Computer model for vehicle dynamic simulation
worked out by Stanislaw Walczak, PhD Eng.


CarDyn v.1.0 is a software designed for simulation of vehicle dynamics in off-board concept studies. The program consists of 3D vehicle model, a 3-D road model, various maneuver controls and a virtual test driver (driver model).

The vehicle model is essentially the application of a multi-body system structure (MBS) with additional features (elastokinematics, tyre model, ...) and equations of motion, based on a modelling concept by Prof. Rill1. The applied vehicle model, consists of 9 rigid bodies joined by leading, elastic and dumping elements, including:
  • rigid car body - 6 degrees of freedom,
  • front suspension with the steering system - 5 degrees of freedom,
  • rear suspension system - 2 degrees of freedom,
  • car wheels - 1 degree of freedom each wheel,
  • tyre model - 2 half degrees of freedom each tyre.
The semi-empirical tyre model TM-Easy is used to compute stationary and dynamic tyre forces (first order dynamics).

The simulation results can be animated by means of a graphical user interface, which is part of the CarDyn software. An off-line visualisation of simulated vehicle motion is possible. The animation can be recorded and saved. Vehicle dynamics simulations with CarDyn can be used for virtual handling tests to evaluate the de-sign of suspension or steering systems by means of concept variations of parameter studies, and ride comfort analysis. The CarDyn running together with Matlab.

1. Equations of Motion

The equations of motion describing the dynamics of the vehicle model have been obtained by means of multibody formalisms. The state of the multibody system is described by state variables, which are divided into two groups: generalized coordinates and generalized speeds. For constrained systems, some of the degrees of freedom might be eliminated (starting with an unconstrained system of bodies) by holonomic constraints, imposed by geometry. The equations of motion are ordinary differential equations involving the state variables, their derivatives, and known functions of time. These differential equations are commonly classified into two groups: kinematical and dynamical. The kinematical equations are used to compute derivatives of the generalized coordinates, and are developed from the definitions of the state variables. The dynamical equations are used to compute derivatives of the independent speeds (accelerations), and are derived from first principles of the dynamics of rigid bodies. Kinematical equations define derivatives of the generalized coordinates as linear combinations of the generalized speeds. They never include influences of masses, forces, or moments. The kinematical equations are simply:

where: K - is the kinematical matrix.

Dynamical equations are derived from Jourdain's principle (i.e., the virtual power associated with constraint forces and torques must vanish):

where:
- are virtual linear and angular velocities,
- are internal force and moment vectors.

Internal force and moment vector are computed using Newton-Euler equations:


For a system with f dynamical degrees of freedom, a set of scalar equations can be obtained that has the form:

where: M - mass matrix, Q - is a column vector of length f, called the force vector.


- are the linear and angular acceleration remainders.
The linear acceleration and angular acceleration remainder is the part of the acceleration that is put on the right-hand side of the equal sign in the equations of motion, in the force vector.

The above analysis method immediately applies several of the simplification methods:
  • permits the introduction of natural state variables, including generalized speeds that are not derivatives of the generalized coordinates. If there is reason to think that a certain set of variables is in fact optimal, the analyst is free to use that set,
  • potential simplification occurs because non-working forces and moments are never introduced.


Fig. 1. The view of modelled vehicle. Front suspension: McPherson suspension, rear suspension: trailing arm,

2. Results of experimental verification

The experimental test covered the tests:
  • step steering input (open loop),
  • severe double-lane change manoeuvre (closed loop),
The measured parameters were as follows:
  • steering wheel angle,
  • longitudinal velocity,
  • yaw velocity,
  • roll velocity,
  • front wheels steering angles
The results of calculation and road tests are presented in fig.2-4 A good correspondence of the calculation results and experiments were achieved.


Fig. 2. The run of a yaw velocity, simulated (black), measured (red), v = 15 [m/s]


Fig. 3. The run of a yaw velocity, simulated (black), measured (red), v = 20 [m/s]


Fig. 4. The run of a yaw velocity, simulated (black), measured (red), v = 25 [m/s]

3. The dynamical forces acting in the suspensions elements


Fig. 5. The dynamical forces and moments acting on the front suspension


Fig. 6. The dynamical forces and moments acting on the rear suspension


Fig. 7. Force acting in steering rod during the test with step steering input, (v = 15 [m/s]),
E - measured, S - simulated.


Fig. 8. Force acting in steering rod during the test of double lane change ISO manoeuvre
(v = 15 [m/s]), E - measured, S - simulated.

4. Demonstrations

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Mov. 1. Car going over the single obstacle

{mp4}ehamowanie2{/mp4}

Mov. 2. Braking the car with asymmetrical braking torque

{mp4}epodw_24{/mp4}

Mov. 3. Severe double-lane change manoeuvre (closed loop)

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Mov. 4. Vehicle under a sudden crosswind


1Georg Rill: Simulation von Kraftfahrzeugen, Braunschweig/Wiesbaden, Vieweg 1994. ISBN 3 528 8931 8

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