2. Benchmark solution#

2.1. Calculation method used for the reference solution#

First, the natural frequencies \({f}_{i}\) and the associated eigenvectors \({\phi }_{\text{Ni}}\) normalized with respect to the mass matrix are calculated. The generalized response of the mono-excited system is then calculated by analytically solving the Duhamel integral [bib1]. Finally, the displacement relative to point \(D\) is restored on a physical basis.

Natural frequency calculation

The mass and stiffness matrices are as follows:

\(M=\left[\begin{array}{ccc}m& 0& 0\\ 0& m& 0\\ 0& 0& m\end{array}\right]\), \(K=k\left[\begin{array}{ccc}2& -1& 0\\ -1& 2& -1\\ 0& -1& 1\end{array}\right]\)

Natural frequencies \(\omega\) are the solution of equation \(\mathrm{det}[K-{\lambda }^{2}M]=0\), i.e. \({\lambda }^{3}-5{\lambda }^{2}+6\lambda -1=0\) where \(\lambda =\frac{\omega }{{\omega }_{0}}\) and \(\omega =\sqrt{\frac{k}{m}}\).

Calculation of the generalized response of the mono-excited system

\(\gamma (t)={\mathrm{a.t}}^{2}\) with \(a={2.10}^{5}\).

In the absolute coordinate system, the fundamental equation for the dynamics of the undamped mass-spring system is written: \(M\ddot{{X}_{a}}+K{X}_{a}=0\).

Absolute displacement \({X}_{a}\) breaks down into a uniform translational training displacement \({X}_{e}\) and into a relative displacement \({X}_{r}\): \({X}_{a}={X}_{r}+{X}_{e}\).

The equation of motion in the relative coordinate system is then written: \(M\ddot{{X}_{r}}+K{X}_{r}=-M\Psi \ddot{{X}_{s}}=Q\)

with \(\ddot{{X}_{s}}=\gamma (t)={\mathrm{a.t}}^{2}\) and \(\Psi =\left[\begin{array}{}1\\ 1\\ 1\end{array}\right]\) and therefore \(Q={\mathrm{a.t}}^{2}m\left[\begin{array}{}1\\ 1\\ 1\end{array}\right]\).

The equation of motion projected on the basis of dynamic modes normalized with respect to the mass matrix is written as:

\(\ddot{{\alpha }_{i}}(t)+{\omega }_{i}^{2}{\alpha }_{i}(t)=\frac{{\Phi }_{i}^{T}\mathrm{.}M\mathrm{.}\Psi }{{\Phi }_{i}^{T}\mathrm{.}M\mathrm{.}{\Phi }_{i}}\gamma (t)=-{p}_{i}(t)\gamma (t)\).

The answer of this linear system, at instant \(t\), is given by the Duhamel integral:

\(\ddot{{\alpha }_{i}}(t)=\frac{1}{{\omega }_{i}}{\int }_{0}^{t}-{p}_{i}(t)\gamma (t)\mathrm{.}\mathrm{sin}{\omega }_{i}(t-\tau )d\tau =-\frac{{p}_{i}(t)}{{\omega }_{i}}{\int }_{0}^{t}{\mathrm{a.t}}^{2}\mathrm{sin}{\omega }_{i}(t-\tau )d\tau\).

However, according to [bib1], \({\int }_{0}^{t}a\mathrm{.}{t}^{2}\mathrm{sin}{\omega }_{i}(t-\tau )d\tau =\frac{\alpha }{{\omega }_{i}}\left[{t}^{2}+\frac{2}{{\omega }_{i}}(\mathrm{cos}{\omega }_{i}t-1)\right]\).

So \({X}_{r}={\Phi }_{i}\cdot {\alpha }_{i}=-\underset{i}{\Sigma }\frac{a\mathrm{.}{p}_{i}(t)\mathrm{.}{\Phi }_{i}}{{\omega }_{i}^{2}}\left[{t}^{2}+\frac{2}{{\omega }_{i}}(\mathrm{cos}{\omega }_{i}t-1)\right]\).

2.2. Benchmark results#

The three natural frequencies of the system and the relative displacement \({x}_{r}\) at the point \(D\) are taken as reference results, for various times between \(0\) and \(\mathrm{0,1}s\).

2.3. Uncertainty about the solution#

None if we calculate the Duhamel integral analytically [bib1], [bib2].

2.4. Bibliographical references#

  1. J.S. PRZEMIENIECKI: Theory of matrix structural analysis. New York, MacGraw-Hill, 1968, p.351-357

  2. S.P. TIMOSHENKO, D.H. YOUNG and W. WEAVER: Vibrations problems in engineering 4th edition, New York, Wiley & Sons, 1974, pp. 284-321