2. Benchmark solution#
2.1. 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 \(\mathrm{0.001s}\).
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:
\(\begin{array}{}\left[\frac{1}{{\Delta t}^{2}}[M]+\frac{1}{2\Delta t}[C]+\frac{1}{3}[K]\right]({u}_{n+2})\\ =\frac{1}{3}([:ref:`{P}_{n+2} <{P}_{n+2}>\)] + [{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} {2Delta t} [C] -frac {1} {3} [K]right] ({u} _ {n})end {array} `
The indices \(n\), \(n+1\), \(n+2\) respectively designate the calculations performed at time \({t}_{n}\), \({t}_{n+1}={t}_{n}+\Delta t\) and \({t}_{n+2}={t}_{n}+2\Delta t\), where \(\Delta t\) is the time increment used. \([M]\), \([C]\) and \([K]\) are the mass, damping, and stiffness matrices respectively, \((u)\) is the displacement vector and \((P)\) the associated force vector.
Point 4: movement as a function of time
2.2. Benchmark results#
Displacement at point \(\mathrm{P4}\) as a function of time, as shown in the graph above.
2.3. Uncertainty about the solution#
position of the extremes: \(\Delta t<0.015\)
maximum amplitude: \(\Delta u/u<0.5\text{\%}\)
2.4. Bibliographical references#
VPCS commission sheet SDLD29 /90