Approximations for motion of the oscillators with a non-negative real-power restoring force

Oscillators with a non-negative real-power restoring force are considered in this paper. This type of restoring force is related to systems with a quasi-zero stiffness characteristic or those in which the restoring force is purely nonlinear in nature. Examples of these types of restoring force are grounded in real physical and engineering systems. Periodic motion of such conservative oscillators is described first in a novel way by means of the elliptic function the parameters of which are obtained from the energy conservation law and Hamilton's variational principle. Then, the approach is extended to non-conservative oscillators by adjusting the elliptic Krylov-Bogoliubov method. The methods proposed for the conservative and non-conservative systems under consideration have wider applications than the existing one with respect to the power of the restoring force. Several examples, the majority of which are so far unsolved, are given to illustrate the methods proposed and to demonstrate their generality, which permits unforeseen solutions for motion, containing higher harmonics and assuring consistent accuracy regardless of the value of the power of the restoring force. The results obtained are compared with numerical results and have excellent accuracy.     

Constant phase angle response with fractal electrodes

An analytical solution is obtained for the dielectric response of an electrode composed of pores whose cross-sectional areas possess a fractal arrangement. When the surface dimension of the pore system is greater than unity the divergent constant phase angle response of previous computations is recovered. However when the surface dimensionality falls below unity the capacitance saturates and a peak is obtained in the loss component which exhibits a low frequency power law dependence on frequency.     

Influence of potential function asymmetries on the performance of nonlinear energy harvesters under white noise

To improve the broadband transduction capabilities of vibratory energy harvesters (VEHs) under random and non-stationary excitations, many researchers have resorted to purposefully introducing nonlinearities into the restoring force of the harvester. While performing this task, it is often very challenging to maintain a perfectly symmetric restoring force which yields a VEH with an asymmetric potential energy function. This paper investigates the influence of potential function asymmetries on the performance of nonlinear VEHs under white noise inputs. To that end, a quadratic nonlinearity is introduced into the restoring force and its influence on the mean output power of the harvester for mono- and bi-stable quartic potentials is investigated. It is shown that, for VEHs with a mono-stable quartic potential function, the mean output power increases with the degree of potential function asymmetry. On the other hand, for energy harvesters with a bi-stable quartic potential function, asymmetries in the restoring force appear to worsen performance especially for low to moderate noise intensities. When the noise intensity becomes sufficiently large, the influence of the potential function?s asymmetry on the mean power diminishes. Results also reveal that a VEH with a symmetric bi-stable quartic potential function produces higher mean power levels than the one with the most asymmetric mono-stable potential. As such, it is concluded that a VEH with a symmetric bi-stable potential is most desirable to improve performance under white noise.     

AgNbO3拉曼光谱随激发功率变化的研究

AgNbO3是一种可见光响应型光催化剂, 具有简单的结构和化学组成, 以及合适的光学带隙, 并且在光催化分解水制氢和光催化降解有机污染物两方面具有较大的开发潜力。本文采用溶胶凝胶法合成了AgNbO3样品, 以457.5 nm固体激光器为激发光源, 首次在不同激发功率下, 获得了AgNbO3样品的拉曼谱图和谱峰参量随激光功率的变化关系。研究发现AgNbO3随激光功率的变化产生了可逆相变。     

Excitation of Rayleigh and Stoneley surface acoustic waves by distributed seismic sources

Using the integral Fourier-transform technique, we obtain a solution in integral form to the problem of excitation of elastic waves in a homogeneous isotropic solid half-space and the bordering homogeneous gas by the time-dependent forces which are arbitrarily distributed in a solid over the plane parallel to the interface of the media. Different configurations of the force sources are analyzed from the viewpoint of excitation of different types of seismoacoustic waves. Expressions for the time-averaged radiated powers of the Stoneley wave at the gas–solid interface and the Rayleigh wave at the solid–vacuum interface as well as analytical expressions for the Rayleigh wave displacements, which are valid for large distances from the source, are obtained for the harmonic dependence of forces on time. Excitation of a Rayleigh wave by the point sources oriented vertically, i.e., along the normal to the surface of elastic half-space, and horizontally, i.e., parallel to this surface, is analyzed in detail. Analytical expressions for the Rayleigh-wave radiated power are obtained. The dependences of these powers on the source orientation and depth are derived. It is shown that the Rayleigh-wave radiated power decreases with distance between the point of the force application and the boundary and turns to zero for a source depth of about 17.5% of the wavelength of the transverse wave in the case of a horizontally oriented subsurface source and a medium with identical Lamé parameters λ and μ. This power increases and reaches a relative maximum when the source depth becomes equal to about 42.4% of the wavelength of the transverse wave and then exponentially falls off as the source depth increases. This maximum is about 5.5% of the surface-source radiated power.     

Excitation of seismoacoustic waves by a harmonic force source acting at the boundary of a liquid layer and an elastic halfspace

A problem on the excitation of seismoacoustic waves in a system of a homogeneous isotropic elastic halfspace covered with a liquid layer is solved in the case of action of a source of point harmonic force on the surface of an elastic medium. Integral expressions are obtained for the radiation powers averaged over a wave period for longitudinal and transverse waves in a solid. Mode excitation is analyzed in detail. Expressions describing parts of the mode powers radiated into a liquid layer and an elastic medium are obtained. Numerical analysis of radiation powers is conducted for spherical longitudinal and transverse waves as well as for the radiation powers of seismoacoustic modes in a solid halfspace and a liquid layer. It is determined that in the conditions characteristic of bottom rocks in the case, where the basin depth is several times and more larger than the sound’s wavelength, about 2/3 of the total power is radiated into a liquid.     

Analytical solutions to rate equations including losses describing threshold pumped fiber lasers

An exact analytical expression to the threshold pump power has been deduced for rate equations including loss coefficients applicable to fiber lasers doped with quasi two-energy-level ions. Following the derivation of a closed-form expression for the variation of the pump power along the fiber, together with analytical solutions to the position dependent atom population density and laser power in terms of the pump power, we have analytically and fully solved rate equations describing threshold pumped fiber lasers. A comparison has also been made between the exact analytical result with the previous reported results, which indicates that various approximation have underestimated the threshold pump powers.     

Electron Transfer through a One-Dimensional Fractal Structure

An adequate model of electron tunneling through a self-similar fractal potential (SFP) defined on a Cantor set is extended to a generalized Cantor set. It is demonstrated that, as in a specific case, the Schrödinger equation for the SFP is reduced to a functional equation for the transfer matrix which admits solutions of three types. Two of them are single-parameter solutions corresponding to SFP barriers and lacunas with arbitrary powers. In both cases, the transfer matrices are nonanalytic in the long-wavelength region and have fractal dimensionalities there. The third solution type includes a unique solution corresponding to the SFP barrier with fixed power for a given barrier width. The corresponding transfer matrix is analytic at the point k = 0. It is shown that generally the SFP possesses only the property of approximate scale invariance on the generalized Cantor set in the long- and short-wavelength regions. Only the limiting SFP, whose fractal dimensionality is equal to unity, possesses the property of rigorous scale invariance irrespective of its power. It is shown that SFPs with identical fractal dimensionalities but different lacunas are described by different transfer matrices.     

Comparison of restoring force models for the identification of structures with hysteresis and degradation

When subjected to events such as earthquakes, engineering structures typically exhibit a nonlinear and hysteretic behaviour with stiffness and strength degradations. Though a reliable evaluation of safety conditions should take into account the nonlinear dynamic and evolutionary nature of the structural response, the experimental identification of a nonlinear behaviour under dynamic and seismic loading is, to date, an open problem. The present research aims at evaluating the potential of different restoring force models for simulating the seismic response of hysteretic structural systems, with special emphasis on the two main problems encountered when applying this approach to full-scale structures under intense excitation: (a) a markedly time-dependent behaviour; (b) need to compare among different restoring force models, either expressed in a parametric or polynomial form. In particular, polynomial models will be formulated both in terms of restoring force and its derivative, in order to present a comprehensive discussion of different strategies. The nonlinear identification technique employed in this paper is required to account for a time-dependent behaviour. In fact, in presence of degradation or any other time-varying characteristics, instantaneous identification certainly constitutes an enhancement of the classical restoring force based approach, and may as well provide checks on the consistency of the assumed models.     

Experimental investigation of asymptotic modal analysis for a rectangular plate

Experimental investigations of the response of a rectangular plate under a point random force have been performed to verify the asymptotic behavior predicted by Asymptotic Modal Analysis (AMA). Measurements have been made for various frequency bandwidths, center frequencies, and locations of the point force. The experimental results approach the results predicted by AMA as the frequency bandwidth becomes large. Moreover, experimental results show that the responses at all points of the plate except for some special areas become the same as the frequency bandwidth becomes large. However, the ratio of experimental results to AMA results has a greater variation from unity when the location of the point force is near the edge of the plate, than when the location of the point force is at the center of the plate. All experimental results show good agreement with the expected results from AMA.     

On the motion of non-linear oscillators with a fractional-order restoring force and time variable parameters

An analytical approach to determine the approximate solution for the periodic motion of non-conservative oscillators with a fractional-order restoring force and slowly varying parameters is presented. The solution has the form of the first-order differential equation for the amplitude and phase of motion. The method used is based on the combination of the Krylov-Bogoliubov method with Hamilton's variational principle with the uncommutative rule for the variation of velocity. The conservative systems with slowly varying parameters are also considered. The corresponding adiabatic invariant is obtained. Two examples are given to illustrate derived theoretical results.     

Moment induced structure-borne sound emission of resiliently mounted machines

I.IntroductionInmanypractica1cnginccringsituations,suchasinbui1dings,carsandships,machinesaremountedonseatingstructurcsthroughcompliantiso1atorsinordertoreducethetransmissionofunwantedmachinevibrations.Thercsidualtransmissionresu1tsfromacombinationofforcesandmomcntsatthccontactpointsandinordcrtoidcntifytheimportanttransmissionmechanjsms,eachshou1didea1lybcasscsscdbymcasurcmcntorprediction.MuchofpreviousworkontranRmissionatmountingshasconccntratcdonvcrtica1forceinduccdpower["l.Itmaybethatsucha…     

Modeling of Multi-Wavelength Brillouin Fiber Laser Based on Analytical Solutions in a Linear Cavity

Abstract

In this article, an accurate model of multi-wavelength Brillouin fiber laser is developed by analytically solving cascade-stimulated Brillouin scattering equations in a linear cavity. The dependence of Stokes signal power on the incident pump power and the successive reflected transmitted powers is calculated. Furthermore, an analytical approach that can be used to predict the generated Stokes signals power is derived. Finally, this derived model is checked with experimental results that show significant agreement with the analytical solution of the model.     

Nature of transitions in augmented discrete nonlinear Schrödinger equations

We investigate the nature of the transitions between free and self-trapping states occurring in systems described by augmented forms of the discrete nonlinear Schr?dinger equation. These arise from an interaction between a moving quasiparticle (such as an electron or an exciton) and lattice vibrations, when the effects of nonlinearities in interaction potential and restoring force are included. We derive analytic conditions for the stability of the free state and the crossover between first- and second-order transitions. We demonstrate our results for different types of nonlinearities in the interaction potential and restoring force. We find that, depending on the type of nonlinearity, it is possible to have both first- and second-order transitions. We discuss possible hysteresis effects.     

Can a transmission coefficient be greater than unity?

In studying the sound transmission through panels, it is usual to consider a model based on an infinite panel. The sound transmission coefficient is defined as the ratio of transmitted power to incident power and must, therefore, be less than or equal to unity due to energy conservation. However, it is shown here that, for a finite panel, it is possible to find values of transmission coefficient that are greater than unity at certain frequencies due to the normalisation used for the incident power. This occurs at the fundamental resonance of the fluid-loaded panel. Example results are presented for a simple circular piston and then for a flexible rectangular panel and an explanation for this phenomenon is provided. In effect, the panel is excited by a sound field that is much larger than its own surface and attracts sound power from a wide region of the incident field. This result stands as an interesting test case for calculation methods where the occurrence of transmission coefficients greater than unity should not necessarily be seen as an error in the calculation.     

Estimating ensemble average power delivered by a piezoelectric patch actuator to a non-deterministic subsystem

Engineering systems such as aircraft, ships and automotive are considered built-up structures. Dynamically they are taught of as being fabricated from many components that are classified as ‘deterministic subsystems’ (DS) and ‘non-deterministic subsystems’ (Non-DS). Structures' response of the DS is deterministic in nature and analysed using deterministic modelling methods such as finite element (FE) method. The response of Non-DS is statistical in nature and estimated using statistical modelling technique such as statistical energy analysis (SEA). SEA method uses power balance equation, in which any external input to the subsystem must be represented in terms of power. Often, input force is taken as point force and ensemble average power delivered by point force is already well-established. However, the external input can also be applied in the form of moments exerted by a piezoelectric (PZT) patch actuator. In order to be able to apply SEA method for input moments, a mathematical representation for moment generated by PZT patch in the form of average power is needed, which is attempted in this paper. A simply-supported plate with attached PZT patch is taken as a benchmark model. Analytical solution to estimate average power is derived using mobility approach. Ensemble average of power given by the PZT patch actuator to the benchmark model when subjected to structural uncertainties is also simulated using Lagrangian method and FEA software. The analytical estimation is compared with the Lagrangian model and FE method for validation. The effects of size and location of the PZT actuators on the power delivered to the plate are later investigated.     

An analytical and experimental study of the vibration response of a clamped ribbed plate

An analytical solution is presented in this paper for the vibration response of a ribbed plate clamped on all its boundary edges by employing a traveling wave solution. A clamped ribbed plate test rig is also assembled in this study for the experimental investigation of the ribbed plate response and to provide verification results to the analytical solution. The dynamic characteristics and mode shapes of the ribbed plate are measured and compared to those obtained from the analytical solution and from finite element analysis (FEA). General good agreements are found between the results. Discrepancies between the computational and experimental results at low and high frequencies are also discussed. Explanations are offered in the study to disclose the mechanism causing the discrepancies. The dependency of the dynamic response of the ribbed plate on the distance between the excitation force and the rib is also investigated experimentally. It confirms the findings disclosed in a previous analytical study [T.R. Lin, J. Pan, A closed form solution for the dynamic response of finite ribbed plates, Journal of the Acoustical Society of America 119 (2006) 917–925] that the vibration response of a clamped ribbed plate due to a point force excitation is controlled by the plate stiffness when the source is more than a quarter plate bending wavelength away from the rib and from the plate boundary. The response is largely affected by the rib stiffness when the source location is less than a quarter bending wavelength away from the rib.     

On the description of quadratic and cubic optical effects of propagation of femtosecond pulses within the framework of the model of the saturable potential

It is shown that, for describing the simultaneous manifestation of nonlinear effects of different orders (quadratic, cubic, etc.) in propagation of high-intensity femtosecond pulses, it is sufficient to use the saturable potential of the restoring force. With the use of computer simulation, it is demonstrated that the spectrum of the response of a medium calculated on the basis of the Duffing equation with quadratic and cubic nonlinearity is in qualitative agreement with the spectrum calculated on the basis of the equation with a saturable restoring force. The dependence of the frequency corresponding to the maximum spectral radiance on the duration of the interacting pulse is discussed. It is noted that, for ultrashort pulses, even in the case of their propagation in a linear medium, the response can occur at the natural frequency rather than at the frequency of the propagating pulse.