Multi-Body Dynamics Modeling and Simulation Analysis of a Vehicle Suspension Based on Graph Theory

Jun Zhang, Xin Li and Renjie Li

(School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China)

Abstract: Multi-body dynamics, relative coordinates and graph theory are combined to analyze the structure of a vehicle suspension. The dynamic equations of the left front suspension system are derived for modeling. First, The pure tire theory model is used as the input criteria of the suspension multibody system dynamic model in order to simulate the suspension K&C characteristics test. Then, it is important to verify the accuracy of this model by comparing and analyzing the experimental data and simulation results. The results show that the model has high precision and can predict the performance of the vehicle. It also provides a new solution for the vehicle dynamic modeling.

Key words: multi-body dynamics; Matlab; suspension; graph theory

With the development of new energy technology, electric vehicles (EVs) have recently emerged and thrived. The differences in structure layout between electric vehicles and traditional vehicles have prompted new requirements for suspension design and development. A suspension system is one of the core systems of the vehicle, and has an important impact on comfortability ,safety and handling of the vehicle. In order to study the effect of suspension design on vehicle performance, a vehicle chassis simulation development platform including a suspension system needs to be established. The key to this platform is to build a vehicle model that can reflect the movement of suspension.

Common classic car models include a linear 2-DOF, 4-DOF, 7-DOF, 14-DOF models, etc[1-3]. These models are sophisticated and provide mathematical models for automotive researchers to study car performance. However, due to avoidance or simplification of the complex movement of the suspension system, these models have little effect on the suspension design and development.

The traditional approach to establishing a model that considers suspension motion is to build a multi-body dynamic model. At present, the research methods for multi-body dynamics are mainly composed of the following methods: vector and analytical mechanics method based on Newton-Euler equations[4], variational method based on the Gaussian minimum bound principle[5], Kane method based on generalized rate[6], spin method[7], and natural coordinate method[8]. However, those solutions are complex.

Co-simulation modeling can be established by combining multi-body dynamics simulation software (such as ADAMS, etc.) and control simulation software MATLAB[9]. However, this method has a large workload and low generality. Moreover, the complicated control simulation is difficult and affects the development cycle due to problems such as interface and packaged hybrid simulation software.

Based on multi-body dynamics theory, establishing models using the Matlab platform grants powerful advantages in control simulation and can overcome the limitations of classical mechanics theory modeling. This article uses the Roberson and Wittenberg (R-W) theory. R-W introduces graph theory into a multi-body system to describe the relationship between different components of a complex system, which makes it easy for computers to identify the ever-changing linkages in a multi-body system and makes it easier to formulate computations. The advantage of graph theory is that characterizing the mutual articulation of the components in a multi-body system is a more intuitive process. Secondly, the velocity and acceleration at the centroid of a series of components can be deduced by recursive method.

In this paper, taking the left front suspension as an example, the method of establishing the multibody dynamic model of the suspension by using multi-body dynamic R-W theory is introduced. Then, the simulation development platform of the vehicle chassis based on the multi-body dynamic model of the suspension is established, and finally verification of the model through simulations and test results.

1 Graph Theory

A complex system of rigid bodies that are specifically connected and interact with each other is called as a multi-body system. In the process of this study, we first consider the deformation of each sub-body, and then consider the multi-body system under study to be a multi-rigid body mechanical system. The method of linking between the rigid bodies is called as the hinge; the rigid body of the hinge joint acts as the adjoining rigid body which is symmetrical to the hinge. Although the actual hinge has a specific shape and quality, for simplification, a geometric point can be used to indicate the position of the hinge, called the hinge point. In kinetic calculations, the quality of the hinge is not considered.

For the application of graph theory in the multi-body system, the following definition is adopted: The vertex of the structure diagram represents a rigid body, denoted asBi(i=1,2,…), and the indexiis the serial number of a rigid body. The arc connecting the vertices represents the hinge, denoted byOj(j=1,2,…), and the indexjis the number of the hinge. The directionality of the arc is to determine which of the adjacent rigid bodies (connected to the hinges) is selected as a reference to determine the relative motion of the other rigid body and also to determine the direction of the interaction between the rigid bodies. This kind of vertex and arc constitute the description of the structural features of the system, called the multi-body system structure diagram. As shown in the structural diagram of the system shown in Fig.1, the directed line segment in the figure represents the articulation relation between rigid bodies, and the objects with directed line segments represent the rigid bodies in the multi-body system.

Fig.1 Multi-body system structure

This paper will take the left front suspension as an example to introduce the application of graph theory in multi-body modeling. The various components in the entire vehicle structural system are assumed to be rigid bodies. The articulation between the components is a rigid connection. For the rubber bushings and hinges, department and other friction will not be considered. The front suspension system in the entire vehicle multi-body system adopts the McPherson independent suspension, as shown in Fig.2. The left front suspension system consists of four rigid bodies: knuckles, struts, lower arm and steering rod. The lower arm and the body are connected by a rotating hinge, and the lower arm and steering knuckle are connected by a ball joint. The upper end of the slide column and the vehicle body are connected by a universal joint. The lower end of the slide column and the steering knuckle are connected by prism hinges. The steering tie rod is connected to the body through a universal joint and is connected to the steering knuckle through a ball joint. The wheel and the knuckle are connected by a rotating hinge.

Fig.2 Left front suspension model

For 1/4 body, front suspension structure topology shown in Fig.3.

Fig.3 Left front suspension topology

As shown in Fig.3, the left front suspension multibody system is a non-tree system with a total of 6 rigid bodies (excluding zero rigid bodies) and 8 hinges. Among them are the left front tireB0, left knucklesB1, left strutsB2, bodyB3, left steering tie rodsB4, left lower A-shaped armsB5. It is therefore necessary to cut the hinge so that the non-tree system becomes a derived tree system where the dotted line represents the severed hinge. For the above non-tree system, lists the system integer function pairs as Tab.1.

From Tab.1, we derive the total correlation matrix of the non-tree system as

(1)

Then derive the path matrix of the derivation tree system and the loop matrix of the non-tree system.

(2)

2 Suspension Model

2.1 Freedom analysis

For the left front suspension derived tree system, the specific form of each hinge in the multi-body system is first established, then the relative degree of freedom between adjacent rigid bodies is determined by the concrete form of the hinge to determine the degree of freedom of the derived tree system. In the derived tree system, the hinge is not considered at first. There are 6 hinges, namely:O1rotating hinges,O2prism hinges,O3universal joints,O4ball hinges,O5ball hinges. The degrees of freedom of each hinge are 1,1,2,3, 3.

Therefore, for the left front suspension multibody system, the number of degrees of freedom existing in the system is 1+1+2+3+3=10. The specific generalized coordinates are as follows:

q=[q1q2q3q4q5]T

(3)

(4)

2.2 Rigid body of the local coordinate system and parameter information

Before the suspension of multi-body system modeling, the local coordinate system needs to be established for each rigid body in the multi-body system to facilitate the mutual conversion of the position coordinates between the rigid bodies and the conversion relative to the inertial coordinate system. In general, the origin of a local coordinate system is established at its rigid body center of mass, and a local coordinate system is established according to its rigid body shape and other rigid body hinge forms.

After the coordinate system is established, the inertial tensor of the rigid body relative to its local coordinate system needs to be measured to facilitate the system modeling.

The relationship between the steering knuckle and the wheel is a rotating hinge. Therefore, a local coordinate system can be established in which the origin of the coordinate system is located at the center of mass, the azimuth is consistent with the tire coordinate system, and the posture is rotated about the axis of the tire coordinate system.

As the hinged relationship between the two is a rotating hinge, the direction of the cosine matrix between the wheel coordinate system and the local coordinate system of the knuckle at the hinge point is

(5)

(6)

The knuckle inertia tensor matrix for the center of mass point is shown above. The unit for inertia is 104kg·mm2. For the convenience of subsequent calculations, an appropriate constant matrix can be chosen to simplify the inertia tensor matrix to a diagonal matrix consisting of three main moments of inertiaJxx,Jyy,Jzzcalled the center main inertia matrix.

The coordinate of the knuckle center of mass in the wheel coordinate system is (14, -625, 7.3). The appropriate constant matrix is selected., the knuckle relative to the mass center point of inertia tensor matrix into the center of the main inertia matrix, as shown in Tab.2.

Tab.2Knuckle center inertia parameters

kg·m2

The constant matrix A1is

(7)

The same can be applied to stroked column, the lower arm, steering tie rod, the body part of the coordinate system and related parameters.

2.3 Derived tree system kinematics equations

For the previous derivation tree system analysis, the generalized coordinate matrix of the derivation tree system and the corresponding vector matrix of the shaft and slip axis are obtained.

(8)

Relative angular velocity and angular acceleration between rigid bodies is derived.

(9)

The absolute angular velocity and angular acceleration of each rigid body in the derived tree system is produced as

(10)

(11)

In Eqs.(10)-(11), the coefficient matrix and array elements are as follows:

(12)

(13)

For the front suspension multi-body system, the body hinge vector and the path vector diagram of the sliding hinge system are established, as shown in Fig.4.

Fig.4 Left front suspension multi-body system body hinge vector and path vector

Fig.4 shows the body hinge vector of each rigid body mass center to each hinge point in the left front suspension system. Expressing the path vector through the body hinge vector can produce the expression of velocity and acceleration at each rigid body center of mass.

Using Fig.4, the body-hinge vector matrix of the left front suspension multi-body system is derived.

By

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Cij=Sijcij,i=0,1,…,n;j=1,2,…n

Then

(14)

The multi-body system vector of the path vector

(15)

Then, expressions of the velocity and acceleration for each rigid body in the left front suspension system are derived.

(16)

(17)

2.4 Cut hinge constraints

In order to facilitate the derivation of the system dynamics equations and the modeling of the program, the non-tree system is cut off as a derivative tree system. The restraining effect on the rigid body of the cutting hinge is expressed by the binding force and the restraining moment.

In the left front suspension system, there are two cutting hingesO19,O20, the hinged forms are rotary hinges and universal joints, which respectively represent the two hinged relationships between ① the tie rod and the vehicle body ② the lower arm and the vehicle body. The virtual cutting of the cutting hinge does not affect the degree of freedom of the actual system. Therefore, constraints need to be added to ensure that the speed and angular velocity of the adjacent rigid bodies associated with the cutting hinge before and after the cutting remain the same.

Since the type of the cut hinge in the left front suspension system is a rotating hinge, there is no relative angular velocity and relative slip velocity of the sliding hinge and the adjacent rigid body.

(18)

(19)

According to the constraints of rotating hinge and universal joint, the constraint of the cutting hinge is deduced.

(20)

2.5 Spring-damper force element constraints

According to the whole body vector and the body hinge vector in Fig.4, the articulation relationship between the steering knuckle and the strut is a prism hinge. According to the characteristics of the spring force, the force and moment acting on the adjacent rigid body by the spring are deduced.

(21)

(22)

According to Fig.4 the shock absorber on the adjacent rigid body and the role of moment are derived through characteristics of the shock absorber the body vector and the body hinge vector.

(23)

(24)

2.6 Non-tree system dynamics equations

The angular velocities, centroid displacements, constraint equations of the cut-off joints and force-element constraint equations of the rigid bodies in the multi-body system have been previously deduced. The dynamics equations of the non-tree system are deduced according to the Lagrange multiplier method.

(25)

where

A=αTmα+βTJβ

(26)

B=αT(Fg-mu1)+βT(Mg-Jσ-ε)+
γTFe+pMa+kFa

(27)

(28)

In Eqs.(26)-(28), each element of the coefficient matrix and the constraint system matrix is defined as follows:

α=-(pT·d+kT)T，β=-(pT)T

(29)

m=diag(m1,m2,m3,m4,m5)

(30)

J=diag(J1,J2,J3,J4,J5)

(31)

Fg=[m1gm2gm3gm4gm5g]T

(32)

Mg=0

(33)

σ=[σ1σ2σ3σ4σ5]T

(34)

ε=[ε1ε2ε3ε4ε5]T=
(ω1×(J1ω1),ω2×(J2ω2),
ω3×(J3ω3),ω4×(J4ω4),ω5×(J5ω5))

(35)

γ=-SeTα+CeT×β

(36)

(37)

(38)

The above matrix symbols have been derived in the previous derivation process, thus completing the left front suspension multi-body dynamics modeling.

2.7 MATLAB simulation model

Due to the characteristics of the non-rigid tire body, the model of the ground tire suspension body should be used to build the left-front suspension non-tree system model. Using the multi-body dynamics model of the suspension, the displacement, velocity and acceleration of the key points in the suspension system are calculated to reflect the force of each rigid body in the suspension system. The rigid bodies are connected to each other or to the vehicle body. The role of the body in this context is to reflect the body’s movement. The specific modeling concept for the left front suspension is shown in Fig.5.

Fig.5 Left front suspension multi-body system modeling ideas

3 Simulation and Results

Suspension K & C features the kinematic suspension and suspension elastic kinematics. Due to the limited test conditions, this paper only provides the data about the variation in positioning parameters of suspension under the vertical loading test. The vertical loading test refers to two rounds of synchronous reciprocating excitation, in which the change in suspension position parameters is studied.

In studying the change in the kinematic parameters of the suspension, the change in characteristic curves depicting the suspension positioning parameters and the wheel jump are primarily investigated. For the suspension positioning parameters, the relationship between the toe angle of the wheel, the camber angle of the wheel, the kingpin inclination angle, the kingpin caster angle and the wheel jump are mainly investigated.

The left front suspension is subjected to the vertical loading test simulation according to the introduction of the K & C characteristics test on the front suspension. The test platform applies a vertical displacement input from -50 mm to 50 mm to the tire. The change in the positioning parameters of the suspension under this input is studied. The results are subsequently compared against the experimental data provided.

Vehicle parameters and tire parameters are shown in Tab.3 and Tab.4.

Test results are shown in Fig.6-Fig.10.

Tab.3 Vehicle structural parameters

Tab.4 Tire parameters

The suspension K & C characteristics test is based on a full load static balance used as a zero point. The test data will include the following: the full load static balance position as a zero, the static suspension under the positioning parameters for differential processing, the test data curve as a whole to shift, observations of the changes occurring. Therefore, before dealing with simula-tion data and test data, the simulation results need to be processed in the same manner as the test data, that is, the overall simulation result must be subtracted from the static value.

Fig.6 Wheel vertical force experimental data and simulation results contrast

Fig.7 Toe angle experimental data and simulation results

Fig.8 Camber test data and simulation results

Fig.9 Kingpin angle test data and simulation results

Fig.10 Caster test data and simulation results

Tab.5 shows the comparison between the simulation results and the experimental data. The comparison reference object is the variation of each positioning parameter. It can be seen from the comparison results shown in Tab.5 that there is a certain deviation between the experimental data and the simulation results. The error rate of the main pin is more than 20%. The error rate of wheel camber is above 10%.The error rate of other parameters is small, which is in an acceptable range.

Through the comparison of the K & C characteristics test data and simulation results, it is evident that the simulation of the suspension positioning parameters (wheel toe angle, wheel camber angle, kingpin inclination and caster angle) and the variation trend in the tire vertical force as well as the experimental data are all basically the same. However, there is a certain deviation between the experimental data and the simulation results with regard to the magnitude of change and the data extrema. The main reasons for the deviation are as follows.

① In the establishment of tire model, using the Gim pure theoretical model, the tire model output considers only the longitudinal, lateral and vertical forces applied to the tire.The self-aligning moment, overturning moment and yaw moment are not considered, resulting in a slight deviation from the actual input to the left-front horsepower multi-body system.

② When establishing the multi-body dynamic model of the left front suspension, all the rigid bodies in the multi-body system are set as rigid

Tab.5 K & C characteristics of suspension test data and simulation results

bodies. The modeling process does not consider the rubber bushings of the elastic and damping characteristics, therefore resulting a certain deviation in the experimental results and conditions.

③ In the analysis of spring-damper pairs, the mass of the elastic element is neglected. To a certain extent, this affects the simulation results, resulting in a deviation between the two.

Notably, the left front suspension multi-body dynamics model is established in this paper. The rigid body and the body of the suspension are rigidly connected. The magnitude of change in factors such as inclination, caster angle and kingpin inclination are reduced more so in the case of the rubber bush connected suspension system as opposed to the suspension with rigid connections. Moreover, the stability and steering ease has a certain degree of improvement as well. In response to this difference, the multi-body dynamics model established in this paper is more in line with the actual tire dynamics with regard to variation trends of and the direction of deviation. Therefore, the accuracy of multi-body dynamics model is verified in this paper.

4 Conclusion

In this paper, the vehicle suspension model is established by using graph theory and the relative coordinate method in multi-body dynamics. The proposed model is different from the traditional 2-DOF, 5-DOF and 7-DOF models. Taking into account the articulation and structure of the various components in the suspension system, the multi-body dynamics model can better reflect the kinematic changes and force characteristics of the suspension under different input conditions, so as to more accurately reflect the changes of vehicle performance. This model provides a more accurate MATLAB simulation platform for considering the influence of suspension motion when studying vehicle handling performance.