An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the ‘cubication’ of the initial nonlinear differential equation. In this cubication method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude, A. The replacement of the original nonlinear equation by an approximate Duffing equation allows us to obtain explicit approximate formulas for the frequency and the solution as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function, respectively. These explicit formulas are valid for all values of the initial amplitude and we conclude this cubication method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is as low as 0.071%. Unlike other approximate methods applied to this oscillator, which are not capable to reproduce exactly the behaviour of the approximate frequency when A tends to zero, the cubication method used in this Letter predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closed-form expression for the approximate frequency is obtained in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean as well as Legendre's formula to approximately obtain this mean are used.     

Approximate analytical solutions to a conservative oscillator using global residue harmonic balance method

In the current research paper, a conservative system comprising of a mass grounded by linear and nonlinear springs in series connection is studied. The equation of motion for the aforementioned system has been derived as a nonlinear ordinary differential equation with inertia and static–type cubic nonlinearities. The global residue harmonic balance method is applied to obtain an approximate analytical frequency and periodic solution of the problem. Using the obtained analytical expressions, the influences of the hardening and softening nonlinear spring on the non–dimensional frequency are investigated. The results show that developing the system nonlinearity leads the displacement of the mass and the deflection of linear spring to approach each other. Moreover, comparison of the results obtained using the proposed procedure with those achieved by other methods such as numerical method, variational iteration method and harmonic balance approach demonstrates the accuracy and advantages of the current approach.     

Solution procedure of residue harmonic balance method and its applications

This paper presents a simple and rigorous solution procedure of residue harmonic balance for predicting the accurate approximation of certain autonomous ordinary differential systems. In this solution procedure, no small parameter is assumed. The harmonic residue of balance equation is separated in two parts at each step. The first part has the same number of Fourier terms as the present order of approximation and the remaining part is used in the subsequent improvement. The corrections are governed by linear ordinary differential equation so that they can be solved easily by means of harmonic balance method again. Three kinds of different differential equations involving general, fractional and delay ordinary differential systems are given as numerical examples respectively. Highly accurate limited cycle frequency and amplitude are captured. The results match well with the exact solutions or numerical solutions for a wide range of control parameters. Comparison with those available shows that the residue harmonic balance solution procedure is very effective for these autonomous differential systems. Moreover, the present method works not only in predicting the amplitude but also the frequency of bifurcated period solution for delay ordinary differential equation.     

Solution of a mixed parity nonlinear oscillator: Harmonic balance

The method of harmonic balance is used to calculate first-order approximations to the periodic solutions of a mixed parity nonlinear oscillator. First, the amplitude in the negative direction is expressed in terms of the amplitude in the positive direction. Then the two auxiliary equations, where the restoring force functions are odd, are solved by using a first harmonic term (without a constant). Therefore, we obtain the two approximate solutions to the mixed parity nonlinear oscillator. One is expressed in terms of the exact amplitude in the negative direction, the other in terms of the approximate amplitude. These solutions are more accurate than the second approximate solution obtained by the Lindstedt–Poincaré method for large amplitudes.     

阻尼谐振子到简谐振子的变换

在牛顿力学、拉格朗日力学和哈密顿力学3种形式中,利用变量变换将阻尼谐振子变换成简谐振子.     

Bifurcation and chaos of a harmonically excited oscillator with both stiffness and viscous damping piecewise linearities by incremental harmonic balance method

In this paper, an explicit formulation of the incremental harmonic balance (IHB) scheme for computation of periodic solutions of a harmonically excited oscillator which is asymmetric with both stiffness and viscous damping piecewise linearities is derived. Analysis of dynamical behavior as bifurcation and chaos of the non-linear vibration system considered is effectively carried out by the IHB procedure, showing that the system exhibits chaos via the route of period-doubling bifurcation, with coexistence of multiple periodic attractors observed and analyzed by the interpolated cell mapping method. In addition, numerical simulation by the IHB method is compared with that by the fourth order Runge-Kutta numerical integration routine, which shows that this method is in many respects distinctively advantageous over classical approaches, and especially excels in performing parametric studies.     

菲涅耳变换求解含时CK谐振子模型的演化

通过对波函数的菲涅耳变换来求解含时CK谐振子的演化问题,所用方法新颖,独特.     

Gauge covariant construction of the coherent states for a time-dependent harmonic oscillator by algebraic dynamical method

Time-dependent coherent states for a time-dependent harmonic oscillator are constructed in the framework of algebraic dynamics. These coherent states are gauge-covariant, and its time evolution is governed only by the solutions of a linear differential equation which describes the motion of the corresponding classical timedependent harmonic oscillator. Its non-classical and quantum statistical properties can thus be controlled by a proper choice of the frequency of the harmonic oscillator. Our coherent states reduce to Glauber coherent states in the case as the frequency is independent of time.     

Probability operator of a harmonic oscillator

A unique linear rule of constructing quantum operators defined by the probability operator for coordinates and momenta, is considered. is assumed to be a normalized, positive definite operator, establishing a dynamical correspondence between the classical and quantum Poisson brackets. It is shown that such an operator exists in the case of a harmonic oscillator. The principal implications of the suggested rule of constructing the operators of physical quantities are determined, in comparison with the corresponding results of conventional quantum mechanics.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 89–93, October, 1982.     

Feynman's formula for a harmonic oscillator

Feynman's original formula for the Green's function of a harmonic oscillator includes the Maslov correction. It also leads to the known formula for caustics for the time intervals that are equal to integer multiplets of the half period.     

Harmonic balance approach to the periodic solutions of the (an)harmonic relativistic oscillator

The first-order harmonic balance method via the first Fourier coefficient is used to construct two approximate frequency-amplitude relations for the relativistic oscillator for which the nonlinearity (anharmonicity) is a relativistic effect due to the time line dilation along the world line. Making a change of variable, a new nonlinear differential equation is obtained and two procedures are used to approximately solve this differential equation. In the first the differential equation is rewritten in a form that does not contain a square-root expression, while in the second the differential equation is solved directly. The approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first due to the fact that, in the second procedure, application of the harmonic balance method produces an infinite set of harmonics, while in the first procedure only two harmonics are produced. Both approximate frequencies are valid for the complete range of oscillation amplitudes, and excellent agreement of the approximate frequencies with the exact one are demonstrated and discussed. The discrepancy between the first-order approximate frequency obtained by means of the second procedure and the exact frequency never exceeds 1.6%. We also obtained the approximate frequency by applying the second-order harmonic balance method and in this case the relative error is as low 0.31% for all the range of values of amplitude of oscillation A.     

Perturbation of the spectrum of three-dimensional harmonic oscillator by a δ-potential

The perturbed spectrum of a three-dimensional harmonic oscillator by a δ-potential is determined at large separations between the potential centres. Two groups of levels are found with quite different behaviour.     

Brownian motion of a quantum harmonic oscillator

The problem of describing the Brownian motion of a quantum harmonic oscillator or free particle is treated in the formalism of quantum dynamical semigroups. Certain inequalities involving the friction and diffusion coefficients and Planck's constant are derived. The nature of the quantum Langevin equation is discussed.     

Coherent states of a damped harmonic oscillator

We construct here the coherent states (annihilation operator eigenstates) of a damped harmonic oscillator. These coherent states, which are normalizable, have the desired behaviour in the classical limit (ℏ→0).     

A q-deformation of the harmonic oscillator

The q-deformed harmonic oscillator is studied in the light of q-deformed phase space variables. This allows a formulation of the corresponding Hamiltonian in terms of the ordinary canonical variables x and p. The spectrum shows unexpected features such as degeneracy and an additional part that cannot be reached from the ground state by creation operators. The eigenfunctions show lattice structure, as expected.     

Non-Heisenberg states of the harmonic oscillator

The effects of the vacuum electromagnetic fluctuations and the radiation reaction fields on the time development of a simple microscopic system are identified using a new mathematical method. This is done by studying a charged mechanical oscillator (frequency ₀)within the realm of stochastic electrodynamics, where the vacuum plays the role of an energy reservoir. According to our approach, which may be regarded as a simple mathematical exercise, we show how the oscillator Liouville equation is transformed into a Schrödinger-like stochastic equation with a free parameter h with dimensions of action. The role of the physical Planck's constant h is introduced only through the zero-point vacuum electromagnetic fields. The perturbative and the exact solutions of the stochastic Schrödinger-like equation are presented for h>0. The exact solutions for which h<h are called sub-Heisenberg states. These nonperturbative solutions appear in the form of Gaussian, non-Heisenberg states for which the initial classical uncertainty relation takes the form (x ²)(p) ² =(h/2) ²,which includes the limit of zero indeterminacy (h 0). We show how the radiation reaction and the vacuum fields govern the evolution of these non-Heisenberg states in phase space, guaranteeing their decay to the stationary state with average energy h ₀ /2 and (x) ² (p) ² =h ² /4 at zero temperature. Environmental and thermal effects-are briefly discussed and the connection with similar works within the realm of quantum electrodynamics is also presented. We suggest some other applications of the classical non-Heisenberg states introduced in this paper and we also indicate experiments which might give concrete evidence of these states.     

A new method based on the harmonic balance method for nonlinear oscillators

The harmonic balance (HB) method as an analytical approach is widely used for nonlinear oscillators, in which the initial conditions are generally simplified by setting velocity or displacement to be zero. Based on HB, we establish a new theory to address nonlinear conservative systems with arbitrary initial conditions, and deduce a set of over-determined algebraic equations. Since these deduced algebraic equations are not solved directly, a minimization problem is constructed instead and an iterative algorithm is employed to seek the minimization point. Taking Duffing and Duffing-harmonic equations as numerical examples, we find that these attained solutions are not only with high degree of accuracy, but also uniformly valid in the whole solution domain.     

Some aspects of the harmonic balance method applied to the clarinet

The clarinet has been extensively studied by various theoretical and experimental techniques. In this paper, the harmonic balance method (HBM), a numerical method mainly working in the frequency domain, has been applied to solve a simple nonlinear clarinet model consisting of a linear exciter (for the reed) nonlinearly coupled to a linear resonator with visco-thermal losses (for the pipe). A recent and improved implementation of the HBM for self-sustained instruments has allowed us to study the model theoretically when including dispersion in the pipe or mass and damping terms in the reed model. The resulting periodic solutions for the internal pressure spectrum and the corresponding playing frequency are shown to align well with previous theoretical and experimental knowledge of the clarinet. Finally, we present and briefly discuss a few (probably unstable) oscillation regimes both with the HBM and with a real clarinet.