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.

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.

Dynamic Behavior of an AMB/Supported Rotor Subject to Parametric Excitation

2006 ◽  
Vol 128 (5) ◽  
pp. 646-652 ◽  
Author(s):  
Y. A. Amer ◽  
M. H. Eissa ◽  
U. H. Hegazy ◽  
A. S. Sabbah
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multiple Scales ◽  
Initial Conditions ◽  
Dynamical Behavior ◽  
Approximate Solutions ◽  
Sensitive Dependence ◽  
Response Function Method ◽  
The Stability ◽  
Gyroscopic Effects

The dynamical behavior of a parametrically excited simple rigid disk-rotor supported by active magnetic bearings (AMB) is investigated, without gyroscopic effects. The principal parametric resonance case is considered and studied. The motion of the rotor is modeled by a coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought applying the method of multiple scales. A reduced system of four first-order ordinary differential equations are 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 method. The numerical results show that the system behavior includes multiple solutions, jump phenomenon, and sensitive dependence on initial conditions. It is also shown that the system parameters have different effects on the nonlinear response of the rotor. Results are compared to previously published work.

A Stability Analysis of Constrained Rotating Disks With Different Boundary Conditions

2007 ◽  
Author(s):  
Ramin M. H. Khorasany ◽  
Stanley G. Hutton
Keyword(s):  
Boundary Conditions ◽  
High Speed ◽  
Equations Of Motion ◽  
Order Partial Differential Equation ◽  
Rotating Disks ◽  
Different Boundary Conditions ◽  
The Difference ◽  
The Stability ◽  
The Galerkin Method ◽  
Mass Spring

The vibration behavior of constrained high speed rotating disks is of interest in industries as diverse as: aerospace, computer disk manufacture and saw design and usage. The purpose of this study is to investigate the stability behavior of guided circular disks with different boundary conditions. The equations of motion are developed for circular rotating disks constrained by space fixed linear, mass, spring, damper systems. The resulting equation of motion is a two dimensional fourth order partial differential equation that requires numerical solution. The Galerkin Method is employed using the eigenfunctions of the stationary non-constrained disk as approximation functions. Of interest is the effect on stability of conditions at the inner boundary. In particular the difference in behavior for centrally clamped, and splined disks (those disks that run on a spline arbor) is investigated. Also discussed is the effect of constraints on the flutter and divergence instability boundaries. Preliminary experimental results are presented for constrained splined disks, and these results are compared with the analytical predictions.

Exact Elasticity Solutions for Stresses and Deflections in Curved Beams and Rings of Exponential and T-Sections

1991 ◽  
Author(s):  
Cemil Bagci
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Plane Stress ◽  
Numerical Results ◽  
Neutral Axis ◽  
Curved Beams ◽  
Equilibrium Stress ◽  
Different Boundary Conditions ◽  
Stress Functions ◽  
Elasticity Solutions

Abstract Exact elasticity solutions for stresses and deflections (displacements) in curved beams and rings of varying thicknesses are developed using polar elasticity and state of plane stress. Basic forms of differential equations of equilibrium, stress functions, and differential equations of compatibility are given. They are solved to develop expressions for radial, tangential, and shearing stresses for moment, force, and combined loadings. Neutral axis location for each type of loading is determined. Expressions for displacements are developed utilizing strain-displacement relationships of polar elasticity satisfying boundary conditions on displacements. In case of full rings stresses are as in curved beams with properly defined moment loading, but displacements differ satisfying different boundary conditions. The developments for constant thicknesses are used to develop solutions for curved beams and rings with T-sections. Comparative numerical results are given.

Free and Forced Vibrations of the Strongly Nonlinear Cubic-Quintic Duffing Oscillators

2017 ◽  
Vol 72 (1) ◽  
pp. 59-69 ◽  
Author(s):  
M.M. Fatih Karahan ◽  
Mehmet Pakdemirli
Keyword(s):  
Numerical Simulations ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Free And Forced Vibrations ◽  
Approximate Analytical Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

AbstractStrongly nonlinear cubic-quintic Duffing oscillatoris considered. Approximate solutions are derived using the multiple scales Lindstedt Poincare method (MSLP), a relatively new method developed for strongly nonlinear oscillators. The free undamped oscillator is considered first. Approximate analytical solutions of the MSLP are contrasted with the classical multiple scales (MS) method and numerical simulations. It is found that contrary to the classical MS method, the MSLP can provide acceptable solutions for the case of strong nonlinearities. Next, the forced and damped case is treated. Frequency response curves of both the MS and MSLP methods are obtained and contrasted with the numerical solutions. The MSLP method and numerical simulations are in good agreement while there are discrepancies between the MS and numerical solutions.

scholarly journals Solving nonlinear boundary value problems by the Galerkin method with sinc functions

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Sertan Alkan ◽  
Aydin Secer
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Nonlinear Boundary Value Problems ◽  
Sinc Approximation ◽  
Sinc Functions ◽  
The Galerkin Method

AbstractIn this paper, the sinc-Galerkin method is used for numerically solving a class of nonlinear differential equations with boundary conditions. The importance of this study is that sinc approximation of the nonlinear term is stated as a new theorem. The method introduced here is tested on some nonlinear problems and is shown to be a very efficient and powerful tool for obtaining approximate solutions of nonlinear ordinary differential equations.

The Influence of Nonlinear Boundary Conditions on the Nonplanar Autoparametric Responses of an Inextensible Beam

2001 ◽  
Author(s):  
Haider N. Arafat ◽  
Ali H. Nayfeh
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Nonlinear Behavior ◽  
Nonlinear Boundary Conditions ◽  
Nonlinear Springs ◽  
Out Of Plane

Abstract The nonplanar responses of a beam clamped at one end and restrained by nonlinear springs at the other end is investigated under a primary resonance base excitation. The beam’s geometry and the springs’ linear stiffnesses are such that the system possesses a one-to-one autoparametric resonance between the nth in-plane and out-of-plane modes. The beam is modeled using Euler-Bernoulli theory and includes cubic geometric and inertia nonlinearities. The objective is to assess the influence of the nonlinear boundary conditions on the beam’s oscillations. To this end, the method of multiple scales is directly applied to the integral-partial-differential equations of motion and associated boundary conditions. The result is a set of four nonlinear ordinary-differential equations that govern the slow dynamics of the system. Solutions of these modulation equations are then used to characterize the system’s nonlinear behavior.

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.

scholarly journals Nonlinear Forced Vibration of a Viscoelastic Buckled Beam with 2 : 1 Internal Resonance

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Liu-Yang Xiong ◽  
Guo-Ce Zhang ◽  
Hu Ding ◽  
Li-Qun Chen
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Mode Shapes ◽  
Analytical Approximations ◽  
Buckled Beam ◽  
Validity Range ◽  
The Galerkin Method

Nonlinear dynamics of a viscoelastic buckled beam subjected to primary resonance in the presence of internal resonance is investigated for the first time. For appropriate choice of system parameters, the natural frequency of the second mode is approximately twice that of the first providing the condition for 2 : 1 internal resonance. The ordinary differential equations of the two mode shapes are established using the Galerkin method. The problem is replaced by two coupled second-order differential equations with quadratic and cubic nonlinearities. The multiple scales method is applied to derive the modulation-phase equations. Steady-state solutions of the system as well as their stability are examined. The frequency-amplitude curves exhibit the steady-state response in the directly excited and indirectly excited modes due to modal interaction. The double-jump, the saturation phenomenon, and the nonperiodic region phenomena are observed illustrating the influence of internal resonance. The validity range of the analytical approximations is assessed by comparing the analytical approximate results with a numerical solution by the Runge-Kutta method. The unstable regions in the internal resonance are explored via numerical simulations.

Dynamic Stability of a Spinning Timoshenko Beam Subjected to a Moving Mass-Spring-Damper Unit

2010 ◽  
Author(s):  
T. H. Young ◽  
M. S. Chen
Keyword(s):  
Dynamic Stability ◽  
Timoshenko Beam ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Axial Direction ◽  
Longitudinal Axis ◽  
Moving Mass ◽  
The Stability ◽  
The Galerkin Method ◽  
Mass Spring

This paper investigates the dynamic stability of a finite Timoshenko beam spinning along its longitudinal axis and subjected to a moving mass-spring-damper (MSD) unit traveling in the axial direction. The mass of the moving MSD unit makes contact with the beam all the time during traveling. Due to the moving MSD unit, the beam is acted upon by a periodic, parametric excitation. In this work, the equations of motion of the beam are first discretized by the Galerkin method. The discretized equations of motion are then partially uncoupled by the modal analysis procedure suitable for gyroscopic systems. Finally the method of multiple scales is used to obtain the stability boundaries of the beam. Numerical results show that if the displacement of the MSD unit is equal to only one of the two transverse displacements of the beam, very large unstable regions may appear at main resonances.

Sign in / Sign up

Export Citation Format

Share Document