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.

Closed-Form Solutions of Axisymmetrical Transverse Vibrations of Concave Parabolic Thickness Annular Plates

2009 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equation Of Motion ◽  
Mode Shapes ◽  
Partial Differential ◽  
First Order ◽  
Annular Plates ◽  
Transverse Vibrations

This paper deals with transverse vibrations of axisymmetrical annular plates of concave parabolic thickness. A closed-form solution of the partial differential equation of motion is reported. An approach in which both method of multiple scales and method of factorization have been employed is presented. The method of multiple scales is used to reduce the partial differential equation of motion to two simpler partial differential equations that can be analytically solved. The solutions of the two differential equations are two levels of approximation of the exact solution of the problem. Using the factorization method for solving the first differential equation, which is homogeneous and includes a fourth-order spatial-dependent operator and second-order time-dependent operator, the general solution is obtained in terms of hypergeometric functions. The first diferential equation and the second differential equation (nonhomogeneous) along with the given boundary conditions give so-called zero-order and first-order approximations, respectively, of the natural frequencies and mode shapes. Any boundary conditions could be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes could be further studied using the first-order approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.

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).

Second Order Perturbation Analysis of a Forced Nonlinear Mathieu Equation

2012 ◽  
Author(s):  
Venkatanarayanan Ramakrishnan ◽  
Brian F. Feeny
Keyword(s):  
Perturbation Analysis ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Mathieu Equation ◽  
Second Order ◽  
Order Perturbation ◽  
Cubic Nonlinearity ◽  
First Order ◽  
Order Expansion ◽  
Direct Forcing

The present study deals with the response of a forced nonlinear Mathieu equation. The equation considered has parametric excitation at the same frequency as direct forcing and also has cubic nonlinearity and damping. A second-order perturbation analysis using the method of multiple scales unfolds numerous resonance cases and system behavior that were not uncovered using first-order expansions. All resonance cases are analyzed. We numerically plot the frequency response of the system. The existence of a superharmonic resonance at one third the natural frequency was uncovered analytically for linear system. (This had been seen previously in numerical simulations but was not captured in the first-order expansion.) The effect of different parameters on the response of the system previously investigated are revisited.

scholarly journals An Evolutionary Analytic Method of Multi-DOF Nonlinear Coupling Dynamic Model for Controllable Close-Chain Linkage Mechanism System

2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
Ru-Gui Wang ◽  
Gan-Wei Cai ◽  
Xiao-Rong Zhou
Keyword(s):  
Approximate Solution ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Dynamic Responses ◽  
Analytic Method ◽  
Nonlinear Coupling ◽  
Linkage Mechanism ◽  
First Order ◽  
Coupling System ◽  
Coupling Dynamic

The 2-DOF controllable close-chain linkage mechanism is investigated in this paper. Based on the characteristics of the multi-DOF nonlinear coupling dynamic equation of the system established by the finite element method, an analytic method of multiple-scales Newmark is presented after thinking about the method of perturbation and the method of numerical analysis. Firstly, the first-order approximate solution of the dynamic responses of the system at the time of t is calculated by the multiple scales method. Then, taken the first-order approximate solution as the initialization of the generalized coordinate of the system, the stable dynamic response of the system is obtained by the implicit Newmark method. The simulation and experimental results are given in the end. The studies indicate that the method of multiple-scales Newmark is correct and practicable to study the dynamic characteristics of such kind of multi-DOF nonlinear coupling system.

Effects of Flexural Stiffness, Axial Velocity and Internal Support Locations on Translating Beam’s Natural Frequencies

2014 ◽  
Vol 532 ◽  
pp. 316-319 ◽  
Author(s):  
Ferid Köstekci
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Flexural Rigidity ◽  
Equations Of Motion ◽  
Flexural Stiffness ◽  
Transverse Vibrations ◽  
Internal Support ◽  
Simple Support ◽  
Moving Speed

The aim of this paper is to examine the natural frequencies of beams for different flexural stiffness, internal simple support locations and axial moving speed. In the present investigation, the linear transverse vibrations of an axially translating beam are considered based on Euler-Bernoulli model. The beam is passing through two frictionless guides and has an internal simple support between the guides. The governing differential equations of motion are derived using Hamiltons Principle for two regions of the beam. The method of multiple scales is employed to obtain approximate analytical solution. Some numerical calculations are conducted to present the effects of flexural rigidity, mean translating speed and different internal support locations on natural frequencies.

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 Travelling Beam Subjected to Multi-Frequency Parametric Excitation

2014 ◽  
Vol 592-594 ◽  
pp. 2076-2080 ◽  
Author(s):  
Bamadev Sahoo ◽  
L.N. Panda ◽  
Goutam Pohit
Keyword(s):  
Parametric Excitation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equation Of Motion ◽  
Cubic Nonlinearity ◽  
First Order ◽  
Reduced Equations ◽  
Stable Periodic ◽  
Bifurcation And Stability ◽  
First Order Differential Equations

This paper deals with two frequency parametric excitation in presence of internal resonance. The cubic nonlinearity is inserted into the equation of motion by considering the mid-line stretching of the beam. The perturbation method of multiple scales is applied directly to the governing nonlinear fourth order integro-partial differential equation of motion. This leads to a set of first order differential equations known as the reduced equations or normalized reduced equations, which are utilized to determine the additional instability zones, appeared in the trivial state stability plot, the bifurcation and stability of fixed-points, periodic, quasi-periodic, mixed mode and chaotic responses. The transition of system behaviour from stable periodic to unstable chaotic occurs through intermittency route

On Nonlinear Motions of Two-Degree-of-Freedom Nonlinear Systems with Repeated Linearized Natural Frequencies

2019 ◽  
Vol 29 (10) ◽  
pp. 1950132
Author(s):  
Hua-Zhen An ◽  
Xiao-Dong Yang ◽  
Feng Liang ◽  
Wei Zhang ◽  
Tian-Zhi Yang ◽  
...  
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Phase Portraits ◽  
Chaotic Motions ◽  
Nonlinear Parameters ◽  
Energy Ratios ◽  
Two Degree Of Freedom ◽  
Nonlinear Motions ◽  
Phase Differences

In this paper, we investigate systematically the vibration of a typical 2DOF nonlinear system with repeated linearized natural frequencies. By application of Descartes’ rule of signs, we demonstrate that there are 14 types of roots describing different modal motions for varying nonlinear parameters. The method of multiple scales is used to obtain the amplitude-phase portraits by introducing the energy ratios and phase differences. The typical nonlinear in-unison and elliptic out-of-unison modal motions are located for the 14 types of roots and then validated by numerical simulations. It is found that the normal in-unison modal motions, elliptic out-of-unison modal motions are analogous to the polarization of classical optic theory. Further, some kinds of periodic and chaotic motions under out-of-unison and in-unison excitations are investigated numerically. The result of this study offers a detailed discussion of nonlinear modal motions and responses of 2DOF systems with cubic nonlinear terms.

scholarly journals Nonlinear Response of Cantilever Beams to Combination and Subcombination Resonances

1998 ◽  
Vol 5 (5-6) ◽  
pp. 277-288 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Nonlinear Ordinary Differential Equations ◽  
Response Curves ◽  
First Order ◽  
Aspect Ratios

The nonlinear planar response of cantilever metallic beams to combination parametric and external subcombination resonances is investigated, taking into account the effects of cubic geometric and inertia nonlinearities. The beams considered here are assumed to have large length-to-width aspect ratios and thin rectangular cross sections. Hence, the effects of shear deformations and rotatory inertia are neglected. For the case of combination parametric resonance, a two-mode Galerkin discretization along with Hamilton’s extended principle is used to obtain two second-order nonlinear ordinary-differential equations of motion and associated boundary conditions. Then, the method of multiple scales is applied to obtain a set of four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two excited modes. For the case of subcombination resonance, the method of multiple scales is applied directly to the Lagrangian and virtual-work term. Then using Hamilton’s extended principle, we obtain a set of four first-order nonlinear ordinary-differential equations governing the amplitudes and phases of the two excited modes. In both cases, the modulation equations are used to generate frequency- and force-response curves. We found that the trivial solution exhibits a jump as it undergoes a subcritical pitchfork bifurcation. Similarly, the nontrivial solutions also exhibit jumps as they undergo saddle-node bifurcations.

scholarly journals Nonlinear Dynamics of an Electrorheological Sandwich Beam with Rotary Oscillation

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Kexiang Wei ◽  
Wenming Zhang ◽  
Ping Xia ◽  
Yingchun Liu
Keyword(s):  
Dynamic Characteristics ◽  
Loss Factor ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Parametric Instability ◽  
Sandwich Beam ◽  
The Finite Element Method ◽  
Rotating Beam ◽  
Flexible Beams ◽  
Instability Regions

The dynamic characteristics and parametric instability of a rotating electrorheological (ER) sandwich beam with rotary oscillation are numerically analyzed. Assuming that the angular velocity of an ER sandwich beam varies harmonically, the dynamic equation of the rotating beam is first derived based on Hamilton's principle. Then the coupling and nonlinear equation is discretized and solved by the finite element method. The multiple scales method is employed to determine the parametric instability of the structures. The effects of electric field on the natural frequencies, loss factor, and regions of parametric instability are presented. The results obtained indicate that the ER material layer has a significant effect on the vibration characteristics and parametric instability regions, and the ER material can be used to adjust the dynamic characteristics and stability of the rotating flexible beams.

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