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Mathematically, an eigenvalue analysis assumes the modal displacements are infinitesimally small, so the change in stiffness in the cable caused by the vibration is a second order effect which can be ignored. This very-small-amplitude vibration is a good place to start understanding the behaviour of the structure - for example, look at how the frequency ...


3

If the algebraic multiplicity of an eigenvalue is greater than the geometric multiplicity, then the system matrix is not diagonalizable and there are vectors which are not linear combinations of the eigenvectors of the matrix. If you interpret the Lyapunov matrix as a "generalized potential energy function" (but not necessarily the real physical potential ...


3

Karlo mentioned to use the state space of $[q,\dot{q}]^T$ this allows you to write the system as a first order differential equation of state space. If $X$ were the state space, you could write $\dot{X} = f(X)$. As far as the work you need to do, you write that $$ X = \begin{bmatrix} q\\ \dot{q}\end{bmatrix} $$ the derivative of $\frac{dq}{dt}=\dot{q}$ ...


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In the section "The Multiple DOF System" the Comsol document says It has been assumed that the mass matrix normalization of the eigenmodes is used and that the damping matrix can be diagonalized by the eigenmodes. Eigenvectors are assumed to be mass normalized in any mathematical derivation using them, unless somebody wants to be deliberately ...


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I assume you will be doing this in a software, matlab or Abaqus for example, it’s not particularly easy thing to do with your given info, but. Perhaps this example method will help Abaqus Infos at csiamerica Matlab function I am unsure how directly your data types will fit, but these are some links to get you started


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Regarding the different state space formulations. First, M might not be invertible. In many cases it will be invertible, but sometimes it will not be. In your specific case, this would correspond to a gear with zero mass. Obviously that never happens in the real world as all gears would have some finite mass. But there are situations, where the modeling ...


2

For any sinusoidal motion, the $\mathbf{q}$ really represents the motion $\Re(\mathbf{q}\,e^{i \omega t})$ in the time domain. $\mathbf{q}$ is complex because in general, the different elements of the eigenvector have different phase angles. This will be the case for your problem since your first equation seems to contain a gyroscopic term - if the ...


1

The code in your OP seems to be for a 2 DOF system not a 24 DOF system. You could check that by printing the "frequencies" variable from your eigensolution. The reason you are not getting anti-resonances is because you are taking the absolute values in the wrong place. You should be calculating $$X(\omega) = \left|\sum_n \frac{F/M}{\omega_n^2 - \omega^2}\...


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The equation you showed is an ordinary differential equation. It works for solving natural frequencies of discrete systems (ie lumped rigid masses connected by massless springs). Here you have a continuous system. You will need to set up two partial differential equations, one for the cable and one for the beam. The resulting PDE can be discretized into a ...


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You basically want to solve for the homogeneous solutions of the system $$ \textbf{M}\, \ddot{\textbf{q}} + \textbf{D}\, \dot{\textbf{q}} + \textbf{K}\, \textbf{q} = \textbf{0}. $$ It can be shown that each solution will be of the form $$ \textbf{q}_i(t) = \textbf{u}_i\,e^{\mu_i\,t}, $$ where $\textbf{u}_i$ is a vector of the same dimension as $\textbf{q}...


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Your "wall" would be referred as a "plate" in the technical literature. The eigenvalue problem for a plate is significantly more complicated than a beam, but similar ideas. I would suggest that you look for a book on "vibrations of continuous systems", for example: http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471771716.html There are also ...


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From my understanding, mass participation for each mode is determined by the general mass and the eigenvector of the mode, and eigenvector of the mode usually consists of contribution from all the $x$, $y$ and $z$ direction, so how can one tell whether the mode is contributed to lateral or vertical direction? That is correct in general, but for a ...


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