Analysis of Transverse Vibration of Axially Moving Viscoelastic Sandwich Beam with Time-Dependent Velocity

2011 ◽  
Vol 338 ◽  
pp. 487-490 ◽  
Author(s):  
Hai Wei Lv ◽  
Ying Hui Li ◽  
Qi Kuan Liu ◽  
Liang Li
Keyword(s):  
Transverse Vibration ◽  
Multiple Scales ◽  
Sandwich Beam ◽  
Core Layer ◽  
Response Elimination ◽  
Response Curves ◽  
Mean Velocity ◽  
State Response ◽  
Axially Moving ◽  
The Stability

Transverse vibration of an axially moving viscoelastic sandwich beam is investigated in this paper. Based on the Kelvin constitutive equation, transverse controlling equation is established. First of all, the multiple scales method is applied to obtained steady-state response. Elimination of scales terms will give us the amplitude of vibrations. Additionally, the stability conditions of trivial and non-trivial solutions are analyzed using Routh-Hurwitz criterion. Eventually, numerical results are obtained to show the thickness of core layer, mean velocity, the amplitude of fluctuation effects on natural frequencies and response curves.

NONLINEAR VIBRATION OF A CANTILEVER WITH A DERJAGUIN–MÜLLER–TOPOROV CONTACT END

2008 ◽  
Vol 08 (01) ◽  
pp. 25-40 ◽  
Author(s):  
Q.-Q. HU ◽  
C. W. LIM ◽  
L.-Q. CHEN
Keyword(s):  
Steady State ◽  
Multiple Scales ◽  
Excitation Amplitude ◽  
Response Curves ◽  
State Response ◽  
Linearized Stability ◽  
Steady State Responses ◽  
The Stability ◽  
Euler Bernoulli Beam ◽  
State Responses

In this paper, the principal resonance is investigated for a cantilever with a contact end. The cantilever is modeled as an Euler–Bernoulli beam, and the contact is modeled by the Derjaguin–Müller–Toporov theory. The problem is formulated as a linear nonautonomous partial-differential equation with a nonlinear autonomous boundary condition. The method of multiple scales is applied to determine the steady-state response. The equation of response curves is derived from the solvability condition of eliminating secular terms. The stability of steady-state responses is analyzed by using the Lyapunov-linearized stability theory. Numerical examples are presented to highlight the effects of the excitation amplitude, the damping coefficient, and the coefficients related to the contact.

Combination Resonances of Traveling Unidirectional Plates Partially Immersed in Fluid with Time-Dependent Axial Velocity and Axially Varying Tension

2021 ◽  
Author(s):  
Hongying Li ◽  
Xibo Wang ◽  
Shumeng Zhang ◽  
Jian Li
Keyword(s):  
Nonlinear Vibration ◽  
Multiple Scales ◽  
Complex Dynamics ◽  
Axial Direction ◽  
Mean Velocity ◽  
Velocity Amplitude ◽  
Study Results ◽  
Axially Moving ◽  
The Stability ◽  
The Galerkin Method

Abstract Nonlinear vibrations of axially moving plates partially immersed in fluid are investigated in this paper. The system has time dependency in velocity as well as tension in axial direction. The Galerkin method is used to solve the nonlinear vibration differential equation. The method of multiple scales and Runge-Kutta method are applied to solve the nonlinear vibration response of the system. Additionally, the stability conditions of trivial and nontrivial solutions are analyzed using the Routh-Hurwitz criterion. The effects of mean velocity, amplitude of pulsating velocity, mean tension, amplitude of pulsating tension and pulsating frequency on the complex dynamics of the system are obtained. The study results reveal rich dynamic behaviors of fluid-structure coupling system.

scholarly journals Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Bamadev Sahoo ◽  
L. N. Panda ◽  
G. Pohit
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Time History ◽  
Mean Value ◽  
Phase Portraits ◽  
Axially Moving Beam ◽  
Internal Resonances ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability

The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beam velocity is assumed to be comprised of a constant mean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of second mode is approximately three times that of first mode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance.

Nonlinear Vibrations of a Thin Plate Under Simultaneous Internal and External Resonances

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
Usama H. Hegazy
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Thin Plate ◽  
Nonlinear Response ◽  
Phase Plane ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
State Response ◽  
The Stability ◽  
Parametric And External Excitations

The dynamic behavior of a rectangular thin plate under parametric and external excitations is investigated. The motion of the thin plate is modeled by coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought by applying the method of multiple scales. A reduced system of four first-order ordinary differential equations is determined to describe the time variation of the amplitudes and phases of the vibration in the horizontal and vertical directions. The steady-state response and the stability of the solutions for various parameters are studied numerically, using the frequency-response function and the phase-plane methods. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. Moreover, the chaotic motion of the thin plate is found by numerical simulation.

scholarly journals Nonlinear Models for Transverse Forced Vibration of Axially Moving Viscoelastic Beams

2011 ◽  
Vol 18 (1-2) ◽  
pp. 281-287 ◽  
Author(s):  
Hu Ding ◽  
Li-Qun Chen
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Steady State ◽  
Transverse Vibration ◽  
Nonlinear Models ◽  
Transverse Motion ◽  
Steady State Response ◽  
Partial Differential ◽  
State Response ◽  
Axially Moving

Nonlinear models of transverse vibration of axially moving viscoelastic beams subjected external transverse loads via steady-state periodical response are numerically investigated. An integro-partial-differential equation and a partial-differential equation of transverse motion can be derived respectively from a model of the coupled planar vibration for an axially moving beam. The finite difference scheme is developed to calculate steady-state response for the model of coupled planar and the two models of transverse motion under the simple support boundary. Numerical results indicate that the amplitude of the steady-state response for the model of coupled vibration and two models of transverse vibration predict qualitatively the same tendencies with the changing parameters and the integro-partial-differential equation gives results more closely to the coupled planar vibration.

scholarly journals Transverse Vibrations and Stability of Axially Traveling Sandwich Beam With Soft Core

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Xiao-Dong Yang ◽  
Wei Zhang ◽  
Li-Qun Chen
Keyword(s):  
Critical Speed ◽  
Natural Frequencies ◽  
Sandwich Beam ◽  
Axial Loading ◽  
Core Layer ◽  
Soft Core ◽  
Axially Moving Beam ◽  
Transverse Vibrations ◽  
Axially Moving ◽  
The Galerkin Method

The transverse vibrations and stability of an axially moving sandwich beam are studied in this investigation. The face layers are assumed to be in the membrane state, which bears only axial loading but no bending. Only shear deformation is considered for the soft core layer. The governing partial equation is derived using Newton's second law and then transferred into a dimensionless form. The Galerkin method and the complex mode method are employed to study the natural frequencies. In comparison with the classical homogenous axially moving beam, the gyroscopic matrix is no longer skew-symmetric because of the introduction of the soft core. The critical speed for the divergence of the axially moving sandwich beam is analytically obtained. The contribution of the core layer shear modulus to the natural frequencies and critical speed is discussed.

scholarly journals Temperature Effects on Nonlinear Vibration Behaviors of Euler-Bernoulli Beams with Different Boundary Conditions

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Yaobing Zhao ◽  
Chaohui Huang
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Temperature Effects ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Different Boundary Conditions ◽  
The Stability ◽  
The Galerkin Method ◽  
Close Form

This paper is concerned with temperature effects on the modeling and vibration characteristics of Euler-Bernoulli beams with symmetric and nonsymmetric boundary conditions. It is assumed that in the considered model the temperature increases/decreases instantly, and the temperature variation is uniformly distributed along the length and the cross-section. By using the extended Hamilton’s principle, the mathematical model which takes into account thermal and mechanical loadings, represented by partial differential equations (PDEs), is established. The PDEs of the planar motion are discretized to a set of second-order ordinary differential equations by using the Galerkin method. As to three different boundary conditions, eigenvalue analyses are performed to obtain the close-form eigenvalue solutions. First four natural frequencies with thermal effects are investigated. By using the Lindstedt-Poincaré method and multiple scales method, the approximate solutions of the nonlinear free and forced vibrations (primary, super, and subharmonic resonances) are obtained. The influences of temperature variations on response amplitudes, the localisation of the resonance zones, and the stability of the steady-state solutions are investigated, through examining frequency response curves and excitation response curves. Numerical results show that response amplitudes, the number and the stability of nontrivial solutions, and the hardening-spring characteristics are all closely related to temperature changes. As to temperature effects on vibration behaviors of structures, different boundary conditions should be paid more attention.

scholarly journals The Combined Internal and Principal Parametric Resonances on Continuum Stator System of Asynchronous Machine

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Baizhou Li ◽  
Qichang Zhang
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Defense Industry ◽  
Response Curves ◽  
Shell System ◽  
State Response ◽  
Noise Characteristics ◽  
Runge Kutta Method ◽  
Nonlinear Dynamic Equations ◽  
Parametric Resonances

With the increasing requirement of quiet electrical machines in the civil and defense industry, it is very significant and necessary to predict the vibration and noise characteristics of stator and rotor in the early conceptual phase. Therefore, the combined internal and principal parametric resonances of a stator system excited by radial electromagnetic force are presented in this paper. The stator structure is modeled as a continuum double-shell system which is loaded by a varying distributed electromagnetic load. The nonlinear dynamic equations are derived and solved by the method of multiple scales. The influences of mechanical and electromagnetic parameters on resonance characteristics are illustrated by the frequency-response curves. Furthermore, the Runge-Kutta method is adopted to numerically analyze steady-state response for the further understanding of the resonance characteristics with different parameters.

Steady-State Transverse Response in Coupled Planar Vibration of Axially Moving Viscoelastic Beams

2010 ◽  
Vol 132 (1) ◽  
Author(s):  
Li-Qun Chen ◽  
Hu Ding
Keyword(s):  
Steady State ◽  
Transverse Vibration ◽  
Nonlinear Models ◽  
Numerical Results ◽  
State Response ◽  
Planar Vibration ◽  
Transverse Loads ◽  
Steady State Responses ◽  
Axially Moving ◽  
Transverse Response

Steady-state periodical response is investigated for planar vibration of axially moving viscoelastic beams subjected external transverse loads. A model of the coupled planar vibration is established by introducing a coordinate transform. The model can reduce to two nonlinear models of transverse vibration. The finite difference scheme is developed to calculate steady-state response numerically. Numerical results demonstrate there are steady-state periodic responses in transverse vibration, and resonance occurs if the external load frequency approaches the linear natural frequencies. The effect of material parameters and excitation parameters on the amplitude of the steady-state responses are examined. Numerical results also indicate that the model of coupled vibration and two models of transverse vibration predict qualitatively the same tendencies with the changing parameters, and the two models of transverse vibration yield satisfactory results.

scholarly journals Nonlinear Parametric Vibration and Chaotic Behaviors of an Axially Accelerating Moving Membrane

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Mingyue Shao ◽  
Jimei Wu ◽  
Yan Wang ◽  
Qiumin Wu
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibration ◽  
Plate Theory ◽  
Velocity Variation ◽  
Mean Velocity ◽  
Thin Plate Theory ◽  
Time Histories ◽  
Axially Moving ◽  
The Stability ◽  
The Galerkin Method

Nonlinear vibration characteristics of a moving membrane with variable velocity have been examined. The velocity is presumed as harmonic change that takes place over uniform average speed, and the nonlinear vibration equation of the axially moving membrane is inferred according to the D’Alembert principle and the von Kármán nonlinear thin plate theory. The Galerkin method is employed for discretizing the vibration partial differential equations. However, the solutions concerning to differential equations are determined through the 4th order Runge–Kutta technique. The results of mean velocity, velocity variation amplitude, and aspect ratio on nonlinear vibration of moving membranes are emphasized. The phase-plane diagrams, time histories, bifurcation graphs, and Poincaré maps are obtained; besides that, the stability regions and chaotic regions of membranes are also obtained. This paper gives a theoretical foundation for enhancing the dynamic behavior and stability of moving membranes.

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