Large Amplitude Vibration of a Cantilever Beam with Tip Mass under Random Base Excitation

2002 ◽  
Vol 4 (4) ◽  
pp. 203-210 ◽  
Author(s):  
Guangfeng Cheng ◽  
Chuh Mei ◽  
Raymond Y. Y. Lee
Keyword(s):  
Cantilever Beam ◽  
Large Amplitude ◽  
Beam Theory ◽  
Single Mode ◽  
Rotary Inertia ◽  
System Response ◽  
Base Excitation ◽  
Random Response ◽  
Lumped Mass ◽  
Tip Mass

Nonlinear large amplitude random vibration of cantilever beam with lumped mass and rotary inertia under zero mean, stationary, Gaussian random base excitation is studied, using the inextensional beam theory. Single-mode approximation is employed to discretize the Lagrange's equation. The resulting nonlinear governing modal equation of motion is solved with application of the stochastic linearization method. Two examples, a cantilever beam with/without tip mass, are analyzed as application of the developed methodology. Effects of mass and rotary inertia variation on system response are investigated in detail. Results showed that increasing rotary inertia could reduce the random response of the beam structure and the random response of the structure is quite sensitive to the tip mass variation. The nonlinearities of the inextensional beam vibration result in a spring hardening system.

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.

Dynamic Simulation and Vibration Analysis of a Mechanical Piano Key Actuator

2011 ◽  
Author(s):  
Ramin Masoudi ◽  
Stephen Birkett ◽  
Armaghan Salehian
Keyword(s):  
Cantilever Beam ◽  
Contact Force ◽  
Beam Theory ◽  
Contact Dynamics ◽  
Hysteretic Behavior ◽  
Contact Period ◽  
Actuation System ◽  
Loading And Unloading ◽  
Tip Mass ◽  
Smooth Transitions

Dynamic modeling of a flexible hub-beam system with an eccentric tip mass including nonlinear hysteretic contact is studied in this paper. In reality, the model is intended to represent the mechanical finger of an actuator for a piano key. Developing a device to achieve a desired finger-key contact force profile that realistically replicates that of a real pianist’s finger is the main objective of this research. The proposed actuation system consists of a flexible arm which is attached to a DC brushless rotary motor thorough a hub. The compliant arm behaves as a cantilever beam to which an eccentric tip mass has been attached at its free end. During the piano key stroke, the contact force input from the tip causes the key to rotate and impact the ground through an interface lined with stiff felt to suppress vibrations and noise. Euler-Bernoulli beam theory in conjunction with Lagrange’s method is utilized to obtain the governing equations of motion for the system. The finite element method is used as the discretization procedure for the flexible cantilever beam, which can be considered a distributed parameter system. To include contact dynamics at the stop, the nonlinear hysteretic behavior of felt under compression is modeled in such a way that smooth transitions between loading and unloading stages are produced, thus ensuring accurate response under dynamic conditions, and particularly with partial loading and unloading states that occur during the contact period. Simulation results show excessive vibration is produced due to the arm flexibility and also the rigid-body oscillations of the arm, especially during the period of key-felt contact. To eliminate these vibrations, the system was supplemented with various dashpot models, including a simple grounded rotational dashpot, and a grounded rotational dashpot with a one-sided relation. The results of simulations are presented showing the effect on vibration behavior attributed to these additional components.

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