Nonlinear Dynamics 34: 113–131, 2003. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Optimization of Vehicle Suspension Systems for Improved Comfort of Road Vehicles Using Flexible Multibody Dynamics JOÃO P. C. GONÇALVES Departamento de Física, Universidade de Évora, Colégio Luis Vernay, P-7000 Évora, Portugal; E-mail:
[email protected]
JORGE A. C. AMBRÓSIO IDMEC, Instituto Superior Técnico, Av. Rovisco Pais, 1, P-1049-001 Lisboa, Portugal; E-mail:
[email protected] (Received: 24 January 2003; accepted: 24 April 2003) Abstract. Methods that account for the flexibility of multibody systems extend the range of applications to areas such as flexible robots, precision machinery, vehicle dynamics or space satellites. The method proposed here for flexible multibody models allows for the representation of complex-shaped bodies using general finiteelement discretizations which deform during the dynamic loading of the system, while the gross rigid body motion of these bodies is still captured using fixed-body coordinate frames. Components of the system for which the deformations are relatively unimportant are represented with rigid bodies. This method is applied to a road vehicle where flexibility plays an important role in its ride and handling dynamic behavior. Therefore, for the study of the limit behavior of the vehicles, the use of flexible multibody models is of high importance. The design process of these vehicles, very often based on intuition and experience, can be greatly enhanced through the use of generalized optimization techniques concurrently with multibody codes. The use of sparse matrix system solvers and modal superposition, to reduce the number of flexible coordinates, in a computer simulation, assures a fast and reliable analysis tool for the optimization process. The optimum design of the vehicle is achieved through the use of an optimization algorithm with finite-difference sensitivities, where the characteristics of the vehicle components are the design variables on which appropriate constraints are imposed. The ride optimization is achieved by finding the optimum of a ride index that results from a metric that accounts for the acceleration in several key points in the vehicle properly weighted in face of their importance for the comfort of the occupant. Simulations with different road profiles are performed for different speeds to account for diverse ride situations. The results are presented and discussed in view of the different methods used with emphasis on models and algorithms. Keywords: Vehicle comfort optimization, road vehicle ride, flexible multibody dynamics.
1. Introduction The need for more accurate models to describe the complex behavior of flexible systems experiencing large motion while undergoing small elastic deformations motivated the development of many powerful analysis techniques. The most popular formulations use time-variant mass matrices to describe the inertia coupling between the rigid body gross motion and the system elastodynamics [1]. Some coefficients of these inertia-coupling matrices are dependent on the type of finite elements used in the model. They do not appear in standard finite-element developments and consequently need to be specially derived when these methods are used. These formulations have been mostly applied to models using beam elements or, at the most, using a particular type of plate element [2].
114 J. P. C. Gonçalves and J. A. C. Ambr´osio For methods where a geometric and/or material nonlinear behavior is described, this problem is of utmost importance because not only the flexible body mass matrix is time variant but also mode component synthesis methods have restrictions [3]. Within the framework of the modeling of flexible multibody systems experiencing nonlinear deformations, Ambrósio and Nikravesh [4] presented a method where the flexible body has both a rigid and a flexible part. Moreover, the mass matrix is formulated using a lumped mass approach and the finite-element nodal coordinates are referred to the inertial frame. In this manner, not only the mass matrix of the flexible body turns out to be diagonal and constant but also all the terms that appear in the equations of motion, related to the body flexibility, are absolutely standard finite-element terms. The single most important feature of the formulation used in this paper is that the flexibility of the system components is described exclusively by standard finite-element terms, available in any commercial finite-element package. The coupling between the linear elastodynamics of the flexible bodies and their large rigid body motion is still fully preserved. Furthermore it is shown that the multibody models can use any type of linear finite elements available with no other restriction than a good finite-element modeling practice [5]. For the application of the proposed method to complex systems, the number of generalized variables that can be integrated has to be greatly reduced. For this purpose, the mode component synthesis technique is applied to the equations of motion of each flexible body. The modes of vibration used are consistent with the boundary conditions between the flexible components and the rigid body frame. A set of kinematics joints is defined between flexible and rigid bodies using the generalized coordinates. The application to complex mechanical systems leads to models with a very large number of integration variables associated with the flexible coordinates’ degrees of freedom. In order to achieve reasonable computer efficiency, some nodal degrees-of-freedom reduction techniques must be used. In this work the component mode synthesis [6] is applied. Boundary conditions consistent with the location of the floating frames, are applied in the determination of the natural vibration modes and frequencies. The use of structural damping, as long as it is kept small, has the advantage of improving the numerical efficiency of the numerical integrators without leading to major variations in the results [7]. In order to minimize numerical problems, a proper choice of integration algorithms and solvers of the system equations must be made. The equations of motion derived by using Cartesian and generalized flexible coordinates lead to very sparse matrix forms which are suitable for solving using sparse matrix solvers [8, 9]. One of the first works in vehicle-dynamics response optimization was presented by Haug and Arora [10], where a 2D model of a car was optimized, using the adjoint variable method. Another alternative is the direct differentiation method [11], which has proven to be more suitable for the dynamic analysis problems. It is known that search methods converge to the nearest minimum, depending on the initial values. The most efficient methods are the local search type, which require the objective function and constraints gradients with respect to the design variables. An optimal solution is obtained, but it most probably corresponds to a local minimum. Approximation methods have been applied with success in the vehicle dynamic response [12]. They have the advantage of being faster in function evaluations and gradient calculations. The search for a global minimum is of major interest. The use of global optimization methods, such as the case of genetic algorithms, is required for this purpose. Genetic algorithms, as a stochastic global optimization technique, have been used in the design optimization of vehicle suspensions [13]. But due to the higher number of function evaluations
Optimization of Vehicle Suspension Systems 115 that they require, when compared to gradient projection methods, makes them suitable only for simple vehicle models. Another possibility is the use of a globally convergent method, such as the min-max dynamic response optimization with the ALM method presented by Kim and Choi [14]. The optimization procedure is a highly time-consuming process due to the high number of design variables and constraints that must be imposed. In optimization procedures, the system sensitivities must be calculated analytically or by finite differences methods. Both the adjoint variable method and direct differentiation are valuable procedures to calculate these gradients in a reliable and efficient manner [15, 16, 32]. Using numerical sensitivities, attention must also be paid to the numerical performance of the methods used in the analysis. The procedure proposed in this work evaluates the sensitivities by finite differences. Dynamic systems have to be optimized with respect to a number of conflicting specifications and different designs may be acceptable as optimal with respect to the same set of specifications. In the case of multiple objective problems, the simultaneous minimization of all objectives is not possible in general. Individual optimization can be done on one objective function at a time, while the other objective functions are treated as constraints. A promising multicriteria optimization approach has been proposed by Eberhard and Bestle [17]. Several optimization applications to general mechanical systems and methods are reported in [18]. The design and optimization of vehicle-dynamic response involves a mathematical representation of road excitations, the development of a vehicle model, a selection of performance criteria, and the use of optimization techniques to achieve optimal design, meeting the desired performance criteria [17, 19]. The use of a complex 3D model of the vehicle, with a detailed description of all suspension systems and road/tire interaction, is necessary to investigate the problem. However, such models are computationally expensive especially when used in an iterative design process. Simple models have been used in the industry in order to improve and optimize the vehicle response. A good alternative is the optimization of a subsystem of a complex model. The suspension subsystem is of utmost importance in terms of vehicle dynamics, being the spring-damper and suspension geometry parameter sets of basic design variables [20, 21]. Several measures, such as the maximum acceleration value or the power spectral densities comparison, can be used for the comfort optimization. The ISO 2631 proposes a vibration tolerance criterion, by using weighted sum rms acceleration values [22]. In situations in which the time period of analysis is small (less than one minute) a good choice is the measure of the vibration dose values [23]. The formulation proposed here for the optimization of complex flexible multibody systems is demonstrated with an application to the study of the optimal ride characteristics of a sports car in a profiled road. In the process, it is shown that besides the implications of the vehicle chassis flexibility on the ride quality, it is also crucial for the model performance to consider reliable tire models and tire/road contact descriptions [21, 24, 25]. 2. Flexible Multibody Equations of Motion The description of the flexibility of multibody systems must have no dependency in the type of coordinates used to describe the body fixed coordinate systems. In this work, Cartesian coordinates are used to describe the position and orientation of the system components. The position and orientation of rigid body i is defined by a set of translation coordinates
116 J. P. C. Gonçalves and J. A. C. Ambr´osio
Figure 1. Global position of node k.
ri = [x y z]T and rotational coordinates described by Euler parameters pi = [e0 e1 e2 e3 ]T . To minimize the number of variables for integration, while improving the numerical stability of the computer implementation, the rigid body time derivatives of the Euler parameters are transformed in angular velocities and accelerations [26]. 2.1. F LEXIBLE B ODY M OTION Let it be assumed that a flexible body is composed of a coordinate system rigidly attached to a point on the flexible body, as depicted in Figure 1. Also assume that the flexible body is represented, using the finite-element method, by a lumped formulation for the mass matrix that results from the diagonalization of the consistent finite-element mass matrix, and where the reference frame, described by vector ri , is located in the body center of mass. Let the position of a node k of the flexible body be represented by dk = ri + Ai bk ,
(1)
where bk = xk + δk is the position of node k in the body-fixed frame, δ k being its deformation, xk its initial position, also measured in the local frame, and Ai is the transformation matrix between the local and the inertial frames. Deriving the position equation with respect to time results in the node velocity, given by [5], d˙ k = r˙ i + Ai ω˜ i bk + Ai δ˙ k ,
(2)
where ωi is the angular velocity of the body fixed frame expressed in the local frame. The lumped procedure used here implies that a rotational inertia is associated to each node. Therefore, the evaluation of the kinetic energy involves the nodal angular velocity given by α k = ωi + θ˙ k ,
(3)
where α k is the generalized nodal angular velocity and θ˙ k is the local nodal deformation velocity. The kinetic energy of node k with a mass mk and a rotational inertia µk is
Tk =
1 ˙ kT ˙ k 1 mk d d + µk α T k αk . 2 2
(4)
Optimization of Vehicle Suspension Systems 117 Finally, the kinetic energy of the complete flexible body is evaluated by summing the contribution of all n nodes 1 ˙ kT ˙ k 1 mk d d + µk α T k αk . 2 k=1 2 k=1 n
T =
n
(5)
Before proceeding, let the coordinates used to describe the flexible body motion be represented by vector qi containing coordinates associated to the motion of the body-fixed frame and the coordinates associated to the nodal displacements and rotations with respect to the local frame. The velocity vector for the flexible body, represented by q˙ i , is written as r˙ i (6) q˙ i = ωi , u˙ where u˙ = [δ˙ T θ˙ T ]T is the vector of the local nodal velocities which contains the nodal translation and angular velocities. The flexible body kinetic energy is now expressed in terms of the defined coordinates. Substituting the definition given by Equation (2) into Equation (5) and using the definition provided by Equation (6), the result leads to 1 Ti = q˙ Ti Mi q˙ i , 2 where the flexible body mass matrix M is now given by − mk Ab˜ k 0 mk AITk mk I ˜ T mk bk A − mk b˜ k + µk I mk b˜ k ITk µk ITk M= − mk Ik b˜ k 0 mk Ik AT mk Ik ITk 0 µk Ik ITk 0 µk Ik
(7) .
(8)
Note that the mass matrix M has 4 × 4 submatrices associated with the use of u˙ = [δ˙ T θ˙ T ]T in the definition given by Equation (6). In methods where the consistent mass formulation is used in the finite-element description of the flexible body, the inertia coupling terms depend on the particular finite-element shape functions used. The structure of the mass matrix given by Equation (8) is independent of the formulation used to describe the flexible body deformations and no special inertia-coupling coefficients have to be derived. As all coordinates defined in Equation (7) are independent, the Lagrange equations of motion for the flexible body are given by ∂Ti ∂Ui d ∂Ti (9) − + − gi = 0, dt ∂ q˙ i ∂qi ∂qi where the elastic energy Ui is written in a generic form as Ui =
1 T q Ki q i . 2 i
(10)
The stiffness matrix Ki used in the equation for the elastic energy can be the finite-element stiffness matrix or any other equivalent matrix, depending on the description of the flexibility
118 J. P. C. Gonçalves and J. A. C. Ambr´osio adopted. Here it is assumed that matrix Ki is not dependent on the deformation of the flexible body, implying that only linear elastic deformations are modeled and that the deformation of the finite-element mesh does not involve large rotations with respect to the body-fixed coordinate frame. The definitions of the kinetic and elastic energy, given by Equations (7) and (10) respectively, are now substituted in Equation (9), leading to Mi q¨ i = gi + si − Ki qi .
(11)
In this equation, gi is the vector of external applied forces and vector si contains the quadratic velocity terms written here as − mk Aω˜ ω˜ bk − 2 mk Aω˜ δ˙ k ˙ ˜ ˜ . (12) s= − mk bk ω˜ ω˜ bk − 2 mk bk ω˜ δ k − mk Ik ω˜ ω˜ bk − 2 mk I˜k ω˜ AT δ˙ k A set of reference conditions is required to ensure the uniqueness of the displacement field. It is assumed here that a set of nodes of the flexible body is fixed to the fixed-body coordinate system, i.e., the nodal displacements of these nodes with respect to the local frame are null. Other sets of reference conditions, such as the mean axis condition [27], can be used. 2.2. F LEXIBLE M ULTIBODY S YSTEM E QUATION OF M OTION For a multibody system, it is necessary to define a set of kinematic constraints describing the joints that restrict the relative motion between the system components. By using Lagrange multipliers, this set of constraints is added to the system equation of motion resulting in Mr Mrf Tqr q¨ r gr sr 0 Mf r Mff Tq u¨ = gf − sf − Kff u . (13) f λ γ 0 0 qr qf 0 Note that the coupling between the rigid-body motion and the system elastodynamics is fully preserved. If the finite-element method is used to represent the flexibility of the system components all terms used in Equation (13) are readily available from any commercial finiteelement package, regardless of the particular type of linear finite elements used in the model. The only restrictions are related to good finite-element modeling practice. In particular, due to the use of a lumped mass formulation to evaluate the finite-element mass matrix, higher-order finite elements should be avoided, as they may lead to incorrect models. 2.3. F LEXIBLE C OORDINATES R EDUCTION BY C OMPONENT M ODE S YNTHESIS The equations of motion for the flexible multibody systems, in the form described by Equation (13), lead to an inefficient numerical implementation due to the large number of generalized coordinates necessary to describe complex models. This problem is overcome by using a component-mode synthesis method. Though only the natural modes of vibration are used in this formulation, other modes, such as the static correction modes [28], could also be considered in order to improve numerical precision and efficiency.
Optimization of Vehicle Suspension Systems 119 Let the nodal displacements of the flexible part of the body be described by a weighted sum of the modes of vibration associated with the lower natural frequencies of the flexible bodies u = Xw,
(14)
where the vector w represents the contributions of the vibration modes towards the nodal displacements and X is the modal matrix. Note that, due to the reference conditions, the modes of vibration used in this formulation are constrained modes. Moreover, due to the assumption of linear elastic deformations, the modal matrix is invariant. A simpler system of equations is obtained by the orthonormality of the modes of vibration with respect to the mass matrix implies that XT Mff X = I and XT Kff X = , where is a diagonal matrix containing the squares of the flexible body natural frequencies Mr Mrf X Tqr q¨ r gr sr 0 XMf r I XT Tqf w ¨ (15) = XT gf − XT sf − w . γ 0 0 λ qr qf X 0 The vector q¨ r contains the translational and angular accelerations of the fixed-body coordinate systems only. All terms required for Equation (15), related to a finite-element model, are obtained directly from commercial finite-element codes. The solution of the system equations of motion depicted by Equation (13) or (15), can represent 70 or 80% of the total computational time in a general multibody code analysis. A close look at the system equation matrix shows a great number of null elements and zero blocks with fixed size. The sparsity of the system matrix is very high and can be exploited in order to improve computational efficiency. The Gauss elimination scheme needed to solve the system of equations has a high number of arithmetic operations, proportional to the cube of the matrix dimension. With the use of a sparse matrix solver [8], the number of arithmetic operations is greatly reduced, becoming almost proportional to the number of nonzero elements in the system matrix. The use of a sparse matrix-solver performs the Gauss elimination only on the nonzero elements of the matrix. Here, the Markowitz scheme with full pivot search is employed, reducing the number of fill-ins needed and showing good numerical stability [9]. 3. Road Vehicle Comfort Automobiles travel at various speeds, experiencing a large spectrum of vibrations. Due to road irregularities and vehicle vibration, occupants are subjected to stress, causing discomfort. Such vibrations are transmitted to the vehicle passengers as a result of their contact with the seat, steering wheel, and foot rest. Therefore, occupant comfort is directly associated with the ride performance of a road vehicle, making this an important criteria in the design and construction of a car. The term ride is commonly used in reference to tactile and visual vibrations (0– 25 Hz). The 25 Hz boundary point is approximately the upper frequency of the common vibrations in road vehicles. Among the multiple sources leading to vehicle ride vibration, road roughness is the most important. Vibrations with frequencies higher than 25 Hz, classified as noise, are not used in the evaluation of the ride quality.
120 J. P. C. Gonçalves and J. A. C. Ambr´osio Table 1. Frequency weighting values. Weighting
Frequency f (Hz)
Value
Wb
0.5 < f < 2 2
