Modal Analysis

With the modal analysis, the fundamental vibrational properties of structures or systems can be determined. It shows at which frequencies the structure can vibrate freely and determines the vibration pattern following free vibration for a chosen frequency.

The modal analysis is based on the superposition of eigenmodes (eigenvectors) $\begin{array}{l}\phi_i\end{array}$ , that are scaled with time-dependent factors $\begin{array}{l}y_i (t)\end{array}$ . It is only applicable for linear vibrations, since only then the principle of superposition is valid.

If the damping matrix is diagonalizable, which can, e.g., be obtained by Rayleigh damping or a modal damping approach, the equations of motion become decoupled. Thus, in the case of an MDOF system with $\begin{array}{l}n\end{array}$ degrees of freedom, $\begin{array}{l}n\end{array}$ different and decoupled ordinary differential equations in the form of the SDOF differential equation are obtained. Depending on the load, the solution for each $\begin{array}{l}y_i (t)\end{array}$ can then be found by any of the methods for the forced SDOF system.

Theory

In general the equation of motion for an undamped linear multi degree of freedom system with $\begin{array}{l}n\end{array}$ degrees of freedom is given by:

$\begin{array}{l}\displaystyle \mathbf{M} \ddot{\mathbf{u}} (t) + \mathbf{K} \mathbf{u} (t) = \mathbf{f} (t)\end{array}$

We first consider the unloaded case $\begin{array}{l}\mathbf{f} (t) = \mathbf{0}\end{array}$ , i.e the homogeneous equation of motion of the undamped MDOF system.

(1)	$\begin{array}{l}\displaystyle \mathbf{M} \ddot{\mathbf{u}} (t) + \mathbf{K} \mathbf{u} (t) = \mathbf{0}\end{array}$

In order to solve (1) we use the following approach for displacement $\begin{array}{l}\mathbf{u} (t)\end{array}$ :

(2)	$\begin{array}{l}\displaystyle \mathbf{u} (t) = \pmb{\phi} \mathrm{e}^{\mathrm{i} \omega t}\end{array}$

where $\begin{array}{l}\pmb{\phi}\end{array}$ is the displacement vector and $\begin{array}{l}\omega\end{array}$ the circular frequency. Equation (2) is then entered into (1)

$\begin{array}{l}\displaystyle - \omega^2 \mathbf{M} \pmb{\phi} \mathrm{e}^{\mathrm{i} \omega t} + \mathbf{K} \pmb{\phi} \mathrm{e}^{\mathrm{i} \omega t} = \mathbf{0}\end{array}$

We finally arrive at

(3)	$\begin{array}{l}\displaystyle \left( - \omega^2 \mathbf{M} + \mathbf{K} \right) \pmb{\phi} \mathrm{e}^{\mathrm{i} \omega t} = \mathbf{0}\end{array}$

This equation is satisfied either for $\begin{array}{l}\pmb{\phi} = \pmb{0}\end{array}$ , which is the trivial solution (no vibration), or for:

(4)	$\begin{array}{l}\displaystyle \det \left( - \omega^2 \mathbf{M} + \mathbf{K} \right) = 0\end{array}$

(4) is called characteristic equation. It is a polynomial of order $\begin{array}{l}n\end{array}$ in terms of $\begin{array}{l}\omega^2\end{array}$ . For convenience we can substitute $\begin{array}{l}\lambda = \omega^2\end{array}$ , where $\begin{array}{l}\lambda\end{array}$ is called eigenvalue of the system.

(5)	$\begin{array}{l}\displaystyle \det \left( - \lambda \mathbf{M} + \mathbf{K} \right) = 0\end{array}$

We obtain $\begin{array}{l}n\end{array}$ solutions for $\begin{array}{l}\lambda\end{array}$ from (5), from which we find the circular eigenfrequencies of the system as

$\begin{array}{l}\displaystyle \omega_i = \sqrt{\lambda_i} \, , \enspace i = 1,\ldots,n\end{array}$

Each circular eigenfrequency $\begin{array}{l}\omega_i\end{array}$ has a corresponding eigenvector $\begin{array}{l}\pmb{\phi}_i\end{array}$ . The $\begin{array}{l}n\end{array}$ circular eigenfrequencies $\begin{array}{l}\omega_i\end{array}$ and eigenvectors $\begin{array}{l}\pmb{\phi_i}\end{array}$ are also called eigensolutions of the vibration problem. In order to determine the eigenvector, we insert $\begin{array}{l}\omega_i\end{array}$ in . By this, we obtain a singular system of equations for the unknown eigenvector.

$\begin{array}{l}\displaystyle \left( - \omega_i^2 \mathbf{M} + \mathbf{K} \right) \pmb{\phi}_i = \mathbf{0}\end{array}$

The scaling of the eigenvectors is arbitrary. For the remainder, we assume that the eigenvectors are normalized with respect to the system's mass, i.e.

$\begin{array}{l}\displaystyle \pmb{\phi}_i^T \mathbf{M} \pmb{\phi}_i = 1 \, .\end{array}$

We now collect the set of eigenvectors in the modal matrix

(6)	$\begin{array}{l}\displaystyle \pmb{\Phi} = \left[ \pmb{\phi}_1, \pmb{\phi}_2, \ldots, \pmb{\phi}_n \right]\end{array}$

Eigenvalues and eigenvectors have a physical meaning for the system: The eigenvalues are the squared circular eigenfrequencies of the system. A system vibrating at one of its eigenfrequencies is resonant. The eigenvectors are the mode shapes at their corresponding eigenfrequency. Each entry of the eigenvector corresponds to the value of the respective degree of freedom.

It can be shown that the mass-normalized eigenvectors obey the following conditions:

$\begin{array}{l}\displaystyle \pmb{\phi}_i^T \mathbf{M} \pmb{\phi}_j = \delta_{ij}\end{array}$

and

$\begin{array}{l}\displaystyle \pmb{\phi}_i^T \mathbf{K} \pmb{\phi}_j = \omega_i^2 \delta_{ij}\end{array}$

where $\begin{array}{l}\delta_{ij}\end{array}$ is Kronecker's symbol:

$\begin{array}{l}\delta_{ij} = \begin{cases} 1 & \text{for} \enspace i = j \\ 0 & \text{for} \enspace i \neq j \end{cases}\end{array}$

Thus, we can state that the eigenvectors are orthogonal with respect to the mass and stiffness matrix.

The above relations in matrix notation read:

$\begin{gather} \pmb{\Phi}^T \mathbf{M} \pmb{\Phi} = \pmb{I} \\ \pmb{\Phi}^T \mathbf{K} \pmb{\Phi} = \pmb{\Lambda} \end{gather}$

Damped modal analysis

We now consider the equation of motion for an damped linear multi degree of freedom system with $\begin{array}{l}n\end{array}$ degrees of freedom. It is given by:

$\begin{array}{l}\displaystyle \mathbf{M} \ddot{\mathbf{u}} (t) + \mathbf{C} \dot{\mathbf{u}} (t) + \mathbf{K} \mathbf{u} (t) = \mathbf{f} (t)\end{array}$

For an arbitrary damping matrix $\begin{array}{l}\mathbf{C}\neq 0\end{array}$ , the transformation

$\begin{array}{l}\displaystyle \mathbf{C^*}=\pmb{\Phi}^T\mathbf{C}\pmb{\Phi}\end{array}$

using the undamped eigenmodes does not yield a diagonal generalized damping matrix $\begin{array}{l}\mathbf{C^*}\end{array}$ and the $\begin{array}{l}n\end{array}$ equations of motion do not decouple.

In this case, different solution strategies exist:

Solve for an arbitrary damping matrix by solving the quadratic eigenvalue problem
Assume a diagonalizable damping matrix

Rigid body modes

For certain systems we obtain eigenvalues $\begin{array}{l}\lambda = 0\end{array}$ , which correspond to an eigenfrequency of $\begin{array}{l}\omega = 0\end{array}$ . The eigenvector corresponding to an eigenvalue $\begin{array}{l}\lambda=0\end{array}$ contains the mode shape of a rigid body movement of the system. Let $\begin{array}{l}\phi_r\end{array}$ be a rigid body mode shape, then $\begin{array}{l}\omega\end{array}$ has to be zero in order to satisfy

$\begin{array}{l}\displaystyle \mathbf{K}\phi_r = \omega^2\mathbf{M}\phi_r,\end{array}$

because $\begin{array}{l}\mathbf{K}\phi_r=0\end{array}$ for rigid body movement.

The number of zero eigenvalues is equal to the number of unrestrained displacements or rotations. A system without any supports has six rigid body modes, three displacements along each axis and three rotations around each axis.

Mode superposition

The eigenvectors of the system span a complete and orthogonal basis to represent any dynamic solution. Thus, we can represent the system response $\begin{array}{l}\mathbf{u} (t)\end{array}$ in modal coordinates as a linear combination of the modes

(7)	$\begin{array}{l}\displaystyle \mathbf{u} (t) = \sum_{i=1}^{n} \pmb{\phi}_i q_i (t) = \pmb{\Phi} \pmb{q} (t)\end{array}$

Here, $\begin{array}{l}q_i (t)\end{array}$ are time-dependent amplitudes, that are the coordinates of the system response in the basis spanned by the eigenvectors. We now insert this ansatz into (1) and obtain:

$\begin{array}{l}\displaystyle \mathbf{M} \pmb{\Phi} \ddot{\pmb{q}} (t) + \mathbf{K} \pmb{\Phi} \pmb{q} (t) = 0\end{array}$

Multiplying by $\begin{array}{l}\pmb{\phi}_j^T\end{array}$ from the left yields:

$\begin{array}{l}\displaystyle \underbrace{\pmb{\Phi}^T \mathbf{M} \pmb{\Phi}}_{\pmb{I}} \ddot{\pmb{q}} (t) + \underbrace{\pmb{\Phi}^T \mathbf{K} \pmb{\Phi}}_{\pmb{\Lambda}} \pmb{q} (t) = 0\end{array}$

Under consideration of the orthogonality conditions, we obtain $\begin{array}{l}n\end{array}$ decoupled equations of motion for each eigenmode:

$\begin{array}{l}\displaystyle \ddot{q}_i (t) + \omega_i^2 q_i (t) = 0\end{array}$

The solution for the $\begin{array}{l}i\end{array}$ th mode is given by

$\begin{array}{l}\displaystyle q_i(t) = q_{0i}\cos\left(\omega_i t\right) + \frac{\dot{q}_{0i}}{\omega_i}\sin\left(\omega_i t\right).\end{array}$

$\begin{array}{l}q_{0i}\end{array}$ and $\begin{array}{l}\dot{q}_{0i}\end{array}$ represent the initial conditions for displacement and velocity at $\begin{array}{l}t=0\end{array}$ . They can be transformed into modal coordinates using the orthogonality conditions

$\begin{align} \mathbf{q}_0 &= \pmb{\Phi}^T\mathbf{M}\mathbf{u}_0\\ \mathbf{\dot{q}}_0 &= \pmb{\Phi}^T\mathbf{M}\mathbf{v}_0 \end{align}$

The first four modes of a simply supported beam are shown in the following. This web app shows the first few modes of beams with different boundary conditions and their components.

Examples

Examples and exercises how to apply Modal Analysis can be found here:

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