1. Reference problem#

1.1. Geometry#

_images/10000200000001E4000001A6EC7B0AFDC6CE4BE3.png

Square plate :

Length: \(l=1.0m\)

Thickness: \(e=0.1m\)

1.2. Material properties#

Young’s module, \(E=4.388{10}^{10}N/{m}^{2}\)

Poisson’s ratio, \(\nu =0.0\)

Density, \(\rho =2500.0\mathrm{kg}/{m}^{3}\)

1.3. Boundary conditions and loads#

Boundary conditions:

L location

Blocked components

A1

DX, DY, DZ, DRX, DRY, DRZ

A

DX, DZ, DRX, DRY, DRZ

C

DZ, DRX, DRY, DRZ

Loads:

We apply the linear force on side B in the direction \(x\), which depends on time like,

\(F(t)={Q}_{0}EKe\mathrm{cos}(\mathrm{Kl})\mathrm{sin}(\omega t)\),

where the following parameters are used:

  • \({Q}_{0}\) (\(={10}^{-4}m\)) - load amplitude

  • \(E\) — Young’s modulus defined above (in \(N/{m}^{2}\))

  • \(e\) — the thickness defined above (in \(m\))

  • \(l\) — the plate size defined above (in \(m\))

  • \(K\) (\(=\frac{\pi }{\mathrm{8l}}\)) the wave number of the analytical solution (in \({m}^{-1}\))

  • \(\omega\) — frequency (times \(2\pi\)), linked to the wave number \(K\), \(K=\omega /c\), \(c\) being the speed of the waves in the structure, \(c=\sqrt{\frac{E}{\rho }}\)

The parameterization introduced makes it possible to apply the load just to obtain the analytical solution, determined simply by the parameters \({Q}_{0}\) and \(K\), and then by other parameters of the dimensions and material properties of the structure.

1.4. Initial conditions#

Initially, the movements are zero everywhere and the speeds obey the following spatial distribution,

\({v}_{0}(x,y)=\omega {Q}_{0}\mathrm{sin}(\mathrm{K.x})\)