Response of Three Oscillators Coupled by a Weak Nonlinearity

1977 ◽  
Vol 44 (1) ◽  
pp. 141-146 ◽  
Author(s):  
N. Yen ◽  
R. E. Kronauer
Keyword(s):  
Perturbation Expansion ◽  
Phase Variation ◽  
Transient Behavior ◽  
Conservative System ◽  
Crystal Plate ◽  
High Frequency Oscillation ◽  
Piezoelectric Crystal ◽  
Weak Nonlinearity ◽  
Resonance Modes ◽  
Piezoelectric Crystal Plate

As a simplified model of the exchange processes occurring among resonance modes in physical systems, such as a piezoelectric crystal plate or an acoustic interferometer, a study is made of the response of three oscillators that are coupled by a weak nonlinearity and whose frequencies satisfy the condition ω1 + ω2 ≅ ω3. The transient behavior is obtained by a perturbation expansion. There exist three integral constraints on the amplitude and phase variation of the oscillations for a conservative system, and the solution of the response can be reduced to quadrature. The phase diagram describing the motion indicates that the high frequency oscillation is unstable; the energy associated with it, under certain conditions, can be diverted to lower frequency oscillations. For nonconservative systems, the effects of dissipation and detuning are examined for their role in limiting the energy exchange among the oscillations and in determining the steady-state response to forcing. Predictions from this analysis are compared with results of a reported experiment in which a piezoelectric crystal plate is forced to oscillate at amplitudes sufficient to generate coupled subharmonics.

scholarly journals Flexoelectric effect on thickness-shear vibration of a rectangular piezoelectric crystal plate

2021 ◽  
Author(s):  
Yang Zheng ◽  
Bin Huang ◽  
Ji Wang
Keyword(s):  
Boundary Conditions ◽  
Displacement Function ◽  
Crystal Plate ◽  
Symmetric Mode ◽  
Piezoelectric Crystal ◽  
Flexoelectric Effect ◽  
Shear Vibration ◽  
Shear Modes ◽  
Piezoelectric Crystal Plate ◽  
Thickness Shear

Abstract Thickness-shear (TSh) vibration of a rectangular piezoelectric crystal plate is studied with the consideration of flexoelectric effect in this paper. The developed theoretical model is based on the assumed displacement function which includes the anti-symmetric mode through thickness and symmetric mode in length. The constitutive equation with flexoelectricity, governing equations and boundary conditions are derived from the Gibbs energy density function and variational principle. For the effect of flexoelectricity, we only consider the shear strain gradient in the thickness direction so as to simply the mathematical model. Thus, two flexoelectric coefficients are used in the present model. The electric potential functions are also obtained for different electric boundary conditions. The present results clearly show that the flexoelectric effect has significant effect on vibration frequencies of thickness-shear modes of thin piezoelectric crystal plate. It is also found that the flexoelectric coefficients and length to thickness ratio have influence on the thickness-shear modes. The results tell that flexoelectricity cannot be neglected for design of small size piezoelectric resonators.

Transient Chaos in Coupled Oscillators with Shape Deformable Potential

2003 ◽  
Vol 13 (06) ◽  
pp. 1459-1474 ◽  
Author(s):  
Serge Bruno Yamgoué ◽  
Timoléon Crépin Kofané
Keyword(s):  
Saddle Points ◽  
Three Dimensional ◽  
Perturbation Expansion ◽  
Transient Behavior ◽  
Unperturbed System ◽  
Dimensional Manifold ◽  
Regular Perturbation ◽  
Stable And Unstable Manifolds ◽  
Perturbed System ◽  
Melnikov Theory

The dynamics of a perturbed system consisting of a particle embedded in a strongly nonlinear potential V(ϕ; r, η) whose shape can be varied continuously as a function of r in the range -1 < r < 1, and coupled to an harmonic oscillator is analyzed. The perturbations are made of the coupling and damping forces. When they are removed, the subsystem formed by the anharmonic oscillator contains saddle points connected by homoclinic loops. Thus, the whole unperturbed system has saddle-centre points whose stable and unstable manifolds coincide in three-dimensional manifold. We concentrate our analysis of the perturbed system in the regions near these saddle-centre points. First, the Melnikov theory is used to investigate the presence of horseshoes chaos in the dynamics. Due to some discrepancies of the Melnikov theory, we next rederive the boundary between the regions of regular and irregular motions in the parameters space by using regular perturbation expansion. It is found here that for small values of the natural pulsation ω of the linear oscillator, complicated behavior intensifies in the system as the absolute value of the shape parameter r approaches zero. On the contrary, chaotic motion intensifies according as r decreases from values close to 1 to values close to -1, for large ω. The numerical analysis, which includes computation of maximal Lyapunov exponent, bifurcation diagrams and Poincaré sections, shows that the system exhibits transient stochastic behavior, but ultimately settles down on a simple set which is either a fixed point or a limit cycle. The dependence of this transient behavior on the system parameters agrees qualitatively well with the analytical predictions.

A Device for Magnification Measurement

1972 ◽  
Vol 30 ◽  
pp. 622-623
Author(s):  
J.A. Eades ◽  
A. van Dun
Keyword(s):  
Electron Microscope ◽  
Small Displacement ◽  
Private Communication ◽  
Piezoelectric Crystal ◽  
Double Exposure ◽  
Standard Methods ◽  
Elegant Method ◽  
The Right ◽  
Piezo Electric

The measurement of magnification in the electron microscope is always troublesome especially when a goniometer stage is in use, since there can be wide variations from calibrated values. One elegant method (L.M.Brown, private communication) of avoiding the difficulties of standard methods would be to fit a device which displaces the specimen a small but known distance and recording the displacement by a double exposure. Such a device would obviate the need for changing the specimen and guarantee that the magnification was measured under precisely the conditions used.Such a small displacement could be produced by any suitable transducer mounted in one of the specimen translation mechanisms. In the present case a piezoelectric crystal was used. Modern synthetic piezo electric ceramics readily give reproducible displacements in the right range for quite modest voltages (for example: Joyce and Wilson, 1969).

Imaging polymers, proteins and dna in aqueous solutions with the atomic force microscope

1989 ◽  
Vol 47 ◽  
pp. 32-33
Author(s):  
S.A.C. Gould ◽  
B. Drake ◽  
C.B. Prater ◽  
A.L. Weisenhorn ◽  
S.M. Lindsay ◽  
...  
Keyword(s):  
Amino Acids ◽  
Dna Sequencing ◽  
Aqueous Solutions ◽  
Atomic Force Microscope ◽  
Piezoelectric Crystal ◽  
Atomic Force ◽  
Structure Of Molecules ◽  
Optical Lever

The atomic force microscope (AFM) is an instrument that can be used to image many samples of interest in biology and medicine. Images of polymerized amino acids, polyalanine and polyphenylalanine demonstrate the potential of the AFM for revealing the structure of molecules. Images of the protein fibrinogen which agree with TEM images demonstrate that the AFM can provide topographical data on larger molecules. Finally, images of DNA suggest the AFM may soon provide an easier and faster technique for DNA sequencing.The AFM consists of a microfabricated SiO2 triangular shaped cantilever with a diamond tip affixed at the elbow to act as a probe. The sample is mounted on a electronically driven piezoelectric crystal. It is then placed in contact with the tip and scanned. The topography of the surface causes minute deflections in the 100 μm long cantilever which are detected using an optical lever.

High-Frequency Oscillation and Chronic Lung Disease in Very Low Birth Weight Infants

PEDIATRICS ◽  
2001 ◽  
Vol 108 (1) ◽  
pp. 212-214
Author(s):  
J. P. Shenai; ◽  
P. Rimensberger; ◽  
U. Thome ◽  
F. Pohlandt; ◽  
P. Rimensberger
Keyword(s):  
Birth Weight ◽  
Low Birth Weight ◽  
Lung Disease ◽  
Chronic Lung Disease ◽  
High Frequency ◽  
Very Low Birth Weight ◽  
High Frequency Oscillation ◽  
Frequency Oscillation ◽  
Low Birth Weight Infants

An Analytical Transient Tire Model

2001 ◽  
Vol 29 (2) ◽  
pp. 108-132 ◽  
Author(s):  
A. Ghazi Zadeh ◽  
A. Fahim
Keyword(s):  
Transient Behavior ◽  
Stability Control ◽  
Vehicle Stability ◽  
Tire Model ◽  
Steering Angle ◽  
First Order ◽  
Vehicle Stability Control ◽  
Proposed Model ◽  
Depth Study ◽  
And Performance

Abstract The dynamics of a vehicle's tires is a major contributor to the vehicle stability, control, and performance. A better understanding of the handling performance and lateral stability of the vehicle can be achieved by an in-depth study of the transient behavior of the tire. In this article, the transient response of the tire to a steering angle input is examined and an analytical second order tire model is proposed. This model provides a means for a better understanding of the transient behavior of the tire. The proposed model is also applied to a vehicle model and its performance is compared with a first order tire model.

A Modular Race Tire Model Concerning Thermal and Transient Behavior using a Simple Contact Patch Description

2013 ◽  
Vol 41 (4) ◽  
pp. 232-246
Author(s):  
Timo Völkl ◽  
Robert Lukesch ◽  
Martin Mühlmeier ◽  
Michael Graf ◽  
Hermann Winner
Keyword(s):  
Load Distribution ◽  
Thermal Condition ◽  
Transient Behavior ◽  
Wheel Load ◽  
Contact Patch ◽  
Fundamental Parameters ◽  
Dry Conditions ◽  
Simple Contact ◽  
Tire Forces ◽  
Descriptive Set

ABSTRACT The potential of a race tire strongly depends on its thermal condition, the load distribution in its contact patch, and the variation of wheel load. The approach described in this paper uses a modular structure consisting of elementary blocks for thermodynamics, transient excitation, and load distribution in the contact patch. The model provides conclusive tire characteristics by adopting the fundamental parameters of a simple mathematical force description. This then allows an isolated parameterization and examination of each block in order to subsequently analyze particular influences on the full model. For the characterization of the load distribution in the contact patch depending on inflation pressure, camber, and the present force state, a mathematical description of measured pressure distribution is used. This affects the tire's grip as well as the heat input to its surface and its casing. In order to determine the thermal condition, one-dimensional partial differential equations at discrete rings over the tire width solve the balance of energy. The resulting surface and rubber temperatures are used to determine the friction coefficient and stiffness of the rubber. The tire's transient behavior is modeled by a state selective filtering, which distinguishes between the dynamics of wheel load and slip. Simulation results for the range of occurring states at dry conditions show a sufficient correlation between the tire model's output and measured tire forces while requiring only a simplified and descriptive set of parameters.

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