
In our first analyses of MDoF systems in section 3, we used direct integration of the equations of motion to determine the response of the system. In those early analyses we assumed arbitrary values of the damping coefficients, purely for demonstration purposes.

Since then, in section 4, we have learned that MDoF dynamic analysis is greatly simplified with a modal superposition approach. However, the damping matrix of a MDoF system is not generally uncoupled by the same modal coordinate transformation that uncouples the mass and stiffness matrices.

So, if we want to use modal superposition in a damped system, we need to take special care to construct a damping matrix that will be diagionalised by modal coordinate transformation. Alternatively, we can uncouple the equations of motion for an equivalent undamped system, and simply apply damping at the modal level before recombining the modal responses. This approach of directly applying modal damping is actually quite common and a more intuitive approach since it is difficult to relate frictional damping to a physical property of the structure.

However, if we cannot use modal superposition, for example if the structural behaviour is non-linear, we must directly solve the equations of motion. In which case, it will be helpful if we can specify a damping matrix that would yield specified modal damping ratios. Fundamentally, it is easier to specify a damping ratio based on past empirical observations, than it is to specify a damping coefficient based on a property of the structure.

In this section we'll work out how to **derive a damping matrix that will yield specific levels of modal damping in one or more vibration modes** and be diagionalised by modal coordinate transformation.

In the final lecture of this section, we'll compare direct integration with modal superposition for a damped system, confirming the validity of our derived damping matrix.

### Mass proportional damping

If we use the mass matrix as a basis for developing our damping matrix, the resulting damping matrix will be diagonalised by the same means as the underlying mass matrix. This is the first approach we'll take to building a suitable damping matrix and is referred to as **mass proportional damping**. We can start by assuming a damping matrix proportional to the mass matrix,

$$
\begin{bmatrix}
C
\end{bmatrix} = \alpha_0
\begin{bmatrix}
M
\end{bmatrix}
\tag{1}
$$

Our objective now is to relate the value of $\alpha_0$ to **one** modal damping ratio, $\xi_i$. This way, we can specify the level of damping in mode $i$ and determine the corresponding damping matrix via the constant $\alpha_0$.

Now we need to premultiply both sides by $\begin{Bmatrix}\phi\end{Bmatrix}_j^T$ and post multiply by $\begin{Bmatrix}\phi\end{Bmatrix}_i$,

$$
\begin{align}
\begin{Bmatrix}
\phi
\end{Bmatrix}_j^T
\begin{bmatrix}
C
\end{bmatrix}
\begin{Bmatrix}
\phi
\end{Bmatrix}_i &= \alpha_0
\begin{Bmatrix}
\phi
\end{Bmatrix}_j^T
\begin{bmatrix}
M
\end{bmatrix}
\begin{Bmatrix}
\phi
\end{Bmatrix}_i\\
\tag{3}
&= 0\:\:(i\neq j)
\end{align}
\tag{2}
$$

The damping ratio for mode $i$ is therefore,

$$
\begin{Bmatrix}
\phi
\end{Bmatrix}_i^T
\begin{bmatrix}
C
\end{bmatrix}
\begin{Bmatrix}
\phi
\end{Bmatrix}_i =c_i = 2\xi_i\omega_i
\tag{4}
$$

or,

$$
\alpha_0
\begin{Bmatrix}
\phi
\end{Bmatrix}_i^T
\begin{bmatrix}
M
\end{bmatrix}
\begin{Bmatrix}
\phi
\end{Bmatrix}_i = 2\xi_i\omega_i
\tag{5}
$$

If the modal matrix has been mass-normalised, $\begin{Bmatrix}
\phi\end{Bmatrix}_i^T\begin{bmatrix}M\end{bmatrix}\begin{Bmatrix}\phi\end{Bmatrix}_i = 1$, therefore,

$$
\alpha_0 = 2\xi_i\omega_i
\tag{6}
$$

With this equation, we can specify a desired value for $\xi_i$ and determine the required value of $\alpha_0$ to determine the mass proportional damping matrix (which will be decoupled by modal coordinate transformation).

Note that this approach only contained one free variable, $\alpha_0$ and thus we have the ability to specify only one modal damping ratio. The other modes will have damping ratios given by,

$$
\xi_j = \frac{\alpha_0}{2\omega_j} = \xi_i\frac{\omega_i}{\omega_j}
\tag{7}
$$

### Stiffness proportional damping

We can repeat the same process and determine a stiffness proportional damping matrix in much the same way,

$$
\begin{bmatrix}
C
\end{bmatrix} = \alpha_1
\begin{bmatrix}
K
\end{bmatrix}
\tag{8}
$$

Again we premultiply both sides by $\begin{Bmatrix}\phi\end{Bmatrix}_j^T$ and post multiply by $\begin{Bmatrix}\phi\end{Bmatrix}_i$,

$$
\begin{align}
\begin{Bmatrix}
\phi
\end{Bmatrix}_j^T
\begin{bmatrix}
C
\end{bmatrix}
\begin{Bmatrix}
\phi
\end{Bmatrix}_i &= \alpha_1
\begin{Bmatrix}
\phi
\end{Bmatrix}_j^T
\begin{bmatrix}
K
\end{bmatrix}
\begin{Bmatrix}
\phi
\end{Bmatrix}_i
\end{align}
\tag{9}
$$

And by a similar means to above, we have,

$$
\begin{Bmatrix}
\phi
\end{Bmatrix}_i^T
\begin{bmatrix}
C
\end{bmatrix}
\begin{Bmatrix}
\phi
\end{Bmatrix}_i = c_i = 2\xi_i\omega_i = \alpha_1\omega_i^2
\tag{10}
$$

Therefore,

$$
\alpha_1 = \frac{2\xi_i}{\omega_i}
\tag{11}
$$

The stiffness proportional damping matrix obtained using $\alpha_1$ (giving the specified damping $\xi_i$ in mode $i$) will yield the following levels of damping in other modes,

$$
\xi_j = \frac{\omega_j \alpha_1}{2} = \xi_i\frac{\omega_j}{\omega_i}
\tag{12}
$$

### Rayleigh damping

Both mass and stiffness proportional damping will, individually, allow us to specify a single modal damping ratio during the construction of our damping matrix. However, if we build our damping matrix from the linear combination of mass and stiffness matrices, we have two free variables and therefore the ability to specify 2 modal damping ratios. This is known as **Rayleigh Damping**,

$$
\begin{bmatrix}
C
\end{bmatrix} =
\alpha_0
\begin{bmatrix}
M
\end{bmatrix} +
\alpha_1
\begin{bmatrix}
K
\end{bmatrix}
\tag{13}
$$

So, specifying damping ratios for modes $i$ and $j$, we can say,

$$
\begin{Bmatrix}
\phi
\end{Bmatrix}_i^T
\begin{bmatrix}
C
\end{bmatrix}
\begin{Bmatrix}
\phi
\end{Bmatrix}_i = 2\xi_i\omega_i = \alpha_0 + \alpha_1\omega_i^2
\tag{14}
$$

$$
\begin{Bmatrix}
\phi
\end{Bmatrix}_j^T
\begin{bmatrix}
C
\end{bmatrix}
\begin{Bmatrix}
\phi
\end{Bmatrix}_j = 2\xi_j\omega_j = \alpha_0 + \alpha_1\omega_j^2
\tag{15}
$$

In matrix form we have,

$$
\frac{1}{2}
\begin{bmatrix}
\frac{1}{\omega_i} & \omega_i\\
\frac{1}{\omega_j} & \omega_j\\
\end{bmatrix}
\begin{Bmatrix}
\alpha_0\\
\alpha_1
\end{Bmatrix} =
\begin{Bmatrix}
\xi_i\\
\xi_j
\end{Bmatrix}
\tag{16}
$$

which can be solved to obtain the scale factors corresponding to $\xi_i$ and $\xi_j$,

$$
\begin{Bmatrix}
\alpha_0\\
\alpha_1
\end{Bmatrix}=
2
\begin{bmatrix}
\frac{1}{\omega_i} & \omega_i\\
\frac{1}{\omega_j} & \omega_j\\
\end{bmatrix}^{-1}
\begin{Bmatrix}
\xi_i\\
\xi_j
\end{Bmatrix}
\tag{17}
$$

If we specify the same level of damping, $\xi$ in both modes of interest, we can solve for the $\alpha_0$ and $\alpha_1$ as follows,

$$
\alpha_0 = \frac{2\xi\omega_i\omega_j}{\omega_i + \omega_j}
\tag{18}
$$

$$
\alpha_1 = \frac{2\xi}{\omega_i + \omega_j}
\tag{19}
$$


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