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

The reference solution comes from guide VPCS.


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

The numerical integration selected to obtain this solution is based on a finite difference integration diagram, such as the improved :math:`\beta` -Newmark method, with a :math:`\mathrm{0.001s}` [:ref:`bib2 <bib2>`] time step.

:math:`\left[\frac{1}{\Delta {t}^{2}}M+\frac{1}{2\Delta t}C+\frac{1}{3}K\right]{u}_{n+2}\mathrm{=}\frac{1}{3}({F}_{n+2}+{F}_{n+1}+{F}_{n})+\left[\frac{2}{\Delta {t}^{2}}M\mathrm{-}\frac{1}{3}K\right]{u}_{n+1}+\left[\frac{1}{\Delta {t}^{2}}M+\frac{1}{2\Delta t}C\mathrm{-}\frac{1}{3}K\right]{u}_{n}`

The movement of point :math:`4` as a function of time looks like this:


.. image:: images/1003165E00002DE4000015CF04B80C1E767FDAB9.svg
    :width: 591
    :height: 281


.. _RefImage_1003165E00002DE4000015CF04B80C1E767FDAB9.svg:

**Figure 2.1-a: Point 4: displacement as a function of time**


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

Move along :math:`x` from point :math:`{P}_{4}`.


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

Accuracy of the Newmark schema.


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


1. File SDLD22 /90 from the VPCS commission.


2. NEWMARK N.M.: "A method of computation for structural dynamics", proceeding ASCE J.Eng. Mech. Div E-3, July 1959, pp 67-94.