Reference solution
=====================


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

The reference solution is the one given in sheet SDLL08 /89 of the guide VPCS, which presents the calculation method as follows:

A Rayleigh-Ritz method makes it possible to perform the calculation with two degrees of freedom based on the following symmetric deformation hypotheses:


* for the abscissa point :math:`y` of the side members :math:`\mathrm{AC}` and :math:`\mathrm{DF}` of length :math:`\mathrm{L1}`


.. image:: images/Object_4.svg
    :width: 162
    :height: 58


.. _RefImage_Object_4.svg:


* for the abscissa point :math:`x` of the crosspiece :math:`\mathrm{BE}` of length :math:`\mathrm{L2}`


.. image:: images/Object_5.svg
    :width: 162
    :height: 58


.. _RefImage_Object_5.svg:


.. image:: images/10002B82000022EF00001089B38A4BC80FC21EA6.svg
    :width: 162
    :height: 58


.. _RefImage_10002B82000022EF00001089B38A4BC80FC21EA6.svg:


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

The first two natural frequencies and **symmetric** natural modes (the other natural frequencies of this system are not studied). For the natural modes, we have the following value:

.. image:: images/Object_6.svg
    :width: 162
    :height: 58


.. _RefImage_Object_6.svg:

In harmonic response we have:


*

    .. image: images/Object_7.svg
        :width: 162
        :height: 58


    .. _RefImage_Object_7.svg:

,


*

    .. image: images/Object_8.svg
        :width: 162
        :height: 58


    .. _RefImage_Object_8.svg:

at point :math:`G`.


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

Analytical solution.


Bibliographical references
---------------------------


1. J.M. BIGGS. Introduction to Structural Dynamics. New York: McGraw Hill, p.184 (1964).