2. Benchmark solution#
2.1. Calculation method used for the reference solution#
A detailed analytical solution is available in reference [bib1].
The following notations are adopted:
\(E\) |
: |
Young’s modulus |
\(\rho\) |
: |
density |
\(L\) |
: |
bar length |
\(A\) |
: |
bar section |
\(N\) |
: |
normal force directed along the \(X\) axis |
\(\alpha ,\beta\) |
: |
Rayleigh damping coefficients |
We also ask:
\({\omega }_{n}=(\mathrm{2n}-1)\frac{\pi }{2}\) where \(n=\mathrm{1,2}\mathrm{,3},\mathrm{...}\)
\({\varepsilon }_{n}=\frac{1}{2}(\alpha {\omega }_{n}+\beta /{\omega }_{n})\)
The displacement to any point \(M(x)\) is given by:
\(u(x,t)=\frac{\mathrm{Nx}}{\mathrm{EA}}+\frac{\mathrm{8NL}}{{\pi }^{2}\mathrm{EA}}\underset{n=1}{\overset{\infty }{\Sigma }}{(-1)}^{n}\frac{{e}^{-{\omega }_{n}{\varepsilon }_{n}t}}{{(\mathrm{2n}-1)}^{2}}\left\{\mathrm{cos}(\sqrt{1-{\varepsilon }_{n}^{2}}{\omega }_{n}t)+\frac{{\varepsilon }_{n}}{\sqrt{1-{\varepsilon }_{n}^{2}}}\mathrm{sin}(\sqrt{1-{\varepsilon }_{n}^{2}}{\omega }_{n}t)\right\}\)
2.2. Benchmark results#
The values of the displacement fields, speed and acceleration fields of the free end (node \(\mathrm{N10}\)) are valid at time \(t=0.0195s\):
Movement \((m)\) |
Speed \((\mathrm{m.}{s}^{-1})\) |
Acceleration \((\mathrm{m.}{s}^{-2})\) |
|
Calculation without depreciation |
—8.3766x10—7 |
1.6753 x 10—3 |
0 |
Calculation with structural depreciation |
—1.00462x10—6 |
1.20384x10—3 |
—1.21564 |
2.3. Uncertainty about the solution#
Analytical solution.
2.4. Bibliographical references#
ROBERT: Analytical solutions in structural dynamics. Samtech Report No. 121, March 1996.