2. Benchmark solution

2.1. Calculation method used for the reference solution

A detailed analytical solution is available in reference [bib2].

Let’s use the following notations:

\(E\)	:	Young’s modulus
\(L\)	:	bar length
\(A\)	:	bar section
\(N\)	:	normal force directed along the \(X\) axis
\(\alpha ,\beta\)	:	Rayleigh damping coefficients
\(\Omega\)	:	excitation frequency

And let’s put

\(r\mathrm{=}\sqrt{\frac{1+{\beta }^{2}\mathrm{/}{\Omega }^{2}}{1+{\alpha }^{2}\mathrm{/}{\Omega }^{2}}}\)

\(k\mathrm{=}p+\mathit{iq}\mathrm{=}\Omega \sqrt{\frac{p}{2E}}\left[\sqrt{r\mathrm{-}\frac{1\mathrm{-}\alpha \beta }{1+{\alpha }^{2}{\Omega }^{2}}}+i\sqrt{r+\frac{1\mathrm{-}\alpha \beta }{1+{\alpha }^{2}{\Omega }^{2}}}\right]\)

The displacement to any point \(M(x)\) is given by:

\(V(x)\mathrm{=}\frac{N}{\mathit{EA}}\frac{A}{(p+\mathit{iq})(1+i\Omega \alpha )}\frac{\mathit{sh}\mathit{px}\mathrm{cos}\mathit{qx}+i\mathit{ch}\mathit{px}\mathrm{sin}\mathit{qx}}{\mathit{ch}L\mathrm{cos}\mathit{qL}+i\mathit{sh}\mathit{pL}\mathrm{sin}\mathit{qL}}\)

	Movement \((m)\)	Speed \((m/s)\)	Acceleration \((m/{s}^{2})\)
Real game	—7.00 10—11	—3.18 10—6	2.76 10—5
Imaginary part	5.07 10—9	—4.40 10—8	—2.00 10—3

2.2. Benchmark results

Fields of movement, speed, and acceleration of the free end of the bar.

2.3. Uncertainty about the solution

Digital solution.

2.4. Bibliographical references

1. 20. KERBER « Harmonic substructuring in Code_Aster », Report EDF, HP‑61/93‑104.

2. 7. ROBERT, Analytical solutions in structural dynamics, Samtech Report No. 121, March 1996.

3. 16. RICHARD, Substructuring methods in Code_Aster, Internal Report EDF - DER, HP-61/92-149.