1. Reference issues

1.1. Geometry

We consider a system with one degree of freedom, consisting of a point mass \(m\), a spring with a stiffness \(k\) oriented along the axis \(x\), and a shock absorber. The movement is done along the \(x\) axis.

Figure 1: Diagram of the mass-spring-shock absorber system.

1.2. Basic characteristics

The mass is assumed to be \(m=1 \ \text{kg}\). The system’s own period is assumed to be equal. to \(T=1 \ \text{s}\), which corresponds to a natural pulsation \(\omega_0 = 2\pi/T\), and therefore to a stiffness \(k = m \omega_0^2 = 2\pi \ \text{N}\cdot\text{s}^{-1}\).

Consider depreciation \(c = 2 \xi \omega_0\), where \(\xi \in [0, 1[\) refers to reduced depreciation of the system.

1.3. Boundary conditions and loads

The base of the spring is embedded, the only degree of freedom is therefore the following movement \(x\) of the point mass \(m\) which is fixed to the other end of the spring.

1.4. Initial conditions

We consider an initial displacement \(u_0 = 1.0 \ \text{m}\) and an initial speed \(v_0 = 1.0 \ \text{m}\cdot\text{s}^{-1}\).