Parametric Resonance of Axially Accelerating Unidirectional Plates Partially Immersed in Fluid

2020 ◽  
Author(s):  
Hongying Li ◽  
Shumeng Zhang ◽  
Jian Li ◽  
Xibo Wang
Keyword(s):  
Differential Equations ◽  
Multiple Scales ◽  
Chaotic Behavior ◽  
Plate Theory ◽  
Three Dimension ◽  
Bifurcation Diagrams ◽  
Moving Plate ◽  
Fluid Structure ◽  
Coupled Equations ◽  
Motion State

Abstract This paper investigates the nonlinear vibration of an axially accelerating moving plate considering fluid-structure interaction. Nonlinear coupled equations of motion are derived by means of Kármán plate theory, the Galerkin method is then applied to transform the nonlinear partial differential equations into nonlinear ordinary differential equations. The steady-state response, various bifurcations and chaotic behavior of the system are studied by the multiple scales method and Runge-Kutta method. The dynamical characteristics of the system are examined via response curves and bifurcation diagrams of Poincaré maps. By three-dimension bifurcation diagrams, change of motion state can be easily observed along with the variation of system parameters during the whole parametric space, meanwhile, it is found that fluctuation amplitude plays a most significant role in the change of motion state for the fluid-structure coupling system.

Effect of Fluid–Structure Interaction on Vibration of Moving Sandwich Plate With Balsa Wood Core and Nanocomposite Face Sheets

2020 ◽  
Vol 12 (07) ◽  
pp. 2050078
Author(s):  
Elham Haghparast ◽  
Amirabbas Ghorbanpour-Arani ◽  
Ali Ghorbanpour Arani
Keyword(s):  
Sandwich Plate ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Fluid Structure Interaction ◽  
Two Phase ◽  
Moving Plate ◽  
Fluid Structure ◽  
Balsa Wood ◽  
Structure Interaction ◽  
Axially Moving

This research presents theoretical investigation to analyze vibration of axially moving sandwich plate floating on fluid. This plate is composed of balsa wood core and two nanocomposite face sheets where the three layers vibrated as an integrated sandwich. The fluid–structure interaction (FSI) effects on the stability of moving plate are considered for both ideal and viscous fluid. Halpin–Tsai model is utilized to determine the material properties of two-phase composite consist of uniformly distributed and randomly oriented carbon nanotubes (CNTs) through the PmPV (poly{(m-phenylenevinylene)-co-[(2,5-dioctoxy-p-phenylene)vinylene]}) matrix. The governing equations are derived based on sinusoidal shear deformation plate theory (SSDT) which is more accurate than the conventional theories, and significantly, it does not require a shear correction factor. Employing Hamilton’s principle, the equations of motion are obtained and solved by the semi-analytical method. Results indicated that the dimensionless frequencies of moving sandwich plate decrease rapidly with increasing the water level and they are almost independent of fluid level when it is higher than 50% of the plate length. The results of this investigation can be used in design and manufacturing of marine vessels and aircrafts.

Nonlinear Dynamics of Axially Moving Plates With Rotational Springs

2013 ◽  
Author(s):  
Mergen H. Ghayesh ◽  
Marco Amabili
Keyword(s):  
Nonlinear Dynamics ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Plate Theory ◽  
Lagrange Equation ◽  
Global Dynamics ◽  
Nonlinear Ordinary Differential Equations ◽  
Moving Plate ◽  
Time Histories ◽  
Axially Moving

In this paper, the in-plane and out-of-plane nonlinear dynamics of an axially moving plate with distributed rotational springs at boundaries is examined numerically. The Von Kármán plate theory along with the Kirchhoff’s hypothesis are employed to construct the kinetic and potential energies of the system. The Lagrange equation is used so as to obtain the equations of motion which are in the form of a set of second-order nonlinear ordinary differential equations. This set is recast into a set of first-order nonlinear ordinary differential equations with coupled terms. Gear’s backward-differentiation-formula is employed to integrate this set of equations numerically, yielding the generalized coordinates of the system as a function of time. The bifurcation diagrams of Poincaré maps are then constructed by sectioning these time histories in every period of the external excitation force. The results are shown in the form of time histories, phase-plane portraits, and Poincaré sections. The effect of the stiffness of the rotational springs on the global dynamics of the system is also investigated.

Nonlinear forced vibration of rectangular plates by modified multiple scale method

2021 ◽  
pp. 095440622110527
Author(s):  
Farzaneh Rabiee ◽  
Ali Asghar Jafari
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Forced Vibration ◽  
Multiple Scales ◽  
Plate Theory ◽  
Multiple Scales Method ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Classical Plate Theory ◽  
Nonlinear Forced Vibration

In the present study, the nonlinear forced vibration of a rectangular plate is investigated analytically using modified multiple scales method for the first time. The plate is subjected to transversal harmonic excitation, and the boundary condition is assumed to be simply supported. The von Karman nonlinear strain displacement relations are used. The extended Hamilton principle and classical plate theory are applied to derive the partial differential equations of motions. This research focuses on resonance case with 3:1 internal resonance. By applying Galerkin method, the nonlinear partial differential equations are transformed into time dependent nonlinear ordinary differential equations, which are then solved analytically by modified multiple scales method. This proposed approach is very simple and straightforward. The obtained results are then compared with both the traditional multiple scales method and previous studies, and excellent compatibility is noticed. The effect of some of the main parameters of the system is also examined.

Chaos in Robot Control Equations

1997 ◽  
Vol 07 (03) ◽  
pp. 707-720 ◽  
Author(s):  
Shrinivas Lankalapalli ◽  
Ashitava Ghosal
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
System Of Nonlinear Equations ◽  
Nonlinear Ordinary Differential Equations ◽  
Bifurcation Diagrams ◽  
Chaotic Motions ◽  
Repetitive Motion ◽  
Route To Chaos

The motion of a feedback controlled robot can be described by a set of nonlinear ordinary differential equations. In this paper, we examine the system of two second-order, nonlinear ordinary differential equations which model a simple two-degree-of-freedom planar robot, undergoing repetitive motion in a plane in the absence of gravity, and under two well-known robot controllers, namely a proportional and derivative controller and a model-based controller. We show that these differential equations exhibit chaotic behavior for certain ranges of the proportional and derivative gains of the controller and for certain values of a parameter which quantifies the mismatch between the model and the actual robot. The system of nonlinear equations are non-autonomous and the phase space is four-dimensional. Hence, it is difficult to obtain significant analytical results. In this paper, we use the Lyapunov exponent to test for chaos and present numerically obtained chaos maps giving ranges of gains and mismatch parameters which result in chaotic motions. We also present plots of the chaotic attractor and bifurcation diagrams for certain values of the gains and mismatch parameters. From the bifurcation diagrams, it appears that the route to chaos is through period doubling.

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

scholarly journals Chaotic Behavior in Gear System. Adaptation of Bifurcation Diagrams and Method of Statistical Mechanics.

1995 ◽  
Vol 61 (587) ◽  
pp. 3108-3115
Author(s):  
Keijin Sato ◽  
Sumio Yamamoto ◽  
Kazutaka Yokota ◽  
Toshihiro Aoki ◽  
Shu Karube
Keyword(s):  
Statistical Mechanics ◽  
Chaotic Behavior ◽  
Gear System ◽  
Bifurcation Diagrams ◽  
System Adaptation

Nonlinear Coupled Transverse and Axial Vibration of Variable Cross-Section Beam Flexures Interconnecting Rigid Body

2016 ◽  
Author(s):  
Hasan Malaeke ◽  
Hamid Moeenfard ◽  
Amir H. Ghasemi
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Cross Section ◽  
Multiple Scales ◽  
Nonlinear Behavior ◽  
Single Mode ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Closed Form Solutions ◽  
Dynamic Performances

The objective of this paper is to analytically study the nonlinear behavior of variable cross-section beam flexures interconnecting an eccentric rigid body. Hamilton’s principle is utilized to obtain the partial differential equations governing the nonlinear vibration of the system as well as the corresponding boundary conditions. Using a single mode approximation, the governing equations are reduced to a set of two nonlinear ordinary differential equations in terms of end displacement components of the beam which are coupled due to the presence of the transverse eccentricity. The method of multiple scales are employed to obtain parametric closed-form solutions. The obtained analytical results are compared with the numerical ones and excellent agreement is observed. These analytical expressions provide design insights for modeling and optimization of more complex flexure mechanisms for improved dynamic performances.

Analyzing Bifurcation, Stability and Chaos for a Passive Walking Biped Model with a Sole Foot

2018 ◽  
Vol 28 (09) ◽  
pp. 1850113 ◽  
Author(s):  
Maysam Fathizadeh ◽  
Sajjad Taghvaei ◽  
Hossein Mohammadi
Keyword(s):  
Dynamic Models ◽  
Chaotic Behavior ◽  
Bifurcation Diagrams ◽  
Behavior Simulation ◽  
Passive Walking ◽  
Low Energy Consumption ◽  
Nonlinear Dynamic Models ◽  
Soles Of The Feet ◽  
The Stability ◽  
Intrinsic Characteristic

Human walking is an action with low energy consumption. Passive walking models (PWMs) can present this intrinsic characteristic. Simplicity in the biped helps to decrease the energy loss of the system. On the other hand, sufficient parts should be considered to increase the similarity of the model’s behavior to the original action. In this paper, the dynamic model for passive walking biped with unidirectional fixed flat soles of the feet is presented, which consists of two inverted pendulums with L-shaped bodies. This model can capture the effects of sole foot in walking. By adding the sole foot, the number of phases of a gait increases to two. The nonlinear dynamic models for each phase and the transition rules are determined, and the stable and unstable periodic motions are calculated. The stability situations are obtained for different conditions of walking. Finally, the bifurcation diagrams are presented for studying the effects of the sole foot. Poincaré section, Lyapunov exponents, and bifurcation diagrams are used to analyze stability and chaotic behavior. Simulation results indicate that the sole foot has such a significant impression on the dynamic behavior of the system that it should be considered in the simple PWMs.

Bifurcation and Chaos in a Periodically Driven Quartic Oscillator at Relativistic Energies

1997 ◽  
Vol 07 (04) ◽  
pp. 945-949
Author(s):  
Sang Wook Kim ◽  
Hai-Woong Lee
Keyword(s):  
Chaotic Behavior ◽  
Relativistic Effects ◽  
Critical Value ◽  
Nonrelativistic Limit ◽  
Classical Dynamics ◽  
Bifurcation Diagrams ◽  
Bifurcation And Chaos ◽  
Force Amplitude ◽  
Regular Motion ◽  
Periodically Driven

The classical dynamics of a damped quartic oscillator driven by a sinusoidal force is investigated, with particular attention to the effects that arise when the motion of the oscillator becomes relativistic. Bifurcation diagrams constructed numerically indicate that, as relativistic effects become strong, chaotic behavior exhibited by the oscillator in the nonrelativistic limit at large force amplitude is replaced by a period-1 regular motion. At relativistic energies, therefore, a transition from chaos to a regular motion occurs as the force amplitude is increased beyond a critical value. The transition is seen to occur abruptly with a slight increase of the force amplitude from below to above the critical value. The sudden destruction of the chaotic attractor is probably triggered by the mechanism of crisis.

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