Dimensionality reduction in mechanical vibratory systems poses challenges for distributed structures including geometric nonlinearities, mainly because of the lack of invariance of the linear subspaces. A reduction method based on direct normal form computation for large finite element (FE) models is here detailed. The main advantage resides in operating directly from the physical space, hence avoiding the computation of the complete eigenfunctions spectrum. Explicit solutions are given, thus enabling a fully non-intrusive version of the reduction method. The reduced dynamics is obtained from the normal form of the geometrically nonlinear mechanical problem, free of non-resonant monomials, and truncated to the selected master coordinates, thus making a direct link with the parametrisation of invariant manifolds. The method is fully expressed with a complex-valued formalism by detailing the homological equations in a systematic manner, and the link with real-valued expressions is established. A special emphasis is put on the treatment of second-order internal resonances and the specific case of a 1:2 resonance is made explicit. Finally, applications to large-scale models of micro-electro-mechanical structures featuring 1:2 and 1:3 resonances are reported, along with considerations on computational efficiency.

The direct parametrisation method for invariant manifolds is a nonlinear reduction technique which derives nonlinear mappings and reduced-order dynamics that describe the evolution of dynamical systems along a low-dimensional invariant-based span of the phase space. It can be directly applied to finite element problems. When the development is performed using an arbitrary order asymptotic expansion, it provides an efficient reduced-order modeling strategy for geometrically nonlinear structures. It is here applied to the case of rotating structures featuring centrifugal effect. A rotating cantilever beam with large amplitude vibrations is first selected in order to highlight the main features of the method. Numerical results show that the method provides accurate reduced-order models (ROMs) for any rotation speed and vibration amplitude of interest with a single master mode, thus offering remarkable reduction in the computational burden. The hardening/softening transition of the fundamental flexural mode with increasing rotation speed is then investigated in detail and a ROM parametrised with respect to rotation speed and forcing frequencies is detailed. The method is then applied to a twisted plate model representative of a fan blade, showing how the technique can handle more complex structures. Hardening/softening transition is also investigated as well as interpolation of ROMs, highlighting the efficacy of the method.

In this work, we address the simulation and testing of MEMS micromirrors with hardening and softening behaviour excited with patches of piezoelectric materials. The forces exerted by the piezoelectric patches are modelled by means of the theory of ferroelectrics developed by Landau–Devonshire and are based on the experimentally measured polarisation hysteresis loops. The large rotations experienced by the mirrors also induce geometrical nonlinearities in the formulation up to cubic order. The solution of the proposed model is performed by discretising the device geometry using the Finite Element Method, and the resulting large system of coupled differential equations is solved by means of the Harmonic Balance Method. Numerical results were validated with experimental data collected on the devices.

Resonant modal interaction is a nonlinear dynamic phenomenon observed in structures excited at large vibration amplitudes. In the present work, we report the experimental evidence of a 1:1 internal resonance in a symmetric resonant micromirror. The experiments are complemented with a reduced model obtained from the 3D finite element discretization of the device by parametrizing the system motion along a low dimensional invariant set of the phase space. The presence of coupling monomials in the governing equations makes the resulting dynamics non-linearizable. Both model and experimental data show the existence of a complex pattern of multiple stable solutions for a given value of the excitation frequency.

The effects of temperature and environmental moisture on the viscoelastic behavior of polyurethane foams were investigated both theoretically and experimentally. It was shown that the effect of the environmental parameters can be explained in terms of a variation of the free volume of the solid fraction of the foams, thus allowing the use of the superposition principle to predict their influence on the viscoelastic behavior of the materials. Dynamic mechanical analyses were performed to measure the dependence on frequency, temperature, and relative humidity of the complex modulus of two different polyurethane foams, differing in terms of their glass transition temperature. The time-temperature-humidity superposition principle was proved to be applicable for the tested materials. Next, the relaxation spectra and their dependence on the relative humidity were adopted to assess its effects on the large strain behavior of the foams.The effects of temperature and environmental moisture on the viscoelastic behavior of polyurethane foams were investigated both theoretically and experimentally. It was shown that the effect of the environmental parameters can be explained in terms of a variation of the free volume of the solid fraction of the foams, thus allowing the use of the superposition principle to predict their influence on the viscoelastic behavior of the materials. Dynamic mechanical analyses were performed to measure the dependence on frequency, temperature, and relative humidity of the complex modulus of two different polyurethane foams, differing in terms of their glass transition temperature. The time-temperature-humidity superposition principle was proved to be applicable for the tested materials. Next, the relaxation spectra and their dependence on the relative humidity were adopted to assess its effects on the large strain behavior of the foams.