Vehicle Planar Dynamics

  • Reza N. Jazar


In this chapter we study the planar model of vehicles to examine maneuvering by steering as well as the wheel torque control. The wheel torque and steer angle are the inputs and the longitudinal velocity, lateral velocity, and yaw rate are the main output variables of the planar vehicle dynamics model. The planar vehicle dynamic model is the simplest applied modeling in which we assume the vehicle remains parallel to the ground and has no roll, no pitch, and no bounce motions. The planar motion of vehicles has three degrees of freedom: translation in the x and y directions, and a rotation about the z-axis. The longitudinal velocity vx along the x-axis, the lateral velocity vy along the y-axis, and the yaw rate \(r=\dot {\psi }\) about the z-axis are the outputs of the dynamic equations of motion.

By ignoring the roll motion as well as the lateral load transfer between left and right wheels, we define a simplified two-wheel model for the vehicle.

The four-wheel planar vehicle model is an extension to the two-wheel planar vehicle model to include the lateral weight transfer. The four-wheel planar model provides us with better simulation of drifting vehicles. This model is capable to simulate drift of vehicles as well as simulation of different tire-wheel interaction for all four tires of a vehicle.


Tire force system Vehicle dynamics Vehicle kinematics Bicycle model Four-wheel vehicle model Steady-state dynamics Drift model 


  1. Beatty, M. F. (1986). Principles of engineering mechanics: Kinematics—the geometry of motion (Vol. 1). New York: Plenum Press.CrossRefGoogle Scholar
  2. Bottema, O., & Roth, B. (1979). Theoretical kinematics. Amsterdam: North-Holland Publication.zbMATHGoogle Scholar
  3. Fenton, J. (1996). Handbook of vehicle design analysis. Warrendale, PA: Society of Automotive Engineers International.Google Scholar
  4. Goldstein, H., Poole, C., & Safko, J. (2002). Classical mechanics (3rd ed.). New York: Addison Wesley.zbMATHGoogle Scholar
  5. Jazar, R. N. (2011). Advanced dynamics: Rigid body, multibody, and aerospace applications. New York: Wiley.CrossRefGoogle Scholar
  6. Jazar, R. N. (2017). Vehicle dynamics: Theory and application (3rd ed.). New York: Springer.CrossRefGoogle Scholar
  7. Karnopp, D. (2013). Vehicle dynamics, stability, and control (2nd ed.). London, UK: CRC Press.CrossRefGoogle Scholar
  8. MacMillan, W. D. (1936). Dynamics of rigid bodies. New York: McGraw-Hill.zbMATHGoogle Scholar
  9. Marzbani, H., & Jazar, R. N. (2015). Steady-state vehicle dynamics. In L. Dai & R. N. Jazar (Eds.), Nonlinear approaches in engineering applications (Vol. 3). New York: Springer.Google Scholar
  10. Milliken, W. F., & Milliken, D. L. (1995). Race car vehicle dynamics. Warrendale, PA: SAE.Google Scholar
  11. Milliken, W. F., & Milliken, D. L. (2002). Chassis design. Warrendale, PA: SAE.CrossRefGoogle Scholar
  12. Schiehlen, W. O. (1982). Dynamics of high-speed vehicles. Wien-New York: Springer.CrossRefGoogle Scholar
  13. Yang, S., Chen, L., & Li, S. (2015). Dynamics of vehicle-road coupled system. Berlin: Springer.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Reza N. Jazar
    • 1
  1. 1.Aerospace, Mechanical and Manufacturing EngineeringRMIT UniversityMelbourneAustralia

Personalised recommendations