scholarly journals Anomalous Nonlinear Dynamics Behavior of Fractional Viscoelastic Beams

2021 ◽  
Author(s):  
Jorge L. Suzuki ◽  
Ehsan Kharazmi ◽  
Pegah Varghaei ◽  
Maryam Naghibolhosseini ◽  
Mohsen Zayernouri
Keyword(s):  
Nonlinear Dynamics ◽  
Differential Equation ◽  
Free Vibration ◽  
Multiple Scales ◽  
Viscoelastic Model ◽  
Primary Resonance ◽  
Decay Rates ◽  
Distribution Functions ◽  
Analytical Scheme ◽  
Nonlinear Viscoelastic

Abstract Fractional models and their parameters are sensitive to intrinsic microstructual changes in anomalous materials. We investigate how such physics-informed models propagate the evolving anomalous rheology to the nonlinear dynamics of mechanical systems. In particular, we study the vibration of a fractional, geometrically nonlinear viscoelastic cantilever beam, under base excitation and free vibration, where the viscoelasticity is described by a distributed-order fractional model. We employ Hamilton's principle to obtain the equation of motion with the choice of specific material distribution functions that recover a fractional Kelvin-Voigt viscoelastic model of order $\alpha$. Through spectral decomposition in space, the resulting time-fractional partial differential equation reduces to a nonlinear time-fractional ordinary differential equation, where the linear counterpart is numerically integrated through a direct L1-difference scheme. We further develop a semi-analytical scheme to solve the nonlinear system through a method of multiple scales, yielding a cubic algebraic equation in terms of the frequency. Our numerical results suggest a set of $\alpha$-dependent anomalous dynamic qualities, such as far-from-equilibrium power-law decay rates, amplitude super-sensitivity at free vibration, and bifurcation in steady-state amplitude at primary resonance.

Effects of Rotary Inertia and Shear Deformation on Nonlinear Free Vibration of Microbeams

2006 ◽  
Vol 128 (5) ◽  
pp. 611-615 ◽  
Author(s):  
Asghar Ramezani ◽  
Aria Alasty ◽  
Javad Akbari
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Free Vibration ◽  
Shear Deformation ◽  
Large Amplitude ◽  
Multiple Scales ◽  
Rotary Inertia ◽  
Partial Differential ◽  
Amplitude Vibration ◽  
Large Amplitude Vibration

In this paper, the large amplitude free vibration of a doubly clamped microbeam is considered. The effects of shear deformation and rotary inertia on the large amplitude vibration of the microbeam are investigated. To this end, first Hamilton’s principle is used in deriving the partial differential equation of the microbeam response under the mentioned conditions. Then, implementing the Galerkin’s method the partial differential equation is converted to an ordinary nonlinear differential equation. Finally, the method of multiple scales is used to determine a second-order perturbation solution for the obtained ODE. The results show that nonlinearity acts in the direction of increasing the natural frequency of the doubly clamped microbeam. Shear deformation and rotary inertia have significant effects on the large amplitude vibration of thick and short microbeams.

Nonlinear Free Vibrations of a Timoshenko Beam Using Multiple Scales Method

Volume 2 ◽  
2004 ◽  
Author(s):  
Asghar Ramezani ◽  
Mehrdaad Ghorashi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Free Vibration ◽  
Timoshenko Beam ◽  
Large Amplitude ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Free Vibrations ◽  
Partial Differential ◽  
Cantilevered Beam

In this paper, the large amplitude free vibration of a cantilever Timoshenko beam is considered. To this end, first Hamilton’s principle is used in deriving the partial differential equation of the beam response under the mentioned conditions. Then, implementing the Galerkin’s method the partial differential equation is converted to an ordinary nonlinear differential equation. Finally, the method of multiple scales is used to determine a second order perturbation solution for the obtained ODE. The results show that nonlinearity acts in the direction of increasing the natural frequency of the thick-cantilevered beam.

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.

Electrostatically Actuated M/NEMS With Casimir Effect: Primary Resonance — Comparison Between Three Methods

2017 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Julio S. Beatriz ◽  
Christian Reyes
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Multiple Scales ◽  
Casimir Effect ◽  
Boundary Value ◽  
Primary Resonance ◽  
Initial Value ◽  
Electrostatically Actuated ◽  
Mems Resonator ◽  
Modes Of Vibration

This paper deals with electrostatically actuated micro- and nano-electromechanical systems (M/NEMS) cantilever resonator under electrostatic actuation. The model includes Casimir effect. Three different methods are used to investigate the primary resonance of the MEMS resonator. The first two methods are based on a Galerkin approach in which the initial value and boundary value problem, given by the partial differential equation (PDE) of motion and the initial and boundary conditions, is transformed into an initial value problem of one ordinary differential equation (ODE) or a system of ODEs depending on how many modes of vibrations are considered in the model. The first method used is the Method of Multiple Scales (MMS) which is an approximate analytical method used to solve the model using one mode of vibration. The second method referred to as Reduced Order Model (ROM) solves the model using two to five modes of vibration using numerical integration. The third method is different than the first two in the sense that the initial value and boundary value problem describing the MEMS resonator is transformed into a boundary value problem (BVP) by using finite differences tor time derivatives. For this Matlab built-in function bvp4c is used to solve the problem. This built-in function is used for two different versions of the same equation. One which involves Taylor expansions of the nonlinear terms, and the other which does not. Results between the methods are in agreement. Thus any of these methods can be used to accurately predict the behavior of the MEMS resonator. For the ROM, two equations are also used, one for Casimir and one for without. The influence of damping, Casimir, and voltage parameter are also shown.

Dynamic Behavior of Carbon Nanotubes Using Nonlocal Rayleigh Beam

2014 ◽  
Author(s):  
Hassan Askari ◽  
Ebrahim Esmailzadeh ◽  
Davood Younesian
Keyword(s):  
Differential Equation ◽  
Carbon Nanotubes ◽  
Carbon Nanotube ◽  
Forced Vibration ◽  
Multiple Scales ◽  
Surface Effect ◽  
Primary Resonance ◽  
Beam Theory ◽  
Winkler Foundation ◽  
Rayleigh Beam

Forced vibration of carbon nanotubes based on the Rayleigh beam theory in conjunction with Eringen’s nonlocal elasticity is investigated. The governing equation of vibration of carbon nanotube using the above theories is developed. The carbon nanotube is rested on a nonlinear Winkler and Pasternak foundation with the simply-supported boundary conditions. The Gelerkin procedure is utilized to find the nonlinear ordinary differential equation of vibration of system. The differential equation is solved using the multiple scales method in order to investigate the primary resonance of the considered system. The frequency response of the system is obtained and the effects of different parameters, such as the surface effect, position and magnitude of applied force and Pasternak and Winkler foundation, on the vibration behavior of the system are studied. The sensitivity of the amplitude of oscillation of carbon nanotube is depicted with respect to the surface effect. It is shown that the surface effect plays an important role in the forced vibration of the nano-scale structure.

Asymmetric Large Deformation Superharmonic and Subharmonic Resonances of Spiral Stiffened Imperfect FG Cylindrical Shells Resting on Generalized Nonlinear Viscoelastic Foundations

2020 ◽  
Vol 12 (05) ◽  
pp. 2050052
Author(s):  
Kamran Foroutan ◽  
Habib Ahmadi ◽  
Mohammad Shariyat
Keyword(s):  
Multiple Scales ◽  
Cylindrical Shells ◽  
Viscoelastic Model ◽  
Excitation Amplitude ◽  
Functionally Graded ◽  
Function Concept ◽  
Nonlinear Viscoelastic ◽  
Foundation Type ◽  
Nonlinear Elastic ◽  
Subharmonic Resonances

This paper is devoted to superharmonic and subharmonic behavior investigation of imperfect functionally graded (FG) cylindrical shells with external FG spiral stiffeners under large amplitude excitations. The structure is embedded within a generalized nonlinear viscoelastic foundation, which is composed of a two-parameter Winkler–Pasternak foundation augmented by a Kelvin–Voigt viscoelastic model with a nonlinear cubic stiffness, to account for the vibration hardening/softening phenomena and damping considerations. The von Kármán strain-displacement kinematic nonlinearity is employed in the constitutive laws of the shell and stiffeners. The external spiral stiffeners of the cylindrical shell are modeled according to the smeared stiffener technique. The coupled governing equations are solved by using Galerkin’s method in conjunction with the stress function concept. The multiple scales method is utilized to detect the subharmonic and superharmonic resonances and the frequency–amplitude relations of the 1/3 and 1/2 subharmonic and 3/1 and 2/1 superharmonic resonances of the system. Finally, the influences of the stiffeners helical angles, foundation type, coefficient of the nonlinear elastic foundation, material distribution, and excitation amplitude on the system resonances are investigated comprehensively.

On the Primary Resonance of an Electrostatically Actuated MEMS Using the Homotopy Perturbation Method

2009 ◽  
Author(s):  
Mahdi Mojahedi ◽  
Mahdi Moghimi Zand ◽  
Mohammad Taghi Ahmadian
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Multiple Scales ◽  
Electrostatic Force ◽  
Nonlinear Partial Differential Equation ◽  
Primary Resonance ◽  
Analytic Solutions ◽  
Design Parameters ◽  
Homotopy Perturbation

In this paper, primary resonance of a double-clamped microbeam has been investigated. The Microbeam is predeformed by a DC electrostatic force and then driven to vibrate by an AC harmonic electrostatic force. Effects of midplane stretching, axial loads and damping are considered in modeling. Galerkin’s approximation is utilized to convert the nonlinear partial differential equation of motion to a nonlinear ordinary differential equation. Afterward, a combination of homotopy perturbation method and the method of multiple scales are utilized to find analytic solutions to the steady-state motion of the microbeam, far from pull-in. The effects of different design parameters on dynamic behavior are discussed. The results obtained by the presented method are validated by comparing with literature.

scholarly journals Uniform Energy Decay Rates for the Fuzzy Viscoelastic Model with a Nonlinear Source

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Fengyun Zhang
Keyword(s):  
Viscoelastic Model ◽  
Energy Decay ◽  
Decay Rates ◽  
Polynomial Form ◽  
Computational Technique ◽  
Exponential Form ◽  
Nonlinear Source ◽  
Priori Estimates ◽  
Energy Decay Rates ◽  
Mathematica Software

This paper considers the fuzzy viscoelastic model with a nonlinear source u t t + L u + ∫ 0 t g t − ζ Δ u ζ d ζ − u γ u − η Δ u t = 0 in a bounded field Ω. Under weak assumptions of the function g t , with the aid of Mathematica software, the computational technique is used to construct the auxiliary functionals and precise priori estimates. As time goes to infinity, we prove that the solution is global and energy decays to zero in two different ways: the exponential form and the polynomial form.

Free volume based nonlinear viscoelastic model for polyurea over a wide range of strain rates and temperatures

2021 ◽  
Vol 152 ◽  
pp. 103650
Author(s):  
Chencheng Gong ◽  
Yan Chen ◽  
Ting Li ◽  
Zhanli Liu ◽  
Zhuo Zhuang ◽  
...  
Keyword(s):  
Free Volume ◽  
Viscoelastic Model ◽  
Strain Rates ◽  
Nonlinear Viscoelastic ◽  
Wide Range ◽  
Nonlinear Viscoelastic Model
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