Dr. Y.P. Daniel Chang Weidong Zhang Velocity Transformation Based Multi-Body Approach for Vehicle Dynamics Abstract: An automobile is a complex close loop multi-body system, which includes the chassis, front suspension, rear suspension, wheels etc. In such a close loop topology, the chassis is modeled as a rigid body, which is linked to low arms, upper arms, tire rods, and tie bars in SLA suspensions by kinematics joints. The dampers and springs in suspensions are considered as velocity based and position based force elements separately. The magic formula is adopted to describe tire’s slip behavior. Based on the virtual power principle, the complete equations of motion including Langrage multipliers are formulated. This formulation is based on the velocity and acceleration vectors of the center of gravity and the angular velocity and acceleration vectors of each element. Velocity transformation is adopted to build the relationships between the dependent velocities the independent velocities, which consist of translational velocities at the center of gravity and angular velocities of the base body and at the kinematics joints. By using Runge-Kutta method, position variables for base body translation and rotation at kinematics joint can be directly acquired. Euler quaternion four parameters are solved from angular velocity and previous quaternion, and they are used to orientate the base body and spherical joints. In terms of these position variables, position and velocity of the rigid body are easily obtained recursively at a specific time. After a series of computations, the vehicle dynamic behavior time history is eventually reached. Velocity based multi-body approach highly increase computational efficiency, and provide an effective solver for vehicle dynamics and tire’s dynamics studies. Multi-body Vehicle Model Vehicle model sketchMulti-Body Position Relation The center of gravity in each body is evaluated here, each body has 6 DOF, three transnational and three rotational corresponding to the global reference frame. The projection matrices R1,which defines the null space of,can be directly obtained through velocity computations, without the need of forming and factoring the Jacobian matrix. The velocities of a body b can be computed from the velocities of base body, and relative velocities of kinematics Joint. corresponding to the constraint of the Open-chain system by cutting close joint. corresponding to the constraint of the close constraint. Orientation and Recursive Kinematics Euler quaternion 4 parameters, are used to describe the orientation of the base body. The quaternion to the rotation matrics Considering a pair of rigid bodies shown in the left figure, the orientation metrics is represented by: Where are the rotational matrics of body j and k respectively. is the rotational matrics between body k and j when a revolute joint is considered. Velocity Analysis Acceleration Analysis Vehicle Model Start Calculate force from damper and spring Quaternion for base body t>tfinal End