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**n degree of freedom systems**

Having discussed single and two degree of freedom systems, and introduced the concept of generalised forces we can now consider the general case of an n degree of freedom system.

A virtual displacement must be consistent with the constraints on the system. The motion can be described by n independent, generalised co-ordinates, q_{1} q_{2} q_{n} , ,...., . Hence a virtual displacement can be represented by small changes in these co-ordinates:-

Suppose only one co-ordinate, qi ( 1 ≤i ≤n) i 1 is given a small, imaginary displacement, . δ _{qi} As a result every particle in the system will be, in general, displaced a certain amount. The virtual work done will be of the form δW = Q_{i}δ_{q}i where Q_{i} is an expression relating directly to the forces acting on the system. Q_{i} is the generalised force associated with q_{i} .

From the principle of virtual work

Since, δ_{qi} is finite, we get

Q_{i}=0

This must be true for i = 1,2,...,n.

The generalised forces have component parts

1) inertial forces (mass x acceleration)

2) elastic or restraining forces

3) damping forces (energy dissipation)

4) external forces

5) constraint forces

(Noting, as before, that the constraint forces do no virtual work)

Then the equations of motion are:

These are the n equations of motion

We will examine each of these components now, in more detail. The aim is to relate these component forces to the generalised co-ordinates q_{1},q_{2}...q_{n}.

Inertial Forces (See also Handout)

The position of the i^{th} particle of mass, in the system, is, in general, related to the n generalised co-ordinates, and time (if the constraints are independent of time) then the position of the i^{th} particle depends only on the n generalised coordinates. Thus

(1)

Now we suppose that the system is in motion and that we represent the inertial force on the i^{th }particle (using D’Alembert’s Principle) as

(2)

We now give the system an arbitrary virtual displacement – this can be

represented in terms of generalised co-ordinates by δ_{q1} δ_{q2} δ_{qn} , ,..., . The virtual displacement of the Ithparticle can be represented by

(3)

and the virtual work done by the inertia force on the I^{th} particle is simply

(4)

(note that this is a scalar product). From this result we get the total virtual work as

(5)

Using equation (1) we have

(6)

Hence

(7)

and re-arranging

(8)

However, the generalised inertial forces, Q_{j}, are effectively defined by

(9)

Comparing (8) and (9) we have

(10)

It is shown in the handout notes that

(11)

where T is the total KE

(12)

**Elastic Forces**

Consider a simple spring:

For static equilibrium

F_{E} = F_{S}

(external force) = (internal spring force)

Suppose we define the POTENTIAL ENERGY, V, as the work done by the external force to extend the spring a distance χ

external work done =1/2kx^{2} = V

∴ V=f(x)=V(x)− a function of x

Here V=1/2kx^{2}

The work done by the internal spring force, W, is equal and opposite to V

Now consider a small, virtual displacement, δx . Corresponding changes to W and V are as follows:

Comparing with standard form

Generally

Compare with

**Lagrange’s equation**

Suppose that no damping forces are present, and there are no externally applied forces. Then

We have found that

If we collect these results we get

This is LAGRANGE’s EQUATION

If we define

L=T-V

And assume that V does not depend on the q_{i} & ’s, then Lagrange’s equation can be written as:

**Example 1- mass/spring system**

This is a single degree of freedom system. Here

The Lagrange equation is (n=1 so only one equation)

**Example 2-simple pendulum**

Hence

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