example
=====

It is [:ref:`Figure 6-a <Figure 6-a>`] the plane pendulum movement of an extendable bar :math:`\mathrm{AB}` of unit length, rotated in :math:`A` to a fixed support, free in :math:`B` and abandoned without speed in the Earth's gravity field from a position defined by the angle :math:`{\theta }_{0}`. All mechanical dissipation phenomena are overlooked.

As the amplitude :math:`{\theta }_{0}` can be large — we will take it from :math:`90°` — the point :math:`B` undergoes large displacements and the problem is non-linear.

The theoretical period is:

:math:`T=\mathrm{1,6744}s`

The calculation of the pendulum movement using diagram HHT (:math:`\alpha` -method) with :math:`\alpha =0` ("trapezium rule") constitutes test case SDNL100.

Figure [:ref:`Figure 6-b <Figure 6-b>`] represents the evolution over a period of time in the dimension of point :math:`B` calculated by the "modified mean acceleration" scheme with three values of :math:`\alpha`. The period is divided into 40 equal time steps.

The solid line curve is relative to :math:`\alpha =0`. Practically no errors are observed.


.. image:: images/10000000000001F100000191905D2D51D5F3AB0D.png
    :width: 3.5756in
    :height: 2.7846in


.. _RefImage_10000000000001F100000191905D2D51D5F3AB0D.png:

**Figure 6-a: Large amplitude pendulum**


.. image:: images/10000000000002E40000020C38F549970901750E.png
    :width: 5.3244in
    :height: 3.639in


.. _RefImage_10000000000002E40000020C38F549970901750E.png:

**Figure 6-b: "Modified mean acceleration" diagram, 40 steps of 0.0419 s**

The centreline curve is relative to :math:`\alpha =-\mathrm{0,1}`. We observe a depreciation rate of around 2% while figure [:ref:`Figure 6-a <Figure 6-a>`], for :math:`\frac{\mathrm{\Delta }t}{T}=\mathrm{0,025}`, forecasts 0.8%. This is because this curve was established linearly, while the movement of our pendulum is non-linear.

The dashed curve is relative to :math:`\alpha =-\mathrm{0,3}`. The depreciation rate is around 5.8%, while that predicted by figure [:ref:`Figure 5.3-a <Figure 5.3-a>`] is around 2.2%. The discrepancy is still due to the non-linearity of the problem.

Finally, the centreline and especially dashed curves reveal a shortening of the calculated period compared to the theoretical period.