ON THE DERIVATION OF THE EQUATIONS OF MOTION FOR A PARAMETRICALLY EXCITED CANTILEVER BEAM

2001 ◽  
Vol 245 (1) ◽  
pp. 165-177 ◽  
Author(s):  
D.I.M. FOREHAND ◽  
M.P. CARTMELL
Keyword(s):  
Cantilever Beam ◽  
Equations Of Motion ◽  
Parametrically Excited

Dynamics of Spinning Disc Parametrically Excited by Noise

1999 ◽  
Author(s):  
Narayanan Ramakrishnan ◽  
N. Sri Namachchivaya
Keyword(s):  
Equations Of Motion ◽  
Planck Equation ◽  
Stochastic Averaging ◽  
Transverse Motion ◽  
Probability Density Distribution ◽  
Integral Of Motion ◽  
Parametrically Excited ◽  
One Dimensional ◽  
Small Intensity ◽  
Spinning Disc

Abstract The nonlinear dynamics of a circular spinning disc parametrically excited by noise of small intensity is investigated. The governing PDEs are reduced using a Galerkin reduction procedure to a two-DOF system of ODEs which, govern the transverse motion of the disc. The dynamics is simplified by exploiting the S1 invariance of the equations of motion of the reduced system and further, reduced by performing stochastic averaging. The resulting one-dimensional Markov diffusive process is studied in detail. The stationary probability density distribution is obtained by solving the Fokker-Planck equation along with the appropriate boundary conditions. The boundary behaviour is studied using an asymptotic approach. Some aspects of dynamical and phenomenological bifurcations of the stationary solution are also investigated. The scheme of things presented here can be applied in principle to a four-dimensional Hamiltonian system possessing one integral of motion in addition to the hamiltonian and having one fixed point.

Dynamic Analysis of Elastic Beam-Like Mechanism Systems

1989 ◽  
Vol 111 (4) ◽  
pp. 626-629
Author(s):  
W. Ying ◽  
R. L. Huston
Keyword(s):  
Dynamic Analysis ◽  
Dynamic Behavior ◽  
Cantilever Beam ◽  
Equations Of Motion ◽  
Elastic Beam ◽  
Order Perturbation ◽  
First Order ◽  
Geometric Stiffness ◽  
Stiffness Matrices ◽  
First Order Perturbation

In this paper the dynamic behavior of beam-like mechanism systems is investigated. The elastic beam is modeled by finite rigid segments connected by joint springs and dampers. The equations of motion are derived using Kane’s equations. The nonlinear terms are linearized by first order perturbation about a system balanced configuration state leading to geometric stiffness matrices. A simple numerical example of a rotating cantilever beam is presented.

Free and Forced Vibration of a Kinked Cantilever Beam

1999 ◽  
Author(s):  
B. S. Reddy ◽  
K. R. Y. Simha ◽  
A. Ghosal
Keyword(s):  
Cantilever Beam ◽  
Mode Shape ◽  
Forced Vibration ◽  
Equations Of Motion ◽  
Element Formulation ◽  
Kink Angle ◽  
Order Polynomial ◽  
Relationship Of ◽  
Tip Mass ◽  
Beam Mode

Abstract In this paper we use the assumed modes method to derive an analytical model of a kinked cantilever beam of unit mass carrying a kink mass (mk) and a tip mass (mt). The model is used to study the free and forced vibration of such a beam. For the free vibration, we obtain the mode shape of the complete beam by solving an eight order polynomial whose coefficients are functions of the kink mass, kink angle and tip mass. A relationship of the form f(mk,mt,δ)=mk+mt(4+103cosδ+23cos2δ)=constant appears to give the same fundamental frequency for a given kink angle, δ, and different combinations of kink mass and tip mass. To derive the dynamic equations of motion, the complete kinked beam mode shape is used in a Lagrangian formulation. The equations of motion are numerically integrated with a torque applied at the base and the tip response for various kink angles are presented. The results match those obtained from a traditional finite element formulation.

Active Damping of Vibrations of a Cantilever Beam by Axial Force Control

2007 ◽  
Author(s):  
Thomas Pumho¨ssel ◽  
Horst Ecker
Keyword(s):  
Cantilever Beam ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Open Loop ◽  
Active Damping ◽  
Nonlinear Feedback ◽  
Euler Beam ◽  
Bending Vibrations ◽  
Loop Control ◽  
Damping Of Vibrations

In several fields, e.g. aerospace applications, robotics or the bladings of turbomachinery, the active damping of vibrations of slender beams which are subject to free bending vibrations becomes more and more important. In this contribution a slender cantilever beam loaded with a controlled force at its tip, which always points to the clamping point of the beam, is treated. The equations of motion are obtained using the Bernoulli-Euler beam theory and d’Alemberts principle. To introduce artificial damping to the lateral vibrations of the beam, the force at the tip of the beam has to be controlled in a proper way. Two different methods are compared. One concept is the closed-loop control of the force. In this case a nonlinear feedback control law is used, based on axial velocity feedback of the tip of the beam and a state-dependent amplification. By contrast, the concept of open-loop parametric control works without any feedback of the actual vibrations of the mechanical structure. This approach applies the force as harmonic function of time with constant amplitude and frequency. Numerical results are carried out to compare and to demonstrate the effectiveness of both methods.

A Parametrically Broadband Nonlinear Energy Harvester

2016 ◽  
Vol 139 (3) ◽  
Author(s):  
Tanju Yildirim ◽  
Mergen H. Ghayesh ◽  
Thomas Searle ◽  
Weihua Li ◽  
Gursel Alici
Keyword(s):  
Energy Harvesting ◽  
Cantilever Beam ◽  
Nonlinear Response ◽  
Energy Harvester ◽  
Shaking Table ◽  
Experimental Investigations ◽  
Dynamical Response ◽  
Parametrically Excited ◽  
Nonlinear Dynamical ◽  
Electrodynamic Shaker

In this work, for the first time, an energy harvester based on the nonlinear dynamical response of a parametrically excited cantilever beam in contact with mechanical stoppers has been fabricated and tested; a 145% increase in the bandwidth at which energy can be effectively harvested has been observed. Experimental and theoretical investigations have been performed in order to assess the increased operating bandwidth of the energy harvester fabricated; for the experimental investigations, an electrodynamic shaker connected to a shaking table has been used to parametrically stimulate the energy harvesting device. Results showed that the parametric energy harvester without stoppers displayed a weak softening-type nonlinear response; however, with the addition of mechanical stoppers the energy harvester displayed a strong hardening-type nonlinear response which is ideal for capturing kinetic energy over larger bandwidths. The addition of mechanical stoppers on a parametrically excited cantilever beam has significant qualitative and quantitative effects on the nonlinear parametric energy harvesting; the energy harvesting bandwidth was increased in the range of 35–145% by adjusting the stoppers.

Experimental Verification of the Importance of The Nonlinear Curvature in the Response of a Cantilever Beam

1996 ◽  
Vol 118 (1) ◽  
pp. 21-27 ◽  
Author(s):  
T. J. Anderson ◽  
A. H. Nayfeh ◽  
B. Balachandran
Keyword(s):  
Analytical Model ◽  
Cantilever Beam ◽  
Theoretical Investigation ◽  
Experimental Verification ◽  
Dominant Role ◽  
Experimental Results ◽  
Parametrically Excited ◽  
Second Mode ◽  
Quadratic Damping ◽  
Theoretical Results

An experimental and theoretical investigation into the first- and second-mode responses of a parametrically excited slender cantilever beam is presented. Inclusion of quadratic damping in the analytical model significantly improves the agreement between the experimental and theoretical results. In addition, the experimental results verify that the often ignored nonlinear curvature terms play a dominant role in the response of the first mode and that the nonlinear inertia terms play a dominant role in the response of the second mode.

scholarly journals An MSPT of a parametrically excited cantilever beam

1993 ◽  
Vol 47 (2) ◽  
pp. 219-239
Author(s):  
S.A. El-Serafi ◽  
F.Z. El-Halafawy ◽  
M.H. Eissa ◽  
M.M. Kamel
Keyword(s):  
Cantilever Beam ◽  
Parametrically Excited

The Udwadia–Kalaba Trajectory Control Applied to a Cantilever Beam—Experimental Results

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Raphael Pereira Spada ◽  
Rodrigo Nicoletti
Keyword(s):  
Cantilever Beam ◽  
Tracking Control ◽  
Equations Of Motion ◽  
Constrained Systems ◽  
Mode Shapes ◽  
Laser Vibrometer ◽  
Successful Implementation ◽  
Out Of Plane ◽  
Control Forces

The Udwadia–Kalaba methodology is a possible way of explicitly obtaining the equations of motion of constrained systems. From these equations of motion, one can estimate the necessary forces in the constraint to keep the system in a given motion. Hence, the Udwadia–Kalaba methodology can also apply to active tracking control of subsystems or the control of points of a structure. In this work, one investigates experimentally the benefits and drawbacks of such control strategy by applying it to the control of out-of-plane vibrations of a cantilever beam. The beam is excited by a shaker mounted near the clamped end of the beam. A second shaker applies the control forces in the free end of the beam, where an accelerometer is used for feedback. The vibration behavior of the beam under excitation/control is measured by a laser vibrometer. Results show that the methodology changes the dynamic behavior of the structure by changing its boundary conditions at the point of control, thus shifting natural frequencies and mode shapes. Results also show that the successful implementation of the method experimentally is sensitive to the quality of modeling of the structure.

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