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.

Mass Sensitivity of Nonuniform Microcantilever Beams

Abstract: Microelectromechanical systems (MEMS) based cantilever beams have been widely used in various sensing applications. Previous studies have aimed at increasing the sensitivity of biosensors by reducing the size of cantilever beams to nanoscale. However, the influence of nonuniform cantilever beams on mass sensitivity has rarely been investigated. In this paper, we discuss the mass sensitivity with respect to linear and nonlinear response of nonuniform cantilever beam with linear and quartic variation in width. To do the analysis, we use the nonlinear Euler-Bernoulli beam equation with harmonic forcing. Subsequently, we derive the mode shape corresponding to linear, undamped, free vibration case for different types of beams with a tip mass at the end. After applying the boundary conditions, we obtain the resonance frequencies corresponding to various magnitudes of tip mass for different kinds of beams. To do the nonlinear analysis, we use the Galerkin approximation and the method of multiple scales (MMS). Analysis of linear response indicates that the nondimensional mass sensitivity increases considerably by changing the planar geometry of the beam as compared to uniform beam. At the same time, sensitivity further increases when the nonuniform beam is actuated in higher modes. Similarly, the frequency shift of peak amplitude of nonlinear response for a given nondimensional tip mass increases exponentially and decreases quadratically with tapering parameter, a, for diverging and converging nonuniform beam with quartic variation in width, respectively. For the converging beam, we also found an interesting monotonically decreasing and increasing behavior of mass sensitivity with tapering parameter a giving an extremum point at alpha = 0.5. Overall analysis indicates a potential application of the nonuniform beams with quartic converging width for biomass sensor.

Nonlinear electrohydrodynamic stability of a fluid layer : effect of a tangential electric field

Abstract: The weakly nonlinear electrohydrodynamic stability of fluid layer sandwiched between two semi-infinite fluids is investigated. The nonlinear theory of perturbation is applied for symmetric and anti-symmetric modes. The method of multiple scales is used to expand the various perturbation quantities to yield the linear and successive nonlinear partial differential equations of the various orders. The solutions of these equations are obtained. The application of the boundary conditions leads to two nonlinear Schrodinger equations. It is found that the presence of the tangential field plays a stabilizing role and can be used to suppress the instability of the system at a given wavenumber which is unstable linear stability. Numerical calculations show a global stability for certain wavenumbers. A local instability is also observed in the graphs. The field plays a dual role. It is observed that the change of the layer thickness redistributes the stable areas.

Using the extended Melnikov method to study multi-pulse chaotic motions of a rectangular thin plate

Abstract: This paper investigates the multi-pulse global heteroclinic bifurcations and chaotic dynamics for the nonlinear vibrations of a simply supported rectangular thin plate by using an extended Melnikov method in the resonant case. The rectangular thin plate is subjected to spatially and temporally varying transversal and in-plane excitations, simultaneously. The equations of motion for the rectangular thin plate are derived from the von Karman equation. Applying the method of multiple scales and Galerkin’s approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1:2 internal resonance and primary parametric resonance. From the averaged equations obtained, the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary Eigenvalues. Based on the explicit expressions of normal form, the extended Melnikov method is used to analyze the multi-pulse heteroclinic bifurcations and chaotic dynamics of the rectangular thin plate. The contribution of the paper is the simplification of the extended Melnikov method. The extended Melnikov function can be simplified in the resonant case and does not depend on the perturbation parameter. The necessary conditions of the existence for the Shilnikov type multi-pulse chaotic dynamics of the rectangular thin plate are analytically obtained. Numerical simulations also display that the Shilnikov type multi-pulse chaotic motions can occur in the rectangular thin plate. Overall, both theoretical and numerical studies demonstrate that the chaos for the Smale horseshoe sense exists in the rectangular thin plate.