# Computing virtual work

Hi all,

I've been trying to understand the vibration of a cantilever beam subjected to a forcing function using Lagrange's equation, but have got stuck at the virtual work part. I would appreciate your inputs here.

Using the assumed modes method, the transverse deformation is written as

$\xi$(x,t) = $\sum$$\psi$i(x) qi(x)

If we use 'n' trial functions ($\psi$), then we have 'n' generalized coordinates (q) that we can use to specify the configuration of the beam at any instant of time.

If we have a distributed loading given by f(x,t) [Load/unit length], then the generalized forces associated with the generalized coordinates are written using the principle of virtual work as

Qi = $\frac{\partial {(\delta W)}}{\partial {(\delta q)}}$

Qi = $\int$ f(x,t) $\psi$i(x) dx

My question is : Does the virtual work have to be computed with variation in inertial positions? For example, if the beam is not fixed, but can translate (say up and down), if the deformations are measured wrt an axis fixed to the root of the beam, how would the virtual work be computed in this case?

Thanks
yogesh

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