Summation Resonance of Parametrically Excited Moving Viscoelastic Belts

2003 ◽  
Author(s):  
Tao Chen ◽  
Zhichao Hou
Keyword(s):  
Order Approximation ◽  
Nonlinear Dynamic Analysis ◽  
Material Behavior ◽  
Nontrivial Solutions ◽  
Parametrically Excited ◽  
Kelvin Model ◽  
Closed Form Solutions ◽  
The Stability ◽  
First Order Approximation ◽  
Moving Speed

The nonlinear dynamic analysis is performed on parametrically excited, viscoelastic moving-belts at summation resonance. The belt material behavior is described by a Voigt-Kelvin model. Closed-form solutions are derived at the first order approximation. Focus is put on the stability of the nontrivial solutions. The explicit expressions on the stability conditions are obtained, and then simplified through numerical simulations. The influences of moving speed and tension fluctuation on the stability of the nontrivial solutions are also demonstrated.

Parametric Vibration of Viscoelastic Moving Belts Using Standard Linear Solid Model

2001 ◽  
Author(s):  
Zhichao Hou ◽  
Jean W. Zu
Keyword(s):  
Order Approximation ◽  
Dynamic Responses ◽  
Solid Model ◽  
Nontrivial Solutions ◽  
Closed Form Solutions ◽  
Standard Linear Solid Model ◽  
Standard Linear Solid ◽  
Standard Linear ◽  
The Stability ◽  
First Order Approximation

Abstract By using a standard linear solid model to describe the viscoelasticity of the belt material, a vibration analysis of a parametrically excited moving belt is performed. Closed-form solutions at principal resonance and summation resonance are derived at the first order approximation. The existence conditions and stability are discussed for the nontrivial solutions, yielding explicit expressions of the existence and the stability conditions in terms of the detuning parameter. Numerical examples clearly show the effects of tension fluctuations and translating speeds on the amplitudes of dynamic responses, the corresponding existence domains and the stability of the solutions. It is also demonstrated that the stability domains of the nontrivial solutions are different from those corresponding to elastic models.

scholarly journals FIRST-ORDER ASYMPTOTICS OF PATH-DEPENDENT DERIVATIVES IN MULTISCALE STOCHASTIC VOLATILITY ENVIRONMENT

2018 ◽  
Vol 21 (03) ◽  
pp. 1850024 ◽  
Author(s):  
YURI F. SAPORITO
Keyword(s):  
Stochastic Volatility ◽  
Path Dependence ◽  
Order Approximation ◽  
Asian Options ◽  
Order Condition ◽  
Closed Form Solutions ◽  
First Order ◽  
Itô Calculus ◽  
Path Dependent ◽  
First Order Approximation

In this paper, we extend the first-order asymptotics analysis of Fouque et al. to general path-dependent financial derivatives using Dupire’s functional Itô calculus. The main conclusion is that the market group parameters calibrated to vanilla options can be used to price to the same order exotic, path-dependent derivatives as well. Under general conditions, the first-order condition is represented by a conditional expectation that could be numerically evaluated. Moreover, if the path-dependence is not too severe, we are able to find path-dependent closed-form solutions equivalent to the first-order approximation of path-independent options derived in Fouque et al. Additionally, we exemplify the results with Asian options and options on quadratic variation.

Stability and Limit Cycles of Parametrically Excited, Axially Moving Strings

1996 ◽  
Vol 118 (3) ◽  
pp. 346-351 ◽  
Author(s):  
E. M. Mockensturm ◽  
N. C. Perkins ◽  
A. Galip Ulsoy
Keyword(s):  
Limit Cycles ◽  
Parametric Instability ◽  
Parametrically Excited ◽  
Weakly Nonlinear ◽  
Closed Form Solutions ◽  
Belt Drives ◽  
Existence And Stability ◽  
Instability Regions ◽  
Axially Moving ◽  
The Stability

Tension fluctuations are the dominant source of excitation in automotive belts. In particular designs, these fluctuations may parametrically excite large amplitude transverse belt vibrations and adversely impact belt life. This paper evaluates an efficient discrete model of a parametrically excited translating belt. The efficiency derives from the use of translating string eigenfunctions as a basis for a Galerkin discretization of the equations of transverse belt response. Accurate and low-order models lead to simple closed-form solutions for the existence and stability of limit cycles near parametric instability regions. In particular, simple expressions are found for the stability boundaries of the general nth-mode principal parametric instability regions and the first summation and difference parametric instability regions. Subsequent evaluation of the weakly nonlinear equation of motion leads to an analytical expression for the amplitudes (and stability) of nontrivial limit cycles that exist around the nth-mode principal parametric instability regions. Example results highlight important conclusions concerning the response of automotive belt drives.

On Parametric Stability of a Nonconstant Axially Moving String Near Resonances

2016 ◽  
Vol 139 (1) ◽  
Author(s):  
Rajab A. Malookani ◽  
Wim T. van Horssen
Keyword(s):  
Order Approximation ◽  
Equations Of Motion ◽  
First Order ◽  
Axially Moving String ◽  
Transverse Vibrations ◽  
Fluctuation Frequency ◽  
Axially Moving ◽  
The Stability ◽  
First Order Approximation ◽  
Speed Fluctuations

The stability of an axially moving string system subjected to parametric excitation resulting from speed fluctuations has been examined in this paper. The time-dependent velocity is assumed to be a harmonically varying function around a (low) constant mean speed. The method of characteristic coordinates in combination with the two timescales perturbation method is used to compute the first-order approximation of the solutions of the equations of motion that governs the transverse vibrations of an axially moving string. It turns out that the system can give rise to resonances when the velocity fluctuation frequency is equal (or close) to an odd multiple of the natural frequency of the system. The stability conditions are investigated analytically in terms of the displacement-response and the energy of the system near the resonances. The effects of the detuning parameter on the amplitudes of vibrations and on the energy of the system are also presented through numerical simulations.

scholarly journals Vibration Control in MEMS Resonator Using Positive Position Feedback (PPF) Controller

2016 ◽  
Vol 12 (11) ◽  
pp. 6821-6834
Author(s):  
Y A Amer ◽  
A.T EL Sayed ◽  
A.M. Salem
Keyword(s):  
Order Approximation ◽  
Perturbation Technique ◽  
Multiple Scale ◽  
Steady State Solution ◽  
Positive Position Feedback ◽  
Mems Resonator ◽  
Position Feedback ◽  
Resonator System ◽  
The Stability ◽  
First Order Approximation

In this paper, the vibration of a micro-electromechanical resonator with positive position feedback controller is studied. The analytical results are obtained to the first order approximation by using the multiple scale perturbation technique. The stability of the steady-state solution is presented and studied applying frequency response equations near the simultaneous primary and internal resonance cases. The effects of the controller and some system parameters on the vibrating system are studied numerically. The main result of this paper indicates that it is possible to reduce the vibration for the resonator system.

scholarly journals Generalised Reliability On Hydro-Geo Objects

2015 ◽  
Vol 9 (1) ◽  
pp. 498-503
Author(s):  
Wang Ya-jun ◽  
Wang Jun
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Shear Failure ◽  
Order Approximation ◽  
Stochastic Finite Element Method ◽  
Stochastic Finite Element ◽  
Safety Performance Function ◽  
The Stability ◽  
First Order Approximation ◽  
Element Method

This study established the fuzzy logic modeling of the stochastic finite element method based on the first-order approximation theorem. Fuzzy mathematical models of safety repertories were incorporated into the stochastic finite element method to analyze the stability of embankments and foundations in order to describe the fuzzy failure procedure for the random safety performance function. The fuzzy models were developed with membership functions with half depressed gamma distribution, half depressed normal distribution, and half depressed echelon distribution. The result shows that the middle region of the dike is the principal zone of concentrated failure due to local fractures. There is also some local shear failure on the dike crust. This study provides a referential method for solving complex multi-uncertainty problems in engineering safety analysis.

Improved first-order approximation of eigenvalues and eigenvectors

AIAA Journal ◽  
1998 ◽  
Vol 36 ◽  
pp. 1721-1727
Author(s):  
Prasanth B. Nair ◽  
Andrew J. Keane ◽  
Robin S. Langley
Keyword(s):  
Order Approximation ◽  
Eigenvalues And Eigenvectors ◽  
First Order ◽  
First Order Approximation

Analytical solution for unsteady flow behind ionizing shock wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field

2021 ◽  
Vol 76 (3) ◽  
pp. 265-283
Author(s):  
G. Nath
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Analytical Solution ◽  
Axial Magnetic Field ◽  
Order Approximation ◽  
Ideal Gas ◽  
Field Region ◽  
First Order ◽  
First Order Approximation ◽  
Flow Variables

Abstract The approximate analytical solution for the propagation of gas ionizing cylindrical blast (shock) wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field is investigated. The axial and azimuthal components of fluid velocity are taken into consideration and these flow variables, magnetic field in the ambient medium are assumed to be varying according to the power laws with distance from the axis of symmetry. The shock is supposed to be strong one for the ratio C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ to be a negligible small quantity, where C 0 is the sound velocity in undisturbed fluid and V S is the shock velocity. In the undisturbed medium the density is assumed to be constant to obtain the similarity solution. The flow variables in power series of C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ are expanded to obtain the approximate analytical solutions. The first order and second order approximations to the solutions are discussed with the help of power series expansion. For the first order approximation the analytical solutions are derived. In the flow-field region behind the blast wave the distribution of the flow variables in the case of first order approximation is shown in graphs. It is observed that in the flow field region the quantity J 0 increases with an increase in the value of gas non-idealness parameter or Alfven-Mach number or rotational parameter. Hence, the non-idealness of the gas and the presence of rotation or magnetic field have decaying effect on shock wave.

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