2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The reference solution is obtained analytically for a Timoshenko beam, taking into account the shear force deformation and the rotary inertia of the sections.
The solution is developed in a series of clean modes. Theoretical aspects are developed in the reference given in 2.4.
2.1.1. Modal basis#
Define the following dimensionless quantities:
\({\lambda }_{n}={k}_{n}L\) wavelengths
\({\Omega }_{n}=\frac{\rho A{L}^{4}}{\mathit{EI}}{\omega }_{n}^{2}\) eigenvalues
\(j=\frac{I}{A{L}^{2}}\) rotary inertia
\(g=\frac{\mathit{EI}}{k\text{'}AG{L}^{2}}\) shear coefficient
Each specific mode of wave number \({k}_{n}\) is characterized by the following quantities:
Natural frequencies:
\({\Omega }_{\mathrm{1,2}}=\frac{\rho A{L}^{4}}{\mathit{EI}}{\omega }_{\mathrm{1,2}}^{2}=\frac{(g+j){\lambda }_{n}^{2}+1\pm \sqrt{{(g-j)}^{2}{\lambda }_{n}^{4}+2(g+j){\lambda }_{n}^{2}+1}}{2gj}\)
with
\({\lambda }_{n}=n\pi\), \(n=\mathrm{1,}\mathrm{2,}\mathrm{3,}\mathrm{...}\),
(the indices 1 and 2 correspond to the + and — signs in front of the root) .
The generalized masses:
\({\mu }_{\mathrm{1,2}}=\rho A={\left({k}_{n}-\frac{{\omega }_{\mathrm{1,2}}^{2}\rho }{{k}_{n}k\text{'}G}\right)}^{2}\rho I\).
Critical depreciation percentages:
\({ϵ}_{\mathrm{1,2}}=\frac{1}{2}\left(\alpha {\omega }_{\mathrm{1,2}}+\frac{\beta }{{\omega }_{\mathrm{1,2}}}\right)\)
2.1.2. Harmonic response#
The amplitude and the phase of the arrow W are given by
\(W(x)=\sum _{n=1}^{+\infty }{p}_{n}\left[\sum _{m=1}^{2}\frac{1}{{\mu }_{m}({\omega }_{m}^{2}-{\omega }^{2}+2i{ϵ}_{m}{\omega }_{m}\omega )}\right]\mathrm{sin}({k}_{n}x)\)
with
\({p}_{n}=\frac{2{p}_{0}}{n\pi }\left[1-{(-1)}^{n}\right]\)
2.2. Benchmark results#
Position |
Arrow \(W\) |
|
Amplitude (\(m\)) |
Phase |
|
\(x=\frac{L}{4}\) |
\(2.136\times {10}^{-5}\) |
\(22.4\text{°}\) |
\(x=\frac{L}{2}\) |
\(1.342\times {10}^{-5}\) |
\(-121.5\text{°}\) |
\(x=3\frac{L}{4}\) |
\(2.136\times {10}^{-5}\) |
\(22.4\text{°}\) |
The quantities actually tested in the test case are the parts real and imaginary whose values are given below.
Position |
Arrow \(W\) |
|
Real part (\(m\)) |
Imaginary part (\(m\)) |
|
\(x=\frac{L}{4}\) |
\(-1.9599467159360556\times {10}^{-5}\) |
\(-8.4917893914738073\times {10}^{-6}\) |
\(x=\frac{L}{2}\) |
\(-6.9993870268574731\times {10}^{-6}\) |
\(-1.1450108350939712\times {10}^{-5}\) |
\(x=3\frac{L}{4}\) |
\(-1.9599467159360556\times {10}^{-5}\) |
\(-8.4917893914738073\times {10}^{-6}\) |
2.3. Uncertainty about the solution#
Analytical solution.
2.4. Bibliographical references#
ROBERT G., Analytical solutions in structural dynamics, Samtech Report No. 121, Liège, 1996.