Translational Mechanical Systems

  • Translational systems involve movement in 1 dimension
  • For example, a the suspension in a car going over bumps going up and down
  • System diagrams can be used to represent systems

  • Diagrams include:
    • Masses
    • Springs
    • Dampers


There are element laws to model each of the three elements involved in mechanical systems. They are modelled using two key variables:

  • Force in newtons ()
  • Displacement in meters ()
    • Also sometimes velocity in meters per second ()

When modelling systems, some assumptions are made:

  • Masses are all perfectly rigid
  • Springs and dampers have zero mass
  • All behaviour is assumed to be linear


  • Stores kinetic/potential energy
  • Energy storage is reversible
    • Can put energy in OR take it out

Elemental equation (Newton's second law):

Kinetic energy stored:


  • Stores potential energy
  • Also reversible energy store
    • Can be stretched/compressed

Elemental equation (Hooke's law):

The spring constant k has units . Energy Stored:

In reality, springs are not perfectly linear as per hooke's law, so approximations are made. Any mechanical element that undergoes a change in shape can be described as a stiffness element, and therefore modelled as a spring.


Dampers are used to reduce oscillation and introduce friction into a system.

  • Dissapates energy as heat
  • Non reversible energy transfer
  • Takes energy out of the system

Elemental equation:

B is the damper constant and has units

Interconnection Laws

Compatibility Law

  • Elemental velocities are identical at points of connection

Equilibrium Law

  • Sum of external forces acting on a body equals mass x acceleration
  • All forces acting on a body in equilibrium equals zero

Fictitious/D'alembert Forces

D'alembert principle is an alternative form of Newtons' second law, stating that the force on a body is equal to mass times acceleration: . is the inertial, or fictitious force. When modelling systems, the inertial force always opposes the direction of motion.


Form a differential equation describing the system shown below.

4 forces acting on the mass:

  • Spring:
  • Damper:
  • Inertial/Fictitious force:
  • The force being applied,

The forces all sum to zero: