Dynamic Analysis of Composite Laminated Circular Plate with Circular Delamination

2016 ◽  
Vol 32 (6) ◽  
pp. 683-692 ◽  
Author(s):  
D.-L. Chen
Keyword(s):  
Differential Equations ◽  
Bessel Function ◽  
Natural Frequency ◽  
Circular Plate ◽  
Nonlinear Dynamic ◽  
Multiple Scales ◽  
Dynamic Equilibrium ◽  
Primary Resonance ◽  
Equilibrium Equations ◽  
Generalized Displacements

AbstractIn this paper, the effect of delamination on free vibration and primary resonance behaviors of composite circular plate with circular delamination is investigated. Through Reissner Variational Principle, the nonlinear dynamic equilibrium equations, the generalized displacements continuity conditions and the generalized forces equilibrium conditions across delamination front and consistent boundary conditions of delaminated circular plate are obtained. In the work, by introducing Bessel Function and Modified Bessel Function and using Galerkin discretization method, the nonlinear dynamic partial differential equations of delaminated circular plate are transferred into a set of nonlinear ordinary differential equations. Then by using semi-analytic method and multiple scales method, the effects of delamination radius and delamination depth in the thickness-wise on the natural frequency and primary resonance behaviors of delaminated circular plate are presented. The Results show that delamination has considerable effects on the natural frequency and its primary resonance behaviors of delaminated plate.

Free vibration of summation and difference resonance of the vertical cable and other coupled structural members

2017 ◽  
Vol 21 (5) ◽  
pp. 707-720
Author(s):  
Chunguang Dong ◽  
Ronghui Wang ◽  
Xiaoxia Zhen ◽  
Haonan Ni
Keyword(s):  
Free Vibration ◽  
Nonlinear Dynamic ◽  
Multiple Scales ◽  
Dynamic Equilibrium ◽  
Order Approximation ◽  
Coupled System ◽  
Superposition Method ◽  
Structural Members ◽  
Equilibrium Equations ◽  
Strongly Nonlinear

Free vibration of summation and difference resonance of the vertical cable and other coupled structural members were investigated in this article. A model of a vertical cable and two mass–springs was built, with the sling considered to be geometrically nonlinear, and the upper and lower connecting structural members were taken as two mass–springs. Assuming the displacement of the sling, modal superposition method and D’Alembert principle were used to derive the dynamic equilibrium equations of the coupled structure. The nonlinear dynamic equilibrium equations were studied by means of multiple scales method, and the second-order approximation solutions of single-modal motion of the system were obtained. Numerical examples were presented to discuss the amplitude responses as functions of time of free vibration, with and without damping, respectively. Additionally, fourth-order Runge–Kutta method was directly used for the nonlinear dynamic equilibrium equations to complement and verify the analytical solutions. The results show that the coupled system performs strongly nonlinear and coupled characteristics, which is useful for engineering design.

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.

Primary Resonance of MEMS/NEMS Circular Plate Biosensors

2014 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Reynaldo Oyervides
Keyword(s):  
Natural Frequency ◽  
Circular Plate ◽  
Dna Detection ◽  
Van Der Waals ◽  
Primary Resonance ◽  
Van Der Waals Forces ◽  
Electrostatic Actuation ◽  
Circular Plates ◽  
Saddle Node Bifurcation ◽  
Saddle Node

This investigation deals with M/NEMS circular plates under electrostatic actuation. Such structures can be used as resonator sensors for medicine and biology applications such as virus, bacteria or DNA detection. The system consists of a clamped circular plate over a ground. The actuation of the plate is done through an AC voltage whose frequency is near half natural frequency of the plate. This produces a primary resonance to be used afterwards for sensing purposes. It is showed that a saddle-node bifurcation occurs. The effects of damping, voltage, Casimir, and van der Waals forces are predicted.

Free vibration and buckling of heavy column with regular polygon cross-section

2021 ◽  
pp. 095440622110293
Author(s):  
Joon Kyu Lee ◽  
Byoung Koo Lee
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Natural Frequency ◽  
Cross Section ◽  
Buckling Load ◽  
Mode Shapes ◽  
Equilibrium Equations ◽  
Direct Integration Method ◽  
Solution Methods ◽  
Buckling Length

This paper deals with the free vibration and buckling of heavy column, considering its own self-weight. The column has a regular polygonal cross-section with a constant area. The column is applied to an external axial load as well as the self-weight. The five end conditions of the column are considered. Based on equilibrium equations of the column element, differential equations governing the vibrational and buckled mode shapes of column are derived. In solution methods, differential equations are numerically integrated by the direct integration method and eigenvalues of the natural frequency, buckling load and self-weight buckling length are calculated by the determinant search method. The numerical results of this study were in good agreement with those of the reference. Parametric study of the end condition, side number and self-weight on the natural frequency and buckling load was carried out.

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.

scholarly journals Nonlinear Dynamic Response of an Axially Functionally Graded (AFG) Beam Resting on Nonlinear Elastic Foundation Subjected to Moving Load

2019 ◽  
Vol 8 (1) ◽  
pp. 250-260 ◽  
Author(s):  
Mehdi Alimoradzadeh ◽  
Mehdi Salehi ◽  
Sattar Mohammadi Esfarjani
Keyword(s):  
Differential Equations ◽  
Dynamic Response ◽  
Nonlinear Vibration ◽  
Natural Frequency ◽  
Nonlinear Dynamic ◽  
Vibration Analysis ◽  
Functionally Graded ◽  
Harmonic Load ◽  
Nonlinear Dynamic Response ◽  
Moving Harmonic Load

Abstract In recent years, structures made of Functionally Graded materials (FGMs) are used in industries due to the continuously compositional variation of the constituents in FGMs along different directions. In order to develop FGMs, nonlinear vibration analysis to study dynamic behavior is needed. This study proposes nonlinear vibration analysis of a simply supported axially functionally graded (AFG) beam subjected to a moving harmonic load as an Euler-Bernoulli beam utilizing Green’s strain tensor. Axial variation of material properties of the beam is based on the power law. The governing equations of motion are derived via Hamilton’s principle. The Galerkin’s method is implemented to reduce the nonlinear partial differential equations of the system to a number of nonlinear ordinary differential equations. He’s variational method is applied to obtain approximate analytical expressions for the nonlinear frequency and the nonlinear dynamic response of the AFG beam. The effect of some parameters such as the power index and stiffness coefficients, among others, on the nonlinear natural frequency has been investigated. The influence of above mentioned parameters as well as the velocity of the moving harmonic load on the nonlinear dynamic response has been studied. The results indicate that these parameters have a considerable effect on both nonlinear natural frequency and response amplitude.

scholarly journals Nonlinear Dynamic Response of Ropeway Roller Batteries via an Asymptotic Approach

2021 ◽  
Vol 11 (20) ◽  
pp. 9486
Author(s):  
Andrea Arena
Keyword(s):  
Nonlinear Dynamic ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Resonance Excitation ◽  
Dynamic Features ◽  
Response Curves ◽  
Internal Resonances ◽  
The Third ◽  
Fold Bifurcation

The nonlinear dynamic features of compression roller batteries were investigated together with their nonlinear response to primary resonance excitation and to internal interactions between modes. Starting from a parametric nonlinear model based on a previously developed Lagrangian formulation, asymptotic treatment of the equations of motion was first performed to characterize the nonlinearity of the lowest nonlinear normal modes of the system. They were found to be characterized by a softening nonlinearity associated with the stiffness terms. Subsequently, a direct time integration of the equations of motion was performed to compute the frequency response curves (FRCs) when the system is subjected to direct harmonic excitations causing the primary resonance of the lowest skew-symmetric mode shape. The method of multiple scales was then employed to study the bifurcation behavior and deliver closed-form expressions of the FRCs and of the loci of the fold bifurcation points, which provide the stability regions of the system. Furthermore, conditions for the onset of internal resonances between the lowest roller battery modes were found, and a 2:1 resonance between the third and first modes of the system was investigated in the case of harmonic excitation having a frequency close to the first mode and the third mode, respectively.

Influence of van der Waals Forces on Electrostatically Actuated MEMS/NEMS Circular Plates

2011 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Iris Alvarado
Keyword(s):  
Circular Plate ◽  
Multiple Scales ◽  
Van Der Waals ◽  
Primary Resonance ◽  
Circular Plates ◽  
Weakly Nonlinear ◽  
Electrostatically Actuated ◽  
Ground Plate ◽  
Scale Surface ◽  
Two Bodies

This paper deals with electrostatically actuated micro and nano-electromechanical (MEMS/NEMS) circular plates. The system under investigation consists of two bodies, a deformable and conductive circular plate placed above a fixed, rigid and conductive ground plate. The deformable circular plate is electrostatically actuated by applying an AC voltage between the two plates. Nonlinear parametric resonance and pull-in occur at certain frequencies and relatively large AC voltage, respectively. Such phenomena are useful for applications such as sensors, actuators, switches, micro-pumps, micro-tweezers, chemical and mass sensing, and micro-mirrors. A mathematical model of clamped circular MEMS/NEMS electrostatically actuated plates has been developed. Since the model is in the micro- and nano-scale, surface forces, van der Waals and/or Casimir, acting on the plate are included. A perturbation method, the Method of Multiple Scales (MMS), is used for investigating the case of weakly nonlinear MEMS/NEMS circular plates. Two time scales, fast and slow, are considered in this work. The amplitude-frequency and phase-frequency response of the plate in the case of primary resonance are obtained and discussed.

Application of a Weakly Nonlinear Absorber to Suppress the Resonant Vibrations of a Forced Nonlinear Oscillator

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
J. C. Ji
Keyword(s):  
Natural Frequency ◽  
Nonlinear Oscillator ◽  
Vibration Suppression ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Integrated System ◽  
Frequency Interval ◽  
Weakly Nonlinear ◽  
Nonlinear Absorber ◽  
Forced Nonlinear Oscillator

A weakly nonlinear vibration absorber is used to suppress the primary resonance vibrations of a single degree-of-freedom weakly nonlinear oscillator with periodic excitation, where the two linearized natural frequencies of the integrated system are not under any internal resonance conditions. The values of the absorber parameters are significantly lower than those of the forced nonlinear oscillator, as such the nonlinear absorber can be regarded as a perturbation to the nonlinear primary oscillator. The characteristics of the nonlinear primary oscillator change only slightly in terms of its new linearized natural frequency and the frequency interval of primary resonances after the nonlinear absorber is added. The method of multiple scales is employed to obtain the averaged equations that determine the amplitudes and phases of the first-order approximate solutions. Selection criteria are developed for the absorber linear stiffness (linearized natural frequency) and nonlinear stiffness in order to achieve better performance in vibration suppression. Illustrative examples are given to show the effectiveness of the nonlinear absorber in suppressing nonlinear vibrations of the forced oscillator under primary resonance conditions.

Magneto-Elastic Combination Resonance of Rotating Circular Plate with Varying Speed Under Alternating Load

2018 ◽  
Vol 18 (03) ◽  
pp. 1850032
Author(s):  
Hu Yuda ◽  
Li Zhe ◽  
Du Guojun ◽  
Wang Yanan
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Frequency Response ◽  
Circular Plate ◽  
Multiple Scales ◽  
Sufficient Conditions ◽  
Global Bifurcation ◽  
Axisymmetric Vibration ◽  
Response Curves ◽  
Amplitude Frequency Response

Nonlinear magneto-elastic combined resonance of parametric and forced excitations is investigated for a rotating circular plate with a variable speed under alternating load. According to the magneto-elastic vibration equations of a conductive rotating thin circular plate, the axisymmetric vibration differential equations of the rotating circular plate under transverse magnetic field are obtained through the application of the Galerkin integral method. The method of multiple scales is applied to solve the differential equations of the circular plate under alternating magnetic field, and the resonance states of the system under combined parametric and forced excitations are obtained by analyzing secular terms. The respective amplitude–frequency response equations are also derived, as well as the necessary and sufficient conditions of the system to make it stable. A numerical method is adopted to acquire amplitude–frequency response curves, bifurcation diagrams of amplitude and the variation pattern of amplitude with magnetic induction intensity and radial force. The influence of parameter variation on stability of the system is also investigated. Based on the global bifurcation diagram of the system, the influence of the change of bifurcation parameters on the system dynamics is discussed.

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