Computing Virtual Work: Assumed Modes Method & Generalized Forces

[boeing_737]

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) q_i(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

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

Q_i = $\int$ 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

 

[eljose79]

Dear Yogesh,

Thank you for your question. The principle of virtual work is a powerful tool for analyzing the dynamics of a cantilever beam subjected to a forcing function. In order to calculate the virtual work, we do not need to consider variations in inertial positions. Instead, we can consider variations in the generalized coordinates (q), which represent the deformations of the beam. These variations can be calculated using the assumed modes method, as you have mentioned in your post.

To answer your specific question, if the beam is not fixed and can translate, we can still use the principle of virtual work by considering the variations in the generalized coordinates. The deformations measured with respect to an axis fixed to the root of the beam can still be expressed in terms of the generalized coordinates. Therefore, the virtual work can still be computed in the same way as before.

I hope this helps clarify your understanding of the virtual work in this scenario. Please let me know if you have any further questions or need additional clarification.