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, and where rigid modes are not fixed, can be carried out by numerical integration in real space:
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To do this, the response was calculated using two industrial codes:
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Newmark integration diagram (\(\alpha =\mathrm{0,25}\), \(\delta =\mathrm{0,5}\)), \(\Delta t={10}^{-\mathrm{4s}}\), |
Integration diagram with Hermite cubic interpolation [bib1], \(\Delta t={10}^{-\mathrm{4s}}\), |
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Hilber-Hughes-Taylor integration diagram [bib2] (\(\alpha =-\mathrm{0,05}\)), \(\Delta t={10}^{-\mathrm{4s}}\), |
and the improved \(\beta\) -Newmark integration method [bib3]:
where \(n\), \(n+1\), \(n+2\) respectively designate the calculations performed at times \({t}_{n}\), \({t}_{n+1}={t}_{n}+\Delta t\) and \({t}_{n+2}={t}_{n}+2\Delta t\) where \(\Delta t\) is the time increment used.
To start, we take:
The adopted time step is \(\Delta t={10}^{-\mathrm{5s}}\).
2.2. Benchmark results#
Displacement, speed and acceleration of point \({P}_{3}\).
Displacement differential between points \({P}_{3}\) and \({P}_{1}\).
2.3. Uncertainty about the solution#
Average of digital solutions.
2.4. Bibliographical references#
J.H. ARGYRIS, P.C. DUNNE and T. ANGELOPOULOS « Non-linear oscillations using the finite technical element » Comp. Meth. Call. Mech. Engng., Vol.2, 1972, pp. 203-254
H.M. HILBER, T.J.R. HUGHES 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
N.M. NEWMARK « A method of computation for structural dynamics » Proceeding ASCE J.Eng.Mech. Div E-3, July 1959, pp. 67-94