Modeling Geometric Nonlinearities in the Free Vibration of a Planar Beam Flexure With a Tip Mass

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Hamid Moeenfard ◽  
Shorya Awtar
Keyword(s):  
Differential Equations ◽  
Perturbation Technique ◽  
Dynamic Performance ◽  
Nonlinear Partial Differential Equations ◽  
Computational Time ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Analytical Technique ◽  
Numerical Solution Methods ◽  
Tip Mass

The objective of this work is to analytically study the nonlinear dynamics of beam flexures with a tip mass undergoing large deflections. Hamilton's principle is utilized to derive the equations governing the nonlinear vibrations of the cantilever beam and the associated boundary conditions. Then, using a single mode approximation, these nonlinear partial differential equations are reduced to two coupled nonlinear ordinary differential equations. These equations are solved analytically using the multiple time scales perturbation technique. Parametric analytical expressions are presented for the time domain response of the beam around and far from its internal resonance state. These analytical results are compared with numerical ones to validate the accuracy of the proposed analytical model. Compared with numerical solution methods, the proposed analytical technique shortens the computational time, offers design insights, and provides a broader framework for modeling more complex flexure mechanisms. The qualitative and quantitative knowledge resulting from this effort is expected to enable the analysis, optimization, and synthesis of flexure mechanisms for improved dynamic performance.

Modeling Geometric Non-Linearities in the Free Vibration of a Planar Beam Flexure With a Tip Mass

2012 ◽  
Author(s):  
Hamid Moeenfard ◽  
Shorya Awtar
Keyword(s):  
Differential Equations ◽  
Closed Form ◽  
Dynamic Performance ◽  
Single Mode ◽  
Resonance State ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Linear Ordinary Differential Equations ◽  
Non Linear ◽  
Tip Mass

The objective of this work is to create an analytical framework to study the non-linear dynamics of beam flexures with a tip mass undergoing large deflections. Hamilton’s principal is utilized to derive the equations governing the non-linear vibrations of the cantilever beam and the associated boundary conditions. Then, using a single mode approximation, these non-linear partial differential equations are reduced to two coupled non-linear ordinary differential equations. These equations are solved analytically using combination of the method of multiple time scales and homotopy perturbation analysis. Closed-form, parametric analytical expressions are presented for the time domain response of the beam around and far from its internal resonance state. These analytical results are compared with numerical ones to validate the accuracy of the proposed closed-form model. We expect that the qualitative and quantitative knowledge resulting from this effort will ultimately allow the analysis, optimization, and synthesis of flexure mechanisms for improved dynamic performance.

A beam–ring circular truss antenna restrained by means of the negative speed feedback procedure

2021 ◽  
pp. 107754632110036
Author(s):  
Ashraf T EL-Sayed Taha ◽  
Hany S Bauomy
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Nonlinear Partial Differential Equations ◽  
Time Frame ◽  
Ring Structure ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Internal Resonances ◽  
Averaged Equations ◽  
Fractional Differential

The present article contemplates the nonlinear powerful exhibitions of affecting dynamic vibration controller over a beam–ring structure for demonstrating the circular truss antenna exposed to mixed excitations. The dynamic controller comprises the included negative speed input added to the framework’s idea. By using the statue, Hamilton, the nonlinear fractional differential administering conditions of movement and the limit conditions have inferred for the shaft ring structure. Through Galerkin’s method, the nonlinear partial differential equations referred to overseeing the movement of the shaft ring structure have diminished to a coupled normal differential equations extending the nonlinearities square terms. Multiple time scales have helped in acquiring (getting) the four-dimensional averaged equations for measuring the primary and 1:2 internal resonances. This article’s controlled assessment is useful for controlling the nonlinear vibrations of the considered framework. Likewise, the controller dispenses with the framework’s oscillations in a brief time frame. The demonstrations of the numerous coefficients and the framework directed at the examined resonance case have been determined. Using MATLAB 7.0 programs has aided in completing the simulation results. At last, the numerical outcomes displayed an admirable concurrence with the methodical ones. A comparison with recently available articles has also indicated good results through using the presented controller.

Flexural-Torsional Post Critical Behavior of a Cantilever Beam Dynamically Excited: Theoretical Model and Experimental Tests

2003 ◽  
Author(s):  
Daniele Zulli ◽  
Rocco Alaggio ◽  
Francesco Benedettini
Keyword(s):  
Differential Equations ◽  
Cantilever Beam ◽  
Perturbation Technique ◽  
Mass Matrix ◽  
Dynamical Behavior ◽  
Galerkin Approximation ◽  
Experimental Tests ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Vertical Force

The 3D dynamics of a cantilever beam undergoing large displacements under a sinusoidally varying, concentrated, vertical force at its free end are analyzed in this paper. The Partial Differential Equations (PDEs) of the motion are obtained by using the Principle of Virtual Power. Then a reduced 4 degrees-of-freedom model is obtained using, in a Galerkin approximation, four eigenfunctions of the linearized model. The obtained four Ordinary Differential Equations (ODEs) of the motion are expanded by means of a 3rd order Multiple Time Scales perturbation technique to obtain Amplitude and Phase Modulation Equations (APMEs). The role of the inertial-elastic nonlinear terms, responsible for the coupling of the mass matrix, and of the viscous-elastic nonlinear terms, both usually neglected in the literature, is discussed. A path following procedure applied to the APMEs is used to describe the global dynamical behavior in the plane of the excitation control parameters. The results obtained using the 4 d.o.f. analytical model are compared with those of an experimental aluminium model of the cantilever. The regions of instability of the 1-modal planar solution, in which the nonlinear modal coupling excites out of plane and/or torsional components, are studied.

On the Equivalence of Normal Form Theory and Multiple Time Scale Method

2007 ◽  
Author(s):  
Fengxia Wang ◽  
Anil K. Bajaj
Keyword(s):  
Nonlinear Systems ◽  
Normal Form ◽  
Differential Equations ◽  
Time Scales ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Perturbation Scheme ◽  
Normal Form Theory ◽  
Weakly Nonlinear ◽  
Form Theory

Multiple time scales technique has long been an important method for the analysis of weakly nonlinear systems. In this technique, a set of multiple time scales are introduced that serve as the independent variables. The evolution of state variables at slower time scales is then determined so as to make the expansions for solutions in a perturbation scheme uniform in natural and slower times. Normal form theory has also recently been used to approximate the dynamics of weakly nonlinear systems. This theory provides a way of finding a coordinate system in which the dynamical system takes the “simplest” form. This is achieved by constructing a series of near-identity nonlinear transformations that make the nonlinear systems as simple as possible. The “simplest” differential equations obtained by the normal form theory are topologically equivalent to the original systems. Both methods can be interpreted as nonlinear perturbations of linear differential equations. In this work, the equivalence of these two methods for constructing periodic solutions is proven, and it is explained why some studies have found the results obtained by the two techniques to be inconsistent.

On Stability of a Nonlinear, Periodically Time Varying, Rotating Blade

2011 ◽  
Author(s):  
Fengxia Wang
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Multiple Time ◽  
Time Varying ◽  
Rotating Blade ◽  
Stability And Bifurcation ◽  
Geometric Stiffening ◽  
Rotational Motions ◽  
The Stability

This paper discusses the stability of a periodically time-varying, spinning blade with cubic geometric nonlinearity. The modal reduction method is adopted to simplify the nonlinear partial differential equations to the ordinary differential equations, and the geometric stiffening is approximated by the axial inertia membrane force. The method of multiple time scale is employed to study the steady state motions, the corresponding stability and bifurcation for such a periodically time-varying rotating blade. The backbone curves for steady-state motions are achieved, and the parameter map for stability and bifurcation is developed. Illustration of the steady-state motions is presented for an understanding of rotational motions of the rotating blade.

scholarly journals Critical Parameters and Influence on Dynamic Behaviours of Nonlinear Electrostatic Force in a Micromechanical Vibrating Gyroscope

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Shuying Hao ◽  
Yulun Zhu ◽  
Yuhao Song ◽  
Qichang Zhang ◽  
Jingjing Feng ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Bias Voltage ◽  
Electrostatic Force ◽  
Dynamic Performance ◽  
Great Influence ◽  
Critical Parameters ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Element Analysis ◽  
Dc Bias

The electrostatic force nonlinearity caused by fringe effects of the microscale comb will affect the dynamic performance of the micromechanical vibrating gyroscopes (MVGs). In order to reveal the influence mechanism, a class of four-degree-of-freedom (4-DOF) electrostatically driven MVG is considered. The influence of DC bias voltage and comb spacing on the nonlinearity of electrostatic force and the dynamic response of the MVG by using multiple time scales method and numerical simulation are discussed. The results indicate that the electrostatic force nonlinearity causes the system to show stiffness softening. The softening characteristics of the electrostatic force cause the offset of the resonance frequency and a decrease in sensitivity. Although the electrostatic nonlinearity has a great influence on the dynamic behaviour, its influence can be avoided by the reasonable design of the comb spacing and DC bias voltage. There exists a critical value for comb spacing and DC bias voltage. In this paper, determining the critical values is demonstrated by nonlinear dynamics analysis. The results can be supported by the finite element analysis and numerical simulation.

scholarly journals On Multi-Laplace Transform for Solving Nonlinear Partial Differential Equations with Mixed Derivatives

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Abdon Atangana ◽  
Suares Clovis Oukouomi Noutchie
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Nonlinear Partial Differential Equations ◽  
Order Derivative ◽  
Partial Differential ◽  
Analytical Technique ◽  
Novel Approach ◽  
Mixed Derivatives ◽  
The Stability

A novel approach is proposed to deal with a class of nonlinear partial equations including integer and noninteger order derivative. This class of equations cannot be handled with any other commonly used analytical technique. The proposed method is based on the multi-Laplace transform. We solved as an example some complicated equations. Three illustrative examples are presented to confirm the applicability of the proposed method. We have presented in detail the stability, the convergence and the uniqueness analysis of some examples.

Active Control of a Rectangular Thin Plate Via Negative Acceleration Feedback

2016 ◽  
Vol 11 (4) ◽  
Author(s):  
H. S. Bauomy ◽  
A. T. EL-Sayed
Keyword(s):  
Differential Equations ◽  
Thin Plate ◽  
Nonlinear Differential Equations ◽  
Perturbation Technique ◽  
Approximate Solutions ◽  
Second Order ◽  
Multiple Time ◽  
Rectangular Thin Plate ◽  
Negative Acceleration ◽  
Acceleration Feedback

In this paper, the dynamic oscillation of a rectangular thin plate under parametric and external excitations is investigated and controlled. The motion of a rectangular thin plate is modeled by coupled second-order nonlinear ordinary differential equations. The formulas of the thin plate are derived from the von Kármán equation and Galerkin's method. A control law based on negative acceleration feedback is proposed for the system. The multiple time scale perturbation technique is applied to solve the nonlinear differential equations and obtain approximate solutions up to the second-order approximations. One of the worst resonance case of the system is the simultaneous primary resonances, where Ω1≅ω1 and  Ω2≅ω2. From the frequency response equations, the stability of the system is investigated according to the Routh–Hurwitz criterion. The effects of the different parameters are studied numerically. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. The simulation results are achieved using matlab 7.0 software. A comparison is made with the available published work.

scholarly journals An Efficient Approach for Solving Fractional Dynamics of a Predator-Prey System

2019 ◽  
Vol 13 (11) ◽  
pp. 116
Author(s):  
Hegagi Mohamed Ali ◽  
Ismail Gad Ameen
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Equilibrium Points ◽  
Nonlinear Partial Differential Equations ◽  
Fractional Dynamics ◽  
Predator Prey ◽  
Analytical Technique ◽  
Biological Population ◽  
Fractional Differential ◽  
The Stability

In this work, we execute a generally new analytical technique, the modified generalized Mittag-Leffler function method (MGMLFM) for solving nonlinear partial differential equations containing fractional derivative emerging in predator-prey biological population dynamics system. This dynamics system are given by a set of fractional differential equations in the Caputo sense. A new solution is constructed in a power series. The stability of equilibrium points is studied. Moreover, numerical solutions for different cases are given and the methodology is displayed. We conducted a comparing between the results obtained by our method with the results obtained by other methods to illustrate the reliability and effectiveness of our main results.

Equivalence of the MTS Method and CMR Method for Differential Equations Associated with Semisimple Singularity

2014 ◽  
Vol 24 (01) ◽  
pp. 1450003 ◽  
Author(s):  
Pei Yu ◽  
Yuting Ding ◽  
Weihua Jiang
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Delay Differential Equations ◽  
Normal Forms ◽  
Functional Differential Equations ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Center Manifold Reduction ◽  
Functional Differential ◽  
Delay Differential

In this paper, the equivalence of the multiple time scales (MTS) method and the center manifold reduction (CMR) method is proved for computing the normal forms of ordinary differential equations and delay differential equations. The delay equations considered include general delay differential equations (DDE), neutral functional differential equations (NFDE) (or neutral delay differential equations (NDDE)), and partial functional differential equations (PFDE). The delays involved in these equations can be discrete or distributed. Particular attention is focused on dynamics associated with the semisimple singularity, and both the MTS and CMR methods are applied to compute the normal forms near the semisimple singular point. For the ordinary differential equations (ODE), we show that the two methods are equivalent up to any order in computing the normal forms; while for the differential equations with delays, we obtain the conditions under which the normal forms, derived by using the MTS and CMR methods, are identical up to third order. Different types of practical examples with delays are presented to demonstrate the application of the theoretical results, associated with Hopf, Hopf-zero and double-Hopf singularities.

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