scholarly journals Parameters Approach Applied on Nonlinear Oscillators

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Fatima Riaz ◽  
Nadeem Alam Khan
Keyword(s):  
Natural Frequency ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Solution Procedure ◽  
Power Exponent ◽  
Restoring Force ◽  
Arbitrary Power ◽  
Collocation Points ◽  
Approximate Frequency

We applied an approach to obtain the natural frequency of the generalized Duffing oscillatoru¨+u+α3u3+α5u5+α7u7+⋯+αnun=0and a nonlinear oscillator with a restoring force which is the function of a noninteger power exponent of deflectionu¨+αu|u|n−1=0. This approach is based on involved parameters, initial conditions, and collocation points. For any arbitrary power ofn, the approximate frequency analysis is carried out between the natural frequency and amplitude. The solution procedure is simple, and the results obtained are valid for the whole solution domain.

Accurate numerical solutions of conservative nonlinear oscillators

2014 ◽  
Vol 3 (4) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Khan Nasir Uddin ◽  
Khan Nadeem Alam
Keyword(s):  
Differential Equation ◽  
Plasma Physics ◽  
Nonlinear Oscillator ◽  
Numerical Solutions ◽  
Arbitrary Order ◽  
Nonlinear Oscillators ◽  
Higher Order ◽  
Algebraic Equations ◽  
Restoring Force ◽  
Arbitrary Power

AbstractThe objective of this paper is to present an investigation to analyze the vibration of a conservative nonlinear oscillator in the form u" + lambda u + u^(2n-1) + (1 + epsilon^2 u^(4m))^(1/2) = 0 for any arbitrary power of n and m. This method converts the differential equation to sets of algebraic equations and solve numerically. We have presented for three different cases: a higher order Duffing equation, an equation with irrational restoring force and a plasma physics equation. It is also found that the method is valid for any arbitrary order of n and m. Comparisons have been made with the results found in the literature the method gives accurate results.

Laplace–Pade Parametric Homotopy Perturbation Method for Uncertain Nonlinear Oscillators

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
S. Chakraverty ◽  
N. R. Mahato
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Nonlinear Oscillator ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Parametric Form ◽  
Homotopy Perturbation

Nonlinear oscillators have wide applicability in science and engineering problems. In this paper, nonlinear oscillator having initial conditions varying over fuzzy numbers has been initially taken into consideration. Here, the fuzziness in the uncertain nonlinear oscillators has been handled using parametric form. Using parametric form in terms of r-cut, the nonlinear uncertain differential equations are reduced to parametric differential equations. Then, based on classical homotopy perturbation method (HPM), a parametric homotopy perturbation method (PHPM) is proposed to compute solution enclosure of such uncertain nonlinear differential equations. A sufficient convergence condition of parametric solution obtained using PHPM is also proved. Further, a parametric Laplace–Pade approximation is incorporated in PHPM for retaining the periodic characteristic of nonlinear oscillators throughout the domain. The efficiency of Laplace–Pade PHPM has been verified for uncertain Duffing oscillator. Finally, Laplace–Pade PHPM is also applied to solve other uncertain nonlinear oscillator, viz., Rayleigh oscillator, with respect to fuzzy parameters.

scholarly journals Frequency formula for a class of fractal vibration system

2022 ◽  
Vol 3 (1) ◽  
pp. 55-61
Author(s):  
Yi Tian ◽  
Keyword(s):  
Variational Principles ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Vibration System ◽  
Transform Method ◽  
Inverse Transform ◽  
Simple Tool ◽  
Fractal Derivative ◽  
Fractal Space ◽  
Approximate Frequency

Four fractal nonlinear oscillators (The fractal Duffing oscillator, fractal attachment oscillator, fractal Toda oscillator, and a fractal nonlinear oscillator) are successfully established by He’s fractal derivative in a fractal space, and their variational principles are obtained by semi-inverse transform method. The approximate frequency of the four fractal oscillators are found by a simple frequency formula. The results show the frequency formula is a powerful and simple tool to a class of fractal oscillators.

Amplitudes Decay in Different Kinds of Nonlinear Oscillators

2015 ◽  
Vol 137 (3) ◽  
Author(s):  
Mohammad A. Al-Shudeifat
Keyword(s):  
Nonlinear Oscillator ◽  
Initial Conditions ◽  
A Priori ◽  
Nonlinear Oscillators ◽  
Transformation Method ◽  
Equation Of Motion ◽  
Time Varying ◽  
Initial Displacement ◽  
Restoring Force ◽  
First Order

A transformation is employed to obtain expressions for the decay of the displacement, the velocity, and the energy for various forms of nonlinear oscillators. The equation of motion of the nonlinear oscillator is transformed into a first-order decay term plus an energy term, where this transformed equation can be decoupled into a set of two analytically solvable equations. The decoupled equations can be solved for the decay formulas. Unlike other methods in the literature, this transformation method is directly applied to the equation of motion, and an approximate solution is not required to be known a priori. The method is first applied to a purely nonlinear oscillator with a non-negative, real-power restoring force to obtain the decay formulas. These decay formulas are found to behave similarly to those of a linear oscillator. In addition, these formulas are employed to obtain an accurate formula for the frequency decay. Based on this result, the exact frequency formula given in the literature for this oscillator is generalized by substituting the initial values of the envelopes for the actual initial conditions. By this modification, the formulas for the initial and time-varying frequencies become valid for any combination of the initial displacement and velocity. Furthermore, a generalized nonlinear oscillator for which the transformation is always valid is introduced. From this generalized oscillator, the proposed transformation is applied to analyze various types of oscillators.

Waves and turbulence on a beta-plane

1975 ◽  
Vol 69 (3) ◽  
pp. 417-443 ◽  
Author(s):  
Peter B. Rhines
Keyword(s):  
Weak Interaction ◽  
Initial Conditions ◽  
Computer Experiments ◽  
Zonal Flow ◽  
Similarity Solutions ◽  
Large Reynolds Number ◽  
Two Dimensional ◽  
Homogeneous Fluid ◽  
Restoring Force ◽  
Model Equations

Two-dimensional eddies in a homogeneous fluid at large Reynolds number, if closely packed, are known to evolve towards larger scales. In the presence of a restoring force, the geophysical beta-effect, this cascade produces a field of waves without loss of energy, and the turbulent migration of the dominant scale nearly ceases at a wavenumber kβ = (β/2U)½ independent of the initial conditions other than U, the r.m.s. particle speed, and β, the northward gradient of the Coriolis frequency.The conversion of turbulence into waves yields, in addition, more narrowly peaked wavenumber spectra and less fine-structure in the spatial maps, while smoothly distributing the energy about physical space.The theory is discussed, using known integral constraints and similarity solutions, model equations, weak-interaction wave theory (which provides the terminus for the cascade) and other linearized instability theory. Computer experiments with both finite-difference and spectral codes are reported. The central quantity is the cascade rate, defined as \[ T = 2\int_0^{\infty} kF(k)dk/U^3\langle k\rangle , \] where F is the nonlinear transfer spectrum and 〈k〉 the mean wavenumber of the energy spectrum. (In unforced inviscid flow T is simply U−1d〈k〉−1/dt, or the rate at which the dominant scale expands in time t.) T is shown to have a mean value of 3·0 × 10−2 for pure two-dimensional turbulence, but this decreases by a factor of five at the transition to wave motion. We infer from weak-interaction theory even smaller values for k [Lt ] kβ.After passing through a state of propagating waves, the homogeneous cascade tends towards a flow of alternating zonal jets which, we suggest, are almost perfectly steady. When the energy is intermittent in space, however, model equations show that the cascade is halted simply by the spreading of energy about space, and then the end state of a zonal flow is probably not achieved.The geophysical application is that the cascade of pure turbulence to large scales is defeated by wave propagation, helping to explain why the energy-containing eddies in the ocean and atmosphere, though significantly nonlinear, fail to reach the size of their respective domains, and are much smaller. For typical ocean flows, $k_{\beta}^{-1} = 70\,{\rm km} $, while for the atmosphere, $k_{\beta}^{-1} = 1000\,{\rm km}$. In addition the cascade generates, by itself, zonal flow (or more generally, flow along geostrophic contours).

scholarly journals Effect of Nonlinear Dissipation on the Basin Boundaries of a Driven Two-Well Modified Rayleigh–Duffing Oscillator

2015 ◽  
Vol 25 (02) ◽  
pp. 1550024 ◽  
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
J. B. Chabi Orou
Keyword(s):  
Initial Conditions ◽  
Duffing Oscillator ◽  
Excitation Amplitude ◽  
Nonlinear Damping ◽  
Homoclinic Bifurcation ◽  
Nonlinear Dissipation ◽  
Transition To Chaos ◽  
Melnikov Criterion ◽  
Quadratic Nonlinearities ◽  
Basin Boundaries

This paper considers the effect of nonlinear dissipation on the basin boundaries of a driven two-well modified Rayleigh–Duffing oscillator where pure cubic, unpure cubic, pure quadratic and unpure quadratic nonlinearities are considered. By analyzing the potential, an analytic expression is found for the homoclinic orbit. The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos. Unpure quadratic parameter and parametric excitation amplitude effects are found on the critical Melnikov amplitude μ cr . Finally, the phase space of initial conditions is carefully examined in order to analyze the effect of the nonlinear damping, and particularly how the basin boundaries become fractalized.

A Numerical Algorithm for Solving Stiff ODEs and DAEs in Multibody Mechanics

1999 ◽  
Author(s):  
Hajrudin Pasic
Keyword(s):  
Algebraic System ◽  
Numerical Solutions ◽  
Solution Procedure ◽  
Differential Algebraic Equations ◽  
Good Precision ◽  
Algebraic Equations ◽  
Constant Matrix ◽  
Fully Implicit ◽  
Collocation Points ◽  
Collocation Technique

Abstract Presented is an algorithm suitable for numerical solutions of multibody mechanics problems. When s-stage fully implicit Runge-Kutta (RK) method is used to solve these problems described by a system of n ordinary differential equations (ODE), solution of the resulting algebraic system requires 2s3 n3 / 3 operations. In this paper we present an efficient algorithm, whose formulation differs from the traditional RK method. The procedure for uncoupling the algebraic system into a block-diagonal matrix with s blocks of size n is derived for any s. In terms of number of multiplications, the algorithm is about s2 / 2 times faster than the original, nondiagonalized system, as well as s2 times in terms of number of additions/multiplications. With s = 3 the method has the same precision and stability property as the well-known RADAU5 algorithm. However, our method is applicable with any s and not only to the explicit ODEs My′ = f(x, y), where M = constant matrix, but also to the general implicit ODEs of the form f (x, y, y′) = 0. In the solution procedure y is assumed to have a form of the algebraic polynomial whose coefficients are found by using the collocation technique. A proper choice of locations of collocation points guarantees good precision/stability properties. If constructed such as to be L-stable, the method may be used for solving differential-algebraic equations (DAEs). The application is illustrated by a constrained planar manipulator problem.

Influence of Bistable Plunge Stiffness On Nonlinear Airfoil Flutter

2021 ◽  
Author(s):  
Renan F. Corrêa ◽  
Flávio D. Marques
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Vibration ◽  
Nonlinear Behavior ◽  
Limit Cycle Oscillation ◽  
Restoring Force ◽  
Aeroelastic System ◽  
Technical Literature ◽  
Practical Applications

Abstract Aeroelastic systems have nonlinearities that provide a wide variety of complex dynamic behaviors. Nonlinear effects can be avoided in practical applications, as in instability suppression or desired, for instance, in the energy harvesting design. In the technical literature, there are surveys on nonlinear aeroelastic systems and the different manners they manifest. More recently, the bistable spring effect has been studied as an acceptable nonlinear behavior applied to mechanical vibration problems. The application of the bistable spring effect to aeroelastic problems is still not explored thoroughly. This paper contributes to analyzing the nonlinear dynamics of a typical airfoil section mounted on bistable spring support at plunging motion. The equations of motion are based on the typical aeroelastic section model with three degrees-of-freedom. Moreover, a hardening nonlinearity in pitch is also considered. A preliminary analysis of the bistable spring geometry’s influence in its restoring force and the elastic potential energy is performed. The response of the system is investigated for a set of geometrical configurations. It is possible to identify post-flutter motion regions, the so-called intrawell, and interwell. Results reveal that the transition between intrawell to interwell regions occurs smoothly, depending on the initial conditions. The bistable effect on the aeroelastic system can be advantageous in energy extraction problems due to the jump in oscillation amplitudes. Furthermore, the hardening effect in pitching motion reduces the limit cycle oscillation amplitudes and also delays the occurrence of the snap-through.

scholarly journals A new autogenous mobile system driven by vibration without impacts, excited by an impulse periodic force

2018 ◽  
Vol 148 ◽  
pp. 04005 ◽  
Author(s):  
The-Hung Duong ◽  
Van-Du Nguyen ◽  
Ngoc-Tuan La
Keyword(s):  
Duffing Oscillator ◽  
Mobile Systems ◽  
Main Body ◽  
Rectilinear Motion ◽  
Mobile System ◽  
Restoring Force ◽  
Periodic Force ◽  
Nonlinear Restoring Force ◽  
Novel Structure ◽  
Excitation Force

This report describes a new proposed design for autogenous mobile systems which can move without any external mechanisms such as legs or wheels. A Duffing oscillator with a cubic spring, which is excited by an impulse periodic force, is utilized to drive the whole system. The rectilinear motion of the system is performed employing the periodically oscillation of the internal mass interacting without collisions with the main body. Utilizing the nonlinear restoring force of the cubic spring, the system can move in desired directions. When the ratio between the excitation force and the friction force is smaller than 2.5, backward or forward motion can be easily achieved by applying an excitation force in the same desired direction. Different from other vibro-impact drifting devices, no impact needed to drive the new proposed system. This novel structure allows to miniaturize the device as well as to simplify the control algorithm thus can significantly expand applicability of the proposed system.

A Study of Noise Impact on the Stability of Electrostatic MEMS

2020 ◽  
Vol 15 (11) ◽  
Author(s):  
Yan Qiao ◽  
Wei Xu ◽  
Hongxia Zhang ◽  
Qin Guo ◽  
Eihab Abdel-Rahman
Keyword(s):  
Noise Intensity ◽  
Initial Conditions ◽  
Basins Of Attraction ◽  
Low Noise ◽  
Intensity Level ◽  
Lumped Mass ◽  
Mass Model ◽  
Restoring Force ◽  
Electrostatic Mems ◽  
The Stability

Abstract Noise-induced motions are a significant source of uncertainty in the response of micro-electromechanical systems (MEMS). This is particularly the case for electrostatic MEMS where electrical and mechanical sources contribute to noise and can result in sudden and drastic loss of stability. This paper investigates the effects of noise processes on the stability of electrostatic MEMS via a lumped-mass model that accounts for uncertainty in mass, mechanical restoring force, bias voltage, and AC voltage amplitude. We evaluated the stationary probability density function (PDF) of the resonator response and its basins of attraction in the presence noise and compared them to that those obtained under deterministic excitations only. We found that the presence of noise was most significant in the vicinity of resonance. Even low noise intensity levels caused stochastic jumps between co-existing orbits away from bifurcation points. Moderate noise intensity levels were found to destroy the basins of attraction of the larger orbits. Higher noise intensity levels were found to destroy the basins of attraction of smaller orbits, dominate the dynamic response, and occasionally lead to pull-in. The probabilities of pull-in of the resonator under different noise intensity level are calculated, which are sensitive to the initial conditions.

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