Multiple-scale analysis

About: Multiple-scale analysis is a research topic. Over the lifetime, 1360 publications have been published within this topic receiving 27530 citations.

Nonlinear dynamic analyses on a magnetopiezoelastic energy harvester with reversible hysteresis

Abstract: This paper presents a systematic approach through a series of nonlinear analyses to predict and design the nonlinear resonance characteristics and energy-harvesting performance of a magnetopiezoelastic vibration energy harvester with reversible hysteresis. To this end, a mathematical model of the energy harvester system, composed of a bimorph cantilever beam along with three permanent magnets, is first derived. With this model, frequency response analyses are conducted using the methods of multiple scales and harmonic balance. In the case of weak excitation, the system’s stationary forced response is obtained through multiple-scale analysis (MSA). With this MSA solution, analytical design criteria in terms of the position parameters of the magnets are derived to determine the hysteresis type of the nonlinear resonance (stiffness hardening or stiffness softening). However, in the case of a relatively strong excitation, the high-dimensional harmonic balance analysis (HDHBA) shows that the stiffness-softening effect tends to strengthen for the oscillation amplitude in a specific condition, and this effect can even lead to hysteresis transition or potential well escape. Using the HDHBA, design criteria are set up and evaluated in terms of source vibration strength to detect the hysteresis transition or the condition required to determine the potential well escape. Finally, based on the present systematic analyses, the complicated resonant responses and energy-harvesting performance of the present system are examined with respect to several design parameters, and the results are discussed in detail.

The method of multiple scales for forced oscillators with some real-power nonlinearities in the stiffness and damping force

Abstract: The steady-state response of forced damped nonlinear oscillators is considered, the restoring force of which has a non-negative real power-form nonlinear term and the linear term of which can be negative, zero or positive. The damping term is also assumed in a power form, thus covering polynomial and non-polynomial damping. The method of multiple scales with a new expansion parameter is presented in order to cover the cases when the nonlinearity is not necessarily small. Amplitude-frequency equations and approximate solutions for the steady-state response at the frequency of excitation are obtained and compared with numerical results, showing good agreement.

Theoretical development and closed-form solution of nonlinear vibrations of a directly excited nanotube-reinforced composite cantilevered beam

Abstract: The nonlinear equations of motion of planar bending vibration of an inextensible viscoelastic carbon nanotube (CNT)-reinforced cantilevered beam are derived. The viscoelastic model in this analysis is taken to be the Kelvin–Voigt model. The Hamilton principle is employed to derive the nonlinear equations of motion of the cantilever beam vibrations. The nonlinear part of the equations of motion consists of cubic nonlinearity in inertia, damping, and stiffness terms. In order to study the response of the system, the method of multiple scales is applied to the nonlinear equations of motion. The solution of the equations of motion is derived for the case of primary resonance, considering that the beam is vibrating due to a direct excitation. Using the properties of a CNT-reinforced composite beam prototype, the results for the vibrations of the system are theoretically and experimentally obtained and compared.

Non-linear flutter of a buckled shear-deformable composite panel in a high-supersonic flow

Abstract: The non-linear dynamic behavior of a uniformly compressed, composite panel subjected to non-linear aerodynamic loading due to a high-supersonic co-planar flow is analyzed The effects of in-plane edge restraints, small initial geometric imperfections, transverse shear deformation, and transverse normal stress are considered in the structural model which satisfies the traction-free condition on the panel faces The panel flutter equations, derived via Galerkin's Method, are solved using Arclength Continuation for the static solution and a predictor-corrector type Shooting Technique to obtain periodic solutions and their bifurcations The possibility of hard flutter is demonstrated when considering non-linear aerodynamics Furthermore, edge compression could yield multiple buckled states or coexistence of multiple periodic solutions with the stable static solution, that is, the panel could either remain buckled or flutter Edge restraints normal to the flow appear to stabilize the panel, whereas those parallel to the flow may result in a buckled-flutter-buckled transition Quasi-periodic and chaotic motions and associated Lyapunov exponents are also obtained For perfect panels, results obtained by the Shooting Technique and the Method of Multiple Scales are in agreement only within the immediate post-flutter regime Results indicate that a shear deformation theory is required for moderately thick composite panels