2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The system of coupled second-order differential equations is of the form:

\(M\ddot{u}+C\dot{u}+Ku=F\)

with \(M=\left[\begin{array}{ccccc}10& & & & \\ & 10& & & \\ & & \mathrm{.}& & \\ & & & & 10\end{array}\right]\) \(C=50\left[\begin{array}{cccccc}2& -1& & & & \\ -1& 2& -1& & & \\ & -1& 2& \mathrm{.}& & \\ & & \mathrm{.}& \mathrm{.}& \mathrm{.}& \\ & & & \mathrm{.}& \mathrm{.}& -1\\ & & & & -1& 2\end{array}\right]\)

\(K={10}^{+5}\left[\begin{array}{cccccc}2& -1& & & & \\ -1& 2& -1& & & \\ & -1& 2& \mathrm{.}& & \\ & & \mathrm{.}& \mathrm{.}& \mathrm{.}& \\ & & & \mathrm{.}& \mathrm{.}& -1\\ & & & & -1& 2\end{array}\right]\)

The \(\omega\) solution to \(F={F}_{0}{e}^{j\omega t}({j}^{2}=-1)\) harmonic excitation is of the form \(u={u}_{0}{e}^{j\omega t}\), which leads to: \((K-M{\omega }^{2}+j\omega C){u}_{0}={F}_{0}\)

This system can be solved for any \(\omega\), either directly or by using the modal transformation from the real eigenmodes obtained by the associated conservative system \((K-M{\omega }^{2})\phi =0\).

It admits \(n\) eigensolutions (8 in this case) \({\omega }_{i}^{2}\) and associated vectors \({\phi }_{i}\) grouped together in the spectral matrix \(\Lambda =[{\omega }_{i}^{2}]\) and the modal matrix \(\Phi =[{\phi }_{i}]\).

The modal transformation consists in writing:

_images/Object_13.svg

which leads to:

\(\left[\Lambda -{\omega }^{2}I+j\omega \xi \right]q={}^{t}\Phi {F}_{0}\)

\(I\) is identity,

here \(\xi\) is diagonal \(\xi =\left[{\xi }_{\mathrm{ii}}\right]\) because the damping is proportional \((C=\alpha K)\).

The answer is written as: \({u}_{0}=\underset{i=1}{\overset{n}{\Sigma }}\frac{{\phi }_{i}^{t}{\phi }_{i}}{{\omega }_{i}^{2}-{\omega }^{2}+j\omega {\xi }_{\mathrm{ii}}}{F}_{0}\)

The exact solution is obtained by taking all the proper modes.

We deduce: \(\dot{{u}_{0}}=j\omega {u}_{0}\) and \(\ddot{{u}_{0}}=-{\omega }^{2}{u}_{0}\)

2.2. Benchmark results#

Displacement along \(x\) of point \({P}_{4}\) for some frequencies.

2.3. Uncertainty about the solution#

Semi-analytical solution.

2.4. Bibliographical reference#

    1. PIRANDA: Instructions for using the modal analysis software MODAN - Version 0.2 (1990). Laboratory of Applied Mechanics - University of Franche Comté - Besançon (France).