Dec 10, 2015 - Finally, I want to thank Hankook Tire for their financial support ... CHAPTER 2 MULTIBODY SYSTEM DYNAMICS . .... 9.5 The first component of ub of rear left tire from drop test with h = 0. ..... The reaction forces can be ..... numerical solution (Ëqn, Ëv)n is obtained for t = tn by an ODE solver. ...... 1m1)/4 + L2.
Hybrid Numerical Integration Scheme for Highly Oscillatory Dynamical Systems
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Authors
Gil, Gibin
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The University of Arizona.
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HYBRID NUMERICAL INTEGRATION SCHEME FOR HIGHLY OSCILLATORY DYNAMICAL SYSTEMS by Gibin Gil
= BY:
Creative Commons Attribution-No Derivative Works 3.0
A Dissertation Submitted to the Faculty of the DEPARTMENT OF AEROSPACE AND MECHANICAL ENGINEERING In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY WITH A MAJOR IN MECHANICAL ENGINEERING In the Graduate College THE UNIVERSITY OF ARIZONA
2013
2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE
As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Gibin Gil entitled Hybrid Numerical Integration Scheme for Highly Oscillatory Dynamical Systems and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy.
Date: 9/3/2013 Parviz E. Nikravesh
Date: 9/3/2013 Ara Arabyan
Date: 9/3/2013 Ricardo G. Sanfelice
Date: 9/3/2013 Mikhail Stepanov
Date: 9/3/2013
Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.
Date: 9/3/2013 Dissertation Director: Parviz E. Nikravesh
3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. This work is licensed under the Creative Commons Attribution-No Derivative Works 3.0 United States License. To view a copy of this license, visit http://creativecommons.org/licenses/bynd/3.0/us/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.
SIGNED:
Gibin Gil
4
ACKNOWLEDGEMENTS
I like to express my deepest appreciation to Professor Parviz E. Nikravesh for his guidance towards the research. Without his advice, this dissertation would not have been possible. It has been a great experience to work with him for the last four years. I thank my committee members Prof. Ara Arabyan, Prof. Ricardo Sanfelice, and Prof. Mikhail Stepanov for their time and patience in reviewing this dissertation. I gained the fundamental knowledge and ideas for my research from their wonderful lectures. I also thank my colleagues Omid Kazemi and Adrijan Ribaric for their support and helpful discussions. I wish them the very best for their future career. I would like to express my gratitude to my wife Miyoung, whose support and sacrifice were essential in this work. I also thank my daughter Lena for giving me the joys of fatherhood. Finally, I want to thank Hankook Tire for their financial support throughout this work.
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TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 12 1.1 Survey of Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2 Scopes and Objectives of Dissertation . . . . . . . . . . . . . . . . . . 20 CHAPTER 2 MULTIBODY SYSTEM DYNAMICS . . . . 2.1 Equations of motion of multibody systems . . . . . 2.1.1 Body-Coordinate Formulation . . . . . . . . 2.1.2 Joint-Coordinate Formulation . . . . . . . . 2.1.3 Canonical Form of Equations of Motion . . . 2.2 Contact-Impact Model in Multibody Systems . . . 2.2.1 Discontinuous Method . . . . . . . . . . . . 2.2.2 Continuous Method . . . . . . . . . . . . . . 2.3 Numerical Analysis of Multibody System Dynamics 2.3.1 Integration Variables . . . . . . . . . . . . . 2.3.2 Constraints Violation . . . . . . . . . . . . .
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23 24 24 28 31 34 34 34 36 36 38
CHAPTER 3 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Adams Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Backward Differentiation Formula . . . . . . . . . . . . . . . . . . . . 3.4 St¨ormer-Verlet Method . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Local Linearization Method . . . . . . . . . . . . . . . . . . . . . . .
42 43 45 46 47 49
CHAPTER 4 APPLICATION OF LLM IN HIGHLY OSCILLATORY MULTIBODY SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1 A Simple Mechanical System . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Contact Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6 TABLE OF CONTENTS – Continued CHAPTER 5 NUMERICAL ERROR ESTIMATION METHOD FOR LLM . 71 5.1 Local Error Estimation Method . . . . . . . . . . . . . . . . . . . . . 72 5.2 Adaptive Step Size Control . . . . . . . . . . . . . . . . . . . . . . . . 75 CHAPTER 6 DEVELOPMENT OF A HYBRID NUMERICAL INTEGRATION SCHEME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.1 Singular Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . 80 6.2 Hybrid Numerical Integration Scheme . . . . . . . . . . . . . . . . . . 85 6.3 Absolute Stability Region of The Hybrid Scheme . . . . . . . . . . . 91 6.4 Accuracy Analysis of The Hybrid Scheme . . . . . . . . . . . . . . . . 96 6.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 CHAPTER 7 TIRE MODEL . . . . . . . . . 7.1 Nodal Positions and Velocities . . . . . 7.2 Non-rotating Reference Frame . . . . . 7.3 Modal Transformation and Truncation 7.4 Equations of Motion . . . . . . . . . .
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109 110 112 115 116
CHAPTER 8 VEHICLE MODEL . . 8.1 Functional Suspension Model 8.2 Velocity Transformation . . . 8.3 Equations of motion . . . . .
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119 120 126 128
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CHAPTER 9 VEHICLE SIMULATION SCHEME . . . . . . . . . . . . . . . . 9.1 Characteristics of Vehicle System 9.2 Simulation Results . . . . . . . .
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USING HYBRID INTEGRATION . . . . . . . . . . . . . . . . . . . . 133 . . . . . . . . . . . . . . . . . . . . 133 . . . . . . . . . . . . . . . . . . . . 138
CHAPTER 10 CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . 152 10.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 10.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7
LIST OF FIGURES
3.1
Detection of the instant that a contact begins . . . . . . . . . . . . . 56
4.1 4.2 4.3
59 62
Configuration of double pendulum . . . . . . . . . . . . . . . . . . . . Rotation angles of double pendulum with body-coordinate formulation Total energy variation of double pendulum with body-coordinate formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Constraint violation of double pendulum with body-coordinate formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Computation time for double pendulum with body-coordinate formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Rotation angles of double pendulum with joint-coordinate formulation 4.7 Computation time for double pendulum with joint-coordinate formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 System of two bouncing balls . . . . . . . . . . . . . . . . . . . . . . 4.9 Simulation results for two bouncing balls . . . . . . . . . . . . . . . . 4.10 Total energy variation for two bouncing balls . . . . . . . . . . . . . . 4.11 Computation time for two bouncing balls . . . . . . . . . . . . . . . . 5.1 5.2
62 63 63 66 66 68 69 69 70
Comparison of exact and estimated local error . . . . . . . . . . . . . 76 Step sizes and estimated errors with tol = 0.1 and tol = 0.05 . . . . . 79
6.1 6.2 6.3
Diagram of the hybrid scheme . . . . . . . . . . . . . . . . . . . . . . 91 The absolute stability region of the fourth-order Runge-Kutta method 92 The absolute stability region of the hybrid scheme for different values of α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.4 The absolute stability region of the hybrid scheme for different values of β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.5 Local error versus step size for different values of α . . . . . . . . . . 98 6.6 Local error versus step size for different values of β . . . . . . . . . . 99 6.7 Configuration of pendulum system . . . . . . . . . . . . . . . . . . . 102 6.8 Comparison of simulation results for pendulum . . . . . . . . . . . . . 103 6.9 Total energy of pendulum . . . . . . . . . . . . . . . . . . . . . . . . 104 6.10 Spring-mass system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.11 Numerical error in u2 for the fourth-order Runge-Kutta Method . . . 107 6.12 Numerical error in u2 for the Hybrid Scheme . . . . . . . . . . . . . . 108 7.1
Nodes of finite element tire model . . . . . . . . . . . . . . . . . . . . 111
8 LIST OF FIGURES – Continued 7.2
Undeformed and deformed configuration of a system . . . . . . . . . . 113
8.1 8.2 8.3 8.4 8.5 8.6 8.7
Suspension Parameter Measurement Machine . . . . . . . . . . . . . 120 Reference frames used in vehicle model . . . . . . . . . . . . . . . . . 121 Camber angle change with respect to wheel vertical displacement . . 123 Toe angle change with respect to wheel vertical displacement . . . . . 123 Wheel center longitudinal position with respect to wheel vertical displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Wheel center lateral position with respect to wheel vertical displacement124 Wheel center vertical force versus wheel travel . . . . . . . . . . . . . 132
9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13 9.14 9.15 9.16 9.17
Eigenvalues of Vehicle-Tire System . . . . . . . . . . . . . . . . . . . 135 Eigenvalues of Vehicle-Tire System (zoomed) . . . . . . . . . . . . . . 136 Vehicle body vertical position for drop test . . . . . . . . . . . . . . . 139 Wheel travels for drop test . . . . . . . . . . . . . . . . . . . . . . . . 139 The first component of ub of rear left tire from drop test with h = 0.01141 The first component of ub of rear left tire from drop test with h = 0.002142 Normal contact force distribution for drop test . . . . . . . . . . . . . 142 Ground geometry for pothole test . . . . . . . . . . . . . . . . . . . . 143 Tire deflection when passing over a pothole . . . . . . . . . . . . . . . 144 The first four components of ub for pothole test . . . . . . . . . . . . 144 Vertical force acting on the wheel center for pothole test . . . . . . . 145 Configuration of ground for ramp test . . . . . . . . . . . . . . . . . . 146 Vehicle center vertical position for ramp test . . . . . . . . . . . . . . 147 Vehicle pitch angle for ramp test . . . . . . . . . . . . . . . . . . . . 147 Front left and rear left wheel travel for ramp test . . . . . . . . . . . 148 The first four components of ub of front left tire for ramp test . . . . 149 The first four components of ub of rear left tire for ramp test . . . . . 150
9
LIST OF TABLES
3.1 3.2 3.3
Coefficients of explicit Adams methos up to order 6 . . . . . . . . . . 46 Coefficients of implicit Adams methos up to order 6 . . . . . . . . . . 47 Coefficients of BDF methos up to order 6 . . . . . . . . . . . . . . . . 48
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Parameters of double pendulum . . . . . . . . . . . . . . . . . . . . . 59
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Spring coefficients in spring-mass system . . . . . . . . . . . . . . . . 105
8.1 8.2
Fitting coefficients of Eq. (8.4) . . . . . . . . . . . . . . . . . . . . . 125 Mass and inertia for the vehicle model of a luxury sedan . . . . . . . 129
9.1 9.2 9.3
The number of steps and computation time for drop test . . . . . . . 140 The number of steps and computation time for pothole test . . . . . . 145 The number of steps and computation time for ramp test . . . . . . . 151
10 ABSTRACT
Computational efficiency of solving the dynamics of highly oscillatory systems is an important issue due to the requirement of small step size of explicit numerical integration algorithms. A system is considered to be highly oscillatory if it contains a fast solution that varies regularly about a slow solution. As for multibody systems, stiff force elements and contacts between bodies can make a system highly oscillatory.
Standard explicit numerical integration methods
should take a very small step size to satisfy the absolute stability condition for all eigenvalues of the system and the computational cost is dictated by the fast solution. In this research, a new hybrid integration scheme is proposed, in which the local linearization method is combined with a conventional integration method such as the fourth-order Runge-Kutta. In this approach, the system is partitioned into fast and slow subsystems. Then, the two subsystems are transformed into a reduced and a boundary-layer system using the singular perturbation theory. The reduced system is solved by the fourth-order Runge-Kutta method while the boundary-layer system is solved by the local linearization method. This new hybrid scheme can handle the coupling between the fast and the slow subsystems efficiently.
Unlike other multi-rate or multi-method schemes,
extrapolation or interpolation process is not required to deal with the coupling between subsystems. Most of the coupling effect can be accounted for by the reduced (or quasi-steady-state) system while the minor transient effect is taken into consideration by averaging. In this research, the absolute stability region for this hybrid scheme is derived
11 and it is shown that the absolute stability region is almost independent of the fast variables. Thus, the selection of the step size is not dictated by the fast solution when a highly oscillatory system is solved, in turn, the computational efficiency can be improved. The advantage of the proposed hybrid scheme is validated through several dynamic simulations of a vehicle system including a flexible tire model. The results reveal that the hybrid scheme can reduce the computation time of the vehicle dynamic simulation significantly while attaining comparable accuracy.
12 CHAPTER 1 INTRODUCTION
Multibody System Dynamics is a subject concerned with modeling and analyzing systems that consist of several bodies connected by kinematic joints and/or force elements like spring-dampers.
Examples of multibody systems range from the
very simple systems like a slider-crank mechanism to the very complex systems like a suspension of a vehicle or a manufacturing machinery. The equations of motion for a multibody system are obtained in the form of a system of ordinary differential equations (ODEs) or differential algebraic equations (DAEs). Due to the nonlinearity, the multibody equations should be solved numerically. In many applications, the deformation of bodies are negligible and bodies are assumed rigid. However, some systems contain bodies that can undergo significant deformations which can not be treated as rigid. In this case, it is necessary to model such flexible bodies by the finite element method (FEM) [1]. A finite element model normally introduces a large number of nodes and, therefore, the number of degrees of freedom of a flexible body becomes much larger than that of a rigid body. As a result, the size of the equations of motion of the multibody system is increased significantly. The large number of degrees of freedom associated with flexible bodies deteriorates the computational efficiency of numerical solution of the equations of motion and can make the dynamic analysis of a flexible-rigid multibody system impractical. Conventionally, the number of degrees of freedom of a finite element model of a flexible body in a multibody setup is reduced by modal reduction techniques, such as the Craig-Bamption method [2]. In these methods, the FE model is transformed into the modal space and high frequency modes are truncated. Then, the equations of motion are written in terms of the modal coordinates associated with a few of the lowest modes of the flexible body. This process can reduce the size of
13 the equations of motion significantly. However, the resultant system consists of solutions of very different time scales because the flexible body usually has higher natural frequencies than the rigid bodies. Such systems are referred to as highly oscillatory. For highly oscillatory systems, the computational efficiency is an issue because the conventional numerical methods should take a very small step size to capture the dynamics of the fast components. Therefore, the computational efficiency is dictated by the fast solution. One typical example of a highly oscillatory system is the vehicle-tire. The vehicle body and suspension linkages can be assumed rigid while the tire should be treated as a flexible body due to its deformation. Typically, the natural frequency of the vehicle body is about 1-2Hz [3]. On the other hand, the natural frequencies of a tire are in much higher range. For a ride comfort analysis, it is known that the tire model should be able to represent the dynamic responses of up to 150Hz [4]. When a conventional numerical method is used, the step size should be very small to satisfy the absolute stability condition for the dynamics of the tire. At the same time, the dynamic analysis should be performed for a time span that is long enough to show the dynamic behavior of the rigid bodies. Consequently, the number of steps required for the dynamic simulation of the vehicle-tire system can be large and the computational efficiency can become an issue. In the research reported in this dissertation, we develop a hybrid numerical integration scheme to improve the computational efficiency of highly oscillatory dynamical systems. This method can take larger step sizes compared to most conventional numerical methods when a highly oscillatory system is solved. Although the cpu speed has increased, the computation time for numerically solving the ordinary differential equations has been decreased, the computational cost is still an important issue when the size of the system is large and the system has high frequency dynamics. Especially, the computation time is a significant factor when a large number of simulation is required as in the Monte-Carlo methods [5] or in
14 the Design Optimization processes [6]. Monte-Carlo methods are widely used in engineering for reliability analysis and quantitative probabilistic analysis. This is the method to estimate the probability of failure by repeating the experiment many times with random samples of the variables. In many applications, a large number of actual experiment is not feasible and the experiment is replaced by the numerical simulation. Typically, the number of simulation required for Monte-Carlo methods is in the order of thousands. Design optimization is the process to find the set of design variables that is superior to all others while satisfying the given constraints. For a complex system, the design optimization rely on iterative simulations with different set of design variables. In this case, the required number of simulations can be large and the computation time for solving the dynamics of the system may be an important issue. In this research, we focus on the equations of motion of a multibody system and its numerical solution. The advantage of the hybrid scheme is demonstrated through several examples. The primary application of the hybrid scheme is a vehicle dynamic simulation with a flexible tire model. However, the application of the proposed hybrid scheme is not limited to the multibody systems. Highly oscillatory dynamical systems can be found in other science and engineering fields such as molecular systems [7, 8], electrical circuits [9] and planetary systems [10]. The developed hybrid scheme can used for such problems to improve the computational efficiency.
15 1.1 Survey of Literature A large amount of research on the numerical solution of highly oscillatory dynamical systems has been conducted and many numerical methods has been developed. A good overview of this subject can be found in [11].
The main problem of
numerical solutions of highly oscillatory systems is that the standard explicit numerical methods need very small step size, causing the number of integration steps, required to solve those systems, to become extremely large. Standard explicit methods are conditionally stable, or in other words their absolute stability regions are limited. Therefore, the step size should be small enough to satisfy the absolute stability condition. Obviously the computational efficiency is restricted by the fast components of the system. This restriction of the step size can be relaxed if implicit methods, like BDF or implicit Runge-Kutta methods, are used because the absolute stability region of implicit methods covers much larger area. In the case of implicit methods, however, the Newton iteration is required to solve the system of nonlinear equation at each integration time step. This increases the computational cost significantly if the size of the problem is large. Furthermore, the step size may still be restricted due to difficulties in convergence of the Newton iteration for larger step sizes [12]. The implicit numerical methods can give stable numerical solutions with larger step size, but it does not mean that the accuracy of the numerical solution is guaranteed. The implicit numerical methods tend to damp out high frequency oscillations when a large step size is used. In some applications, damping the oscillation can destroy important properties of the solution [11]. Exponentially fitted Runge-Kutta (EFRK) methods have been suggested for the numerical solution of ODEs which have periodic or oscillating solutions [13-17] . These methods obtain the approximated numerical solutions belonging to the linear space spanned by a set of functions of type {sin(ωt), cos(ωt), sin(2ωt), cos(2ωt), ...} instead of polynomial functions where ω is a prescribed frequency. The coefficients of EFRK methods are frequency-dependent and they are determined such that the
16 local error of the numerical solution is minimized. A more detailed review on EFRK methods can be found in [18]. Similar approaches can be found in [19, 20] where they are referred to as Adapted RK methods. EFRK and Adapted RK methods can handle some oscillatory problems better than standard Runge-Kutta methods, but they require some prior knowledge about the main frequency of the solution. In the case of multiple timescale ODE problems, there have been numerous attempts to exploit the different timescales of the system for better computational efficiency.
We can find two research directions on this subject: multi-method
schemes and multi-rate schemes. In multi-method schemes, the system is partitioned into the fast and the slow subsystems. Then different methods are applied for those subsystems respectively while the step sizes of the integrators for different subsystems are the same. Andrus [21] proposed a numerical method in which the slow subsystem is integrated by the fourth-order Runge-Kutta and the fast subsystem is solved by any sufficiently accurate method or some closed form solution if available. In a later paper [22], Andrus considered a numerical method which consists of integrating the slow subsystem and the fast subsystem with fourth-order and third-order Runge-Kutta methods, respectively.
He derived the absolute
stability condition of this method and showed that the region of stability is nearly as large as that of the embedded methos if the subsystems are weakly coupled. In references [23, 24], multi-method schemes in which an explicit method is used for the slow subsystem and an implicit method is used for the fast subsystem are suggested. Those schemes are based on the assumption that the user can partition the entire system into the slow and the fast subsystems with some prior information about the system. Weiner et al. [25] suggested an algorithm for dynamic partitioning of the system which is based on the local information of the system Jacobian. Multi-rate schemes utilize different step sizes for each subsystems, while using the same numerical methods for all subsystems. A smaller step size is used for the subsystem of fast dynamics and a much larger step size is applied for other parts
17 of the system to achieve better computational efficiency. The step sizes for the fast and the slow subsystems are referred to as microstep and macrostep, respectively [26]. The first attempt to use different step sizes can be found in [27]. Multi-rate integration schemes based on Runge-Kutta methods were discussed in [26, 28, 29]. On the other hand, Gear [30] and Socia [31] suggested multi-rate schemes based on BDF. Skelboe [32] presented the stability properties of multi-rate schemes based on BDF. These multi-rate schems rely on interpolation/extrapolation algorithms for handling the coupling between subsystems. As a result, multi-rate schemes are susceptible to stability problems. Recent work of Bartel [33] deals with the coupling between subsystems by using internal stages instead of interpolation/extrapolation to overcome this problem. Buzdugan [34] developed a multi-rate scheme which is based on predictor-corrector Adams-Bashforth-Moulton method incorporating variable step and order. Application areas of multi-rate schemes include simulations of electric circuit [9], molecular dynamics [7, 8], vehicle models containing flexible tire [29], aero-elastic models of helicopter [31] and planetary problems [10]. Multi-method and multi-rate schemes have similarity with the waveform relaxation (WR) methods in the sense that the system is partitioned into several subsystems according to their time scales. WR methods were originally introduced by Lelarasmee [35, 36] for solving very large scale integrated (VLSI) circuits. There were many attempts of applying WR methods to the simulation of electric circuits [37-40]. The main concept of WR methods is that each subsystem is integrated independently over a number of iterative step-sweeps with information from the other subsystems being input only at the end of each step-sweep. WR methods can be seen as a generalization of algebraic relaxation schemes like Jacobi and GaussSeidel methods for solving systems of linear equations. For the partitioned coupled differential equations y˙ 1 = f1 (y1 , y2 ), y1 (0) = y10
(1.1)
y˙ 2 = f2 (y1 , y2 ), y2 (0) = y20
(1.2)
18 Jacobi WR method takes the form y˙ 1k+1 = f1 (y1k+1 , y2k ), y1k+1 (0) = y10
(1.3)
y˙ 2k+1 = f1 (y1k , y2k+1 ), y2k+1 (0) = y20
(1.4)
so that two decoupled differential equations can be solved in parallel and independent of one another. There is no limitation of selecting numerical methods and step sizes for each subsystem. The motivation of the WR methods is to take advantage of parallel computing for very large systems. There are other variations of WR methods such as Gauss-Seidel WR and SOR WR [41]. Important issues of the WR methods are the stability and the rate of convergence of iteration, which have been discussed in [42-46]. Another alternative for efficient numerical solution of multiple timescale problems is averaging. Averaging can be useful if the fast dynamics is not of primary interest and, therefore feeding an average of the fast dynamics into the differential equation of the slow subsystem would suffice. Heterogeneous multiscale methods (HMM) [47-53] provide a general framework of linking models at different time scales through averaging. It consists of two main components: The macroscale solver and a procedure for estimating the missing numerical data from the microscale model [50]. The missing information from the microscale model is estimated through averaging instead of the pointwise evaluation of oscillatory functions. Initially, the main application area of HMM was molecular dynamics simulation which involves the positions and momenta of the nuclei (macroscale) and the effects of the electrons (microscale). Calvo [54] applied HMM for the numerical integration of the equations of motion of mechanical systems subjected to fast vibrations. Recently, Calvo [55] introduced the stroboscopic averaging method (SAM) as a technique to solve highly oscillatory differential systems with a single high frequency. This is a numerical form of implementing the analytical technique of stroboscopic averaging [56]. Another approach to handle highly oscillatory systems is the Local Linearization
19 Method (LLM). LLM was initially suggested as an alternative to Runge-Kutta methods for achieving real-time simulation of time-invariant systems [57]. However, its application is not limited to time-invariant systems and it can be used for general non-autonomous ODEs [58]. The LLM is an exponential method which is based on the piecewise linear approximation of the equation through a first-order Taylor expansion at each time step. The solution at the next time step is determined by the analytic solution of the approximated linear system. This approach also can be found in the literature with other names, such as matricial exponentially fitted method [59], exponential Euler method [60], piece-wise linearized method [61] and exponentially fitted Euler method [62]. Dynamic properties of the LLM have been analyzed in [63]. Jimenez compared several algorithms for the numerical implementation of the LLM and investigated the order of convergence of the LLM [58], and a modified version of LLM was proposed for achieving the higher order of convergence [64]. There have been efforts to apply the LLM to more general kinds of equations; e.g., stochastic differential differential equations [65], random differential equations [66] and delay differential equations [67]. There is an equivalence between stiff and singularly perturbed differential equations [68]. From this observation, there have been attempts to obtain numerical solutions of stiff differential equations using the singular perturbation theory [69, 70]. In reference [69], the singular perturbation method was applied for the numerical method to solve stiff differential equations. Unlike conventional numerical methods, the singular perturbation method performs better as the stiffness of the system increases. This approach was extended to the -Independent Method [70], where the small parameter does not need to be identified. In these methods, however, the transient part of the solution is neglected with the assumption that they decay exponentially. So, they can not be used for highly oscillatory systems where the transient dynamics of the system is important. The transient behavior of the system can not be obtained accurately by these methods even though the asymptotic behavior of the system can be well reconstructed.
20 1.2 Scopes and Objectives of Dissertation The objective of this research is to develop a hybrid numerical integration scheme to increase the efficiency of the dynamic analysis of highly oscillatory dynamical systems. In this scheme, the LLM is combined with a standard ODE solver, such as the Runge-Kutta method. This dissertation is organized as follow. Chapter 2 presents an overview of multibody system dynamics. Multibody system dynamics provides systematic methodologies for modeling a multibody system and can be applied to a variety of engineering problems from vehicle design to biomechanics. Several techniques to derive the equations of motion and analysis methods for contact problems will be discussed. Then, we address the general procedure of numerical analysis of multibody equations. The concepts and equations presented in this chapter will be used for discussions in the following chapters. The equations of motion of a multibody system are in the form of ordinary differential equations (ODEs) or differential algebraic equations (DAEs). These multibody equations should be integrated numerically due to their nonlinearity. Chapter 3 reviews some commonly used numerical integration methods such as Runge-Kutta, Adams, and BDF method. Then, the formula and properties of the LLM is discussed. It is also discussed how the LLM can be used for solving the multibody equations. There have been few attempts to apply the LLM to solve the equations of motion of a multibody system. The LLM can be a better candidate for solving highly oscillatory multibody systems than other conventional methods.
Con-
ventional methods show poor computational efficiency in solving such systems due to very small step sizes. Because the LLM is an A-stable method, a larger step size can be used while ensuring a stable numerical solution. In chapter 4,
21 the LLM is applied to simple multibody systems having high frequency dynamics and its computational efficiency is compared to that of other numerical methods. The LLM is based on the linear approximation of the system and its numerical error is caused by nonlinearity of the system. So far, the methods to estimate the numerical error of the LLM have not been reported. In chapter 5, the methods to estimate the local error of the LLM are proposed. The proposed local error estimation method is validated through numerical examples and utilized for the implementation of the adaptive step size control. In chapter 6, we propose a new hybrid numerical integration scheme. The efficiency of the LLM deteriorates when the size of system becomes too large. The proposed hybrid scheme aims to handle this problem. In this method, the LLM is combined with the fourth-order Runge-Kutta, where the LLM takes care of the subsystem of fast dynamics instead of the entire system. This scheme is based on the concepts of slow manifold and boundary layer system in singular perturbation theory.
This scheme can be classified as a multi-method scheme in the sense
that two different integration methods are combined. Chapter 6 presents a brief overview of singular perturbation theory and derives the mathematical model of the hybrid scheme. The absolute stability condition for the hybrid scheme is derived and the absolute stability region is obtained. Then, the performance of the hybrid scheme is investigated through numerical experiments and compared to that of the fourth-order Runge-Kutta method. Vehicle dynamic simulation is one of the areas that the principles of multibody system dynamics are widely used. In chapter 7, we review existing tire modeling methods and discuss the modal tire model that will be employed in the vehicle dynamic simulation of this study. Chapter 8 gives a brief overview of existing vehicle models and discusses a vehicle model based on the functional suspension model. Then, the equations of motion for the vehicle system are derived, which will
22 be solved numerically in chapter 9. The vehicle model is highly oscillatory because the dynamic responses of the tires are much faster than the responses of the vehicle and its suspension. The hybrid scheme proposed in chapter 6 is applied to the dynamic simulations of the vehicle system. The simulation results from the hybrid scheme and other numerical methods will be compared in terms of accuracy and computational efficiency in chapter 9. Then, in chapter 10, concluding remarks are made and areas of future investigation are addressed.
23 CHAPTER 2 MULTIBODY SYSTEM DYNAMICS
Mechanical systems consisting of several moving bodies are referred to as multibody systems. The bodies of a multibody system may be connected by kinematic joints and force elements such as springs and dampers. Kinematic joints remove relative degrees of freedom between bodies, which should be included in the derivation of the equations of motion. There are several systematic methodologies to formulate the equations of motion of multibody systems.
Body-coordinates
formulation [71] uses the coordinates of rigid bodies to define the motion of and the constraints on the bodies. Body-coordinates formulation is a method suitable for general purpose program development because the method can systematically generate the kinematic and dynamic equations for generic mechanical systems. Conventionally, the equations of motion of multibody systems are derived in the form of Newton-Euler equations.
Joint-coordinate formulation [72] uses a set
of relative joint coordinates that are associated with the kinematic joints of the system. The equations of motion are derived in terms of the joint accelerations. The accelerations of body coordinates or joint coordinates are obtained by solving the corresponding equations of motion. Alternatively, the equations of motion for a multibody system can be derived in terms of the total momenta of the system [73]. This type of equations of motion is referred to as the canonical form. Contact or impact between bodies causes an abrupt change in the velocities. The analysis of this intermittent motion is an important subject of multibody system dynamics. In a broad sense, the methods for the analysis of contact can be classified into discontinuous methods [74] and continuous methods [75]. Discontinuous methods are based on the assumption that an impact occurs instantaneously. In this approach, the velocities after an impact are determined through a momentum
24 balance process. Continuous methods use contact force models to represent the repulsive forces between bodies. This chapter discusses several formulation methods of the equations of motion for multibody systems. These formulation methods will be used for numerical examples in the following chapters. Then, the methods for the analysis of contact problems in multibody systems are reviewed. Finally, the general procedure of numerical analysis of multibody dynamics is addressed. 2.1 Equations of motion of multibody systems The equations of motion of multibody systems can be systematically derived by several formulation techniques. This section presents a review of formulation methods for multibody equations. 2.1.1 Body-Coordinate Formulation Let us consider a single body of a multibody system. We denote this body as body i and we define two reference frames. The first one is a global frame which is fixed in the space. The global frame is also called the non-moving or the inertial frame. The other one is a local frame which is fixed to the centroid of a body. The local frame rotates with the body and is also called the body-fixed frame. The position and the orientation of body i can be described by body coordinates qi as r r i i qi = or qi = φ e i
(2.1)
i
where ri are the Cartesian coordinates of the origin of the local frame in the global frame. The orientation of the body can be defined by Euler angles φi or Euler parameters ei . The velocity of body i is defined as r˙ i vi = ω i
(2.2)
25 where ω i is the angular velocity of body i expressed in the global frame. It should be noted that in spatial dynamics q˙ i 6= vi
(2.3)
Newton-Euler equations for body i can be written as (c)
mi¨ri = fi + fi
(2.4a)
(c)
˜ i Ji ω i Ji ω˙ i = ni + ni − ω
(2.4b)
where mi is the mass, Ji is the rotational inertia matrix, fi is the external force (c)
acting on body i, fi
is the reaction force from the kinematic joints, ni is the sum of
the moments caused by external forces and any external torques acting on body i, (c)
˜ i is the skew-symmetric ni is the reaction moment from the kinematic joints and ω matrix defined as
0
˜ = ω ωz −ωy
−ωz 0 ωx
ωy
−ωx
for ω =
ωx ωy
0
ωz
(2.5)
In a compact form, Newton-Euler equations for body i are expressed as (c)
Mi v˙ i = hi + hi with
(c) mi I 0 fi fi (c) , hi = Mi = , hi = ni − ω n(c) ˜ i Ji ω i 0 Ji i
(2.6)
(2.7)
The inertia matrix Ji is obtained with respect to a reference frame attached to the mass center of the body but remaining parallel to the global frame. Therefore the components of Ji vary with changing orientation of the body.
26 The rotational equations of motion (2.4b) can also be expressed in the local frame as 0(c)
J0i ω˙ 0i = n0i + ni
˜ 0i J0i ω 0i −ω
(2.8)
where 0 indicates that the components of a vector are expressed in the local frame. With a different definition of vi as vi =
r˙ i
(2.9)
ω 0 i
Newton-Euler equations for body i can be written as (c)
Mi v˙ i = hi + hi with
(2.10)
(c) mi I 0 fi fi (c) , hi = , hi = Mi = n0(c) n0 − ω ˜ 0i J0i ω 0i 0 J0i i i
(2.11)
In this case, the inertia matrix J0i is invariant. The array of velocity of body i, denoted by vi , can be defined as either Eq. (2.2) or Eq. (2.9) in the following discussion. The formulations are valid for either definitions of vi . The distinction between Eq. (2.2) and Eq. (2.9) vanishes if a planar multibody system is considered. In the case of a planar system, the position and the orientation of body i can be defined by two Cartesian coordinates and one rotational coordinate as qi =
r i
φi
,
ri =
x i
(2.12)
yi
Furthermore, we have q˙ i = vi for a planar system. Let us consider a system of nb bodies. The equations of motion of the system can be written as Mv˙ = h + h(c)
(2.13)
27 where M denotes mass matrix of the system, h denotes the array of applied forces and moments and h(c) represents the array of reaction forces and moments of the system. The mass matrix M is formed as M1 .. M= .
(2.14)
Mnb The array of applied forces and moments h is defined as h1 .. h= . hnb
(2.15)
Kinematic joints in a multibody system impose constraints on the bodies. Mathematically, these constraints can be expressed by the constraint equations as Φ(q) = 0
(2.16)
The velocity and acceleration constraints are expressed as ˙ = Dv = 0 Φ
(2.17)
¨ = Dv˙ + Dv ˙ =0 Φ
(2.18)
where D is the Jacobian matrix of the constraints. The reaction forces can be represented with the aid of Lagrange multipliers λ as h(c) = D> λ
(2.19)
By using Eq. (2.19), the equations of motion of the system are derived as Mv˙ = h + D> λ
(2.20)
28 Together with Φ(q) = 0, Eq. (2.20) forms a system of differential-algebraic equations (DAEs). Appending the acceleration constraints to Eq. (2.20) yields M −D> v˙ h ˙ = , γ = −Dv λ γ D 0
(2.21)
Assuming that all the coordinates and velocities are known, we can determine v˙ and λ by solving the system of linear equations (2.21). The body-coordinates formulation is suitable for a general purpose multibody dynamics program because adding bodies or joints is straightforward. However, this method requires a large number of coordinates. For a system of nb bodies, the number of body coordinates is 6nb if Euler angle is used, which is much larger than the number of system’s degree of freedom if kinematic joints are present. 2.1.2 Joint-Coordinate Formulation Joint-coordinate formulation uses a set of joint coordinates θ that are associated with the kinematic joints of a system. The number of joint coordinates can be equal or slightly larger than the number of system’s degree of freedom. Therefore, the number of equations generated by this method is smaller than that generated by the body-coordinate formulation. A joint coordinate is defined such that it represents the motion permitted by a joint. For instance, the joint coordinate for a revolute joint is defined as a relative angle between the two connected bodies, and for a sliding joint, the joint coordinate is defined as a relative distance between two points on the involving bodies along the joint axis.
29 Open-chain systems For an open-chain system having ndof degrees of freedom, we define ndof joint coordinates denoted as θ 1 .. θ= . θndof
(2.22)
Then, we can express the body coordinates as nonlinear functions of the joint coordinates: q = q(θ)
(2.23)
The time deriavative of Eq. (2.23) yields a linear relationship between the body velocities and the joint velocities as v = Bθ˙
(2.24)
where B is the velocity transformation matrix. An important property of matrix B is that the rows of D and the columns of B are orthogonal, which means DB = 0
(2.25)
The acceleration transformation is obtained as ¨+B ˙ θ˙ v˙ = Bθ
(2.26)
¨ as Then, the equations of motion of Eq. (2.13) can be transformed in terms of θ ¨+B ˙ θ˙ = h + D> λ M Bθ
(2.27)
Pre-multiplying both sides of this equation by B> yields ¨+B ˙ θ˙ = B> h + B> D> λ B> M Bθ
(2.28)
Because of DB = 0, the equations of motion becomes ¨=h ¯ ¯θ M
(2.29)
30 where ¯ = B> MB M ¯ = B> h − M B ˙ θ˙ h It should be noted that the Jacobian matrix D and the array of Lagrange multipliers λ disappear from the equations of motion. The size of the equations of motion is equal to ndof , which is much smaller that 6ndof of the body-coordinate formulation. Closed-chain systems The first step to apply the joint coordinate formulation to a closed-chain system is to cut one joint per closed chain and create an open-chain system. For the resultant open-chain systems, the transformation expressions are obtained as q = q(θ)
(2.30a)
v = Bθ˙
(2.30b)
¨+B ˙ θ˙ v˙ = Bθ
(2.30c)
The kinematic constraints for the cut joints are expressed in body-coordinate formulation as
where an
∗
Φ∗ (q(θ)) = 0
(2.31a)
˙ ∗ = D∗ Bθ˙ = D ¯ θ˙ = 0 Φ
(2.31b)
¨+D ¨∗ = D ¯θ ¯˙ θ˙ = 0 Φ
(2.31c)
¯ D∗ B ≡ D
(2.31d)
indicates a cut joint. Applying the transformation expressions and pre-
multiplying by B> yield the equations of motion for the closed-chain system as ¨=h ¯ +D ¯θ ¯ > λ∗ M where ¯ = B> MB M > ¯ ˙ ˙ h = B h − MBθ
(2.32)
31 Equations (2.31) and (2.32) represent a system os DAE’s. If Eq. (2.31c) is appended to Eq. (2.32), we obtain θ > ¨ h ¯ ¯ −D ¯ M = , λ ∗ γ ∗ ¯ D 0
¯˙ θ˙ γ ∗ = −D
(2.33)
¨ and λ∗ . that can be solved for θ 2.1.3 Canonical Form of Equations of Motion In the previous sections, the body-coordinate and the joint-coordinate formulations are presented in the standard form of equations of motion. In the standard form, the equations of motion are written in terms of the corresponding accelerations. The body or joint accelerations are obtained by solving these equations. Alternatively, the equations of motion for a multibody system can be derived in terms of the total momenta of the system. If the body coordinates are used, the momentum equations are written as Mv = p + D> σ
(2.34)
where p denotes the momenta of the system and σ denotes the integral of Lagrange multipliers λ. The time derivative p is determined by ˆ −D ˙ >σ p˙ = h ˆ is defined as where the force array h ˆ h1 .. ˆ h= , . ˆ hnb
ˆi = h
(2.35)
f i
(2.36)
ni
ˆ the velocity v is defined as Eq. (2.2). It should be noted For this definition of h, ˆ i . Differentiating Eq. (2.34) with that gyroscopic moments are not included in h respect to time yields ˙ > σ + D> σ˙ Mv˙ + hgyro = p˙ + D
(2.37)
32 where
hgyro
0 ˜ ω J ω 1 1 1 .. = . 0 ω ˜ nb Jnb ω nb
(2.38)
ˆ − hgyro = h. If we apply σ˙ = λ and the expression of p˙ It should be noted that h to Eq. (2.37), we obtain Mv˙ = h + D> λ
(2.39)
which is exactly the same as Eq. (2.20). In the canonical form of the equations of motion, the body velocities are computed by solving a system of linear equations M −D> v p = (2.40) σ 0 D 0
If we use the joint coordinates, the canonical form of equations of motion for open-chain systems is obtained as ¯ B> MBθ˙ = p
(2.41)
¯ is the transformed momenta of the system. Differentiating Eq. (2.41) yields where p ˙ > MBθ˙ + B> d MBθ˙ ¯˙ = B p (2.42) dt where d d ˆ + D> λ MBθ˙ = (Mv) = Mv˙ + hgyro = h dt dt Therefore, Eq. (2.42) becomes > > ˆ > ˙ ˙ ˙p ¯ = B MBθ + B h + D λ
(2.43)
(2.44) ˆ ˙ > MBθ˙ + B> h =B
33 A similar procedure can be applied to derive the canonical form of equations of motion for closed-chain systems. The momentum equations of a closed-chain system are written as Mv = p + D> σ + D∗> σ ∗
(2.45)
where D∗ is the Jacobian matrix for cut joints and D is the Jacobian matrix for other joints. If we apply the velocity transformation of Eq. (2.24) and pre-multiply both sides of this equation by B> , we obtain B> MBθ˙ = B> p + B> D> σ + B> D∗> σ ∗
(2.46)
Since we have DB = 0, this equation becomes ¯ >σ∗ ¯ +D B> MBθ˙ = p
(2.47)
where ¯ = B> p, p
¯ > = B> D∗> D
(2.48)
Taking the time derivative of Eq. (2.47) yields ¯˙ > σ ∗ + D ¯ > λ∗ ˙ > MBθ˙ + B> d MBθ˙ = p ¯˙ + D B dt
(2.49)
d d ˆ + D> λ + D∗> λ∗ MBθ˙ = (Mv) = Mv˙ + hgyro = h dt dt
(2.50)
where
Therefore, Eq. (2.49) becomes ˆ −D ˙ > MBθ˙ + B> h ¯˙ > σ ∗ ¯˙ = B p
(2.51)
¯˙ . that can be solved for p Generally, the acceleration array of a multibody system has larger fluctuations than its velocity array.
Since the dynamics of a system is represented by its
velocities, the canonical form has advantages over the standard form in terms of accuracy and stability when the equations of motion are solved numerically [73].
34 In chapter 7, the equations of motion of a vehicle will be derived in the form of Eq. (2.41) and Eq. (2.44). The procedure to solve the multibody equations of motion numerically will be discussed in section 2.3 in more detail. 2.2 Contact-Impact Model in Multibody Systems Colliding bodies experience sudden change in their velocities during an impact. The analysis of this intermittent motion is an important subject of multibody system dynamics. In a broad sense, the methods for the analysis of contact can be classified into discontinuous and continuous methods. 2.2.1 Discontinuous Method In discontinuous methods, the impact is assumed to occur instantaneously. Within this approach, the integration of the equations of motion is stopped at the time of impact and the post-impact velocities are determined through a momentum balance analysis. Then, the integration of the equations of motion is resumed with the updated velocities until the next impact occurs. The restitution coefficient comes into play in determining the post-impact velocities with the energy dissipation taken into account. This method, commonly referred to as piecewise analysis, has some limitations in the case that the contact period is relatively long, where the assumption of instantaneous impact does not hold. As a result, this method is more suitable for an analysis of impact between rigid bodies where impulsive forces appear during the period of contact. More information and discussion about the discontinuous method can be found in [74]. 2.2.2 Continuous Method Continuous methods, which are also referred to as the penalty methods, are based on a continuous contact force model [73, 75, 76] where it is assumed that the contact forces and deformations change in a continuous manner. The simplest contact force model is the Kelvin-Voigt viscous-elastic model, which utilizes a parallel spring-
35 damper element [77]. In this model, the spring element accounts for the elasticity of the contacting bodies and the damper represents the loss of kinetic energy during the impact. The spring coefficient K depends on the material and geometric properties of the contacting bodies. For example, in the case of two spherical surfaces in contact, the spring coefficient can be determined as [78] 1/2 Ri Rj 4 K= 3(σi + σj ) Ri + Rj
(2.52)
where the material parameters σi and σj are given by σk =
1 − νk2 , Ek
(k = i, j)
(2.53)
In these equations νk and Ek are the Poisson’s ratio and the Young’s modulus of each sphere. Regarding the damping in the impact, Zhu et al. [79] proposed a theoretical formula for calculating damping with the assumption that both the spring and the damper are linear. When the contact bodies are separating from each other, the energy loss can be taken into account by multiplying the rebound force with a coefficient of restitution. The normal Kelvin-Voigt contact force FN is calculated for a given penetration depth, δ, as Kδ if δ > 0, vN > 0 (loading phase) FN = Kδce if δ > 0, vN < 0 (unloading phase)
(2.54)
where K is spring coefficient, δ is the relative penetration depth, ce is the restitution coefficient, and vN is the relative normal velocity of the contacting bodies. In this research, we use the contact force model of Eq. (2.54) to represent the tire-ground contact, which will be discussed in chapter 7.
36 2.3 Numerical Analysis of Multibody System Dynamics This section presents the general procedure for numerically solving the equations of motion of a multibody system. Except for some simple systems, the equations of motion of a multibody system are nonlinear differential equations. Thus, numerical analysis is the only possible option for solving the equations. It is possible to solve the multibody equations by any ordinary differential equation (ODE) solver. The performance of an ODE solver highly depends on the characteristics of the multibody system. Numerical integration methods for ODEs will be discussed in detail in Chapter 3. A generic ODE solver assumes that the differential equation has the form y˙ = f (t, y)
(2.55)
where y is the integration variable. A function to evaluate f (t, y) must be provided to the ODE solver. In addition, the initial value of the integration variable y0 should be given. With these inputs, for a set of prescribed time steps t1 , . . . , tN , the ODE solver generates the array of numerical solutions yn ,
n = 1, . . . , N
(2.56)
Unlike other methods, the local linearization method, which is of particular interest in this research, additionally requires the system Jacobian ∂f /∂y as an input. The reason for this will be clarified in the next chapter. Some implicit methods also take the system Jacobian as an optional input. Providing the system Jacobian can accelerate the computation by improving the efficiency of the Newton method in solving the required nonlinear equations. 2.3.1 Integration Variables Depending on the formulation method, the array of integration variables is formed differently for multibody equations. If the body-coordinate formulation is used, the
37 array of integration variables and the function f (t, y) are defined as q q˙ y= , f= v v˙
(2.57)
The body velocities v of y is used to compute q˙ of f . Then, the body accelerations v˙ are computed by solving Eq. (2.21). In the body-coordinate formulation, it is important to ensure that the initial condition q 0 y0 = v 0
(2.58)
satisfies the constraint equations Φ(q0 ) = 0,
D(q0 )v0 = 0
(2.59)
Otherwise, the subsequent solutions will not satisfy the constraints. Because the equations of motion in Eq. (2.21) contains the constraint equations in the acceleration level only, a solution may satisfy Φ(q(t)) = Φ(q0 ) + (t − t0 )D(q0 )v0
(2.60a)
D(q(t))v(t) = D(q0 )v0
(2.60b)
and the constraint equations at position and velocity level may not be satisfied. Similarly, the array of integration variables for the joint-coordinate formulation is defined as
θ y= , θ˙
f=
θ˙ θ ¨
(2.61)
¨ are computed by solving Eq. (2.29). The joint accelerations θ If the canonical form of equations of motion is used, the array of integration variables consists of the coordinates and the momenta of the system. In the case of
38 the canonical form with the body coordinates, the array of integration variables y and the function f are defined as y=
q p
,
f=
q˙
(2.62)
p˙
The components of f are computed by solving Eqs. (2.34) and (2.35). The initial condition for the momenta, p0 , should be determined from q˙ 0 by setting σ = 0. In the case of the canonical form with the joint coordinates, the array of integration variables y and the function f are defined as θ θ˙ y= , f= p p ¯ ¯˙
(2.63)
The components of f are computed by solving Eqs. (2.41) and (2.44) if an openchain system is considered. For a closed-chain system, the component of f are computed by solving Eqs. (2.47) and (2.51). The initial condition for the momenta, ¯ 0 , should be determined from θ˙ 0 by setting σ ∗ = 0. p 2.3.2 Constraints Violation The equations of motion generated by the body-coordinate formulation contain constraint equations from the kinematic joints. An ODE solver takes an input q˙ (2.64) y˙ = v˙ and at each time step it computes a numerical solution as q y= v
(2.65)
Due to the numerical error introduced by the ODE solver, the numerical solution may not satisfy the constraint equations, namely, Φ(q) 6= 0,
D(q)v 6= 0
(2.66)
39 The amount of the constraint violation may accumulate over time. The constraint violation can be an important issue when dynamic analysis for a long period of time is attempted. Several techniques to reduce or eliminate the constraints violation have been proposed in the past decades. The most widely used technique is the Baumgarte stabilization technique [80]. In this method, Dv˙ = γ of Eq. (2.21) is replaced by ˙ − β 2Φ Dv˙ = γ − 2αΦ
(2.67)
where α and β are positive constants. This equation is equivalent to ¨ + 2αΦ ˙ + β 2Φ = 0 Φ
(2.68)
The idea of this technique is to choose free parameters α and β such that Eq. (2.68) is an asymptotically stable equation and Φ converges. Typically, the parameter β is set to be β = α. The difficulty of this technique lies in a good choice for α. Another technique is the Projection on Constraint [81]. In this approach, the ˜ )n is obtained for t = tn by an ODE solver. The q ˜ n is numerical solution (˜ qn , v projected on the position constraint by solving nonlinear system ˜ n ) + D> (˜ M (˜ qn ) (qn − q qn )λ = 0
(2.69a)
Φ(qn ) = 0
(2.69b)
With the solution qn of Eq. (2.69), the projected velocity vn is obtained by solving the system of linear equations ˜ n ) + D> (qn )λ = 0 M (qn ) (vn − v
(2.70a)
D(qn )vn = 0
(2.70b)
The projected solution (qn , vn ) becomes the final numerical solution. This process is performed at each time step.
40 The coordinate partitioning method [71] makes use of the fact that the body coordinates are not independent when constraints are present. The body coordinates are partitioned into independent and dependent set as q(k) q= q(u)
(2.71)
where q(k) denotes a set of independent coordinates and q(u) denotes a set of dependent coordinates. The number of independent coordinates should be equal to the number of degrees of freedom. Similarly, the body velocity is partitioned as v(k) (2.72) v= v(u) Then, the array of integration variables is defined as q(k) y= v(k)
(2.73)
and the numerical integration is performed only for the independent coordinates and velocities. The dependent coordinates q(u) can be determined by solving the constraint equations: Φ q(k) , q(u) = 0
(2.74)
Equation (2.74) is a system of nonlinear equations and should be solved by the Newton method. The velocity constraint equations can be written as Dk v(k) + Du v(u) = 0
(2.75)
where Dk is the constraint Jacobian matrix associated with the independent coordinates and Du is the constraint Jacobian matrix associated with the dependent coordinates. If the partitioning is chosen such that Du is non singular, the dependent velocities can be determined by (k) v(u) = −D−1 u Dk v
(2.76)
41 The coordinate partitioning method, in principle, can remove the constraint violation. However, the main issue of this method is how to choose the set of independent coordinates. If the independent coordinates are poorly selected, the Newton method for solving Eq. (2.74) may fail to converge. When an open-chain system is consider, the joint-coordinate formulation can solve the problem of constraint violation completely. The equations of motion for an open-chain system generated by the joint-coordinate formulation do not have any constraint equations. As a result, the joint-coordinate formulation is free from the issue of constraints violation. For a closed-chain system, the equations of motion still contain constraints for cut joints and the constraint violation can occur. However, other kinematic joints except for the cut joints are free from the constraint violation when the joint-coordinate formulation is used. In the case of the canonical form of equations of motion in Eq. (2.40), the constraint conditions are imposed at the velocity level. Therefore, the velocity constraints Dv = 0 is exactly satisfied, but the position constraints may not be satisfied due to the numerical error introduced by an ODE solver. However, the amount of the constraint violation will be smaller than that of the standard form.
42 CHAPTER 3 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS
The analysis of a dynamical system involves solving ordinary differential equations which describe the dynamics of the system. The analytic solution for an ordinary differential equation can be obtained by integrating the differential equation with respect to time.
However, the analytic integration of the differential equation
can be achieved for some exceptional cases, where in most cases, obtaining a numerical solution is the only possible option.
The advent of modern digital
computers enables us to solve a large system of differential equations in reasonable computation time. This chapter presents a brief review of numerical integration methods for ordinary differential equations. Developing numerical integration methods has been a major research subject in applied mathematics and computational dynamics community for several decades. The first numerical method dates back to 1770, when Leonhard Euler suggested the Euler method in his book Institutionum calculi integralis [82]. The Euler method is the most basic numerical method for numerical integration of ordinary differential equations. The solution of an ordinary differential equation y˙ = f (t, y) can be written by the Taylor series expansion y(tn ) = y(tn−1 ) + hy(t ˙ n−1 ) + O(h2 )
(3.1)
where h = tn − tn−1 . The forward Euler method is derived by dropping the higher order terms, O(h2 ). The formula for the forward Euler method is yn = yn−1 + hf (tn−1 , yn−1 )
(3.2)
The forward Euler method is a first-order explicit method. Explicit methods refer to the methods which compute the numerical solution without iteration at each time
43 step. Unlike the forward Euler method, the backward Euler method is an implicit method. The backward Euler method has the formula as yn = yn−1 + hf (tn , yn )
(3.3)
Equation (3.3) contains the unknown yn on both sides in a nonlinear expression. Consequently, a nonlinear system of algebraic equations needs to be solved at each time step. Since the Euler methods are first-order methods, a very small step size is required to obtain a numerical solution of a good accuracy. As a result, the computational efficiency is degraded. This problem inspired the development of higher-order methods like the Runge-Kutta and Adams methods. 3.1 Runge-Kutta Methods The Runge-Kutta methods are an important family of numerical methods and were developed around 1900 by the German mathematicians C. Runge and M.W. Kutta [83, 84]. The solution of an ordinary differential equation y˙ = f (t, y) should satisfy Z tn y(tn ) = y(tn−1 ) + f (t, y)dt (3.4) tn−1
The formula of the Runge-Kutta methods can be derived by applying quadrature R tn rules to the integral tn−1 f (t, y)dt. For example, applying the trapezoidal quadrature rule to Eq. (3.4) yields yn = yn−1 +
h (f (tn , yn ) + f (tn−1 , yn−1 ) 2
(3.5)
This scheme is called the Trapezoidal method, which is a second-order implicit Runge-Kutta method. To obtain an explicit method based on this scheme, we can approximate yn in f (tn , yn ) by the forward Euler method as yˆ = yn−1 + hf (tn−1 , yn−1 )
(3.6)
Replacing f (tn , yn ) by f (tn , yˆn ) yields the explicit trapezoidal method: yn = yn−1 +
h (f (tn , yˆn ) + f (tn−1 , yn−1 )) 2
(3.7)
44 which is an explicit two-stage Runge-Kutta method of order two. The classical fourth-order Runge-Kutta method is based on Simpson’s quadrature rule and its formula is given by Y1 = yn−1 h Y2 = yn−1 + f (tn−1 , Y1 ) 2 h Y3 = yn−1 + f (tn−1/2 , Y2 ) 2
(3.8)
Y3 = yn−1 + hf (tn−1/2 , Y3 ) yn = yn−1 +
h f (tn−1 , Y1 ) + 2f (tn−1/2 , Y2 ) + 2f (tn−1/2 , Y3 ) + f (tn , Y4 ) 6
The general structure of a Runge-Kutta method of s-stages is given by Yi = yn−1 + h yn = yn−1 + h
s X j=1 s X
aij f (tn−1 + hcj , Yj ),
i = 1, . . . , s (3.9)
bj f (tn−1 + hcj , Yj )
j=1
where Y1 , . . . , Ys represent the intermediate approximations to the solution at tn−1 + c1 h, . . . , tn−1 + cs h. Such a method can be represented by the so-called Butcher tableau as c
A b>
where
(3.10)
c b a a · · · a 1 1 11 12 1s c b a a · · · a 2 2 21 22 2s c= . , b= . , A= . .. . . .. . . . . . . . . . cs bs as1 as2 · · · ass
In the case of an explicit method, A is strictly lower triangular and c1 = 0. As an
45 example, the explicit trapezoidal method is represented by 0 1/2
0
0
1/2 0 0
(3.11)
1
The Runge-Kutta methods are widely used for general ordinary differential equations. The Runge-Kutta methods are usually implemented together with an adaptive step size selection scheme. Matlab’s standard ordinary differential equation solver, ODE45, is based on an explicit Runge-Kutta formula, the Dormand-Prince pair [85]. The Dormand-Prince pair provides the way of estimating the numerical error, which will be explained in chapter 5 in more detail. The step size is adaptively selected in ODE45 so that the numerical error is maintained within a predefined tolerance. In general, ODE45 is the best solver to apply as a first try for most problems. 3.2 Adams Methods Adams methods are multi-step methods, which use information from previous integration steps to construct higher-order approximations. The Runge-Kutta methods are one-step methods in a sense that they do not use any information from previous steps. The basic idea of Adams methods is to approximate the integrand f (t, y) of Eq. (3.4) by an interpolating polynomial through previously computed values of f (t, y) [86]. As an example, the second-order explicit Adams method (AdamsBashforth) is obtained by interpolating f (t, y) through the two previous points as 3 1 yn = yn−1 + h fn−1 − fn−2 (3.12) 2 2 The k-step explicit Adams method is obtained by interpolating f (t, y) through the k previous points and its formula is yn = yn−1 + h
k X j=1
βj f (tn−j , yn−j )
(3.13)
46 Table 3.1 gives the coefficients of the explicit Adams methods up to k = 6. The k-step explicit Adams method has the k-th order of accuracy. The problem of the explicit Adams methods is the small regions of absolute stability. The implicit Adams method (Adams-Moulton) [87] is introduced to overcome this issue. In the case of the implicit Adams method, the unknown value f (tn , yn ) is included in the interpolating polynomial. The formula of the k-step implicit Adams method is yn = yn−1 + h
k X
βj f (tn−j , yn−j )
(3.14)
j=0
It should be noted that the order of the k-step implicit Adams method is p = k + 1 except for the case of k = 1. The implicit Adams methods use one fewer step for the same order and have larger stability region than the explicit Adams method. However, the nonlinear algebraic equations should be solved at each time step within the implicit Adams methods. Frequently, the implicit Adams methods are used together with the explicit Adams methods, where the initial guess for solving nonlinear equation is provided by the explicit Adams methods. Such an approach is called a predictor-corrector scheme. 3.3 Backward Differentiation Formula Another family of multi-step methods is the backward differentiation formula (BDF) [88] family. In the case of the Adams methods, a polynomial interpolates the previous values of f . The BDF methods are constructed by differentiating the polyTable 3.1: Coefficients of explicit Adams methos up to order 6 k j→ 1 2 3 4 5 6 1 βj 1 2 2βj 3 -1 3 12βj 23 -16 5 4 24βj 55 -59 37 -9 5 720βj 1901 -2774 2616 -1274 251 6 1440βj 4277 -7923 9982 -7298 2877 -475
47 nomial which interpolates the previous values of y, and setting the derivatives at tn to f (tn , yn ). The BDF methods are also known as Gear algorithms. The BDF methods is the most popular multi-step methods for stiff systems. A stiff system is referred to as a system in which the complete solution consists of fast and slow components. The k-step BDF methods can be written as k X
αi yn−i = hβ0 f (tn , yn )
(3.15)
i=0
The k-step BDF method has order p = k. The coefficients for the BDF methods are listed in Table 3.3. Since the BDF methods are implicit methods, it is necessary to solve a nonlinear equation at each time step. To this end, the BDF methods are usually implemented together with the Newton method. The BDF methods are implemented in Matlab as ODE15s [85]. It is Matlab’s standard solver for stiff problems or differential-algebraic problems. The order of the method is adaptively changed according to the dynamics of the solution in ODE15s. 3.4 St¨ormer-Verlet Method Some scientific and engineering problems can be modelled as a Hamiltonian system. These problems have the form p˙ = −Hq (p, q),
q˙ = Hp (p, q)
(3.16)
where the Hamiltonian H(p, q) represents the total energy of the system, q denotes the position and p denotes the momenta. Hp and Hq are the vectors of partial Table 3.2: Coefficients of implicit Adams methos up to order 6 k j→ 1 2 3 4 5 6 1 βj 1 2 2βj 1 1 3 12βj 5 8 -1 4 24βj 9 19 -5 1 5 720βj 251 646 -264 106 -19 6 1440βj 475 1427 -798 482 -173 27
48 derivatives. The Hamiltonian is an invariant and the solution of Eq. (3.16) should satisfy H (p(t), q(t)) = Const
(3.17)
In addition, the exact flow of a Hamiltonian system (3.16) is area preserving. For Hamiltonian systems, preserving such properties of the system is a primary concern when numerical solutions are obtained. Numerical integration methods for this purpose are referred to as geometric integration methods. A detailed discussion about geometric integration methods can be found in [89]. One example of geometric integration methods is the St¨ormer-Verlet method. The ordinary differential equation of the form p˙ = f (q),
q˙ = p
(3.18)
is equivalent to a second order differential equation q¨ = f (q)
(3.19)
Replacing the second derivative by the central second-order difference yields the discretization formula qn+1 − 2qn + qn−1 = h2 f (qn ) Table 3.3: k β0 1 1 2 32 6 3 11 4 12 25 60 5 137 60 6 147
Coefficients of BDF methos up to order 6 α0 α1 α2 α3 α4 α5 α6 1 -1 1 1 − 43 3 18 9 2 1 − 11 − 11 11 36 3 1 − 48 − 16 25 25 25 25 300 300 200 75 12 1 − 137 − 137 − 137 137 137 360 450 400 225 72 10 1 − 147 147 − 147 147 − 147 147
(3.20)
49 Alternatively, Eq. (3.20) can be written in terms of p and q by h pn+1/2 = pn + f (qn ) 2 (3.21)
qn+1 = qn + hpn+1/2
h pn+1 = pn+1/2 + f (qn+1 ) 2 This method is called the St¨ormer method in astronomy, the Verlet method in molecular dynamics, the leap-frog method in the context of partial differential equations, and it has further names in other areas [89]. 3.5 Local Linearization Method The Local Linearization method (LLM) was initially suggested as an alternative to the Runge-Kutta methods for achieving real-time simulation of time-invariant systems [57]. However, its application is not limited for time-invariant systems. The LLM can be used for general non-autonomous ODEs [58]. The LLM is an exponential method which is based on the piecewise linear approximation of an ordinary differential equation through a first-order Taylor expansion at each time step. The solution at the next time step is determined by the analytic solution of the approximated linear system. For a linear differential equation y˙ = Ay + b(t),
y(0) = y0
(3.22)
eA(t−τ ) b(τ )dτ
(3.23)
the analytical solution is given by [90] At
Z
y(t) = e y0 +
t
0
Let us consider the non-autonomous differential equation y˙ = f (t, y). We take the Taylor series expansion of f (t, y) around t = tn as ∂f ∂f f (t, y) = f (tn , y(tn )) + (y − y(tn )) + ∂y ∂t t=tn
(t − tn ) + HOT
(3.24)
t=tn
Then, we approximate f (t, y) by a linear system f (t, y) ≈ An y + bn (t)
∀t ∈ [tn , tn+1 ]
(3.25)
50 where ∂f An = ∂y
t=tn
∂f bn (t) = f (tn , y(tn )) − ∂y
∂f y(tn ) + ∂t
t=tn
(t − tn ) t=tn
We denote the solution of the approximated system at t = tn by yˆn and we have the expression of yˆn+1 as hn fy
yˆn+1 = e
Z
hn
e(hn −u)fy (f (tn , yˆn ) − fy yˆn + ft u) du
yˆn +
(3.26)
0
where ∂f fy = ∂y
(t=tn ,y=ˆ yn )
∂f , ft = ∂t
, hn = tn+1 − tn (t=tn ,y=ˆ yn )
Equation (3.26) can be rearranged as Z Z hn (hn −u)fy hn fy e fy du yˆn + yˆn+1 = e −
hn
e(hn −u)fy (f (tn , yˆn ) + ft u) du (3.27)
0
0
From the equality d (hn −u)fy e = −e(hn −u)fy fy du we obtain Z 0
hn
hn e(hn −u)fy fy du = −e(hn −u)fy 0 = −I + ehn fx
By plugging Eq. (3.29) into Eq. (3.27) we obtain the expression of yˆn+1 as Z hn yˆn+1 = yˆn + e(hn −u)fy (f (tn , yˆn ) + ft u) du
(3.28)
(3.29)
(3.30)
0
As a result, for the non-autonomous differential equation y˙ = f (t, y), y ∈ Rm , the LLM formula can be represented as the following [58] equation yˆn+1 = yˆn + φ(tn , yˆn ; hn ),
yˆn ∈ Rm
(3.31)
51 starting with y0 = y(t0 ), where Z hn efy (tn ,ˆyn )(hn −u) (f (tn , yˆn ) + ft (tn , yˆn )u) du, φ(tn , yˆn ; hn ) =
(3.32)
0
It should be noted that the forward Euler method computes the numerical solution at t = tn+1 by yˆn+1 = yˆn +hn f (tn , yˆn ). In the case of the LLM, hn f (tn , yˆn ) is replaced by a function φ(tn , yˆn ; hn ), which involves integral of matrix exponential over time. Equation (3.32) can be evaluated by applying Theorem 1 in reference [91]. This theorem proves that integrals involving matrix exponentials can be computed by the exponential of a block triangular matrix. More precisely, if the m × m block triangular matrix C is defined as A1 B1 C1 0 A B 2 2 C= 0 0 A3 0 0 0 then for t > 0
eCt
D1
C2 B3 A4
F1 (t) G1 (t) H1 (t) 0 F2 (t) G2 (t) = 0 0 F3 (t) 0 0 0
K1 (t) H2 (t) G3 (t) F4 (t)
(3.33)
(3.34)
where Fj (t) = eAj t j = 1, 2, 3, 4 Z t Gj (t) = eAj (t−s) Bj eAj+1 s ds j = 1, 2, 3 0 Z t Z tZ s Aj (t−s) Aj +2s Hj (t) = e Cj e ds + eAj (t−s) Bj eAj+1 (s−r) Bj+1 eAj+2 r drds j = 1, 2 0
0
0
and Z K1 (t) =
t A1 (t−s)
e 0
Z tZ s D1 e ds+ eA1 (t−s) C1 eA3 (s−r) B3 + B1 eA2 (s−r) C2 eA4 r drds Z t Z 0s Z 0 r + eA1 (t−s) B1 eA2 (s−r) B2 eA3 (r−w) B3 eA4 w dwdrds A4 s
0
0
0
52
To use Eq. (3.34) for evaluating Eq. (3.31), Eq. (3.32) can be rewritten in the form of H1 (t) as Z
hn
e
φ(tn , yˆn ; hn ) =
fy (tn ,ˆ yn )(hn −u)
Z
Z
f (tn , yˆn )du + 0
0
hn
u
efy (tn ,ˆyn )(hn −u) ft (tn , yˆn )dvdu
0
Then, Eq. (3.35) can be obtained from F (tn , yˆn ; hn ) g1 (tn , yˆn ; hn ) φ(tn , yˆn ; hn ) ehn Cn = 0 1 g (t , y ˆ ; h ) 2 n n n 0 0 1
(3.35)
(3.36)
where fy (tn , yˆn ) ft (tn , yˆn ) f (tn , yˆn ) ∈ R(m+2)×(m+2) Cn = 0 0 1 0 0 0 For autonomous ODEs, we do not need to consider ft , and the function φ(tn , yˆn ; hn ) can be evaluated as ehn Cn where
F (tn , yˆn ; hn ) φ(tn , yˆn ; hn ) = 0 1
(3.37)
fy (tn , yˆn ) f (tn , yˆn ) ∈ R(m+1)×(m+) Cn = 0 0
An advantage of the LLM can be seen in the analysis of absolute stability. For a given numerical method, the region of absolute stability is the region of the complex ξ-plane such that applying the method for the test equation y˙ = λy, with ξ = hλ from within this region, yields an approximate solution satisfying the absolute stability requirement, |yn+1 | ≤ |yn | [86]. If the absolute stability region
53 of a numerical method covers the entire left half ξ-plane, the numerical method is called A-stable method. For the test equation y˙ = λy, the LLM gives an exact solution regardless of the step size h because the test equation is linear. Thus, the LLM is an explicit A-stable method. It should be noted that there is no explicit Runge-Kutta method satisfying A-stable property. Another advantage of the LLM is that the numerical solution at any time t ∈ [tn , tn+1 ] can be computed easily. The linear approximation at t = tn is valid for t ∈ [tn , tn+1 ] and the numerical solution at t can be computed as yˆ(t) = yˆn + φ(tn , yˆn ; t − tn )
t ∈ [tn , tn+1 ]
(3.38)
Thus, it is possible to generate a denser output for a specific time domain of interest. This process can be performed in the post-processing if it is desired. Having dense output capability is important for detecting event locations such as the beginning of contact between bodies in a multibody system. A numerical solution by the LLM is obtained through the computation of the matrix exponential as in Eq. (3.36). Computation of this matrix can be an expansive numerical operation, but its computational efficiency has been substantially improved by the development of new algorithms. This is why exponential type numerical integrators have been receiving more attention. Standard algorithm of matrix exponential is the Pad´e approximation with scaling and squaring [92], which is implemented in Matlab as the command expm [85].
System Jacobian The LLM requires the system Jacobian ∂f /∂y as an input. The system Jacobian may be computed by the finite difference method. However, the finite difference process can take a long time when the size of problem becomes large. In the case of the multibody equations, the explicit expression for the system Jacobian can be
54 derived by taking the partial derivatives of f with respect to the integration variable y. Using the explicit expression of the system Jacobian can improve the efficiency of the LLM. When the body-coordinate formulation is used, the system Jacobian ∂f /∂y has the following form:
0 ∂f = ∂ q¨ ∂y
∂q
I ¨ ∂q ∂ q˙
(3.39)
By taking the partial derivatives of Eq. (2.20), we can obtain the expressions of ¨ /∂q and ∂ q ¨ /∂ q˙ as ∂q ¨ ∂q ∂q
−1
=M
¨ ∂q ∂ q˙
∂D ∂q
>
D> ∂λ ∂q
∂h ∂q
+ − λ+ = M−1 D> ∂λ + ∂h ∂ q˙ ∂ q˙
∂M ¨ q ∂q
(3.40) (3.41)
The expression of λ is obtained from (2.21) as DM−1 D> λ = γ − DM−1 h
(3.42)
The partial derivatives of Eq. (3.42) yields the expressions of ∂λ/∂q and ∂λ/∂ q˙ as
−1 ∂γ ∂D ∂h ∂λ ¨ − DM−1 = DM−1 D> − q ∂q ∂q ∂q ∂q ∂D> ∂M ¨ − DM−1 λ + DM−1 q ∂q ∂q −1 ∂λ = DM−1 D> ∂ q˙
∂γ ∂h − DM−1 ∂ q˙ ∂ q˙
!
(3.43)
(3.44)
If the joint-coordinate formulation is used for a open-chain system, the system Jacobian has the following form:
0
I
∂θ
∂ θ¨ ∂ θ˙
∂f = ¨ ∂θ ∂y
(3.45)
55 The components of this matrix can be obtained by taking the partial derivatives of ˙ respectively. Taking the partial derivatives of Eq. (2.29) with respect to θ and θ, Eq. (2.29) yields ¨ ∂θ ¯ −1 =M ∂θ
¯ ¯ ∂h ∂M − , ∂θ ∂θ
¨ ¯ ∂θ ¯ −1 ∂ h =M ∂ θ˙ ∂ θ˙
(3.46)
The system Jacobian for other formulations can be obtained through a similar procedure.
Depending on the configuration of the system, some terms may
disappear. Furthermore, the expression of partial derivative in each term may be obtained with the help a symbolic manipulator.
Contact Instant Detection In section 2.2, several methods for the analysis of contact problem were discussed. In this research we use the continuous method because the primary interest is the contact of flexible bodies such as tires. The continuous method is more suitable than the discontinuous method if the contact between bodies occur for a long period of time. The Kelvin-Voigt contact force model of Eq. (2.54) is considered in the numerical examples having contact conditions. Specifically, we consider the simple linear contact force model as
FN =
Kδ
if δ > 0
0
if δ 6= 0
(3.47)
where δ is the penetration depth. The LLM is based on the successive linear approximation of the system, where it is assumed that the system Jacobian is constant during one time step. In the case of the contact problem modelled by the penalty method, the system Jacobian may change significantly when bodies come into or out of contact. To avoid this problem, time step should be adjusted such that the new step starts exactly when the contact begins or ends. To detect the instant when the contact begins or ends, we define the function Ψ(t) that represents the contact
56 condition as Ψ(t) = δ(y(t))
(3.48)
When the numerical solution yˆn+1 is obtained, it is checked if sign(Ψ(tn )) = sign(Ψ(tn+1 )). When the signs are different, it indicates that bodies have come in or out of contact during the steps tn and tn+1 . The contact instant t∗ can be determined by solving the nonlinear equation Ψ(t∗ ) = 0. The Newton-Raphson or Bi-section method can be used for solving this nonlinear equation. Figure 3.1 illustrates the case that the contact occurs between tn and tn+1 . The penetration depth at any time between time steps δ(ˆ y (t)) can be evaluated without additional numerical error from interpolation since the LLM formula gives the numerical solution yˆ(t) for t ∈ [tn , tn+1 ] as yˆ(t) = yˆn + φ(tn , yˆn ; t − tn )
for t ∈ [tn , tn+1 ]
(3.49)
Once t∗ is determined, the step size hn is reset to h∗ = t∗ − tn . This process is repeated until the end of the time domain. If the system contains k contact candidates, the penetration depth δk for each contact candidate should be checked and the contact instant t∗k should be determined as mentioned above for each contact candidate respectively. Then, the step size hn is reset to h∗ = mink (t∗k ) − tn .
tn1
tn
t
h hn Figure 3.1: Detection of the instant that a contact begins
57 This chapter has provided a review of the numerical integration methods for ordinary differential equations. The formula for the LLM was introduced and the properties of the LLM were addressed. The matters which need to be considered when the LLM is used for solving the multibody equations were also discussed. The next chapter will present the application of the LLM in the solution of highly oscillatory multibody systems. It will be demonstrated that the LLM can solve such systems more efficiently than other methods through simple numerical examples.
58 CHAPTER 4 APPLICATION OF LLM IN HIGHLY OSCILLATORY MULTIBODY SYSTEMS
The equations of motion of multibody systems can be solved successfully by the explicit integration methods like the fourth-order Runge-Kutta as long as the system is not stiff. For stiff or highly oscillatory multibody systems, the efficiency of the explicit methods is severely degraded. A spring element of high stiffness or a contact force element can make a multibody system highly oscillatory. For this type of systems, explicit methods should take very small step size to satisfy the absolute stability condition. Standard explicit methods are conditionally stable, in other words, their absolute stability regions are limited. Therefore, the step size should be small enough to satisfy the absolute stability condition. This restriction of the step size can be relaxed if implicit methods like BDF or implicit Runge-Kutta methods are used because the absolute stability region of implicit methods covers much larger area. In the case of implicit methods, however, the Newton iteration is required to solve the system of nonlinear equations at each integration time step. This increases the computational cost significantly if the size of the problem is large. In addition, the step size may still be restricted due to difficulties in the convergence of the Newton iteration for larger step sizes. The LLM may be a good option for solving highly oscillatory multibody systems because it is an A-stable explicit method. The LLM can take a much larger step size than other numerical methods. In this chapter, the LLM is applied to highly oscillatory multibody systems and its efficiency is compared to other numerical methods.
59 4.1 A Simple Mechanical System In this section, we consider a simple planar mechanical system containing kinematic joints. This example is chosen for simplicity and clarity. The same methodology of formulating and solving the equations of motion can be applied to general spatial multibody systems. Figure 4.1 shows a double pendulum. This system contains two revolute joints and a torsional spring between the two links. The system parameters are listed in Table. 4.1. The high spring coefficient is chosen intentionally to make the system highly oscillatory. The only external force is the gravity that acts in the vertical direction.
Revolute Joint
Torsional Spring
Revolute Joint
Torsional Spring
𝑘
𝑘
Figure 4.1: Configuration of double pendulum
Table 4.1: Parameters of double pendulum Parameter Value Description m1 100 mass of the upper link (kg) J1 100 inertia of the upper link (kg · m2 ) m2 1 mass of the lower link (kg) J2 1 inertia of the lower link (kg · m2 ) L1 1 length of the upper link (m) L2 1 length of the lower link (m) k 3000 stiffness of the torsional spring (N m/deg)
60 The equations of motion of the system are derived by the body-coordinate and the joint-coordinate formulation as presented in chapter 2. Then, the equations of motion are solved by using the LLM, ODE45, ODE15s. Body-Coordinate Formulation The body coordinate q for the double pendulum is defined as r1 θ 1 q= r2 θ 2 The mass matrix M and the m1 I 0 M= 0 0
(4.1)
force array h can be written as m1 g 0 0 0 k(θ2 − θ1 ) J1 0 0 , h = m2 g 0 m2 I 0 k(θ1 − θ2 ) 0 0 J2
where g = [0, −9.81]> . The kinematic constraints from the two revolute joints are written as
Φ=
r1x − L1 /2 sin θ1 r1y + L1 /2 cos θ1
−r1x − L1 /2 sin θ1 + r2x − L2 /2 sin θ2 −r + L /2 cos θ + r + L /2 cos θ 1y 1 1 2y 2 2
The constraint Jacobian matrix 1 0 0 1 D= −1 0 0 −1
=0
(4.2)
D is −L1 /2 cos θ1 0 0
1 0 L2 /2 cos θ2 0 1 −L2 /2 sin θ2
−L1 /2 sin θ1 0 0 −L1 /2 cos θ1 −L1 /2 sin θ1
0
0
(4.3)
61
The initial conditions for the system are set as 0 L1 /2 −L /2 0 1 0 1 q0 = , q˙ 0 = 0 L1 − L2 /2 −L − L /2 0 1 2 0 −1
(4.4)
which satisfy the constraint equations. The equations of motion are constructed in the form of Eq. (2.21). The equations of motion are solved by the LLM, ODE45 and ODE15s. In the case of the LLM, the step size is h = 0.02. Other two methods choose the step size adaptively. The numerical solutions and computation times for the three methods are compared. Figure 4.2 shows the rotation angles of the double pendulum obtained by these methods. It can be seen that all three methods produce almost identical solutions. This system of double pendulum does not contain any damping, thus the total energy of the system should be constant. Figure 4.3 shows the total energy variations for the three methods, and Fig. 4.4 shows the norm of the constraint violations at position level. These results reveal that the LLM produces slightly better numerical solution than ODE45 and ODE15s in terms of total energy conservation and constraint violation with the particular choice of step size. The computation times for the three numerical methods are shown in Fig. 4.5. It can be seen that the LLM shows better computational efficiency compared to other two methods.
62
Rotation Angles
θ1 (deg)
0.5 LLM ODE45 ODE15s 0
−0.5
0
1
2
3 Time (s)
4
5
6
0
1
2
3 Time (s)
4
5
6
1
θ2 (deg)
0.5 0 −0.5 −1
Figure 4.2: Rotation angles of double pendulum with body-coordinate formulation
Total Energy Variation 1.2
LLM ODE45 ODE15s
1
δE
0.8
0.6
0.4
0.2
0
−0.2
0
1
2
3 Time (s)
4
5
6
Figure 4.3: Total energy variation of double pendulum with body-coordinate formulation
63
Constraint Violation 0.035 LLM ODE45 ODE15s
0.03
0.025
|Φ|
0.02
0.015
0.01
0.005
0
0
1
2
3 Time (s)
4
5
6
Figure 4.4: Constraint violation of double pendulum with body-coordinate formulation
Computation Time 1.4
1.2
CPU Time (s)
1
0.8
0.6
0.4
0.2
0
LLM
ODE45
ODE15s
Figure 4.5: Computation time for double pendulum with body-coordinate formulation
64 Joint-Coordinate Formulation An obvious choice for the joint coordinates for the system of double pendulum is θ 1 θ= (4.5) θ2 The transformation from the joint-coordinate to the body-coordinate is L1 /2 sin θ1 −L /2 cos θ 1 1 θ1 q= L1 sin θ1 + L2 /2 sin θ2 −L cos θ − L /2 cos θ 1 1 2 2 θ
(4.6)
2
Then, the velocity transformation matrix B of Eq. (2.24) is obtained as L /2 cos θ1 0 1 0 L1 /2 sin θ1 1 0 B= L1 cos θ1 L2 /2 cos θ2 L1 sin θ1 L2 /2 sin θ2 0 1 ¯ for this system are obtained as ¯ and h Using Eq. (2.29), M 2 2 J + (L1 m1 )/4 + L1 m2 (L1 L2 m2 cos(θ1 − θ2 ))/2 ¯ = 1 M (L1 L2 m2 cos(θ1 − θ2 ))/2 (m2 L22 )/4 + J2
(4.7)
(4.8)
K(θ − θ ) − (L gm sin(θ ))/2 − L gm sin(θ ) − (L L m θ˙2 sin(θ − θ ))/2 2 1 1 1 1 1 2 1 1 2 2 2 1 2 ¯= h 2 ˙ (L1 L2 m2 sin(θ1 − θ2 )θ )/2 + K(θ1 − θ2 ) − (L2 gm2 sin(θ2 ))/2 1
(4.9)
65 The equations of motion are solved by the LLM, ODE45 and ODE15s. Figure 4.6 shows the rotation angles of the double pendulum obtained by these numerical methods. As in the case of the body formulation, it can be seen that the three methods produce almost identical solutions. The computation times for the three numerical methods are shown in Fig.
4.7.
This figure reveals that the LLM
shows better computational efficiency compared to other two methods. It should be noted that the computation time of the joint coordinate formulation is much smaller than that of the body coordinate formulation for all those numerical methods. The joint coordinate formulation provides better efficiency than the body coordinate formulation because the number of equations are reduced from 6 to 2 and the computation of Lagrange multiplier is avoided. In addition, the constraint equations are always satisfied in the case of the joint coordinate formulation. This example demonstrates that the LLM can be useful when the equations of motion of highly oscillatory multibody system are numerically solved. Good computational efficiency can be achieved since a much larger step size can be used compared to other methods. This advantage is magnified as the system contains higher frequency responses.
66
Rotation Angles
θ1 (deg)
0.5 LLM ODE45 ODE15s 0
−0.5
0
1
2
3 Time (s)
4
5
6
0
1
2
3 Time (s)
4
5
6
1
θ2 (deg)
0.5 0 −0.5 −1
Figure 4.6: Rotation angles of double pendulum with joint-coordinate formulation
Computation Time 0.4 0.35
CPU Time (s)
0.3 0.25 0.2 0.15 0.1 0.05 0
LLM
ODE45
ODE15s
Figure 4.7: Computation time for double pendulum with joint-coordinate formulation
67 4.2 Contact Problem In section 2.2, several methods for the analysis of contact problem were discussed. The methods can be classified into discontinuous and continuous methods. Within the discontinuous method, the integration of the equations of motion is stopped at the time of impact and the post-impact velocity is determined by the momentum balance. Then, the integration of the equations of motion is resumed with the updated velocities until the next impact occurs. The period of impact is assumed to be instantaneous and is not taken into account when solving the equations of motion. Consequently, conventional numerical integration methods can be used for this approach without the issue of a small step size. On the other hand, the continuous method in contact problem introduces a fictitious spring of high stiffness to represent the contact force. As a result, the system becomes highly oscillatory during the period of contact.
Conventional
numerical methods may have difficulty in dealing with the fictitious spring due to its high stiffness. The LLM may be a good candidate for solving the contact problem when the continuous method is employed. In this section, the LLM is applied to a simple contact problem and its performance is compared to that of other conventional numerical methods. Let us consider two bouncing balls moving in vertical direction under gravity as shown in Fig. 4.8. The contact between the balls and between the ball and the ground are modeled by Eq. (2.54). The spring coefficient for the contact force model is chosen to be 105 N/m and the restitution coefficient ce is chosen to be 1. Each body has only one degree of freedom in the vertical movement. Assuming that both bodies have unit mass, the equations of motion can be easily constructed as x¨ −9.81 + F 1 1 (4.10) = x¨2 −9.81 − F1 + F2 where F1 represents the contact force between two balls and F2 represents the
68
g x1 x2
Figure 4.8: System of two bouncing balls contact force between second ball and the ground. The dynamic simulation of the two bouncing balls are performed by the LLM, ODE45 and ODE15s. Figure 4.9 shows the position of the balls with respect to time and Fig. 4.10 shows the total energy variation of the system. The result of ODE45 is quite different from the results of the other two numerical methods The results show large total energy variation. The total energy of the system should be constant since there is no damping. In contrast, the total energy of the numerical solution by the LLM is well conserved. With the step size adjustment discussed in section 2.5, the LLM can solve this problem without error because the dynamics of the system is piecewise-linear. The computation times for these methods are shown in Fig. 4.11. The LLM and ODE15s give similar results but the computation time of the LLM is much smaller than that of ODE15s. These results demonstrate that the LLM can be a good option for solving contact problems when a linear contact force method is employed. In this section, a very simple example of contact problem is considered. However, the same approach can be used for more complicated problems. In chapter 7, this contact force model is used for the contact between a tire and the ground.
69
Simulation results for two bouncing balls 15
x1
10
5
0
0
1
15
2
3
4
5 Time (s)
6
7
8
9
10
3
4
5 Time (s)
6
7
8
9
10
LLM ODE45 ODE15s
x2
10
5
0
0
1
2
Figure 4.9: Simulation results for two bouncing balls
Total Energy Variation 10 LLM ODE45 ODE15s
8 6 4 2 0 −2 −4 −6 −8 −10
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 4.10: Total energy variation for two bouncing balls
70
Computation time 12
10
8
6
4
2
0
LLM
ODE45
ODE15s
Figure 4.11: Computation time for two bouncing balls
71 CHAPTER 5 NUMERICAL ERROR ESTIMATION METHOD FOR LLM
The adaptive step size selection scheme is a common feature of conventional numerical integration methods. Such scheme changes the step size hn at each time step instead of using a constant step size h, so that the number of steps required to solve the ODEs are reduced. The step size hn should be determined such that the local error of numerical solution is maintained within a given tolerance. To this end, a procedure for estimating the local error introduced by the numerical method at each time step is essential. One approach to estimate the local error is to compute two solutions of different orders and compare them. This approach is called Embedded method. For the Runge-Kutta methods, the Dormand-Prince 4(5) pair [93] is the most popular embedded method and is implemented in ODE45. The Dormand-Prince method computes the numerical solution of order 4 and 5. Then, the local error is estimated by comparing these two solutions. This method has seven stages, but the last stage is the same as the first stage for the next step. Therefore, this method has the cost of a six-stage method. Two numerical solutions share internal stage computations, so the cost of computing two solutions is not far from that of computing single solution. Embedded methods require a set of coefficients which is derived for this purpose. Another approach is step doubling. The idea of step doubling is to estimate the local error by subtracting the solution obtained with the step size h from the solution obtained using the step size 2h. The local error introduced between tn and
72 tn+1 can be written as ln+1 = ψ (tn , yn ) hp+1 + O hp+2
(5.1)
where the function ψ is the principal error function [94] and p is the order of the numerical method. Now, let yn+1 be the solution using two steps of size h, and let y˜n+1 be the solution using one step of size 2h. Then, the two local errors satisfy 2ln+1 (h) = 2hp+1 ψ + O(hp+2 ) ln+1 (2h) = (2h)p+1 ψ + O(hp+2 ) by assuming that the local error after two steps is twice the local error after one step. The difference between yn+1 and y˜n+1 gives |˜ yn+1 − yn+1 | ≈ 2hp+1 (2p − 1) |ψ (tn , yn )| + O(hp+2 )
(5.2)
The local error can be estimated by 2|ln+1 | ≈ 2hp+1 |ψ (tn , yn )| ≈ |˜ yn+1 − yn+1 |/ (2p − 1)
(5.3)
The step doubling procedure is general and can be applied to any numerical method. However, the step doubling requires more computational time compared to the embedded methods. The step doubling can be used for the LLM, but it is not an efficient way. This chapter presents a new method to estimate the numerical error of the LLM. This method can be used for the implementation of the adaptive step size control. In section 5.2, the implementation of the adaptive step size control is discussed and the result is presented. 5.1 Local Error Estimation Method In this section, we propose a method to estimate the local error of the LLM and validate this method through a numerical example. For simplicity, we consider the
73 autonomous ordinary differential equation y˙ = f (y),
y(tn ) = yˆn
The linear approximation of Eq. (5.4) is given as ∂f ˙yˆ = g(ˆ y ) = f (ˆ yn ) + (ˆ y − yˆn ) , ∂y y=ˆyn
(5.4)
∂f An ≡ ∂y y=ˆyn
(5.5)
We define the local error as E = y − yˆ. Then, the dynamics of E can be expressed as E˙ = f (y) − g(ˆ y ) = f (y) − g(y) + g(y) − g(ˆ y ) = r(t) + An E
(5.6)
where r = f (y) − g(y),
E(tn ) = 0
The solution for this system can be expressed as Z t E(t) = eAn (t−τ ) r(τ )dτ
for t ≥ tn
(5.7)
tn
We need the expression for r(t) to estimate the local error. Let us consider the Taylor series of r(t) r(t) = r(tn ) +
d2 r dr (t − tn ) + 2 (t − tn )2 + HOT dt t=tn dt t=tn
(5.8)
By the definition of r(t) we have, using Eq. (5.5), r(tn ) = f (y(tn )) − g(y(tn )) = f (ˆ yn ) − f (ˆ yn ) = 0 dr df (y) dg(y) ∂f ∂f = − = y(t ˙ n) − y(t ˙ n) = 0 dt t=tn dt t=tn dt t=tn ∂y t=tn ∂y t=tn
(5.9) (5.10)
So, r(t) can be approximated around t = tn as a quadratic function by neglecting the higher order terms. 2
r(t) ≈ vn · (t − tn ) ,
d2 r v = 2 dt t=tn
(5.11)
Then, using Eq. (5.11), we approximate vn as vn =
r(tn+1 ) f (y(tn+1 )) − g(y(tn+1 )) f (ˆ yn+1 ) − g(ˆ yn+1 ) = ≈ (tn+1 − tn )2 (tn+1 − tn )2 (tn+1 − tn )2
(5.12)
74 The local error between tn and tn+1 is obtained as Z tn+1 eAn (tn+1 −τ ) vn · (τ − tn )2 dτ ln+1 = E(tn+1 ) = tn
Z
hn
eAn (hn −u) vn u2 du 0 Z hn Z u Z r eAn (hn −u) 2vn dsdrdu =
=
0
0
(5.13)
0
The integral in Eq. (5.13) can be computed by using Eq. (3.34) as hn An e G1 H1 ln+1 1 2 0 I h h n n 2 ehn L = 0 0 I h n 0 0 0 I
(5.14)
where
An 2vn 0 0
0 0 1 0 L= ∈ R(m+3)×(m+3) , 0 0 0 1 0 0 0 0 Z hn Z hn An (t−s) H1 = 2vn e sds, G1 = 2vn eAn (t−s) ds 0
(5.15)
0
When the state variable y is a scalar, the integral of Eq. (5.13) can be computed analytically as Z
hn
eAn (hn −u) vn u2 du 0 2vn An hn (An hn )2 = 3 e − 1 − An hn − An 2 vn = (An hn )3 + O(h4n ) 3A3n
ln+1 =
(5.16)
This result indicates that the order of convergence of the LLM is two, which is consistent with the analysis of [58].
75 To check the validity of the proposed local error estimate method, we consider the system of differential equation x˙ 1 = −
x2 , x23
x˙ 2 =
x1 , x23
x˙ 3 = 1
(5.17)
with an initial condition x1 (1/π) = 0, x2 (1/π) = −1, x3 (1/π) = 1/π, The analytic solution for this ODEs is given as sin (1/t) x(t) = cos (1/t) , t
t ≥ 1/π
(5.18)
This system of differential equation is solved with the LLM for one time step and the local error is estimated using the proposed local error estimate method. The estimated local error and the exact local error are compared for the various values of hn in Fig. 5.1. The results show that the proposed method can provide a good estimate for the local error. The proposed local error estimation method can be used for any ordinary differential equation and provides the local error for each integration variable. Thus, we can specify the local error tolerance for individual integration variable. 5.2 Adaptive Step Size Control Using the proposed local error estimation method, the adaptive step size control is implemented so that the step size can be adjusted adaptively and the computational efficiency can be maximized while satisfying the accuracy requirement. If the order of the numerical integration method is p, the local error and the step size have an asymptotic relationship ln+1 = ψ(tn , yn )hp+1 + O(hp+2 n n )
(5.19)
76
−4
1.2
−4
x 10
0 Exact local error Estimated local error
−0.5 Local Error in x2
Local Error in x1
1
x 10
0.8 0.6 0.4
−1
−1.5 Exact local error Estimated local error
0.2 0
0
0.002
0.004 0.006 Step Size h
0.008
0.01
−2
0
0.002
0.004 0.006 Step Size h
0.008
0.01
0.008
0.01
−4
x 10
1 Exact local error Estimated local error
Exact local error Estimated local error Norm of local error
Local Error in x3
0.5
0
2
1
−0.5
−1
0
0.002
0.004 0.006 Step Size h
0.008
0.01
0
0
0.002
0.004 0.006 Step Size h
Figure 5.1: Comparison of exact and estimated local error
77 where ψ is the principal error function [94]. Based on this relationship, a standard algorithm for step size control is given as [95] 1/k tol ∗ h =γ hn rn+1
(5.20)
where hn is the step size, h∗ is the new step size, tol is the tolerance specified by user, k is a positive parameter of the algorithm, and γ ∈ (0, 1) is a safety factor for reducing the risk of a rejected step. Conventionally, the estimated error r is defined as rn+1 =
|ln+1 |,
error per step (EPS)
(5.21)
|ln+1 |/hn , error per unit step (EPUS) The reason for using EPUS is to have the same accumulated global error for the same integration time regardless of the number of steps taken [95].
The
parameter k is determined as k = p + 1 (for EPS) or k = p (for EPUS), where p is the order of the numerical integration method.
At every time step, the
local error of the numerical solution is estimated and the condition rn+1 ≤ tol is checked. If this condition is not satisfied then the obtained numerical solution is rejected and the new solution is computed again with a smaller step size. The new step size h∗ is determined by Eq. (5.20) and this process is repeated until the condition rn+1 ≤ tol is satisfied. Once this condition is satisfied, the step size for the next step is set as hn+1 = h∗ and the numerical method proceeds to the next step. This algorithm is applied to the numerical solution of the double pendulum problem discussed in section 4.2. We use the EPUS definition of rn+1 and the parameter k of Eq. (5.20) is set to k = 2 since the order of the LLM is two. The safety factor γ is set to 0.9. The numerical solution is obtained by the LLM with the tolerance is specified as tol = 0.1 and tol = 0.05, respectively. Figure 5.2 shows the change of step size and the estimated error for the given tolerances. The results indicate that as the more strict tolerance is imposed smaller step size is taken, i.e., the number of steps increases . The plot of the estimated error confirms that the error is maintained within the specified value. With the help of the adaptive step
78 size scheme, the user does not need to choose the step size by trial-and-error. The step size is chosen automatically as large as possible while the error tolerance is satisfied. In this research, an elementary algorithm for step size control as given in Eq. (5.20) is considered. There have been some attempts to apply other technique of control theory to adaptive step size selection [95, 96]. In [96], the PI control strategy was used for the step size selection in explicit Runge-Kutta methods. The PI controller takes the form: hn+1 =
tol rn+1
kI
rn rn+1
kP hn
(5.22)
This controller was suggested to overcome the issue of oscillating step size sequence, which often occurs when stability rather than accuracy dictates the step size. This issue can be seen when the system is stiff and the step size yields ξ = hn λ which is located around the boundary of absolute stability region of the numerical method. It was shown that the PI controller can produce a smoother step size sequence and avoid frequent step size rejection. The LLM is an A-stable method and its absolute stability region covers the entire left half ξ-plane. Thus, the LLM does not suffer from the issue of the oscillating step size sequence even for very stiff systems. For the double pendulum example, the elementary algorithm of Eq. (5.20) turns out to perform well and the problem of the oscillating step size is not observed.
79
Stepsize h
The estimated error r
0.02 0.018
tol=0.1 tol=0.05
0.016
tol=0.1 tol=0.05
Estimated Error rn
Stepsize hn
0.014 0.012 0.01 0.008 0.006
0.1
0.05
0.004 0.002 0
0
20
40
60 Steps
80
100
0
0
0.1
0.2 0.3 Time (s)
0.4
Figure 5.2: Step sizes and estimated errors with tol = 0.1 and tol = 0.05
0.5
80 CHAPTER 6 DEVELOPMENT OF A HYBRID NUMERICAL INTEGRATION SCHEME
In chapter 4, it was shown that the Local Linearization Method (LLM) has the advantage over conventional numerical methods in terms of computational efficiency when a highly oscillatory dynamical system is solved. However, the efficiency of the LLM may deteriorate when the size of the system is large. The reason is that the computational efficiency of the matrix exponential operation highly depends on the size of the system Jacobian. Quite often, some sub-parts of a multibody system contain high frequency dynamics while other parts have slow dynamics. Therefore, we should expect better efficiency if the LLM can be applied only to the subsystem with fast dynamics. This observation motivates the development of a hybrid numerical integration scheme, combines the LLM and a standard ODE solver, such as the fourth-order Runge-Kutta method, to take advantage of the LLM and reduce the size of the matrix which involves in the matrix exponential operation. This scheme can be categorized as a Multi-method scheme since two different methods are used with the same step size. This chapter first presents an overview of the singular perturbation theory. Then, the algorithm of the proposed hybrid scheme is addressed and its absolute stability region is analyzed. Finally, the usefulness of the proposed scheme is demonstrated by numerical examples. 6.1 Singular Perturbation Theory The singular perturbation model associated with a dynamical system is a state model where the derivatives of some of the states are multiplied by a small positive
81 parameter [97]: x˙ = f (t, x, z, )
(6.1a)
z˙ = g(t, x, z, )
(6.1b)
It is assumed that the functions f and g are continuously differentiable and x ∈ Rn , z ∈ Rm . By setting = 0, Eq. (6.1b) becomes 0 = g(t, x, z, 0)
(6.2)
If Eq. (6.2) has k ≥ 1 isolated real roots z = hi (t, x),
i = 1, 2, . . . , k
(6.3)
then Eqs. (6.1a)-(6.1b) are in standard form and they reduce to x˙ = f (t, x, h(t, x), 0)
(6.4)
This model is called the slow, reduced, or quasi-steady-state model. With the new time variable τ = (t − t0 )/, the so-called boundary-layer model is defined as dy = g(t, x, y + h(t, x), 0), dτ
where y = z − h(t, x)
(6.5)
A geometric view of the singular perturbed system and the reduced model can be obtained by the concept of invariant manifolds. For simplicity, we consider the autonomous singularly perturbed system x˙ = f (x, z)
(6.6a)
z˙ = g(x, z)
(6.6b)
Let z = h(x) be an isolated root of 0 = g(x, z). Then, the equation z = h(x) is an invariant manifold for the system x˙ = f (x, z)
(6.7a)
0 = g(x, z)
(6.7b)
82 When = 0, any trajectory starting in the manifold z = h(x) will remain in that manifold for all positive time. The dynamics in this manifold can be described by the reduced model as x˙ = f (x, h(x))
(6.8)
Extending this concept to the case of nonzero , the invariant manifold for > 0 can be found in the following form: z = H(x, ) = H0 (x) + H1 (x) + 2 H2 (x) + · · ·
(6.9)
By differentiating z = H(x, ) with respect to t, we obtain ∂H 1 g(x, z) = x˙ ∂x
(6.10)
and, after some rearrangement we obtain 0 = g(x, H(x, )) −
∂H f (x, H(x, )) ∂x
(6.11)
which is called the manifold condition. The function H(x, ) must satisfy the manifold condition for all x in the region of interest. The invariant manifold z = H(x, ) is called a slow manifold for Eqns. (6.6a)-(6.6b). By setting = 0, it can be seen that H0 (x) = h(x)
(6.12)
The first order term H1 (x) can be determined from the manifold condition as follows. The manifold condition is g(x, H(x, )) =
∂H f (x, H(x, )) ∂x
(6.13)
We can obtain the following expression through series expansion of : g(x, h(x)) +
∂g (x, h(x))H1 (x) + O(2 ) ∂z
∂h = f (x, h(x)) + O(2 ) ∂x Since g(x, h(x)) = 0, H1 (x) can be obtained as −1 ∂g ∂h H1 (x) = (x, h(x)) f (x, h(x)) + O(2 ) ∂z ∂x
(6.14)
(6.15)
83 if the Jacobian [∂g/∂z] is nonsingular. The invariant manifolds can be used for the design of control systems, where the reduced system is considered instead of the full system. In [98], the control problem of multibody systems with rigid links and flexible joints was considered. It was shown that any control law that stabilizes the rigid system would stabilize the dynamics of a flexible system on the invariant manifold. The stability analysis is based on Lyapunov functions for the reduced system and the boundary-layer system as described in [97, 99]. Similar results were shown for hybrid control systems. In [100], it was shown that hybrid control can be achieved based on a simple plant model that ignores stable, fast actuator dynamics. Another application of the singular perturbation theory is the computational singular perturbation (CSP) method that is widely used in combustion modelling and chemical kinetics analysis [101, 102, 103]. The CSP is essentially an algorithm to find the reduced system and match the initial conditions to the dynamics on the invariant manifolds. The purpose of the CSP is to reduce the dimension of the problem and obtain the approximated long-term dynamics of the system more efficiently. There have been attempts to obtain numerical solutions of stiff differential equations using the singular perturbation theory. Some equivalence can be found between stiff and singularly perturbed differential equations [68]. If x and z are assumed to be scalars and Eqs.(6.1a)-(6.1b) are linearized along its trajectory, they may be expressed as x˙
x = , z˙ gx / gz / z
fx
fz
x, z ∈ R
(6.16)
Examination of the Jacobian eigenvalues indicates that they spread more as becomes smaller - one eigenvalue approaches zero while the other grows large in absolute value. Thus Eq. (6.16) can be regarded as the linearized representation of a stiff system with widely separated eigenvalues. Here, x corresponds to the slow
84 variable and z corresponds to the fast variable. In [69], the singular perturbation method was applied in a numerical method to solve stiff differential equations. This method utilizes the formal expansions of the type x(t) ∼
∞ X
∞
xr (t)
r=0
r r X + Xr (τ ) r! r=0 r!
∞ X
(6.17a)
∞
r X r z(t) ∼ zr (t) + Zr (τ ) r! r=0 r! r=0
(6.17b)
where τ = t/ The first sums in Eq. (6.17) are the reduced model solutions, and the second sums are the boundary layer model solutions. In compact form, the solution at t = h can be written as u(h) = u0 (h) + u1 (h) + U0 (h/) + U1 (h/) + O(2 )
(6.18)
where u = (x, z) ,
U = (X, Z)
With the assumption that U0 (h/) and U1 (h/) are near zero since h , we approximate u(h) by u0 (h) + u1 (h), the approximation being O(2 ). The components of the solution, u0 (h) and u1 (h), are determined numerically. Unlike conventional numerical methods, this method performs better as the stiffness of the system increases. This approach was extended to the -Independent Method [70], where the small parameter does not need to be identified. However, in these methods, the transient part of the solution is neglected with the assumption that they decay exponentially. So, they can not be used for highly oscillatory systems that contain eigenvalues of large imaginary part. The transient behavior of the system can not be obtained accurately even though the asymptotic behavior of the system can be reconstructed well by this method.
85 6.2 Hybrid Numerical Integration Scheme In this section, a new numerical integration scheme for highly oscillatory systems is discussed.
The scheme is based on the concept of slow manifold in
singular perturbation theory. Unlike the method of [69], this scheme does not neglect the boundary layer model solution, thus it can capture the transient dynamics of highly oscillatory systems. Furthermore, there is no need to identify the value of the singular perturbation parameter in order to apply this scheme. Let us consider a system of ordinary differential equations as: x˙ = f (x, z),
x(0) = ξ
(6.19a)
z˙ = g(x, z),
z(0) = η
(6.19b)
The state variable x corresponds to the slow part of the system and the variable z represents the fast part. We take the linear approximation for g(x, z) as z˙ = g(x, z) ≈ gx · (x − ξ) + gz · (z − η) + g(ξ, η)
(6.20)
where gx =
∂g ∂g , gz = ∂x (ξ,η) ∂z (ξ,η)
Now, z is decomposed into z¯ and y as z = z¯ + y
(6.21)
where z¯ is the quasi-steady-state reduced model of z and y is the boundary-layer model of z. Furthermore, z¯ is assumed to satisfy the following condition: 0 = g(x, z¯)
(6.22)
If we take the approximation of Eq. (6.20), the condition for z¯ becomes 0 = gx · (x − ξ) + gz · (¯ z − η) + g(ξ, η)
(6.23)
86 where we obtain z¯ = η − (gz )−1 [gx · (x − ξ) + g(ξ, η)] := H(x)
(6.24)
Here, it is assumed that gz is not singular. By plugging Eq. (6.21) into Eq. (6.20), the following equation is obtained: z¯˙ + y˙ = gz · y
(6.25)
It follows that y˙ = gz · y − z¯˙ = gz · y −
∂H x˙ ∂x
(6.26)
≈ gz · y + (gz )−1 gx f (ξ, η) The initial condition for y can be determined as follows. Since z(0) = z¯(0) + y(0)
(6.27)
z¯(0) = H(ξ) = η − (gz )−1 g(ξ, η)
(6.28)
y(0) = η − z¯(0) = (gz )−1 g(ξ, η)
(6.29)
and
we have
The dynamics of x can be approximated as: x˙ = f (x, z) = f (x, z¯ + y) ≈ f (x, H(x)) + fz · y(t)
(6.30)
Then, the numerical solution for x is obtained without the last term, fz ·y(t), where it is denoted by xˆ. Any conventional numerical method can be used since the equation x˙ = f (x, H(x)) ,
x(0) = ξ
(6.31)
87 depends on x and has slow dynamics. The complete numerical solution for x is obtained by adding the effect of fz · y(t) as: Z h fz · y(τ )dτ x(h) = xˆ +
(6.32)
0
The superposition in Eq. (6.32) is possible because the effect of fz · y(t) on x is small. The variable x having slow dynamics implies that the effect of fz is not dominant and the main contribution of z on x is taken into consideration by H(x). The effect of the boundary-layer model y on x is accounted for in the sense of averaging. Due to the transient dynamics of y, evaluating the effect of y at several intermediate points can not capture the effect of y appropriately. Averaging can be a better way of capturing the high frequency dynamics of y. The last term on the right hand side of Eqn. (6.32) involves the integration of y. The dynamics of y is approximated as a linear differential equation: γ := (gz )−1 gx f (ξ, η)
y˙ = gz · y + γ,
The analytic solution for y is given as Z t gz t egz (t−s) γds, y(t) = e σ +
σ = (gz )−1 g(ξ, η)
(6.33)
(6.34)
0
Therefore, the last term of Eq. (6.32) can be computed as Z h Z h fz · y(τ )dτ = fz y(τ )dτ 0 0 Z hZ s Z h gz τ = fz e σdτ + egz (s−r) γdrds 0 0 0 Z hZ s gz (s−r) −1 gz h = fz gz (e − I)σ + e γdrds 0
(6.35)
0
This integration can be performed by using the theorem about the exponential of the block matrix of Eq. (3.34) as
0
I
0
D= 0 g γ z 0 0 0
(6.36)
88
F1 (t) G1 (t) J1 (t)
eDt =
F2 (t) G2 (t) 0 F3 (t)
0 0
(6.37)
where F1 (t) = I F2 (t) = egz t F3 (t) = 1 G1 (t) = tI Z t G2 (t) = egz (t−s) γds Z 0t Z s J1 (t) = egz (s−r) γdrds 0
(6.38)
0
Thus, the solution for x can be obtained as x(h) = xˆ + fz gz−1 (F2 (h) − I) σ + J1 (h)
(6.39)
Using the solution for x in Eq. (6.39), the dynamics of y can be approximated more accurately as y˙ = gz · y − z¯˙ = gz · y −
∂H x˙ ∂x
(6.40)
−1
≈ gz · y + (gz ) gx (f (x, H(x)) + fz · y) ≈ gz + (gz )−1 gx fz y + (gz )−1 gx f (x, H(x)) We take a linear interpolation for f (x, H(x)) and treat it as a function of time for t ∈ [0, h]: f (x, H(x)) ≈
t 1− h
t f (x(0), H(x(0))) + f (x(h), H(x(h))) h
(6.41)
Then, the differential equation of y takes the form of nonautonomous linear equation as y˙ = Ay + u(t)
∀ t ∈ [0, h]
(6.42)
89 where A = gz + (gz )−1 gx fz t t −1 u(t) = (gz ) gx 1− f (x(0), H(x(0))) + f (x(h), H(x(h))) h h and the solution for y can be obtained as Z
At
t
y(t) = e y(0) +
eA(t−τ ) u(τ )dτ
(6.43)
0
The computation for y can be achieved through the exponential of block matrix as discussed in [58]. Finally, the solution for z is determined by z(h) = H(x(h)) + y(h)
(6.44)
This process can be generalized and a numerical solution can be obtained by for each time step. The numerical integration procedure can be summarized as: 1. Compute
R tn+1 tn
y(τ )dτ Z
tn+1
y(τ )dτ = (gz )−1 (F2 − I) σ + J1
(6.45)
tn
where γ = (gz )−1 gx f (xn , zn ) σ = (gz )−1 g(xn , zn ) F2 = egz hn Z hn Z s J1 = egz (s−r) γdrds 0
0
F2 and J1 are obtained from the exponential of block matrix D of Eqn. (6.36):
F1 G1 J1
eDhn = 0 0
F2 G2 0 F3
(6.46)
90 2. Compute xn+1 Here, the fourth order Runge-Kutta method is used to compute xˆ: Z tn+1 y(τ )dτ xn+1 = xˆ + fz
(6.47)
tn
where H(x) = zn − (gz )−1 (gx · (x − xn ) + g(xn , zn )) X 1 = xn hn f (X1 , H(X1 )) 2 hn X3 = xn + f (X2 , H(X2 )) 2 X 2 = xn +
X4 = xn + hn f (X3 , H(X3 )) xˆ = xn +
hn (f (X1 , H(X1 )) + 2f (X2 , H(X2 )) 6
+ 2f (X3 , H(X3 )) + f (X4 , H(X4 ))) 3. Compute zn+1 zn+1 = H(xn+1 ) + yn+1 = H(xn+1 ) + σ + φ
(6.48)
¯
φ is obtained from ehn D as ¯ hn D
e
ehn A
=
R hn 0
eA(t−s) ds
0
1
0
0
φ
hn 1
where
A B C
¯ = 0 D
0
0
0
1 0
A = gz + (gz )−1 gx fz B = Aσ + (gz )−1 gx f (xn , H(xn )) C=
(gz )−1 gx (f (xn+1 , H(xn+1 )) − f (xn , H(xn ))) hn
(6.49)
91 The algorithm of the hybrid scheme is illustrated as a diagram in Fig. 6.1 to clarify the flow of the data.
𝑤 = 𝑓𝑧
𝑥 = 𝑓 𝑥, 𝑧 𝑧 = 𝑔(𝑥, 𝑧) 𝑥 𝑡𝑛 = 𝑥𝑛 𝑧 𝑡𝑛 = 𝑧𝑛
𝑡𝑛+1
𝑦 𝜏 𝑑𝜏
𝑡𝑛
𝑤
RK4 𝑥 = 𝑓 𝑥, 𝐻 𝑥
𝑥
𝑥𝑛+1
+
𝑧 = 𝐻(𝑥)
LLM 𝑦 = 𝐴𝑦 + 𝑢(𝑡)
𝑧𝑛+1 𝑦𝑛+1
+
𝑧𝑛+1
Figure 6.1: Diagram of the hybrid scheme
6.3 Absolute Stability Region of The Hybrid Scheme For a given numerical method, the region of absolute stability is the region of the complex ξ-plane such that applying the method for the test equation y˙ = λy, with ξ = hλ from within this region, yields an approximate solution satisfying the absolute stability requirement, |yn+1 | ≤ |yn | [86]. Generally, all Runge-Kutta methods can be written in the following form when applied to the test equation y˙ = λy: yn+1 = R (hλ) yn
(6.50)
92 Absolute Stability Region in ξ−plane (ξ=hλ) 5 4 3 2
Im(ξ)
1 0 −1 −2 −3 −4 −5 −4
−3
−2
−1 Re(ξ)
0
1
2
Figure 6.2: The absolute stability region of the fourth-order Runge-Kutta method where R is called the stability function of the Runge-Kutta method. For the RungeKutta fourth order method, the stability function is given as 1 1 1 R(ξ) = 1 + ξ + ξ 2 + ξ 3 + ξ 4 2 6 24
(6.51)
The region of absolute stability is the region where |R(ξ)| ≤ 1 is satisfied. The absolute stability region of the fourth-order Runge-Kutta is shown in Fig. 6.2. Due to the symmetricity with respect to the x-axis, only the upper half of the stability region will be exhibited in the upcoming figures. We consider the following test equation to analyze the absolute stability of the hybrid integration scheme. The same test equation was used in [31] to derive the absolute stability region of a multirate integration method based on BDF: x˙ λs µ x = z˙ δ λf z
(6.52)
93 The parameter α and β are defined as follows. The parameter α indicates the ratio of the frequencies of the fast and slow variables. The parameter β indicates the strength of coupling between the fast and the slow variables. Introducing these parameters enables us to identify the absolute stability region in the ξ-plane: α=
λf , λs
β=
δ µ = , λf λs
ξ = hλs
(6.53)
The proposed hybrid scheme is applied to the test equation and the condition for the absolute stability is derived. By applying 0 = g(x, z¯), we obtain the slow manifold as: z¯ = H(x) = −
δ x = −βx λf
(6.54)
The dynamics of the slow variable x is given by x˙ = (λs −
µδ )x + µy = (1 − β 2 )λs x + µy λf
The numerical solution for the slow variable x at tn+1 is determined as Z tn+1 xn+1 = xˆ + µ y(τ )dτ
(6.55)
(6.56)
tn
where xˆ = R((1 − β 2 )ξ)xn R(s) = 1 + s +
s2 s3 s4 + + 2 6 24
Equation (6.56) reduces to the following expression: xn+1 = M11 xn + M12 zn where β2 β2 M11 = R (1 − β 2 )ξ + (eαξ − 1) − 2 (αξ − eαξ + 1) α α 3 β M12 = β(eαξ − 1) − 2 (αξ − eαξ + 1) α
(6.57)
94 The numerical solution of z is determined as: zn+1 = H(xn+1 ) + yn+1 = M21 xn + M22 zn
(6.58)
where M21 = −βM11 + βeαξ + M22 = −βM12 + eαξ +
β αξ (e − 1) α
β 2 αξ (e − 1) α
The relationship between the two successive numerical solutions can be written in a matrix form as: x M11 M12 xn n n+1 =M = zn zn+1 M21 M22 zn x
(6.59)
where β2 β2 M11 = R (1 − β 2 )ξ + (eαξ − 1) − 2 (αξ − eαξ + 1) α α 3 β M12 = β(eαξ − 1) − 2 (αξ − eαξ + 1) α β M21 = −βM11 + βeαξ + (eαξ − 1) α 2 β M22 = −βM12 + eαξ + (eαξ − 1) α The proposed hybrid scheme is absolute stable if and only if the spectral radius of M is less than or equal to 1; i.e., ρ (M) ≤ 1
(6.60)
The region of absolute stability is the set of ξ = hλs which satisfies the condition ρ (M) ≤ 1. Figures 6.3 and 6.4 show half of the region of absolute stability for different values of the parameters α and β. When β is zero, the state x and z are completely decoupled. The state z is solved by the analytic solution form of the linear system and the numerical solution is absolutely stable if Re(hλf ) ≤ 0
(6.61)
95 Therefore, the region of absolute stability is the intersection of the region of absolute stability of the fourth-order Runge-Kutta and the left half plane. As the parameter β increase, the region of absolute stability changes slightly. So, the coupling between x and z does not have significant effect on the region of absolute stability. From Fig. 6.3, it can be seen that the region of absolute stability does not change much as the parameter α increases. It means that the region of absolute stability only depends on the eigenvalues of the slow variable and it is almost independent of the eigenvalues of the fast variable. As a result, the restriction on the step size h is weakened and the computational efficiency can be improved by employing a larger step size than the conventional numerical methods when multiple-timescale problems are solved.
Absolute Stability Region (α=10,β=0.1) 4
3
3 Im(ξ)
Im(ξ)
Absolute Stability Region (α=1,β=0.1) 4
2 1
1 −3
0 −4
−2
−1 0 1 Re(ξ) Absolute Stability Region (α=100,β=0.1)
4
4
3
3 Im(ξ)
Im(ξ)
0 −4
2
2 1 0 −4
−3
−2
−1 0 1 Re(ξ) Absolute Stability Region (α=300,β=0.1)
−3
−2
2 1
−3
−2
−1 Re(ξ)
0
1
0 −4
−1 Re(ξ)
0
1
Figure 6.3: The absolute stability region of the hybrid scheme for different values of α
96 Absolute Stability Region (α=100,β=0.1) 4
3
3 Im(ξ)
Im(ξ)
Absolute Stability Region (α=100,β=0) 4
2 1
1 −3
0 −4
−2
−1 0 1 Re(ξ) Absolute Stability Region (α=100,β=0.2)
4
4
3
3 Im(ξ)
Im(ξ)
0 −4
2
2 1 0 −4
−3
−2
−1 0 1 Re(ξ) Absolute Stability Region (α=100,β=0.4)
−3
−2
2 1
−3
−2
−1 Re(ξ)
0
0 −4
1
−1 Re(ξ)
0
1
Figure 6.4: The absolute stability region of the hybrid scheme for different values of β 6.4 Accuracy Analysis of The Hybrid Scheme In this section, we study the local error of the proposed method by using the linear test equation of Eq. (6.52). The exact solutions for Eq. (6.52) are denoted by xˆ(t) and zˆ(t) and they are given as xˆ(t)
= eAt
zˆ(t) where
λs A= δ
x(0)
(6.62)
z(0)
µ λf
The local errors of the states x and z are defined as lx (h) = |x(h) − xˆ(h)|
(6.63a)
lz (h) = |z(h) − zˆ(h)|
(6.63b)
97 We choose λs and the initial conditions as λs = −1 + i, x(0) = 1, z(0) = 1
(6.64)
Then, the local error is numerically computed and plotted with respect to the step size h in Figs. 6.5 and 6.6. The slope of the graph, which is in log scale, represents p + 1, where p is the order of accuracy of the numerical method. The results show that the order of accuracy of the proposed method is one and the local error slightly decreases as the parameter α increases. The order of accuracy of the fourth-order Runge-Kutta is four, but its local error is highly dependent on the parameter α. The accuracy of the fourth-order Runge-Kutta becomes worse as the parameter α increases. In Fig. 6.6, it can be seen that the hybrid scheme shows the fourth-order accuracy for the state x when the coupling parameter β is zero. Without the coupling term, the hybrid scheme maintains the order of accuracy of the fourth-order Runge-Kutta for the variable x. And the error of the state z is in the order of the machine accuracy, which is 10−15 , due to fact that the LLM can solve the linear test equation exactly. When the parameter β is non-zero, the order of accuracy for both variables becomes one. The reason is that the first-order approximation is used for the dynamics of the state x as in Eq. (6.30). Figure 6.6 indicates that the accuracy becomes worse as the parameter β increases. The results suggest that the proposed method will perform better when the system has states of two separate time scales and the coupling between them is weak. It should be noted that the results do not account for the error which will be introduced by local linearization of z˙ = g(x, z) since the test equation is linear. In this study, the order of accuracy of the proposed method is estimated numerically. More rigorous analysis of the accuracy and other properties of the hybrid scheme will be attempted in the future.
98
0
Local Error in x (The Proposed Method)
0
lz(h)
−10
10
−10
10
−5
10
−4
−3
10
10
−2
−5
10
10
h Local Error in x (RK4)
0
−4
−3
10
10
−2
10
h Local Error in z (RK4)
0
10
10
lz(h)
α=1,β=0.1 α=10,β=0.1 α=100,β=0.1 α=300,β=0.1
x
l (h)
Local Error in z (The Proposed Method)
10
x
l (h)
10
−10
10
−10
10
−5
10
−4
−3
10
10 h
−2
10
−5
10
−4
−3
10
10 h
Figure 6.5: Local error versus step size for different values of α
−2
10
99
0
Local Error in x (The Proposed Method)
0
lz(h)
−10
10
−10
10
−5
10
−4
−3
10
10
−2
−5
10
10
h Local Error in x (RK4)
0
10
−4
−3
10
10
−2
10
h Local Error in z (RK4)
0
10
lz(h)
α=100,β=0 α=100,β=0.05 α=100,β=0.1 α=100,β=0.2
x
l (h)
Local Error in z (The Proposed Method)
10
x
l (h)
10
−10
10
−10
10
−5
10
−4
−3
10
10 h
−2
10
−5
10
−4
−3
10
10 h
Figure 6.6: Local error versus step size for different values of β
−2
10
100 6.5 Numerical Examples Example 1: In this section, we consider a highly oscillatory system consisting of a pendulum and a particle in order to illustrate how the proposed numerical scheme integrates the differential equation of motion. The results will demonstrate that the proposed method can solve the equations of motion more efficiently compared to the explicit Runge-Kutta methods. This is due to relaxing the requirement for the absolute stability and using a larger step size when the proposed method is employed. Figure 6.7 shows the configuration of the pendulum system. The meaning of the state variables and the physical parameters and their values are: r1 : position of mass center of the bar θ : rotation angle of the bar m1 , J1 : mass and inertia of the bar(m1 = 100, J1 = 100) L : length of the bar(L = 1) r2 : position of the particle m2 : mass of the particle(m2 = 1e − 5) k : spring coefficient(k = 5) The equations of motion of the system are derived by using the body-coordinate formulation presented in chapter 2 as f ¨r1 mI 0 1 T 1 =D λ+ n1 0 J1 θ¨1 m2¨r2 = f2
(6.65a) (6.65b)
101 where f1 = m1 g + fs f2 = m2 g − fs n1 = s × fs fs = k (r2 − (r1 + s)) L/2 sin θ s= −L/2 cos θ 1 0 −L/2 cos θ D= 0 1 −L/2 sin θ 0 g= −9.81 The constraint equation for the revolute joint is given as Φ = r1 − s = 0
(6.66)
The position of the particle must exhibit high frequency dynamics due to the small value of m2 , which corresponds to the parameter in the standard singular perturbation model of Eq. (6.1). Based on this observation, the state of the system is partitioned into the slow state x and the fast state z as r1 θ r 2 x= , z= r˙ 2 r˙ 1 θ˙ Then, the differential equation for z can be written as r˙ 2 z˙ = g(x, z) = − k (r2 − (r1 + s)) − g m2
(6.67)
(6.68)
The slow manifold of the system is obtained by imposing 0 = g(x, z¯) , which gives r2 = r1 + s −
m2 g ≈ r1 + s k
(6.69)
102
𝜃
𝑟1
𝑚1 , 𝐽1 𝑘 𝑟2
𝑚2
Figure 6.7: Configuration of pendulum system The slow manifold corresponds to the quasi-steady-state case that the particle is rigidly attached to the end of the pendulum. This accounts for the limiting case that m2 goes to zero. The boundary-layer model y = z − z¯ represents the displacement of the particle relative to the end point of the pendulum. This system of differential equation is solved by the hybrid scheme, the fourthorder Runge-Kutta, and ODE45.
For the hybrid scheme and the fourth-order
Runge-Kutta, the step size is chosen to be h = 0.005.
Figure 6.8 compares
the numerical solutions from the methods. It can be seen that the fourth-order Runge-Kutta yields an unstable solution, since the step size h does not satisfy the absolute stability condition. The eigenvalue of the fast variable is approximately λf ≈ (k/m2 )i = 5 × 105 i. With the choice of h = 0.005, ξ = hλf ≈ 2.5 × 103 i is located outside the region of absolute stability of the fourth-order Runge-Kutta that is shown in Fig. 6.2. The hybrid scheme gives a stable solution with the step size h = 0.005. This is possible since the region of absolute stability of the hybrid scheme only depends on
103 102
1
RK4 (h=0.005)
x 10
Proposed method (h=0.005) 1 r1x
0
0.5
r2x
−1
0
−2
−0.5
−3
0
1
2
3
−1
Time (s) ODE45 1
2
0.5
1.5
0
−1
0
1
2 Time (s) Computation Time
3
0.5
r2x 0
r2x
1
r1x
−0.5
r1x
1
2
3
0
ODE45
Proposed method
Time (s)
Figure 6.8: Comparison of simulation results for pendulum the eigenvalues of the slow variable x. The hybrid scheme can maintain the absolute stability with a larger step size and it leads to better computational efficiency. Figure 6.8 shows that the computation time of the hybrid scheme is about 60% of that of ODE45. The total energy of the system should be constant because the system has no damping. The total energy of the system can be computed as 1 1 1 E = m1 |˙r1 |2 + J1 θ˙2 + m2 |˙r2 |2 2 2 2 1 + 9.81 (m1 r1y + m2 r2y ) + k|r2 − (r1 + s)|2 2
(6.70)
Figure 6.9 compares the total energy of the numerical solutions obtained by ODE45 and the hybrid scheme. It can be seen that the hybrid scheme provides comparable accuracy with larger step size than ODE45.
104 Total Energy of the numerical solution using ODE45 −420
Total Energy
−422 −424 −426 −428 −430
0
0.5
1
1.5
2
2.5
3
Total Energy of the numerical solution using the proposed method −420
Total Energy
−422 −424 −426 −428 −430
0
0.5
1
1.5 Time (sec)
2
2.5
3
Figure 6.9: Total energy of pendulum Example 2: To study the property of the hybrid scheme, let us consider the spring-mass system shown in Fig. 6.10. It is assumed that both bodies have unit mass.
The dynamics
of this system can be written by a linear differential equation as u˙ = Ku where
0 −(k + k ) 2 3 K= 0 k3
1 0 0 0
0 0 k3 0 , 0 1 −(k1 + k3 ) 0
(6.71)
u=
u1 u˙ 1
u2 u˙ 2
(6.72)
105
k2
k3
k1 (1)
(2)
u1
u2
Figure 6.10: Spring-mass system Because the system is purely linear, we can compute the exact solution of this system as u(t) = exp(Kt)u(0)
(6.73)
The spring coefficients are assigned in three different sets as shown in Table 6.1. In all cases, high coefficient k1 makes body (1) move much faster than body (2). Therefore, the state variables for body (1) are assigned to the fast variable z of the hybrid scheme. Specifically, the variable x and z of Eq. (6.19) are assigned as u u 1 2 (6.74) , z= x= u˙ 1 u˙ 2 For each set, numerical solutions are obtained by the proposed hybrid scheme and the fourth-order Runge-Kutta. In all cases, the step size is chosen to be h = 0.01. The error of the numerical solutions can be computed because the exact solution of the system is handy as Eq. (6.73).
Table 6.1: Spring coefficients in spring-mass system
set 1 set 2 set 3
k1 103 104 105
k2 1 1 1
k3 0.1 0.1 0.1
106 Figures 6.11-6.12 show the numerical error from the methods. It can be seen that the error of the Runge-Kutta increases as the magnitude of k1 becomes larger. When k1 = 105 , the Runge-Kutta method can not solve the system with h = 0.01 and the numerical solution becomes unstable. The Runge-Kutta method needs a smaller step size as the system becomes more oscillatory. This holds for other conventional numerical integration methods. The computational efficiency is restricted by the requirement for the step size when conventional numerical integration methods are used. On the other hand, the error of the hybrid scheme decreases as the magnitude of k1 becomes larger. This means that the larger step size can be used when the system is more oscillatory. The larger k1 corresponds to the smaller singular parameter and the error of the hybrid scheme decreases when k1 is larger because the invariant manifold to describe the dynamics of the system better. Therefore, the accuracy of the hybrid scheme improves as the system becomes more oscillatory. This is a unique property of the hybrid scheme and it makes the method useful for solving highly oscillatory dynamical systems.
107
Error in u (RK4, k =1000)
−8
Error in u2
4
2
x 10
2 0 −2 −4
0
0.5
1
−7
Error in u2
2
1.5
2 2.5 Time (s) Error in u (RK4, k =10000) 2
x 10
3
3.5
4
3
3.5
4
3
3.5
4
1
1 0 −1 −2
0
0.5
1
121
1 Error in u2
1
1.5
2 2.5 Time (s) Error in u (RK4, k =100000) 2
x 10
1
0 −1 −2
0
0.5
1
1.5
2 Time (s)
2.5
Figure 6.11: Numerical error in u2 for the fourth-order Runge-Kutta Method
108
Error in u (Hybrid, k =1000)
−6
Error in u2
2
2
x 10
1 0 −1
0
0.5
1
−7
Error in u2
2
1.5
2 2.5 Time (s) Error in u (Hybrid, k =10000) 2
x 10
3
3.5
4
3
3.5
4
3
3.5
4
1
1 0 −1
0
0.5
1
−8
2 Error in u2
1
1.5
2 2.5 Time (s) Error in u (Hybrid, k =100000) 2
x 10
1
1 0 −1 −2
0
0.5
1
1.5
2 Time (s)
2.5
Figure 6.12: Numerical error in u2 for the Hybrid Scheme
109 CHAPTER 7 TIRE MODEL
A tire model plays an important role in a vehicle’s ride and handling simulations. Tires are the only components of a vehicle that are in contact with the ground. Most of external forces and moments that act on a vehicle are generated by tires. For an accurate simulation, a good tire model is crucial. One simple approach to the tire modeling is to use a set of empirical formula to compute the forces and moments of the tire. This kind of models are referred to as empirical tire models. The Magic Formula by Bakker and Pacejka [104] is a typical example. In this method, coefficients of the empirical formula are obtained by fitting the measurement data. During a simulation, the forces and moments of the tire are computed by feeding the driving conditions, such as slip angle and normal load, into this empirical formula. The empirical tire models require a large set of measurement data to determine the set of fitting coefficients. The empirical tire models are computationally efficient but can not represent the dynamic response of the tire accurately. Another approach is to model a tire using mechanical components such as elastic strings, beams or rigid ring supported by springs. Some examples are the rigid ring models by Bruni at el. [105] and MF-SWIFT model by Pacejka et al. [106]. This approach also requires an extensive amount of experiments to obtain the properties of the mechanical components.
In addition, the high frequency
response of the tire can not be represented well by these models. A tire model can also be constructed using the finite element method. Tire companies use commercial finite element packages like ABAQUS or ANSYS to model and design tires. Unlike the aforementioned models, the finite element tire models are capable of representing a tire in much greater detail. However, the large number
110 of degrees of freedom of a finite element tire model is the obstacle that prevents it from being adopted in the vehicle dynamic simulation. As a remedy to this problem, a modal tire model was developed in [107, 108] to reduce the number of degree of freedom of the finite element tire models so that the vehicle dynamic simulation could be performed with a reasonable computation time. The number of degrees of freedom of the tire model is reduced through the Craig-Bampton method [109] while the important modal characteristics of the tire are preserved. On top of that, this tire model is formulated with a non-rotating reference frame using the rotational invariant characteristic of the tire. As a result, the number of boundary nodes can be minimized and the artificial vibration at the tire contact patch can be prevented. This chapter gives an overview of the modal tire model which will be used for the full vehicle simulation in chapter 9. The hybrid integration scheme presented in chapter 6 will be applied to the full vehicle simulation and its advantage will be demonstrated. 7.1 Nodal Positions and Velocities A finite element tire model consists of nodes. Each node has three degrees of freedom. First, we partition these nodes into three groups as shown in Fig. 7.1. The w-nodes are located on the boundary of the wheel. These nodes are assumed to be rigidly fixed to the wheel. The b-nodes are a collection of nodes that may become in contact with the ground. The rest of the nodes are classified as the f -nodes. Accordingly, all associated matrices are split. For instance, the stiffness matrix Kt and the mass matrix Mt of the tire are split as bb bf bb bf K K M M , Kt = Mt = fb ff fb ff K K M M
(7.1)
We define a reference frame which is attached to the wheel center and it will be referred to as the wheel frame. In Fig. 7.1, the wheel frame is denoted by ξ − η − ζ
111
𝑤-nodes 𝑏-nodes 𝑓-nodes
rw
Figure 7.1: Nodes of finite element tire model axes while the global frame is denoted by x − y − z axes. The absolute position of a typical node can be expressed in terms of the nodal deflection as di = rw + bi = rw + si + δ i
(7.2)
where si and bi represent the nodal position vectors with respect to the wheel center in the undeformed and deformed states respectively. rw is the position of wheel center expressed in the global frame. δ i denotes the deformation of the i-th node expressed in the global frame and can be expressed as δ i = Aw δ i0
(7.3)
where Aw denotes the rotation matrix of the wheel frame and δ i0 is the deformation of i-th node expressed in the wheel frame. The time derivative of Eq. (7.2) is written as i ˜ w δ i + δ˙ rel d˙ i = r˙ w + s˙ i + ω
(7.4)
i In this notation, δ˙ rel denotes the time derivative of δ i relative to the wheel frame
and ω w denotes the angular velocity of the wheel frame expressed in the global frame.
112 7.2 Non-rotating Reference Frame Conventionally, a local frame translates and rotates with the body that it is attached to. If this concept is applied to the tire, the wheel frame should translate and rotate with the tire. As the tire rolls on the ground, all the nodes on the outer part of the tire may get into contact with the ground, which means that they should be classified as b-nodes even though actual contact should occur at the lower part of the tire. In the modal tire model, the degrees of freedom associated with the f -nodes are reduced significantly through truncation in modal space. However, the degrees of freedom associated with the b-nodes cannot be truncated because the contact condition should be imposed. So, minimizing the number of b-nodes is very important for reducing the total number of degrees of freedom. When the wheel frame rotates with the tire, the b-nodes should include all the nodes on the outer part of the tire although most of them are not in contact. In addition, the roll of the tire makes some of the nodes at the leading edge of the contact patch come into contact while some other nodes at the other end leave the contact area. This transition causes an artificial vibration at the contact patch because the mass assigned to a node experiences a sudden change in its velocity. This artificial vibration occurs even when the tire is in a steady-state rolling. To overcome this problem, the Arbitrary-Lagrangian-Eulerian formulation [110] is employed in the derivation of the equations of motion of the tire. The Lagrangian description is widely used in solid mechanics. In the Lagrangian description, the motion of a continuum body is expressed by the mapping function χ x = χ(X, t)
(7.5)
where x is the position of the material in the deformed state and X is the position of the material in the undeformed position shown in Fig. 7.2. A typical physical and kinematic property P is expressed as continuous functions of position and time; i.e., P = P (X, t)
(7.6)
113
x ( X, t )
Undeformed configuration
0
p
P
X3
Deformed configuration
X
x
t
X2 X1 Figure 7.2: Undeformed and deformed configuration of a system In the Lagrangian description, the material derivative of P = P (X, t) is simply the partial derivative with respect to time as d ∂ [P (X, t)] = [P (X, t)] dt ∂t
(7.7)
because the position vector X is constant. In the Eulerian description, we focus to what is occurring at a fixed point in space as time progresses, instead of giving attention to individual particles as they move through space and time. This approach is commonly used in the fluid dynamics. Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function X = χ−1 (x, t)
(7.8)
which relates the particle that now occupies the position x in the deformed configuration to its original position X in the undeformed configuration. In the Eulerian description, a physical property P is expressed as P = P (X, t) = P χ−1 (x, t) = p(x, t)
(7.9)
114 where the functional form of P in the Lagrangian description is not the same as the form of p in the Eulerian description. The material derivative of P , using the chain rule, is then ∂ ∂ dxk d [p(x, t)] = [p(x, t)] + [p(x, t)] dt ∂t ∂xk dt
(7.10)
In the modal tire model, the Lagrangian description is used for the motion of the wheel except for the rotation about the axis of rotation of the tire. Meanwhile, the Eulerian description is used for the rotation of the wheel, which means that the wheel frame does not rotate with the tire. If the nodal deflection δ i0 is expressed in the wheel frame, the time derivative of δ i0 relative to the wheel frame can be written, using Eq. (7.10), as ∂δ i0 ∂δ i0 ˙ i0 δ˙ rel = + φ ∂t ∂φ
(7.11)
where φ is the rotation angle of the tire. The mesh points of the tire model do not rotate with the tire even though actual material points of the tire rotate. i0 ˙ The mass flow induced by the rotation is accounted for by the convective term ∂δ φ. ∂φ
In the non-rotating frame formulation, the b-nodes need to cover only the lower part of the tire. As a result, the number of b-nodes can be reduced substantially compared to the standard formulation where the wheel frame and mesh points rotate with the tire. In addition, the artificial high-frequency vibration at the contact patch can be avoided because the same set of b-nodes stays in contact with the ground when the tire is in a steady-state rolling. A new contact occurs only when the size or shape of the contact patch is changed due to a transient response of a tire.
115 7.3 Modal Transformation and Truncation We transform the nodal deflections into the modal space using the Craig-Bampton method of reference [109]. The modal transformation is given as δ b0 Ψb 0 ub = δ f 0 Ψs Ψb Ψf uf
(7.12)
where the modal matrix for f -nodes, Ψf , is obtained using the sub-matrices Mf f and Kf f . Similarly, the modal matrix for b-nodes, Ψb , is obtained using the submatrices Mbb and Kbb . The sub-matrix Ψs is determined by Ψs = − Kf f
−1
Kf b
(7.13)
and its columns vectors represent the static modes [111]. The number of degrees of freedom is reduced by truncating high-frequency modes associated with the f nodes. This process is done by deleting Ψfd and ufd of ub δ b0 Ψb 0 0 f = (7.14) uk δ f 0 Ψs Ψb Ψfk Ψfd f ud Meanwhile, all modes associated with the b-nodes should be kept because the contact between the tire and the ground should be accurately considered. Normally, we keep the first 20 modes associated with the f nodes and delete other high-frequency modes since the first 20 modes can cover the dynamic responses of the tire up to 150Hz. It should be noted that deleting high-frequency modes has no effect on the b-nodes. Through this process, the number of degrees of freedom is reduced from tens of thousand to a few hundreds. Let us use a compact notation for Eq. (7.14) as δ 0 = Ψu
(7.15)
116 The modal transformation matrix Ψ is independent of time and a function of the rotation angle φ. Thus, we have δ 0 = Ψ(φ)u(t)
(7.16)
∂u ∂Ψ ˙ 0 δ˙ rel = Ψ + φu ∂t ∂φ
(7.17)
Consequently, Eq. (7.11) becomes
This expression is used for the derivation of the equations of motion. 7.4 Equations of Motion The canonical form of the equations of motion can be written as Mt d˙ = pd
(7.18)
p˙ d = fd
(7.19)
where Mt is the tire mass matrix, pd is the momenta of nodal points, and fd is the array of forces acting on nodal points. Equation (7.18) can be written in the wheel frame as Mt d˙ 0 = p0d
(7.20)
ˆ > to both sides of Eq. (7.20) yields Pre-multiplying Ψ ˆ > Mt d˙ 0 = Ψ ˆ > p0 = π d Ψ d where
I
0
(7.21)
ˆ = Ψ 0 Ψf
(7.22)
This process projects the system of equations to the vector space spanned by the ˆ If we denote the number of b-nodes by nb−nodes and the number of columns of Ψ. kept free modes by nf −modes , the size of the equations is 3nb−nodes + nf −modes . Using Eqs. (7.4) and (7.17), we can express the nodal velocities in the wheel frame as ˙ ¯˜ 0w Ψu + Ψu˙ + ∂Ψ φu d˙ 0 = ˆI˙r0w − ˆ˜s0 (ω 0w + ω 0o ) + ω ∂φ
(7.23)
117 where I . ˆI = .. , I
˜s10 ˆ˜s0 = ... , ˜sn0
˜ 0w ω ... ¯˜ 0w = ω
˜ 0w ω
ω 0o =
,
0 0 ˙ φ
Applying Eq. (7.23) to Eq. (7.21) yields > > ∂Ψ ˙ 0 0 0 0 ˆ ˆ ˆ ¯ φu Ψ Mt Ψ u˙ = π d + Ψ Mt ˜s (ω w + ω o ) − ω ˜ w Ψu − ∂φ
(7.24)
(7.25)
˙ It should be noted that r˙ 0w , ω 0w and φ˙ should be which can be solved for u. determined in prior by solving the equations of motion of a vehicle. The equations of motion of a vehicle will be discussed in detail in chapter 8. The time derivative of π d is given by π˙ d =
> −1 > d ˆ> 0 ˆ >f 0 − Ψ ˆ Ψ ˆ ˆ ω ˆ d ¯˜ 0w Ψπ Ψ pd = Ψ Ψ d dt
(7.26)
where fd0 is the array of forces expressed in the wheel frame. This array of forces can be determined by 0 0 0 fd0 = fgravity + felastic + fdamping +
contact
0
f 0
+
f riction
0
f 0
(7.27)
where 0 felastic = −KΨu ∂Ψ ˙ 0 0 ¯ fdamping = −C Ψu˙ + φu − ω ˜ o Ψu ∂φ
(7.28) (7.29)
ˆ > KΨ and the In the computation of Eq. (7.26), the transformed stiffness matrix Ψ ˆ > CΨ are much smaller than the original stiffness and transformed damping matrix Ψ damping matrices. In addition, these matrices remain constant during a simulation, so they can be prepared in a pre-processing step for a better efficiency. The normal 0 contact forces should be taken into account by fcontact . The linear contact force
118 model of Eq. (2.54) is used in this study. The tangential friction forces acting on b-nodes are accounted for by ff0 riction . The tangential friction forces are modeled by the viscous friction as ffi riction = −cvti
(7.30)
where c is a positive coefficient and vti is the velocity of i-th b-node on the tangential plane of the ground. The real friction phenomenon of the tire is much more complicated than the viscous friction model. In this research, the simple friction model is employed because the ride comfort simulation is the primary interest, where the effect of the friction force is not dominant. For a cornering simulation, an accurate friction model will be needed.
119 CHAPTER 8 VEHICLE MODEL
Vehicle dynamics is one of the areas that the principles of multibody system dynamics are widely used.
Computer simulation of vehicle systems is getting
more popular among automobile and tire companies. The simulation can replace expensive and time consuming field tests. In addition, a simulation can provide detailed information about the dynamic response of the vehicle, such as the camber angle and the wheel load during various driving situations. Such information is hard to obtain from direct measurements. Within the computer simulation, some of the system parameters can be changed easily. Thus, the vehicle simulation can be used for Monte-Carlo analysis or design optimization. Such analyses usually require an extensive number of simulations. So the computational efficiency of vehicle simulation is an important issue. A vehicle system consists of a large number of mechanical components, such as suspension links and kinematic joints. A vehicle model can be constructed by considering all of the suspension links and kinematic joints, such as ADAMS model reported in reference [112]. In this model, the geometry of suspension is taken into account accurately. However, constructing an ADAMS model requires lots of informations about the vehicle design. Typically, an ADAMS model can be constructed only when vehicle CAD data is available. Another approach is to use the measured data of a vehicle’s suspension in the computational model. The kinematic properties of the suspension can be obtained using the Suspension Property Measuring Machine (SPMM) [113] or Chassis and Suspension Testing Systems of MTS [114].
These test machines can measure
the change of position and orientation of the wheel with respect to the vertical
120
Figure 8.1: Suspension Parameter Measurement Machine movement of the wheel. The functional suspension model [115] and CarSim [116] utilize this measurement data to model the suspension indirectly.
Specifically,
the functional suspension model uses a polynomical equation to represent the suspension kinematics. The vehicle model of this study is based on the functional suspension model that has been reported in reference [115]. This model consists of five bodies – four wheels and the vehicle’s main body. Each wheel has two degrees of freedom: one is the vertical movement of the wheel and the other is the rotation of the wheel. Let us denote the vertical movement of the wheel by µ and the rotation of the wheel by φ. In this model, the effect of the steering input is not considered since the ride comfort simulation is the primary interest. The total number of degrees of freedom in this model is 14. 8.1 Functional Suspension Model The Suspension Property Measuring Machine (SPMM) is a test equipment to measure kinematic characteristics of a vehicle’s suspension. Figure 8.1 shows the configuration of this test machine. This machine measures the relative position and orientation of the wheel with respect to the main body when the wheel moves up
121 and down. For later discussion, we define reference frames as shown in Fig. 8.2. The vehicle body frame is attached to the center of the vehicle body and moves with it. The wheel frame is attached to the wheel center and moves with the wheel except for the rotation along the ζ-axis.
y Vehicle Body frame
z
Wheel frame
rw / b
x
rb
Y
rw
Z
X
Global (Inertial) frame
Figure 8.2: Reference frames used in vehicle model
One example of the measured data is the camber angle γ
1
versus µ shown in
Fig. 8.3. This curve shows how the camber angle of a wheel changes as the wheel moves up and down. In most cases, a suspension is designed such that the camber angle becomes negative when the suspension is compressed in order to compensate for the body roll 2 of a vehicle and keep the tire vertical to the ground during severe cornering. Similarly, the toe angle α 3 is measured when the wheel moves in vertical direction as shown in Fig. 8.4. The camber angle γ and the toe angle α define the 1
Camber is the angle between the vertical axis of a wheel and the vertical axis of the vehicle
when viewed from the front or rear. If the top of the wheel is farther out than the bottom (that is, away from the axle), it is called positive camber; if the bottom of the wheel is farther out than the top, it is called negative camber. 2 The leaning of a vehicle’s body to one side when turning sharply. 3 Toe is the angle that a wheel makes with the longitudinal axis of the vehicle. Positive toe, or toe in, is the front of the wheel pointing in towards the centerline of the vehicle.
122 orientation of the wheel frame relative to the vehicle body frame. The rotation of the wheel in this direction is described by the wheel spin angle φ. If we use the Euler angles (Y-X-Z convention) to define the relative orientation of the wheel frame with respect to the vehicle body frame, the toe angle α corresponds to the rotation about y-axis in the vehicle body frame and the camber angle γ corresponds to the rotation about the rotated x-axis, i.e. ξ-axis in the wheel frame. The relative rotation matrix Aw/b for the wheels on the left-hand side is obtained as
(lef t)
Aw/b
cos(−γ) − sin(−γ) 0 sin(−γ) cos(−γ) 0 = 0 1 0 − sin(−α) 0 cos(−α) 0 0 1 cos(−α)
0 sin(−α)
and for the wheels on the right-hand side we have cos(α) 0 sin(α) cos(γ) − sin(γ) 0 (right) sin(γ) cos(γ) 0 Aw/b = 0 1 0 − sin(α) 0 cos(α) 0 0 1
(8.1)
(8.2)
This difference is due to the sign convention of the camber and toe angles. Figures 8.5 and 8.6 also show the change of x and z coordinate of the wheel center versus the wheel travel µ. We denote the wheel center position relative to the vehicle body center expressed in the vehicle body frame by r0w/b , which can be written as r0w/b =
xw/b
0 yw/b −µ zw/b
(8.3)
0 where yw/b is the relative y position of the wheel with respect to the vehicle body
center in the static equilibrium state. And xw/b is the relative x position of the wheel, and zw/b is the relative z position of the wheel with respect to the vehicle body center.
123
SPMM Data Front Left Wheel Camber Angle 3 2.5
Camber angle γ (deg)
2 1.5 1 0.5 0 −0.5 −1 −80
Measured Data Fitting Equation −60 −40 −20 Compression
80 100 Extension
120
Figure 8.3: Camber angle change with respect to wheel vertical displacement
SPMM Data Front Left Wheel Toe Angle 0.5 0.45
Toe angle α (deg)
0.4
Measured Data Fitting Equation
0.35 0.3 0.25 0.2 0.15 0.1 0.05 −80
−60 −40 −20 Compression
80 100 Extension
120
Figure 8.4: Toe angle change with respect to wheel vertical displacement
124
SPMM Data Front Left Wheel Longitudinal Position 1378.4
Relative x position x
w/b
(mm)
1378.2 Measured Data Fitting Equation
1378 1377.8 1377.6 1377.4 1377.2 1377
−80
−60 −40 −20 Compression
80 100 Extension
120
Figure 8.5: Wheel center longitudinal position with respect to wheel vertical displacement
SPMM Data Front Left Wheel Lateral Position −734
Relative z position zw/b (mm)
−736
Measured Data Fitting Equation
−738 −740 −742 −744 −746 −748 −750 −80
−60 −40 −20 Compression
80 100 Extension
120
Figure 8.6: Wheel center lateral position with respect to wheel vertical displacement
125 As shown in Figs. 8.3-8.6, the measured data can be fitted by fourth-order polynomials: α = a4 µ4 + a3 µ3 + a2 µ2 + a1 µ1 + a0
(8.4a)
γ = b4 µ 4 + b3 µ 3 + b2 µ 2 + b1 µ 1 + b0
(8.4b)
xw/b = c4 µ4 + c3 µ3 + c2 µ2 + c1 µ1 + c0
(8.4c)
zw/b = d4 µ4 + d3 µ3 + d2 µ2 + d1 µ1 + d0
(8.4d)
In this process, we can obtain empirical equations representing the motion of the wheel relative to the main body as a function of the vertical displacement of the wheel µ. The fitting coefficients for the measured data from a luxury sedan are listed in Table. 8.1. These empirical equations define the path that a wheel can move along. Thus, the suspension kinematics can be characterized by these equations without specific information about the suspension geometries like the length of the links or position of the joints. Table 8.1: Fitting coefficients of j→ 4 3 2 aj 2.2757e-9 -5.4902e-7 2.6997e-5 bj -1.8849e-10 1.0371e-8 1.2508e-4 cj 1.3768e-8 -2.8402e-6 5.4035e-5 dj -5.0308e-10 3.1314e-7 1.3441e-3
Eq. (8.4) 1 0 -5.1672e-5 0.1878 1.7699e-2 -0.2789 6.4131e-3 1377.46 -4.8231e-3 -749.14
126 8.2 Velocity Transformation From the empirical equations (8.4c) and (8.4d), we can obtain the relationship between the relative velocity of the wheel and the time derivative of the wheel travel µ as r˙ 0w/b
3 2 4c µ + 3c µ + 2c µ + c 4 3 2 1 = µ˙ = T(µ)µ˙ −1 4d4 µ3 + 3d3 µ2 + 2d2 µ + d1
(8.5)
where r˙ 0w/b is the relative velocity of the wheel with respect to the vehicle body center expressed in vehicle body frame. Similarly, the relative angular velocity of the wheel frame can be obtained using Eqs. (8.4a) and (8.4b) as 0 −γ˙ ω 0w/b = −α˙ + Aw/b 0 0 0 3 2 4b µ + 3b µ + 2b µ + b 0 3 2 1 4 3 2 µ˙ = − 4a4 µ + 3a3 µ + 2a2 µ + a1 µ˙ − Aw/b 0 0 0 = R(µ)µ˙ (8.6) where ω 0w/b is the angular velocity of the wheel with respect to the vehicle body expressed in the vehicle body frame. Equation (8.6) is applied for the wheels of the left-hand side. For the wheels of the right-hand side, −α˙ and −γ˙ should be replaced by α˙ and γ˙ respectively. Using Eqs. (8.5) and (8.6), the absolute velocity of a wheel can be written as r˙ w = r˙ b + Ab r˙ 0w/b = r˙ b + Ab Tµ˙ 0 0 0 > ˙ ω 0w = A> w/b ω b + ω w/b = Aw/b (ω b + Rµ)
(8.7a) (8.7b)
The expression of the absolute velocity of Eq. (8.7) can be applied to all four wheels. We use the joint-coordinate formulation presented in chapter 2 with the
127 joint velocity array as r˙ b 0 ω b µ ˙ f l θ˙ = µ˙ f r µ ˙ rl µ ˙ rr φ˙
with φ˙ =
φ˙ f l φ˙ fr
(8.8)
φ˙ rl φ˙ rr
Then, the velocity transformation constructed as ˙ r I b 0 I ωb I A b Tf l r˙ f l 0 > A> ωf l f l/b Af l/b Rf l ˙ r Ab Tf r f r I ω 0f r = A> A> f r/b Rf r f r/b I Ab Trl r˙ rl A> ω 0rl A> rl/b Rrl rl/b I r˙ rr Ab Trr ω 0rr A> A> rr/b rr/b Rrr φ˙ I
r˙ b 0 ω b µ ˙ f l µ˙ f r µ ˙ rl µ ˙ rr φ˙ (8.9)
128 Thus, the velocity transformation matrix B is defined as I I I Ab Tf l > A> f l/b Af l/b Rf l I Ab Tf r B= A> A> f r/b f r/b Rf r Ab Trl I A> A> rl/b rl/b Rrl I Ab Trr A> A> rr/b Rrr rr/b
(8.10)
I 8.3 Equations of motion The equations of motion of the vehicle model is derived in canonical form. The momentum equation is written as ¯ B> MBθ˙ = p
(8.11)
where mb I J0b mf l I J0f l ... M= mrr I J0rr Jfφl .. .
φ Jrr
(8.12)
129 In this expression mb is the mass of the vehicle main body, J0b is the inertia matrix of the vehicle main body in the vehicle body frame, mf l denotes the mass assigned to the front left wheel center which includes the mass of suspension links, knuckle, and the wheel. The actual mass center will not be at the wheel center. However, in this model it is assumed that the masses of the suspension linkages and the knuckle are lumped to the wheel center because the effect of them is minor. Similarly, J0j denotes the combined inertia matrix of the suspension linkages, the knuckle and the wheel. However, J0j does not include the inertia of the wheel in the direction of tire rotation, namely, ζ-axis. The inertia in this direction is taken into account by Jjφ
j = f l, . . . , rr
(8.13)
In practice, the mass and the inertia of the vehicle are difficult to obtain. It requires a special test rig like the Centre of Gravity And Inertial Measurement System [117]. The mass and inertia of the knuckle and the wheel can be measured after disassembled from the vehicle. Alternatively, CAD data can be used for estimating the mass and inertia of the suspension parts. The mass and inertia information of a luxury vehicle is given in Table 8.2, where the mass and inertia of the main body are obtained from a measurement and the mass and inertia of the wheel are roughly estimated from CAD data. Table 8.2: Mass and inertia for the Parameter mb Jbxx Jbyy Jbzz Jbxy mf l , mf r mrl , mrr xx Jj , j = f l, . . . , rr Jjyy , j = f l, . . . , rr Jjzz , j = f l, . . . , rr Jjφ , j = f l, . . . , rr
vehicle model of a luxury sedan Value Unit 1768.6 kg 560.2 kgm2 3276.2 kgm2 3095.9 kgm2 −57.7 kgm2 40 kg 35 kg 0.7 kgm2 0.7 kgm2 0.2 kgm2 0.3 kgm2
130 The time derivative of the array of momenta is given as
where
ˆ +B ˙ > MBθ˙ ¯˙ = B> h p
(8.14)
fb 0 n b f f l 0 n fl . . . ˆ= f h rr 0 n rr φ n f l φ nf r φ n rl φ n
(8.15)
rr
The forces and moments acting on the main body is given as fb = fbgravity + fbspring−damper
(8.16)
n0b = nspring−damper0 b
(8.17)
The forces and moments acting on the wheel is given as fj = fjgravity + fjspring−damper + fjtire ,
j = f l, . . . , rr
n0j = ntire0 j
(8.18) (8.19)
The forces from the spring and damper of the suspension are modeled as functions of µ and µ˙ , respectively, as
f spring0
0 = fs (µ) , 0
f damper0
0 = fd (µ) ˙ 0
(8.20)
The forces from the spring and damper are assumed to be in vertical direction of the main body and act on the wheel center. SPMM can measure the vertical force
131 acting on the wheel center with respect to the wheel travel µ as shown in Fig. 8.7. The function fs (µ) can be obtained either by fitting this measured data or using it as a look-up table. In this study, the function fs (µ) is defined as a look-up table and the magnitude of spring force is determined through an interpolation of the data. The damping force function fd (µ) ˙ also can be defined as a look-up table if the damping force curve with respect to the piston-speed is available. In this study, a simple linear function is used as fd (µ) ˙ = −cd µ˙
(8.21)
where cd is a damping coefficient. The term fjspring−damper of Eq. (8.18) can be written as fjspring−damper = −Ab fjspring0 + fjdamper0 ,
j = f l, . . . , rr
(8.22)
The forces and moments on the main body are given as spring−damper fbspring−damper = −(ffspring−damper + · · · + frr ) l damper0 0 spring0 damper0 ˜ nbspring−damper0 = ˜r0f l/b ffspring0 + f + · · · + r f + f rr/b rr rr l fl
(8.23) (8.24)
The effect of the force of the tire is taken into account by the term fjtire and the effect and nφj . The of the moment from the tire is taken into account by the term ntire0 j moment along the axis of rotation should not be included in ntire0 , but in nφj . The j forces and moments from tires are provided by a tire model which was discussed in chapter 7. The equations of motion of the vehicle consist of Eqs. (8.11) and (8.14). The total number of first-order differential equations for the vehicle is 28.
132
Wheel Rate Measurement Data 9000
Wheel Center Vertical Force (N)
8000 7000 6000 5000 4000 3000 2000 1000 0 −80
−60 −40 −20 Compression
80 100 Extension
120
Figure 8.7: Wheel center vertical force versus wheel travel
133 CHAPTER 9 VEHICLE SIMULATION USING HYBRID INTEGRATION SCHEME
This chapter presents the results of a vehicle dynamic simulation using the vehicle and tire models discussed in chapters 7 and 8. This is the first attempt to apply the modal tire model to a full vehicle simulation. The vehicle model is constructed using the data of HMC Genesis and the tire model is generated using the ABAQUS model of a P245/50R181 tire. The tire model contains 25 b-nodes and the number of kept normal modes is 20. As a result, the number of nodal and modal coordinates of each tire is 3 × 25 + 20 = 95, which is much smaller than the number of degrees of freedom of the original ABAQUS model – 82842. In this chapter, eigenvalues of the vehicle system are investigated and the dynamic characteristics of the vehicle system are discussed. The equations of motion of the system are solved by the hybrid scheme and conventional methods. The numerical solutions from these methods are compared in terms of accuracy and computational efficiency. 9.1 Characteristics of Vehicle System Eigenvalues of a system provide information about the dynamic characteristics of the system. The imaginary part of the eigenvalue indicates the frequency of oscillation and the real part of the eigenvalue is associated with the damping of the system. Eigenvalues are solutions of the characteristic equation det (A − λI) = 0 1
(9.1)
‘P’ indicates a passenger car tire; ‘245’ is the nominal width of the tire; ‘50’ indicates that the
height of the sidewall of the tire is 50% of the width; ‘R’ indicates that it is a radial tire; and ‘18’ means this tire fits 18in wheels.
134 where A is the Jacobian of the system.
We can obtain A by linearizing the
equations of motion at the static equilibrium. The eigenvalues of the vehicle system are obtained and shown in Fig. 9.1. Figure 9.2 shows the eigenvalues near the origin more closely. From this figure, two groups of eigenvalues can be identified. One group is associated with the vehicle’s degrees of freedom while the other group is associated with the free modes of tires. Other eigenvalues which are not shown in Fig. 9.2 are related to the b-nodes of tires. The eigenvalues of tire’s degrees of freedom have much larger imaginary part than the eigenvalues of vehicle’s degrees of freedom. This means that the tires have much faster dynamics compared to the vehicle and that the vehicle system is highly oscillatory. Therefore, we can expect the hybrid scheme presented in chapter 6 to improve the computational efficiency of solving this system. Standard numerical methods should take a very small step size in order to satisfy the absolute stability condition when the dynamics of the vehicle system is solved. For a stable numerical solution, the step size should be chosen such that all ξ = hλ are located inside the absolute stability region. In the case of the hybrid scheme, the absolute stability region is almost independent of the eigenvalues associated with the fast variables and it only depends on the eigenvalues associated with the slow variables. As a result, the absolute stability condition can be satisfied with a much larger step size compared to conventional methods and the computational efficiency can be improved. In order to apply the hybrid scheme, a set of differential equations containing slow and fast variables should be written in the form x˙ = f (x, z)
(9.2a)
z˙ = g(x, z)
(9.2b)
The observation of the eigenvalues indicates that the states of the vehicle correspond to the slow variable x and the states of the tires correspond to the fast variable z. In addition, there is no direct coupling between the tires. Thus, The system of
135 4
3
x 10
Eigenvalues of Vehicle−Tire System
2
Im(λ)
1
0
−1
−2
−3 −1800 −1600 −1400 −1200 −1000 −800 −600 Re(λ)
−400
−200
0
200
Figure 9.1: Eigenvalues of Vehicle-Tire System differential equation can be expressed as x˙ = f (x, z 1 , z 2 , z 3 , z 4 )
(9.3a)
z˙ 1 = g1 (x, z 1 )
(9.3b)
z˙ 2 = g2 (x, z 2 )
(9.3c)
z˙ 3 = g3 (x, z 3 )
(9.3d)
z˙ 4 = g4 (x, z 4 )
(9.3e)
where z i , i = 1, . . . , 4 correspond to each tire. For this particular system, the hybrid scheme needs minor modifications. The reduced model as in Eq. (6.24) is constructed for each tire as z¯i = Hi (x),
i = 1, . . . , 4
(9.4)
The intermediate solution xˆ of Eq. (6.32) is obtained , with the fourth-order RungeKutta method, by solving x˙ = f (x, H1 (x), H2 (x), H3 (x), H4 (x)) ,
x(tn ) = xn
(9.5)
136
Eigenvalues of Vehicle−Tire System (zoomed) 1000 800 600 400
Im(λ)
200
λ’s associated with free modes of tires
0 λ’s associated with vehicle body
−200 −400 −600 −800 −1000 −400
−300
−200
−100 Re(λ)
0
100
200
Figure 9.2: Eigenvalues of Vehicle-Tire System (zoomed) Then, the complete solution of x is computed as Z tn+1 Z tn+1 Z 1 2 xn+1 = xˆ+fz1 y (τ )dτ +fz2 y (τ )dτ +fz3 tn
tn
tn+1
tn
3
Z
tn+1
y (τ )dτ +fz4
y 4 (τ )dτ
tn
(9.6) where y i , i = 1, . . . , 4 is the boundary-layer model for each tire. The fast solution i is obtained by using Eq. (6.48) for each tire as discussed in chapter 6. zn+1
For the vehicle dynamic simulation, the contact between a tire and the ground should be handled properly. During the simulation, boundary nodes of the tire model may come into or out of contact with the ground. To properly deal with this transition, standard numerical methods should take a very small step size in order to avoid an excessive penetration, which could cause unstable response. When the hybrid scheme is used, the step size adjustment discussed in chapter 3 can be used. In the tire model, the ground contact is modeled by the linear contact force model and its effect is reflected in the system Jacobian. The system Jacobian changes substantially when a b-node comes into or out of contact. The hybrid
137 integration scheme is based on a successive local linearization where the system Jacobian matrices are assumed to be constant during each time step. In order to handle this contact problem properly, the step size should be adjusted such that a new time step to begin at the instant when a contact between a b-node and the ground begins or ends. To this end, the contact instant detection discussed in section 3.5 can be applied to the hybrid scheme. For detecting the instant when the contact begins or ends, we define the function Φk that represents the contact condition for the k-th b-node as Φk = dky − η
(9.7)
where dky denotes the vertical position of the b-node expressed in the global frame and η is a function representing the height of the ground. Specifically, η can be defined as a function of the horizontal position of the b-node, namely, η = η(dkx , dkz )
(9.8)
When the numerical solution at t = tn+1 is obtained by the hybrid scheme, it is checked if sign (Φk (tn )) = sign (Φk (tn+1 )). If the signs are different, that is an indication that the node has come into or out of contact between tn and tn+1 . The contact instant t∗ can be determined by solving the nonlinear equation Φk (t∗k ) = 0. The Newton-Raphson or Bi-section method can be used for solving this nonlinear equation. The numerical solution at time t between tn and tn+1 is required for evaluating the function Φk . The slow solution x(t) can be estimated by a linear interpolation as x(t) = xn +
t − tn (xn+1 − xn ) , tn+1 − tn
for t ∈ [tn , tn+1 ]
(9.9)
The fast solution z i (t) for i-th tire can be obtained by z i = Hi (x(t)) + σ + φ(t − tn ) where σ=
∂gi ∂z i
−1
gi (xn , zn1 , zn2 , zn3 , zn4 )
(9.10)
(9.11)
138 ¯
and φ(t − tn ) is a sub-matrix of e(t−tn )D as defined in Eq. (6.49). If several contact instants are detected between tn and tn+1 , the step size should be adjusted to the one which occurs first. The step size hn is reset to h∗ = (min t∗k ) − tn
(9.12)
and the numerical solution is determined at tn+1 = tn + h∗ . 9.2 Simulation Results In this section, we present the simulation results for several scenarios that reveal the dynamic response of the vehicle system and the capabilities of the vehicle and the tire model. These simulations are performed by the proposed hybrid scheme and conventional numerical methods, then, the simulation results are compared in terms of accuracy and computational efficiency. Drop Test First, we consider a case where the vehicle system is dropped from a height to the ground. Initially, the tires are not in contact with the ground and there is no tire deflection. The gravitational force makes the vehicle move downward and the contacts between tires and the ground are initiated. The main body of the vehicle oscillates until the energy is dissipated by the suspension dampers and the entire system reaches a static equilibrium. This simulation is performed by the hybrid scheme, ODE45, and ODE15s on a laptop with dual core 2.10GHz CPU. For the hybrid scheme, the step size is set to h = 0.01 initially and it is adjusted according to the contact instant when necessary. The chosen step size h = 0.01 is the one that satisfies the absolute stability condition and attains an acceptable accuracy for this problem. Figure 9.3 shows the vertical position of the vehicle body and Fig. 9.4 shows the wheel travel of each wheel obtained by these numerical methods. The results indicate that these three methods produce almost identical numerical solutions.
139
Vehicle Center Vertical Position 0.62 Hybrid ODE15s ODE45
0.6
0.58
ry (m)
0.56
0.54
0.52
0.5
0.48
0
0.5
1
1.5
2
2.5 Time (s)
3
3.5
4
4.5
5
Figure 9.3: Vehicle body vertical position for drop test
Front Left Wheel Travel Wheel Travel µ (m)
0.15 Hybrid ODE15s ODE45
0.1 0.05 0 −0.05
0
0.5
1
1.5
0
0.5
1
1.5
2
2.5 3 3.5 Time (s) Rear Left Wheel Travel
4
4.5
5
4
4.5
5
Wheel Travel µ (m)
0.15 0.1 0.05 0 −0.05
2
2.5 Time (s)
3
3.5
Figure 9.4: Wheel travels for drop test
140
Table 9.1: The number of steps and computation time for drop test The number of steps CPU time Hybrid Scheme 574 25min ODE15s 212118 2h 42min ODE45 239925 2h 34min Table 9.1 compares the number of steps and the computation time taken by these numerical methods. The comparison reveals that the hybrid scheme can solves this problem with much fewer steps and less computation time than other methods. A fairly large step size h = 0.01 can be employed for the hybrid scheme since the absolute stability region only depends on the slow components of the system.
In addition, the contact between the tires and the ground is handled
efficiently by detecting the contact instants and taking a smaller step size when it is necessary. Figure 9.5 shows the first component of ub of rear left tire from t = 0 to t = 0.06. It can be seen that the hybrid scheme computes the numerical solution with the interval of 0.01 until t = 0.04 because no contact is initiated up to this point. After t = 0.04, numerical solutions are obtained with different intervals since the step size is adjusted according to the contact between the boundary nodes and the ground. The numerical solutions from ODE15s and ODE45 are very accurate since very small step sizes are used due to the absolute stability condition. From Fig. 9.5, it can be seen that the solution from the hybrid scheme contains discrepancy around 0.05sec from that of other methods. There is a small time delay in the numerical solution from the hybrid scheme and it is believed that this delay is caused by the numerical error of the hybrid scheme. If we take a smaller time step, this discrepancy diminishes as seen in Fig. 9.6. However, as seen in Figs. 9.3 and 9.4, the effect of this discrepancy is very small and the dynamic response of the vehicle can be computed with an acceptable accuracy by the hybrid scheme with the step size h = 0.01. The contact phenomenon between tires and the ground also can be captured well by the hybrid scheme, which can be observed in Fig. 9.7. The first
141 −6
2
x 10
The first component of ub of rear left tire
1.5
1
ub(1)
0.5
0
−0.5 Hybrid ODE15s ODE45
−1
−1.5
0
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Time (s)
Figure 9.5: The first component of ub of rear left tire from drop test with h = 0.01 two plots of Fig. 9.7 show the contact force distributions obtained from the hybrid scheme and ODE15s at t = 0.045. This is right after the contact between the tire and the ground begins. Higher normal contact forces are observed at the center of the contact patch because the contact starts to form in that area. The last two plots shows the contact force distributions at t = 3.26, when the vehicle system converges to the static equilibrium. These results demonstrate that the contact between the tires and the ground can be captured well by the hybrid scheme with the help of the step size adjustment.
142
The first component of ub of rear left tire
−6
2.5
x 10
Hybrid ODE15s ODE45
2 1.5
ub(1)
1 0.5 0 −0.5 −1 −1.5
0
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Time (s)
Figure 9.6: The first component of ub of rear left tire from drop test with h = 0.002
ODE15s, T=0.04588 600
400
400 FN (N)
FN (N)
Hybrid Scheme, T=0.04588 600
200
200
0
0
0.7 0.8 0.9
0.7 0.8 0.9 1.6
1.55
1.5
1.45
1.4
1.35
1.3
1.25
1.6
600
400
400
200 0
0 0.7 0.8 0.9 1.55
1.5
1.45
1.4
1.45
1.4
1.35
1.3
1.25
1.4
1.35
1.3
1.25
200
0.7 0.8 0.9 1.6
1.5
ODE15s, T=3.2639
600
FN (N)
FN (N)
Hybrid Scheme, T=3.2639
1.55
1.35
1.3
1.25
1.6
1.55
1.5
1.45
Figure 9.7: Normal contact force distribution for drop test
143 Pothole Test The second example is the case that one tire drops into a pothole at a speed of 30kph. We consider the ground profile as shown in Fig. 9.8. When a tire reaches the pothole, the leading edge loses contact with the ground temporarily, then, the contact is formed again as the tire deforms.
Y
X
H 0.01m
Figure 9.8: Ground geometry for pothole test Figure 9.9 shows the deformation of the tire computed by the hybrid scheme when the tire drops into the pothole. From this figure, it can be seen that the tire is deformed such that contact area is conformed to the ground geometry. Figure 9.10 compares the first four components of ub obtained by the hybrid scheme and ODE45. The plots of this figure show that the numerical solution from both methods are close to each other. The vertical force acting on the wheel center is shown in Fig. 9.11. The solid line indicates the result from ODE45 and the circles indicate the result from the hybrid scheme. For the hybrid scheme, the default step size is h = 0.01 and smaller step sizes are used during the tire passes the transition zone to detect the contact instant accurately. This plot shows that the vertical force decreases as the leading edge of the tire loses contact with the ground. As the tire recovers its contact with the ground, the force acting on the wheel increases again and converges to the level of the static equilibrium. With a fairly large step size h = 0.01, the hybrid scheme can give the result that is well matched to that from ODE45.
144
Figure 9.9: Tire deflection when passing over a pothole
FL Tire ub(1)
−5
8
x 10
FL Tire ub(2)
−5
6 Hybrid ODE45
6
x 10
5
3
b
u (2)
ub(1)
4 4 2
2 1
0 −2
0 0
0.05
0.1
0.2
0.25
0.3
FL Tire ub(3)
−5
−1
0.15 Time (s)
−1
x 10
0.1
0.15 Time (s)
0.2
0.25
0.3
0.2
0.25
0.3
FL Tire ub(4)
x 10
0 −1 u (4)
−3 −4
−2
b
ub(3)
0.05
−5
1
−2
−5
−3 −4
−6 −7
0
−5 0
0.05
0.1
0.15 Time (s)
0.2
0.25
0.3
−6
0
0.05
0.1
0.15 Time (s)
Figure 9.10: The first four components of ub for pothole test
145 Vertical Front Left Wheel Force 7000 ODE45 Hybrid
6500 6000 5500
Fy (N)
5000 4500 4000 3500 3000 2500 2000
0
0.05
0.1
0.15 Time (s)
0.2
0.25
0.3
Figure 9.11: Vertical force acting on the wheel center for pothole test The number of steps and the computation time for different numerical methods are listed in Table 9.2. The comparison of the computation time reveals that the hybrid scheme can improve the efficiency for the vehicle dynamic simulation including the flexible tire model. Table 9.2: The number of steps and computation time for pothole test The number of steps CPU time Hybrid Scheme 199 7min 15sec ODE15s 11165 18min 8sec ODE45 16081 11min 28sec
146 Ramp Test Let us consider the case that a vehicle passes over a small ramp on the ground ,as shown in Fig. 9.12, at a speed of 50kph. The ramp will excite the system and will bring about a dynamic response. Mathematically, the ground height is modeled as 0, if X < Xstart (9.13) η(X) = H(X−Xstart ) , if Xstart ≤ X < Xend L H, if X ≥ Xend
Y
(X )
H 0.05m
X
X start L 1m X end
Figure 9.12: Configuration of ground for ramp test The numerical solutions are obtained by the hybrid scheme and ODE45. In the case of the hybrid scheme, the step size is set to h = 0.01. Figure 9.13 shows the vertical position of the vehicle center with respect to time. After passing over the ramp, the vehicle center moves upwards and oscillates until the motion is damped out. The vehicle center position from these numerical methods are almost identical as depicted in Fig. 9.13. Figure 9.14 shows the pitch angle of the vehicle with respect to time. The pitch is the angle of the vehicle body along the axis of the lateral direction. Positive pitch angle means that the vehicle body leans backward. From Fig. 9.14, it can be seen that the pitch angles from these numerical are almost identical.
147
Vehicle Center Vertical Position 0.65 Hybrid ODE45
ry (m)
0.6
0.55
0.5
0
0.5
1
1.5 Time (s)
2
2.5
3
Figure 9.13: Vehicle center vertical position for ramp test
Vehicle Pitch Angle 1.5 Hybrid ODE45
Pitch Angle (deg)
1
0.5
0
−0.5
−1
0
0.5
1
1.5 Time (s)
2
2.5
Figure 9.14: Vehicle pitch angle for ramp test
3
148 Figure 9.15 shows the travels of the front left wheel and the rear left wheel. Because the front wheels pass the ramp before the rear wheels, the compression starts first at the front suspension. Although the compression of the rear suspension starts later, the motion of the rear suspension synchronize with that of the front suspension quickly.
It is important to keep the motion of both axle in-phase
for better ride comfort. Otherwise, the pitch motion will last longer and cause discomfort. To this end, the suspension springs should be chosen such that the natural frequency of the rear suspension is slightly higher than that of the front suspension [118]. The first four components of ub for the front left tire and the rear left tire are shown in Figs. 9.16 and 9.17. From these figures, it can be seen that the result of the hybrid scheme is well agreed to that of ODE45.
Front Left Wheel Travel Wheel Travel µ (m)
0.04 Hybrid ODE45
0.02 0 −0.02 −0.04
0
0.5
1
0
0.5
1
1.5 2 Time (s) Rear Left Wheel Travel
2.5
3
2.5
3
Wheel Travel µ (m)
0.04 0.02 0 −0.02 −0.04
1.5 Time (s)
2
Figure 9.15: Front left and rear left wheel travel for ramp test
149
FL Tire ub(1)
−5
2
x 10
FL Tire ub(2)
−5
2 Hybrid ODE45
1
x 10
1
ub(2)
ub(1)
0 0 −1
−1 −2
−2 −3
−3 0
0.5
1
2
2.5
−4
3
FL Tire ub(3)
−5
−4
1.5 Time (s)
x 10
0
0.5
1
2
2.5
3
2
2.5
3
FL Tire ub(4)
−5
−3
1.5 Time (s)
x 10
−4
−5
ub(4)
ub(3)
−5 −6 −7
−6 −7
−8 −9
−8 0
0.5
1
1.5 Time (s)
2
2.5
3
−9
0
0.5
1
1.5 Time (s)
Figure 9.16: The first four components of ub of front left tire for ramp test
150
RL Tire ub(1)
−5
2
x 10
Hybrid ODE45
ub(2)
ub(1)
0
−1
0
0.5
1
1.5 Time (s)
2
2.5
0
−2
3
RL Tire ub(3)
x 10
−4
−3
−5
−4
−6 −7 −8
0
0.5
1
1.5 Time (s)
2
2.5
3
2
2.5
3
RL Tire ub(4)
−5
−2
ub(4)
ub(3)
1
−1
−5
−3
x 10
2
1
−2
RL Tire ub(2)
−5
3
x 10
−5 −6
0
0.5
1
1.5 Time (s)
2
2.5
3
−7
0
0.5
1
1.5 Time (s)
Figure 9.17: The first four components of ub of rear left tire for ramp test
151
Table 9.3: The number of steps and computation time for ramp test The number of steps CPU time Hybrid Scheme 454 27min ODE45 145773 1hr 45min The number of steps and the computation time for the hybrid scheme and ODE45 are compared in Table 9.3. It turns out that the computation time for the hybrid scheme is about one-fourth of that for ODE45. This result demonstrate that the proposed hybrid scheme can improve the computational efficiency while retaining comparable accuracy. The accuracy of the numerical solution from the hybrid method may be improved if a smaller step size is used.
However, the
efficiency will be sacrificed. The choice of step size depends on the purpose of the simulation and user’s priority. The benefit of the hybrid scheme is that the user can have flexibility in choosing the step size when solving a highly oscillatory dynamical system. Other numerical methods can not give this flexibility because the step size is restricted by the absolute stability condition. In this chapter, the results of a vehicle dynamic simulation for several different scenarios are presented. These results demonstrate that the vehicle and tire model discussed in previous chapters can represent the dynamic response of the vehicle system well. Since the contact is modeled by a large number of boundary nodes, the interaction between the tire and the ground having arbitrary profile can be considered. This chapter presented the simulation results for several simple ground profiles, but, we can consider any other ground profiles in the simulation. The comparison of computational times reveals that the proposed hybrid scheme can improve the computational efficiency of the vehicle simulation. With the hybrid scheme, the simulation takes about 20 minutes while other conventional methods require couple of hours. The improvement of efficiency will make the vehicle dynamic simulation more attractive option over the actual field test. With less computation time, it is more feasible to conduct the design optimization and reliability analysis using the vehicle dynamic simulation.
152 CHAPTER 10 CONCLUSIONS AND RECOMMENDATIONS
10.1 Conclusions Computational efficiency of solving the dynamics of highly oscillatory systems is an important issue due to the requirement of small step size of explicit numerical integration methods. Standard explicit numerical methods need to take a very small step size to satisfy the absolute stability condition for the fastest component of the solution. To overcome this problem, in this study a hybrid numerical integration scheme is developed.
Within this hybrid scheme, two
different numerical integration algorithms are utilized while the same step size is used for both methods. A highly oscillatory system is partitioned into the fast and the slow subsystems and these subsystems are transformed into a reduced (quasi-steady-state) and a boundary-layer system using the singular perturbation theory. The obtained reduced system is solved by the fourth-order Runge-Kutta method and the boundary-layer system is solved by the Local Linearization Method. The absolute stability region for this hybrid scheme is derived in chapter 6 where it is shown that the absolute stability region is almost independent of the fast variables. Thus, the selection of the step size is not dictated by the fast solution when a highly oscillatory system is solved, hence, the computational efficiency can be improved. This hybrid scheme handles the coupling between the fast and the slow subsystems using the concept of the singular perturbation theory. Thus, unlike other multi-rate or multi-method schemes, extrapolation and interpolation process are not required to deal with the coupling between subsystems. Most of the coupling effect can be accounted for by the reduced (or quasi-steady-state) system while the minor transient effect is taken into consideration by averaging. As
153 a system becomes more oscillatory, the proposed hybrid scheme can yield a more accurate numerical solution with the same step size. This property is demonstrated by the numerical example presented in chapter 6. In this regard, the proposed hybrid scheme has an advantage over other multi-rate methods. Multi-rate methods use different step sizes for the fast and the slow subsystems. A smaller step size should be used for the fast subsystem as the system gets more oscillatory. Hence, the computational cost is still dictated by the fastest component of the solution. The proposed hybrid scheme is not useful for every system, but it is meaningful when a system that consists of distinctly separate frequencies is analyzed. It is also required that the system should be split into a slow and a fast subsystems by the user with prior knowledge of the dynamics of the system. Typically, the fast components of the system can be identified by investigating the distribution of the eigenvalues of the system. It is also assumed that the partitioning into the slow and the fast subsystems is valid throughout the time span of the numerical integration. The hybrid scheme will perform better when the coupling between subsystems is weak and the size of the fast subsystem is small. To reduce the size of the fast subsystem, the user may make use of any existing model reduction techniques. To validate the performance of the hybrid scheme, several vehicle dynamic simulations were conducted. The concepts of the tire and vehicle models are discussed and the equations of motion are derived in chapter 7 and 8. The vehicle model contains 14 degrees of freedom, in which the kinematics of the suspension is characterized by empirical equations. The tire model is based on the Craig-Bampton reduction technique and the equations of motion are derived in a non-rotating reference frame. Since the tire has much higher natural frequency than the rigid bodies of the vehicle, this system is an example of highly oscillatory dynamical systems. This system was analyzed by the hybrid scheme and other conventional methods, and then the results were compared in terms of computational efficiency and accuracy. The comparision results showed that the hybrid scheme can reduce the computation
154 time significantly while attaining comparable accuracy. 10.2 Recommendations For future investigation, the following works can be recommended: • In this study, the fourth-order Runge-Kutta method is considered as one component of the hybrid scheme. It will be possible to employ other conventional integration methods for solving the reduced system. Combination with other integration methods will lead to different absolute stability regions and the computational cost per single step will be changed. Depending on the characteristics of the problem, optimal performance may be achieved by other combination of numerical methods. • One possible application of the hybrid scheme is in the dynamic simulation of flexible-rigid multibody systems. Unlike the vehicle-tire system presented in this study, in general, a flexible body is connected to other bodies by kinematic joints. In this case, the coupling between the flexible body and the rigid body becomes more complicated. More study is needed regarding how to split a general flexible-rigid multibody system into subsystems and how to compute the required system Jacobian. • In dynamic analysis of a multibody system with kinematic constraints, a range of constraints violation stabilization methods can be used to avoid an excessive accumulation of violation for a long simulation periods. These stabilization methods may introduce artificial high-frequency responses, which are separate from the physically significant high-frequency responses. Conventional numerical methods may have difficulty in dealing with the artificial high-frequency responses. However, there is a possibility that the hybrid scheme could serve as a better option to handle this problem. To this end, more study is needed regarding how to define the fast subsystem and how to revise the algorithm of the scheme, if necessary.
155 • The application of the hybrid scheme is not limited to the analysis of mechanical systems. The hybrid scheme will be useful for the simulation of large size electrical circuits containing fast components. There have been many attempts to apply multi-rate methods or waveform relaxation methods to this area. The proposed hybrid scheme may serve as another option for improving the computational efficiency in solving such problems.
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