1. Reference problem#

1.1. Geometry#

We calculate the response of a linear system composed of three masses and three springs to an acceleration imposed at its anchor point \((A)\):

_images/1000000000000330000000D440684DD880BB8CF9.png

1.2. Material properties#

  • connection stiffness: \({k}_{1}=1000N/m\) and \({k}_{2}={k}_{3}=100N/m\);

  • point masses: \(m={m}_{1}={m}_{2}={m}_{3}=1\mathrm{kg}\).

  • \(\text{5 \%}\) modal damping for all modes

1.3. Boundary conditions and loads#

Boundary conditions

The only authorized movements are translations according to axis \(x\).

Point \(A\) is embedded: \(\mathit{dx}\mathrm{=}\mathit{dy}\mathrm{=}\mathit{dz}\mathrm{=}\mathit{drx}\mathrm{=}\mathit{dry}\mathrm{=}\mathit{drz}\mathrm{=}0\).

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The anchor point \(A\) is subject to a harmonic acceleration of frequency \({f}_{\mathrm{ex}}\). The calculation is done from \(0\) to \(20s\).

1.4. Initial conditions#

The system is initially at rest: at \(t=0\), \(\mathrm{dx}(0)=0\), and \(\mathrm{dx}/\mathrm{dt}(0)=0\) at all points.