Response statistics of discretized structures to non-stationary random excitation

Closed-form time-dependent response statistics such as (i) the variance of displacement response and variance of velocity response, (ii) the covariance of displacement and velocity responses, and (iii) covariances of displacement responses and velocity responses of structures, discretized by the finite element method, subjected to a wide class of non-stationary random excitations are presented. The response statistics include the contribution due to the evolutionary coincident and quadrature spectral density functions. Application of the expressions obtained has been made to a quarter-scale physical model of a class of mast antenna structures excited at the base. Computed results indicate that the contribution due to the evolutionary quadrature spectral density function is insignificant.     

Response of uni-modal duffing-type harvesters to random forced excitations

Linear energy harvesters have a narrow frequency bandwidth and hence operate efficiently only when the excitation frequency is very close to the fundamental frequency of the harvester. Consequently, small variations of the excitation frequency around the harvester's fundamental frequency drops its small energy output even further making the energy harvesting process inefficient. To extend the harvester's bandwidth, some recent solutions call for utilizing energy harvesters with stiffness-type nonlinearities. From a steady-state perspective, this hardening-type nonlinearity can extend the coupling between the excitation and the harvester to a wider range of frequencies. In this effort, we investigate the response of such harvesters, which can be modeled as a uni-modal duffing-type oscillator, to White Gaussian and Colored excitations. For White excitations, we solve the Fokker-Plank-Kolmogorov equation for the exact joint probability density function of the response. We show that the expected value of the output power is not even a function of the nonlinearity. As such, under White excitations, nonlinearities in the stiffness do not provide any enhancement over the typical linear harvesters. We also demonstrate that nonlinearities in the damping and inertia may be used to enhance the expected value of the output power. For Colored excitations, we use the Van Kampen expansion and long-time numerical integration to investigate the influence of the nonlinearity on the expected value of the output power. We demonstrate that, regardless of the bandwidth or the center frequency of the excitation, the expected value of the output power decreases with the nonlinearity. With such findings, we conclude that energy harvesters modeled as uni-modal duffing-type oscillators are not good candidates for harvesting energy under forced random excitations. Using a linear transformation, results can be extended to the base excitation case.     

Stationariness provided by filtration to a periodic non-stationary random process

The concept of generalized spectral density is used for evaluating the variance of the response of a simple linear oscillator excited by periodic non-stationary random functions. The generalized Wiener-Khintchine relations are slightly reshaped, so that the generalized spectral density of the excitation function could be expressed as the sum of Dirac delta functions, whose coefficients could be given in the form of closed analytical expressions. The results show that, apart from a few exceptions, the response of a lightly damped system is practically stationary. It is noted in general that filtration brings about stationarization.     

Response of a strongly non-linear oscillator to narrowband random excitations

The principal resonance of a van der Pol-Duffing oscillator subject to narrowband random excitations has been studied. By introducing a new expansion parameter the method of multiple scales is adapted for the strongly non-linear system. The behavior of steady state responses, together with their stability, and the effects of system damping and the detuning, and magnitude of the random excitation on steady state responses are analyzed in detail. Theoretical analyses are verified by some numerical results. It is found that when the random noise intensity increases, the steady state solution may change form a limit cycle to a diffused limit cycle, and the system may have two different stable steady state solutions for the same excitation under certain conditions. The results obtained for the strongly non-linear oscillator complement previous results in the literature for weakly non-linear systems.     

The covariance response of linear systems to locally stationary random excitation

A simple method is given for calculating the covariance response of linear, time-invariant systems to random excitation processes which are locally stationary, or approximately so. As an illustration, the method is used to estimate the response of an idealized model of a ten-storey building to non-stationary ground acceleration; the accuracy of the estimated response is assessed by a comparison with the results of a less approximate, but lengthier, general calculation method, previously published.     

Non-linear responses of suspended cables to primary resonance excitations

We investigate the non-linear forced responses of shallow suspended cables. We consider the following cases: (1) primary resonance of a single in-plane mode and (2) primary resonance of a single out-of-plane mode. In both cases, we assume that the excited mode is not involved in an autoparametric resonance with any other mode. We analyze the system by following two approaches. In the first, we discretize the equations of motion using the Galerkin procedure and then apply the method of multiple scales to the resulting system of non-linear ordinary-differential equations to obtain approximate solutions (discretization approach). In the second, we apply the method of multiple scales directly to the non-linear integral-partial-differential equations of motion and associated boundary conditions to determine approximate solutions (direct approach). We then compare results obtained with both approaches and discuss the influence of the number of modes retained in the discretization procedure on the predicted solutions.     

A new formulation of the linear dynamic response to random excitations

With the damped harmonic oscillator as a model, a new formulation is presented of the stochastic response of linear dynamic systems under random excitation. The starting point of the method is the Fourier-Stieltjes representation for a general random process first introduced by Priestley. A unique feature of the present formulation is analytical compactness. This becomes a significant advantage over existing methods when one attempts to compute the covariance matrix elements which are often necessary for crossing problems.     

Diffractive responses to optical wave-train excitations in nanotube-doped nematics

Diffractive properties of temporary holographic gratings were studied in cells of the homogeneously aligned nematic E7 doped with a trace of carbon nanotubes. These phase gratings were induced by interference modulation of the pulsed wave-train excitation of two coherent writing beams, in conjunction with an externally applied dc voltage. The effects of the optical excitation on the diffraction efficiencies and the decay time constants were observed to be strongly dependent on the pulse width of the writing beams. PACS 42.40.Lx; 42.70.Df; 42.70.Ln; 81.05.Tp     

Time-dependent random walks and the theory of complex adaptive systems

Motivated by novel results in the theory of complex adaptive systems, we analyze the dynamics of random walks in which the jumping probabilities are time dependent. We determine the survival probability in the presence of an absorbing boundary. For an unbiased walk, the survival probability is maximized in the case of large temporal oscillations in the jumping probabilities. On the other hand, a random walker who is drifted towards the absorbing boundary performs best with a constant jumping probability. We use the results to reveal the underlying dynamics responsible for the phenomenon of self-segregation and clustering observed in the evolutionary minority game.     

有界噪声和谐和激励联合作用下一类非线性系统的混沌研究

研究一类有界噪声和谐和激励联合作用下的非线性系统，首先用多尺度方法将该系统约化，针对约化后的平均系统，利用随机Melnikov过程方法结合均方值准则导出随机系统可能产生混沌运动的临界条件，结果表明在一定的参数范围内，随着Weiner过程强度参数值的增大，混沌的临界激励幅值先递减继而递增. 同时，用两类数值方法即最大Lyapunov指数法和Poincare截面法验证了解析结果. 关键词： 有界噪声 多尺度 随机Melnikov过程 混沌     

Time-dependent Hilbert spaces, geometric phases, and general covariance in quantum mechanics

We investigate consequences of allowing the Hilbert space of a quantum system to have a time-dependent metric. For a given possibly nonstationary quantum system, we show that the requirement of having a unitary Schrödinger time-evolution identifies the metric with a positive-definite (Ermakov–Lewis) dynamical invariant of the system. Therefore the geometric phases are determined by the metric. We construct a unitary map relating a given time-independent Hilbert space to the time-dependent Hilbert space defined by a positive-definite dynamical invariant. This map defines a transformation that changes the metric of the Hilbert space but leaves the Hamiltonian of the system invariant. We propose to identify this phenomenon with a quantum mechanical analogue of the principle of general covariance of general relativity. We comment on the implications of this principle for geometrically equivalent quantum systems and investigate the underlying symmetry group.     

A unified model for non-stationary and/or non-Gaussian random processes

An analytical model for non-stationary and/or non-Gaussian random processes described in the paper is based on a normal stationary random process. The non-stationarity is introduced as a deterministic dependence of the parameters of the marginal distribution function or those of the correlation function upon the argument t. Consideration that the mentioned parameters are random variables or stationary random processes results in generating non-Gaussian distributions of the unconditioned process. By combining deterministic and random components of the parameters' dependencies, non-stationary and simultaneously non-Gaussian random processes may be easily specified. The model described may be useful for analytical treatment, for identification of experimentally obtained realizations of random processes and for simulation of random processes on computers as well as in the laboratory.     

Nonlinearity of magnetoacoustic excitations in planar structures

This paper reports on the results of experimental investigations into the threshold power of the onset of nonlinearity of magnetoacoustic vibrations in planar structures (such as a ferrite film-dielectric substrate structure) in the range of phase matching of the higher bulk magnetostatic and acoustic modes. Under the experimental conditions, the wavelength of the higher bulk magnetostatic modes is of the order of 1 μm and shorter. On this basis, the energy of these vibrations with respect to the origin of the magnetostatic wave spectrum is determined by the energy of the inhomogeneous exchange interaction. The standing magnetoacoustic waves are examined in conventional yttrium iron garnet films with free surface spins in which, under standard conditions, only dipole magnetostatic vibrations are excited in planar resonators. Consideration is given to the threshold power of the onset of precession instability of the dipole exchange acoustic modes which, as was shown earlier by the authors, are excited in the range of the phase matching of the exchange and acoustic modes. A comparative analysis is performed for the threshold powers of dipole magnetostatic, exchange acoustic, and dipole exchange acoustic modes. It is demonstrated that the threshold power of the instability of magnetostatic modes decreases significantly when the natural frequencies of the dipole modes coincide with those of the exchange acoustic modes. __________ Translated from Fizika Tverdogo Tela, Vol. 44, No. 7, 2002, pp. 1285–1289. Original Russian Text Copyright ? 2002 by Bugaev, Gorsky.     

An iterative method for non-stationary response analysis of non-linear random systems

A method for calculating the non-stationary response of non-linear systems subjected to random excitations is formulated. The time-dependent equivalent linear system is considered and an iterative procedure for evaluating the non-stationary mean-square responses is developed. Several examples are presented and applicability of the technique is illustrated.