Benchmark solution
=====================


Calculation method
-----------------

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


Benchmark solution 
---------------------

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

.. math::
 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 :math:`t \ge 0`, with :math:`\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. 

Reference quantities and results
-----------------------------------

The comparisons will focus on the displacement, speed, and acceleration of the point mass :math:`m` at several points in time.