2. Benchmark solution#
We are interested in the calculation of periodic solutions of the system thus characterizing the non-linear mode [1].
2.1. Calculation method#
If we note \(u\) the displacement of the mass and \(F\) the reaction force, the system equation is written as:
: label: EQ-None
ddot {u} (t) +text {ku} (t)mathrm {=}mathrm {-} F (u (t))
Where:
\(F(u)=0\) if \(u\le e\)
\(F(u)=K(u-e)\) if \(u>e\)
The following initial conditions are assumed:
\(\dot{u}(0)=-\sqrt{\frac{2E}{m}}\) and \(u(0)=0\)
Where \(E\) refers to stored mechanical energy.
The problem is solved by considering two linear problems. The first corresponds to free flight (excluding impact), and the second corresponds to the system with the mass attached to the wall (during impact).
After calculation, we obtain the duration \({T}_{1}\) of the free flight phase:
\({T}_{1}=2\sqrt{\frac{m}{k}}\mathrm{arccos}(-e\sqrt{\frac{k}{2E}})\)
And the duration of impact \({T}_{2}\) is given by the following relationship:
\({T}_{2}=2\sqrt{\frac{m}{K+k}}\mathrm{arccos}(ek\sqrt{\frac{1}{2E(K+k)}})\)
The frequency of the periodic solution is therefore equal to: \(N(E)\mathrm{=}\frac{1}{{T}_{1}+{T}_{2}}\)
The stability of the periodic solution is calculated based on Floquet’s theory, by a Newmark diagram and an eigenvalue calculation.
2.2. Reference quantities and results#
The reference quantities chosen are the frequency-energy pair and the stability of the periodic solution obtained.
The periodic solution is stable for the frequency-energy pair such as:
\(0.644\mathit{Hz}<f<0.6475\mathit{Hz}\) and \(6.2{10}^{\mathrm{-}3}J<E<6.9{10}^{\mathrm{-}3}J\)
2.3. Uncertainties about the solution#
The relationship between frequency and energy is obtained analytically.
2.4. Bibliographical references#
MOUSSI, Analysis of vibrating structures with localized nonlinearities at play using nonlinear modes. Doctoral thesis 2013.