# 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, .

## Example

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: