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

Analytical solution.


Benchmark results
----------------------

Displacement of the middle sheet

Middle sheet constraints, upper and lower sheets.

In modeling :math:`I` we calculate the mass, the coordinates of the center of gravity and the terms of the inertia matrix. Analytic expressions are given in the documentation [:ref:`R3.07.02 <R3.07.02>`].


Calculation method used for the reference solution in displacements and constraints
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**In incompressible:**


.. image:: images/Object_7.svg
    :width: 382
    :height: 67


.. _RefImage_Object_7.svg:


.. image:: images/Object_8.svg
    :width: 382
    :height: 67


.. _RefImage_Object_8.svg:


.. image:: images/Object_9.svg
    :width: 382
    :height: 67


.. _RefImage_Object_9.svg:

Passage through the Cartesian system:


.. image:: images/Object_10.svg
    :width: 382
    :height: 67


.. _RefImage_Object_10.svg:


Determination of masses, center of gravity and inertia tensor
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For shell-type modeling of revolution around an axis :math:`\mathrm{OZ}`


1. the mass is equal to:

    .. image:: images/Object_11.svg
        :width: 382
        :height: 67


    .. _RefImage_Object_11.svg:

;


1. the coordinates of the center of gravity are:

    .. image:: images/Object_12.svg
        :width: 382
        :height: 67


    .. _RefImage_Object_12.svg:

;


2. the inertia tensor with respect to :math:`O` is equal to:


.. image:: images/Object_13.svg
    :width: 382
    :height: 67


.. _RefImage_Object_13.svg:


1. the inertia tensor with respect to :math:`G` is equal to:


.. image:: images/Object_14.svg
    :width: 382
    :height: 67


.. _RefImage_Object_14.svg:

**Note:**

*In practice, we overlook the terms en*

.. image:: images/Object_19.svg
    :width: 382
    :height: 67


.. _RefImage_Object_19.svg:

*in these expressions.*