Modelling of Torsion Beam Rear Suspension by Using Multibody Method

Transcription

1 Multibody System Dynamics 12: , C 2004 Kluwer Academic Publishers. Printed in the Netherlands. 303 Modelling of Torsion Beam Rear Suspension by Using Multibody Method G. FICHERA, M. LACAGNINA and F. PETRONE Dipartimento di Ingegneria Industriale e Meccanica (DIIM), Facoltà diingegneria, Università dicatania, Viale A. Doria 6, Catania, Italy; gabriele.fichera@diim.unict.it (Received 7 July 2003; accepted in revised form 27 April 2004) Abstract. The multibody systems analysis has become one of the main simulation techniques to calculate the elasto-kinematics characteristics of a car suspension under wheel loads or to realize complex full vehicle models in order to predict the handling performances or the NVH quality. The modelling of torsion beam rear suspensions widely adopted in cars belonging to B or C class presents some problems arising from the structural behaviour of this component. A linear method based on component mode synthesis was used to represent the flexible torsion beam within the multibody model. This kind of approach was compared with a non-linear FE analysis. The elastokinematics analysis of the suspension was performed by using SIMPACK multibody code. The main suspension parameters (toe angle, camber angle, wheelbase and track variation) were calculated by changing wheel travel and loads. Static analyses, involving great displacements, were performed and a different number of modes were considered in the modal condensation of the torsion beam. The results of multibody simulations were compared with those obtained from a non-linear FE model. Different stiffness values of the bushings that connect the torsion beam to the vehicle chassis were taken into account. Key words: passenger car, torsion beam suspension, elasto-kinematic analysis 1. Introduction The elasto-kinematic analysis of vehicle suspension systems is usually performed by means of multibody system simulations (MBS). As a matter of fact, a numerical suspension model must take into account the non-linear effects due to great translational and rotational displacements and to non-linear forces. The multibody model for the elasto-kinematic analysis of a suspension system is typically assembled by rigid bodies since the static deformation of structural components is often negligible in comparison with the deformation of connection elements (springs, bumpstops, bushings, etc.). Rigid bodies cannot represent components like anti-roll bars or specific structural components whose elastic deformations have a relevant influence on elastokinematic characteristics of the suspension. This is the typical case of the torsion beam rear suspension. The torsion beam has to be modelled as a flexible body since

2 304 G. FICHERA ET AL. its torsional deformation has the purpose to uncouple the vertical motions of the wheels connected by the beam. The elastic deformation of the torsion beam, which depends on wheel travels and applied loads, affects the variation of suspension geometric characteristics. A multibody model for the elasto-kinematic analysis of a rear torsion beam suspension was built up by means of SIMPACK code. The torsion beam was introduced inside the model as a flexible body by using a linear approach based on Component Mode Synthesis. The modal reduction was operated starting from a linear finite element (FE) model of the torsion beam. Several wheel travel and static load analyses were performed, varying the stiffness of chassis connection bushings. The results of multibody simulations were compared with those obtained from a non-linear FE model in order to evaluate the possible limits of the multibody model, where the elastic deformation of the torsion beam is based on a linear modal superposition approach. 2. The Finite Element Models First of all, the mesh of the torsion beam was created (Figure 1). The same mesh was used to build both a non-linear FE (ABAQUS) and a linear FE model (NASTRAN). The structure was modelled by means of shell elements and rigid elements in the points where the beam is attached to the chassis and other suspension parts. The attachment points (shown in Figure 1) are: chassis bushing attachments: points 1, 2; springs lower attachments: points 3, 4; shock absorbers lower attachments: points 5, 6; wheel bearing centres: points 7, 8. The non-linear FE model includes the other suspension elements, such as: springs, bumpstops, chassis attachment bushings, wheel bearings and shock Figure 1. Torsion beam mesh with shell and rigid elements.

3 MODELLING OF TORSION BEAM REAR SUSPENSION BY USING MULTIBODY METHOD 305 Table I. Eigenvalues from free-free modal analysis. Mode no. Frequency (Hz) absorbers elastic forces. It also includes the static subcases to run the elastokinematic analyses. These subcases consist of the commands which assign vertical displacements or applied loads to the wheel centres. A free-free modal analysis of the structure was performed by means of FE linear model. The first ten eigenvalues (excluding rigid body modes with zero frequency) are shown in Table I. The linear FE model was also used to perform the dynamic condensation in order to introduce the torsion beam as a flexible body inside the multibody model. 3. The Multibody Model 3.1. GENERAL DESCRIPTION The multibody model for the elasto-kinematic analysis of the suspension includes: inertial system which represents the vehicle chassis; torsion beam as a flexible body; other suspension parts considered as rigid bodies, such as: shock absorbers, rims and tyre rings; connecting forces: springs, dampers, bumpstops, reboundstops, wheel bearings, chassis attachment bushings and tyre vertical stiffness; testrig which assigns the vertical displacements or applied loads (lateral loads, aligning torques, braking or traction forces) to the wheel centres THE TORSION BEAM AS A FLEXIBLE BODY Aflexible body can be introduced into a SIMPACK multibody model by means of the interface program (FEMBS) with the finite element codes. Starting from the results of a specific FE analysis of the structure, the interface program generates input data for flexible bodies. These data are stored in a so-called standard input data (SID) file in text format.

4 306 G. FICHERA ET AL. The representation of flexible bodies in SIMPACK, like other commercial multibody codes, is based on: floating frame of reference formulation [1 3]: large overall non-linear body motion accompanied by small deformations u(c, t), with c as vector from the body fixed reference frame to any point of the elastic body in its undeformed state; Ritz method (modal approach): elastic displacements u(c, t) are expressed as a linear combination of space-dependent shape functions and time-dependent coordinates: u(c, t) = [Φ(c)]q(t). (1) The equations of motion of a flexible body in a multibody system require information on: location of attachment points and observation points (where markers will be located); rigid body mass properties; mode shapes Φ(c); modal mass, stiffness and damping matrices, coupling matrices with coordinates describing rigid body motion. The calculation of modal mass and stiffness matrices is based on the component mode synthesis method [4 6]. By dividing the displacement vector u of a flexible body into a vector u m including the displacements of the attachment points (master nodes) and a vector u i including the displacements of the remaining nodes (internal nodes), the equations of motion of the flexible body can be written as: [ [M mm ][M mi ) [ ] ](üm [K mm ][K mi ] [M im ] [M ii + ] [K im ] [K ii ] ü i ]( um u i ) = ( Pm 0 ). (2) The displacements of internal nodes can be expressed as linear combinations of the displacements of master nodes and a certain number of generalized degrees of freedom: u i = [K ii ] 1 [K im ]u m + [Φ]y. (3) The generalized degrees of freedom are related to a special modal matrix [ ] = [ui 1,...,uq i ] including a number q of eigenvectors for the virtually constrained full system. The matrix is rectangular (q < i = number of internal DOFs) if the modes related to very high frequencies are excluded (their contribution is often negligible). The modal matrix [Φ] iscalculated by solving the eigenvalue equation obtained by setting to zero the master nodes displacements (u m = 0). The transformation

5 MODELLING OF TORSION BEAM REAR SUSPENSION BY USING MULTIBODY METHOD 307 matrix [T], calculated by using Equation (3), is used to obtain the mass and stiffness matrices of the reduced system: u = ( um u i ) = [T] ( um y ) [ ] [I] [0] [T] = [K ii ] 1 [K im ] [Φ] (4) [M m ] = [T] T [M][T] (5) [K m ] = [T] T [K][T]. (6) The mode shapes and corresponding frequencies of the reduced virtually unconstrained system are calculated by solving the following eigenvalue equation: [[K m ] ω 2 m [Mm ]] u m = 0. (7) The mathematical operations needed for matrix calculations are performed by the FE code. The process is divided into two steps, both performed in one single analysis: 1. The displacements of master nodes u m (collected in the b-set) are set equal to zero and the modal analysis for the full virtually constrained system is performed, in order to calculate the modal matrix [Φ]. The number, q, of significant eigenmodes is chosen in the q-set.a certain number of internal nodes (collected in the c-set) can be chosen as measurement points and points for graphical representations. 2. A modal analysis for the reduced virtually unconstrained system is performed in order to get the eigenvectors [Φ m ] and the corresponding eigenvalues ω m, which should be close to those of the original system. For the torsion beam, the b-set of master nodes includes the attachment points from 1 to 8 (Figure 1) where forces and constraints have to be applied within the multibody model. The c-set contains some internal nodes for the graphical representation of deformations. The number of dynamic DOFs in the q-set was put equal to 16 (= 6 rigid body modes + 10 eigenmodes up to 500 Hz) in order to consider a frequency range up to max = 250 Hz for future dynamic analysis. In Table 2 the eigenvalues of the reduced unconstrained system (excluding rigid body modes) are shown together with those of the original system. The representation of the flexible body must take into account the interactions with the other parts of the multibody system, given by forces and constraints applied to the flexible body. In order to obtain a good representation, it is recommended to include the so-called static modes, which are obtained by applying static loads to

6 308 G. FICHERA ET AL. Table II. Eigenvalues of unconstrained torsion beam. Mode no. Original Reduced frequency (Hz) frequency (Hz) attachment points (master nodes) and by calculating the corresponding deformations. Instead of the static modes approach, an alternative general method is based on the calculation of another particular set of modes: the frequency response modes (FRM). The frequency response modes are calculated by applying dynamic loads p m (depending harmonically on time with a given excitation frequency 0 )tothe attachment points: [ [K m ] 2 0 [Mm ] ] u p m = p m (8) The FRM approach is valid if the excitation frequency differs from the natural frequencies of the flexible body. The number of FRM depends on the type of forces and constraints applied to the flexible body. The combination of eigenmodes and frequency response modes requires some additional transformations in order to decouple the frequency response modes with respect to rigid body modes, to eigenmodes and to each other [7]. The calculation of FRM and required additional transformations are performed directly by FEMBS. The major advantage of FRM approach is just given by the opportunity to change the boundary conditions without performing new FE runs. Of course, the set of master nodes has to include all the points where forces can be applied. The dynamic loads are defined in specific load cases where modulus and direction of the applied forces (or moments) are chosen for the desired attachment points. Concerning the torsion beam, 36 different load cases were created in order to calculate FRM. The load cases for chassis bushing attachments and wheel bearing centres (nodes 1, 2, 7, 8) consist of 3 unit forces and 3 unit torques applied independently to each node in all directions of the inertial system. The load cases for spring and shock absorbers attachments (nodes 3, 4, 5, 6) consist of 3 unit forces only. An excitation frequency of 1 Hz was chosen for FRM calculation.

7 MODELLING OF TORSION BEAM REAR SUSPENSION BY USING MULTIBODY METHOD 309 Figure 2. Full multibody model. In conclusion, the damping of the flexible body was introduced by considering a natural damping for each mode: r ii = 2 h i kii m ii (9) where k ii and m ii are the stiffness and the mass related to generic mode number i. The value of h i was put equal to 0.03 for each mode, which is a typical value for steel structures THE FULL MODEL FOR THE ELASTO-KINEMATIC ANALYSIS OF THE SUSPENSION After the torsion beam was implemented as a flexible body, the other elements of the suspension model were created: shock absorbers, springs, bumpstops, reboundstops, wheel bearings and chassis attachment bushings. At first, the bushings were assimilated to uni-ball joints with very high translational stiffness (K = 50,000 N/mm) and zero rotational stiffness in all directions. This choice was operated in order to evaluate the effect of torsion beam deformation on suspension parameters since the bushing deformations are negligible. The bumpstops have also constant stiffness. In a second time, the bushings were modelled considering feasible non-linear force-deformation curves with increasing stiffness (Figure 2) and constant rotational stiffness in all directions [8, 9]. The bumpstops have also non-linear force deformation curves. At the end, the testrig for elasto-kinematic analysis was generated. It includes parts, joints and numerical functions to assign the vertical displacements to the wheels or to apply wheel loads. The vertical motion can be assigned to the wheel centres directly or to the tyre contact patch, including the tyre vertical static stiffness. The full model is shown in Figure 2. Two kinds of simulation were submitted: wheel travel analysis and static load analysis. Wheel travel analyses are described below (+=bump travel, =rebound travel):

8 310 G. FICHERA ET AL. parallel wheel travel analysis, z 0 = 0 mm: equal vertical displacements are assigned to both wheel centres in the range ±80 mm from the initial nominal position; opposite wheel travel analysis, z 0 = 0 mm: opposite vertical displacements are assigned to both wheel centres in the range ±80 mm from the initial nominal position; opposite wheel travel analysis, z 0 = 40 mm: opposite vertical displacements are assigned to both wheel centres in the range ±40 mm from 40 mm initial position; opposite wheel travel analysis, z 0 =+40 mm: opposite vertical displacements are assigned to both wheel centres in the range ±40 mm from +40 mm initial position. Static load analyses are described below: lateral load: applied to the contact patch of left wheel, the range is ±4000 N; aligning torque: applied around the vertical axis of left wheel, the range is ±100 Nm; braking force: a longitudinal force is applied to left wheel centre and a reaction torque (= product between the force and loaded tyre radius) is applied to the upright, the range of the force is ±3000 N; driving force: a longitudinal force is applied to left wheel centre, the range of the force is ±3000 N. The main suspension elasto-kinematic characteristics were calculated: vertical load, toe angle, camber angle, half track and wheel base variations. The results of multibody analysis have been compared with non-linear FE analysis. The results obtained from models with linear bushings and bumpstops are shown in Sections from 4.1 to 4.3, while those obtained with non-linear bushings and bumpstops are shown in Sections 5.1 and Elasto-Kinematic Analysis With Linear Bushings and Bumpstops 4.1. PARALLEL WHEEL TRAVEL ANALYSIS During parallel wheel travel there is no torsional deformation in the beam. There are only small bending deformations due to vertical wheel loads, spring and bumpstop forces. As a consequence, the results of multibody analysis are in accordance with the non-linear FE analysis (see Figure 3) OPPOSITE WHEEL TRAVEL ANALYSES During opposite wheel travel, the torsional deformation of the beam occurs: it increases while increasing the vertical travel. As a consequence, the multibody

9 MODELLING OF TORSION BEAM REAR SUSPENSION BY USING MULTIBODY METHOD 311 Figure 3. Parallel wheel travel analysis: z 0 = 0 mm; -----multibody analysis; FE analysis. non-linear analysis is less accurate in the evaluation of toe angle and wheel base variations (see Figure 4). Because of the high stiffness of the bushings, these errors are due to the linear approach used to represent the flexible body inside the multibody model. In particular, the bigger differences in toe angle variation curves are observed when the beam is subjected to higher torsional deformation, that is when the wheels reach the limits of wheel travel. Opposite travel analyses with different initial position (+40, Figure 5 or 40, Figure 6) give better results since the relative vertical displacement of the wheels is lower (±40 mm). The analysis z 0 = 40 mm gives the best results since the bumpstops does not act. The curves of wheel-base variation present large errors, especially when the wheels start to move from the initial nominal position. The multibody model is not able to calculate correctly the longitudinal motions of wheel centres due to the torsional deformation of the beam. As a matter of fact, the curves differ a lot when the maximum vertical displacements are reached. In order to include the multibody suspension model in a full vehicle model for handling or ride-comfort evaluation, the revealed errors do not seem to be critical.

10 312 G. FICHERA ET AL. Figure 4. Opposite wheel travel analysis: z 0 = 0 mm; -----multibody analysis; FE analysis. non-linear Figure 5. Opposite wheel travel analysis: z 0 non-linear FE analysis. = +40 mm; multibody analysis; As a matter of fact, the maximum values of vertical displacements are reached only at maximum lateral acceleration, when the vehicle behaviour is mainly related to the tyre non-linear behaviour. The errors in the calculation of wheel base variation could affect only those ride-comfort analyses which involve opposite displacements of the wheels, such as asymmetrical obstacles or holes.

11 MODELLING OF TORSION BEAM REAR SUSPENSION BY USING MULTIBODY METHOD 313 Figure 6. Opposite wheel travel analysis: z 0 non-linear FE analysis. = 40 mm; multibody analysis; Figure 7. Static load analysis: lateral load; multibody analysis; analysis. non-linear FE 4.3. STATIC LOAD ANALYSES The results of static load analyses confirmed that the multibody model is in accordance with the non-linear FE model. The curves are always superimposed except for toe angle variation with lateral load (Figure 7). 5. Elasto-Kinematic Analysis with Non-Linear Bushings and Bumpstops The following results were obtained by taking into account feasible non-linear force deformation curves both for chassis connection bushings and suspension bumpstops. The results obtained from parallel wheel travel analysis reveal a perfect accordance between multibody model and non-linear FE model, as shown in Section 4.1. The graphics have been omitted, since all the curves are superimposed OPPOSITE WHEEL TRAVEL ANALYSES By decreasing the stiffness of connection bushings, the multibody model gives better results in the calculation of toe angle variation (Figures 8 10). The bigger

12 314 G. FICHERA ET AL. Figure 8. Opposite wheel travel analysis: z 0 = 0 mm; -----multibody analysis; FE analysis. non-linear Figure 9. Opposite wheel travel analysis: z 0 non-linear FE analysis. = +40 mm; multibody analysis; Figure 10. Opposite wheel travel analysis: z 0 = 40 mm; multibody analysis; non-linear FE analysis. errors remain in the calculation of wheel-base variation; that confirms the limits of the multibody model to calculate accurately the longitudinal motion of wheel centres when the torsional deformation of the beam increases. In particular, the left wheel centre seems to move rearward (positive wheel-base variation in Figure 8) because of the reduced stiffness of attachment bushings along the longitudinal axis STATIC LOAD ANALYSES The results of static load analyses show higher value of displacements, since the reduced stiffness of bushings. The multibody model and the non-linear FE model

13 MODELLING OF TORSION BEAM REAR SUSPENSION BY USING MULTIBODY METHOD 315 Figure 11. Static load analysis: lateral load; multibody analysis; analysis. non-linear FE are still in good accordance. The only curves which present appreciable differences are still those relating to lateral load analysis (Figure 11). 6. Conclusions The elasto-kinematic analysis of a torsion beam rear suspension was performed by means of multibody systems simulation. The torsion beam was implemented inside the multibody model as a flexible body by using a linear approach based on Component Mode Synthesis. Only the first 10 eigenmodes of the unconstrained reduced structure were considered. Frequency response modes were also calculated to take into account forces and torques applied to attachment points. Wheel travel analysis involving great displacements and static load analysis were performed. The results of multibody simulations were compared with those obtained from a non-linear FE model. The aim of the comparison was to evaluate the possible limits of multibody approach which uses a linear modal superposition method to calculate elastic deformations. At first, the chassis connection bushings were assimilated to uni-ball joints with very high translational stiffness. In this case, all the variations of suspension parameters depend on the torsion beam structural behaviour. The results of simulations have shown that the multibody model gives bad results only for the wheel base variation with opposite wheel travel. The toe angle variation presents also some errors that become relevant only when the maximum values of opposite wheel travels or lateral loads are reached. By considering more feasible force deformation curves for the bushings and the bumpstops, the toe angle variation curves become quite coincident during wheel travel. On the contrary, the multibody model still gives poor results in the calculation of the wheel base variation with opposite wheel travel. The multibody model is not able to calculate accurately the longitudinal motion of wheel centres when the torsional deformation of the beam increases. In spite of the above mentioned limits, the multibody model needs very short simulation times to run a complete elasto-kinematic analysis of the suspension. So, it is possible to run parametric analyses in order to test different configurations.

14 316 G. FICHERA ET AL. Moreover, the revealed errors do not seem to be crucial if the suspension model will be included in a full multibody vehicle model for handling or ride-comfort analysis. The maximum values of vertical wheel displacements are reached only at maximum lateral acceleration, when the vehicle behaviour is mainly influenced by the tyre non-linear forces. The errors in the calculation of the wheel base variation could affect only ride-comfort analyses which involve opposite displacements of the wheels, such as asymmetrical obstacles or holes. References 1. Schwertassek, R., Wallrapp, O. and Shabana, A.A., Flexible multibody simulation and choice of shape functions, Nonlinear Dynamics 20(4), 1999, Schiehlen, W., Multibody system dynamics Roots and perspectives, Multibody System Dynamics 1, 1997, Shabana, A.A., Flexible multibody dynamics: review of past and recent developments, Multibody System Dynamics 1, 1997, Dietz, S., Vibration and fatigue analysis of vehicle systems using component modes, Fortschritt- Berichte VDI, Reihe 12, Nr. 401, Likins, P.W., Modal method for analysis of free rotations of spacecraft, AIAA Journal 5, 1967, Craig, R.R. and Bampton, M.C., Coupling of substructures for dynamic analysis, AIAA Journal 6, 1968, Dietz, S., Wallrapp, O. and Wiedemann, S., Nodal vs. modal representation in flexible multibody system dynamics, in proceedings of Multibody Dynamics 2003, IDMEC/IST, Lisbon, Portugal, July 1 4, H. Shimatani, Murata, S. and Watanabe, K., Development of torsion beam rear suspension with toe control links, SAE , Kuo E.Y., Testing and characterization of elastomeric bushings for large deflection behaviour, SAE , 1999.