2. Benchmark solution

2.1. Calculation method

The numerical solution of the problem is calculated via diagrams HHT and NOHHT of « DYNA_NON_LINE », considering a \(\Delta t = 5\cdot 10^{-3} \ \text{s}\) time step. We also consider a coefficient \(\alpha=-0.3\) with respect to these diagrams. The two formulations available for these schemes, namely formulations in motion and in acceleration, are tested.

2.2. Benchmark solution

The problem presents an analytical solution. In fact, the displacement of the point mass subjected to initial conditions for movement and speed \(u_0, v_0 \in \mathbb{R}\) can be written as follows:

\[u_x (t) =\ exp {(-\ xi\ omega t)}\ left [u_0\ cos {(\ omega_d t) +\ frac {1} {\ omega_d} (v_0} (v_0 +\ xi\ omega_0 u_0)}\ right],\]

for all \(t \ge 0\), with \(\omega_d = \omega_0 \sqrt{1-\xi^2}\).

The solutions obtained with Code_Aster are also compared to the results obtained with a Python implementation diagrams HHT and NOHHT.

2.3. Reference quantities and results

The comparisons will focus on the displacement, speed, and acceleration of the point mass \(m\) at several points in time.