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 analysis of an elastic cable under harmonic excitation

Abstract: The nonlinear behavior of an elastic cable subjected to harmonic excitation is studied and solved. The method of multiple scales perturbation is applied to analyze the response of the nonlinear system near the simultaneous principle primary and internal resonance. The stability of the proposed analytic nonlinear solution near the simultaneous primary-internal resonance is studied and the stability condition is investigated. The effect of different parameters on the steady state responses of the vibrating system is studied and discussed using frequency response equations. The numerical solutions and chaotic response of the nonlinear system of the elastic cable for different parameters are also studied.

A geometrically exact approach to the overall dynamics of elastic rotating blades—part 2: flapping nonlinear normal modes

Abstract: The geometrically exact equations of motion about the prestressed state discussed in part 1 (i.e., the nonlinear equilibrium under centrifugal forces) are expanded in the Taylor series of the incremental displacements and rotations to obtain the third-order perturbed form. The expanded form is amenable to a perturbation treatment to unfold the nonlinear features of free undamped flapping dynamics. The method of multiple scales is thus applied directly to the partial-differential equations of motion to construct the backbone curves of the flapping modes and their nonlinear approximations when they are away from internal resonances with other modes. The effective nonlinearity coefficients of the lowest three flapping modes of elastic isotropic blades are investigated when the angular speed is changed from low- to high-speed regimes. The novelty of the current findings is in the fact that the nonlinearity of the flapping modes is shown to depend critically on the angular speed since it can switch from hardening to softening and vice versa at certain speeds. The asymptotic results are compared with previous literature results. Moreover, 2:1 internal resonances between flapping and axial modes are exhibited as singularities of the effective nonlinearity coefficients. These nonlinear interactions can entail fundamental changes in the blade local and global dynamics.

Dynamics of an electrically actuated resonant microsensor

Abstract: We present a nonlinear model of an electrically actuated microbeam-based resonant microsensor encompassing the electrostatic force of an air gap capacitor, the restoring force of the microbeam, and the axial load applied to the microbeam. The model accounts for moderately large deflections, dynamic loads, and the coupling between the mechanical and electrical forces. It accounts for the nonlinearity in the elastic restoring forces and the electric forces. A perturbation method, the method of multiple scales, is applied to the distributed-parameter system to study the local dynamics of the sensor under primary, superharmonic, and subharmonic excitations. In each case, we obtain two first-order nonlinear ordinary-differential equations that describe the modulation of the amplitude and phase of the response and its stability, and hence the bifurcations of the response. The perturbation results are validated by comparing them to experimental results. The DC electrostatic load affects the qualitative and quantitative nature of the frequency-response curves, resulting in either a softening or a hardening behavior. The results also show that an inaccurate representation of the system nonlinearities may lead to a qualitatively and quantitatively erroneous prediction of the frequency-response curves. The results provide an analytical tool to predict a microsensor response to primary, superharmonic, and subharmonic excitations, specifically the locations of sudden jumps and regions of hysteretic behavior allowing designers to examine the impact of the design parameters on the device behavior.