Three-to-one internal resonances in a general cubic non-linear continuous system

A general continuous system with an arbitrary cubic non-linearity is considered. The non-linearity is expressed in terms of an arbitrary cubic operator. Three-to-one internal resonance case is considered. A general approximate solution is presented for the system. Amplitude and phase modulation equations are derived. Steady state solutions and their stability are discussed in the general sense. The sufficiency condition for such resonances to occur is derived. Finally the algorithm is applied to a beam resting on a non-linear elastic foundation.     

A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions

The subject of this paper is the development of a general solution procedure for the vibrations (primary resonance and nonlinear natural frequency) of systems with cubic nonlinearities, subjected to nonlinear and time-dependent internal boundary conditions—this is a commonly occurring situation in the vibration analysis of continuous systems with intermediate elements. The equations of motion form a set of nonlinear partial differential equations with nonlinear, time-dependent, and coupled internal boundary conditions. The method of multiple timescales, an approximate analytical method, is applied directly to each partial differential equation of motion as well as coupled boundary conditions (i.e. on each sub-domain and the corresponding internal boundary conditions for a continuous system with intermediate elements) which ultimately leads to approximate analytical expressions for the frequency-response relation and nonlinear natural frequencies of the system. These closed-form solutions provide direct insight into the relationship between the system parameters and vibration characteristics of the system. Moreover, the suggested solution procedure is applied to a sample problem which is discussed in detail.     

Nonlinear vibration of a curved beam under uniform base harmonic excitation with quadratic and cubic nonlinearities

This paper presents nonlinear vibration analysis of a curved beam subject to uniform base harmonic excitation with both quadratic and cubic nonlinearities. The Galerkin method is employed to discretize the governing equations. A high-dimensional model that can take nonlinear model coupling into account is derived, and the incremental harmonic balance (IHB) method is employed to obtain the steady-state response of the curved beam. The cases investigated include softening stiffness, hardening stiffness and modal energy transfer. The stability of the periodic solutions for given parameters is determined by the multi-variable Floquet theory using Hsu's method. Particular attention is paid to the anti-symmetric response with and without excitation, as the excitation frequency is close to the first and third natural frequencies of the system. The results obtained with the IHB method compare very well with those obtained via numerical integration.     

Nonlinear free transverse vibrations of in-plane moving plates: Without and with internal resonances

In this paper, nonlinear free transverse vibrations of in-plane moving plates subjected to plane stresses are investigated. The Hamilton principle is applied to derive the governing equation and the associated boundary conditions. The method of multiple scales is employed to analyze the nonlinear partial differential equation. The solvability conditions are established in the cases without internal resonance and with 3:1 or 1:1 internal resonances. Some numerical examples are presented to demonstrate the effects of in-plane moving speeds on the frequencies. The nonlinear frequencies of the in-plane moving plate without internal resonances are numerically calculated. The relationship between the nonlinear frequencies and the initial amplitudes is showed at different in-plane moving speeds and the nonlinear coefficients, respectively. It is feasible to investigate resonances without the modes not involved in the resonances. The effects of the related parameters are demonstrated for the case of 3:1 and 1:1 internal resonances, respectively. The differential quadrature scheme is developed to solve numerically the governing equation and confirm results via the method of multiple scales.     

A theorem on the representation of the field of forced vibrations of a composite elastic system

A theorem on the representation of a vibration field of an elastic system comprising two subsystems is proved in a general form. It is assumed that the system is linear and the subsystems are rigidly connected and interact along a continuous surface S. According to the theorem, forced vibrations of the system can be represented in the form of the sum of two components, which are the solutions of two simpler auxiliary boundary-value problems. The first component is the field of vibrations of the isolated (separated or blocked along S) subsystems under the effect of preset external forces. The second component represents the forced vibrations of the junction of the subsystems, where the external forces are taken equal to zero and only the reaction forces obtained in solving the first auxiliary problem act at the surface S. The theorem is applied to the problem on the reflection and transmission of elastic waves through a junction of two media. It is demonstrated that the theorem utilization reduces the amount of calculations. Other applications are discussed.     

A2B模型阻尼受迫振动方程组的通解

导出了A2B模型阻尼受迫振动方程组的通解,并讨论其算例和图像.     

On the forced vibrations of an oscillator with a periodically time-varying mass

In this paper the forced vibrations of a linear, single degree of freedom oscillator (sdofo) with a time-varying mass will be studied. The forced vibrations are due to small masses which are periodically hitting and leaving the oscillator with different velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator will periodically vary in time. Additionally, an external harmonic force will be applied to the oscillator with a time-varying mass. Not only solutions of the oscillator equation will be constructed, but also stability properties for the forced vibrations will be presented for various parameter values.     

Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber

In a single degree-of-freedom weakly nonlinear oscillator subjected to periodic external excitation, a small-amplitude excitation may produce a relatively large-amplitude response under primary resonance conditions. Jump and hysteresis phenomena that result from saddle-node bifurcations may occur in the steady-state response of the forced nonlinear oscillator. A simple mass-spring-damper vibration absorber is thus employed to suppress the nonlinear vibrations of the forced nonlinear oscillator for the primary resonance conditions. The values of the spring stiffness and mass of the vibration absorber are significantly lower than their counterpart of the forced nonlinear oscillator. Vibrational energy of the forced nonlinear oscillator is transferred to the attached light mass through linked spring and damper. As a result, the nonlinear vibrations of the forced oscillator are greatly reduced and the vibrations of the absorber are significant. The method of multiple scales is used to obtain the averaged equations that determine the amplitude and phases of the first-order approximate solutions to primary resonance vibrations of the forced nonlinear oscillator. Illustrative examples are given to show the effectiveness of the dynamic vibration absorber for suppressing primary resonance vibrations. The effects of the linked spring and damper and the attached mass on the reduction of nonlinear vibrations are studied with the help of frequency response curves, the attenuation ratio of response amplitude and the desensitisation ratio of the critical amplitude of excitation.     

Three-state quantum systems: A procedure for the solution

Summary An iterative method to obtain a solution of the differential equation , with Ĥ a 3×3 Hermitian matrix anda the unknown vector, is proposed. The procedure is particularly suitable for computer implementation and, as an example, has been applied to find the excitation probability of a three-level atom after the synchronous passage of two laser pulses each almost resonant with a pair of atomic levels.

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Riassunto Si presenta un metodo iterativo per ottenere una soluzione esatta dell'equazione differenziale , con Ĥ una matrice 3×3 hermitiana eda il vettore incognito. La procedura è particolarmente adatta ad un trattamento numerico e, come esempio, è stata applicata per trovare la probabilità d'eccitazione di un atomo a tre livelli dopo il passaggio simultaneo di due impulsi laser di cui ognuno quasi risonante con una coppia di livelli atomici.

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Резюме Предлагается итерационный метод получения решения дифференциального уравнения , где Ĥ есть эрмитова матрица 3×3a-неизвестный вектор. Предложенная процедура решеня является удобной для компьтерных вычислений. Рассматривается пример: определение вероятности возбуждения атома с тремя уровнями после синхронного прохождения двух лазерных импульсов, каждый из которых имеет почти резонансную частоту с парой атомных уровней.

     

A mathematical solution for the parameters of three interfering resonances

The multiple-solution problem in determining the parameters of three interfering resonances from a fit to an experimentally measured distribution is considered from a mathematical viewpoint. It is shown that there are four numerical solutions for a fit with three coherent Breit-Wigner functions. Although explicit analytical formulae cannot be derived in this case, we provide some constraint equations between the four solutions. For the cases of nonrelativistic and relativistic Breit-Wigner forms of amplitude functions, a numerical method is provided to derive the other solutions from that already obtained, based on the obtained constraint equations. In real experimental measurements with more complicated amplitude forms similar to Breit-Wigner functions, the same method can be deduced and performed to get numerical solutions. The good agreement between the solutions found using this mathematical method and those directly from the fit verifies the correctness of the constraint equations and mathematical methodology used.     

A note on forced vibrations of a rectangular plate embedded in a non-homogeneous foundation

The title problem is solved for the case where the plate is subjected to a uniformly distributed, sinusoidally varying excitation. The plate dynamic amplitude is approximated by means of a polynomial co-ordinate function which contains an undetermined parameter as an exponent. The governing functional is minimized with respect to the unknown coefficient of the coordinate function and with respect to the exponential parameter. Good agreement with known results is shown to exist.     

Crystals with sublattices belonging to the cubic system and special features of their elementary excitation spectra

Genesis of spectra of elementary excitations of crystals comprising sublattices belonging to the cubic system is investigated. Using the methods of group theory, the Brillouin sublattice zones are transformed into the Brillouin crystal zones, and degeneration is established caused by the convolution of branches of the spectrum. The degeneration is removed with allowance for hybridization of sublattice states. The special features of the band spectra of some crystals belonging to the cubic system are discussed. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 21–27, May, 2006.     

A search for narrow resonances in the φφ system

Inclusive φφ production has been studied in π^?Be interactions at 85 GeV/c incident momentum, and a signal of 4327 φφ events was found in a sample of data recorded with the aim of studying the low mass φφ system. We estimate that ～ 75% of this signal can be attributed to the correlated production of two φ′s. A search for narrow enhancements leads to upper limits at the 99.7% confidence level for σ(X) × BR(X → φφ) varying from ? 562 nb/Be nucleus at a mass of 2.1 GeV/c² to ? 221 nb/Be nucleus at 3.1 GeV/c², where X represents a possible narrow state. These upper limits represent an increase in sensitivity by a factor 3 to 5 when compared to currently available data.     

Multi-pulse homoclinic orbits and chaotic dynamics of a parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities

In this paper,the complicated dynamics and multi-pulse homoclinic orbits of a two-degree-of-freedom parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities are studied.The damping,parametrical excitation and the nonlinearities are regarded as weak.The averaged equation depicting the fast and slow dynamics is derived through the method of multiple scales.The dynamics near the resonance band is revealed by doing a singular perturbation analysis and combining the extended Melnikov method.We are able to determine the criterion for the existence of the multi-pulse homoclinic orbits which can form the Shilnikov orbits and give rise to chaos.At last,numerical results are also given to illustrate the nonlinear behaviors and chaotic motions in the nonlinear nano-oscillator.