2. Benchmark solution

2.1. Calculation method used for the reference solution

The discretized problem verifies:

,

with clue \(l\): degree of free freedom

Clue \(d\): imposed degree of freedom

\({F}_{d}(t)\) external loads applied to the end nodes and leading to the imposed displacements \({u}_{d}\) are unknown, so we eliminate these equations and we get:

.

The only non-zero terms in the second member of this system are related to the kinematic variables relating to the end node where the displacement is imposed. Now, at \(t\mathrm{=}0\), \(\ddot{{u}_{\mathit{dC}}}\) and \(\dot{{u}_{\mathit{dC}}}\) are not defined but at \(t\mathrm{=}{0}^{\text{-}}\) and \(t\mathrm{=}{0}^{\text{+}}\), \(\ddot{{u}_{\mathit{dC}}}\) and \(\dot{{u}_{\mathit{dC}}}\) are zero. All the complexity of the problem comes from that.

To obtain a reference solution, we considered \(\ddot{{u}_{\mathit{dC}}}\) and \(\dot{{u}_{\mathit{dC}}}\) to be uniformly zero, which is the same as considering only the elastic internal forces at the \(C\) end. This is questionable from a physical point of view but, by adopting the same assumptions when modeling the problem, the validation of Code_Aster can be successfully completed.

The reference solution is calculated by dealing with the following problem:

\(\mathrm{[}{M}_{\text{ll}}\mathrm{]}\mathrm{\{}\ddot{{u}_{l}}\mathrm{\}}+\mathrm{[}{C}_{\text{ll}}\mathrm{]}\mathrm{\{}\dot{{u}_{l}}\mathrm{\}}+\mathrm{[}{K}_{\text{ll}}\mathrm{]}\mathrm{\{}{u}_{l}\mathrm{\}}\mathrm{=}\mathrm{-}\mathrm{[}{K}_{\mathit{ld}}\mathrm{]}\mathrm{\{}{u}_{d}(t)\mathrm{\}}\) with \(\mathrm{\{}{u}_{l}(0)\mathrm{\}}\mathrm{=}0\) and \(\mathrm{\{}\dot{{u}_{l}}(0)\mathrm{\}}\mathrm{=}0\).

To do this, we transport the problem into the modal base of the system which verifies:

\(\mathrm{[}{M}_{\text{ll}}\mathrm{]}\mathrm{\{}\ddot{{u}_{l}}\mathrm{\}}+\mathrm{[}{K}_{\text{ll}}\mathrm{]}\mathrm{\{}{u}_{l}\mathrm{\}}\mathrm{=}0\).

Since the damping is diagonal, the diagonal system is obtained:

,

with \(\mathrm{\{}X(0)\mathrm{\}}\mathrm{=}0\) and \(\mathrm{\{}\dot{X}(0)\mathrm{\}}\mathrm{=}0\).

In modal space, we therefore solve three differential equations (3 free degrees of freedom) of the second order (3 free degrees of freedom) and then we return to physical space. The displacement of the midpoint is then obtained:

\({u}_{B}(t)\mathrm{=}\mathrm{\sum }_{i+1}^{3}{e}^{\mathrm{-}\lambda it}({a}_{i}\mathrm{cos}(\tilde{{\omega }_{i}}t)+{b}_{i}\mathrm{sin}(\tilde{{\omega }_{i}}t))\),

with \(\tilde{{\omega }_{i}}\): \(i\) th pseudo-pulsation proper to the damped system.

2.2. Benchmark results

Displacement, speed, and acceleration of the \(B\) midpoint of the beam.

2.3. Uncertainty about the solution

Analytical solution of the discretized problem into four elements of equal length by considering speed and acceleration uniformly zero at point \(C\) where the displacement is imposed.