EFFECT OF ADDED TIP MASS ON THE NONLINEAR FLAPWISE AND CHORDWISE VIBRATION OF CANTILEVER COMPOSITE BEAM UNDER BASE EXCITATION

2012 ◽  
Vol 12 (02) ◽  
pp. 285-310 ◽  
Author(s):  
M. EFTEKHARI ◽  
M. MAHZOON ◽  
S. ZIAEI-RAD
Keyword(s):  
Cantilever Beam ◽  
Multiple Scales ◽  
Equilibrium Points ◽  
Laminated Composite ◽  
Base Excitation ◽  
System A ◽  
Autoparametric Resonance ◽  
The Stability ◽  
Tip Mass ◽  
Made In

In this paper, a comparative study is performed for a symmetrically laminated composite cantilever beam with and without a tip mass under harmonic base excitation. The base is subjected to both flapwise and chordwise excitations tuned to the primary resonances of the two directions and conditions of 2:1 autoparametric resonance. In the literature, the governing nonlinear equations of the same problem without tip mass have been derived using the extended Hamilton's principle. Extension is made in this study to include the effect of a tip mass on the response of the beam. The natural frequencies are obtained numerically using the diversity guided evolutionary algorithm (DGEA). Next, the multiple scales method is applied to determine the nonlinear response and stability of the system. A set of four first-order differential equations describing the modulation of the amplitudes and phases of interacting modes are derived for the perturbation analysis. For verification, the above equations are reduced to the special case of the cantilever beam without tip mass for comparison with existing results. Finally, the effect of the tip mass on the stability of the fixed points and on the amplitude of oscillation about the equilibrium points in both the frequency and force modulation responses is examined.

Response variations of a cantilever beam–tip mass system with nonlinear and linearized boundary conditions

2018 ◽  
Vol 25 (3) ◽  
pp. 485-496 ◽  
Author(s):  
Vamsi C. Meesala ◽  
Muhammad R. Hajj
Keyword(s):  
Boundary Conditions ◽  
Cantilever Beam ◽  
Nonlinear Boundary ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Boundary Conditions ◽  
Distributed Parameter ◽  
Governing Equations ◽  
Mass System ◽  
Tip Mass

The distributed parameter governing equations of a cantilever beam with a tip mass subjected to principal parametric excitation are developed using a generalized Hamilton's principle. Using a Galerkin's discretization scheme, the discretized equation for the first mode is developed for simpler representation assuming linear and nonlinear boundary conditions. The discretized governing equation considering the nonlinear boundary conditions assumes a simpler form. We solve the distributed parameter and discretized equations separately using the method of multiple scales. Through comparison with the direct approach, we show that accounting for the nonlinear boundary conditions boundary conditions is important for accurate prediction in terms of type of bifurcation and response amplitude.

Modeling and Dynamic Analysis of Cantilever Beam Coupled with Moving Mass

2011 ◽  
Vol 105-107 ◽  
pp. 250-253
Author(s):  
Xing Wei Zhao ◽  
Zhen Dong Hu
Keyword(s):  
Cantilever Beam ◽  
Numerical Solutions ◽  
Multibody System ◽  
Variable Coefficients ◽  
Dynamic Responses ◽  
Moving Mass ◽  
Time Varying ◽  
System A ◽  
Varying Mass ◽  
Made In

A system of cantilever beam coupled with moving mass is investigated in this paper. Based on the time varying feature of multibody system, a dynamic model of the mass moving along the cantilever beam is established. The time varying motion equation including time varying mass and stiffness is derived by using the generalized Hamiltion’s principle, which results to a partial differential equation with variable coefficients. The nonlinearity of the equation is featured by those variable coefficients. To deal with the nonlinearity, methods of assumed modal shapes, Runge-Kutta are occupied. Then, a time varying system of a cantilever beam with stepwise sections coupled with a moving mass is given as an example. Numerical solutions of the dynamic responses of the system are achieved and certain analyses are made in the form of graphs.

Nonlinear Vibration of a Magneto-Elastic Cantilever Beam With Tip Mass

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Barun Pratiher ◽  
Santosha K. Dwivedy
Keyword(s):  
Magnetic Field ◽  
Frequency Response ◽  
Cantilever Beam ◽  
Multiple Scales ◽  
Flexible Structures ◽  
Nonlinear Damping ◽  
Equation Of Motion ◽  
Response Curves ◽  
Temporal Form ◽  
Tip Mass

In this work the effect of the application of an alternating magnetic field on the large transverse vibration of a cantilever beam with tip mass is investigated. The governing equation of motion is derived using D’Alembert’s principle, which is reduced to its nondimensional temporal form by using the generalized Galerkin method. The temporal equation of motion of the system contains nonlinearities of geometric and inertial types along with parametric excitation and nonlinear damping terms. Method of multiple scales is used to determine the instability region and frequency response curves of the system. The influences of the damping, tip mass, amplitude of magnetic field strength, permeability, and conductivity of the beam material on the frequency response curves are investigated. These perturbation results are found to be in good agreement with those obtained by numerically solving the temporal equation of motion and experimental results. This work will find extensive applications for controlling vibration in flexible structures using a magnetic field.

scholarly journals Resonance in the Cart-Pendulum System—An Asymptotic Approach

2021 ◽  
Vol 11 (23) ◽  
pp. 11567
Author(s):  
Wael S. Amer ◽  
Tarek S. Amer ◽  
Roman Starosta ◽  
Mohamed A. Bek
Keyword(s):  
Multiple Scales ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Pendulum System ◽  
Damped System ◽  
Stability And Instability ◽  
Parametric Pendulum ◽  
The Stability ◽  
Fundamental Equations

The major objective of this research is to study the planar dynamical motion of 2DOF of an auto-parametric pendulum attached with a damped system. Using Lagrange’s equations in terms of generalized coordinates, the fundamental equations of motion (EOM) are derived. The method of multiple scales (MMS) is applied to obtain the approximate solutions of these equations up to the second order of approximation. Resonance cases are classified, in which the primary external and internal resonance are investigated simultaneously to establish both the solvability conditions and the modulation equations. In the context of the stability conditions of these solutions, the equilibrium points are obtained and graphically displayed to derive the probable steady-state solutions near the resonances. The temporal histories of the attained results, the amplitude, and the phases of the dynamical system are depicted in graphs to describe the motion of the system at any instance. The stability and instability zones of the system are explored, and it is discovered that the system’s performance is stable for a significant number of its variables.

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