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 \((A)\):

1.2. Material properties

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

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

1.3. Boundary conditions and loads

Boundary conditions

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

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

Loading

Anchor point \(A\) is subject to acceleration, an increasing function of time, in the direction \(x\): \(\gamma (t)\mathrm{=}2\mathrm{.}{10}^{5}\mathrm{.}{t}^{2}\) (\(t\) varies from \(0\) to \(\mathrm{0,1}s\)).

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.