Interaction of fundamental parametric resonances with subharmonic resonances of order one-half

The interaction of fundamental parametric resonances with subharmonic resonances of order one-half in a single-degree-of-freedom system with quadratic and cubic nonlinearities is investigated. The method of multiple scales is used to derive two first-order ordinary differential equations that describe the modulation of the amplitude and the phase of the response with the non-linearity and both resonances. These equations are used to determine the steady state solutions and their stability. Conditions are derived for the quenching or enhancement of a parametric resonance by the addition of a subharmonic resonance of order one-half. The degree of quenching or enhancement depends on the relative amplitudes and phases of the excitations. The analytical results are verified by numerically integrating the original governing differential equation.     

采用曲线拟合消除探测器阵列非线性对吸收光谱的影响

在微型光谱分析系统中,光电探测器阵列像元的非线性响应对光谱信号有着很大的影响,尤其是吸收谱的测量中,将导致吸收谱的畸变和假谱线的出现。文章对这一影响进行了深入的分析,并提出了根据线性像元输出信号,利用曲线拟合的方法对线性较差的像元输出进行矫正,减小非线性对光谱信号的影响。采用该方法对实验中获取的吸收谱信号进行了矫正,得到了理想的效果。     

Breathing pulses and beams innon-linear media

Oscillating solutions are known to exist for the (1+1)-dimensional non-linear Schrödinger equations (NLSE) with Kerr non-linearity and for (3+1)-dimensional and (2+1)-dimensional NLSE with saturable non-linearity. In them, there is an interplay between the amplitude and the widths of the solutions. Using the variational method, we show that non-linear Schrödinger equations with Kerr non-linearity support breather-like solutions in the form of oscillating spatiotemporal pulses and beams but unlike the cases mentioned above, the oscillating behaviour takes place between the widths, with the amplitude kept almost constant. For the space–time NLSE these solutions exist only in a medium with anomalous dispersion.     

Measurement of non-linear distortions in the vibration of acoustic transducers and acoustically driven membranes

Different methods exist to quantify non-linearity, and recently a new method was proposed that uses multisines to determine output response level, noise level, and level of non-linear distortions simultaneously from one single short experiment. We used heterodyne vibrometry to measure the vibration of a membrane while it is stimulated acoustically, and to determine the non-linear distortions in its response with this new method of analysis. We demonstrate the method on an electrically driven earphone. Analysis of the sound output and optically measured vibration deliver comparable results, and clearly detect non-linear distortions. Next, we apply the method on an acoustically driven elastic membrane. At vibration amplitudes of the same magnitude as on the earphone, non-linear distortions are much smaller, as to be expected on an elastic membrane, but they can still be detected with the vibrometer. This technique can be used to investigate the non-linearity of acoustically driven systems such as the middle ear.     

非定常可压等熵流非线性方程显式解析解的推导

本文对作者以前凭试凑、灵感、运气与经验得出的一系列非定常可压流动显式解析解,寻找线索,总结出其可能的推导途径,并以非定常可压等熵一维流为例,具体给出了四种新的求解方法。这些方法会对今后寻找工程热物理领域的非线性主控方程的解析解有所帮助。本文同时还给出了两个新的解析解。     

Response of a bar constrained by a non-linear spring to a harmonic excitation

An analysis is presented of the longitudinal response of a bar constrained by a non-linear spring to a harmonic excitation. The method of multiple scales is used to determine equations describing the evolution of the amplitudes and phases with damping, non-linearity and the cases of primary, subharmonic, superharmonic, combination and ultrasubharmonic resonances. These equations are used to determine the steady state responses and their stability.     

Non-linear response of a gyrostabilized platform to transient input

In this paper the response of a gyrostabilized platform subjected to a transient torque has been analyzed by deliberately introducing non-linearity into the command of the servomotor. The resulting third-order non-linear differential equation has been solved by using a transformation technique involving the displacement variable. The condition under which platform oscillations may grow with time or die with time are important from the point of view of platform stabilization. The effect of deliberate addition of non-linearity with a view to achieving the ideal response—that is, to bring the platform back to its equilibrium position with as few oscillations as possible—has been investigated. The conditions under which instability may set in on account of the small transient input and small non-linearity has also been discussed. The analysis is illustrated by means of a numerical example. The results of analysis are compared with numerical solutions obtained on a digital computer.     

Periodic solution of the generalized Rayleigh equation

The periodic solutions of a strongly cubic nonlinear oscillator whose motion is described with the generalized Rayleigh equation are studied. Approximate analytic solving methods are introduced. A new method based on homotopy and averaging is developed to determine the limit cycle motion. The obtained analytical solutions are compared with those calculated by the elliptic harmonic balance method with generalized Fourier series and Jacobian elliptic functions. Three types of cubic nonlinearity are considered: the coefficients of the linear and cubic terms are positive, the coefficient of the linear term is positive and that of the cubic term is negative and the opposite case. Comparisons of the analytical solution and numerical solution, obtained by using the Runge-Kutta method, are illustrated with examples.     

The extended (G′/G)-expansion method and travelling wave solutions for the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity

In this paper, we construct the travelling wave solutions to the perturbed nonlinear Schrödinger’s equation (NLSE) with Kerr law non-linearity by the extended (G′/G)-expansion method. Based on this method, we obtain abundant exact travelling wave solutions of NLSE with Kerr law nonlinearity with arbitrary parameters. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions and rational functions.     

Analytical solutions for the spin-1 Bose-Einstein condensate in a harmonic trap

The homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions for the Gross-Pitaevskii equations, a set of nonlinear Schrödinger equation used in simulation of spin-1 Bose-Einstein condensates trapped in a harmonic potential. We investigate the one-dimensional case and get the approximate analytical solutions successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are in agreement well with each other when the atomic interaction is weakly. We also find a class of exact solutions for the stationary states of the spin-1 system with harmonic potential for a special case.     

Analytical Solutions for the Two-Dimensional Gross-Pitaevskii Equation with a Harmonic Trap

Both the homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions of the two-dimensional and time-independent Gross-Pitaevskii equation,a nonlinear Schrdinger equation used in describing the system of Bose-Einstein condensates trapped in a harmonic potential.The approximate analytical solutions are obtained successfully.Comparisons between the analytical solutions and the numerical solutions have been made.The results indicate that they are agreement very well with each other when the atomic interaction is not too strong.     

On solutions of the fifth-order dispersive equations with porous medium type non-linearity

In this work, we focus on obtaining the exact solutions of the fifth-order semi-linear and non-linear dispersive partial differential equations, which have the second-order diffusion-like (porous-type) non-linearity. The proposed equations were not studied in the literature in the sense of the exact solutions. We reveal solutions of the proposed equations using the classical Riccati equations method. The obtained exact solutions, which can play a key role to simulate non-linear waves in the medium with dispersion and diffusion, are illustrated and discussed in details.     

Unsteady analytical solutions of the spherical shallow water equations

A new class of unsteady analytical solutions of the spherical shallow water equations (SSWE) is presented. Analytical solutions of the SSWE are fundamental for the validation of barotropic atmospheric models. To date, only steady-state analytical solutions are known from the literature. The unsteady analytical solutions of the SSWE are derived by applying the transformation method to the transition from a fixed cartesian to a rotating coordinate system. Fundamental examples of the new unsteady analytical solutions are presented for specific wind profiles. With the presented unsteady analytical solutions one can provide a measure of the numerical convergence in the case of a temporally evolving system. An application to the atmospheric model PLASMA shows the benefit of unsteady analytical solutions for the quantification of convergence properties.     

Theory of nonlinear directional couplers

A method is proposed for obtaining exact analytical solutions for the system of nonlinear equations for the propagation of electromagnetic waves in directional couplers with an arbitrary nonlinearity in the propagation constant. The resulting solutions can be used to determine the operating characteristics of nonlinear directional couplers. Zh. Tekh. Fiz. 69, 69–71 (August 1999)     

Vibration analysis of a wheel composed of a ring and a wheel-plate modelled as a three-parameter elastic foundation

The free in-plane vibrations of circular rings with wheel-plates as generalised elastic foundations are studied using analytical methods and numerical simulations. The three-parameter Winkler elastic layer is proposed as a mathematical model of the foundation. The effects of rotary inertia and shear deformation are included in the analytical model of the system. The motion equations of systems are derived on the basis of the thin ring theory and Timoshenko?s theory. The separation of variables method is used to find general solutions to the free vibrations. Elaborated analytical models are used to determine the natural frequencies and the natural mode shapes of vibrations of an arbitrarily chosen set of simplified models of aviation gears and railway wheels. The eigenvalue problem is formulated and solved by using a finite element representation for each simplified model. The results for these models are discussed and compared. The proposed solutions are verified by experimental investigation. It is important to note that the solutions proposed here could be useful to engineers dealing with the dynamics of aviation gears, railway wheels and other circular ring systems.     

On the performance and resonant frequency of electromagnetic induction energy harvesters

This paper investigates the linear response of an archetypal energy harvester that uses electromagnetic induction to convert ambient vibration into electrical energy. In contrast with most prior works, the influence of the circuit inductance is not assumed negligible. Instead, we highlight parameter regimes where the inductance can alter resonance and derive an expression for the resonant frequency.The governing equations consider the case of a vibratory generator directly powering a resistive load. These equations are non-dimensionalized and analytical solutions are obtained for the system's response to single harmonic, periodic, and stochastic environmental excitations. The presented analytical solutions are then used to study the power delivered to an electrical load.     

Fundamental frequency of clamped plates with circularly periodic boundaries

A boundary perturbation method is developed to determine the fundamental frequency of vibrating plates. The method is then applied to wavy, star shape and polygonal plates with clamped boundary conditions. Approximate analytical solutions of the fundamental frequency are obtained with an accuracy of O(ε⁴), where ε is the deviation from the unit circle.     

Analytical close-form solutions to the elastic fields of solids with dislocations and surface stress

The concept of eigenstrain is adopted to derive a general analytical framework to solve the elastic field for 3D anisotropic solids with general defects by considering the surface stress. The formulation shows the elastic constants and geometrical features of the surface play an important role in determining the elastic fields of the solid. As an application, the analytical close-form solutions to the stress fields of an infinite isotropic circular nanowire are obtained. The stress fields are compared with the classical solutions and those of complex variable method. The stress fields from this work demonstrate the impact from the surface stress when the size of the nanowire shrinks but becomes negligible in macroscopic scale. Compared with the power series solutions of complex variable method, the analytical solutions in this work provide a better platform and they are more flexible in various applications. More importantly, the proposed analytical framework profoundly improves the studies of general 3D anisotropic materials with surface effects.     

A non-linear analysis of the interactions between parametric and external excitations

Solutions of a non-linear equation having a form often encountered in the study of structural dynamics and in which both parametric and external excitation is present have been obtained analytically and numerically. All the asymptotic (analytical) results can be verified by numerical integration for small values of ϵ. Some other solutions, obtained numerically, cannot be approximated by conventional asymptotic methods. Both types of solutions can exist simultaneously; in these situations the initial conditions determine which solution describes the actual response of the system. The solutions that can be predicted only by numerical means exhibit period multiplications (i.e., the response has a period that is an integral multiple of the period of the excitation) and chaos. In fact, in a certain range of ϵ, one set of initial conditions can lead to period doubling in the response while another set can lead to a response very close to the asymptotic prediction. Very flexible structures can exhibit non-linear characteristics in their response at very small amplitudes of the excitation. The present results serve to illustrate some of the extraordinary behavior that such structures can exhibit and suggest that it may be extremely difficult to develop control systems for them.     

Application of Hamilton's law to forced,damped, stationary systems

The algebraic equations for the forced, damped, periodic, axisymmetric motion of circular plates, solid and annular, are derived directly through the application of Hamilton's law of varying action. The simplicity, for many problems, of direct analytical solutions by means of Hamilton's law has previously been demonstrated. The method is called the Hamilton-Ritz method. In this paper, direct analytical solutions from Hamilton's law are shown to be exactly the same as direct analytical solutions from the ancient and fundamental principle of virtual work. The Hamilton-Ritz formulation is compared to the Galerkin formulation. Results from one- and two-term solutions by direct virtual work (Hamilton-Ritz) are compared to results from the exact solution and to results from the Galerkin method.