Academic Year 2023/2024 - Teacher: ALESSANDRO CAMMARATA

Expected Learning Outcomes

The first part of the course intends to provide the basic concepts for formulating the dynamic equations of motion of rigid and deformable bodies. All computational aspects for the computer-aided analysis of general multibody systems will be provided. Starting from the kinematic analysis of constrained systems the computational methods in kinematics will be discussed using different formulations. The numerical implementation of several dynamic formulations, with emphasis on the Differential-Algebraic Equations, will be described. The main numerical integration schemes will be also investigated and applied to general multibody systems.

The second part of the course aims to provide the students with the main concepts of vehicle dynamics through a deep analysis of forces that govern their motion and determine handling performance (i.e. acceleration, braking and cornering) and ride-comfort. Theory and applications of suspensions kinematics and compliance are provided to identify main parameters related both to handling and ride-comfort. Basic knowledge and numerical methods for tire modeling are provided too. Principles of Multibody dynamics are used to create specific models to simulate K&C analysis of suspension systems and full vehicle dynamic maneuvers.

Course Structure

Lectures: 28 hours

Exercises: 65 hours

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

Required Prerequisites

Rational Mechanics, Applied Mechanics, Modeling and Simulation of Mechanical Systems

Detailed Course Content

1. – Introduction to multibody systems

2. – Fundamentals of planar and spatial kinematics

  • Rotation matrices, invariants, and parametrization
  • Angular velocity and instant screw
  • Type of Coordinates
  • Rigid Body kinematics

3. - Kinematic analysis

  • Joints and constraints equations
  • The Newton-Raphson algorithm

4. - Fundamentals of planar and spatial dynamics

  • Review of rigid body dynamics
  • Constrained equations of motion and DAE
  • Reaction forces and Lagrange multipliers
  • Inverse dynamics and examples

5. - Direct dynamics

  • The Baumgarte stabilization method
  • Penalty and Augmented Lagrangian formulations
  • Basic concepts on the numerical integration of the equations of motion
  • The Runge-Kutta algorithm and applications
  • The Newmark algorithm

6. - Introduction to vehicle dynamics.

  • Road vehicle’s performance: handling, ride-comfort, safety
  • Forces applied to the vehicle
  • Tyre performance

7. - Longitudinal dynamics

  • Driveline layouts
  • Longitudinal motion at a constant speed, calculation of max speed
  • Acceleration performance: basic equations, maximum acceleration and slip limit
  • Braking performance: basic equations, maximum longitudinal deceleration

8. – Suspension and steering systems

  • Main suspension parameters, kinematics, roll stiffness
  • Kinematics and Compliance analysis (K&C)
  • Multibody simulation of K&C tests

9. – Cornering and handling

  • Single track model: basic equations, steady-state cornering, understeer/oversteer, static margin, force derivatives and vehicle stability, transient dynamics
  • Vertical load transfer, suspension effects on cornering, roll motion and roll stiffness distribution
  • Multibody simulation of vehicle handling

Textbook Information

[1] Nikravesh, P. E. (2007). Planar multibody dynamics: formulation, programming and applications. CRC press

[2] Genta G., Morello L. (2007). The automotive chassis Vol. 1 – Components design; Vol. 2 – System design. Springer.

[3] Shabana, A. A. (2009). Computational dynamics. John Wiley & Sons.

[4] De Jalon, J. G., & Bayo, E. (2012). Kinematic and dynamic simulation of multibody systems: the real-time challenge. Springer Science & Business Media.

[5] Shabana, A. A. (2013). Dynamics of multibody systems. Cambridge university press.

[6] Pennestrì, E. (2001). Dinamica tecnica e computazionale: sistemi lineari (Vol. 2). Casa Editrice Ambrosiana.

[7] Lecture notes.

[8] Jorge Angeles, Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms-Springer International Publishing (2014).

[9] Paulo Flores, Concepts and Formulations for Spatial Multibody Dynamics-Springer International Publishing (2015)

Course Planning

 SubjectsText References
1Rotation matrices, invariants, rotation parameters, screw motion[8] Chapter 2
2Angular velocity, instant screw axis[8] Chapter 3

Learning Assessment

Learning Assessment Procedures

Written exam (3 hours)

Learning assessment may also be carried out on line, should the conditions require it.

Examples of frequently asked questions and / or exercises


  • rotation matrices
  • Kinematics of constrained systems
  • Constrained equations of motion and AED
  • Reaction forces and Lagrange multipliers
  • The stabilization of Baumgarte


  • Tire forces at contact patch and examples 
  • maximum acceleration and deceleration on flat roads 
  • understeer plot and gradient
  • suspension characteristic angles and lengths, numerical examples