The Control of Parametric Vibration in a Simply Supported Beam

1987 ◽  
Vol 109 (3) ◽  
pp. 315-318
Author(s):  
J. S. Burdess
Keyword(s):  
Numerical Simulation ◽  
Theoretical Analysis ◽  
Perturbation Method ◽  
Natural Frequency ◽  
Multiple Scales ◽  
Equation Of Motion ◽  
Control Law ◽  
Parametric Vibrations ◽  
Simply Supported ◽  
Multiple Scales Perturbation Method

The paper shows how unstable parametric vibrations of a uniform beam can be controlled. A control law is proposed and it is shown that the beam can be made to vibrate at a present amplitude at its natural frequency. The beam is modelled by its first mode and a solution to the governing equation of motion is derived by applying the multiple scales perturbation method. The results of the theoretical analysis are verified by a numerical simulation.

Nonplanar Motion of a String due to Two-Frequency Excitation

2001 ◽  
Author(s):  
Kimihiko Yasuda ◽  
Keisuke Kamiya
Keyword(s):  
Numerical Simulation ◽  
Theoretical Analysis ◽  
Perturbation Method ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Harmonic Excitation ◽  
Chaotic Motions ◽  
Frequency Excitation

Abstract It is known that, under certain conditions, a stretched string subjected to a planar harmonic excitation executes nonplanar motions due to the instability of the palanar motion. In recent years, studies on bifurcations of such nonplanar motions to amplitude modulated quasiperiodic motions and chaotic motions have been reported. However no literatures on the problem of nonplanar motions due to a multi-frequency excitation are found. In this paper, the possibility of nonplanar motions in a string due to a two-frequency excitation is studied. For this purpose two cases are considered, i.e. one in which both components of the excitation are in a plane, and one in which they are perpendicular to each other. In both cases the sum of the frequencies of the components is supposed to near to twice one of the natural frequencies of the string. Theoretical analysis using the perturbation method of multiple scales and numerical simulation are carried out to show that nonplanar motions occur.

Natural Frequency and Vibration Modal Analysis of Composite Laminated Plate

2013 ◽  
Vol 711 ◽  
pp. 396-400
Author(s):  
Long Sheng Zhao ◽  
Jin Wu Wu
Keyword(s):  
Natural Frequency ◽  
Good Accuracy ◽  
Natural Frequencies ◽  
Equation Of Motion ◽  
Theory Method ◽  
Laminated Plate ◽  
Simply Supported ◽  
Composite Laminated Plate ◽  
Experiment Method ◽  
Vibration Modal

In this paper,a layer-wise theory is used to analyze the natural frequency and vibration modal of the composite laminated plate. Layer-wise theory assumes that displacement is continue through thickness direction and has good accuracy to analyze free vibration. The frequency and vibration modal are acquired while building the equation of motion according to layer-wise theory. Through comparing layer-wise theory and other theories, numerical results show that layer-wise theory is credible to analyze composite laminated plate. At the same time, experiment is used in this paper to acquire the natural frequencies and vibration modal of a simply supported composite laminated plate. Lastly, combination of the theory method and experiment method canprobably predict the natural frequencies and vibration modal.

scholarly journals Stability of the dynamic shifting process in the vehicle transmission with the input shaper

2018 ◽  
Vol 211 ◽  
pp. 02007 ◽  
Author(s):  
Alexander Taratorkin ◽  
Victor Derzhanskii ◽  
Igor Taratorkin
Keyword(s):  
Numerical Simulation ◽  
Natural Frequency ◽  
Control Algorithm ◽  
Control Law ◽  
Single Node ◽  
Input Shaper ◽  
Inertial Parameters ◽  
Wheel Drive ◽  
And Performance ◽  
The Stability

This paper investigates stability of the dynamic process of gear shift in the vehicle transmission with the input shapers (IS) taking in consideration uncertainty of the natural frequency of the mechanical system. The proposed control algorithm increases the stability of the researched system with variation of its elastic and inertial parameters. The control law with guaranteed asymptotic stability is obtained for full load of the vehicle when the all-wheel drive is turned on. The monitoring of the state and performance of the required parameters for the regulation of input shapers are tested by means of numerical simulation. Analyzing the results it is established that the best stability is reached by the adaptive setting of the input shapers in accordance with controlled value of the natural frequency of the lowest single-node mode.

Sensitivity of MEMS/NEMS Biosensors Near Twice Natural Frequency

2010 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Martin W. Knecht
Keyword(s):  
Natural Frequency ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Casimir Effect ◽  
Electrostatic Force ◽  
Equation Of Motion ◽  
Direct Approach ◽  
Mass Detection ◽  
Electrostatically Actuated ◽  
Phase Amplitude

Bio-MEMS/NEMS resonator sensors near twice natural frequency for mass detection are investigated. Electrostatic force along with fringe correction and Casimir effect are included in the model. They introduce parametric nonlinear terms in the system. The partial-differential equation of motion of the system is solved by using the method of multiple scales. A direct approach of the problem is then used. Two approximation problems resulting from the direct approach are solved. Phase-amplitude relationship is obtained. Numerical results for uniform electrostatically actuated micro resonator sensors are reported.

A New Kind of Energy Transfer From the High-Frequency to Low-Frequency Mode in a Composite Laminated Plate

2010 ◽  
Author(s):  
Wei Zhang ◽  
Xiang-Ying Guo ◽  
Qian Wang ◽  
Cui-Cui Liu ◽  
Yun-cheng He
Keyword(s):  
Numerical Simulation ◽  
Natural Frequency ◽  
Thin Plate ◽  
High Frequency ◽  
Low Frequency ◽  
Frequency Mode ◽  
Rectangular Thin Plate ◽  
Chaotic Motions ◽  
Simply Supported ◽  
Low Frequency Mode

This paper focuses on the analysis on a new kind of nonlinear resonant motion with the low-frequency large-amplitude, which can be induced by the high-frequency small-amplitude mode through the mechanism of modulation of amplitude and phase. The system investigated is a simply supported symmetric cross-ply composite laminated rectangular thin plate subjected to parametric excitations. Experimental research has been carried out for the first time. The test plate was excited near the first natural frequency with parametric forces and the above mentioned high-to-low frequency mode has been observed, whose frequency is extremely lower than the first natural frequency. Theoretical job goes to analysis the above phenomenon accordingly. Based on the Reddy’s third-order shear deformation plate theory and the von Karman type equation, the nonlinear governing equations of the simply supported symmetric cross-ply composite laminated rectangular thin plate subjected to parametric excitations are formulated. The Galerkin method is utilized to discretize the governing partial differential equations into a two-degree-of-freedom nonlinear system. Numerical simulation is conducted to investigate this non-autonomous system subsequently. The results of numerical simulation demonstrate that there is a qualitative agreement between the experimental observation and the theoretical result. Besides, the multi-pulse chaotic motions are also reported in numerical simulations.

scholarly journals Application of Homotopy Perturbation Method with an Auxiliary Term for Nonlinear Dropping Equations Arisen in Polymer Packaging System

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Xiang Hong ◽  
Jun Wang ◽  
Li-xin Lu
Keyword(s):  
Numerical Simulation ◽  
Perturbation Method ◽  
Analytical Solutions ◽  
Homotopy Perturbation Method ◽  
Equation Of Motion ◽  
High Accuracy ◽  
Second Order ◽  
Packaging System ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions

The homotopy perturbation method (HPM) with an auxiliary term was applied to obtain approximate analytical solutions of polymer cushioning packaging system. The second-order solution of the equation of motion was obtained and compared with the numerical simulation solution solved by the Runge-Kutta algorithm. The results showed the high accuracy of this modified HPM with convenient calculation.

Response of Electrostatically Actuated Sensors Near Half of the Natural Frequency

2009 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Martin W. Knecht
Keyword(s):  
Natural Frequency ◽  
Governing Equation ◽  
Multiple Scales ◽  
Equation Of Motion ◽  
Casimir Forces ◽  
Order Model ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Micro Sensor ◽  
Phase Amplitude

A cantilever micro-resonator electrostatically actuated near half of the natural frequency is investigated. Hamilton’s principle is used to derive the partial-differential equation of motion for a general non-uniform sensor. Nonlinearities arise due to the electrostatic and Casimir forces. The electrostatic actuation introduces parametric coefficients in both linear and nonlinear parts of the governing equation. A direct approach is taken using the method of multiple scales resulting in a phase-amplitude relationship for the system. Numerical results for a uniform capacitive resonator micro-sensor are provided and tested numerically using a reduced-order model of the governing equation of motion.

Numerical Simulation Study on Healthy Monitoring of the Framework Based on Aseismatic Behavior

2012 ◽  
Vol 479-481 ◽  
pp. 1253-1257
Author(s):  
Xiao Ming Yang ◽  
Jun Yi Wang
Keyword(s):  
Neural Network ◽  
Numerical Simulation ◽  
Theoretical Analysis ◽  
Natural Frequency ◽  
Simulation Study ◽  
Structural Damage ◽  
Frame Structure ◽  
Earthquake Loading ◽  
The Neural Network ◽  
Frequency Changes

Considering back and forth and the complexity of earthquake loading, according to theoretical analysis, this paper is verified theoretically that the natural frequency changes of structure contain the information such as location and degree of the damage in earthquake, based on this theory, applying the large common software – ANSYS, and an experiment to simulate the damage of a six-storey-high frame structure is used, extracting the natural frequency changes and forming the input vectors of the neural network, the structural damage is detected by the neural network trained, the result showed the effectiveness of this method in the engineering.

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