Nonlinear Flexural Vibrations of Unshearable Elastic Rings

2013 ◽  
Author(s):  
Walter Lacarbonara ◽  
Andrea Arena ◽  
Stuart S. Antman
Keyword(s):  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equations Of Motion ◽  
Cosserat Theory ◽  
Flexural Mode ◽  
Curved Rods ◽  
Deformation Modes ◽  
Nonlinear Motions ◽  
Planar Motions

Refined theories are employed to study nonlinear vibrations of elastic rings in the mth flexural mode away from autoparametric resonances involving other flexural modes. A top-down modeling approach is followed to describe the rings undergoing all deformation modes in space by the Special Cosserat theory of curved rods. The specialization to extensional-flexural-shearing, then to extensional-flexural, and finally to purely flexural planar motions is illustrated. Free undamped extensional-flexural nonlinear motions involving the mth mode and its companion mode are investigated via a direct asymptotic approach based on the method of multiple scales applied to the geometrically exact equations of motion and it is shown that these motions are softening for linearly elastic rings while there are thresholds in the constitutive laws separating softening from hardening behaviors.

An Experimental and Theoretical Investigation of Nonlinear Vibrations of Piezoelectrically-Driven Microcantilevers

2006 ◽  
Author(s):  
S. Nima Mahmoodi ◽  
Nader Jalili
Keyword(s):  
Natural Frequency ◽  
Piezoelectric Layer ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Galerkin Approximation ◽  
Closed Form Solution ◽  
Cubic Form ◽  
Dynamic Representation ◽  
Microcantilever Beam

The nonlinear vibrations of a piezoelectrically-driven microcantilever beam are experimentally and theoretically investigated. A part of the microcantilever beam surface is covered by a piezoelectric layer, which acts as an actuator. Practically, the first resonance of the beam is of interest, and hence, the microcantilever beam is modeled to obtain the natural frequency theoretically. The bending vibrations of the beam are studied considering the inextensibility condition and the coupling between electrical and mechanical properties in piezoelectric materials. The nonlinear term appears in the form of quadratic due to presence of piezoelectric layer, and cubic form due to geometry of the beam (mainly due to the beam's inextensibility). Galerkin approximation is utilized to discretize the equations of motion. The obtained equation is simulated to find the natural frequency of the system. In addition, method of multiple scales is applied to the equations of motion to arrive at the closed-form solution for natural frequency of the system. The experimental results verify the theoretical findings very closely. It is, therefore, concluded that the nonlinear approach could provide better dynamic representation of the microcantilever than previous linear models.

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).

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.

Theoretical and Experimental Investigation of an L-Shape Beam Structure

2008 ◽  
Author(s):  
Dong-Xing Cao ◽  
Wei Zhang ◽  
Ming-Hui Yao
Keyword(s):  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Space Station ◽  
Dynamic Responses ◽  
Flexible Beam ◽  
Beam Structure ◽  
Large Space ◽  
Chaotic Motions ◽  
Experimental Apparatus

Flexible multi-beam structures are significant components of large space station, architecture engineering and other structural systems. The understanding of the dynamic characteristics of these structures is essential for their design and control of vibrations. In this paper, the planar nonlinear vibrations and chaotic dynamics of an L-shape flexible beam structure will be investigated using theoretical and experimental methods. The L-shape beam structure considered here is composed of two flexible beams with right-angle. The governing equations of motion for the L-shape beam structure are established firstly. Then, the method of multiple scales is utilized to obtain a four-dimensional averaged equation. Numerical method is used to analyze the nonlinear dynamic responses and chaotic motions. Finally, The experimental apparatus and schemes for measuring the amplitude of nonlinear vibrations for the L-shape beam structure are introduced briefly. Then, the detailed analysis for experimental data and signals which represent the nonlinear responses of the beam structure are given.

Primary Resonance of the Current-Carrying Beam in Thermal-Magneto-Elasticity Field

2010 ◽  
Vol 29-32 ◽  
pp. 16-21 ◽  
Author(s):  
Xiao Yan Xi ◽  
Zhian Yang ◽  
Li Li Meng ◽  
Chang Jian Zhu
Keyword(s):  
Response Curve ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Response Curves ◽  
Bending Vibration ◽  
Approximation Solution ◽  
System Parameters ◽  
Practical Engineering

On base of the electro-magneto-elastic theory and the theory of the bending vibration of the electric beam, nonlinear vibration equation of current-carrying beam subjected to thermal-magneto-elasticity field is studied. The Lorentz force and thermal force on the beam are derived. According to the method of multiple scales for nonlinear vibrations the approximation solution of the primary resonance of the system is obtained. Numerical analysis results show that the amplitude changed with the system parameters. With the decrease of magnetic intensity, the amplitude increases rapidly. The response curve occurs bending phenomenon and soft features is increased gradually. Increasing current, the amplitudes increase. With the decrease of temperature, the peak of response curves decrease. With the increase of temperature, natural frequency decreased. It is useful in practical engineering.

Internal Resonance and Nonlinear Vibrations of Eccentric Rotating Composite Laminated Circular Cylindrical Shell

2019 ◽  
Author(s):  
Tao Liu ◽  
Wei Zhang ◽  
Yan Zheng ◽  
Yufei Zhang
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Shear Deformation Theory ◽  
Internal Resonances

Abstract This paper is focused on the internal resonances and nonlinear vibrations of an eccentric rotating composite laminated circular cylindrical shell subjected to the lateral excitation and the parametric excitation. Based on Love thin shear deformation theory, the nonlinear partial differential equations of motion for the eccentric rotating composite laminated circular cylindrical shell are established by Hamilton’s principle, which are derived into a set of coupled nonlinear ordinary differential equations by the Galerkin discretization. The excitation conditions of the internal resonance is found through the Campbell diagram, and the effects of eccentricity ratio and geometric papameters on the internal resonance of the eccentric rotating system are studied. Then, the method of multiple scales is employed to obtain the four-dimensional nonlinear averaged equations in the case of 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance. Finally, we study the nonlinear vibrations of the eccentric rotating composite laminated circular cylindrical shell systems.

On Nonlinear Motions of Two-Degree-of-Freedom Nonlinear Systems with Repeated Linearized Natural Frequencies

2019 ◽  
Vol 29 (10) ◽  
pp. 1950132
Author(s):  
Hua-Zhen An ◽  
Xiao-Dong Yang ◽  
Feng Liang ◽  
Wei Zhang ◽  
Tian-Zhi Yang ◽  
...  
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Phase Portraits ◽  
Chaotic Motions ◽  
Nonlinear Parameters ◽  
Energy Ratios ◽  
Two Degree Of Freedom ◽  
Nonlinear Motions ◽  
Phase Differences

In this paper, we investigate systematically the vibration of a typical 2DOF nonlinear system with repeated linearized natural frequencies. By application of Descartes’ rule of signs, we demonstrate that there are 14 types of roots describing different modal motions for varying nonlinear parameters. The method of multiple scales is used to obtain the amplitude-phase portraits by introducing the energy ratios and phase differences. The typical nonlinear in-unison and elliptic out-of-unison modal motions are located for the 14 types of roots and then validated by numerical simulations. It is found that the normal in-unison modal motions, elliptic out-of-unison modal motions are analogous to the polarization of classical optic theory. Further, some kinds of periodic and chaotic motions under out-of-unison and in-unison excitations are investigated numerically. The result of this study offers a detailed discussion of nonlinear modal motions and responses of 2DOF systems with cubic nonlinear terms.

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.

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