Nonlinear Free Vibration of a Beam With Off-Axis Attached Mass

2013 ◽  
Author(s):  
Pezhman A. Hassanpour
Keyword(s):  
Free Vibration ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Beam Theory ◽  
Governing Equations ◽  
Attached Mass ◽  
Lumped Mass ◽  
Nonlinear Free Vibration ◽  
Euler Bernoulli Beam

A model of a clamped-clamped beam with an attached lumped mass is presented in this paper. The system is modeled using the Euler-Bernoulli beam theory. In the models presented in literature, it is assumed that the center of mass of the attached mass is located on the neutral axis of the beam. In this paper, this assumption is relaxed. The governing equations of motion are derived. It has been shown that the off-axis center of mass of the attached mass generates an amplitude-dependent transverse force in the beam, which introduces a quadratic nonlinearity. The nonlinear governing equations of motion are solved using the Multiple Scales method. The nonlinear free vibration frequencies are determined.

Nonlinear Forced Vibration of a Beam-Type Resonator With Attached Eccentric Mass

2014 ◽  
Author(s):  
Pezhman A. Hassanpour
Keyword(s):  
Forced Vibration ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Nonlinear Resonance ◽  
Beam Theory ◽  
Attached Mass ◽  
Beam Type ◽  
Nonlinear Forced Vibration ◽  
Euler Bernoulli Beam

In this paper, the nonlinear model of an asymmetric micro-bridge resonator with an attached eccentric mass is investigated. The resonator is treated using the Euler-Bernoulli beam theory. The attached mass represents the electrostatic comb-drive actuator in micro-electromechanical applications. The center of mass of the actuator is assumed to be off the neutral axis of the beam. The governing equations of motion are derived assuming that a concentrated harmonic force is applied to the attached mass. The nonlinear forced vibration of the system is studied using the method of multiple scales. It has been demonstrated that the eccentricity of the mass may lead to different types of nonlinear resonance, e.g., superharmonic and internal resonance. The end application of the structure under investigation is in resonant sensing and energy harvesting applications.

Dynamic Analysis of a Beam-Type Resonator With Off-Axis Attached Mass

2012 ◽  
Author(s):  
Pezhman A. Hassanpour ◽  
Kamran Behdinan
Keyword(s):  
Equations Of Motion ◽  
Center Of Mass ◽  
Beam Theory ◽  
Numerical Approach ◽  
Governing Equations ◽  
Bernoulli Beam ◽  
Comb Drive ◽  
Comb Drives ◽  
Beam Type ◽  
Euler Bernoulli Beam

In this paper, the model of a micro-machined beam-type resonator is presented. The resonator is a micro-bridge which is modeled using Euler-Bernoulli beam theory. A comb-drive electrostatic actuator is attached to the micro-bridge for the excitation/detection of vibrations. In the models presented in the literature, it is assumed that the center of mass of the comb-drive is located on the neutral axis of the beam. In this paper, it is demonstrated that this assumption can not be applied for asymmetric-shaped comb-drives. Furthermore, the governing equations of motion are derived by relaxing the above assumption. It has been shown that the off-axis center of mass of the comb-drive generates an amplitude-dependent transverse force in the beam, which is essentially a nonlinear effect. The governing equations of motion are solved using a hybrid analytical-numerical approach. The end application of the structure under investigation is in resonant sensing and energy harvesting applications.

Large Amplitude Free Vibration of Nanobeams Subjected to Magnetic Field Based on Nonlocal Elasticity Theory

2015 ◽  
Vol 764-765 ◽  
pp. 1199-1203 ◽  
Author(s):  
Tai Ping Chang
Keyword(s):  
Magnetic Field ◽  
Free Vibration ◽  
Nonlocal Elasticity ◽  
Beam Theory ◽  
Nonlinear Frequency ◽  
Bernoulli Beam ◽  
Nonlinear Free Vibration ◽  
Vibration Behavior ◽  
Bernoulli Beam Theory ◽  
Euler Bernoulli Beam

In the present study, nonlinear free vibration behavior of nanobeam subjected to magnetic field is investigated based on Eringen's nonlocal elasticity and Euler–Bernoulli beam theory. The Hamilton's principle is adopted to derive the governing equations together with Euler–Bernoulli beam theory and the von-Kármán's nonlinear strain–displacement relationships. An approximate analytical solution is obtained for the nonlinear frequency of the nanobeam under magnetic field by using the Galerkin method and He's variational method. In the numerical results, the ratio of nonlinear frequency to linear frequency is presented. The effect of nonlocal parameter on the nonlinear frequency ratio is studied; furthermore, the effect of magnetic field on the nonlinear free vibration behavior of nanobeam is investigated.

Geometrically Nonlinear Free Vibration of Composite Materials: Clamped-Clamped Functionally Graded Beam with an Edge Crack Using Homogenisation Method

2017 ◽  
Vol 730 ◽  
pp. 521-526 ◽  
Author(s):  
Mohcine Chajdi ◽  
El Bekkaye Merrimi ◽  
Khalid El Bikri
Keyword(s):  
Free Vibration ◽  
Edge Crack ◽  
Volume Fraction ◽  
Crack Depth ◽  
Beam Theory ◽  
Functionally Graded ◽  
Geometrically Nonlinear ◽  
Nonlinear Free Vibration ◽  
Functionally Graded Beam ◽  
Euler Bernoulli Beam

The problem of geometrically nonlinear free vibration of a clamped-clamped functionally graded beam containing an open edge crack in its center is studied in this paper. The study is based on Euler-Bernoulli beam theory and Von Karman geometric nonlinearity assumptions. The cracked section is modeled by an elastic spring connecting two intact segments of the beam. It is assumed that material properties of the functionally graded composites are graded in the thickness direction and estimated through the rule of mixture. The homogenisation method is used to reduce the problem to that of isotropic homogeneous cracked beam. Direct iterative method is employed for solving the eigenvalue equation for governing the frequency nonlinear vibration, in order to show the effect of the crack depth and the influences of the volume fraction on the dynamic response.

scholarly journals Boundary Characteristic Bernstein Polynomials Based Solution for Free Vibration of Euler Nanobeams

2019 ◽  
Vol 3 (4) ◽  
pp. 99
Author(s):  
Somnath Karmakar ◽  
Snehashish Chakraverty
Keyword(s):  
Free Vibration ◽  
Bernstein Polynomials ◽  
Ritz Method ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Computationally Efficient ◽  
Governing Equations ◽  
Classical Boundary ◽  
Convergence Study ◽  
Euler Bernoulli Beam

This paper is concerned with the free vibration problem of nanobeams based on Euler–Bernoulli beam theory. The governing equations for the vibration of Euler nanobeams are considered based on Eringen’s nonlocal elasticity theory. In this investigation, computationally efficient Bernstein polynomials have been used as shape functions in the Rayleigh-Ritz method. It is worth mentioning that Bernstein polynomials make the computation efficient to obtain the frequency parameters. Different classical boundary conditions are considered to address the titled problem. Convergence of frequency parameters is also tested by increasing the number of Bernstein polynomials in the simulation. Further, comparison studies of the results with existing literature are done after fixing the number of polynomials required from the said convergence study. This shows the efficacy and powerfulness of the method. The novelty of using the Bernstein polynomials is addressed in detail and solutions obtained by this method provide a better representation of the vibration behavior of Euler nanobeams.

Nonlinear Free Vibration Analysis of a Fluid-Conveying Microtube

2014 ◽  
Author(s):  
Shamim Mashrouteh ◽  
Mehran Sadri ◽  
Davood Younesian ◽  
Ebrahim Esmailzadeh
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Variational Iteration Method ◽  
Solution Technique ◽  
Nonlinear Free Vibration ◽  
Nonlinear Equations Of Motion ◽  
Analytical Expressions

Nonlinear free vibration of a microstructure has been analyzed in this study. A fluid-conveying microtube is mathematically modeled using non-classical beam theory. Partial differential equation of the model is considered in non-dimensional form. Simply-supported boundaries are taken into account and assuming three vibrating modes, an analytical method is employed to obtain the nonlinear equations of motion. Variational iteration method has been utilized as an analytical solution technique. In order to obtain the nonlinear natural frequencies of the system, analytical expressions are found based on this method. A parametric study is also carried out to investigate the effect of different parameters on the vibration characteristics of the microstructure.

Analytical Solutions for Nonlinear Free Vibration of Micro-Scale Timoshenko Beams

2016 ◽  
Author(s):  
Zia Saadatnia ◽  
Ebrahim Esmailzadeh ◽  
Davood Younesian
Keyword(s):  
Free Vibration ◽  
Analytical Solutions ◽  
Equations Of Motion ◽  
Variational Iteration Method ◽  
Strain Gradient Theory ◽  
Physical Parameters ◽  
Gradient Theory ◽  
Beam Model ◽  
Governing Equations ◽  
Nonlinear Free Vibration

Nonlinear free vibration of micro-beams based on the Timoshenko beam model is studied. The governing equations of motion using the strain gradient theory, Von Kármán strain tensors and Hamiltonian principle, are developed. The Galerkin method is applied to the governing equations and the coupled nonlinear ordinary differential equations of system are obtained. The variational iteration method is utilized to determine the time responses of the micro-beam and also a close form expression for the frequency-amplitude is found. The analytical solutions obtained for different values of parameters are compared with those found from different numerical methods. The effects of geometrical and physical parameters on the dynamics of micro-beam are also examined. Moreover, the analytical formulation for frequency ratio, i.e., the ratio of nonlinear natural frequency to the linear one is obtained and the sensitivity of this ratio to the variations of various parameters is evaluated. It is proved that the proposed solution methods and the results obtained are accurate and reliable when dynamics of such micro structures are studied.

scholarly journals Vibration and Instability of Rotating Composite Thin-Walled Shafts with Internal Damping

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Ren Yongsheng ◽  
Zhang Xingqi ◽  
Liu Yanghang ◽  
Chen Xiulong
Keyword(s):  
Virtual Work ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Dynamical Analysis ◽  
Design Parameters ◽  
Internal Damping ◽  
Analysis Method ◽  
Governing Equations ◽  
Thin Walled

The dynamical analysis of a rotating thin-walled composite shaft with internal damping is carried out analytically. The equations of motion are derived using the thin-walled composite beam theory and the principle of virtual work. The internal damping of shafts is introduced by adopting the multiscale damping analysis method. Galerkin’s method is used to discretize and solve the governing equations. Numerical study shows the effect of design parameters on the natural frequencies, critical rotating speeds, and instability thresholds of shafts.

Nonlinear Forced Vibration Analysis of Smart Curved CNTs Conveying Fluid

2021 ◽  
Author(s):  
Vahid Mohamadhashemi ◽  
Amir Jalali ◽  
Habib Ahmadi
Keyword(s):  
Carbon Nanotube ◽  
Velocity Vector ◽  
Multiple Scales ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Maximum Amplitude ◽  
Beam Theory ◽  
Physical Parameters ◽  
Conveying Fluid

In this study, the nonlinear vibration of a curved carbon nanotube conveying fluid is analyzed. The nanotube is assumed to be covered by a piezoelectric layer and the Euler–Bernoulli beam theory is employed to establish the governing equations of motion. The influence of carbon nanotube curvature on structural modeling and fluid velocity vector is considered and the slip boundary conditions of CNT conveying fluid are included. The mathematical modeling of the structure is developed using Hamilton’s principle and then, the Galerkin procedure is employed to discretize the equation of motion. Furthermore, the frequency response of the system is extracted by applying the multiple scales method of perturbation. Finally, a comprehensive study is carried out on the primary resonance and piezoelectric-based parametric resonance of the system. It is shown that consideration of nanotube curvature may lead to an increase in nonlinearity. Implementing the fluid velocity vector in which nanotube curvature is included highly affects the maximum amplitude of the response and should not be ignored. Furthermore, different system parameters have evident impacts on the behavior of the system and therefore, selecting the reasonable geometrical and physical parameters of the system can be very useful to achieve a favorable response.

Investigation on Free Vibration of Buckled Beams

2012 ◽  
Vol 433-440 ◽  
pp. 41-44 ◽  
Author(s):  
Ming Hsu Tsai ◽  
Wen Yi Lin ◽  
Kuo Mo Hsiao ◽  
Fu Mio Fujii
Keyword(s):  
Free Vibration ◽  
Virtual Work ◽  
Beam Theory ◽  
Taylor Series Expansion ◽  
Element Formulation ◽  
Governing Equations ◽  
Nonlinear Beam ◽  
Buckled Beams ◽  
Buckled Beam ◽  
D’Alembert Principle

The objective of this study is to investigate the deformed configuration and free vibration around the deformed configuration of clamped buckled beams by co-rotational finite element formulation. The principle of virtual work, d'Alembert principle and the consistent second order linearization of the nonlinear beam theory are used to derive the element equations in current element coordinates. The governing equations for linear vibration are obtained by the first order Taylor series expansion of the equation of motion at the static equilibrium position of the buckled beam. Numerical examples are studied to investigate the natural frequencies of clamped buckled beams with different slenderness ratios under different axial compression.

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