MBS Benchmark A05: Stiff Flyball Governor
=========================================

Benchmark Objective
-------------------
The **A05** MBS benchmark problem is an example of a stiff mechanical system.


Benchmark Description
---------------------

The **A05** benchmark problem is also known as stiff flyball governor and was invented by J. Watt in the 18th century.

In this stiff mechanical system, coupler rods are substituted by spring-damper elements.

.. figure:: ../images/5MBS_Flyball.png
   :align: center
   :height: 250pt
   :alt: Flyball Governor
   :figclass: align-center

Shaft, rods and slider can be modeled as prismatic bodies with the characteristics provided in the next table.
Two 5 kilograms point mass are placed on points colored in purple in the previous figure.



+----------------------------------+--------------------------------------+
|         **System Properties and Configuration**                         |
+==================================+======================================+
|  Shaft, Rods dimensions          |   :math:`1.0\times0.01\times0.01 m`  |
+----------------------------------+--------------------------------------+
|  Slider dimensions               |   :math:`0.1\times0.1\times0.1 m`    |
+----------------------------------+--------------------------------------+
|  Density :math:`\rho`            |   :math:`3000 kg/m^3`                |
+----------------------------------+--------------------------------------+
|  Point masses                    |    5 kg                              |
+----------------------------------+--------------------------------------+
|  Spring stiffness :math:`K`      |    8e5 N/m                           |
+----------------------------------+--------------------------------------+
|  Spring damping :math:`C`        |    4e4 Ns/m                          |
+----------------------------------+--------------------------------------+
|  Spring rest length              |    0.5 m                             |
+----------------------------------+--------------------------------------+
|  ::math:`s`                      |    0.5 m                             |
+----------------------------------+--------------------------------------+
|  ::math:`\alpha`                 |    30 degree                         |
+----------------------------------+--------------------------------------+
|  ::math:`\dot{\omega}`           |   :math:`2\pi rad/s`                 |
+----------------------------------+--------------------------------------+


Results
-------

The dynamic simulation of the **A05** benchmark was executed for 10 s.
The starting position of the system in shown in the previous figure and numerical values are reported in the previous table.

:math:`s` values estimated with the OpenSim simulation are compared with the values provided as reference :cite:`2006:gonzales`.


.. figure:: ../images/A05_kinematics.png
   :align: center
   :height: 300pt
   :alt: A05 kinematics.
   :figclass: align-center

   Comparison between OpenSim simulation results for coordinate :math:`s` (colored line) and MBS benchmark reference value (black dashed line).

Computational Performances
~~~~~~~~~~~~~~~~~~~~~~~~~~
To compute the simulation the developed implementation required 0.233 seconds.

Results obtained using an Intel Core i5-4570 @ 3.2 GHz computer running Windows 8.1 64-bit.
OpenSim official release used: 3.2.

Comparision with other dynamic engines has been conducted submitting the obtained results to the benchmark collaborative library.
Click `here <http://goo.gl/ylkXzN>`_ to see the comparison results.

Videos
------

.. only:: html

    .. youtube:: pNLlsh6tIQU

.. only:: latex

  Video of the problem simulated in OpenSim is available at https://youtu.be/pNLlsh6tIQU

Download
--------

* Stiff flyball governor on MBS Benchmark library available at: http://goo.gl/ylkXzN
* OpenSim implementation available at: http://goo.gl/R9tl3z
* Video of stiff flyball governor simulated in OpenSim available at: https://youtu.be/pNLlsh6tIQU