Perturbation Analysis of a Nonlinear Resonator

2012 ◽  
Vol 8 (1) ◽  
Author(s):  
Shahin S. Nudehi ◽  
Umar Farooq
Keyword(s):  
Perturbation Analysis ◽  
Nonlinear Differential Equations ◽  
Primary Resonance ◽  
Nonlinear Behavior ◽  
Original System ◽  
Approximate Solutions ◽  
Singular Perturbation Theory ◽  
First Order ◽  
Nonlinear Resonator ◽  
Resonator System

A perturbation analysis of a Helmholtz-type resonator with one of the resonator ends replaced by a membrane is studied in this work. A membrane is known to exhibit nonlinear behavior under certain conditions; thus, when attached to a resonator system, it modifies the dynamic characteristics of the original system. This modified resonator system is modeled by coupled nonlinear differential equations and investigated by using the singular perturbation theory. The resonant frequency of the nonlinear resonator in the primary resonance case is analytically obtained using first-order approximate solutions. A good agreement is seen when the frequency response of the first-order approximate system is compared with the numerically simulated results.

Perturbation Analysis of a Nonlinear Resonator

2009 ◽  
Author(s):  
Shahin S. Nudehi ◽  
Umar Farooq ◽  
Eliot Motato
Keyword(s):  
Perturbation Analysis ◽  
Nonlinear Differential Equations ◽  
Primary Resonance ◽  
Helmholtz Resonator ◽  
Original System ◽  
Approximate Solutions ◽  
Singular Perturbation Theory ◽  
First Order ◽  
Nonlinear Resonator ◽  
Resonator System

A perturbation analysis of a Helmholte type resonator with one of the resonator ends replaced by a membrane is studied in this work. A membrane is known to exhibit nonlinear behavior under certain conditions, and thus when attached to a resonator system modifies the dynamic characteristics of the original system. This modified Helmholtz resonator system modeled by coupled nonlinear differential equations is investigated by using the singular perturbation theory. The resonant frequency of nonlinear resonator in the primary resonance case is analytically obtained by using the first order approximate solutions. A good agreement is seen when the frequency response of first order approximate system is compared with the numerically simulated results.

scholarly journals A Study on Vibration Characteristics and Stability of the Ambulance Nonlinear Damping System

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Meng Yang ◽  
Xinxi Xu ◽  
Chen Su
Keyword(s):  
Approximate Solution ◽  
Primary Resonance ◽  
Nonlinear Behavior ◽  
Singularity Theory ◽  
Nonlinear Damping ◽  
Nonlinear Stiffness ◽  
First Order ◽  
Damping Effect ◽  
Vibration Model ◽  
The Impact

Considering the impact of the nonlinear stiffness, a 2-DOF vibration model with cubic terms was established according to the structural feature and nonlinear behavior. Ignoring the impact of nonlinear terms, the system was linearly analyzed. In the case of primary resonance and 1 : 1 internal resonance, a multiscale method was used to obtain a first-order approximate solution. Taking the parameters of one tracked ambulance for instance, the approximate solution was corroborated and the influence of the parameters on damping effect was investigated. Finally, motion stability of the damping system was analyzed with singularity theory. The theoretical bases for improving efficiency of the damping system were provided.

Designing Experiments for Model Identification of the Nitrification Process

1991 ◽  
Vol 24 (6) ◽  
pp. 9-16 ◽  
Author(s):  
P. J. Ossenbruggen ◽  
H. Spanjers ◽  
H. Aspegren ◽  
A. Klapwijk
Keyword(s):  
Mechanistic Model ◽  
Nonlinear Differential Equations ◽  
Model Identification ◽  
Reaction Rates ◽  
Variable Coefficients ◽  
Model Specification ◽  
First Order ◽  
Interactive Behavior ◽  
Batch Tests ◽  
Nitrification Process

A series of batch tests were performed to study the competition for oxygen by Nitrosomonas and Nitrobacter in the nitrification of ammonia in activated sludge. Oxygen uptake rate (OUR) and dynamic (compartment) models describing the process are proposed and tested. The OUR model is described by a Monod relationship and the biogradation process by a set of first order nonlinear differential equations with variable coefficients. The results show a mechanistic model and ten reaction rates are sufficient to capture the interactive behavior of the nitrification process. Methods for model specification, calibrating, and testing the model and the design of additional experiments are described.

Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

First Order Analysis for Automotive Body Structure Design-Part 2: Joint Analysis Considering Nonlinear Behavior

2004 ◽  
Author(s):  
Yasuaki Tsurumi ◽  
Hidekazu Nishigaki ◽  
Toshiaki Nakagawa ◽  
Tatsuyuki Amago ◽  
Katsuya Furusu ◽  
...  
Keyword(s):  
Nonlinear Behavior ◽  
Structure Design ◽  
Joint Analysis ◽  
Body Structure ◽  
Order Analysis ◽  
First Order ◽  
Automotive Body

Radiant Heat Transfer From Isothermal Dispersions With Isotropic Scattering

1967 ◽  
Vol 89 (4) ◽  
pp. 300-308 ◽  
Author(s):  
R. H. Edwards ◽  
R. P. Bobco
Keyword(s):  
Heat Transfer ◽  
Approximate Solutions ◽  
Radiant Heat ◽  
Second Order ◽  
Isotropic Scattering ◽  
Radiant Heat Transfer ◽  
Approximate Methods ◽  
First Order ◽  
Order Solution ◽  
Radiant Transfer

Two approximate methods are presented for making radiant heat-transfer computations from gray, isothermal dispersions which absorb, emit, and scatter isotropically. The integrodifferential equation of radiant transfer is solved using moment techniques to obtain a first-order solution. A second-order solution is found by iteration. The approximate solutions are compared to exact solutions found in the literature of astrophysics for the case of a plane-parallel geometry. The exact and approximate solutions are both expressed in terms of directional and hemispherical emissivities at a boundary. The comparison for a slab, which is neither optically thin nor thick (τ = 1), indicates that the second-order solution is accurate to within 10 percent for both directional and hemispherical properties. These results suggest that relatively simple techniques may be used to make design computations for more complex geometries and boundary conditions.

CAUSAL BULK VISCOUS COSMOLOGIES IN CONFORMALLY FLAT SPACETIME

2000 ◽  
Vol 09 (04) ◽  
pp. 475-493 ◽  
Author(s):  
M. K. MAK ◽  
T. HARKO
Keyword(s):  
Large Time ◽  
Viscosity Coefficient ◽  
Approximate Solutions ◽  
Conformally Flat ◽  
First Order ◽  
Flat Spacetime ◽  
Bulk Viscous ◽  
Type Differential Equation ◽  
Bulk Viscosity Coefficient ◽  
Large Time Limit

The evolution of a causal bulk viscous cosmological fluid filled open conformally flat spacetime is considered. By means of appropriate transformations the equation describing the dynamics and evolution of the very early Universe can be reduced to a first order Abel type differential equation. In the case of a bulk viscosity coefficient proportional to the square root of the density, ξ~ρ1/2, an exact and two particular approximate solutions are obtained. The resulting cosmologies start from a singular state and generally have a noninflationary behavior, the deceleration parameter tending, in the large time limit, to zero. The thermodynamic consistency of the results is also checked.

ESTIMATES FOR APPROXIMATE SOLUTIONS TO A FUNCTIONAL DIFFERENTIAL EQUATION MODEL OF CELL DIVISION

2021 ◽  
pp. 1-20
Author(s):  
STEPHEN TAYLOR ◽  
XUESHAN YANG
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cell Division ◽  
Nonnegative Function ◽  
Approximate Solutions ◽  
Equation Model ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
First Order ◽  
Functional Differential

Abstract The functional partial differential equation (FPDE) for cell division, $$ \begin{align*} &\frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t))\\ &\quad = -(b(x,t)+\mu(x,t))n(x,t)+b(\alpha x,t)\alpha n(\alpha x,t)+b(\beta x,t)\beta n(\beta x,t), \end{align*} $$ is not amenable to analytical solution techniques, despite being closely related to the first-order partial differential equation (PDE) $$ \begin{align*} \frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)+F(x,t), \end{align*} $$ which, with known $F(x,t)$ , can be solved by the method of characteristics. The difficulty is due to the advanced functional terms $n(\alpha x,t)$ and $n(\beta x,t)$ , where $\beta \ge 2 \ge \alpha \ge 1$ , which arise because cells of size x are created when cells of size $\alpha x$ and $\beta x$ divide. The nonnegative function, $n(x,t)$ , denotes the density of cells at time t with respect to cell size x. The functions $g(x,t)$ , $b(x,t)$ and $\mu (x,t)$ are, respectively, the growth rate, splitting rate and death rate of cells of size x. The total number of cells, $\int _{0}^{\infty }n(x,t)\,dx$ , coincides with the $L^1$ norm of n. The goal of this paper is to find estimates in $L^1$ (and, with some restrictions, $L^p$ for $p>1$ ) for a sequence of approximate solutions to the FPDE that are generated by solving the first-order PDE. Our goal is to provide a framework for the analysis and computation of such FPDEs, and we give examples of such computations at the end of the paper.

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