2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The search for the transitory answer to this problem with non-proportional damping can be carried out by numerical integration in real space:

\([M]\{\ddot{{u}_{n}}\}+[C]\{\dot{{u}_{n}}\}+[K]\{{u}_{n}\}=\{F\}\)

To do this, the response was calculated using two industrial codes:

  • PERMAS: Newmark integration diagram (\(\alpha =\mathrm{0,25}\) and \(\delta =\mathrm{0,5}\)) \(\Delta t={10}^{-\mathrm{4s}}\);

  • ABAQUS: Hilbert-Hugues-Taylor integration diagram [bib1] (\(\alpha =–\mathrm{0,05}\)) \(\Delta t={10}^{-\mathrm{4s}}\);

and the improved \(\beta\) -Newmark integration method [bib2]:

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

where \(n\), \(n+1\), \(n+2\) respectively designate the calculations performed at times \({t}_{n}\), \({t}_{n+1}={t}_{n}+\Delta t\) and \({t}_{n+2}\mathrm{=}{t}_{n}+2\Delta t\) where \(\Delta t\) is the time increment used.

To start, we take:

  • \({u}_{0}\) and \({u}_{-1}={u}_{0}-\Delta t\dot{{u}_{0}}\)

  • \({F}_{-1}=2{F}_{0}-{F}_{1}\)

The adopted time step is \(\Delta t={10}^{-\mathrm{5s}}\).

2.2. Benchmark results#

Displacement and speed of the endpoint \(B\).

2.3. Uncertainty about the solution#

Average of digital solutions.

2.4. Bibliographical references#

  1. H.M. HILBERT, T.J.R HUGUES and R.L. TAYLOR « Improved numerical dissipation for time integration algorithms in structural dynamics » Earthquake Engineering and Structural Dynamics, Vol.5, 1977, 1977, pp. 283-292

  2. N.M. NEWMARK « A method of computation for structural dynamics » Proceeding ASCE J.Eng.Mech. DIV E-3, July 1959, pp. 67-94