Nonlinear Dynamics of a Taut Spatial String With Material Nonlinearities

1999 ◽  
Author(s):  
Michael J. Leamy ◽  
Oded Gottlieb
Keyword(s):  
Nonlinear Dynamics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Response ◽  
Natural Frequency ◽  
Continuum Mechanics ◽  
Finite Deformation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
String Model

Abstract A spatial string model incorporating a nonlinear (and non-conservative) material law is proposed using finite deformation continuum mechanics. The influence of material nonlinearities on the string’s dynamic response to excitation near a transverse natural frequency is shown to be small due to their appearance at high orders only. Material nonlinearities appear at low order in the equations for excitation near a longitudinal natural frequency and a solution for this case is developed by applying the method of multiple scales directly to the partial differential equations. An example string is considered to explore the influence of material nonlinearities on the dynamic response. Rich modal content is found, which can not be predicted by simpler models. Additionally, the material nonlinearities are shown to exert their greatest, influence away from resonance, where they serve to limit the response amplitudes.

Nonlinear Dynamics of a Taut String with Material Nonlinearities

2000 ◽  
Vol 123 (1) ◽  
pp. 53-60 ◽  
Author(s):  
M. J. Leamy ◽  
O. Gottlieb
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamic Response ◽  
Natural Frequency ◽  
Continuum Mechanics ◽  
Finite Deformation ◽  
Multiple Scales ◽  
String Model ◽  
Taut String ◽  
Linear Material ◽  
Nonlinear String

A spatial string model incorporating a nonlinear (and nonconservative) material law is proposed using finite deformation continuum mechanics. The resulting model is shown to reduce to the classical nonlinear string when a linear material law is used. The influence of material nonlinearities on the string’s dynamic response to excitation near a transverse natural frequency is shown to be small due to their appearance at high orders only. Material nonlinearities appear at low order in the equations for excitation near a longitudinal natural frequency, and a solution for this case is developed by applying a second order multiple scales method directly to the partial differential equations. The material nonlinearities are found to influence both the degree of nonlinearity in the response and its softening or hardening nature.

Numerical and Experimental Analysis of the Nonlinear Dynamics due to Impacts of a Continuous Overhung Rotor

1997 ◽  
Author(s):  
Mohammed F. Abdul Azeez ◽  
Alexander F. Vakakis
Keyword(s):  
Nonlinear Dynamics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Experiments ◽  
The Other ◽  
Experimental Setup ◽  
Partial Differential ◽  
Qualitative Comparison ◽  
Set Up ◽  
Theory And Experiments

Abstract This work is aimed at obtaining the transient response of an overhung rotor when there are impacts occurring in the system. An overhung rotor clamped on one end, with a flywheel on the other and impacts occurring in between, due to a bearing with clearance, is considered. The system is modeled as a continuous rotor system and the governing partial differential equations are set up and solved. The method of assumed modes is used to discretize the system in order to solve the partial differential equations. Using this method numerical experiments are run and a few of the results are presented. The different numerical issues involved are also discussed. An experimental setup was built to run experiments and validate the results. Preliminary experimental observations are presented to show qualitative comparison of theory and experiments.

Nonlinear forced vibration of rotating composite laminated cylindrical shells under hygrothermal environment

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Xiao Li ◽  
Wentao Jiang ◽  
Xiaochao Chen ◽  
Zhihong Zhou
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Multiple Scales ◽  
Cylindrical Shells ◽  
Nonlinear Partial Differential Equations ◽  
Dynamic Responses ◽  
Partial Differential ◽  
Moisture Concentration ◽  
Hygrothermal Environment ◽  
Laminated Cylindrical Shells

Abstract This work aims to study nonlinear vibration of rotating composite laminated cylindrical shells under hygrothermal environment and radial harmonic excitation. Based on Love’s nonlinear shell theory, and considering the effects of rotation-induced initial hoop tension, centrifugal and Coriolis forces, the nonlinear partial differential equations of the shells are derived by Hamilton’s principle, in which the constitutive relation and material properties of the shells are both hygrothermal-dependent. Then, the Galerkin approach is applied to discrete the nonlinear partial differential equations, and the multiple scales method is adopted to obtain an analytical solution on the dynamic response of the nonlinear shells under primary resonances of forward and backward traveling wave, respectively. The stability of the solution is determined by using the Routh–Hurwitz criterion. Some interesting results on amplitude–frequency relations and nonlinear dynamic responses of the shells are proposed. Special attention is given to the combined effects of temperature and moisture concentration on nonlinear resonance behavior of the shells.

Effects of Nonlinear Damping Washers on the Automatic Ball Balancer for Optical Disk Drives

2005 ◽  
Author(s):  
Cheng-Kuo Sung ◽  
Paul C. P. Chao ◽  
Ben-Cheng Yo
Keyword(s):  
Nonlinear Dynamics ◽  
Steady State ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Dynamic Performance ◽  
Nonlinear Damping ◽  
Disk Drives ◽  
Optical Disc ◽  
Scaled Model ◽  
Jump Phenomena

This study is devoted to explore the effect of nonlinear dynamics of damping washers on the dynamic performance of automatic ball balancer (ABB) system installed in optical disc drives. The ABB is generally used on rotational system to reduce vibration. Researches have been conducted to study the performance of the ABB by investigating the nonlinear dynamics of the system; however, the model adopted often consider the damping washer in a typical ABB suspension system as a linear one, which does not reflect the fact that the practical washers are inevitably exhibit nontrivial nonlinear dynamics at some range of operation, deviating the ABB performance away from the expecteds. In this study, a complete dynamic model of the ABB including a detailed nonlinear model of the damping washers based on experimental data for practical wahers is established. The method of multiple scales is then applied to formulate a scaled model to find all possible steady-state ball positions and analyze stabilities. It is found that with reasonable level of nonlinearity, the balancing balls of the ABB are still reside at the desired positions at steady state, rendering expected vibration reduction; however, jump phenomena also occurs as the spindle operated through natural frequency of the suspension, causing unwanted system vibrations. Numerical simulations and experiments are conducted to verify the theoretical findings. The obtained results are used to predict the level of residual vibration, with which the guidelines on choices of the nonlinear damping washers are distilled to achieve desired performance.

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).

Bifurcation and Chaos in Externally Excited Circular Cylindrical Shells

1996 ◽  
Vol 63 (3) ◽  
pp. 565-574 ◽  
Author(s):  
Char-Ming Chin ◽  
A. H. Nayfeh
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Multiple Scales ◽  
Period Doubling ◽  
Single Mode ◽  
Equilibrium Solutions ◽  
Multiple Attractors ◽  
Partial Differential ◽  
Quadratic Nonlinearities ◽  
Mode Solution

The nonlinear response of an infinitely long cylindrical shell to a primary excitation of one of its two orthogonal flexural modes is investigated. The method of multiple scales is used to derive four ordinary differential equations describing the amplitudes and phases of the two orthogonal modes by (a) attacking a two-mode discretization of the governing partial differential equations and (b) directly attacking the partial differential equations. The two-mode discretization results in erroneous solutions because it does not account for the effects of the quadratic nonlinearities. The resulting two sets of modulation equations are used to study the equilibrium and dynamic solutions and their stability and hence show the different bifurcations. The response could be a single-mode solution or a two-mode solution. The equilibrium solutions of the two orthogonal third flexural modes undergo a Hopf bifurcation. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos, multiple attractors, explosive bifurcations, and crises.

Multi-Frequency Multi-Modal Excitation of a Heated Annular Plate

2006 ◽  
Author(s):  
Haider N. Arafat ◽  
Ali H. Nayfeh
Keyword(s):  
Nonlinear Dynamics ◽  
Internal Resonance ◽  
Radial Direction ◽  
Natural Frequencies ◽  
Annular Plate ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Plate Theory ◽  
Frequency Excitation ◽  
The Difference

The forced nonlinear dynamics of a pre-buckled thermally loaded annular plate are investigated. The plate is modeled using the von Ka´rma´n plate theory and the heat equation. The heat, which is generated by the difference between the uniformly distributed temperatures at the inner and outer boundaries, is assumed to symmetrically flow in the radial direction. The amount of heat affects the natural frequencies, which may give rise to different internal resonance conditions. The method of multiple scales is used to examine the system axisymmetric responses when it is driven by an external multi-frequency excitation. The plate responses could be very complex exhibiting Hopf and cyclic-fold bifurcations, quasi-periodicity, chaos, and multiplicity of attractors.

On Nonlinear Dynamics of Nanotweezers

2016 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Bin Liu
Keyword(s):  
Nonlinear Dynamics ◽  
Frequency Response ◽  
Parametric Resonance ◽  
Taylor Series ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Van Der Waals ◽  
Van Der Waals Forces ◽  
The Other ◽  
Amplitude Frequency Response

This paper deals with amplitude-frequency response of electrostatic nanotube nanotweezer device system. Soft alternating current (AC) of frequency near natural frequency actuates the nanotubes. This leads the system into parametric resonance. The Method of Multiple Scales (MMS) in which the nonlinear electrostatic and van der Waals forces are expanded in Taylor series is used to compare two expansions, one up to third power and the other up to fifth power. The frequency response of the system is reported and the effects of van der Waals forces, electrostatic forces, and damping forces on the frequency response are investigated.

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