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


Calculation method used for the reference solution
--------------------------------------------------------

The reference solution is the one given in sheet SDLD02 /89 of the guide VPCS, which presents the calculation method as follows:

The problem leads to the search for the eigenvalues and eigenvectors of:

:math:`(K\mathrm{-}M{\omega }_{i}){\Phi }_{i}\mathrm{=}0`

:math:`K\mathrm{=}\left[\begin{array}{cccccc}k& \mathrm{-}k& & & & \\ \mathrm{-}k& \mathrm{2k}& \mathrm{-}k& & & \\ & & \mathrm{..}& & & \\ & & & \mathrm{-}k& \mathrm{2k}& \mathrm{-}k\\ & & & & \mathrm{-}k& k\end{array}\right]` :math:`M\mathrm{=}\left[\begin{array}{ccccc}0& & & & \\ & m& & & \\ & & \mathrm{..}& & \\ & & & m& \\ & & & & 0\end{array}\right]`

from where:

:math:`{f}_{i}\mathrm{=}\frac{1}{\pi }\sqrt{\frac{k}{m}}\mathrm{cos}(\frac{n+1\mathrm{-}i}{(n+1)}\frac{\pi }{2})`

:math:`i\mathrm{=}\mathrm{1,}\mathrm{2,}\mathrm{...},n`

:math:`n` = number of masses

:math:`{\Phi }_{i}^{t}` calculated by solving the linear system.


Benchmark results
----------------------

8 first natural frequencies and the first and eighth normalized eigenvectors such as:

Test VPCS provides modes that are standardized to :math:`{\Phi }^{t}M\Phi \mathrm{=}10`. Standard modes are presented:


* at the generalized unit mass: :math:`{\Phi }^{t}M\Phi \mathrm{=}1`; the reference components are divided by :math:`\sqrt{10}`,


* to the generalized stiffness, which is the same as dividing the previous components by

    .. image:: images/Object_10.svg
        :width: 18
        :height: 22


    .. _RefImage_Object_10.svg:

,


* to the largest travel component.


Uncertainty about the solution
---------------------------

Analytical solution.


Bibliographical references
---------------------------


1. M. LALANNE, P. BERTHIER, J. DERHAGOPIAN. Mechanics of linear vibrations. Paris: MASSON, 2nd edition, chapter 3, p. 100-101 (1986)