Analytical Approach to Vibration Analysis of a Pendulum Wrapping on a Cylinder

2011 ◽  
Author(s):  
H. Mazaheri ◽  
A. Hosseinzadeh ◽  
M. T. Ahmadian ◽  
Ahmad Barari
Keyword(s):  
Analytical Solution ◽  
Nonlinear Oscillation ◽  
Vibration Analysis ◽  
Analytical Approach ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Nonlinear Frequency ◽  
Initial Amplitude ◽  
System Frequency ◽  
Good Agreement

In this paper, nonlinear oscillation of a pendulum wrapping and unwrapping around two cylindrical bases is studied and an analytical solution is obtained using multiple scales method. Equations of motion are derived based on energy conservation technique. Applying perturbation method on the equations, nonlinear natural frequency of the system is calculated along with its time response. Analytical results are compared with numerical findings and good agreement is found. Effect of nonlinearity due to large amplitude and radius of cylinders on the system frequency is evaluated. Results indicate that as the radius of cylinder is increased, nonlinear frequency is enhanced. Initial amplitude plays a dual role on the frequency. As initial amplitude increases up to a certain point, frequency increases and decreases after wards.

scholarly journals Nonlinear vibration analysis of the nanobeams subjected to magneto-electro-thermo loading based on a novel HSDT

2021 ◽  
Author(s):  
Reza Mohammadi
Keyword(s):  
Nonlinear Vibration ◽  
Frequency Ratio ◽  
Vibration Analysis ◽  
Multiple Scales ◽  
Numerical Technique ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Nonlinear Frequency

Abstract In this paper, the nonlinear vibration analysis of the nanobeams subjected to magneto-electro-thermo loading based on a novel HSDT is studied. Nonlocal elasticity theory is applied to consider the small scale effect. The nonlinear equations of motion are derived using Hamilton’s principle. First, a Galerkin-based numerical technique is applied to reduce the nonlinear governing equation into a set of Duffing-type time-dependent differential equations. Afterward, the analytical solutions are derived based on the method of multiple scales (MMS) and perturbation technique. All of the mechanical properties of the beam are temperature dependent. The impacts of the several variables are investigated on the nonlinear frequency ratio of the nanobeams. The results illustrate that when maximum deflection is smaller/ greater than 0.2, its impact on the nonlinear frequency ratio will decrease/increase.

Vibration Characteristics of Straight Blades of Asymmetrical Aerofoil Cross-Section

1969 ◽  
Vol 20 (2) ◽  
pp. 178-190 ◽  
Author(s):  
W. Carnegie ◽  
B. Dawson
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Limited Range ◽  
First Order ◽  
Special Cases ◽  
Theoretical Results ◽  
Theoretical Procedure ◽  
Good Agreement

SummaryTheoretical and experimental natural frequencies and modal shapes up to the fifth mode of vibration are given for a straight blade of asymmetrical aerofoil cross-section. The theoretical procedure consists essentially of transforming the differential equations of motion into a set of simultaneous first-order equations and solving them by a step-by-step finite difference procedure. The natural frequency values are compared with results obtained by an analytical solution and with standard solutions for certain special cases. Good agreement is shown to exist between the theoretical results for the various methods presented. The equations of motion are dependent upon the coordinates of the axis of the centre of flexure of the beam relative to the centroidal axis. The effect of variations of the centre of flexure coordinates upon the frequencies and modal shapes is shown for a limited range of coordinate values. Comparison is made between the theoretical natural frequencies and modal shapes and corresponding results obtained by experiment.

Nonlinear Vibration Absorber Design: An Asymptotic Approach

2015 ◽  
Author(s):  
Arnaldo Casalotti ◽  
Walter Lacarbonara
Keyword(s):  
Nonlinear Vibration ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Sensitivity Analyses ◽  
Vibration Absorber ◽  
Asymptotic Approach ◽  
Nonlinear Vibration Absorber ◽  
The One ◽  
Two Degree Of Freedom ◽  
Good Agreement

The one-to-one internal resonance occurring in a two-degree-of-freedom (2DOF) system composed by a damped non-linear primary structure coupled with a nonlinear vibration absorber is studied via the method of multiple scales up to higher order (i.e., the first nonlinear order beyond the internal/external resonances). The periodic response predicted by the asymptotic approach is in good agreement with the numerical results obtained via continuation of the periodic solution of the equations of motion. The asymptotic procedure lends itself to manageable sensitivity analyses and thus to versatile optimization by which different optimal tuning criteria for the vibration absorber can possibly be found in semi-closed form.

Accurate modeling of preload discontinuity in the analytical approach of the nonlinear free vibration of beams

2012 ◽  
Vol 226 (10) ◽  
pp. 2474-2484 ◽  
Author(s):  
Hamid M Sedighi ◽  
Kourosh H Shirazi ◽  
Arash Reza ◽  
Jamal Zare
Keyword(s):  
Boundary Condition ◽  
Analytical Solution ◽  
Vibration Analysis ◽  
Analytical Approach ◽  
Dynamical Analysis ◽  
Suspension Bridges ◽  
Nonlinear Free Vibration ◽  
Equivalent Function ◽  
Realistic Problem ◽  
Computational Difficulty

This article attempts to investigate the dynamical analysis of beam vibrations in the presence of preload discontinuity and proposes an innovative accurate equivalent function for this well-known nonlinearity. This approach enables us to overcome the inherent computational difficulty of the preload nonlinearity in the analytical investigations. At first, the nonlinear equation of beam vibration with preload boundary condition is considered and analytical solution is obtained using He’s parameter expanding method. The precision of the proposed equivalent function has been elucidated by comparison of our results with the obtained solutions using numerical method. Finally, the accuracy of the obtained results in the vibration analysis of suspension bridges as a realistic problem, verifies the strength of the presented modeling.

scholarly journals Targeted Energy Transfer Under Harmonic Forcing With a Vibro-Impact Nonlinear Energy Sink: Analytical and Experimental Developments

2015 ◽  
Vol 137 (3) ◽  
Author(s):  
Etienne Gourc ◽  
Guilhem Michon ◽  
Sébastien Seguy ◽  
Alain Berlioz
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Nonlinear Energy Sink ◽  
Small Mass ◽  
Harmonic Excitation ◽  
Mass Ratio ◽  
Electrodynamic Shaker ◽  
Energy Sink ◽  
Nonlinear Energy ◽  
Good Agreement

Recently, it has been demonstrated that a vibro-impact type nonlinear energy sink (VI-NES) can be used efficiently to mitigate vibration of a linear oscillator (LO) under transient loading. The objective of this paper is to investigate theoretically and experimentally the potential of a VI-NES to mitigate vibrations of an LO subjected to a harmonic excitation (nevertheless, the presentation of an optimal VI-NES is beyond the scope of this paper). Due to the small mass ratio between the LO and the flying mass of the NES, the obtained equations of motion are analyzed using the method of multiple scales in the case of 1:1 resonance. It is shown that in addition to periodic response, system with VI-NES can exhibit strongly modulated response (SMR). Experimentally, the whole system is embedded on an electrodynamic shaker. The VI-NES is realized with a ball which is free to move in a cavity with a predesigned gap. The mass of the ball is less than 1% of the mass of the LO. The experiment confirms the existence of periodic and SMR regimes. A good agreement between theoretical and experimental results is observed.

scholarly journals Analytical Solution for Free Vibrations of a Moderately Thick Rectangular Plate

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Ivo Senjanović ◽  
Marko Tomić ◽  
Nikola Vladimir ◽  
Dae Seung Cho
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Thick Plate ◽  
Mixed Boundary ◽  
Separation Of Variables ◽  
Equations Of Motion ◽  
Mixed Boundary Conditions ◽  
Free Vibrations ◽  
Force Displacement ◽  
Good Agreement

In the present thick plate vibration theory, governing equations of force-displacement relations and equilibrium of forces are reduced to the system of three partial differential equations of motion with total deflection, which consists of bending and shear contribution, and angles of rotation as the basic unknown functions. The system is starting one for the application of any analytical or numerical method. Most of the analytical methods deal with those three equations, some of them with two (total and bending deflection), and recently a solution based on one equation related to total deflection has been proposed. In this paper, a system of three equations is reduced to one equation with bending deflection acting as a potential function. Method of separation of variables is applied and analytical solution of differential equation is obtained in closed form. Any combination of boundary conditions can be considered. However, the exact solution of boundary value problem is achieved for a plate with two opposite simply supported edges, while for mixed boundary conditions, an approximate solution is derived. Numerical results of illustrative examples are compared with those known in the literature, and very good agreement is achieved.

Effect of the Transverse Shear Deformation on the Free Vibration of Rotating Blades Modeled as Thin-Walled Composite Beams

2015 ◽  
Vol 798 ◽  
pp. 119-124
Author(s):  
Serhat Yilmaz ◽  
Seher Eken ◽  
Metin Orhan Kaya
Keyword(s):  
Transverse Shear ◽  
Fiber Orientation ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Transverse Shear Deformation ◽  
Combined Effects ◽  
Shear Deformations ◽  
Thin Walled ◽  
Good Agreement

In this paper, vibration analysis of a blade modeled as an anisotropic composite thin-walled beam is carried out. The analytical formulation of the beam is derived for the flapwise bending, chordwise bending and transverse shear deformations. The equations of motion are solved by applying the extended Galerkin method (EGM) for anti-symmetric lay-up configuration that is also referred as Circumferentially Uniform Stiffness (CUS). Consequently, the natural frequencies are validated by making comparisons with the results in literature and it is observed that there is a good agreement between the results. Combined effects of transverse shear, fiber orientation, and rotational speed on the natural frequencies are further investigated.

scholarly journals The Solution to the Task of Spreading the Non-Stationary Two-Dimensional Potential Flow (The Radial Source)

2020 ◽  
Vol 8 (2) ◽  
pp. 21-25
Author(s):  
Olga Burtseva ◽  
Viktor Kochanenko ◽  
Sergej Evtushenko ◽  
Anatoly Kondratenko
Keyword(s):  
Analytical Solution ◽  
Boundary Value Problem ◽  
Wave Front ◽  
Radial Flow ◽  
Equations Of Motion ◽  
Experimental Setup ◽  
Two Dimensional ◽  
Small Perturbations ◽  
Experimental Parameters ◽  
Good Agreement

The equations of motion of a non-stationary radial flow are derived, the boundary value problem is set, and its analytical solution is obtained. The solution of the problem in this paper is in good agreement with the experimental parameters obtained at the experimental setup for small perturbations. The equations for determining the height of the wave front that decreases downstream of the flow are obtained, and the instantaneous velocity of the wave front tends to zero.

Nonlinear Vibrations Analysis of Overhead Power Lines: A Beam With Mass–Spring–Damper–Mass Systems

2018 ◽  
Vol 140 (3) ◽  
Author(s):  
Mohammad A. Bukhari ◽  
Oumar R. Barry
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Nonlinear Frequency ◽  
Response Curves ◽  
Mass System ◽  
Parametric Studies ◽  
Overhead Power Lines ◽  
Nonlinear Equations Of Motion ◽  
Mass Spring ◽  
Stockbridge Damper

This paper examines the nonlinear vibration of a single conductor with Stockbridge dampers. The conductor is modeled as a simply supported beam and the Stockbridge damper is reduced to a mass–spring–damper–mass system. The nonlinearity of the system stems from the midplane stretching of the conductor and the cubic equivalent stiffness of the Stockbridge damper. The derived nonlinear equations of motion are solved by the method of multiple scales. Explicit expressions are presented for the nonlinear frequency, solvability conditions, and detuning parameter. The present results are validated via comparisons with those in the literature. Parametric studies are conducted to investigate the effect of variable control parameters on the nonlinear frequency and the frequency response curves. The findings are promising and open a horizon for future opportunities to optimize the design of nonlinear absorbers.

Coupled Torsional and Transverse Vibration of Unbalanced Rotor

1985 ◽  
Vol 52 (3) ◽  
pp. 701-705 ◽  
Author(s):  
R. Cohen ◽  
I. Porat
Keyword(s):  
Transverse Vibration ◽  
Constant Velocity ◽  
Vibration Analysis ◽  
Equations Of Motion ◽  
Resonance Effect ◽  
Combination Resonance ◽  
Unbalanced Rotor ◽  
Direct Numerical Solution ◽  
Nonlinear Equations Of Motion ◽  
Good Agreement

A model of an unbalanced rotor, driven by a torsion-flexible shaft through a constant velocity joint, is used to investigate the combination-resonance effect in coupled torsional-transverse vibration. Analysis of the nonlinear equations of motion by an asymptotic method yields the instability zones of the system. Results are in very good agreement with those obtained by direct numerical solution of the equations of motion.

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