What are the Benefits of Using Modal Coordinates in Structural Dynamics?

[CivilSigma]

Homework Statement

In structural dynamics of multiple degrees of freedom structures, the solution of the following PDE varies with the respect of the applied load, however in numerous literature I have read, the solution is a combination of modal coordinates and modal shapes:

$$m \ddot v + c \dot v +kv = P(t)$$

The solution to this PDE is:

$$v(t)= \sum_i^\infty \phi_i(t) \cdot \psi_i(t)$$

where phi is the modal coordinate and psi is the modal shape and are obtained from solving the eigenvalue problem of the equation above.

To my understanding a modal coordinate represents an amplitude of the modal shape and the modal shape is the displacement function of unit displacement. Is this correct?

Moreover, I am not clear regarding the following:

1. What is meant by "Modal Coordinate" are we still in the x-y plane or did we leave to another plane?
2. The benefit of introducing "Modal Coordinates" is that we decouple the PDE which allows us to solve 'N' linearly independent equation of motion, and their sum is the true solution. Out of curiosity, what if I did not want to do this, how would I proceed to solve the MDOF equation of motion?

Thank you !

 

[Dr.D]

The modal coordinate is the coefficient for each mode shape in the summation that is the final solution. Your original problem was in the v-t plane, but the modal transformation took you into a different space. Don't spend too much time trying to attach a geometric interpretation to this. Just understand that it works, and it makes life a lot, lot easier.

Regarding the second question, if you don't want to do this, be prepared to spend a lot of time and confusion with a "straight forward" solution.