High-frequency Vibrational Control of Principal Parametric Resonance of a Nonlinear Cantilever Beam: Theory and Experiment

2021 ◽  
pp. 116138
Author(s):  
Pradyumna Kumar Sahoo ◽  
S. Chatterjee
Keyword(s):  
Cantilever Beam ◽  
Parametric Resonance ◽  
High Frequency ◽  
Beam Theory ◽  
Vibrational Control ◽  
Principal Parametric Resonance ◽  
Theory And Experiment

Applications of Holography to Dynamics: High-Frequency Vibrations of Beams

1970 ◽  
Vol 37 (2) ◽  
pp. 287-291 ◽  
Author(s):  
R. Aprahamian ◽  
D. A. Evensen
Keyword(s):  
Cantilever Beam ◽  
Timoshenko Beam ◽  
Holographic Interferometry ◽  
High Frequency ◽  
Beam Theory ◽  
Mode Shapes ◽  
Resonant Frequencies ◽  
Transverse Vibrations ◽  
High Frequency Vibrations ◽  
Time Average

The relatively new experimental technique of holographic interferometry is described, and time-average holography is discussed. Time-average holography has been applied to study high-frequency transverse vibrations of a uniform cantilever beam. Modes from the 5th through the 34th were identified and recorded on holograms, the corresponding resonant frequencies ranged from 500–25,665 cps. In addition, the 77th mode was recorded at 99,395 cps, which demonstrates that the technique is workable at frequencies on the order of 100 kc. The experimental mode shapes and frequencies show good correlation with the Timoshenko beam theory. Other applications of holography to dynamics are briefly discussed.

Analysis of Parametric Resonances in In-Plane Vibrations of Electrostrictive Hyperelastic Plates

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Astitva Tripathi ◽  
Anil K. Bajaj
Keyword(s):  
Electric Field ◽  
Parametric Resonance ◽  
Mechanical Response ◽  
Element Model ◽  
Mode Shapes ◽  
Principal Parametric Resonance ◽  
Parametric Resonances ◽  
Second Mode ◽  
Electrostrictive Polymer ◽  
Large Amplitude Vibrations

Electrostriction is a recent actuation mechanism which is being explored for a variety of new micro- and millimeter scale devices along with macroscale applications such as artificial muscles. The general characteristics of these materials and the nature of actuation lend itself to possible production of very rich nonlinear dynamic behavior. In this work, principal parametric resonance of the second mode in in-plane vibrations of appropriately designed electrostrictive plates is investigated. The plates are made of an electrostrictive polymer whose mechanical response can be approximated by Mooney Rivlin model, and the induced strain is assumed to have quadratic dependence on the applied electric field. A finite element model (FEM) formulation is used to develop mode shapes of the linearized structure whose lowest two natural frequencies are designed to be close to be in 1:2 ratio. Using these two structural modes and the complete Lagrangian, a nonlinear two-mode model of the electrostrictive plate structure is developed. Application of a harmonic electric field results in in-plane parametric oscillations. The nonlinear response of the structure is studied using averaging on the two-mode model. The structure exhibits 1:2 internal resonance and large amplitude vibrations through the route of parametric excitation. The principal parametric resonance of the second mode is investigated in detail, and the time response of the averaged system is also computed at few frequencies to demonstrate stability of branches. Some results for the case of principal parametric resonance of the first mode are also presented.

Vibrational Control of a 2-Link Mechanism

2020 ◽  
Author(s):  
Zakia Ahmed ◽  
Sevak Tahmasian ◽  
Craig A. Woolsey
Keyword(s):  
Translational Control ◽  
High Frequency ◽  
Horizontal Plane ◽  
High Amplitude ◽  
Torsional Stiffness ◽  
Physical Parameters ◽  
Vibrational Control ◽  
Link Mechanism ◽  
Torsional Damping ◽  
Fixed Input

Abstract This paper describes vibrational control and stability of a planar, horizontal 2-link mechanism using translational control of the base pivot. The system is a 3-DOF two-link mechanism that is subject to torsional damping, torsional stiffness, and is moving on a horizontal plane. The goal is to drive the averaged dynamics of the system to a desired configuration using a high-frequency, high-amplitude force applied at the base pivot. The desired configuration is achieved by applying an amplitude and angle of the input determined using the averaged dynamics of the system. We find the range of stable configurations that can be achieved by the system by changing the amplitude of the oscillations for a fixed input angle and oscillation frequency. The effects of varying the physical parameters on the achievable stable configurations are studied. Stability analysis of the system is performed using two methods: the averaged dynamics and averaged potential.

Principal parametric resonance of axially accelerating viscoelastic strings with an integral constitutive law

2005 ◽  
Vol 461 (2061) ◽  
pp. 2701-2720 ◽  
Author(s):  
Li-Qun Chen
Keyword(s):  
Steady State ◽  
Parametric Resonance ◽  
Multiple Scales ◽  
Constitutive Law ◽  
Transverse Motion ◽  
Viscoelastic Parameters ◽  
Principal Parametric Resonance ◽  
Linearized Stability ◽  
Speed Fluctuation ◽  
Axial Speed

The steady-state transverse responses and the stability of an axially accelerating viscoelastic string are investigated. The governing equation is derived from the Eulerian equation of motion of a continuum, which leads to the Mote model for transverse motion. The Kirchhoff model is derived from the Mote model by replacing the tension with the averaged tension over the string. The method of multiple scales is applied to the two models in the case of principal parametric resonance. Closed-form expressions of the amplitudes and the existence conditions of steady-state periodical responses are presented. The Lyapunov linearized stability theory is employed to demonstrate that the first (second) non-trivial steady-state response is always stable (unstable). Numerical calculations show that the two models are qualitatively the same, but quantitatively different. Numerical results are also presented to highlight the effects of the mean axial speed, the axial-speed fluctuation amplitude, and the viscoelastic parameters.

scholarly journals Natural Frequency Characteristics of the Beam with Different Cross Sections Considering the Shear Deformation Induced Rotary Inertia

2020 ◽  
Vol 10 (15) ◽  
pp. 5245
Author(s):  
Chunfeng Wan ◽  
Huachen Jiang ◽  
Liyu Xie ◽  
Caiqian Yang ◽  
Youliang Ding ◽  
...  
Keyword(s):  
Shear Deformation ◽  
Cross Sections ◽  
Timoshenko Beam ◽  
High Frequency ◽  
Natural Frequencies ◽  
Beam Theory ◽  
Equation Of Motion ◽  
Timoshenko Beam Theory ◽  
Rotary Inertia ◽  
Cross Sectional

Based on the classical Timoshenko beam theory, the rotary inertia caused by shear deformation is further considered and then the equation of motion of the Timoshenko beam theory is modified. The dynamic characteristics of this new model, named the modified Timoshenko beam, have been discussed, and the distortion of natural frequencies of Timoshenko beam is improved, especially at high-frequency bands. The effects of different cross-sectional types on natural frequencies of the modified Timoshenko beam are studied, and corresponding simulations have been conducted. The results demonstrate that the modified Timoshenko beam can successfully be applied to all beams of three given cross sections, i.e., rectangular, rectangular hollow, and circular cross sections, subjected to different boundary conditions. The consequence verifies the validity and necessity of the modification.

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.

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