2. Benchmark solution#

2.1. 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:

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

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

from where:

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

\(i\mathrm{=}\mathrm{1,}\mathrm{2,}\mathrm{...},n\)

\(n\) = number of masses

\({\Phi }_{i}^{t}\) calculated by solving the linear system.

2.2. Benchmark results#

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

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

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

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

    _images/Object_10.svg

,

  • to the largest travel component.

2.3. Uncertainty about the solution#

Analytical solution.

2.4. Bibliographical references#

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