2. Benchmark solution

2.1. Calculation method used for the reference solution

• Without damping: the analytical solution of the one-element problem is:

\({x}_{B}(t)=\frac{{F}_{x}}{m{\omega }_{0}^{2}}(1-\mathrm{cos}{\omega }_{0}t)\)

\(m=\frac{1}{3}\rho \mathrm{SI}\), \({\omega }_{0}^{2}=\frac{3E}{\rho {I}^{2}}\), \({T}_{0}=\frac{2\pi }{{\omega }_{0}}\)

where \(S\) is the area of section \((\pi {R}^{2})\).

• With amortization: the analytical solution of the one-element problem is:

\({x}_{B}(t)=\frac{{F}_{x}}{m{\omega }_{0}^{2}}\left[1-\mathrm{exp}(-\frac{\mu +\lambda {\omega }_{0}^{2}}{2}\mathrm{.}t)\mathrm{.}(\frac{\mu +\lambda {\omega }_{0}^{2}}{2{\omega }_{1}}\mathrm{sin}({\omega }_{1}t)+\mathrm{cos}({\omega }_{1}t))\right]\)

\(\lambda ,\mu\) proportional damping coefficient \(C=\lambda K+\mu M\)

\({\omega }_{1}=\frac{\sqrt{(4-2\lambda \mu ){\omega }_{0}^{2}-{\mu }^{2}-{\lambda }^{2}{\omega }_{0}^{4}}}{2}\)

2.2. Benchmark results

Move \({x}_{B}\) to \(t=\frac{i{T}_{0}}{10}\) \(i=\mathrm{1,}\mathrm{...},10\)

with: \({T}_{0}=\frac{2\pi }{{\omega }_{0}}\)

2.3. Uncertainty about the solution

Analytical solution.

Note:

The reference solution corresponds to the solution obtained with single-element discretization and keeping a full mass matrix. This allows the algorithm to be validated but it is not the step to solve the physical problem.