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In my mechanical vibrations class we studied the method to orthonormalize a set of differential equations by the mass matrix (principle coordinates). This is where you take the matrix of eigenvectors from the un-damped system and normalize it by the mass matrix.

Multiplying the mass matrix by the modal matrix gives:

X'MX = I

And multiplying the stiffness matrix by the modal matrix gives:

X'KX = W^2(ii)

Where W^2(ii) is the matrix of squared natural frequencies.

My question is: is it appropriate to try and orthonormalize the damping matrix in the same fashion?

X'CX = 2*zeta*omega(ii)

Should this transform the system's damping into principle coordinates?

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The one-word answer is "maybe".

In real structures, damping is often nonlinear, and hard to model from first principles. Also, a physically realistic but "arbitrary" damping matrix does not have nice mathematical properties - for example the mode shapes will be complex (i.e. different parts of the structure will vibrate out of phase with each other).

As a consequence, in class (and in many real world situations) you will learn about models of damping that are mathematically convenient but which don't have much basis in physics. Your "modal damping" model based on $2\zeta\omega$ is a good choice for lightly damped structures. Instead of trying to model or measure the $N^2$ terms of the complete damping matrix, you only need to estimate $m$ damping coefficients for the $m$ modes in the frequency range of interest (and often $m \ll N$). You will probably also learn about the Rayleigh damping model, which arbitrarily assumes the damping matrix is of the form $a M^\alpha + b K^\beta$ where $a$, $b$, $\alpha$, $\beta$ are scalar parameters, and most often $\alpha = \beta = 1$.

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