1. Reference problem

1.1. Geometry

Characteristics:

Pendulum length	\(L=0.6m\)
Eccentricity	\(a=0.1m\)
Profile height	\(h=0.01m\)
Profile width	\(b=0.004m\)
Section	\(S=\mathrm{bh}\)
Flexural inertia	\({I}_{Z}={\mathrm{bh}}^{3}/12\)

1.2. Material properties

Young’s module	\(E=7.E10N/{m}^{2}\)
Poisson’s Ratio	\(\nu =0.3\)
Density	\(\rho =2700\mathrm{kg}/{m}^{3}\)

1.3. Boundary conditions and loading

The beam is hinged at point \(A\). The axis of the joint is axis \(Y\). The initial state of stress that makes it possible to calculate the geometric and centrifugal stiffness is obtained by imposing a speed of rotation and gravity.

Gravity acceleration	\(g=-9.81m/{s}^{\mathrm{²}}\) (parallel to the \(Z\) axis)
Rotation speed	\(\Omega =10\mathrm{rad}/s\)

The static equilibrium position \({\theta }_{0}\) corresponding to the load is calculated by the relationship:

\(3g\mathrm{cos}{\theta }_{0}={\Omega }^{2}(\mathrm{3a}+\mathrm{2L}\mathrm{cos}{\theta }_{0})\mathrm{sin}{\theta }_{0}\)

We find \({\theta }_{0}=11.269931365°\)

The conditions for travel to point \(A\) are as follows:

\(u=v=w=0\); \({\phi }_{X}={\phi }_{Z}=0\)

It is also considered that the section through \(A\) remains rigid.

1.4. Initial conditions

Not applicable in modal analysis.