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


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

Since numerical integration (approximated) by the direct method using a finite difference numerical integration scheme, the time step used must be small enough to obtain a sufficiently precise solution. With one of the schemes used (improved Newmark method), the time step taken was :math:`\mathrm{0.001s}`.

The improved Newmark method (NEWMARK N.M., "A method of computation for structural dynamics" proceeding ASCE J. Eng. Mech. Div E-3, July 1959, pp 67-94) uses the following integration diagram:

:math:`\begin{array}{}\left[\frac{1}{{\Delta t}^{2}}[M]+\frac{1}{2\Delta t}[C]+\frac{1}{3}[K]\right]({u}_{n+2})\\ =\frac{1}{3}([:ref:`{P}_{n+2} <{P}_{n+2}>`] + [{P} _ {n+1}]] + [{P} _ {n}]]) +\ left [\ frac {2} {{\ Delta t} ^ {2}} [M] -\ frac {1} {3} {3} {3}} [K] {3}} [K] {3} [K] {3} [K]\ right] (K]\ right] ({u} _ {n+1}) +\ left [\ frac {1}} {{\ Delta t} ^ {2}} [M] -\ frac {1}} [M] -\ frac {1} {1}} [M] -\ frac {1} {1}} [M] -\ frac {1} {1} {\ delta t} {1} {[M] +\ frac {1} {2\ Delta t} [C] -\ frac {1} {3} [K]\ right] ({u} _ {n})\ end {array} `

The indices :math:`n`, :math:`n+1`, :math:`n+2` respectively designate the calculations performed at time :math:`{t}_{n}`, :math:`{t}_{n+1}={t}_{n}+\Delta t` and :math:`{t}_{n+2}={t}_{n}+2\Delta t`, where :math:`\Delta t` is the time increment used. :math:`[M]`, :math:`[C]` and :math:`[K]` are the mass, damping, and stiffness matrices respectively, :math:`(u)` is the displacement vector and :math:`(P)` the associated force vector.


.. image:: images/1003287E00002E33000016530091713B691C84FA.svg
    :width: 595
    :height: 288


.. _RefImage_1003287E00002E33000016530091713B691C84FA.svg:

**Point 4: movement as a function of time**


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

Displacement at point :math:`\mathrm{P4}` as a function of time, as shown in the graph above.


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


* position of the extremes: :math:`\Delta t<0.015`


* maximum amplitude: :math:`\Delta u/u<0.5\text{\%}`


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


1. VPCS commission sheet SDLD29 /90