Rotational Mechanical Systems

Dynamic Systems

  • A system is a set of interconnected elements which transfer energy between them
  • In a dynamic system, energy between elements varies with time
  • Systems interact with their environments through:
    • Input
      • System depends on
      • Do no affect environment
    • Output
      • System does not depend on
      • Affects Environment
  • Mathematical models of dynamic systems are used to describe and predict behaviour
  • Models are all, always approximations

Lumped vs Distributed Systems

  • In a lumped system, properties are concentrated at 1 or 2 points in an element
    • For example
      • Inelastic mass, force acts at centre of gravity
      • Massless spring, forces act at either end
    • Modelled as an ODE
    • Time is only independent variable
  • In a distributed system, properties vary throughout an element
    • For example, non-uniform mass
    • Time and position are both independent variables
    • Can be broken down into multiple lumped systems

Linear vs Non-Linear Systems

  • For non-linear systems, model is a non-linear differential equation
  • For linear systems, equation is linear
  • In a linear system, the resultant response of the system caused by two or more input signals is the sum of the responses which would have been caused by each input individually
    • This is not true in non-linear systems

Discrete vs Continuous Models

  • In discrete time systems, model is a difference equation
    • output happens at discrete time steps
  • In continuous systems, model is a differential equation
    • output is a continuous function of the input

Rotational Systems

Rotational systems are modelled using two basic variables:

  • Torque measured in
    • A twisting force
    • Analogous to force in Newtons
  • Angular displacement measured in radians
    • Angular velocity
    • Analogous to displacement in meters

Element Laws

Moment of Inertia

  • Rotational mass about an axis
  • Stores kinetic energy in a reversible form
  • Shown as rotating disc with inertia , units

Elemental equation:

Energy Stored:

The force acts in the opposite direction to the direction the mass is spinning

Rotational Spring

  • Stores potential energy by twisting
  • Reversible energy store
  • Produced torque proportional to the angular displacement at either end of spring

Elemental Equation:

Stored Energy:

Rotational Damper

  • Dissapates energy as heat
  • Non-reversible
  • Energy dissapated angular velocity

Elemental Equation:

Interconnection Laws

Compatibility Law

Connected elements have the same rotational displacement and velocity

Interconnection Law

D'alembert law for rotational systems:

is considered an inertial/fictitious torque, so for a body in equilibrium, .


Form an equation to model the system shown below.

4 torques acting upon the disk:

  • Stiffness element,
  • Friction element,
  • Input torque
  • Inertial force

The forces sum to zero, so: