2. Reference solution

The reference solution comes from guide VPCS.

2.1. 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 \(\beta\) -Newmark method, with a \(\mathrm{0.001s}\) [bib2] time step.

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

The movement of point \(4\) as a function of time looks like this:

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

2.2. Benchmark results

Move along \(x\) from point \({P}_{4}\).

2.3. Uncertainty about the solution

Accuracy of the Newmark schema.

2.4. 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.