Application of numerical integration methods to continuously variable transmission dynamics models a

Transcription

1 ttps://doi.org/ /ssconf/ Application of numerical integration metods to continuously variable transmission dynamics models a Orlov Stepan 1 1 Peter te Great St. Petersburg Polytecnic University Russian Federation, , St.Petersburg, Polytecnicesaya, 29 Abstract. Te expansion of digital engineering tecnologies in te framewor of te fourt industrial revolution relies on core tecnologies for matematical modeling and computer simulation of pysical processes. Time required for a computer simulation and te quality of numerical solutions are ey factors tat ultimately ave impact on product quality and te amount of waste at product design stage. Tis paper is aimed on te reduction computer simulation time for continuously variable transmission (CVT) models. Te modeling of te device as a set of deformable rigid bodies wit numerous contact interactions is an extraordinary problem addressed in [1]. Model dynamics is described in terms of ordinary differential equations (ODE) of motion. Te system of ODEs as about 300 variables (generalized coordinates and speeds). Despite relatively low dimension of te model, computer simulation using traditional numerical integration metods taes long time to run. Te reason for tis is te stiffness of te ODE system. In tis paper, we present te results of an investigation aimed on finding a numerical integration metod appropriate for problems of tis ind. 1 Introduction Te world scientific and industrial society is now entering a new pase of its development, called Industrie 4.0 in Europe. One of te main features of tis pase is te expansion of digital engineering tecnologies, wic dominated until recently, in te direction of te development of virtual engineering concepts based on te latest acievements of virtual and augmented reality tecnologies. Motivating factors for tat are a ig level of competition of industrial enterprises, te searc for new tecnologies for design, modeling, verification of modeling results, te creation of industrial products, mareting and sales of enterprise products. Te newest tecnologies of virtual engineering are of ey importance in te modern industrial world in te design and modeling of complex industrial facilities, since tey allow to significantly reduce te amount of waste at te design stage of te product. Te core of te newest virtual engineering tecnologies are te newest metods of matematical modeling a Tis wor was supported by Russian Science Foundation grant No Corresponding autor: majorsteve@mail.ru Te Autors, publised by EDP Sciences. Tis is an open access article distributed under te terms of te Creative Commons Attribution License 4.0 (ttp://creativecommons.org/licenses/by/4.0/).

2 ttps://doi.org/ /ssconf/ for solving extraordinary problems in te industry. Solving tose problems requires te development of a complete simulation cycle from creating a pysical model, deducing up constitutive equations, selecting or creating numerical scemes for solving te equations and creating an industrial software system for te predictive modeling of product beavior. In tis paper we consider numerical simulations of CVT dynamics. Te system of ODEs of motion contains about 1800 generalized coordinates, plus te same number of generalized speeds, totaling in about 300 variables. Initial value problems for tose equations need to be solved to simulate CVT dynamics. Te dimension of our ODE system is relatively low, compared to tat of ODE systems in oter fields lie structural mecanics, computational fluid dynamics, and oters. However, computer simulation times are ig: one second of real time needs several ours of CPU time to perform a simulation. It is igly desirable to reduce simulation times since tat would allow users to modify CVT parameters in a desired manner muc faster, resulting in better quality product. One obvious approac to tis goal is to parallelize te simulation code. We already ave some success wit te parallelization [2]. But in tis paper we consider te oter approac, focused on a searc for a numerical metod tat would run simulations faster tan te classical Runge Kutta 4-t order metod (RK4) [3] tat we ave been using in production versions of CVT software. A combination of bot approaces promises to ave muc faster simulations. Next sections are as follows. In sec. 2 we provide an overview of CVT model, wic elps reader understand te sources of its numerical stiffness. Section 3 presents te result of te investigation of numerical integration metods applied to CVT dynamics. Section 4 summarizes te results obtained and outlines future wor. 2 CVT Model overview Te model of CVT includes two elastic safts, te input and te output one, on nonlinearly elastic supports. Tere are two pulleys on eac saft, one motionless and one moving (Figure 1). Te pulleys ave toroidal (almost conical) contact surfaces. Tere is a cain consisting of rocer pins and plates (Figure 2). Eac pin as two alves tat roll over eac oter during cain motion. End surfaces of pin alves are in contact wit te pulleys. Te application of clamping forces to pulleys leads to certain cain configuration suc tat pins are at certain contact radius at eac pulley set; te gear ratio can be canged by sifting te moving pulleys along te safts. Te torque is transmitted due to te friction forces at pin-pulley contact points. A number of matematical models of CVT parts for dynamics simulation ave been developed [1]. Most advanced models consider rocer pins bending elasticity and te rolling of pin alves. Cain models ave up to 21 generalized coordinates per lin, totaling in 100+ degrees of freedom per typical cain. Tere are many contact interactions in te CVT model: between pin and pulley; between pins and plates; between pin alves (Figure 3). Elastic normal forces acting at contact pairs are modeled according te Hertz teory [4]. Importantly, tere are also tangential friction forces. Tose are proportional to normal forces and te friction coefficient f depending on relative speed v r. Te friction law f ( v r ) used is close to te Coulomb s one, but is regularized: is constant at relative speeds greater tan te saturation speed v 0, and depends on speed linearly at speeds less tan v 0. Terefore, te friction law is piecewise linear. Te non-smootness of te friction law may affect te beavior of numerical metods investigated in tis paper, tat s wy we also consider an alternative system in wic te friction law is smooted using a parabola connecting straigt parts (Figure 4). 2

3 ttps://doi.org/ /ssconf/ Two factors mae te developed CVT matematical models computationally stiff: elasticity (of model bodies and of normal contact force) and friction in contact pairs. Taing elasticity and inertia into account results in natural frequencies of linearized system up to 10 rad/s. On te oter and, friction results in even more serious numerical stiffness, as we will see in sec Figure 1. CVT model general view. Figure 2. CVT cain. pin-pulley pin-plate pin-pin Figure 3. Types of contact interaction in CVT. original nonsmoot friction law Figure 4. Friction laws used in numerical experiments. smooted friction law, 1 = 2 Differential equations of motion for te CVT model are obtained in te framewor of Lagrangian mecanics, so tey ave te following form: 3

4 ttps://doi.org/ /ssconf/ t d dt L L q q ~ = Q, =1,, n, were is te time, q are generalized coordinates, L is te Lagrange s function, non-potential applied generalized forces, and n is te number of degrees of freedom. Equations of motion can be rewritten in te normal form: q v x = F( t, x), x =, F =, 1 v A P (we too into account tat te system is scleronomous), were vector of generalized coordinates, (q) wit elements A ij i j (2) (1) Q ~ are L ~ P( q, v, t) Q A v q q ] T = [ q 1,, q n is te A symmetric positive-definite matrix of inertia 2 L =, v vector of generalized speeds, P ( q, v, t) essentially q q nonlinear function, in particular due to Q ~ containing friction forces. 3 Investigation of numerical metods Production versions of te CVT simulation software ave been using te classical Runge Kutta numerical integration sceme of fourt order (RK4) [3] to solve te initial value problem of CVT dynamics. It is nown tat te RK4 sceme, as well as oter explicit numerical integration scemes, ave a step size limitation due to te stability requirement: in general, te value, were is te step size and is an eigenvalue of ODE rigt-and side Jacobian matrix, must belong to te stability region of te metod. 8 Due to tat limitation, step sizes tat can be used wit RK4 vary in te range [ 10 s, 10 s], wic leads to long CPU time required for a simulation. In next subsection we analyze ODE system Jacobian matrix and explain te origin of its ig eigenvalues. Ten we proceed wit te investigation of a number of numerical integration metods in application to CVT dynamics equations (2). Te goal of te investigation is to find a metod allowing muc longer step sizes tan RK4 and aving te potential to run significantly faster tan RK Eigenvalues of ODE rigt and side Jacobian matrix To mae te searc for an appropriate numerical metod more purposeful, we ave numerically solved te full eigenvalue problem at some typical state in a stationary regime of CVT operation. Te eigenvalues found split into two groups: firstly, tere are complex eigenvalues corresponding to damped oscillations; secondly, tere are real negative eigenvalues corresponding to non-oscillatory dissipation processes. Absolute values of eigenvalue imaginary parts ave maximum at approx. 10 s 1, wile absolute values of 8 real parts at approx. 10 s 1. Figure 5 (left) sows te computed eigenvalues of te Jacobian matrix. Furter we proved tat te largest negative eigenvalues are caused by friction forces acting between pin alves, and te operating points in te friction law are at te linear part ( v < v 0 ). Te idea of te proof is to cange (e.g., increase 10 times) te saturation speed v 0 4

5 ttps://doi.org/ /ssconf/ in te friction law, ten recompute te Jacobian matrix and find its eigenvalues. Te results of evaluation are sown in Figure 5 (rigt) for te case of v 0 increased 10 times for te friction law between pin alves; canging v 0 for oter contact types (pin-plate, pin-pulley) results in canges of oter eigenvalues, wile tose wit maximum absolute values remain te same. Notice tat te worst eigenvalues are proportional to te tension force in te most loaded straigt part of te cain, and, consequently, vary from one CVT operation regime to anoter. original friction law parameters v 0 increased 10 times for pin-pin contact nonsmoot friction law smooted friction law Figure 5. Eigenvalues of ODE rigt and side Jacobian matrix. 3.2 Numerical experiment setup For eac numerical integration sceme, two tests ave been done. In te first test, te dependency of step local error norm e on te step size is investigated. Te error is computed simply by comparison wit te exact reference solution obtained wit a very 9 small step size of 2 10 s using te RK4 sceme. In te second test, a dynamics simulation is performed during second of real time; a sample istory curve is obtained (namely, te axial force in a pin alf entering a pulley set, P z (t) at certain pin Figure ) as an evident indicator of te acceptability of numerical results. Relative global error E of a numerical solution is estimated as * * E max P ( t) P ( t) / max P ( t), (3) t [ t *, 0.005s] z z t [ t *,0.005s] were P * ( t z ) is te reference curve; t * = 0.001s as been cosen to avoid culling out some numerical solutions due to a small time sift resulting in a large error at t [0, t * ], wic is not representative because te reference curve as a large slope at some pints in te range [0, t * ]. P * z ( t) z nonsmoot smoot Figure. Pin axial force istory (reference curve). t 5

6 ttps://doi.org/ /ssconf/ e nonsmoot smoot e Figure. Step local error for explicit metods. To illustrate te impact of te nonsmootness of friction law on te accuracy of numerical results, we included numerical results for two modifications of te model, differing only by te friction law (nonsmoot and smooted, see Fig. 4, left and rigt respectively). 3.3 Explicit metods Explicit metods covered in tis subsection are te following classical ones. Tree embedded Dormand Prince scemes wit step size control [3]. Te step size control was disabled in test simulations. An embedded sceme provides two solutions of different orders of accuracy at eac step, wic can be used to control te step size; tose orders are encoded in te name of te sceme. Te tree scemes are DOPRI45 (orders 4, 5), DOPRI5 (orders 5, ), and DOPRI8 (orders,8). Gragg Bulirsc Stoer metod wit (GBS p s) or witout (GBS p ) smooting step [3]. It is an extrapolation metod wit te symmetric Gragg s sceme used as te reference sceme. We tried tis metod wit a fixed extrapolation order p (2, 4, ) and te armonic step size sequence. Extrapolated explicit Euler sceme, wit p extrapolation stages and te armonic step size sequence (referred to as EULER-X p below). nonsmoot smoot E E Figure 8. Global error for sample curve, explicit metods.

7 ttps://doi.org/ /ssconf/ Te step local error test (Figure ) sows tat te local error is generally less for smoot friction law; furter data processing also indicates tat te local error beaves according to sceme order only in a limited step size ranges, different for different scemes; some scemes (DOPRI45, DOPRI8, GBS4, GBS) do not sow te expected beavior at all, altoug tey do so in tests wit simple ODEs. Global error in te dynamics test (Figure 8) confirms tat all explicit scemes considered ave severe step size limitation tat is about 10 for nonsmoot friction law and scemes GBS2, DOPRI5, and is less for oter scemes; for smoot friction law, te limit is iger yet it is less tan We can also conclude tat low order scemes (2 4) are preferable in model wit bot smoot and nonsmoot friction low; altoug some iger order scemes (e.g. GBSs) give less error in model wit smoot friction law, tey are not an appropriate coice due to teir complexity. 3.4 Linearly implicit metods Tere was a ope tat a W-metod [5] is capable of producing acceptable numerical solution at steps muc greater tan 10, because tose metods generally ave better stability properties tan explicit ones. However, in our case all W-metods tested failed for some reason, toug tey wored good in tests wit simple ODEs. Te scemes considered in tis subsection are SW24- d [5] (were d is a parameter) and te W1 metod extrapolated according to te Ricardson s procedure [3]. Te W1 sceme is as follows: x x F( t, x ) da( x x ), 1 = 1 (4) were x is te numerical solution vector, te subscript denotes te step number, is te step size, t is te time, F is te ODE rigt-and side vector, d is a parameter (usually between 0 and 1), and A is a matrix approximating te ODE system Jacobian DF/ Dx. W- metods are attractive compared to Rosenbroc metods [] due to te ability to eep A constant during many time steps, tus eliminating te necessity to compute it and factorize te matrix I da at eac time step. Scemes resulting from p extrapolation stages of W1 are furter referred to as W1-X p - d. e local error, nonsmoot E global error, nonsmoot Figure 9. Local step error and global error for sample curve, linearly implicit metods compared to RK4 and explicit Euler metods.

8 ttps://doi.org/ /ssconf/ Figure 9 (left) sows tat all scemes tested ave muc greater local step error tan explicit scemes. Te expected order of scemes is observed at step sizes less tan 10 ; te iger te order, te narrower te range in wic sceme order is obeyed. Figure 9 (rigt) sows tat all W-metod scemes produce inacceptable solution ( E > 0.05) even at step 10. We ave to conclude tat tey didn t wor in our case. 3.5 Trapezoidal Rule Metod Among many implicit metods, we cose te trapezoidal rule (a 2-nd order sceme), furter abbreviated as TRPZ: x 1 = x [ F( t, x ) F( t, x 1)], (5) 2 Figure 10 sows te step local error for te trapezoidal rule. Notice tat it is less tan for any oter sceme tested at steps greater tan Sample curves obtained at step size 2 10 practically coincide wit reference curves; 3 4 global error E is for nonsmoot friction law and for smoot friction law. It is possible to use larger step sizes, but only wit step size control because te Newton s metod used at a time step may fail to converge. e Figure 10. Step local error for trapezoidal rule, compared to RK4 and explicit Euler metods. Notice tat, despite te large step size in te trapezoidal rule metod, it does not currently run faster tan RK4. Te reasons are te necessity to solve a system of nonlinear algebraic equations at eac step using an iterative Newton-lie metod, and te need to compute te Jacobian matrix. We omit details ere, but in sort, te number of Newton s iterations is low (2 3) wen te Jacobian matrix is recomputed at eac iteration, and is significantly iger (10 20) in cases wit approximated Jacobian matrix (e.g., using an update formula similar to te Broyden s one []). 3. Stabilized explicit metod Having investigated te stiffness properties of te ODE system, we decided to consider so called stabilized explicit, or Cebysev Runge Kutta [8, c. IV.2], numerical integration metods. Among tose we piced te one called DUMKA3 [9]. Te coice of tat particular metod was based on public availability of solver implementation code in C programming language. 8

9 ttps://doi.org/ /ssconf/ A stabilized explicit sceme as stability region, defined by te condition ( ) 1, extended into real negative direction of complex plane. Te function s R s R s is a polynomial of degree, called te stability polynomial. Te DUMKA3 solver actually implements a family of -stage Runge Kutta scemes, eac of wic realizes stability polynomial of degree varying from 3 up to 324. Te solver implements automatic step size control, based on step local error estimation, and polynomial degree control, based on te estimation of ODE rigt and side Jacobian matrix spectral radius. Preliminary tests ave sown tat solver performance wit bot control options enabled is far from optimal in our system. Besides, numerical estimation of Jacobian spectral radius don t wor well due to discontinuities of te ODE rigt and side in te case of nonsmoot friction law, resulting sometimes in too large values. We disabled bot control options. Te motivation was to obtain best performance at fixed step for eac fixed polynomial degree. Giving eac polynomial an index from 0 to 13 (DUMKA3 implements 14 polynomials), we denote corresponding scemes by suffixes -P. In particular, we tested degrees s = 21 ( = 4), s = 2 ( = 5 ), s = 3 ( = ), s = 48 ( = ), s = 3 ( = 8 ), s = 81 ( = 9 ). Polynomials 4 and 9 ave sown poor performance: in te first case te stability region is too small in te real negative direction, and in te second case it is too small in te imaginary direction. s s e local error, nonsmoot E global error, smoot Figure 11. Local step error and global error for sample curve for DUMKA3, compared to RK4 and explicit Euler metods. Figure 11 (left) sows te dependency of step local error norm on te step size. DUMKA3 results are compared against RK4 and te trapezoidal rule; te latter one is nown (see sec. 3.5) to give sufficiently accurate solution at = 2 10 s. It can be sown tat te slope for eac curve at steps below 10 corresponds to te order of te sceme (3 for DUMKA3, 4 for RK4, and 2 for TRPZ). Notice also tat at steps above 10 local error for all DUMKA scemes rises sarply at some point; additional error jumps can be seen at step 10 for nonsmoot friction law. At large step sizes, local error for DUMKA scemes is approximately te same as for te trapezoidal rule, in te case of smooted friction law; for nonsmoot fricion law, te trapezoidal rule gives smaller error. Figure 11 (rigt) sows te global error for sample curve. It follows from te figure tat scemes DUMKA3-P5 DUMKA3-P8 give sufficiently accurate solution at step = 2 10, and at step = 4 10 only te sceme DUMKA3-P8 does so. 9

10 ttps://doi.org/ /ssconf/ Table 1. Performance of DUMKA3 scemes compared against RK4. sceme, s n speedup against RHS RK4 RK DUMKA-P DUMKA-P DUMKA-P Comparing simulation times, we can conclude tat te DUMKA3 solver can perform simulations several times faster tan RK4, wic is summarized in Table 1. Notice tat te value n RHS in te table is te total number of ODE rigt and side evaluations in te test simulation (te one used to obtain te sample curve). 4 Conclusions and Future Wor Tis paper represents te results of an investigation of application of different numerical integration metods to te ODEs of CVT dynamics model, in a ope to find a potentially faster metod. Te investigation as sown tat traditional explicit numerical integration scemes and W-metods don t wor in our case. Implicit metods give good results; owever, to mae tose metods run faster tan RK4, additional efforts are required: for example, ODE rigt-and side Jacobian could be computed muc faster but it requires tedious programming (te idea is to combine te approac presented in [10] wit te decomposition of te ODE rigt-and side into a sum and providing faster code for te Jacobian of contact forces). Te attempt to use stabilized explicit Runge Kutta solver, RK4, as proven to be successful in terms of performance. However, te original polynomial order control implemented in te solver doesn t wor: it tends to increase polynomial order, but in tat case Jacobian eigenvalues wit maximum imaginary parts fall outside te stability region. Tis motivates us to consider oter stabilized explicit solvers, first of all RKC and SERK2 [11], and probably oters, constructed according to te approac presented in [12], because tere is a way to construct a metod wit stability region tat best fits te spectrum of ODE rigt and side Jacobian. References 1. N. Sabrov, Yu. Ispolov, S. Orlov, ZAMM 94(11) (2014) 2. S. Orlov, A. Kuzin, N. Sabrov, CCIS 93 (201) 3. E. Hairer, S. Nørsett, G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems (Springer-Verlag, New Yor, 1993) 4. K. Jonson, Contact mecanics (Cambridge University Press, Cambridge, 198) 5. T. Steiaug, A. Wolfbrandt, Mat. Comp. 33(14) (199). H. Rosenbroc, Comput. J. 5 (193). C. Broyden, Mat Comp. 19 (195) 8. E. Hairer, G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential- Algebraic Problems (Springer-Verlag, Berlin Heidelberg, 199) 9. Medoviov, BIT Numerical Matematics 38(2) (1998) 10. T. Ypma, J. Comput. Appl. Mat., 18(1) (198) 11. J. Martín-Vaquero, B. Janssen, Comput. Pys. Commun. 180(10) (2009) 12. M. Torrilon, R. Jeltsc, Numerisce Matemati 10(2) (200) 10