The iterative homotopy harmonic balance method for conservative Helmholtz-Duffing oscillators

An approach that the iterative homotopy harmonic balance method which incorporates salient features of both the parameter-expansion and the harmonic balance is presented to solve conservative Helmholtz-Duffing oscillators. Since the behaviors of the solutions in the positive and negative directions are quite different, the asymmetric equation is separated into two auxiliary equations. The auxiliary equations are solved by proposed method. The results show it works very well for the whole range of initial amplitudes in a variety of cases, and the excellent agreement of the approximate periods and periodic solutions with exact ones have been demonstrated and discussed. And, the proposed method is very simple in its principle and has great potential to be applied to other nonlinear oscillators.     

Global residue harmonic balance method to periodic solutions of a class of strongly nonlinear oscillators

A new approach, namely the global residue harmonic balance method, was advanced to determine the accurate analytical approximate periodic solution of a class of strongly nonlinear oscillators. A class of nonlinear jerk equation containing velocity-cubed and velocity times displacements-squared was taken as a typical example. Unlike other harmonic balance methods, all the former residual errors are introduced in the present approximation to improve the accuracy. Comparison of the result obtained using this approach with the exact one and simplicity and efficiency of the proposed procedure. The method can be easily extended to other strongly nonlinear oscillators.     

First order perturbation theory for harmonic oscillators

Zusammenfassung Eine Störungsrechnung erster Ordnung wird für ein autonomes System von gestörten harmonischen Oszillatoren hergeleitet mittels einer Verallgemeinerung der Methode von Krylow-Bogoljubow.     

Relationship between the homotopy analysis method and harmonic balance method

This paper presents a study of the relationship between the homotopy analysis method (HAM) and harmonic balance (HB) method. The HAM is employed to obtain periodic solutions of conservative oscillators and limit cycles of self-excited systems, respectively. Different from the usual procedures in the existing literature, the HAM is modified by retaining a given number of harmonics in higher-order approximations. It is proved that as long as the solution given by the modified HAM is convergent, it converges to one HB solution. The Duffing equation, the van der Pol equation and the flutter equation of a two-dimensional airfoil are taken as illustrations to validate the attained results.     

Nonlinear transversely vibrating beams by the improved energy balance method and the global residue harmonic balance method

Mathematical modeling of many engineering systems such as beam structures often leads to nonlinear ordinary or partial differential equations. Nonlinear vibration analysis of the beam structures is very important in mechanical and industrial applications. This paper presents the high order frequency-amplitude relationship for nonlinear transversely vibrating beams with odd and even nonlinearities using the improved energy balance method and the global residue harmonic balance method. The accuracy of the energy balance method is improved based on combining features of collocation method and Galerkin–Petrov method, and an improved harmonic balance method is proposed which is called the global residue harmonic balance method. Unlike other harmonic balance methods, all the former global residual errors are introduced in the present approximation to improve the accuracy. Finally, preciseness of the present analytic procedures is evaluated in contrast with numerical calculations methods, giving excellent results.     

Incremental harmonic balance method for nonlinear flutter of an airfoil with uncertain-but-bounded parameters

The limit cycle oscillation of a two-dimensional airfoil with parameter variability in an incompressible flow is investigated using the incremental harmonic balance (IHB) method. The variable parameters, such as the wind speed, the cubic plunge and pitch stiffness coefficients, are modeled as either bounded uncertain or stochastic parameters. In the solution process of the IHB method, the bounded parameters are considered as an active increment. Taking all values over the considered bounded regions of the active parameters provides us with a series of IHB solutions of limit cycle oscillations of the airfoil. With the aid of the attained solutions, the bounds and some statistical properties of the limit cycle oscillations are determined and compared with Monte Carlo simulation (MCS) results. Numerical examples show that the proposed approach is valid and effective for analyzing strongly nonlinear vibration problems with bounded uncertainties.     

Simulation of harmonic oscillators on the lattice

This work deals with the simulation of a two-dimensional ideal lattice having simple tetragonal geometry. The harmonic character of the oscillators give rise to a system of second-order linear differential equations, which can be recast into matrix form. The explicit solutions which govern the dynamics of this system can be expressed in terms of matrix trigonometric functions. For the derivation we employ the Lagrangian formalism to determine the correct solutions, which extremize the underlying action of the system. In the numerical evaluation we develop diverse state-of-the-art algorithms which efficiently tackle equations with matrix sine and cosine functions. For this purpose, we introduce two special series related to trigonometric functions. They provide approximate solutions of the system through a suitable combination. For the final computation an algorithm based on Taylor expansion with forward and backward error analysis for computing those series had to be devised. We also implement several MATLAB programs which simulate and visualize the two-dimensional lattice and check its energy conservation.     

q-analogs of harmonic oscillators and related rings

A ringH _q which is aq-analog of the universal enveloping algebra of the Heisenberg Lie algebraU(h) is constructed, and its ring theoretic properties are studied. It is shown thatH _q has a factor ringA _q which is a simple domain with properties that are compared to the Weyl algebra. A secondq-analogH _q ofU(h) is constructed, andH _q is shown to be a primitive ring.     

Bounded perturbations of forced harmonic oscillators at resonance

Summary Let e be continuous and 2π-periodic, h continuous and bounded, and n>0 an integer. Sufficient conditions for the existence of 2π-periodic solutions of x″+n²x+h(x)= =e(t) are given. The proofs are based on a modification of Cesari's method and the Schauder fixed point theorem. Author is partially supported by N. S. F. under Grant 7447. Entrata in Redazione il 26 agosto 1968.     

On the spectrum of certain noncommutative harmonic oscillators

Abstract Some spectral properties of certain 2×2 globally elliptic systems of ordinary differential operators, a class of vector-valued deformations of the classical harmonic oscillator here called noncommutative harmonic oscillators, will be described, with special emphasis on the Poisson relation and clustering properties of the eigenvalues. Keywords: Clustering theorems, Periodic trajectories, Poisson relations, Noncommutative harmonic oscillators     

On a hidden symmetry of quantum harmonic oscillators

We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg uncertainty principle is presented.     

Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise

We consider a heat conduction model for solids. Nearest neighbour atoms interact as coupled oscillators exchanging velocities in such a way that the total energy is conserved. The system is considered under periodic boundary conditions. We will show that the system has a hydrodynamic limit given by the solution of the heat equation and we discuss some aspects of the model.     

Regularity and decay of solutions of nonlinear harmonic oscillators

We prove sharp analytic regularity and decay at infinity of solutions of variable coefficients nonlinear harmonic oscillators. Namely, we show holomorphic extension to a sector in the complex domain, with a corresponding Gaussian decay, according to the basic properties of the Hermite functions in ${\mathbb{R}}^d$. Our results apply, in particular, to nonlinear eigenvalue problems for the harmonic oscillator associated to a real-analytic scattering, or asymptotically conic, metric in ${\mathbb{R}}^d$, as well as to certain perturbations of the classical harmonic oscillator.     

The Gibbs' phenomenon and harmonic oscillators with periodic forcing

We present an interesting nonlinear equation originating from a geometrical problem. We show how it can be solved numerically, how algorithmic differentiation is useful, and how the problem can also be solved analytically using the computer algebra system Maple.     

Dynamic magnification factors of SDOF oscillators under harmonic loading

The magnification factor for the steady-state response of a SDOF system under harmonic loading is described in many structural dynamics textbooks; the well-known analytical solution is easily obtained from the solution to the damped equation of motion for harmonic loading. The complete and steady-state solutions can differ significantly. An analytical expression for the maximum response to the complete solution (steady state plus transient) remains elusive; however, a simple analytical expression is identified herein for the undamped case. Differences in the magnification factors obtained for the two solutions are discussed.     

Synchronization for two coupled oscillators with inhibitory connection

In this paper we present an oscillatory neural network composed of two coupled neural oscillators with inhibitory connections. Each of the oscillators describes the dynamics of average activities of excitatory and inhibitory populations of neurons. Regarding time delays τ as the bifurcation parameter, we not only obtain the existence of Hopf bifurcations but also investigate the bifurcation direction and stability of bifurcated periodic solutions by employing normal form theory and center manifold reduction. Finally, numerical simulations are provided to illustrate the theoretical results. Copyright © 2009 John Wiley & Sons, Ltd.     

Application of the harmonic balance method to ground moling machines operating in periodic regimes

A new system for ground moling has been patented by the University of Aberdeen and licensed world-wide. This new system is based on vibro-impact dynamics and offers significant advantages over existing systems in terms of penetrative capability and reduced soil disturbance. This paper describes current research into the mathematical modelling of the system. Periodic response is required to achieve the optimal penetrating conditions for the ground moling process, as this results in reduced soil penetration resistance. Therefore, there is a practical need for a robust and efficient methodology to calculate periodic responses for a wide range of operational parameters. Due to the structural complexity of a real vibro-impact moling system, the dynamic response of an idealised impact oscillator has been investigated in the first instance. This paper presents a detailed study of periodic responses of the impact oscillator under harmonic forcing using the alternating frequency-time harmonic balance method. Recommendations of how to effectively adapt the alternating frequency-time harmonic balance method for a stiff impacting system are given.     

Piecewise-linearized methods for single degree-of-freedom problems

A piecewise linearization method based on the Taylor series expansion of the nonlinearities and forcing with respect to time, displacement and velocity for the study of smooth single degree-of-freedom problems, is presented. The method provides piecewise analytical solutions which are smooth everywhere, is second-order accurate in time and yields explicit finite difference formulae for the displacement and velocity. The method is applied to nine single degree-of-freedom problems and its accuracy is assessed in terms of the displacement, velocity and energy as functions of the time step, and its results are compared with those of piecewise linearization methods that use Taylor series expansion of the forcing and nonlinearities with respect to time. It is shown that, for nonlinear problems with unknown free frequency and damping, the linearization method presented here is more accurate and robust than linearization techniques based on Taylor series expansions with respect to time. For linear problems with oscillatory forcing, linearization methods that employ fourth-order expansions in time are more accurate than the linearization method proposed here provided that the time step is sufficiently small.