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: ZW j=1 + A = 0, i = 1,2,3, (1) where 8i]- is the flexibility coefficients, Xj the unknown forces and moment, A, the displacement at i due to the external loading in the direction of the restraint at i. The flexibility coefficients 8ij are determined by the Maxwell-Mohr integral [17]: 8 = f-î-LZ-GI, ■J MMdz- EL I MMl*. EI J Ldz EA ■] KQQdz. GA tKQ.O. J y^y^dz. (2) GA For the considered case (no axial forces nor horizontal bending, the influence of shear forces is neglected), 8has the form: 8„ = f—î-lz-1 GI, ■f MMLdz^ 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 / /2 s+2-//- (i+/j /+/2m )+ 3 EI 2/3 S = 2 22 EI /22 L GI,, /2 L 3 EIx2 6 EIx3 2 GI,- 8 =-ll2 2L EIx 2 GIa 812 = l22 (3lj + 2l2 ) (4) 813 = _ l2 (2l1 +12 ) _ l2L EI GI, 3 3 EL 823 =_ EI As can be seen from Eq. (4), bending stiffness (EI) and torsional stiffness (GI) 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: Sih + S)3 Ix = 2-i-+ 2 x 12 i^Sit+s{„-S) 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 <