Modal reduction-based finite element method for nonlinear FG piezoelectric energy harvesters

In this study, harvesting energy from a nonlinear functionally graded (FG) piezoelectric cantilever beam under harmonic excitation is investigated. The material properties of the piezoelectric are assumed to be a combination of piezo-ceramic and piezo-polymer materials for high performance and flexibility. The neutral axis is obtained in order to eliminate bending–stretching coupling. The geometrical nonlinearity and electromechanical coupling are incorporated in the coupled nonlinear equations that are derived using the generalized Hamilton’s principle and solved using the combination of modal reduction and finite element methods. The shooting method is employed to obtain steady-state periodic response of an FG nonlinear harvester with appropriate initial conditions. Also it is shown that at least two-mode approximation is required for accurate estimation of nonlinear response and harvested power. Using the method of nonlinear modal reduction, the unstable branches for frequency domain solution are estimated and the computational time is reduced considerably compared to full finite element method. A case study is also accomplished in detail to analyze the effects of base amplitude values and material distribution on harvested power and bandwidth.

The nodal-based floating frame of reference formulation with modal reduction

In a recent paper of the authors, a novel nodal-based floating frame of reference formulation (FFRF) for solid finite elements has been proposed. The nodal-based approach bypasses the unhandy inertia shape integrals ab initio, i.e. they neither arise in the derivation nor in the final equations of motion, leading to a surprisingly simple derivation and computer implementation without a lumped mass approximation, which is conventionally employed within commercial multibody codes. However, the nodal-based FFRF has so far been presented without modal reduction, which is usually required for efficient simulations. Hence, the aim of this follow-up paper is to bring the nodal-based FFRF into a suitable form, which allows the incorporation of modal reduction techniques to reduce the overall system size down to the number of modes included in the reduction basis, which further reduces the computational complexity significantly. Moreover, this exhibits a way to calculate the so-called FFRF invariants, which are constant “ingredients” required to set up the FFRF mass matrix and quadratic velocity vector, without integrals and without a lumped mass approximation.

Modal Reduction Procedures for Flexible Multibody System Applications

The comprehensive simulation of flexible multibody systems calls for the ability to model various types of structural components such as rigid bodies, beams, plates, and kinematic joints. Modal components test offer additional modeling versatility by enabling the treatment of complex, three-dimensional structures via modal reduction procedures based on the small deformation assumption. In this paper, the problem is formulated within the framework of the motion formalism. The kinematic description involves simple, straightforward frame transformations and leads to deformation measures that are both objective and tensorial. Derivatives are expressed in the material frame, which results in the remarkable property that the tangent matrices are independent of the configuration of the modal component with respect to an inertial frame. This implies a reduced level of geometric nonlinearity as compared to standard description. In particular, geometrically nonlinear problems can be solved with the constant tangent matrices of the reference configuration, without re-evaluation and re-factorization.

Research Progress of Synthesis and Modification Methods Based on Dynamic Substructures

In the process of model modification of large and complex structures, substructure synthesis method and modal reduction method have been widely used, but there are still some difficulties in precision control and engineering application in the process of model updating. In order to better study the dynamic response of the vibration substructure, the synthesis and correction method of the classical dynamic substructure is described in this paper, which provides a new idea for further engineering development. In the aspect of substructure synthesis method, the modal reduction of substructures, two methods of classical substructure synthesis, mechanical impedance method, singular value decomposition method, rigidity-flexibility equivalence, and transformation of degree of freedom are analyzed. The advantages and disadvantages of the above methods are discussed. In terms of substructure modification, the reference datum method, function dynamic modification method, neural network model modification, and frequency response function modification are analyzed, and the shortcomings of the dynamic substructure modification method are summarized. Finally, the development trend of dynamic substructure synthesis and modification algorithm is proposed.

Extended Modal Reduction for On-Board Rotor With Multifrequency Parametric Excitation

A new reduction method is proposed to investigate the behavior stability of rotor-bearing systems subject to a multifrequency rotational motion of their base. Combining the modal analysis and the construction of specific dynamic Ritz vectors, this method is able to deal with complex rotordynamics characteristics such as nonproportional damping, nonself-adjoint matrices, or time-varying parametric coefficients. This paper focuses first on assessing the accuracy and efficiency of the reduction method by computing time history and spectral responses of full and reduced models due to multifrequency base excitations. Its main potential is then highlighted in the parametric stability analysis through Floquet theory. The proposed numerical examples are composed with academic and industrial rotors, both modeled with one-dimensional Timoshenko beam finite element and supported by hydrodynamic journal bearings.