Strojniški vestnik - Journal of Mechanical Engineering 61(2015)3, 167-175 © 2015 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2014.2294 Original Scientific Paper Received for review: 2014-11-11 Received revised form: 2015-01-16 Accepted for publication: 2015-01-29 Deterministic Mathematical Modelling of Platform Performance Degradation in Cyclic Operation Regimes Nenad Kapor1 - Momcilo Milinovic1- Olivera Jeremic1 - Dalibor Petrovic2 JUniversity of Belgrade, Faculty of Mechanical Engineering, Serbia 2University of Defense, Military Academy, Serbia This paper considers the modelling of extreme-capability working platforms that are operated in periodic cycles, each cycle having a predefined number of operations that affect the working surfaces. A novel hypothesis is introduced about the platform-degrading effects that cause an equivalent decrease in the successful operations after repeated cycles. Deterministic modelling, based on the basic equations of Lanchester and Dinner, is generalized here to include coupling between parameters. The newly developed mathematical model of performance degradation is in good agreement with both experimental measurements and numerical simulations. It is assumed that the new variables and their correlations link the Gaussian distribution and the observed performances of the testing platforms. Relative probability dispersions of affected surfaces are derived, as a new indirect referencing figure of merit, to describe simulations and compare them to experimental test data. The model proves a hypothesis that the degrading effects are a function of the platform capacity, frequency of operations and the number of available cycles. Degradation effects are taken into account through an equivalent decrease of effective operation capacities, reflected on the properties of the affected operating surfaces. The obtained estimations of degradation could be used in the planning of platform capacity as well as in the selection of real affected surfaces in various machining systems and for a wide range of different materials. Keywords: cycles, operations, extreme machine platforms, probabilities, deterministic modelling Highlights • Modelling of extreme-capability working platforms that are operated in periodic cycles. • Proposed methodology for performances degradation caused by operations composed in cycles. • Using Gaussian probability distribution law to predict degradation measures. • Predicting the changes of probability dispersion based on modelling and experimental data. • Determination of an analytical model based on a hypothesis that the degrading effects are a function of the platform capacity, frequency of operations and the number of available cycles. 0 INTRODUCTION Machines operating in cycles and their properties have not been studied in depth in literature and, as such, are not well described by integral mathematical models. If the effects of their operation are actions on the given working surfaces under given constraints, then the quality of the affected surfaces can be described by reliability functions. In this manner, the operating capabilities of the platform can be determined. A majority of the published papers use a standard approach to the measured performances that depend on the machine's designed purposes. Such processes are described in [1] to [3] for the abrasive flow machines (AFM) with which material is hardened by randomly treating the working surface with abrasive particles with polymeric fillers, and dispersed within the flow media. The authors of [1] classified the work piece parameters into three groups based, among others, on the number of cycles (operations) and the machining time. Some of these parameters were determined experimentally in [2], in which the authors recognized that the parameters denoted as the creeping time and the cycles frequency have impact on the quality of the machining process. In [3], the authors experimentally prove that the aforementioned parameters influence the process. Common for all three papers is that they do not include hidden random effects caused by particles affecting the surfaces in cyclic operations, although such effects significantly influence the quality of the surface treatment. In all three papers, there is no mathematical modelling of the process. Another similar type of machine with cyclic operation affecting working surfaces is described in [4] as shot-peening (SP) platforms. They bombard a surface with spherical beads to increase the material fatigue strength. The physical modelling of the influence of the bead shapes on the performance of the surface hardening process is presented in [5]. Random surface effects due to bombing cycles are a result of the quality of the machine's performance. However, the connection between the effects and the particular operations is missing in [5]. Paper [6] utilizes a risk function to consider the example of solar rays hitting a surface as a random process. In fact, the determination of the risk function dumping requires much more precise estimations of the distribution if probabilities *Corr. Author's Address: University of Belgrade, Faculty of Mechanical Engineering, Kraljice Marije 16, 11000 Belgrade, Serbia, mmilinovic@mas.bg.ac.rs 167 Strojniski vestnik - Journal of Mechanical Engineering 61(2015)3, 167-175 of the effect occurrence on the attacked surface. However, this paper lacks a mathematical model of random disturbances of these probabilities. Common for the description of the processes in both AFM and SP machines, as well as the processes described in [6], is that they lack the deterministic or probabilistic mathematical modelling of cycles and their parameters on the efficiency of the final process. Mechanical engineering of extreme machines in defence technologies have particular operations grouped in the cycle regimes. These operations affect the working surfaces or areas, with constrained machine capacities with regard to operation numbers. The modelling of efficiencies in such cases is usually done using the deterministic differential theories of operational research. This approach is based on the so-called Lanchester and/or Dinner equations of particular probabilities and their distribution laws, as presented in detail in [7], as well as in [8]. Their equations use variable attrition rates as the frequencies in operations probabilities, similar to [9], in which surface point effects are taken with variable probabilities. The modelling of cycles' efficiencies in these references is performed with coupled equations, in which two subjects simultaneously affect each other. Their actions are interdependent but different; their efficiencies also evolve differently over time. This approach is not entirely useful to define a standalone efficiency estimation for a single subject. A mathematical model of the equipment with constrained capacities that generates identical repeated operations in a given order is presented in [10] for air platform equipment. The main contribution of this paper is the treatment of the action on the working surfaces as a random process, but the probability distribution laws on the affected surfaces are missing. The two-dimensional Gaussian distribution laws, used for welding processes, as referred in [11], could be useful in the estimation of random processes on surfaces. According to the state-of-the-art analysis as presented in the cited papers, there is no comprehensive mathematical model, proven by experimental data, that would be capable of explaining relations between the machine's cyclic performances, its capacities and the quality of randomly affected surfaces, as well as the designed processing time. This is because the papers do not consider two repeating processes simultaneously acting on a single object (one as working and the second as redundant or parasitic), which together change the quality of the expected performances. The objective of this paper was to develop a general joint mathematical model that includes all pertinent factor that influence the final efficiency of processing, thus enabling the simulation and evaluation of these parallel processes. Based on the specific requirements for sequential processing of the surfaces, a mathematical model is developed using a deterministic approach treating the surface processes as random variables. The objective was to test the efficiency of cyclic operations affecting the working surfaces, essentially by considering differences caused by the capacities and operation rates of the processing machine. This was shown using the experimental data on operation platforms with extreme performances. The novelty of the approach presented in this manuscript is the redesign of coupled deterministic equations done in a new manner. In the literature above, these equations are employed to describe the mutual effects of objects as a function of the elapsed process time. This approach in the literature makes the time functions dependent on the performances of two objects. In our approach, one object executes two operations in parallel, one of them comprising the working process itself, and the other, parasitic, occurring as self-degradation dependent on the first one. Both happen on the same object, i.e. the operation platform performing the same action. The new approach composes deterministic equations to describe this and to measure changes in the platform efficiency. With our approach, the quality of the working process is the convolution of both kinds of operations in one cycle, as well as their frequencies. The number of cycles influences the random arguments and reflects the probabilities of working surface coverage that follow the two-dimensional Gaussian distribution laws. This was taken as the measure of the changes in quality due to self-degradation. 1 EXPLANATION OF THE GENERAL MODEL The model offers the possibility of evaluating the degradation of the platform performance, with regards to the equipment and devices contained within the platform. The quality of the affected surface is regarded as the dimension of probability dispersion. This dimension appears during the execution as the consequence of cycle duration and the operations frequency, as well as of the capacities of the platform. The approach presented in [12], which developed operations frequency coupled with execution probabilities as the combined attrition parameter, was used in developing our general model. Changes 168 Kapor, N. - Milinovic, M - Jeremic, O. - Petrovic, D. Strojniski vestnik - Journal of Mechanical Engineering 61(2015)3, 167-175 of probability dispersions of random values on the affected surface appear in the form of the Gaussian distribution law. The degradation of the platform properties through the operation cycles is represented by changes in the Gaussian distribution. This is valid under the assumption that one particular Gaussian distribution function is valid for each cycle in the working regime. In our approach, this function is distributed in successive cycles in the form of extended probability dispersions of both random arguments in the two surface directions. Consequently, the changing of efficiencies over time is measured by the resulting effects on the new randomly affected surfaces. The decrease in the efficiency with each new cycle in reflected in the new less-affected surfaces. This also results in the degradation of the working platforms' capabilities caused by less effective particular operations in the cycles. The cause of this degradation could be a consequence of rapid high-energy operations realized in short-time sequences (high mechanical power values) in successive, orderly repeated cycles, similar to those described in paper [13]. However, according to that paper, the affected points on the surfaces do not obey any probabilistic law, and thus there is no error distribution as a modelling parameter. In our research, we use the changes of the probability dispersion (PD) after each cycle, due to all the errors in the cycle, as a measure of the platform efficiency. These changes are caused by the generator of the cycles and by its self-degradation, and are reflected in the decreased number of declared operations. This makes the designed operational capacities of the platforms less effective with the number of cycles. In order to estimate the degraded platform performances by means of time-based simulation, new relative parameters have been accepted in the modelling. The deterministic modelling of the estimations of the so-called vulnerability performances is presented in [14] and [15]. The performances considered there are similar to our degrading platforms' performances. The models presented in [16], called Pexpot, Levpot and Dynpot, were also developed as vulnerability considerations based on the attrition rate function and thus indirectly describe the kind of expected degradation capabilities. An essential difference of our model is that the degradation of the system appears as a direct consequence of self-degradation caused by the effects of the repeated cycles. The designed frequencies and functional probabilities, contained in each operation, are reflected in the full platform capacity on the affected surfaces. This effect makes the proposed model more useful in planning the redesigning of platform capacities for required affected surfaces. 2 MATHEMATICAL MODELS In the presented model, the platform has the capacity of Mpa particular operations oriented toward the working surface. These operations occur in dynamic regimes with successive frequencies X and a probability of surface action of approximately p = 0.997. This is provided using the maximum technical dimensions of the surface, which correspond to the 64 PDav2. The width and depth of the surface used eight of the same average probability dispersions PDav, in both surface directions. Average probability dispersions PDav are taken as an equal of the expected Gaussian distribution of two-dimensional random arguments. The probability variations are represented as functions of the cycle number and of the full capacity of operations. The designed properties of these processes are consequently the function of probability changes. The adopted hypothesis is that the degradation of the platform performances is an imaginary effect, able to be explained by the values of the effective and ineffective numbers of operations. This ensures a possibility of considering the ineffective number to be a value increasing with the number of cycles during the exploitation time. In that sense, the increasing number of ineffective operations corresponds to the increase of cycle probabilities dispersions. Operative consumption is realized in cycles with the same sequential probabilities of operations, p, as in [10]. In that case, the frequency of executions of real operations, as the real rate of operation, is: dM dt p = -Xp = -ap. (1) This determines the remaining number of operations as the Mp=Mp (t) in each moment of time t in the cycle duration interval. It is expected that the probability would not have a fixed value but would vary over exploitation time. The changes of probability function p could mean random changeable performances that disturb the rate of real operations on the working surfaces. The changes in probabilities affect the rate of real operations Mp in Eq. (1). This is not really possible because the frequency of operation executions is a designed property of the platform hardware. The present hypothesis has only an imaginary effect. Deterministic Mathematical Modelling of Platform Performance Degradation in Cyclic Operation Regimes 169 Strojniski vestnik - Journal of Mechanical Engineering 61(2015)3, 167-175 The acceptable solution could be to recalculate the influence of the number of ineffective operations on the new probable dispersion PD reflected in a new Gaussian distribution but for the unchanged execution operations probabilities. The consequence is that the model has to consider the extended working surfaces, with new dimensions 64 PD2 engaged in operations after each cycle. The platform degradation, as an imaginary effect, is a process in the real cycle time and is simultaneously parasitic in real operations. A new value of the modified equivalent number of operations mp(t) is diminished by this imaginary effect. This is generated as a current and recalculated capability of working platform. The new value is lower than the real number of the remaining operations Mp(t). At the very beginning, it is equal to the real available capacity mp0 = Mp0. The reason is underpinned by the fact that the model of self-degradation is viewed as a new, fictive rate of equivalent non-effective operations changes mp, which is not equal to the rate of real operation Mp . This orients the mathematical model to consider the share of degraded value on each of the real operations and, by that effect, redesign the remaining number of operations available on the platform. Such a transformation implies that the degrading rate of mp and the real rate Mp during each cycle are proportional to the remaining equivalent dimensionless number of operations 1/mp. The correction coefficient is the portion of one operation within the actual remaining equivalent number mp. Based on the previous concept, the differential equation for the degradation rates, Eq. (1), becomes: dmr 1 dt ■ = -ar (2) Since the model of equivalent numbers is a function of time and the current equivalent number mp , as the instantaneous remaining capacity, the platform performances are degraded continually with each cycle. This always means a new valid number of operations with regard to the remaining capacity. It is inappropriate to use the approach as constant and fixed for any platform capacity since it is dependent on the available number of operations. The relative degradation of the platform capacity is taken as more acceptable in modelling with the functional ratio ^p(t)=mp(t) /m where ^p (t = 0) = 1, as the current relative capacity degradation of the platform. The general differential equation (Eq. (2)) of new functional by methodology given in [7], is: d^p dt ■ = -a P VpM2Pq (3) If the platform, under the same conditions, executes repeating working cycles n times, in equal time intervals for each cycle of At , and the rate of operations in the cycle is the same, then using Eq. (3), any of i cycles (where i = 1, 2, ..., n) is used at the beginning of a new, redesigned equivalent number of operations from the previous cycle. The current relative degrading in cycle is defined as a new differential equation: d Vp. 1 P, . (4) ■ = -ap dt p i = l, 2,3. VP mp pi p(i-i) The solution of Eq. (4) is: Mp.(t) = 1 -- 2a r f(i-i) (5) The function of the current relative capacity degradation of the platform full capacity after (i-1), and during the ith cycle at an instant (i-1)At < t < iAt, similarly to [10], is: i-1 Mp(t) = Mpt (tj)' (6) j=1 with the condition H-"^j) for the i = 1. The estimation of the relative efficiency of the current process is the function of the affected and the initial working surface. This functional is determined for the unaffected, remaining surface at each cycle and the final working surface from the previous cycle, taken as initial in the current one. It is given in the form: Mpi (t) = 1 - Si (t) S(i-1) (7) The differential equation of the relative efficiency of the current process, as the remained relative surface within operation cycle considered as the degraded ones, according to a similar differential equation in [7] and [10], is: dt - = -Ui Vpi Ma , 0 < t < At. (8) If the platform operates in cycles without degradation effect, its = 1= the functional \xp 170 Kapor, N. - Milinovic, M - Jeremic, O. - Petrovic, D. 1 t. m Strojniski vestnik - Journal of Mechanical Engineering 61(2015)3, 167-175 does not affect Eq. (8) and the coupling of Eqs. (4) and (8) is lost. Consequently, the relative efficiency of the current process, denoted by nc., during the un-degrading surface processing in the cycles is described by: d £ TT * dt (9) In both Eqs. (8) and (9), the operation number in one cycle is the designed capability, and could be variable. This depends on the designed cartridge capacity used for continual operations in the short impulse regimes. The well-balanced example between the number of operations and the covered affected surface in one cycle is the referent platform given in [10]. It uses cartridges of maximum Np (At) = 8, and its cycle expires in 4.4 seconds. The accepted functional designed capability of the platform, redesigned for the considered example, is: Ut =lp 1.82a (10) Appropriate values need to be calculated for each platform cartridge with their declared performances regarding the expected affected surface. The solution of Eq. (8) is: „2 U\m H