scholarly journals Nonlinear tuning of microresonators for dynamic range enhancement

2015 ◽  
Vol 471 (2179) ◽  
pp. 20140969 ◽  
Author(s):  
M. Saghafi ◽  
H. Dankowicz ◽  
W. Lacarbonara
Keyword(s):  
Nonlinear Response ◽  
Dynamic Range ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equations Of Motion ◽  
Path Following ◽  
Critical Amplitude ◽  
Inertia Effects ◽  
Response Characteristics ◽  
The Galerkin Method

This paper investigates the development of a novel framework and its implementation for the nonlinear tuning of nano/microresonators. Using geometrically exact mechanical formulations, a nonlinear model is obtained that governs the transverse and longitudinal dynamics of multilayer microbeams, and also takes into account rotary inertia effects. The partial differential equations of motion are discretized, according to the Galerkin method, after being reformulated into a mixed form. A zeroth-order shift as well as a hardening effect are observed in the frequency response of the beam. These results are confirmed by a higher order perturbation analysis using the method of multiple scales. An inverse problem is then proposed for the continuation of the critical amplitude at which the transition to nonlinear response characteristics occurs. Path-following techniques are employed to explore the dependence on the system parameters, as well as on the geometry of bilayer microbeams, of the magnitude of the dynamic range in nano/microresonators.

Nonlinear Dynamics of Electrostatically Actuated Coupled MEMS Parallel Cantilever Resonators

2012 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Kyle N. Taylor
Keyword(s):  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equations Of Motion ◽  
Lagrange Equations ◽  
Electrostatically Actuated ◽  
Frequency Responses ◽  
Beam System ◽  
Ground Plate ◽  
Steady State Solutions

This paper deals with the nonlinear response of a coupled cantilever system composed of two micro beams electrostatically actuated. The AC frequency of actuation is near natural frequency of the cantilevers. The two cantilevers are identical. Lagrange equations are used to develop a mathematical model of the system. These equations of motion are nondimensionalized and subjected to the method of multiple scales in order to find steady state solutions. Alternating Current (AC) and Direct Current (DC) actuation voltages are applied between the first cantilever and ground plate with DC voltage applied between the first and second cantilevers. Amplitude-frequency and phase-frequency responses of the system are provided for typical micro beam system structures.

Electrostatically Actuated Coupled MEMS Resonators

2011 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Kyle N. Taylor
Keyword(s):  
Nonlinear Response ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
System Structure ◽  
Lagrange Equations ◽  
Electrostatically Actuated ◽  
Beam System ◽  
Mems Resonators ◽  
The Mathematical Model ◽  
Steady State Solutions

This paper deals with the nonlinear response of an electrostatically actuated cantilever beam system composed of two micro beam resonators near natural frequency. The mathematical model of the system is obtained using Lagrange equations. The equations of motion are nondimensionalized and then the method of multiple scales is used to find steady state solutions. Both AC and DC actuation voltages of the first beam are considered, while the influence on the system of DC on the second beam is explored. Graphical representations of the influence of the detuning parameters are provided for a typical micro beam system structure.

Reduced Order Model for MEMS Cantilever Resonators Under Soft AC Voltage Near Natural Frequency

2012 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Israel Martinez
Keyword(s):  
Natural Frequency ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Electrostatic Force ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
First Order ◽  
Electrostatically Actuated

The nonlinear response of an electrostatically actuated cantilever beam microresonator is investigated. The AC voltage is of frequency near resonator’s natural frequency. A first order fringe correction of the electrostatic force and viscous damping are included in the model. The dynamics of the resonator is investigated using the Reduced Order Model (ROM) method, based on Galerkin procedure. Steady-state motions are found. Numerical results for the uniform microresonator are compared with those obtained via the Method of Multiple Scales (MMS).

SOLITON-LIKE PHONON LOCALIZED MODES IN DIATOMIC NONLINEAR LATTICE CHAIN

1991 ◽  
Vol 05 (13) ◽  
pp. 2237-2252 ◽  
Author(s):  
ZHU-PEI SHI ◽  
GUOXING HUANG ◽  
RUIBAO TAO
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equations Of Motion ◽  
Localized Modes ◽  
Wave Approximation ◽  
Long Wave ◽  
Model Hamiltonian ◽  
Nonlinear Lattice ◽  
Gap Solitons ◽  
Long Wave Approximation

We have studied the localization of multivibrational states in diatomic nonlinear lattice chain. A simple model Hamiltonian presented here describes a system containing two kinds of phonons. The equations of motion for these boson operators are two partial differential equations with nonlinear coupling in long-wave approximation. With the help of the method of multiple scales, these equations are reduced to the nonlinear Schrödinger equation. It is shown that soliton-like phonon localized modes, multi-phonon localized modes can exist. The possibility of observing the gap solitons (phonon localized modes in the frequency gap) in diatomic nonlinear lattice chain is predicted.

Effects of Flexural Stiffness, Axial Velocity and Internal Support Locations on Translating Beam’s Natural Frequencies

2014 ◽  
Vol 532 ◽  
pp. 316-319 ◽  
Author(s):  
Ferid Köstekci
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Flexural Rigidity ◽  
Equations Of Motion ◽  
Flexural Stiffness ◽  
Transverse Vibrations ◽  
Internal Support ◽  
Simple Support ◽  
Moving Speed

The aim of this paper is to examine the natural frequencies of beams for different flexural stiffness, internal simple support locations and axial moving speed. In the present investigation, the linear transverse vibrations of an axially translating beam are considered based on Euler-Bernoulli model. The beam is passing through two frictionless guides and has an internal simple support between the guides. The governing differential equations of motion are derived using Hamiltons Principle for two regions of the beam. The method of multiple scales is employed to obtain approximate analytical solution. Some numerical calculations are conducted to present the effects of flexural rigidity, mean translating speed and different internal support locations on natural frequencies.

Superharmonic Resonance of Cross-Ply Laminates by the Method of Multiple Scales

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
Hadj Youzera ◽  
Sid Ahmed Meftah ◽  
El Mostafa Daya
Keyword(s):  
Composite Materials ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equations Of Motion ◽  
Viscoelastic Model ◽  
Transversely Isotropic ◽  
Superharmonic Resonance ◽  
Plastic Composite ◽  
Isotropic Composite ◽  
Forced Vibration Analysis

General differential equations of motion in nonlinear forced vibration analysis of multilayered composite beams are derived by using the higher-order shear deformation theories (HSDT's). Viscoelastic properties of fiber-reinforced plastic composite materials are considered according to the Kelvin–Voigt viscoelastic model for transversely isotropic composite materials. The method of multiple scales is employed to perform analytical frequency amplitude relationships for superharmonic resonance. Parametric study is conducted by considering various geometrical and material parameters, employing HSDT's and first-order deformation theory (FSDT).

Nonlinear Response of a Sandwich Plate Subjected to Blast Load

2007 ◽  
Author(s):  
Ali Bas¸ ◽  
Zafer Kazancı ◽  
Zahit Mecitog˘lu
Keyword(s):  
Sandwich Plate ◽  
Virtual Work ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Blast Load ◽  
Thin Plates ◽  
Inertia Effects ◽  
The Time Domain ◽  
The Galerkin Method

Present work includes in-plane stiffness and inertia effects on the motion of a sandwich plate under blast load. The geometric nonlinearity effects are taken into account with the von Ka´rma´n large deflection theory of thin plates. All edges clamped boundary conditions are considered in the analyses. The equations of motion for the plate are derived by the use of the virtual work principle. Approximate solutions are assumed for the space domain and substituted into the equations of motion. Then the Galerkin Method is used to obtain the nonlinear differential equations in the time domain. The finite difference method is applied to solve the system of coupled nonlinear equations. The results of theoretical analyses are obtained.

Dynamic Stability of a Spinning Timoshenko Beam Subjected to a Moving Mass-Spring-Damper Unit

2010 ◽  
Author(s):  
T. H. Young ◽  
M. S. Chen
Keyword(s):  
Dynamic Stability ◽  
Timoshenko Beam ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Axial Direction ◽  
Longitudinal Axis ◽  
Moving Mass ◽  
The Stability ◽  
The Galerkin Method ◽  
Mass Spring

This paper investigates the dynamic stability of a finite Timoshenko beam spinning along its longitudinal axis and subjected to a moving mass-spring-damper (MSD) unit traveling in the axial direction. The mass of the moving MSD unit makes contact with the beam all the time during traveling. Due to the moving MSD unit, the beam is acted upon by a periodic, parametric excitation. In this work, the equations of motion of the beam are first discretized by the Galerkin method. The discretized equations of motion are then partially uncoupled by the modal analysis procedure suitable for gyroscopic systems. Finally the method of multiple scales is used to obtain the stability boundaries of the beam. Numerical results show that if the displacement of the MSD unit is equal to only one of the two transverse displacements of the beam, very large unstable regions may appear at main resonances.

scholarly journals Nonlinear Response of Cantilever Beams to Combination and Subcombination Resonances

1998 ◽  
Vol 5 (5-6) ◽  
pp. 277-288 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Nonlinear Ordinary Differential Equations ◽  
Response Curves ◽  
First Order ◽  
Aspect Ratios

The nonlinear planar response of cantilever metallic beams to combination parametric and external subcombination resonances is investigated, taking into account the effects of cubic geometric and inertia nonlinearities. The beams considered here are assumed to have large length-to-width aspect ratios and thin rectangular cross sections. Hence, the effects of shear deformations and rotatory inertia are neglected. For the case of combination parametric resonance, a two-mode Galerkin discretization along with Hamilton’s extended principle is used to obtain two second-order nonlinear ordinary-differential equations of motion and associated boundary conditions. Then, the method of multiple scales is applied to obtain a set of four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two excited modes. For the case of subcombination resonance, the method of multiple scales is applied directly to the Lagrangian and virtual-work term. Then using Hamilton’s extended principle, we obtain a set of four first-order nonlinear ordinary-differential equations governing the amplitudes and phases of the two excited modes. In both cases, the modulation equations are used to generate frequency- and force-response curves. We found that the trivial solution exhibits a jump as it undergoes a subcritical pitchfork bifurcation. Similarly, the nontrivial solutions also exhibit jumps as they undergo saddle-node bifurcations.

Nonlinear Vibrations of a Beam-Spring-Mass System

1994 ◽  
Vol 116 (4) ◽  
pp. 433-439 ◽  
Author(s):  
M. Pakdemirli ◽  
A. H. Nayfeh
Keyword(s):  
Nonlinear Response ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Partial Differential Equations ◽  
Primary Resonance ◽  
Fast Time ◽  
Lagrange Equations ◽  
Response Curves ◽  
Mass System

The nonlinear response of a simply supported beam with an attached spring-mass system to a primary resonance is investigated, taking into account the effects of beam midplane stretching and damping. The spring-mass system has also a cubic nonlinearity. The response is found by using two different perturbation approaches. In the first approach, the method of multiple scales is applied directly to the nonlinear partial differential equations and boundary conditions. In the second approach, the Lagrangian is averaged over the fast time scale, and then the equations governing the modulation of the amplitude and phase are obtained as the Euler-Lagrange equations of the averaged Lagrangian. It is shown that the frequency-response and force-response curves depend on the midplane stretching and the parameters of the spring-mass system. The relative importance of these effects depends on the parameters and location of the spring-mass system.

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