On Nonlinear Forced Response of Nonuniform Beams

2008 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Multiple Scales ◽  
Rectangular Cross Section ◽  
Constant Width ◽  
Nonlinear Partial Differential Equation ◽  
Primary Resonance ◽  
Single Mode ◽  
Forced Response ◽  
Thickness Variation ◽  
Softening Effect ◽  
Bending Vibration

This paper reports the primary resonance of single mode forced, undamped, bending vibration of nonuniform sharp cantilevers of rectangular cross-section, constant width, and convex parabolic thickness variation. The case of nonlinear curvature, moderately large amplitudes, is considered. The method of multiple scales is applied directly to the nonlinear partial-differential equation of motion and boundary conditions. The frequency-response is analytically determined, and numerical results show a softening effect of the geometrical nonlinearities.

On Subharmonic Resonances of Geometric Nonlinear Vibrations of Nonuniform Beams

2008 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Multiple Scales ◽  
Rectangular Cross Section ◽  
Constant Width ◽  
Phase Evolution ◽  
Factorization Method ◽  
Thickness Variation ◽  
Bending Vibrations ◽  
Order Problem ◽  
First Order ◽  
Subharmonic Resonances

Subharmonic resonances of nonlinear forced bending vibrations in the case of moderately large curvature of nonuniform cantilever beams of rectangular cross section and a sharp end are reported. Cantilevers of constant width and parabolic thickness variation are considered in this research. The method of multiple scales is directly applied to the governing partial-differential equation of motion and boundary conditions. Two problems, zeroth- and first-order, result. Using factorization method, the linear modes of the zeroth-order problem are obtained in terms of hypergeometric functions. The first-order problem provides the amplitude and phase evolution equation and consequently the regions where subharmonic responses exist.

On Superharmonic Resonances of Nonlinear Nonuniform Beams

2008 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Multiple Scales ◽  
Rectangular Cross Section ◽  
Constant Width ◽  
Phase Evolution ◽  
Factorization Method ◽  
Thickness Variation ◽  
Bending Vibrations ◽  
Order Problem ◽  
First Order ◽  
Paper Method

Superharmonic resonances of nonlinear forced bending vibrations of moderately large curvature of nonuniform cantilever beams of rectangular cross section and a sharp end are reported. Cantilevers of constant width and parabolic thickness variation are considered in this paper. Method of multiple scales is directly applied to the governing partial-differential equation of motion and boundary conditions. Two problems, zero- and first-order problem, result. Solving the zero-order problem, the linear modes are obtained in terms of hypergeometric functions by using the factorization method. The first-order problem provides the amplitude and phase evolution equation and consequently the superharmonic frequency response of the nonlinear system.

On Geometric Nonlinear Vibrations of Nonuniform Beams

2007 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Multiple Scales ◽  
Rectangular Cross Section ◽  
Constant Width ◽  
Factorization Method ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Bending Vibrations ◽  
Nonlinear Bending ◽  
Nonlinear Modes ◽  
Weakly Nonlinear

Nonlinear bending vibrations in the case of moderately large curvature are reported for a nonuniform cantilever beam of rectangular cross section and a sharp end. This is a beam of constant width and parabolic thickness variation. The method of multiple scales is directly applied to the governing partial-differential equation of motion and boundary conditions. The linear modes are obtained in terms of hypergeometric functions by using the factorization method. In the absence of internal resonance (weakly nonlinear systems) the nonlinear modes are taken to be perturbed versions of the linear modes. The nonlinear mode shapes and frequencies of the beam are reported.

Primary Resonance of the Current-Carrying Beam in Thermal-Magneto-Elasticity Field

2010 ◽  
Vol 29-32 ◽  
pp. 16-21 ◽  
Author(s):  
Xiao Yan Xi ◽  
Zhian Yang ◽  
Li Li Meng ◽  
Chang Jian Zhu
Keyword(s):  
Response Curve ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Response Curves ◽  
Bending Vibration ◽  
Approximation Solution ◽  
System Parameters ◽  
Practical Engineering

On base of the electro-magneto-elastic theory and the theory of the bending vibration of the electric beam, nonlinear vibration equation of current-carrying beam subjected to thermal-magneto-elasticity field is studied. The Lorentz force and thermal force on the beam are derived. According to the method of multiple scales for nonlinear vibrations the approximation solution of the primary resonance of the system is obtained. Numerical analysis results show that the amplitude changed with the system parameters. With the decrease of magnetic intensity, the amplitude increases rapidly. The response curve occurs bending phenomenon and soft features is increased gradually. Increasing current, the amplitudes increase. With the decrease of temperature, the peak of response curves decrease. With the increase of temperature, natural frequency decreased. It is useful in practical engineering.

A study of the nonlinear primary resonances of a micro-system under electrostatic and piezoelectric excitations

2014 ◽  
Vol 229 (10) ◽  
pp. 1904-1917 ◽  
Author(s):  
Ali K Hoshiar ◽  
Hamed Raeisifard
Keyword(s):  
Multiple Scales ◽  
Primary Resonance ◽  
Nonlinear Behavior ◽  
Nonlinear Effects ◽  
Resonance Excitation ◽  
Smart Device ◽  
Perturbation Approach ◽  
Softening Effect ◽  
Electrostatically Actuated ◽  
The Galerkin Method

In this article, a nonlinear analysis for a micro-system under electrostatic and piezoelectric excitations is presented. The micro-system beam is assumed as an elastic Euler-Bernoulli beam with clamped-free end conditions. The dynamic equations of this model have been derived by using the Hamilton method and considering the nonlinear inertia, curvature, piezoelectric and electrostatic terms. The static and dynamic solutions have been achieved by using the Galerkin method and the multiple-scales perturbation approach, respectively. The results are compared with numerical and other existing experimental results. By studying the primary resonance excitation, the effects of different parameters such as geometry, material, and excitations voltage on the system’s softening and hardening behaviors are evaluated. In an electrostatically actuated micro-system, it was showed that the nonlinear behavior occurs in frequency response as softening effect. In this paper, it is demonstrated that by applying a suitable piezoelectric DC voltage, this nonlinear effects can be controlled and altered to a linear domain. This model can be used to design a nano- or micro-scale smart device.

Electrostatically Actuated MEMS Cantilevers of Linear Thickness Variation

2013 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Mostafa M. Fath El-Den
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Electrostatic Force ◽  
Constant Width ◽  
Thickness Variation ◽  
Electrostatically Actuated ◽  
Phase Amplitude ◽  
Mems Cantilevers ◽  
Relationship Of ◽  
Steady State Solutions

This paper deals with nonuniform linear thickness variation and constant width MEMS cantilever resonators electrostatically actuated through AC voltage near half natural frequency. The frequency response of the structure is investigated. Nonlinearities in the system arise from the electrostatic force. The electrostatic actuation introduces parametric coefficients in both linear and nonlinear parts of the governing equation. The method of multiple scales (MMS) is used to obtain the phase-amplitude relationship of the system, and the steady-state solutions. Parameters’ influences are reported.

Simultaneous Resonances of Geometric Nonlinear Nonuniform Beams

2009 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Multiple Scales ◽  
Constant Width ◽  
Phase Evolution ◽  
Factorization Method ◽  
Thickness Variation ◽  
Bending Vibrations ◽  
Order Problem ◽  
Nonlinear Bending ◽  
First Order ◽  
Simultaneous Resonances

Simultaneous resonances, superharmonic and subharmonic, of two-term excitation nonlinear bending vibrations in the case of moderately large curvature of nonuniform cantilever beams are reported. Cantilevers of constant width and parabolic thickness variation are considered in this research. The method of multiple scales is directly applied to the governing partial-differential equation of motion and boundary conditions. Two problems, zero- and first-order, result. Using factorization method, the linear modes of the zero-order problem are obtained in terms of hypergeometric functions. The first-order problem provides the amplitude-phase evolution relationship and consequently the simultaneous resonances response.

scholarly journals Nonlinear Coupled Vibration of Electrically Actuated Arch with Flexible Supports

Micromachines ◽  
2019 ◽  
Vol 10 (11) ◽  
pp. 729 ◽  
Author(s):  
Ze Wang ◽  
Jianting Ren
Keyword(s):  
Multiple Scales ◽  
Primary Resonance ◽  
Single Mode ◽  
Coupled Vibration ◽  
Order Ordinary Differential Equation ◽  
Electrically Actuated ◽  
Second Mode ◽  
Flexible Supports ◽  
Stable Periodic ◽  
Phase Angles

The nonlinear coupled vibration of an electrically actuated arch microbeam has attracted wide attention. In this paper, we studied the nonlinear dynamics of an electrically actuated arch microbeam with flexible supports. The two-to-one internal resonance between the first and second modes is considered. The multiple scales method is used to solve the governing equation. Four first-order ordinary differential equation describing the modulation of the amplitudes and phase angles were obtained. The equilibrium solution and its stability are determined. In the case of the primary resonance of the first mode, stable periodic motions and modulated motions are determined. The double-jumping phenomenon may occur. In the case of the primary resonance of the second mode, single-mode and two-mode solutions are possible. Moreover, double-jumping, hysteresis, and saturation phenomena were found. In addition, the approximate analytical results are supported by the numerical results.

ANALYTIC SOLUTIONS TO THE OSCILLATORY BEHAVIOR AND PRIMARY RESONANCE OF ELECTROSTATICALLY ACTUATED MICROBRIDGES

2011 ◽  
Vol 11 (06) ◽  
pp. 1119-1137 ◽  
Author(s):  
M. MOJAHEDI ◽  
M. MOGHIMI ZAND ◽  
M. T. AHMADIAN ◽  
M. BABAEI
Keyword(s):  
Differential Equation ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Axial Stress ◽  
Nonlinear Partial Differential Equation ◽  
Primary Resonance ◽  
Dynamic Responses ◽  
Design Parameters ◽  
Electrostatically Actuated ◽  
Homotopy Perturbation

In this paper, the vibration and primary resonance of electrostatically actuated microbridges are investigated, with the effects of electrostatic actuation, axial stress, and mid-plane stretching considered. Galerkin's decomposition method is adopted to convert the governing nonlinear partial differential equation to a nonlinear ordinary differential equation. The homotopy perturbation method (a special case of homotopy analysis method) is then employed to find the analytic expressions for the natural frequencies of predeformed microbridges, by which the effects of the voltage, mid-plane stretching, axial force, and higher mode contribution on the natural frequencies are studied. The primary resonance of the microbridges is also investigated, where the microbridges are predeformed by a DC voltage and driven to vibrate by an AC harmonic voltage. The methods of homotopy perturbation and multiple scales are combined to find the analytic solution for the steady-state motion of the microbeam. In addition, the effects of the design parameters and damping on the dynamic responses are discussed. The results are shown to be in good agreement with the existing ones.

ROM of Primary Resonance Amplitude-Voltage Response of MEMS Cantilevers

2013 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Le Luo ◽  
Israel Martinez ◽  
Mostafa Fathelden ◽  
Christian Reyes
Keyword(s):  
Parametric Excitation ◽  
Constant Width ◽  
Primary Resonance ◽  
Reduced Order Model ◽  
Thickness Variation ◽  
Voltage Response ◽  
Order Model ◽  
Resonance Amplitude ◽  
Reduced Order ◽  
Mems Cantilevers

This paper uses Reduced Order Model (ROM) method to investigate the amplitude-voltage response of MEMS cantilever resonators of linear thickness variation and constant width under AC voltage of frequency near half natural frequency of the resonator. The influences of nonlinearities resulting from AC voltage parametric excitation on the response of the structure are reported.

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