.. _solfec-1.0-validation-pendulum:

Pendulum
========

.. |br| raw:: html

  <br />

+---------------------------------------------------------------------------------------------------------------------------------+
| **Reference:** W. Rubinowicz, W. Królikowski, Mechanika teoretyczna (Theoretical mechanics), Państwowe Wydawnictwo Naukowe,     |
| Warszawa, 1998, pp. 91-99.                                                                                                      |
| |br|                                                                                                                            |
| **Analysis:** Explicit dynamics, bilaterally constrained motion.                                                                |
| |br|                                                                                                                            |
| **Purpose:** Examine the accuracy of an analysis involving rigid rod constraint.                                                |
| |br|                                                                                                                            |
| **Summary:** A mathematical pendulum composed of a mass point and a weightless rod swings with a large amplitude. Pendulum      |
| period, energy conservation, constraint satisfaction and convergence are examined.                                              |
+---------------------------------------------------------------------------------------------------------------------------------+

The period of an oscillatory mathematical pendulum reads

.. math::

  T=2\pi\sqrt{\frac{l}{g_{3}}}\left(1+\left(\frac{1}{2}\right)^{2}k^{2}+\left(\frac{1\cdot3}
  {2\cdot4}\right)k^{2}+\left(\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\right)k^{2}+...\right)
  
where

.. math::

  k=\sin\left(\frac{\theta_{max}}{2}\right)
  
and :math:`l` is the length of the pendulum, :math:`g_{3}` is the vertical component of the gravity acceleration
and :math:`\theta_{max}` is the maximal tilt angle of the pendulum. Let us assume the initial velocity of the
pendulum to be zero. Thus :math:`\theta_{max}=\theta\left(0\right)`. Taking the rest configuration position of
the mass point :math:`\bar{\mathbf{x}}=\left[0,0,0\right]` and considering the swing in the :math:`x-z` plane,
the initial position of the pendulum reads

.. math::

  \bar{\mathbf{x}}\left(0\right)=\left[\begin{array}{c}
  l\sin\left(\theta_{max}\right)\\
  0\\
  l\left(1-\cos\left(\theta_{max}\right)\right)
  \end{array}\right]
  
Without the initial kinetic energy :math:`(E_{k}\left(0\right)=0)`, the energy conservation requires that

.. math::

  E_{k}\left(t\right)+E_{p}\left(t\right)=E_{p}\left(0\right)
  
where

.. math::

  E_{p}\left(0\right)=mg_{3}\bar{x}_{3}\left(0\right)
  
and :math:`m` is the scalar mass.

Input parameters
----------------

+------------------------------------------------------------------------------+-----------------------------------------------+
| Length :math:`\left(m\right)`                                                | :math:`l=1`                                   |
+------------------------------------------------------------------------------+-----------------------------------------------+
| Mass :math:`\left(kg\right)`                                                 | :math:`m=1`                                   |
+------------------------------------------------------------------------------+-----------------------------------------------+
| Initial angle :math:`\theta\left(0\right)=\theta_{max}\,\, \left(rad\right)` | :math:`\theta_{max}=\pi/2`                    |
+------------------------------------------------------------------------------+-----------------------------------------------+
| Gravity acceleration :math:`\left(m/s^{2}\right)`                            | :math:`\mathbf{g}=\left[0,0,-\pi^{2}\right]`  |
+------------------------------------------------------------------------------+-----------------------------------------------+

The gravity acceleration :math:`g_{3}` has been chosen so that for :math:`\theta_{max}=0\deg` there holds :math:`T=2s`.

Results
-------

The table below summarizes the results for the time step :math:`h=0.001`. The solution is accurate and stable after 1
and 10 swings. :numref:`pendulum` illustrates the energy balance over one period of the pendulum. The potential and
kinetic energies sum up to :math:`\pi^{2}`.

+-------------------------------------------------------------+-----------------+-------------+---------+
|                                                             | Target          | Solfec-1.0  | Ratio   |
+-------------------------------------------------------------+-----------------+-------------+---------+
| Pendulum period -- 1 swing :math:`\left(s\right)`           | 2.360           | 2.360       | 1.000   |
+-------------------------------------------------------------+-----------------+-------------+---------+
| Total energy -- 1 swing :math:`\left(J\right)`              | :math:`\pi^{2}` | 9.86960     | 1.000   |
+-------------------------------------------------------------+-----------------+-------------+---------+
| Pendulum period -- 10 swings :math:`\left(s\right)`         | 23.60           | 23.60       | 1.000   |
+-------------------------------------------------------------+-----------------+-------------+---------+
| Total energy -- 10 swings :math:`\left(J\right)`            | :math:`\pi^{2}` | 9.86960     | 1.000   |
+-------------------------------------------------------------+-----------------+-------------+---------+

.. _pendulum:

.. figure:: pendulum/pendulum.png
   :width: 75%
   :align: center

   Energy balance over one period of the pendulum.