Universidade Federal do Rio Grande do Sul (UFRGS)
Programa de Pós-Graduação em Engenharia Civil (PPGEC)

PEC00025: Introduction to Vibration Theory¶

Class 11 - Free vibration of multi degree of freedom systems¶

1. Natural vibration modes and frequency
1.1. The general solution for free vibration
1.2. Natural vibration modes and frequencies
1.3. Orthogonality of vibration modes
2. Examples of modal properties assessment
2.1. Example 1: steel plane truss
2.2. Example 2: beam element
2.3. Example 3: experimental 3-dof model
3. Structural response to initial conditions
4. Assignment

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Prof. Marcelo M. Rocha, Dr.techn. (ORCID)
Porto Alegre, RS, Brazil

1. Natural vibration modes and frequencies ¶

1.1. The general solution for free vibration ¶

Once the stiffness and mass matrices are defined for a given structure, the undamped equilibrium matrix equation results to be a set of coupled equilibrium equations, each one for one of the degrees of freedom. In matrix forms it reads:

$$\mathbf{M} \, \ddot{\vec{u}} + \mathbf{K} \, \vec{u} = \vec{F}(t)$$

where $\vec{F}(t)$ is the (time dependent) external loads vector. In case of free vibration we have:

$$\mathbf{M} \, \ddot{\vec{u}} + \mathbf{K} \, \vec{u} = \vec{0}$$

Let us now assume that there is a solution $\vec{u}_k(t)$ such that:

$$\vec{u}_k(t) = u_k(t) \, \vec{\varphi}_k$$

where $\vec{\varphi}_k$ is not time dependent. This assumption may be understood as a separation of time and space dependence of $\vec{u}_k(t)$. Now the acceleration vector results to be:

$$\ddot{\vec{u}}_k(t) = \ddot{u}_k(t) \, \vec{\varphi}_k$$

and the free vibration equation becomes:

$$\ddot{u}_k(t) \, \mathbf{M} \, \vec{\varphi}_k + u_k(t) \, \mathbf{K} \, \vec{\varphi}_k = \vec{0}$$

Premultiplying this equation by $\mathbf{K}^{-1}$ and dividing by $u_k(t)$ results:

$$\frac{\ddot{u}_k(t)}{u_k(t)} \, \mathbf{D} \, \vec{\varphi}_k + \mathbf{I} \vec{\varphi}_k = \vec{0}$$

where $\mathbf{I}$ is the identity matrix and $\mathbf{D} = \mathbf{K}^{-1} \, \mathbf{M}$ is called the system dynamic matrix. Recalling that $\vec{\varphi}_k$ is not time dependent implies that the equation above is only valid if the coefficient of matrix $\mathbf{D}$ is constant in time. We denote this constant quotient as $-\omega_k^2$ and the condition becomes:

$$\ddot{u}_k(t) + \omega_k^2 u_k(t) = 0$$

The solution for this equation has the general form:

$$u_k(t) = u_{k0} \sin \left( \omega_k t + \theta_k \right)$$

which is the same form found for a single degree of freedom system. However, the time function $u_k(t)$ is only part of the solution for $\vec{u}(t)$, corresponding to its time dependent amplitude. There is still the need of finding the time independent vector, $\vec{\varphi}_k$, and the free vibration frequency, $\omega_k$.

1.2. Natural vibration modes and frequencies ¶

The general amplitude solution above implies that the acceleration vector is:

$$\ddot{\vec{u}}_k(t) = -\omega_k^2 u_{k0} \sin \left( \omega_k t + \theta_k \right) \, \vec{\varphi}_k$$

Replacing this result in the free vibration equation and simplifying gives:

$$\mathbf{K} \, \vec{\varphi}_k = \omega_k^2 \, \mathbf{M} \, \vec{\varphi}_k$$

or, alternativelly, with the dynamic matrix:

$$\mathbf{D} \, \vec{\varphi}_k = \lambda_k \, \vec{\varphi}_k$$

with $\lambda_k = 1\,/\,\omega_k^2$. The two equations above represent an eigenvalue-eigenvector problem, which has as many solutions as the matrices order, $N$, which is also the number of system degrees of freedom. Each solution is a pair $\left( \omega_k, \vec{\varphi}_k \right)$ or, alternatively, $\left( \lambda_k, \vec{\varphi}_k \right)$ if the dynamic matrix is used.

The eigenvalues $\omega_k$ are called the natural vibration frequencies of the strutural system, while the eigenvectors $\vec{\varphi}_k$ are called the vibration modes, or modal shapes. It is very important to keep in mind that the modal shapes have no prespecified scale, what means that they can be multiplied by any scale factor, $\alpha$, and still remain solutions for the eigenproblem:

$$\mathbf{K} \, (\alpha \vec{\varphi}_k) = \omega_k^2 \, \mathbf{M} \, (\alpha \vec{\varphi}_k)$$

Numerical algorithms for solving this eigenproblem are available in many environments, including the best models of HP handheld calculators. In Python, they are available in scipy.linalg module and will be used in section 2 for the three examples provided in the previous class.

1.3. Orthogonality of vibration modes ¶

The eigenvectors $\vec{\varphi}_k$ presents the important property of orthogonality with respect to the stiffness and to the mass matrix. This is a direct consequence of their symmetry, as shown in the following. Let us start by considering two vibration modes $i$ and $j$ that are solutions for the eigenproblem:

$$\begin{align*} \mathbf{M} \, \vec{\varphi}_i &= \lambda_i \mathbf{K} \, \vec{\varphi}_i \\ \mathbf{M} \, \vec{\varphi}_j &= \lambda_j \mathbf{K} \, \vec{\varphi}_j \end{align*}$$

Transposing the equation for mode $i$ above and recognizing that $\mathbf{M} = \mathbf{M}^{\intercal}$ and $\mathbf{K} = \mathbf{K}^{\intercal}$ gives:

$$\vec{\varphi}_i^{\intercal} \, \mathbf{M} = \lambda_i \vec{\varphi}_i^{\intercal} \, \mathbf{K}$$

Now, postmultiplying by $\vec{\varphi}_j$ gives:

$$\vec{\varphi}_i^{\intercal} \, \mathbf{M} \, \vec{\varphi}_j = \lambda_i \vec{\varphi}_i^{\intercal} \, \mathbf{K} \, \vec{\varphi}_j$$

On the other hand, the eigenproblem for mode $j$ above can be premultiplied by $\vec{\varphi}_i^{\intercal}$ to give:

$$\vec{\varphi}_i^{\intercal} \, \mathbf{M} \, \vec{\varphi}_j = \lambda_j \vec{\varphi}_i^{\intercal} \, \mathbf{K} \, \vec{\varphi}_j$$

Subtracting this last equation from the previous one results in:

$$(\lambda_i - \lambda_j) \, \vec{\varphi}_i^{\intercal} \, \mathbf{K} \, \vec{\varphi}_j = 0$$

This condition can be satisfied if and only if:

$$\vec{\varphi}_i^{\intercal} \, \mathbf{K} \, \vec{\varphi}_j = 0, \hspace{1cm} {\rm for} \; i \neq j$$

Starting again this demonstration with the $j$ eigenproblem solution leads also to:

$$\vec{\varphi}_i^{\intercal} \, \mathbf{M} \, \vec{\varphi}_j = 0, \hspace{1cm} {\rm for} \; i \neq j$$

These are the two orthogonality conditions for the eigenvectors $\vec{\varphi}_k$. In the next class they will be used to decouple the matrix equilibrium equation into a set of scalar equations, one for each vibration mode. Once this orthogonality condition has been stated, we observe that the eigenvectors $\vec{\varphi}_k$ constitutes a base of independent vectors (of a linear vector space) that can be linearly combined to represent the complete system response as:

$$\vec{u}(t) = \sum_{k = 1}^{N} u_k(t) \, \vec{\varphi}_k = \mathbf{\Phi}\vec{u}_k(t)$$

where $\mathbf{\Phi}$ is the modal matrix, whose columns are the the eigenvectors $\vec{\varphi}_k$.

2. Examples of modal properties assessment ¶

In the following sections, each of the three examples presented in the last class are subjected to a modal analysis and the corresponding solutions, for both natural frequencies and associated vibration modes, are plotted for visualization.

The eigenvalues and eigenvectors are solved with scipy function eig from module linalg. We define a general function to return natural vibration frequencies, and the associated vibration modes, in ascending order:

2.1. Example 1: steel plane truss ¶

For the steel truss presented last class this is done as follows:

The script below shows the results as nodal displacements at the truss top.

2.2. Example 2: beam element ¶

The interpolation functions could not be dumped with pickle, so we must re-create them for visualizing the modal shapes:

Furthermore, the stiffness matrix is not positive definite, for no boundary conditions have been applied so far (it is a bar "floating in space"). It is necessary to restrain at least two degrees of freedom to suppress a free body motion. For instance, to model a cantilever beam we can restrain $u_1 = 0$ and $u_2 = 0$, what implies that the first two rows and two columns of $\mathbf{K}$ and $\mathbf{M}$ can be removed:

Now the eigenvalues problem can be solved:

For the visualization below, the vibration modes are a linear combination of the interpolation functions (for the remaining degrees of freedom), each one multiplied by the resulting eingenvector coordinate.

2.3.Example 3: experimental 3-dof model ¶

For the experimental model we get:

$f_1 = 5.4$Hz	$f_2 = 15.2$Hz	$f_3 = 22.9$Hz

3. Structural response to initial conditions ¶

Let us recall the free vibration solution for mode $k$, previously stated:

$$u_k(t) = u_{k0} \sin \left( \omega_k t + \theta_k \right)$$

As stated before, the complete solution will be a superposition of solutions for all modes:

$$\vec{u}(t) = \sum_{k = 1}^{N} u_k(t) \, \vec{\varphi}_k = \sum_{k = 1}^{N} u_{k0} \sin \left( \omega_k t + \theta_k \right) \, \vec{\varphi}_k$$

where $N$ is the number of degrees of freedom (length of vector $\vec{\varphi}_k$). Deriving the equation above with respect to time gives the corresponding instantaneous velocity:

$$\dot{\vec{u}}(t) = \sum_{k = 1}^{N} u_{k0} \omega_k \cos \left( \omega_k t + \theta_k \right) \, \vec{\varphi}_k$$

Now we provide the initial conditions $\vec{u}_0$ and $\vec{v}_0$ for time $t = 0$:

$$\begin{align*} \vec{u}_0 = \vec{u}(0) &= \sum_{k = 1}^{N} u_{k0} \sin \left( \theta_k \right) \, \vec{\varphi}_k \\ \vec{v}_0 = \dot{\vec{u}}(0) &= \sum_{k = 1}^{N} u_{k0} \omega_k \cos \left( \theta_k \right) \, \vec{\varphi}_k \end{align*}$$

To separate the conditions equation for each mode, we create the following scalar quantities:

$$\begin{align*} \vec{\varphi}_i^{\intercal} \mathbf{M} \vec{u}_0 &= \vec{\varphi}_i^{\intercal} \mathbf{M} \sum_{k = 1}^{N} u_{k0} \sin \left( \theta_k \right) \, \vec{\varphi}_k = u_{i0} \sin \left( \theta_i \right) \; \vec{\varphi}_i^{\intercal} \mathbf{M} \vec{\varphi}_i \\ \vec{\varphi}_i^{\intercal} \mathbf{M} \vec{v}_0 &= \vec{\varphi}_i^{\intercal} \mathbf{M} \sum_{k = 1}^{N} u_{k0} \omega_k \cos \left( \theta_k \right) \, \vec{\varphi}_k = u_{i0} \omega_i \cos \left( \theta_i \right) \, \vec{\varphi}_i^{\intercal} \mathbf{M} \vec{\varphi}_i \end{align*}$$

Dividing the two expressions above yields the phase angle of each modal solution, $\theta_i$,

$$\tan(\theta_i) = \omega_i \, \left( \frac{\vec{\varphi}_i^{\intercal} \mathbf{M} \vec{u}_0}{\vec{\varphi}_i^{\intercal} \mathbf{M} \vec{v}_0} \right)$$

which can be used to calculate the corresponding amplitudes $u_{i0}$:

$$u_{i0} \sin \left( \theta_i \right) = \left( \frac{\vec{\varphi}_i^{\intercal} \mathbf{M} \vec{u}_0}{\vec{\varphi}_i^{\intercal} \mathbf{M} \vec{\varphi}_i} \right)$$

ou:

$$u_{i0} \cos \left( \theta_i \right) = \frac{1}{\omega_i} \left( \frac{\vec{\varphi}_i^{\intercal} \mathbf{M} \vec{v}_0}{\vec{\varphi}_i^{\intercal} \mathbf{M} \vec{\varphi}_i} \right)$$

We recall that the scalar quantities $\vec{\varphi}_i^{\intercal} \mathbf{M} \vec{\varphi}_i$, in the equations above, are the so-called modal masses, $M_i$. Furthermore, observe that special care must be taken for zero initial velocity, what gives infinity for $\tan(\theta_i)$ implying that $\theta_i = \pi/2$.

As an example, let us calculate the free vibration response of the 3-dof experimental model subjected to a small displacement of 5mm in the top mass only. We start by calculating the modal masses, $M_i$ and the scalar quantities $\vec{\varphi}_i^{\intercal} \mathbf{M} \vec{u}_0$ and $\vec{\varphi}_i^{\intercal} \mathbf{M} \vec{v}_0$:

Then we calculate the free vibration properties $u_{i0}$ and $\theta_i$:

Finally we superpose the modal responses and get the nodal displacements in free vibration:

We can see that the system response contains all natural 3 system natural frequencies, as can be confirmed by taking a look at the periodograms:

4. Assignments ¶

1. Desenvolver no FTool um modelo que tenha uma dimensão predominante (p.e. torre, passarela, edifício alto, etc.). O modelo após simplificação deve contar com pelo menos 10 graus de liberdade em deslocamento (vertical ou horizontal). A massa concentrada por grau de liberdade deve ser tal que a frequência fundamental do sistema esteja abaixo de 2Hz e acima de 0.2Hz.
2. Para o modelo anteriormente desenvolvido, calcular as frequências naturais de vibração livre e as formas modais, plotando estes resultados para visualização.
3. Calcular a resposta e vibração livre para um impulso unitário no grau de liberdade de maior amplitude de vibração no primeiro modo.
4. Relatório com deduções, gráficos e resultados (nome do arquivo P2_T2_xxxxxxxx.pdf).

Prazo: 10 de maio de 2021.

Appendix: Save all matrices to be used next class.¶