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ES480 - Dynamic Analysis of Mechanical Systems

(15 Credits)


Progress in understanding natural mechanical systems and in designing new machines depends upon an ability to model and analyse them by the methods of classical mechanics. Simplified, planar models can adequately predict the behaviour of many practical systems. However, important classes of systems (in e.g. aerospace, automotive, packaging machinery, robotics, biomechanics) require full analysis in 3D and professional engineers working in these fields must be familiar with the more advanced techniques. This module therefore builds directly on ES3C3, which has developed a firm grounding in planar analysis. It introduces students to methods of describing mathematically kinematic and kinetic behaviour in three-dimensional mechanisms. It provides physical insight into their behaviour and shows how the concepts are applied to real systems in applications drawn from a variety of fields including e.g. robotics and biomechanics.

At the end of the module students should have a sound understanding of the underlying physical principles and of methods of modelling 3D mechanisms. In particular, they should be able to use either Newtonian or Lagrangian mechanics to predict the forces needed to establish a specified motion or the motion that will arise from a set of specified forces, since these are important steps in the design process.

Learning Outcomes

By the end of the module the student should be able to:

  • Identify and apply appropriate co-ordinate systems for the modelling of 3D mechanical systems by Newton's or Lagrange's equations.
  • Perform analytic kinematic analysis on 3D mechanisms, appreciating their sensitivity to design parameters and tolerances.
  • Make simplifying approximations to more complex engineering mechanisms to enable efficient analysis with acceptable predictive accuracy.
  • Use either Newtonian mechanics or Lagrangian mechanics to analyse the forces at joints and external forces to move mechanisms in specified ways (3D kinetic analysis).
  • Predict forces and motions of important special cases of general motion such as gyroscopes.
  • Appreciate the application of the methods referred to above to important engineering systems, such as i,c. engines, robots and bio- mechanisms.


Development of classical mechanics in 3D formulation; accelerating co-ordinate systems; vector and matrix formulations; generalised co-ordinates and Lagrange's equation .

Kinematics of 3D systems; joint constraints; 3D Kutzbach criterion; vector and matrix methods for analysis (esp. accelerations); effect of tolerances; analytical techniques for cams (2D systems).

Formalisation of basic equations of kinetics; general equations of motion for a rigid body, (Euler's equations); forces at joints and constraints; applications in e.g. robotics, biomechanics...; rotation of a rigid body about a fixed axis; multicylinder engines; balancing of V engines, gas & inertia forces, torque curve; rotation of a rigid body about a fixed point (Euler's Angles); gyroscopic forces, precession, nutation, motion of satellites, gyroscopic compass, stabilisation.