Benchmark solution
=====================


Calculation method used for the reference solution
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The system of coupled second-order differential equations is of the form:

:math:`M\ddot{u}+C\dot{u}+Ku=F`

with :math:`M=\left[\begin{array}{ccccc}10& & & & \\ & 10& & & \\ & & \mathrm{.}& & \\ & & & & 10\end{array}\right]` :math:`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]`

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

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

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

The modal transformation consists in writing:

.. image:: images/Object_13.svg
    :width: 61
    :height: 19


.. _RefImage_Object_13.svg:

which leads to:

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

:math:`I` is identity,

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

The answer is written as: :math:`{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: :math:`\dot{{u}_{0}}=j\omega {u}_{0}` and :math:`\ddot{{u}_{0}}=-{\omega }^{2}{u}_{0}`


Benchmark results
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Displacement along :math:`x` of point :math:`{P}_{4}` for some frequencies.


Uncertainty about the solution
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Semi-analytical solution.


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