
We saw in the last lecture that we could specify the damping for up to two modes with a combination of mass and stiffness proportional damping, so-called Rayleigh damping. For many practical purposes, this is sufficient since we've also seen that the vibration response of MDoF systems is often dominated by one or two modes.

However, it would nevertheless be helpful to have a method for determining a damping matrix which yielded specified levels of damping in all $N$ modes of an NDoF system. This is what we will focus on in this lecture.

Consider an NDoF system with modal damping specified in each mode, $\xi_1$, $\xi_2$,...,$\xi_N$. Upon transformation into modal coordinates, such a system will have a transformed damping matrix $\begin{bmatrix}\hat{C}\end{bmatrix}$ given by,

$$
\begin{bmatrix}\hat{C}\end{bmatrix} =
\begin{bmatrix}
2\xi_1\omega_1 &0&\ldots &0\\
0 & 2\xi_2\omega_2 &\ldots &0\\
\ldots & \ldots &\ldots &0\\
0 & 0 &\ldots &2\xi_N\omega_N\\
\end{bmatrix}
\tag{1}
$$

If we now imagine that this transformed matrix is obtained from the _original_ damping matrix using the coordinate transformation,

$$
\begin{bmatrix}
\Phi
\end{bmatrix}^T
\begin{bmatrix}
C
\end{bmatrix}
\begin{bmatrix}
\Phi
\end{bmatrix} = \begin{bmatrix}\hat{C}\end{bmatrix}
\tag{2}
$$

The task now is to work backwards to obtain the matrix $\begin{bmatrix}C\end{bmatrix}$. Rearranging,

$$
\begin{bmatrix}
C
\end{bmatrix} =
\left(\begin{bmatrix}
\Phi
\end{bmatrix}^T\right)^{-1}
\begin{bmatrix}
\hat{C}
\end{bmatrix}
\begin{bmatrix}
\Phi
\end{bmatrix}^{-1}
\tag{3}
$$

From the orthogonality expression we have seen previously,

$$
\begin{bmatrix}
\Phi
\end{bmatrix}^T
\begin{bmatrix}
M
\end{bmatrix}
\begin{bmatrix}
\Phi
\end{bmatrix} =
\begin{bmatrix}I\end{bmatrix}
\tag{4}
$$

we can obtain the following expressions,

$$
\left(\begin{bmatrix}
\Phi
\end{bmatrix}^T\right)^{-1} =
\begin{bmatrix}
M
\end{bmatrix}
\begin{bmatrix}
\Phi
\end{bmatrix}
\tag{5}
$$

$$
\begin{bmatrix}
\Phi
\end{bmatrix}^{-1} =
\begin{bmatrix}
\Phi
\end{bmatrix}^T
\begin{bmatrix}
M
\end{bmatrix}
\tag{6}
$$

Using the two preceding expressions, we can obtain the following expression for the damping matrix,

$$
\begin{bmatrix}
C
\end{bmatrix} =
\begin{bmatrix}
M
\end{bmatrix}
\begin{bmatrix}
\Phi
\end{bmatrix}
\begin{bmatrix}
\hat{C}
\end{bmatrix}
\begin{bmatrix}
\Phi
\end{bmatrix}^T
\begin{bmatrix}
M
\end{bmatrix}
\tag{7}
$$

Due to the diagonal nature of $\begin{bmatrix}\hat{C}\end{bmatrix}$, we can represent this in the alternative summation form,

$$
\begin{bmatrix}
C
\end{bmatrix} =
\begin{bmatrix}M\end{bmatrix}
\left(\sum_{n=1}^N 2\xi_n\omega_n
\begin{Bmatrix}\phi\end{Bmatrix}_n\begin{Bmatrix}\phi\end{Bmatrix}_n^T
\right)
\begin{bmatrix}M\end{bmatrix}
\tag{8}
$$

A term must be included in the summation for each mode. Omitting a term is equivalent to setting the damping for that mode to zero. This approach gives us maximum flexibility to determine a damping matrix which results in known damping levels in each mode.


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