Damped Natural Frequencies of a Multiple Degree of Freedom Mechanical System

Question

I was wondering if there are any analytical methods to find damped natural frequencies of a general mechanical system that can be modeled with $$M\vec{\ddot{x}}+C\vec{\dot{x}}+K\vec{x}=0$$ where it is not guaranteed that $M$, $C$, and $K$ are symmetric.

The reason I am asking this is that I would like to find forcing frequencies that would amplify the system's response to a maximum without iterative methods.

To be clear: $M$, $C$, and $K$ are matrices and $x$ is a column vector that contains the displacements of point masses.

Answer 1

1st Method

Let $\vec{x}=\vec{u} \exp (j\omega t)$ where $j=\sqrt{-1}$, so that we have

$$-\mathbf{M}\omega^2\vec{u}\exp (j\omega t)+\mathbf{C}j \omega\vec{u}\exp (j\omega t)+\mathbf{K}\vec{u}\exp (j\omega t)=0 \tag{1}$$

Divide by $\exp (j\omega t)$, noting it is not equal to zero.

$$(-\mathbf{M}\omega^2+\mathbf{C}j \omega+\mathbf{K})\vec{u}=0 \tag{2}$$

Any $\omega$ that would satisfy $\det (-\mathbf{M}\omega^2+\mathbf{C}j \omega+\mathbf{K})=0$ is a damped natural frequency.

2nd Method (better)

Transforming the system into systems of first-order equations, which can quickly be done, will connect the problem with the algebraic eigenvalue problem.

The equation can be represented as

$$\vec{\dot{q}}=A\vec{q} \tag{3}$$

where $$ A=\begin{bmatrix} \mathbf{0} & \mathbf{I}\\ -\mathbf{M}^{-1} \mathbf{K} & -\mathbf{M}^{-1} \mathbf{C} \end{bmatrix} \tag{4} $$

$$ \mathbf{I}=\text{Identitiy Matrix} \tag{5} $$

$$ \vec{q}=\begin{bmatrix} \vec{x}\\ \vec{\dot{x}} \end{bmatrix} \tag{6} $$

Then, let $\vec{q}=\vec{v} \exp (j\omega t)$, so that we have

$$j\omega \vec{v} \exp (j\omega t)=\mathbf{A} \vec{v} \exp (j\omega t) \tag{7}$$

Divide by $\exp (j\omega t)$, noting it is not equal to zero.

$$j\omega \vec{v}=\mathbf{A} \vec{v} \tag{8}$$

This is precisely the statement of the algebraic eigenvalue problem, and the damped natural frequencies are the eigenvalues of $\mathbf{A}$ divided by $j$; i.e., any $\omega$ that would satisfy Equation 8 is a damped natural frequency.

Thanks @Lutz Lehmann, for warning me about an error.

After solving the eigenvalue problem, it turns out $\omega$ is a complex number. The real part of each $\omega$ turns out to be a damped natural frequency, and the imaginary part turns out to be equal to $\zeta_i \omega_{n,i}$, where $\zeta_i$ is a corresponding damping ratio and $\omega_{n, i}$ is a corresponding natural frequency.