# 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

## Elements

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

### Mass

• 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:

### Spring

• 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.

### Damper

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.

## Example:

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: