bolted connections cannot be ignored when the load distribution and carrying capacity of a slewing bearing are analysed. The effects of supporting structure, bolts number and preload, ball-race contact truncation and bolt-hole backlash on the carrying capacity of the slewing bearing are analysed in [11]. Results show that the fatigue life and carrying capacity of the slewing bearing can be enhanced by appropriately decreasing the supporting structure stiffness. Duval et al. [12] proposed fatigue analysis, taking into account the complex multiaxial stress state and the gradient of material properties, due to the surface treatment of the tracks (induction hardened parts). A method for the fatigue testing of the raceway by using a small sample is presented in [13]. Recent research [14] is directed to additional factors that influence the position of the resultant force exerted by the superstructure on the undercarriage (large excavation or loading forces, the mass of the transported material and ground inclination). Smaller deviations of the centre of the gravity accelerate the wear of the bearing raceway and cause overloading of the bolts that connect the bearing to the supporting elements. For that purpose, the experimental determination of the centre of the gravity of opencast mining machines is presented in [15]. There are not many publications describing the influences of geometrical parameters of the large diameter bearing support structure on its stiffness. Namely, the clearance between one of the undercarriage frame supports and the crane track or the terrain may occur. Hence, the problem of missing contact can appear due to irregularities on the base or, more rarely, errors that arise during manufacturing. As a consequence, there is a redistribution of vertical forces at the supports and the deformation of the carrying structure during exploitation. In this paper, attention is directed to the creation of a calculation model, used to define theoretical dependences between the geometrical parameters of box-like girders and the magnitude of additional forces at the undercarriage frame supports. Specifically, the magnitude of additional forces directly influences the functioning of large diameter bearings. The research of those relations was carried out to define such stiffness that the deformation of the support surface of the bearing would not exceed recommended values. Along with the experimental verification of the obtained theoretical results, the preconditions for significant simplification of some optimization methods [16] in the design of carrying structures with box-like girders were established. After the introductory notes and the overview of previous investigations, the calculation model of the undercarriage frame is presented. The next section deals with the box-like section with constant thickness. Firstly, after some approximations, the theoretical dependence between the ratio of bending and torsional stiffness and the height-width ratio of the section is defined. Consequently, after solving the canonical equations and by using MATLAB curve-fitting tools, the influences of change in girders' height-width ratios and change of structure lengths on the additional forces are determined. After this, an experimental verification on a laboratory model of the undercarriage frame is presented. The conclusion section gives the final remarks and the directions for further research. 1 CREATION OF THE CALCULATION MODEL In the carrying structures of loader bridges, portal cranes and excavators, clearance A may occur under one of the supports of undercarriage frames (e.g., under the support C) in relation to the crane track or terrain (Fig. 1). The basic calculation model obtains the form as in Fig. 2b, where we take the reactions X1 as the force redundant at D and redundant reactions X2 and X3 by cutting the structure at an arbitrary interior point. The flexibility coefficients are now interpreted as the relative displacements of the adjacent cross-section. Fig. 1. Undercarriage frames of crane and earth-moving machines Analysis of the Influence Parameters on the Support Structure Stiffness of Large Radial-Axial Bearings 367 Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 366-374 The resulting displacements of the primary structure due to the external loading and redundant reactions are expressed as: ^SjXj +A,. = 0, i = 1,2,3, j=i (1) where Sj is the flexibility coefficients, Xj the unknown forces and moment, Ai the displacement at i due to the external loading in the direction of the restraint at i. The flexibility coefficients Sij are determined by the Maxwell-Mohr integral [17]: j Gl J-dz~ EL j MM*- EL, rN.N, hj N^dZ- EA J K(J Q dz GA j ^Qß^dz. (2) GA For the considered case (no axial forces nor horizontal bending, the influence of shear forces is neglected), 8tj has the form: c M Mtj ö„ = l-^jz-j GI, J M&dz, EL (3) Fig. 2. Schematic presentation of the undercarriage frame: a) position of the undercarriage frame with the clearance A at the support C; b) calculation model of the frame with the unknowns X1, X2 and X3 Integration over the entire contour results in: 4/3 / / S = + 2-//- (1 + IJ / + /J3/! ) + 3 EL S22 = 2/ EL /2 L GI2 /2 L 3 EIx2 6 EL3 2GI2 8 = 33 2L eiz2 GI3 812 = l22 ( + 2l2 ) (4) 813 =- l ( + l ) I2L EI GI, 3 3 EL 823 = -- EI As can be seen from Eq. (4), bending stiffness (EI) and torsional stiffness (GIt) figure in some of them. By establishing the theoretical dependence between them, Eq. (4) would obtain a more concise form, which would simplify the analysis below. 2 THEORETICAL DEPENDENCE OF THE RATIO OF BENDING AND TORSIONAL STIFFNESS OF BOX-LIKE GIRDERS The moment of inertia of the box-like cross-section (Fig. 3) with constant thicknesses of horizontal and vertical plates 8, for the axis x, is defined by the expression: S(h + 5)3 Ix = 2-i->- + 2 x 12 12 v ' 4 (5) Fig. 3. Section of the box-like girder with constant plate thickness Further, if the height of the box-like girder h is expressed through the width b, i.e., if the coefficient k = h / b is introduced, by neglecting the members in which 83 and 84 (8 < F Contribution [%] Model 0.2904 18 0.0161 36.91 0.0000 Vc 0.0097 2 0.0049 11.13 0.0049 3.30 a 0.0121 2 0.0061 13.88 0.0025 4.12 f 0.2545 2 0.1273 291.1 0.0000 86.59 Vcxa 0.0087 4 0.0022 4.99 0.0258 2.96 Vcf 0.0024 4 0.0006 1.35 0.3306 0.82 a*ft 0.0030 4 0.0008 1.69 0.2445 1.02 Error 0.0035 8 0.0004 1.19 Total 0.2939 26 0.0113 100 In Table 4, the contributions of each factor on surface roughness were listed in the last column. It is clear from the results of ANOVA that the most important factor affecting the surface roughness was feed rate (86.59 %). The other factors differently affect surface roughness. The second factor influencing surface roughness was the axial depth of cut (4.12 %). The third factor influencing on the surface roughness was cutting speed (3.3 %). 2.1.2 Regression and Verification of Surface Roughness Model The regression analysis was used to model and analyse the relationship between a dependent variable and one or more independent variables. In this study, three dependent variables are surface roughness and amplitudes of cutting force in three directions (axial, feed, normal), whereas the independent variables are cutting speed (Vc), axial depth of cut (a), and feed rate (ft). By using Intercooled Stata 8.2TM software, the most suitable regression of surface roughness was a quadratic regression, as given in Eq. (1). 'Ra = 0.1143 - 0.0018 x Vc - 0.5515 x a + 4.9692 x f +0.0016 xVc x a - 0.0026 xVc x ft +0.0750 x ax f + 0.00007 x Vc2 +0.6958xa2 -11.3333x f2, (1) R2 = 95.85%, R2Adj = 93.66%, R2Adj is the Adjusted R2 0.6 0 i --------------------------------------------------- 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 Run No. —•— Actual value —►— Predicted value Fig. 3. Measured and predicted results of surface roughness The verification results of the surface roughness model were described in Fig. 3. As seen from this figure, the predicted results were very close to the experimental results. There is a very good relation between predicted values and test values. The R2 values of the equations obtained by the quadratic regression model for surface roughness were found to be 95.85 %. These results showed that the quadratic regression model was shown to be successfully investigated regarding surface roughness in the face-milling processes of SKD61 hard steel. 2.1.3 Parametric Influence on Surface Roughness The variation of surface roughness corresponding to the axial depth of cut, feed rate, and cutting speed was described in Fig 4. The consequences of the axial depth of cut and federate on surface roughness for three cutting speed as 80 m/min, 140 m/min, and 200 m/min, respectively. It is very clear that surface roughness increases with increasing feed rate. This trend can be explained that when feed rate increases, that results in an increase in undeformed chip thickness, and undeformed chip thickness is directly proportional to cutting forces. Then, when 378 Hoang, T.D. - Nguyen, N.T. - Tran, B.Q. - Nguyen, V.T. Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 375-385 the cutting forces increase, the stability and damping characteristics of the machine-tool system will be affected, which makes more vibrations and ultimately affects the surface roughness. This result is similar to the result of the change in the surface roughness that is noted in several works such as [34] and [35]. Fig. 4. Effect of cutting conditions on surface roughness The surface roughness values exhibited decreasing tendency with increasing of axial depth of cut from 0 mm to about 0.3 mm, but when the axial depth of cut increases from about 0.3 mm to 0.5 mm, the tendency of surface roughness was reversed. Similarly, the surface roughness decreases with increasing of cutting speed from 0 m/min to about 130 m/min, and with the cutting speed increases from about 130 m/min to 200 m/min, the tendency of surface roughness is also increasing. So, to improve the surface roughness in the face-milling process of SKD61 hard steel, with regard to machining conditions, it was proposed that as the feed rate decreases, the axial depth of cut is about 0.3 mm, and the cutting speed is about 130 m/min. 2.2 Analysis for the Feed Force's Amplitude 2.2.1 Analysis of Variance for the Feed Force's Amplitude The ANOVA results for the amplitude of the feed force was illustrated in Table 5; the contributions of each factor on the amplitude of feed force were listed in the last column. It seems that the most important factor affecting the amplitude of feed force was the axial depth of cut (72.89 %). The second factor influencing the amplitude of feed force was feed rate (16.78 %). The third factor influencing the surface roughness was cutting speed (4.96 %). Table 5. Results of ANOVA for feed force's amplitude Number of obs: 27 Root MSE: 15.9447 R-squared: 0.9953 Adj R-squared: 0.9846 Source SS DOF MS F-value Prob > F Contr. [%] Model 427244.87 18 23735.83 93.36 0.0000 Vc 21301.93 2 10650.97 41.89 0.0001 4.96 a 312903.54 2 156451.77 615.29 0.0000 72.89 f 72015.59 2 36007.80 141.63 0.0000 16.78 Vcxa 1122.92 4 280.73 1.10 0.4175 0.26 Vcf 2063.01 4 515.75 2.03 0.1831 0.48 a*ft 17837.86 4 4459.47 17.54 0.0005 4.16 Error 2033.86 8 254.23 0.47 Total 429278.72 26 16510.72 100 2.2.2 Regression and Verification of the Feed Force's Amplitude Model The most suitable regression of amplitude of feed force was a quadratic regression as given in Eq. (2). The compared results of measured values and predicted values of feed force's amplitudes were described in Fig. 5. It seems that the predicted results were very close to the measured results. The R2 values of the equations obtained with the quadratic regression model for feed force's amplitude were found to be 98.74 %. So, the quadratic regression model is the most suitable regression of amplitude of the feed force. Af = 44.3740 - 0.9807 xV + 275.4722 x a +617.9998 x ft + 0.3819x Vc x a +3.3500x Vc x ft + 3673.333xax f (2) +0.0039x V/ - 61.8055x a2 - 4655.555x f2 R2 = 98.74 %, R2Adj = 98.08% Fig. 5. Measured and predicted results of feed force's amplitude Cutting Forces and Surface Roughness in Face-Milling of SKD61 Hard Steel 379 Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 375-385 2.2.3 Parametric Influence on Amplitude of Feed Force The variation of feed force's amplitude corresponding to the axial depth of cut and feed rate is described in Fig 6. This figure shows clearly that the feed force's amplitude increases with the increasing axial depth of cut and feed rate. This can be explained by the fact that when the axial depth of cut and feed rate increase, which increases the volume of chip thickness, and volume of chip thickness is directly proportional to cutting forces. Thus, the cutting forces' amplitude increases. Vc = 80 [m/min] Vc = 120 [m/min] Vc = 200 [m/min] Fig. 6. Effect of cutting conditions on feed force's amplitude 2.3 Analysis for Normal Force's Amplitude 2.3.1 Analysis of Variance for Normal Force's Amplitude The values of Table 6 show that the quadratic model is the most suitable of normal force amplitude and the contributions of each factor on the amplitude of feed force are listed in the last column. Table 6. Results of ANOVA for Normal force's amplitude Number of obs: 27 Root MSE: 7.3608 R-squared: 0.9958 Adj R-squared: 0.9864 Source SS DOF MS F-value Prob > F Contr. [%] Model 102808.49 18 5711.58 105.42 0.0000 Vc 273.98 2 136.99 2.53 0.1409 0.27 a 72309.73 2 36154.87 667.29 0.0000 70.04 ft 24673.77 2 12336.88 227.69 0.0000 23.90 Vcxa 518.74 4 129.69 2.39 0.1365 0.50 Kxft 137.79 4 34.45 0.64 0.6514 0.13 axf 4894.47 4 1223.62 22.58 0.0002 4.74 Error 433.45 8 54.18 0.42 Total 103241.94 26 3970.84 100 Clearly, the most important factor affecting on the amplitude of the normal force was the axial depth of cut (70.04 %). The second factor influencing the amplitude of normal force was the feed rate (23.90 %). The cutting speed has little effect on the amplitude of normal force (0.27 %). 2.3.2 Regression and Verification of Normal Force's Amplitude The regression of amplitude of the normal force was a quadratic regression, as given in Eq. (3). The compared results of the measured values and predicted values of normal force's amplitudes are described in Fig. 7, which shows that the predicted results were very close to the measured results. The R2 values of the equations obtained by quadratic regression model for feed force's amplitude were found to be 99.42 %. So, the quadratic regression model is the most suitable regression of amplitude of normal force. Ak = 2.7917 - 0.0818 x Vc + 253.8473 x a +69.4723 x f - 0.5063 xVc x a +0.8194x Vc x f + 2014.1660xax f (3) +0.00075x V2 -112.9167xa2 -240.0002x f2, R2 = 99.42%, RL = 99.11% 1 3 5 7 9 11 13 15 17 19 21 23 25 27 Run No. ---■>— Actual value -*-Predicted value Fig. 7. Measured and predicted results of normal force's amplitude 2.3.3 Parametric Influence on Amplitude of Normal Force The tendency of normal force's amplitude is the same as that of the feed force. The variation of the normal force's amplitude corresponding to the axial depth of cut and feed rate is described in Fig. 8, which shows that the normal force's amplitude increases with increasing of the axial depth of cut and feed rate. This is a result of the fact that when the axial depth of cut and feed rate increase, that results increase in the volume of chip thickness, and if the volume of chip thickness increases, the cutting forces' amplitude will also increase. 380 Hoang, T.D. - Nguyen, N.T. - Tran, B.Q. - Nguyen, V.T. Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 375-385 Fig. 8. Effect of cutting conditions on normal force's amplitude 2.4 Analysis for Axial Force's Amplitude 2.4.1 Analysis of Variance for Axial Force's Amplitude The ANOVA results for the amplitude of axial force are described in Table 7. The contributions of each factor on the amplitude of axial force were calculated and listed in the last column. It seems that the influence of each factor on the amplitude of axial force was quite similar (feed rate: 33.74 %, cutting speed: 30.42 %, and axial depth of cut: 23.93 %). Table 7. Results of ANOVA for Axial force's amplitude Number of obs: 27 Root MSE: 38.5002 R-squared: 0.9698 Adj R-squared: 0.902 Source SS DOF MS F-value Prob > F Contr. [%] Model 381253.26 18 21180.74 14.29 0.0003 Vc 119568.02 2 59784.01 40.33 0.0001 30.42 a 94068.87 2 47034.43 31.73 0.0002 23.93 f 132632.98 2 66316.49 44.74 0.0000 33.74 Vcxa 21406.58 4 5351.65 3.61 0.0577 5.45 Vcf 9411.24 4 2352.81 1.59 0.2676 2.39 a*ft 4165.57 4 1041.39 0.7 0.6118 1.06 Error 11858.14 8 1482.27 3.02 Total 393111.40 26 15119.67 100 2.4.2 Regression and Verification of Axial Force's Amplitude The most suitable regression of axial force' amplitude was a quadratic regression as given in Eq. (4). The R2 values of the equations obtained by quadratic regression model for axial force's amplitude were found to be 93.74 %. The compared results of measured values and predicted values of feed force's amplitudes are described in Fig. 9, which shows that the tendencies and the values of predicted results were very close to the tendencies and the values of measured results. So, the quadratic regression model is the most suitable regression of amplitude of axial force. A^ = 40.9349 + 0.1876 xVc + 206.9375 x a -188.6806 x ft + 2.6045 x Vc x a +9.1569x Vc x ft +1099.5830x ax f (4) -0.0019x V2 -542.5695xa2 +1465.5550x f2, R2 = 93.74% R2AdJ = 90.42% Fig. 9. Measured and predicted results of axial force's amplitude 2.4.3 Parametric Influence on Amplitude of Axial Force The tendency of axial force's amplitude is also the same as that of the feed force. Fig. 10 presents the variation of axial force's amplitude corresponding to axial depth of cut and feed rate. Figure 10 revealed that the axial force's amplitude increases with increase in axial depth of cut and feed rate. The reason for this tendency is the same as for the feed force's amplitude and normal force's amplitude as well. Fig. 10. Effect of cutting conditions on axial force's amplitude 2.5 The Relationship of Cutting Force and Surface Roughness in Milling In this paper, the relationship between cutting force's amplitude and surface roughness was also investigated. It seems that this relationship has not been studied previously. The amplitudes of cutting forces and the surface roughness were drawn in Cutting Forces and Surface Roughness in Face-Milling of SKD61 Hard Steel 381 Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 375-385 one diagram, as shown in Fig. 11. It seems that the tendency of the surface roughness is the same as that of the cutting force's amplitudes. As seen from this figure, if the cutting force's amplitudes increase, the surface roughness increases; if the cutting force's amplitudes decrease, the surface roughness also decreases. These could be explained that in finish-face milling processes, at the high stable cutting conditions with small vibrations, the chip thickness is quite stable, it makes the machining surface quite smooth, and so the surface roughness is often small (good surface). Furthermore, in this process, because the chip thickness is quite stable, the cutting forces in all directions are often quite stable, and these make the amplitudes of cutting forces smaller than those at other machining conditions. In the opposite cases, when the milling conditions are unstable cutting with chatter or large vibrations, the chip thickness will be unstable; it makes the surface and cutting force more unstable as well, and so that the amplitudes of cutting forces and surface roughness will be larger than that of other milling conditions. Thus, in this study, it is seen that there is the same tendency of output parameters in machining processes. By this reason, in this study, the optimal value was determined for surface roughness that is one of the most important values to improve the quality of machining product, as presented in Section 2.6. Fig. 11. The relationship between cutting force's amplitude and surface roughness 2.6 Optimization of Cutting Conditions 2.6.1 Estimation of Optimum Surface Roughness by ANOVA Method The lowest value of surface roughness is very important for quality improvement of the machining product and lowering production costs. In this study, the quadratic regression model of surface roughness as presented by Eq. (1) that was used to find the optimized values of surface roughness and machining parameters. By using MATLAB R2013aTM software, the optimized results of machining parameters were obtained as fval = 0.1531, x= [113.0595, 0.2636, 0.0500], where fvai is the optimal value of the surface roughness as presented by Eq. (1). And, x are the values of variables in Eq. (1) when surface roughness equals to fal. Thus, with the ANOVA method, the optimal parameters of milling process were determined as cutting speed: Vc = 113.0595 m/min, axial depth of cut: a = 0.2636 mm, feed rate: f = 0.0500 mm/flute. And, the optimization value of surface roughness was Ra = 0.1531 ^m. 2.6.2 Analysis of the Signal-to-Noise (S/N) Ratio Using the Taguchi method, the optimal values of control factor were determined by analysis of the signal-to-noise ratio. The lowest value of surface roughness is very important to improve the machining product, so the smaller-the-better equation was used for calculation of the S/N ratio [31]. The values of the S/N response for observations of surface roughness were listed in Table 8 and shown in Fig. 12. Table 8. The S/N response for surface roughness Factor Mean S/N ratio Delta Level 1 Level 2 Level 3 Vc 9.8245 10.0299 9.5829 0.4469 a 10.0552 10.2519 9.1301 1.1218 ft 13.4388 8.5886 7.4099 6.0289 Fig. 80 120 200 0.1 0.3 0.5 0.05 0.10.15 Machining parameters 12. Main effects of each factor on surface roughness With Taguchi techniques, the best level of each control factor was determined according to the highest S/N ratio at the level of that control factor [32]. With these techniques, from the values of Table 7 and Fig. 12, the level and S/N ratios for the factors giving the 382 Hoang, T.D. - Nguyen, N.T. - Tran, B.Q. - Nguyen, V.T. Strojniski vestnik - Journal of Mechanical Engineering 65(2019)6, 375-385 best Ra value were specified as factor cutting speed (level 2, S/N = 10.029881 dB), factor axial depth of cut (level 2, S/N = 10.251945 dB), factor feed rate (level 1, S/N = 13.438819 dB). So, with the Taguchi method, the optimum value of surface roughness was obtained at a cutting speed of 140 m/min, a depth of cut of 0.3 mm, and a feed rate of 0.05 mm/flute. The value of surface roughness with optimized values is 0.1605 ^m. The comparison results of the optimization values of the two methods were described in Table 9. The optimized results between the ANOVA method and Taguchi method are quite close together. However, the ANOVA method gave the better result regarding surface roughness. So, this method can be applied to improve the surface quality in finished face milling the SKD61 hard steel. Table 9. The optimized results of ANOVA and Taguchi method Machining Optimization values Differences parameters ANOVA method Taguchi method [%] Vc 113.0595 140 19.24 a 0.2636 0.3 12.13 f 0.05 0.05 0.00 Ra [um] 0.1605 0.1531 4.58 3 CONCLUSIONS In this study, an experimental method was performed to investigate the influence of cutting conditions on the surface roughness and cutting force's amplitudes. Depending on the analysis of experimental results, the conclusions of this study can be drawn as follows. 1. The most important factor affecting the surface roughness was the feed rate (86.594 %), while the most important factor affecting on the cutting force's amplitudes was the axial depth of cut (72.891 % for feed force amplitude and 70.039 % for normal force amplitude). The other factors affect differently on the surface roughness and cutting force's amplitudes. 2. The cutting force's amplitudes increase with the increasing of the axial depth of cut and feed rate. While the surface roughness decreases with the decreasing of the feed rate, with increasing of axial depth of cut from 0 mm to about 0.3 mm, and with the increasing of cutting speed from 0 m/ min to about 130 m/min. Furthermore, the surface roughness increases with increasing of feed rate, with increasing of axial depth of cut from about 0.3 mm to 0.5 mm, and with increasing of cutting speed from about 130 m/min to 200 m/min. To improve the surface roughness in the face-milling process of SKD61hard steel, with regards to machining conditions, it was proposed that the feed rate decreases, the axial depth of cut is about 0.3 mm, and the cutting speed is about 130 m/ min. 3. The tendency of the surface roughness is the same as that of the cutting force's amplitudes. So, the optimal value was determined for surface roughness, which is one of the most important values to improve the quality of machining product. 4. 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Cutting Forces and Surface Roughness in Face-Milling of SKD61 Hard Steel 385 Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, 386-397 © 2019 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2018.5901 Original Scientific Paper Received for review: 2018-11-30 Received revised form: 2019-05-15 Accepted for publication: 2019-05-31 Numerical Simulation-Based Effect Characterization and Design Optimization of a Micro Cross-Flow Turbine Endashaw T. Woldemariam - Hirpa G. Lemu* University of Stavanger, Faculty of Science and Technology, Norway In this era of high power demand along with a need for sustainability, small and micro-scale hydro turbines are indispensable, especially for off-grid power generation. These hydro-turbines are more appealing, particularly for small-scale enterprises, hospitals, telecommunication facilities, residential buildings, and other public institutions located in remote areas that have rich hydropower potential. Due to their simple design, manufacturability, and maintainability, micro cross-flow turbines are more feasible and applicable for power generation in undeveloped and developing countries with scattered hydropower resources. Such hydropower generation is environmentally friendly as their construction and operations have no significant effect on the surrounding ecosystem. To extract the highest possible power, however, the power generation efficiency of cross-flow turbines are is yet well optimized. In this paper, a numerical-based investigation on 3D numerical models and a design optimization study that is aimed at improving the turbine's performance has been studied and reported. A numerical simulation-based metamodel assisting optimization approach is deployed to carry out the optimization after the characterization of the design parameters, following a previous experimental study on a similar model. The benefits of the approach with regard to computational costs and outcome of the optimization are discussed. Responses from the optimized design are compared against the original design responses at different magnitudes of design parameters. Published experimental test results based on similar conditions are used to verify the models at each stage. The approach is found effective, and the optimized model shows better performance and efficiency than the original models at similar working condition. Keywords: crossflow turbine, micro hydro turbine, numerical simulation, metamodel based optimization Highlights • A numerical simulation based investigation of a 3D model of a micro cross-flow turbine is conducted. • A metamodel-assisted optimization approach is deployed to carry out the design optimization. • The benefits of the numerical simulation approach in terms of computational costs and the outcome of the optimization are discussed. • The response of the optimized design is compared with the original design responses at different magnitudes of design parameters. • The model results at each stage of the turbine are compared with similar experimental works. 0 INTRODUCTION In an era of increasingly growing energy demands and concern regarding the massive consumption of fossil fuel to satisfy energy needs, any alternative energy source is indispensable. Moreover, the growing concern regarding climate change has brought several nations to a common consensus to pursue cutting the amount of greenhouse gas emissions, which demands an urgent reduction of fossil-fuel consumption as a significant energy source and promoting carbon capture [1]. To balance those concerns, sustainable and renewable energy sources are more critical in this period than at any time before. Since hydropower resources are the most accessible sources of renewable energy, small and micro-scale hydropower generations are undeniably essential to contributing a considerable portion to this effort. In addition, these hydro turbines undisputedly play a significant role in providing off-grid power generation to small-scale enterprises, hospitals, telecommunications facilities, residential buildings and other public institutions located in remote areas and are more feasible particularly for developing and less developed countries. As studies by Kaunda et al. [2] and Paish [3] show, however, the benefits of the application of these kinds of turbines are not limited to developing and less-developed countries only but to some extent also for developed countries. Off-grid power generation contributes to the effort of reducing power loss in the power transmission process, most importantly, for scattered power distribution systems in rural areas. Moreover, unlike large-scale hydropower generators, small scale turbines generate power without significant impact on the ecosystem [3]. A micro-scale turbine type called a cross-flow turbine, also known as Michell-Banki's turbine, is economically efficient and simple design compared to other conventional micro-turbines. Fig. 1 illustrates such a turbine called the International Assistance Mission (IAM) [4]. Apart from its simplicity, this 386 *Corr. Author's Address: University of Stavanger, Faculty of Science and Technology, Norway, Hirpa.g.lemu@uis.no Strajniski vestnik - Journal of Mechanical Engineering 65(2019)6, 386-397 turbine type incurs relatively low manufacturing, operation, and maintenance costs [5]. (b) Sectional view (adjustable nozzle) Fig. 1. IAM turbine assembly drawing adopted from [4]; a) 3D isometric view, and b) sectional view with the adjusting part Moreover, its favourable run-of-the-river application and the power generation efficiency at part load conditions attract more research interest. Despite all those benefits over other conventional hydro turbines, however, the power-generating performance and efficiency of the turbine are not yet well optimized. In efficiency research on cross-flow turbines, experimental works dominate earlier studies while recent works mostly adopted numerical methods. This is partly due to the powerful and reliable computational tools and partly due to the accompanying high costs of experimental works. To improve the efficiency, for instance, Olgun [6] conducted an experimental study on effect of geometric parameters of the nozzle and the runner such as diameter ratios, throat-width ratios and gate-opening size on the efficiency and reported that micro cross-flow turbines can be operated with a wider range of gate opening compared with other turbines having maximum efficiency. Acharya et el. [7] conducted numerical research involving the modification of nozzle radius, varying the guide vane angle, and changing the number of runner blades. From this study, it has been observed that the highest efficiency of 76.6 % was obtained with 22 blades. Similarly, Choi et al. [8] studied the efficiency and flow characteristics numerically by varying the shape of the nozzle and the runner blade. However, most of the reported theoretical studies [9] to [11] and experimental investigations [12] and [13] show massive inconsistency in the results of their efficiency computation. Most theoretical research reports, for instance, in [11], [14], and [15], show that the contribution from the second stage ranges from 18 % to 30 % of the total power generated. An overview of achieved efficiencies by different researchers in [16] show variations from 66 % to 88 %. In cross-flow turbines, the water jet from a penstock initially passes through a nozzle, which is controlled by a valve. Once the jet exits the nozzle, it then strikes the rotor blades from the outer and inner periphery for first and second stage power generation, respectively. More details on operating principles of cross-flow turbines and the detail description of the main parts are discussed in a previous related work by the authors [17]. The main objective of this paper is to instigate a numerical simulation-based approach to the study of the effects of selected design parameters on the optimized performance, and thus improved the output of the turbine utilizing the parameters. It aims to obtain a turbine with better performance, and thus with better hydraulic efficiency. Following this introduction section, a brief discussion of the methodology and the approaches employed are presented. Sections 2 and 3 are devoted to the theoretical power computation and numerical models, the chosen governing numerical models and their validation through sensitivity analyses. In Section 4, the optimization approach and brief comparison with other tools are discussed, followed by a discussion of the results in Section 5. Finally, conclusions are drawn in the last section. 1 METHODOLOGIES AND APPROACHES As stated, this paper presents work carried out on a numerical simulation-based investigation and design optimization, aiming to study the effects of some essential design parameters on the performance and to narrow down the massive inconsistency of efficiency results seen in multiple studies of micro cross-flow turbines. The numerical simulation-based design optimization approach uses a metamodel assisted optimization tool to improve the performance of a Numerical Simulation-Based Effect Characterization and Design Optimization of a Micro Cross-Flow Turbine 387 Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, 386-397 selected micro cross-flow turbine model. In a previous related work [17], a similar approach was discussed but with limited study on a 2D numerical model whose results have been published. This extended version of the work focuses on the 3D model analysis of the turbine for numerical model verification and comparison of the performance against the original model using time-dependent (transient) analyses. The numerical models and computational tools are verified using experimental test results in [4]. The thesis reported in [4] aims to study the fluid flow characteristics and measurements of the relative moment using a laboratory setup that enables collecting relative load responses on the rotor blade design (Fig. 1). The experimental test was carried out at different head and rotational speeds of the rotor. Our numerical simulation based study, however, focuses on a 5-meter net fluid head input at rotational speeds of 250 and 350 revolutions per minute [rpm] because stable experimental test results are obtained in these range of load conditions. Table 1. Original turbine geometric parameter values Parameter Description Value [unit] R Outer radius of the rotor 13.5 cm r Inner radius of the rotor 9.366 cm ' c Entry arc curvature 15.3 cm ßl Blade angle at rotor outer circle 30 degrees ß2 Blade angle at rotor inner circle 90 degrees rs Rotor shaft radius 2.25 cm b Depth (width) of the turbine 20.4 cm N Number of blades 24 A detailed description of the deployed numerical simulation-based metamodel-assisted optimization (NSBMAO) approach and the alleviated number of function evaluation reports are introduced in Section 4. A commercial ANSYS workbench tool [18] was interfaced with the optimization tool, which served for modelling and computational fluid dynamics (CFD) simulation in both the 2D and 3D analyses. The single objective global optimization (SOGO) algorithm of the metamodel assisted optimization tool, from Optimization Assisted System Integrated Software (OASIS) tool [19], which is designed for computationally expensive black box problems [20], is deployed to perform the optimization. In the investigations, parameters assumed appealing for the turbine performance improvement from previous studies are considered (Table 1). Two of the parameters, i.e. valve angle and the entry wall arc, are assumed to have a significant impact on the 1st stage power generation, while the rotor shaft diameter has an impact mainly at the 2nd stage. 2 THEORETICAL POWER COMPUTATION IN CROSS-FLOW TURBINES Due to the turbine's configuration and power transfer characteristic condition inside the turbine, crossflow turbines are mostly assumed to be impulse turbines. The theoretical power conversion computation, therefore, is carried out through fluid velocity differences at the rotor's entry and exit blades using Euler's turbomachinery equations. From the given hydraulic head subject to the turbine inlet, Eq. (1), the theoretical power computation, therefore, uses the theoretical fluid jet velocities at four points, two at the first stage, and two at the second stage (Eqs. (2) to (5)). Vn = V * Cv*j2gH, (1) Vui = ^cosa, (2) Vu2 = V2cosa2, (3) Vu 2 = U2 = Vu 3, (4) Vu 4 =q>V1cosa1 -wx R(*p +1). (5) The theoretical torque and power calculations then follow Euler's turbomachinery equations (Eqs. (6) to (9)). i F = pQJdv = pQ(Vi - V2), (6) 2 T = FxR = pQ(V -V2)R, (7) Pout = T x®, (8) Pin = Q(P_in - Pr_„„ V P, (9) where Pr in and Pr out are the pressure at the inlet due to the head of the water and at the outlet of the turbine model, respectively. Fig. 2a shows the peripheral velocities tangent to the outer and inner diameters, ui (i = 1 to 2), the actual fluid velocities, Vj (j = 1 to 4) and the relative velocities, wk (k = 1 to 4), corresponding to points 1 to 4 on the rotor. In addition, the diagrams in Fig. 2b describe the detail velocity components at the rotor using velocity triangles at each power generation location. Similar to most experimental studies, the theoretical studies also show massive inconsistency in the efficiency analysis studies of similar turbine types [9] to [11]. The inconsistencies raised are mainly due to the uncertainty on the fluid angles at the entry 388 Woldemariam, E.T. - Lemu, H. G. Strajniski vestnik - Journal of Mechanical Engineering 65(2019)6, 386-397 a) @ Î ih b) Fig. 2. Fluid velocity components: a) velocity positions at the rotor, and b) velocity triangles at each points and exit of the rotor blades. In the theoretical power calculations, some studies use the entry arc curvature angle, for instance in [21], but others use the nozzle valve angle [22] as an attack angle (location illustrated in Fig. 3 of the fluid at the entry. Fig. 3. 2D sketch of IAM turbine with parameters are utilized (Fig. 4). One section of the turbine, which is one third of the actual depth of the design, is used considering symmetry in that direction and that it could represent the problem well. Each domain of the models is divided into three zones: (1) housing, (2) rotor, and (3) stationary. The interfacing surfaces between the zones are recognized and coupled automatically in the simulation tool while generating the grid to smoothly transfer results between the moving and other zones at the interfaces. The turbulent Navier Stokes equation (Eq. (10)) is simplified in different viscous models and tested for sensitivity analysis [18]. ^V. (pV) = Sm , dt — (pV) + V- (pVVT) = -VP + V- (T) + V- (-pVV ),(10) dt t = ^ (VV + VVT )--V-VI where Sm is the source term and t is the stress tensor given in Eq. (10), the term -pv'v'T is the Reynolds stresses term, V is the velocity vector representing the sum of the average and the fluctuating components (i.e., V = V + V), j and I are the molecular viscosity of the fluid and the unit tensor respectively. 3.1 3D Numerical Model Grid Validation 3 MODEL VALIDATION & SENSETIVITY ANALYSIS To obtain realistic responses from the numerical simulation, it is essential to validate the model grid, the numerical turbulence model, and the algorithms employed to study both the steady-state and transient analysis and the CFD simulation-based optimisation. For the analyses, 3D numerical models with one-to-one scale as the IAM turbine design, except the depth, The grid qualities of all numerical models both in the transient and steady-state analyses are controlled so that the parameters that ensure the quality fall in the acceptable range so as to obtain realistic results. Some of the important parameters considered to ensure the grid qualities are orthogonal skewness, aspect ratio and orthogonal quality of the elements in the grid domain. Table 2 shows the details of the grid node and element sizes of one of the models after the Numerical Simulation-Based Effect Characterization and Design Optimization of a Micro Cross-Flow Turbine 389 Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, 386-397 b) c) Fig. 4. IAM cross-flow turbine 3D numerical model and mesh element quality; a) unmeshed model assembly, b) meshed model (front view), and c) meshed model with detailed view quality validation that is utilized in the 3D numerical analysis. The model grid qualities are also validated while comparing the trends of the numerical moment coefficient responses from one of the rotor blade's front wall against the relative moment signal from experimental test results of the sensitivity analysis in the following section. The voltage signals from the experimental tests are collected from a mechanical strain gage carefully attached to one of the front walls of the rotor's blade. Table 2. Mesh measurements and qualities of the 80 % opened model Housing Rotor Stationary Total Nodes 633368 1967512 320251 2921131 Elements 594920 1620087 307160 2522167 Responses are collected from the gage while the model is subject to a five-metre water head, and the rotor is controlled at 350 rpm [4]. The numerical moment coefficient responses on the front face of a selected blade are collected at each time step from the transient numerical simulation. The post-processing tool of the research version solver tool used, ANSYS Fluent, computes the moment coefficients after solving the turbulent Navier Stokes equation. The moment coefficient computations use the moment response and the reference values set in the tool. The total numerical moment response, Eq. (11), is the combined effect of both the pressure force and viscous force, Eq. (12), from the transient numerical simulation analysis result on one of the rotor's blade's wall. M = r x Fp + rx Fv, (11) P = t (P - Pref )An, (12) ¡=1 where M is the total moment in [N m] due to the pressure and viscous forces, Fp and Fv respectively. Both forces are computed based on Eq. (10). r is the moment arm from the moment centre of the wall to the rotational axis of the rotor. P and Pref are the total and the reference pressures computed, respectively. A 390 Woldemariam, E.T. - Lemu, H. G. Strajniski vestnik - Journal of Mechanical Engineering 65(2019)6, 386-397 and n are the area and the normal vector to the surface. Moreover, to compute the hydraulic efficiency (n) of the turbine model from numerical analysis, the output hydraulic power computed from the total moment response and the rotational speed subject on the rotor. While the input power is computed from the hydraulic input head, the numerical flow rate is computed at the inlet of the turbine (Eq. (13)). n = M Q(Pr in - Pr out V P (13) where m is the rotational speed [rad/s], Q is the mass flow rate; and p is the fluid density. The noisy data collected from the transient analysis per time step at different inputs and boundary conditions are carefully filtered to obtain a smooth average graph without losing the property of the data. After various trials in the MATLAB tool, a cut-off frequency of 80 Hz with 850 Hz data sample frequency with Lowpass Filter gave us a smooth curve that better retain the characteristics of the original data (Fig. 5). Note that all graphs in the paper are generated using the MATLAB programming tool. 3.2 Sensitivity Analysis of Numerical Models for Computational Analysis To make the sensitivity analysis and investigate the model that better represents the turbulent flow problem inside the turbine, three different well-known viscous models have been tested in the numerical computation: 1. standard k-epsilon (k- e) with scalable wall functions for near wall treatment, 2. k-omega (k- m) with a shear flow correction k-omega option, and 3. large eddy simulation (LES) with WALE sub-grid-scale model for high turbulence test purposes. In all the three model tests, the same solution algorithms are employed to solve the pressure-velocity coupling, and the momentum and turbulent kinetic energy in the spatial discretization. The sensitivity test is carried out comparing the trends of the moment coefficient responses from each numerical model against the relative moment signal from the experimental test result on a similar model design at the same valve opening, net inlet fluid head, and rotational speed. Based on the gradient diffusion hypothesis, the k - e model relates the Reynolds stresses to the mean velocity gradients and the turbulent viscosity in solving the kinetic energy and the turbulent dissipation. This model is found to be robust, computationally cheap and widely used even though it performs poorly for complex flows that involve a strong streamline curvature, severe pressure gradient, and separation. In contrast, the k- a model allows for better near-wall treatment and performs well in low Reynolds number flows, but it requires intense mesh resolution near walls [23] and [24]. Unlike the other two models, the LES model better allows explicitly solving large eddies and accounts for the selected small eddies using the sub-grid-scale model. It is more popular in simulating highly turbulent flows but gives only a fair enough solution near walls [25]. Fig. 5. Curves from original unfiltered data and filtered data on 80 % opened valve model at 350 rpm —k-e —ku — LES — Exp. test ill /1 \ IL - Wk / Ss \ ^ i i i Time step [s] Fig. 6. Moment coefficient responses' graph from three viscous models vs. experimental test moment signal from strain gages on a blade face As observed in Fig. 6, the relative moment test signal from experimental test results [4] and the numerical responses show a good agreement with regard to showing the 1st and 2nd stage power generation regions clearly. Moreover, in all cases, the average period of the responses match the experimental test with insignificant differences. Note that the magnitudes of the numerical responses are not Numerical Simulation-Based Effect Characterization and Design Optimization of a Micro Cross-Flow Turbine 391 Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, 386-397 expected to match with the experimental test as the experimental study signal is not calibrated to give the moment response, hence the objective was to study the relative load at the two stages. However, the setup and the result were later calibrated to obtain the total torque on the rotor, as reported in [26]. In both the transient simulation and the experimental test results, the first stage of power generation has shown a better output than the second stage power output did, as was expected. Since the result from the k-e model shows a better stable response cycle in both stages compared to the other models (Fig. 6), it is chosen to better represent the problem for further analyses. 3.3 Sensitivity Analysis of Numerical Solution Methods In this section, two of the widely used solution schemes in CFD, the SIMPLE and the PISO algorithms, are tested. Both algorithms are fundamentally used to drive a pressure correction equation in the Pressure-Velocity Coupling scheme in solving the turbulent Navier-Stokes equation [27]. The fundamental difference between them is that PISO uses the velocity correction from adjacent cells while solving the continuity equation, whereas SIMPLE does not. Thus, SIMPLE leads to a simpler solution procedure that directly uses the pressure correction terms to the velocity correction. Even though the standard SIMPLE algorithm is initially developed for steady analysis, it can provide a realistic solution with smaller time step size in the time discretization. Thus, in both cases, a relatively smaller time step size of 0.001 is used in the transient analyses, as a compromise between the computational cost and accuracy. Second-order upwind methods are employed to solve the momentum, turbulent kinetic energy models and specific dissipation rate in the spatial discretization. Both time-discretized PISO and SIMPLE algorithms — SIMPLE PISO — Exp. test Time step [s] Fig. 7. Moment coefficient responses from different numerical computational algorithms vs. experimental test moment signal gave stable moment coefficient's response cycle and showed a good agreement with experimental test [4] with respect to relative 1st and 2nd stage cycles (Fig. 7). However, the result from the later converged faster than the former without showing a significant difference in the moment coefficient response. Thus, it is used in both the transient and steady-state analyses of the hydraulic efficiency computation. 4 ALGORITHM EMPLOYED IN THE OPTIMIZATION In most simulation-based global optimization problems where the objective function is a result of an expensive implicit modelling and analysis tool or black-box function, applying standard optimization tools is computationally expensive and requires extensive tool-interfacing effort. Intelligent algorithms and optimization techniques are, however, being developed and employed to reduce the costs without significantly reducing the accuracy of the outcome. Thus, metamodel-assisted optimization approaches contribute significantly in this regard. In our numerical simulation-based optimization approach, the objective function returns the average weighted total moment on the rotor walls about the Z-axis passing through the shaft centre from the implicit analysis tool (Workbench 17.1). The selected design variables and objective functions are parametrized prior to the simulation in the design modeller and ANSYS Fluent. Moreover, the design space bounds, the lower bounds (LB) and upper bounds (UB), of each of the three selected design variable are given in Table 3. The bounds are given considering the actual turbine design geometry and their validity on the numerical model. Table 3. Lower and Upper bounds of parameters Valve opening angle Curvature radius Shaft radius [deg] ([rad]) [cm] [cm] LB UB LB UB LB UB 54 (0.942) 70 (1.22) 14.5 19.0 0 4.5 The problem is formulated in a simple form as in Eq. (14). To carry out the optimization, a 2D model is utilized, aiming to further reduce the computational cost. Note that the 2D model is validated using the mesh quality control and response trends before applying the optimization. A metamodel-assisted design optimization (MADO) approach was employed to interface with the 2D numerical model using the OASIS optimization tool, which was developed mainly to 392 Woldemariam, E.T. - Lemu, H. G. Strajniski vestnik - Journal of Mechanical Engineering 65(2019)6, 386-397 solve computationally intensive numerical problems. From the different MADO strategies discussed in [28], this tool employs a direct sampling approach. In this approach, the metamodel is used to generate samples toward the optimum value while the metamodel is adaptively updated using each function evaluation results obtained from the black-box function. The most important benefit of the approach is that it is less dependent on the accuracy of the metamodel. The entire NSBMADO framework is shown in Fig. 8. Fig. 8. An adaptive direct sampling meta-model based design optimization framework using OASIS tool For this particular problem, the Single Objective Global Optimization (SOGO) algorithm is employed to solve the numerical simulation-based design optimization problem. Minimize F(x), Constraints g (x) < 0, x , xu ] (14) where the F(x) is the objective function that returns the total moment on the rotor walls including the shaft wall from the simulation tool as a function of g(x), where x = [a, ra, rs], is the constraint function and XL and Xv are the lower and upper bounds of the design variables in the variable vector X. The optimization was run with a stopping criterion set to 80 function evaluations, which is selected based on the authors' experience on the tool after carrying out optimizations on similar models. Note that since the tool searches minimum values, the output response should be adjusted to give a negative response. 5 RESULTS AND DISCUSSION 5.1 Optimization Results After establishing a clear interface between the MADO and the simulation tool, the optimization elapsed a total of 18 hours and 55 minutes with an average single function evaluation time of around 14 minutes. The total overhead time of the optimization tool is 2 minutes and 47 seconds. In a simple calculation deploying a regular Genetic Algorithm, the tool would take around 116 hours for 1000 iteration analyses on the same model with a simulation that takes 14 minutes on a similar machine assuming the problem converges at 500 iterations. The convergence history of the NSBMADO optimization process per number of function evaluation is given in Fig. 9. The corresponding results of the design parameters of the optimized model are also given in Table 4. Note that the shaft size shows the stepped size of the shaft passing through the rotor, some designs have shafts to support at the sides only attached to the side plates. Fig. 9. Optimum moment's convergence history from OASIS per number of function evaluation Table 4. Values of optimized model parameters Valve angle opening [rad] Shaft diameter [cm] Entry arc curvature [cm] 1.217 0.136 18.28 5.2 Comparative Analysis with Optimization Results Both transient and steady simulation on 3D models of the 80 % and 100 % opened nozzle model and model geometry from the optimization are carried out to verify the optimization results at 250 rpm and 350 rpm. The moment response graphs from the transient analyses both at 250 rpm and 350 rpm show that the Numerical Simulation-Based Effect Characterization and Design Optimization of a Micro Cross-Flow Turbine 393 Strajniski vestnik - Journal of Mechanical Engineering 65(2019)6, 386-397 Fig. 10. Entire rotor's moment response comparison of the optimized, 80 % and 100 % valve opening model at 350 rpm rotation of the rotor Fig. 11. Entire rotor's torque response comparison of the optimized, 80 % and 100 % valve opening model at 250 rpm rotation of rotor optimum model gives better performance especially at most time steps for 250 rpm rotor speed (Figs. 10 and 11). The velocity streamline contour diagrams in Figs. 12 and 13, particularly at 100 % opened the valve, also could give a visual understanding of how the shaft Fig. 12. Velocity streamlines inside the turbines body at 350 rpm at: a) 80 % valve opening, b) 100 % opening, and c) optimized model 394 Woldemariam, E.T. - Lemu, H. G. Strajniski vestnik - Journal of Mechanical Engineering 65(2019)6, 386-397 Fig. 13. Velocity streamlines inside the turbines body at 250 rpm at: a) 80 % valve opening, b) 100 % opening, and c) optimized model Table 5. Numerical steady state analysis response values and hydraulic efficiencies at 350 rpm No. Nozzle opening Pressure at outlet [Nm] Mass flow rate [kg/s] Moment response [N-m] Numerical efficiency [%] 1 80 % 1541.67 37.76 21.831 (31.78 coeff) 44.60 2 100 % 1998.33 47.91 39.02 (63.70 coeff) 60.85 3 Optimized model 2112.34 44.93 37.16 (60.70 coeff) 61.80 Table 6. Numerical steady state analysis response values and hydraulic efficiencies at 250 rpm No. Nozzle valve opening Fluid pressure at outlet [N-m] Mass flow rate [kg/s] Moment response [N-m] Numerical efficiency [%] 1 80 % 878.50 36.09 37.62 (30.51 coeff) 50.62 2 100 % 635.11 34.75 46.56 (76.01 coeff) 71.52 3 Optimized Model 1834.23 49.70 55.01(89.82 coeff) 84.49 at the centre of the rotor could obstruct a significant portion of the fluid passing through the second stage power generation. This obscuring of the shaft would also produce a disturbance on the streamlines of the neighbouring fluids, which would result in more loss on the hydraulic power due to turbulence. Thus, design optimization is required. From the steady analysis, the efficiency computation results show that the optimized model gives better efficiency than the numerical hydraulic efficiencies obtained from the original models with the valve opened 80 % and 100 % both at 250 rpm and 350 rpm (Tables 5 and 6). Unlike the reported experimental results, the original model gives Numerical Simulation-Based Effect Characterization and Design Optimization of a Micro Cross-Flow Turbine 395 Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, 386-397 maximum numerical hydraulic efficiency at 100 % valve opening in both cases. It is discussed in the results from the experimental test in [4] that the maximum efficiency was obtained from the 80 % opening at 350 rpm. Moreover, using the average pick values from the transient graphs for a rough estimation of the relative loads, the second stage contributes 23.42 % to 25.89 % of the total power in both cases. However, results from the experiment shown in the range from 36.4 % to 53.7 %. The later shows a massive deviation from most of the theoretical study reports on this kind of microturbine, which ranges from 26 % to 30 % only. 6 CONCLUSION The numerical simulation based study reported in this article turns out to be effective and appealing for further study on similar turbomachinery. The separate simulation studies, both the transient and steady state, on the numerical models at different parametric values returned valid and sound results. Thus, the investigation significantly attributed in the relative load trend studies on the blades' front wall and the characterization of the flow inside the turbine. It is observed that the entry arc curvature and the nozzle opening angle played a significant role in determining the attack angle of the flow stream at the entry to the first stage. It is also observed, from the velocity streamline diagrams, in three out of four cases, that the shaft diameter obstructed the fluid flow heading to the second stage therewith increasing the turbulence, which results in a relatively lower percentage power contribution from the stage. Thus, the optimization returned a highly reduced shaft size. The optimized design outperformed the original model at different rotational speeds and valve openings including the 80 % opening at which the experimental test result, utilized for the model validation in the text, observed to have the maximum efficiency. As a result, the overall output power and efficiency are improved with the optimized model. Meanwhile, it is studied that the percentage power generated from the second stage fall close to the range other researchers reported through theoretical analysis unlike the experimental test, whose report showed a huge deviation. Using the optimized model, the efficiency is improved by 17.88 % compared to the maximum efficiency obtained from the original model at 250 rpm. Moreover, the adaptive direct sampling metamodel assisted optimization tool deployed, unlike any other global optimization tools, highly reduced computational cost that would otherwise be incurred. The numerical simulation based analysis and optimization approach could thus be employed for further studies on the same and similar turbomachinery applications. 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Numerical Simulation-Based Effect Characterization and Design Optimization of a Micro Cross-Flow Turbine 397 Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6 Vsebina Vsebina Strojniški vestnik - Journal of Mechanical Engineering letnik 65, (2019), številka 6 Ljubljana, junij 2019 ISSN 0039-2480 Izhaja mesečno Razširjeni povzetki (extended abstracts) Lin Zhou, Guoqiang Wang, Kangkang Sun, Xin Li: Študija sledenja trajektorije goseničnega vozila na osnovi modelnega prediktivnega vodenja SI 43 Ozgur Poyraz, Melih Cemal Ku§han: Motnje zaradi zaostalih napetosti pri laserskem spajanju s lojev nikljevih superzlitin v prahu SI 44 Yu Zhang, Hongzhi Yan, Zhiyong Wang, Qing Zhao: Vpliv parametrov rezkarja na kontaktne lastnosti površin zob, obdelanih po postopku petih prehodov in po metodi Duplex Helical SI 45 Spasoje Trifkovic, Nebojša Zdravkovic, Milomir Gašic, Mile Savkovic, Goran Markovic: Analiza vplivnih parametrov na togost podporne konstrukcije velikih radialnoaksialnih ležajev SI 46 Tien Dung Hoang, Nhu-Tung Nguyen, Buc Quy Tran - Van Thien Nguyen: Študija rezalnih sil in površinske hrapavosti pri čelnem rezkanju trdega jekla SKD61 SI 47 Endashaw T. Woldemariam, Hirpa G. Lemu: Karakterizacija vplivov in optimizacija konstrukcije mikroturbine s prečnim tokom na osnovi numeričnih simulacij SI 48 Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, SI 43 © 2019 Strojniški vestnik. Vse pravice pridržane. Prejeto v recenzijo: 2019-01-04 Prejeto popravljeno: 2019-04-14 Odobreno za objavo: 2019-05-07 Študija sledenja trajektorije goseničnega vozila na osnovi modelnega prediktivnega vodenja Lin Zhou - Guoqiang Wang* - Kangkang Sun - Xin Li Univerza Jilin, Šola za strojništvo in letalsko tehniko, Kitajska V današnjem času potekajo obsežne raziskave avtonomnih vozil, ki so osredotočene predvsem na kolesna vozila, manj pa je raziskav na področju samodejne vožnje goseničnih vozil. Večina raziskav sledenja trajektorije goseničnih vozil uporablja kinematične modele ter ne upošteva delovanja motorja in elektromehanske sklopitve. Metode modelnega prediktivnega vodenja se pogosto uporabljajo pri sledenju kolesnih vozil, manj pa je primerov uporabe pri goseničnih vozilih. V članku je predstavljen poskus uporabe metode MPC pri goseničnih vozilih in preverjanje, ali metoda dosega želene rezultate sledenja. Namen je dodajanje elektromagnetne sklopitve goseničnega vozila v proces sledenja trajektorije. V članku je najprej postavljen kinematični model, nato pa še dinamični model vozila. Nato so dodane lastnosti motorja in je vzpostavljen model dinamike elektromehanske sklopitve. Za preizkus učinkovitosti vodenja goseničnega vozila po metodi MPC je bilo opravljenih nekaj simulacij v okolju Simulink, vključno s sledenjem linearne in spiralne trajektorije. Poleg tega je bilo postavljeno nekaj eksperimentalne opreme in uporabljen je bil pomanjšani model goseničnega vozila za izvedbo preizkusov v realnem okolju. Za uporabo algoritma MPC za sledenje trajektorije pri goseničnih vozilih z električnim pogonom je bil postavljen model dinamike elektromagnetne sklopitve vozila in opravljena simulacija algoritma za vodenje MPC v programski opremi za simulacije, nato pa je bilo opravljenih še nekaj eksperimentov v realnem okolju. V simulacijah sta bili upoštevani kinematika in dinamika na osnovi predlaganega algoritma vodenja, delovanje pa je bilo preverjeno za ravno in spiralno trajektorijo. V članku je predstavljena uporaba metode MPC za vodenje goseničnih vozil z električnim pogonom po trajektoriji pri razmeroma majhnih hitrostih vozila in brez upoštevanja pojava zdrsavanja. Metoda MPC je primerna za sledenje trajektorije pri goseničnih vozilih. Z matematičnimi izpeljavami je bilo mogoče določiti model elektromehanske sklopitve za gosenično vozilo. V nadaljnjih simulacijah in eksperimentih je bilo ugotovljeno, da lahko metoda MPC v kratkem času popravi smer vozila za odpravo trenutne napake in ohranitev trajektorije do cilja, tako pri linearni kot pri spiralni trajektoriji. Ko je vozilo na ciljni trajektoriji, se obdrži na njej. Pri premikanju goseničnih vozil po netlakovanem vozišču in med obračanjem prihaja do zdrsov. Zdrsavanje gosenic vpliva na natančnost sledenja trajektorije, kar pa ni bil predmet obravnave tega članka. V prihodnje raziskave sledenja trajektorije bi bilo treba vključiti tudi zdrsavanje. Poleg tega sta bili uporabljeni le razmeroma preprosti trajektoriji - ravna in spiralna. V prihodnjih raziskavah bo mogoče preveriti tudi učinkovitost vodenja pri zahtevnejših trajektorijah ter pri višjih hitrostih vozila. Novost predstavljene študije je uvedba elektromehanske sklopitve v vodenje sledenja trajektorije ter uporaba metode MPC, ki je pogostejša pri kolesnih vozilih, pri vodenju goseničnega vozila. Na osnovi rezultatov simulacij so bili opravljeni tudi eksperimenti v realnem okolju, kar je na tem raziskovalnem področju razmeroma redko. Rezultati predstavljene raziskave so uporabni v praksi in v znanosti ter bodo lahko pripomogli k implementaciji avtonomnih tehnologij v goseničnih vozilih. Ključne besede: sledenje trajektorije, gosenično vozilo, modelno prediktivno vodenje, elektromehanska sklopitev, goseničar, avtonomno vozilo, strojni vid *Naslov avtorja za dopisovanje: Univerza Jilin, Šola za strojništvo in letalsko tehniko, Kitajska, linzhou17@mails.jlu.edu.cn SI 43 Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, SI 44 © 2019 Strojniški vestnik. Vse pravice pridržane. Prejeto v recenzijo: 2019-01-15 Prejeto popravljeno: 2019-05-10 Odobreno za objavo: 2019-05-31 Motnje zaradi zaostalih napetosti pri laserskem spajanju s lojev nikljevih superzlitin v prahu Ozgur Poyraz1* - Melih Cemal Ku§han2 1TEI, Tusa§ Engine Industries Inc., Turkey 2Univerza Eski§ehir Osmangazi, Fakulteta za inženirstvo, Oddelek za strojništvo, Turkey Cilj predstavljene študije je oblikovanje modela za hitre termomehanske simulacije postopka laserskega spajanja slojev praškastega materiala in izvedba eksperimentalne izdelave preizkušancev z novimi prahovi nikljevih superzlitin. Ena od glavnih težav pri laserskem spajanju slojev praškastih materialov so zaostale napetosti in deformacije, ki se pojavijo pri segrevanju finih kovinskih prahov do tališča, ki mu sledi hitro ohlajevanje na začetno temperaturo. Deformacije zaradi zaostalih napetosti prinašajo težave z mehansko integriteto in geometrijo končnega izdelka, nad določeno mejo pa obstaja tudi tveganje prekinitev proizvodnega procesa zaradi drgnjenja med izdelkom in sistemom za nanos prahu. Kot rešitev omenjenega problema je predlagan model za hitre termomehanske simulacije po metodi končnih elementov, ki omogoča napovedovanje deformacij zaradi preostalih napetosti. Najprej je bil za termodinamične simulacije privzet model premikajočega se toplotnega vira z Gaussovo porazdelitvijo na srednji skali za določitev vnosa energije na enoto površine. Nato je bil uporabljen ekvivalenten vnos energije v celotne plasti kot površinski vir toplote med procesom gradnje po plasteh. Upoštevano je bilo tudi dodajanje slojev s tehniko popravljanja mreže. Predstavljeni model je bil eksperimentalno verificiran z laserskim spajanjem prahu zlitine Inconel 718, ki mu je sledila meritev izdelka z laserskim skenerjem na koordinatnem merilnem stroju. Rezultati modela deformacij so dosegli primerljivo stopnjo konvergence (< 10 %) z eksperimentalnimi rezultati. Rezultati za asimetrične preizkušance so razkrili, da imajo od geometrije izdelka odvisni učinki večji vpliv na deformacije zaradi zaostalih napetosti kot pa učinki, ki so odvisni od priprave, npr. nanašanje plasti prahu ali smer pihanja plina. Nadaljnje študije bodo lahko usmerjene v podrobne meritve zaostalih napetosti z rentgensko difrakcijsko analizo (XRD), raziskave drugih nikljevih superzlitin kot je Hastelloy-X, parametrične simulacije reprezentativnih delov geometrije in vključitev drugih tehnik skeniranja oz. vodenja žarka, kot je vzporedno skeniranje s fiksnim kotom ali cik-cak skeniranje. V članku je predstavljena uvedba modela za hitre termomehanske simulacije in eksperimenti s preizkušanci iz nikljevih superzlitin. Rezultati študije so nova spoznanja o deformacijah zaradi zaostalih napetosti in prispevek v bazo znanstvenega znanja z novo zasnovo preizkušancev ter metodologijo izdelave in vrednotenja za preučevanje deformacij. Ključne besede: dodajalna tehnologija, analiza po metodi končnih elementov, Inconel 718, lasersko spajanje slojev praškastega materiala, zasnova preizkušanca, termomehanske simulacije SI 44 *Naslov avtorja za dopisovanje: TEI, Tusa§ Engine Industries Inc., 26210 Eski§ehir, Turčija, ozgur.poyraz@tei.com.tr Strojniški vestnik - Journal of Mechanical Engineering 65(2019)5, SI 45 © 2019 Strojniški vestnik. Vse pravice pridržane. Prejeto v recenzijo: 2018-12-18 Prejeto popravljeno: 2019-03-22 Odobreno za objavo: 2019-05-07 Vpliv parametrov rezkarja na kontaktne lastnosti površin zob, obdelanih po postopku petih prehodov in po metodi Duplex Helical Yu Zhang1* - Hongzhi Yan2 - Zhiyong Wang1 - Qing Zhao1 1 Gozdarska in tehniška univerza srednjega juga, Inštitut za sodobne tehnologije mehanskih prenosov, Kitajska 2 Univerza srednjega juga, Državni laboratorij za visokozmogljivo kompleksno obdelavo, Kitajska Danes se za čelno rezkanje stožčastih zobnikov s spiralnimi zobmi in hipoidnih zobnikov uporabljata dve glavni metodi. Prva je tradicionalna metoda čelnega rezkanja v petih prehodih oz. s fiksno nastavitvijo. Druga metoda se imenuje Duplex Helical in ponuja številne prednosti, kot so večja učinkovitost obdelave, nižji proizvodni stroški, višja trdnost zob in možnost suhe obdelave. Metoda Duplex Helical zato postopoma izriva obdelavo v petih prehodih. Med posplošenimi teorijami obeh metod obstajajo pomembne razlike in korelacije. Raziskovalci zobniških prenosnikov so posvetili veliko pozornosti posplošeni teoriji stožčastih zobnikov s spiralnimi zobmi in hipoidnih zobnikov, izdelanih po metodi petih prehodov. Avtorjem pa niso znane objave o posplošeni teoriji metode Duplex Helical ter o razlikah in povezavah med obema teorijama. Glavni znanstveni problemi metode Duplex Helical še niso razrešeni, kar gotovo ne koristi uveljavljanju in uporabi te metode. Najprej so podrobno analizirana različna načela delovanja mehanizma za nagibanje rezkarja pri obeh metodah. Rezkar pri metodi Duplex Helical zahteva manjši notranji in večji zunanji radij rezila za dostop notranjega in zunanjega rezila glave do pastorka. Primerjava radijev rezkarjev pri obeh metodah pokaže, da zagotavljata pravo debelino zoba in zelo velik dolžinski profil na obeh straneh pastorka. Pri obdelavi stožčastih zobnikov s spiralnimi zobmi in hipoidnih zobnikov z istočasno spremembo kota rezila in nagiba rezkarja je mogoče zmanjšati veliko »naravno« dolžinsko profiliranje. To je tudi glavna zamisel posplošene teorije metode Duplex Helical. Primeren radij ukrivljenosti (radij ukrivljenosti je razdalja med referenčno točko na rezilu in osjo rezkarja, pravokotno na rezalni rob) v referenčni točki je mogoče spremeniti s prilagoditvijo kota rezila in nagibnih kotov rezkarja za ustrezno dolžinsko ukrivljenost na obeh straneh pastorka. Predstavljena je primerjava in numerična analiza vpliva radija rezkarja na lastnosti ubiranja površine zoba pri obeh metodah obdelave. Vpliv kota rezila in radija rezkarja na kontaktne lastnosti površine zoba pri metodi Duplex Helical je ugodnejši kot pri postopku obdelave v petih prehodih. Parametra se zato pogosto uporabljata za optimizacijo ubiranja stožčastih zobnikov s spiralnimi zobmi in hipoidnih zobnikov. Končno je bila opravljena tudi primerjava in numerična analiza vpliva kota rezila na lastnosti ubiranja površine zob pri obeh metodah obdelave. Rezultati analize so: (1) Pri postopku obdelave v petih prehodih imajo velikost, položaj in smer nosilnega kontakta ter vrednost neujemanja površin zob le majhen vpliv na radij rezkarja, radij glavne osi kontaktne elipse se poveča s povečanjem radija rezkarja na dejavni strani, medtem ko je na nedejavni strani ravno nasprotno; sprememba radija manjše osi kontaktne elipse je majhna na obeh površinah. (2) Pri metodi Duplex Helical se velikost nosilnega kontakta povečuje z večanjem radija rezkarja, hitrost sprememb na dejavni in na nedejavni strani pa je različna. Vrednost neujemanja površin zob se zmanjšuje s povečevanjem radija rezkarja. (3) Pri enakih konstrukcijskih parametrih in parametrih rezkarjev imajo zobniki, ki so izdelani po metodi Duplex Helical, večjo upogibno trdnost zob kot zobniki, ki so izdelani po metodi petih prehodov. Razlika lahko znaša do 2,8 %. Predstavljeno raziskovalno delo predstavlja prispevek na področju posplošene teorije metode Duplex Helical ter ima svoj teoretični in tehnični pomen kot osnova za učinkovito izbiro optimalnih nastavitev strojev za izdelavo stožčastih zobnikov s spiralnimi zobniki in hipoidnih zobnikov. Ključne besede: metoda Duplex Helical, stožčasti zobniki s spiralnimi zobniki in hipoidni zobniki, posplošena teorija, parameter rezkarja *Naslov avtorja za dopisovanje: Gozdarska in tehniška univerza srednjega juga, Inštitut za sodobne tehnologije mehanskih prenosov, Shaoshan South Road, No. 498, Changsha 410004, Hunan, Kitajska, zhyu@csuft.edu.cn SI 38 Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, SI 46 © 2019 Strojniški vestnik. Vse pravice pridržane. Prejeto v recenzijo: 2019-01-15 Prejeto popravljeno: 2019-04-13 Odobreno za objavo: 2019-05-16 Analiza vplivnih parametrov na togost podporne konstrukcije velikih radialnoaksialnih ležajev Spasoje Trifkovic1 - Nebojša Zdravkovic2 - Milomir Gašic2 - Mile Savkovic2 - Goran Markovic2* 1 Univerza v Vzhodnem Sarajevu, Fakulteta za strojništvo, Bosna in Hercegovina 2 Univerza v Kragujevcu, Fakulteta za strojništvo in gradbeništvo v Kraljevu, Srbija Pri nekaterih vrstah žerjavov in gradbenih strojev, kot so portalni žerjavi, nakladalniki in bagri, se obremenitve prenašajo na žerjavno progo oz. na podlago preko okvirja podvozja, ki je narejen iz škatlastih nosilcev. Pri takih nosilnih konstrukcijah lahko pride do privzdigovanja ene ali več podpor od žerjavne proge oz. od terena. Izguba stika s podlago je lahko posledica neravne podlage ali pa, kar je bolj redko, napak pri izdelavi. Posledično prihaja med delom do prerazporejanja vertikalnih reakcijskih sil v podporah in do dodatnih deformacij nosilne konstrukcije. Togost velikih radioaksialnih ležajev je majhna in dodatne deformacije nosilne konstrukcije zato neposredno vplivajo na njihovo delovanje. Od togosti je odvisna tudi nagnjenost ležajnih podpornih konstrukcij k deformacijam. V strokovni literaturi ni veliko objav o vplivih geometrijskih parametrov na togost okvirjev podvozij._ V članku je opisana raziskava, ki je bila opravljena z namenom ustvarjanja analitičnega modela značilnega okvirja podvozja za bagre. Model opredeljuje teoretične odvisnosti med geometrijskimi parametri škatlastih nosilcev in velikostjo dodatnih sil v podporah okvirja, ko je ena od podpor izpostavljena določeni začetni deformaciji. Te dodatne sile so značilne za celotno togost podporne konstrukcije velikega radialnoaksialnega ležaja. Statično nedoločena konstrukcija je bila razrešena po metodi sil. Za natančnejšo obliko zapisa vplivnih koeficientov v Maxwell-Mohrovih integralih je bila določena teoretična odvisnost med upogibnimi in torzijskimi vztrajnostnimi momenti škatlastega nosilca. Izkazalo se je, da relativna napaka aproksimacije pri večini običajnih množic geometrijskih parametrov ne presega 6,5 %. Na osnovi te aproksimacije je bil opredeljen količnik upogibne in torzijske togosti škatlastega nosilca. Na ta način je bila pridobljena množica strnjenih enačb, v katerih je ena od neznank dodatna sila v eni od podpor okvirja. Velikost dodatnih sil je v vseh podporah enaka zaradi simetrije modela. Sistem enačb je bil razrešen za različne parametre realnih konstrukcij okvirja podvozja. Podani so rezultati za različne vrednosti dolžine okvirja L (4,0 mm; 4,5 mm; 5,0 mm; 5,5 mm; 6,0 mm), razmerja med višino h in širino b škatlastega prereza k = h/b (območje 1 do 3) in razmerja med upogibnim vztrajnostnim momentom vzdolžnih in prečnih nosilcev e (1,0; 1,5; 2,0). Po numerični analizi z orodji za iskanje krivulj najboljšega prilega v programskem paketu Matlab z zanemarjenim vplivom koeficienta e je bila določena sprememba togosti okvirja v odvisnosti od spremembe dolžine ogrodja L in parametra k, izražena z razmerjem dodatnih sil. Rezultati analitičnega modela so bili tudi eksperimentalno preverjeni, za izvedbo eksperimentov pa je bilo zasnovano in izdelano posebno preizkuševališče. Preizkusni modeli okvirja podvozja so bili narejeni iz kvadratnih cevi z enotno debelino stene 2,8 mm in prerezom b * h = 60 mm * 60 mm oz. b * h = 60 mm * 120 mm. Dimenzije preizkušenih modelov (1200 mm x 1200 mm in 1200 mm* 800 mm) so približno petkrat manjše kot pri realnih okvirjih podvozja. Relativna napaka predstavljenega analitičnega modela v primerjavi z rezultati preizkusov ne presega 6 %. Določeno razmerje med upogibno in torzijsko togostjo škatlastih nosilcev je pomembno za nadaljnje raziskave nosilnih konstrukcij žerjavov in gradbenih strojev, saj znatno poenostavlja analitične modele. To razmerje omogoča ustvarjanje kompaktnih modelov, ki so primerni za uporabo optimizacijskih metod ter zagotavljajo rešitve s povečano upogibno in torzijsko togostjo, kakor tudi zmanjšanje mase. Ugotovljene odvisnosti med geometrijskimi parametri okvirja podvozja in njegovo togostjo omogočajo inženirjem, da poskrbijo za primerno togost konstrukcij, ki je potrebna za pravilno delovanje in dolgo življenjsko dobo velikih radialnoaksialnih ležajev. Ključne besede: ravninski nosilni okvirji, upogibna in torzijska togost, veliki radialnoaksialni ležaji, deformacije, dodatne sile v podporah, eksperimentalna verifikacija SI 46 *Naslov avtorja za dopisovanje: Univerza v Kragujevcu, Fakulteta za strojništvo in gradbeništvo v Kraljevu, Srbija, markovic.g@mfkv.kg.ac.rs Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, SI 47 © 2019 Strojniški vestnik. Vse pravice pridržane. Prejeto v recenzijo: 2019-03-02 Prejeto popravljeno: 2019-05-06 Odobreno za objavo: 2019-05-31 Študija rezalnih sil in površinske hrapavosti pri čelnem rezkanju trdega jekla SKD61 Tien Dung Hoang - Nhu-Tung Nguyen* - Buc Quy Tran - Van Thien Nguyen Tehniška univerza v Hanoju, Vietnam Cilj predstavljene študije je preučitev vpliva rezalnih parametrov na površinsko hrapavost in amplitude rezalnih sil. Namen raziskave je iskanje načinov za izboljšanje kakovosti površine po čelnem rezkanju z različnimi optimizacijskimi metodami. Rezalni parametri postopkov čelnega rezkanja, kot so rezalna hitrost, globina rezanja in podajanje, pogosto vplivajo na obdelano površino in na rezalno silo. S preučitvijo vpliva rezalnih parametrov na hrapavost površine in rezalno silo ter analizo odvisnosti med rezalno silo in hrapavostjo površine lahko izboljšamo kakovost obdelanih površin in zmanjšamo porabo moči. Za eksperimentalno čelno rezkanje je bilo izbrano toplotno obdelano jeklo SKD61 s trdoto 46 HRC. Za tri nadzorovane dejavnike v treh ravneh (rezalna hitrost, aksialna globina rezanja in podajanje) je bilo izbrano najprimernejše ortogonalno polje L27 z meritvami štirih zmogljivosti - amplitude rezalnih sil v treh smereh (podajanje, normalna smer in aksialna smer) in površinske hrapavosti. Vpliv pogojev rezanja na amplitude rezalnih sil in površinsko hrapavost je bil analiziran in modeliran po metodi ANOVA. Optimalna vrednost površinske hrapavosti je bila določena po metodah Taguchi in ANOVA. Najpomembnejši dejavnik vpliva na površinsko hrapavost je podajanje (86,594 %), najpomembnejši dejavnik vpliva na amplitudo rezalnih sil pa aksialna globina rezanja (72,891 % amplitude podajalne sile in 70,039 % amplitude normalne sile). Ostali dejavniki imajo drugačen vpliv na površinsko hrapavost in na amplitudo rezalnih sil. Amplitude rezalnih sil se povečujejo s povečevanjem aksialne globine rezanja in podajanja. Površinska hrapavost se zmanjšuje z zmanjševanjem podajanja, s povečanjem aksialne globine rezanja z 0 mm na 0,3 mm in s povečanjem rezalne hitrosti z 0 m/min na približno 130 m/min. Površinska hrapavost se povečuje s povečanjem podajanja, s povečanjem aksialne globine rezanja z 0,3 mm na 0,5 mm in s povečanjem rezalne hitrosti s 130 m/ min na 200 m/min. Trend površinske hrapavosti je enak trendu amplitude rezalnih sil. Določena je bila optimalna vrednost površinske hrapavosti, ki spada med najpomembnejše vrednosti za izboljšanje kakovosti obdelanega izdelka. Najprimernejša regresijska krivulja za amplitude rezalnih sil in površinsko hrapavost je bila kvadratna krivulja s stopnjo zaupanja nad 93,74 %, kar so potrdili tudi rezultati eksperimentov. Proces optimizacije hrapavosti površine je bil opravljen po metodah Taguchi in Anova, rezultati pa so bili zelo podobni (razlika površinske hrapavosti je približno 4,58 %). Optimizirani rezultati metode ANOVA so bili boljši od rezultatov metode Taguchi. Optimalna vrednost površinske hrapavosti 0,1531 ^m je bila ugotovljena pri rezalni hitrosti 113,0595 m/min, aksialni globini rezanja 0,2636 mm in vrednosti podajanja 0,05 mm/utor. Modeli amplitude rezalnih sil in površinske hrapavosti do zdaj še niso bili uporabljeni za izboljšanje obdelanih površin in zmanjšanje rezalnih sil v CNC-sistemih za adaptivno vodenje rezkanja, zato je to ena od možnih smeri za prihodnje raziskave. Glavni prispevki študije so: (1) vrednotenje vpliva rezalnih parametrov na hrapavost površine in amplitudo rezalnih sil, (2) vrednotenje trendov površinske hrapavosti in amplitude rezalnih sil pri spremembi rezalnih parametrov, in (3) določitev optimalnih pogojev rezanja po različnih metodah. Predstavljeni pristop je mogoče uporabiti v industriji za izboljšanje kakovosti površine trdega jekla SKD61 po čelnem rezkanju. Ključne besede: hrapavost površine, rezalna sila, metoda Taguchi, ANOVA, SKD61, optimizacija, čelni rezkar *Naslov avtorja za dopisovanje: Tehniška univerza v Hanoju, Number 298, Cau Dien Street, Bac Tu Liem Distric, Ha Noi City, Vietnam, tungnn@haui.edu.vn SI 47 Strojniški vestnik - Journal of Mechanical Engineering 65(2019)6, SI 48 © 2019 Strojniški vestnik. Vse pravice pridržane. Prejeto v recenzijo: 2018-11-30 Prejeto popravljeno: 2019-05-15 Odobreno za objavo: 2019-05-31 Karakterizacija vplivov in optimizacija konstrukcije mikroturbine s prečnim tokom na osnovi numeričnih simulacij Endashaw T. Woldemariam - Hirpa G. Lemu Univerza v Stavangru, Znanstvena in tehniška fakulteta, Norveška V članku je predstavljena študija, katere cilj je bil preučitev zmogljivosti mikroturbine s prečnim tokom z numeričnimi simulacijami, izboljšanje pregleda nad vplivom izbranih konstrukcijskih parametrov na zmogljivost in optimizacija zasnove. Cilj je turbina z izboljšano zmogljivostjo kot funkcijo konstrukcijskih parametrov in posledično izboljšanim hidravličnim izkoristkom. Študija poleg tega uvaja optimizacijski okvir na osnovi numeričnih simulacij, ki bo na voljo za podobne aplikacije. Uporabljen je bil pristop s podporo metamodela na osnovi numeričnih simulacij za optimizacijo zasnove vodne turbine s konstrukcijskimi parametri, kot so kot ventila, ukrivljenost vstopnega loka in premer osi rotorja, ki pomembno vplivajo na učinkovitost proizvodnje električne energije. Uporabljeno je bilo optimizacijsko orodje s podporo metamodela, ki je namenjeno računsko zahtevnim problemom vrste črna škatla, in sicer globalni optimizacijski algoritem z enim ciljem. Komercialno orodje ANSYS je bilo povezano z optimizacijskim orodjem za modeliranje in izvedbo simulacij računalniške dinamike fluidov (CFD) po 2D- in 3D-modelu. Odzivi optimizirane zasnove so bili primerjani z odzivi izvirne zasnove pri različnih vrednostih konstrukcijskih parametrov. Nato so bili uporabljeni objavljeni rezultati eksperimentalnih preizkusov s podobnimi začetnimi pogoji za preverjanje modelov v posameznih stopnjah turbine. Uporabljena je bila tudi analiza občutljivosti za preverjanje numeričnih rešitev CFD, oz. algoritma SIMPLE in PISO pri enačbi za popravek tlaka v sklopitvi tlak/hitrost pri reševanju Navier-Stokesovih enačb za turbulentni tok. Ugotovljeno je bilo, da imata ukrivljenost vstopnega loka in kot odprtja šobe pomembno vlogo pri določanju kota natoka vode na vstopu v prvo stopnjo. Analiza tokovnic je tudi pokazala, da premer osi ovira tok fluida proti drugi stopnji in povečuje turbulenco, zaradi česar je prispevek moči od te stopnje razmeroma majhen. Os je bila zato med optimizacijo močno stanjšana. Časovno diskretizirani algoritmi so v analizi dali stabilen odzivni cikel za koeficient momenta in dobro ujemanje s trendom odziva eksperimentalnih preizkusov v obeh stopnjah cikla, zato so bili uporabljeni v izračunih hidravličnega izkoristka za analizo prehodnih in stacionarnih stanj. Rezultati izračunov izkoristka v analizi stacionarnega stanja so pokazali, da je izkoristek optimiziranega modela večji od numeričnih hidravličnih izkoristkov, določenih za izvirne modele pri 80 % in 100 % odprtem ventilu za izbrani vrtilni hitrosti 250 vrt./min in 350 vrt./min. S predlaganim pristopom se je močno skrajšal računski čas. Članek obravnava numerični pristop k optimizaciji zasnove vodne turbine s prečnim tokom. Rezultati tovrstnih simulacij zahtevajo eksperimentalno verifikacijo, zato so bili kot sekundarni vir uporabljeni objavljeni rezultati eksperimentov. Čeprav so bili v simulacijah uporabljeni enaki parametri kot v eksperimentih, bo potrebna še dodatna eksperimentalna validacija in verifikacija modelov. V predstavljeni analizi in optimizaciji turbine s prečnim tokom na osnovi numeričnih simulacij so bila kombinirana orodja za reševanje razmeroma kompleksnih problemov. Pristop se je izkazal za učinkovitega, optimizirani model pa zagotavlja boljšo zmogljivost in izkoristek kot originalni model v podobnih delovnih pogojih. Razvita metoda bo uporabna tudi za prihodnje raziskave turbostrojev s podobnimi delovnimi karakteristikami. Optimizacijsko orodje je še posebej zanimivo za optimizacijske probleme na osnovi visokodimenzijskih numeričnih simulacij, saj zmanjšuje računsko zahtevnost iskanja optimalne zasnove kompleksnih sistemov. Ključne besede: turbina s prečnim tokom, mikro vodna turbina, numerična simulacija, optimizacija na osnovi metamodela SI 48 *Naslov avtorja za dopisovanje: Univerza v Stavangru, Znanstvena in tehniška fakulteta, Norveška, Hirpa.g.lemu@uis.no Information for Authors All manuscripts must be in English. Pages should be numbered sequentially. The manuscript should be composed in accordance with the Article Template given above. The maximum length of contributions is 10 pages. 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Lemu: Numerical Simulation-Based Effect Characterization and Design Optimization of a Micro Cross-Flow Turbine 329 343 351 366 375 386 9770039248001