Direct parametrisation of invariant manifolds for non-autonomous forced systems including superharmonic resonances

Abstract

The direct parametrisation method for invariant manifold is a model-order reduction technique that can be applied to nonlinear systems described by PDEs and discretised e.g. with a finite element procedure in order to derive efficient reduced-order models (ROMs). In nonlinear vibrations, it has already been applied to autonomous and non-autonomous problems to propose ROMs that can compute backbone and frequency-response curves of structures with geometric nonlinearity. While previous developments used a first-order expansion to cope with the non-autonomous term, this assumption is here relaxed by proposing a different treatment. The key idea is to enlarge the dimension of the parametrising coordinates with additional entries related to the forcing. A new algorithm is derived with this starting assumption and, as a key consequence, the resonance relationships appearing through the homological equations involve multiple occurrences of the forcing frequency, showing that with this new development, ROMs for systems exhibiting a superharmonic resonance, can be derived. The method is implemented and validated on academic test cases involving beams and arches. It is numerically demonstrated that the method generates efficient ROMs for problems involving 3:1 and 2:1 superharmonic resonances, as well as converged results for systems where the first-order truncation on the non-autonomous term showed a clear limitation.

Figures (7)

• Figure 1: Clamped-clamped beam. (a) geometry $L=1000\,\mu$m, $B=24\,\mu$m, $H=10\,\mu$m. (b-d) first three bending eigenmodes.
• Figure 2: Frequency response curves corresponding to a 3:1 superharmonic resonance on the doubly clamped beam having a mass-proportional damping $\alpha=\omega_{B1}/500$ and an external excitation with amplitude $\kappa=5\,\mu\text{m}/\mu\text{s}^2$ i.e. $\epsilon=0.1170 [-]$. (a) DPIM solution with different expansion orders compared to the HBFEM solution obtained with 9 harmonics. (b) enlarged view of the FRC peaks. For the ROMs, the stability of the solution branches is reported with solid lines (resp. dashed lines) for stable solutions (resp. unstable solutions). The star markers pinpoint the saddle-node bifurcation points.
• Figure 3: Computational burden to construct the ROMs in terms of time and number of monomials. (a) Increasing orders in the parametrisation with a fixed truncation order in $\varepsilon$. (b) Increasing truncation in $\varepsilon$ order with a fixed order expansion order for the normal coordinate. (c-d) number of monomials to be computed in each case.
• Figure 4: Graphical representation of the whisker motion with time in the case of the 3:1 superharmonic resonance in a straight beam. Single nonlinear normal mode truncation along the first bending mode (B1), parametrised with an order $\mathcal{O}(z^5,\epsilon^5)$. (a)-(c) 3-d and 2-d representation using the second bending mode as slave coordinate. (b)-(d) Using the third bending modal coordinate B3.
• Figure 5: Sine-arched structure. (a) Geometry with length $L = 640\,\mu\text{m}$, rise $R = 3.84\,\mu\text{m}$, width $B=\,32\mu\text{m}$ and thickness $H=\,6.4\mu\text{m}$. (b-d) First three bending eigenmodes.