2. Reference solution

2.1. Calculation method

In the linear static case:

\({E}_{\mathit{pot}}={W}^{\mathit{ext}}=\frac{1}{2}{\sum }_{i\in N}{D}_{i}{F}_{i}^{\mathit{ext}}\) where \(N\) is the set of model nodes.

For a small model, it is therefore easy to calculate the potential energy from the displacements.

In the case of modal calculation:

If \(\Phi\) is an eigenmode of the problem, of natural frequency \(f=\frac{\omega }{2\pi }\), with \(K\) stiffness matrix of \(M\) mass matrix then \((K-{\omega }^{2}M)\Phi =0\), hence \({\Phi }^{T}(K-{\omega }^{2}M)\Phi =0\).

If we norm the modes with respect to the mass matrix \(M\) then we have \({\Phi }^{T}K\Phi ={\omega }^{2}={(2\pi f)}^{2}\).

Gold \({E}_{\mathit{pot}}=\frac{1}{2}{\Phi }^{T}K\Phi\). So all you have to do is check that \({E}_{\mathit{pot}}=2{(\pi f)}^{2}\).

2.2. Reference quantities and results

2.3. Uncertainties about the solution

None.