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


Calculation method used for the reference solution
--------------------------------------------------------

The reference solution is analytical. In the absence of damping, it is a simple sine wave whose period is equal to the natural pulsation of the oscillator, :math:`{\omega }_{0}=\sqrt{\frac{k}{m}}`, and whose amplitude is the initial elongation :math:`({x}_{0})` of the spring. The :math:`x(t)` position of the mass is given by the equation:

     :math:`x(t)={x}_{0}\mathrm{cos}({\omega }_{0}t)` (1)

So the speed of the mass is:

     :math:`v(t)=-{\omega }_{0}{x}_{0}\mathrm{sin}({\omega }_{0}t)` (2)

In the presence of viscous damping :math:`({c}_{[\mathrm{N.s}/m]})`, the oscillations become damped and position :math:`x(t)` is written as:

     :math:`x(t)={x}_{0}{e}^{-\zeta {\omega }_{0}t}[\mathrm{cos}(\omega t)+(\frac{\zeta }{\sqrt{1-{\zeta }^{2}}})\mathrm{sin}(\omega t)]` (3)

where :math:`\zeta` is the reduced depreciation given by :math:`\zeta =\frac{c}{2{\omega }_{0}m}`. :math:`\zeta` is considered to be less than 1 to maintain oscillations. The pulsation is given by the formula :math:`\omega ={\omega }_{0}\sqrt{(1-{\zeta }^{2})}`. It is therefore different from the system's natural pulsation :math:`({\omega }_{0})`.


Results
----------

*Case 1*: conservative system (without amortization)

For this system, the natural pulsation :math:`{\omega }_{0}=\pi \mathrm{rad}/s`. So the natural frequency is :math:`{f}_{0}={\omega }_{0}/2\pi =\mathrm{0,5}\mathrm{Hz}`.

The displacement (in m) and the speed (in m/s) of the mass, given respectively by Eqs.1 and 2 are:

     :math:`x(t)=\mathrm{cos}(\pi t)` and :math:`v(t)=-\pi \mathrm{sin}(\pi t)`


.. image:: images/100000000000034A00000253D0D0B4AE766DB174.jpg
    :width: 5.4307in
    :height: 3.8374in


.. _RefImage_100000000000034A00000253D0D0B4AE766DB174.jpg:

*Case 2*: viscous dissipative system

The reduced depreciation is :math:`\zeta =\mathrm{0,1}`. The pulsation is :math:`\omega =\mathrm{0,995}\pi \mathrm{rad}/s` and the frequency is therefore :math:`f=\omega /2\pi =\mathrm{0,4975}\mathrm{Hz}`.

The displacement (in m) can then be calculated according to Eq.3.


Uncertainty about the solution
---------------------------

Analytical solution.