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**Principle of Virtual Work**

Degrees of Freedom Associated with the concept of the lumped-mass approximation is the idea of the NUMBER OF DEGREES OF FREEDOM. This can be defined as “the number of independent co-ordinates required to specify the configuration of the system”. The word “independent” here implies that there is no fixed relationship between the coordinates, arising from geometric constraints.

**Degrees of Freedom of Special Systems**

A particle in free motion in space has 3 degrees of freedom

particle in free motion in space has 3 degrees of freedom

If we introduce one constraint – e.g. r is fixed then the number of degrees of freedom reduces to 2. note generally: no. of degrees of freedom = no. of co-ordinates –no. of equations of constraint

**Rigid Body**

This has 6 degrees of freedom 3 translation 3 rotation

e.g. for partials P1, P2 and P3 we have 3 x 3 = 9 co-ordinates but the distances between these particles are fixed – for a rigid body – thus there are 3 equations of constraint.

The no. of degrees of freedom = no. of co-ordinates (9) - no. of equations of constraint (3) = 6.

**Formulation of the Equations of Motion**

Two basic approaches:

- application of Newton’s laws of motion to free-body diagrams Disadvantages of Newton’s law approach are that we need to deal with vector quantities – force and displacement. thus we need to resolve in two or three dimensions – choice of method of resolution needs to be made. Also need to introduce all internal forces on free-body diagrams – these usually disappear when the final equation of motion is found.
- use of work with work based approach we deal with scalar quantities – e.g. work – we can develop a routine method – no need to take arbitrary decisions.

**Principle of Virtual Work**

The work done by all the forces acting on a system, during a small virtual displacement is ZERO.

Definition A virtual displacement is a small displacement of the system which is compatible with the geometric constraints.

**e.g.** This is a one-degree of freedom system, only possible movement is a rotation.

work done by P_{1} = P_{1}(- aδθ)

work done by P_{2} = P_{2}(bδθ)

Total work done = P1(- aδθ) + P2(bδθ) = δW

By principle of Virtual Work

δW = 0

therefore:

P_{1} (- aδθ) + P2(bδθ) = 0

- a P1 + bP2 = 0

P1a = P2b

**D’Alembert’s Principle**

Consider a rigid mass, M, with force FA applied

From Newton’s 2^{nd} law of motion

F^{A}= Ma =

or

F^{A}== 0

Now, the term ( − ) can be regarded as a force – we call it an inertial force, and denote it FI – thus

we can then write:

In words – the sum of all forces acting on a body (including the inertial force) is zero – this is a statics principle. In fact all statics principles apply if we include inertial forces, including the Principle of Virtual Work.

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30 videos|72 docs|65 tests

### Test: Conservative Forces

- Test | 15 ques | 30 min
### Introduction to Work and Virtual Work

- Video | 07:51 min
### Test: Work

- Test | 30 ques | 60 min
### Virtual Work and Displacements

- Doc | 1 pages
### Virtual Work and Displacements

- Video | 03:43 min

- Principle of Virtual Work
- Video | 12:08 min