Nonlinear Dynamics of a Parametrically Excited Laminated Beam: Deterministic Excitation

2011 ◽  
Vol 464 ◽  
pp. 260-263 ◽  
Author(s):  
Xiang Jun Lan ◽  
Zhi Hua Feng ◽  
Hong Lin ◽  
Xiao Dong Zhu
Keyword(s):  
Nonlinear Dynamics ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Plate Theory ◽  
Third Order ◽  
Force Amplitude ◽  
Laminated Beam ◽  
Galerkin Approach ◽  
Nonlinear Dynamic Equation ◽  
Nonlinear Behaviors

The nonlinear dynamic equation of a laminated beam subject to parametrically deterministic excitation is derived based on the general von Karman-type equations and the Reddy third-order shear deformation plate theory. The first mode parametric resonance is taken into consideration using Galerkin approach. The modulation equations are obtained with the method of multiple scales. The frequency-amplitude and force-amplitude characters are investigated. Results show that the nonlinear behaviors belong to hardening effect.

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.

Nonlinear Dynamics of a Parametrically Excited Laminated Beam: Narrow-Band Random Excitation

2010 ◽  
Vol 43 ◽  
pp. 257-261
Author(s):  
Xiang Jun Lan ◽  
Zhi Hua Feng ◽  
Xiao Dong Zhu ◽  
Hong Lin
Keyword(s):  
Nonlinear Dynamics ◽  
Finite Difference Method ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Narrow Band ◽  
Joint Probability ◽  
Random Excitation ◽  
Laminated Beam ◽  
Stochastic Jump ◽  
Stochastic Jump And Bifurcation

The first mode parametric resonance of a laminated beam subject to narrow-band random excitation is taken into consideration. The method of multiple scales is used and the stochastic jump and bifurcation have been investigated aiming at the stationary joint probability of the response of the system by using finite difference method. Results show that stochastic jump occurs mainly in the region of triple valued solution. The higher is the frequency, the more probable is the jump from the stationary nontrivial branch to the trivial one, whereas the most probable motion gradually approaches the trivial one when the band width becomes higher.

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.

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.

scholarly journals Free Vibration and Hardening Behavior of Truss Core Sandwich Beam

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
J. E. Chen ◽  
W. Zhang ◽  
M. Sun ◽  
M. H. Yao
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Sandwich Beam ◽  
Third Order ◽  
First Order ◽  
Hardening Behavior ◽  
Truss Core ◽  
Averaged Equations ◽  
Core Height

The dynamic characteristics of simply supported pyramidal truss core sandwich beam are investigated. The nonlinear governing equation of motion for the beam is obtained by using a Zig-Zag theory. The averaged equations of the beam with primary, subharmonic, and superharmonic resonances are derived by using the method of multiple scales and then the corresponding frequency response equations are obtained. The influences of strut radius and core height on the linear natural frequencies and hardening behaviors of the beam are studied. It is illustrated that the first-order natural frequency decreases continuously and the second-order and third-order natural frequencies initially increase and then decrease with the increase of strut radius, and the first three natural frequencies all increase with the rise of the core height. Furthermore, the results indicate that the hardening behaviors of the beam become weaker with the increase of the rise of strut radius and core height. The mechanisms of variations in hardening behavior of the sandwich beam with the three types of resonances are detailed and discussed.

Bifurcations and Chaos of Composite Laminated Piezoelectric Rectangular Plate With One-to-Three Internal Resonance

2008 ◽  
Author(s):  
Zhi-Gang Yao ◽  
Wei Zhang
Keyword(s):  
Rectangular Plate ◽  
Multiple Scales ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Governing Equations ◽  
Third Order ◽  
Chaotic Motions ◽  
Piezoelectric Layers ◽  
Parametric And External Excitations ◽  
First Time

The bifurcations and chaotic motions of a simply supported symmetric cross-ply composite laminated piezoelectric rectangular plate are analyzed for the first time, which are forced by the transverse and in-plane excitations. It is assumed that different layers of symmetric cross-ply composite laminated piezoelectric rectangular plate are perfectly bonded to each other and with piezoelectric actuator layers embedded in the plate. Based on the Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the composite laminated piezoelectric rectangular plate are derived by using the Hamilton’s principle. The excitation loaded by piezoelectric layers is considered. The Galerkin’s approach is employed to discretize partial differential governing equations to a two-degree-of-freedom nonlinear system under combined the parametric and external excitations. The method of multiple scales is utilized to obtain the four-dimensional averaged equation. Numerical method is used to find the periodic and chaotic motions of the composite laminated piezoelectric rectangular plate. The numerical results show the existence of the periodic and chaotic motions in the averaged equation. It is found that the chaotic responses are especially sensitive to the forcing and the parametric excitations. The influence of the transverse, in-plane and piezoelectric excitations on the bifurcations and chaotic behaviors of the composite laminated piezoelectric rectangular plate is investigated numerically.

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 Oscillations of a Composite Laminated Plate with Parametrically and Externally Excitations

2013 ◽  
Vol 699 ◽  
pp. 641-644
Author(s):  
Xiao Li Bian ◽  
Shuang Bao Li
Keyword(s):  
Thin Plate ◽  
Nonlinear Oscillations ◽  
Plate Theory ◽  
Laminated Plate ◽  
Rectangular Thin Plate ◽  
Third Order ◽  
Composite Laminated Plate ◽  
Strain Relationship ◽  
Relationship Of ◽  
Equations Of Motions

Nonlinear oscillations of a simply-supported symmetric cross-ply composite laminated rectangular thin plate are investigated in this paper. The rectangular thin plate is subjected to the transversal and in-plane excitations. Based on the Reddy’s third-order shear deformation plate theory and the stress-strain relationship of the composite laminated plate, a two-degree-of-freedom non-autonomous nonlinear system governing equations of motions for the composite laminated rectangular thin plate is derived by using the Galerkin’s method. Numerical simulations illustrate that there exist complex nonlinear oscillations for composite laminated rectangular thin plate.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

Sign in / Sign up

Export Citation Format

Share Document