Prerequisites
If you are not familiar with this subject you may like to look at the following pages first:
Lagrangian methods
This is a method for analysing mechanical systems based on energy conservation. For many types of mechanical system this method will allow its behavior to be defined with fewer equations than would otherwise be required.
The Lagrangian for classical mechanics is taken to be the difference between the kinetic energy and the potential energy.
This is known as the 'Action' (lookup: principle of least action).
Constraints
A mechanical system may be made up of particles or solid bodies. If unconstraind then each particle has 3 degrees of freedom (x,y and z coordinates) and each solid body has 6 degrees of freedom (3 for position and 3 for orientation) .
However in a mechanism the objects are constraind in various ways so that the position/orientation of each object depends on the other objects in the system. These constrains will therefore allow us to greatly simplify the equations describing the system in most cases.
Examples of these contraints are things like:
- Joints - limits how one object can move relative to another.
- Gears - defines a rotational speed ratio between objects.
- Wheel - rolling contact
Generalised Coordinates, Generalised Forces
Each component is therefore likely to have fewer degrees of freedom than its throretical maximum. It therefore makes sense to define coordinates along the paths that objects are free to move.
Virtual Displacements and Virtual Work
Lagranges equations for a single particle
Lagranges equations for a system of particles
Lagranges equations for rigid bodies
Lagranges equations for a moving coordinate system
D'Alemberts Equation
For a single particle:
m (ax dx + ay dy + az dz ) = Fx dx + Fy dy + Fz dz
where:
- m = mass
- ax,ay,az = second differential (acceleration) in x,y and z dimensions
- dx,dy,dz = partial differential of position in x,y and z dimensions
- Fx,Fy,Fz = force in x,y and z dimensions
