Effects of inertia and surface tension on a power-law fluid flowing down a wavy incline

We consider a thin film of a power-law liquid flowing down an inclined wall with sinusoidal topography. Based on the von Kármán–Pohlhausen method an integral boundary-layer model for the film thickness and the flow rate is derived. This allows us to study the influence of the non-Newtonian properties on the steady free surface deformation. For weakly undulated walls we solve the governing equation analytically by a perturbation approach and find a resonant interaction of the free surface with the wavy bottom. Furthermore, the analytical approximation is validated by numerical simulations. Increasing the steepness of the wall reveals that nonlinear effects like the resonance of higher harmonics grow in importance. We find that shear-thickening flows lead to a decrease while shear thinning flows lead to an amplification of the steady free surface. A linear stability analysis of the steady state shows that the bottom undulation has in most cases a stabilizing influence on the free surface. Shear thickening fluids enhance this effect. The open questions which occurred in the linear analysis are then clarified by a nonlinear stability analysis. Finally, we show the important role of capillarity and discuss its influence on the steady solution and on the stability.     

直立码头前船波浪力耦合计算模型

建立了外域用Boussinesq方程、内域用刚体运动方程的直立码头前二维船剖面波浪力的时 域计算耦合模型,内域与外域在交界面的匹配条件是流量连续和压力相等. 进行了相关模型 实验,并把计算结果与实验结果进行了对比. 推导了船体与水底和直立码头之间间隙内流体 运动的自振频率,研究了间隙内流体运动的共振现象.     

双频内共振系统的Normal Form及其简化

讨论双频内共振系统的 Normal Form及其降维问题.利用发展的 Normal Form直接方法,导出了任意双频内共振系统 Normal Form的一般形式.指出 Poincare共振项分为内共振项和非内共振性两类,并定义了内共振项的阶.提出了一种普遍适用的降维变换,并证明了该变换可将任意双频内共振系统的 Normal Form方程降到3维.应用举例表明,该变换不仅适用于半单问题,也适于非半单问题（即强:1:1内共振系统）.     

A nonlinear analysis of surface acoustic waves in isotropic elastic solids

With the fast evolution of wireless and networking communication technology, applications of surface acoustic wave (SAW), or Rayleigh wave, resonators are proliferating with fast shrinking sizes and increasing frequencies. It is inevitable that the smaller resonators will be under a strong electric field with induced large deformation, which has to be described in wave propagation equations with the consideration of nonlinearity. In this study, the formal nonlinear equations of motion are constructed by introducing the nonlinear constitutive relation and strain components in a standard procedure, and the equations are simplified by the extended Galerkin method through the elimination of harmonics. The wave velocity of the nonlinear SAW is obtained from approximated nonlinear equations and boundary conditions through a rigorous solution procedure. It is shown that if the amplitude is small enough, the nonlinear results are consistent with the linear results, demonstrating an alternative procedure for nonlinear analysis of SAW devices working in nonlinear state.     

An efficient approach for calculation of pitching moment in nonlinear reduced frequency method at low Mach number transonic flows

In this research, an efficient methodology for calculation of pitching moment coefficient at low Mach number transonic flows by using the perturbed nonlinear reduced frequency approach is presented. The proposed approach uses the perturbation technique in the nonlinear frequency domain (NLFD) method to estimate the solution at high harmonics. In this approach, the density and velocity fields at high harmonics are perturbed about those at low harmonics. Perturbing the density and velocity fields, the semi‐linear form of the governing equations is obtained. The resulting solution vector and spatial operator are then approximated by discrete form of Fourier transformation and governing equations are solved by using the pseudo‐spectral approach. Numerical results show that the proposed approach predicts good pitching moment coefficient at low Mach number transonic flows with up to 50% savings in computational time. Copyright © 2011 John Wiley & Sons, Ltd.     

1/3 subharmonic solution of elliptical sandwich plates

IntroductionInrecentyears,withtheessentialadvantageoflightweightandhighrigidity ,sandwichplatesandshellshavebeenusedasanimportantpatternofstructuralelementsinaeronautical,astronauticalandnavalengineering .However,nonlinearproblemsforsandwichplatesandshellsareonlyinvestigatedbyafewbecauseofthedifficultiesofnonlinearmathematicalproblems.LiuRen_huaiandXuJia_chu[1,2 ]andothershavemadesomeinvestigationsinthisfield .Bifurcationofnonlinearvibrationforsandwichplateshasnotyetbeeninvestigated .Inthisp…     

Linear resonance in viscous films on inclined wavy planes

We study viscous gravity-driven films flowing over periodically undulated substrates. Linear analysis describes steady flow along small amplitude corrugations for films of arbitrary thickness. Solving the resulting system numerically, we demonstrate resonance (or, possibly, near resonance) and identify different behaviours for thin, intermediate and thick films. Approximating the leading-order velocity profile by the free surface value allows for an analytic solution, which – in the limit of high Reynolds numbers – recovers the different regimes and reveals the relevant physical mechanisms. Our results support the view that the resonance is associated with an interaction of the undulated film with capillary-gravity waves travelling against the mean flow direction. As a consequence, the resonance peak is attained under conditions that render the wave phase velocity equal to zero in the laboratory reference frame, and thus permit direct exchange of energy between the steadily deformed film and the free surface.     

Nonlinear theory of forced surface waves in a circular basin

A nonlinear theory is developed to study surface waves excited by the prescribed horizontal oscillation of the side wall of a circular basin. It is assumed that the frequency of the forced oscillation is near either one of the resonance frequencies of the water in the basin or twice of it. A multiple-scale asymptotic expansion is constructed to derive an equation for the amplitude of an excited eigenmode and critical points of some parameters are found for primary and subharmonic resonance waves. Across these critical points the eigenmode amplitude increases abruptly but remains bounded except at certain values of the water radius to the depth ratio where internal resonance appears.     

Nonlinear vibrations of a circular cylindrical shell with multiple internal resonances under multi-harmonic excitation

The nonlinear response of a water-filled, thin circular cylindrical shell, simply supported at the edges, to multi-harmonic excitation is studied. The shell has opportune dimensions so that the natural frequencies of the two modes (driven and companion) with three circumferential waves are practically double than the natural frequencies of the two modes (driven and companion) with two circumferential waves. This introduces a one-to-one-to-two-to-two internal resonance in the presence of harmonic excitation in the spectral neighbourhood of the natural frequency of the mode with two circumferential waves. Since the system is excited by a multi-harmonic point-load excitation composed by first and second harmonics, very complex nonlinear dynamics is obtained around the resonance of the fundamental mode. In fact, at this frequency, both modes with two and three circumferential waves are driven to resonance and each one is in a one-to-one internal resonance with its companion mode. The nonlinear dynamics is explored by using bifurcation diagrams of Poincaré maps and time responses.     

The stability of vortex filaments

The article discusses the problem of the nonlinear oscillations of concentrated vortexes, For a vortical annulus and a spiral vortical filament, the article demonstrates the character of the change in the form and the frequency of the oscillations with an increase in the amplitude. In the case of standing waves, the solution is obtained in the form of a series with respect to the amplitude of the perturbations, with an accuracy up to terms of the third order of smallness, inclusive. For running waves, the solution is constructed using a method analogous to the Stewart method. The same problem is solved using direct numerical integration of the starting equations of motion. The values of the critical amplitudes which bring about the breakdown of a vortical annulus are obtained. It is shown that as a result of the nonlinearity of the equations in solution of the problem with the starting data higher harmonics can separate out intensively. For a spiral vortical filament, a study was made of the nonlinear interaction of the perturbations; it is shown, in particular, that a perturbation which is stable according to the linear theory may become unstable as a result of nonlinear interaction with neighboring perturbations along the length of the wave and of perturbations of the frequency.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–49, May–June, 1973.The authors thank G. I. Petrov for his direction of the work, as well as V. Ya. Shkadov for his interest in the work.     

Generation of higher electron cyclotron frequency harmonics in plasma with a beam

We examine nonlinear excitation of the higher electron-cyclotron frequency harmonics for waves propagating perpendicular to an external uniform magnetic field in a Maxwell plasma for the case of low-density electron beam passage through the plasma. It is shown that the nonlinear excitation mechanism leads to the possibility of generating cyclotron harmonics for plasma parameters for which generation does not occur from the linear theory viewpoint. The nonlinear cyclotron harmonic generation increments are calculated for nonlinear scattering by the beam and plasma electrons of the high frequency longitudinal waves excited in the plasma by the beam.Translated from Zhurnal Prikladnoi Mekhaniki Tekhnicheskoi Fiziki, Vol. 10, No. 6, pp. 40–51, November–December, 1969.The author wishes to thank V. N. Tsytovich for posing the problem and for many discussions of the questions touched upon in the article.     

Thermal Fatigue Damage Assessment in an Isotropic Pipe Using Nonlinear Ultrasonic Guided Waves

The use of nonlinear ultrasonic waves has been accepted as a potential technique to characterize the state of material micro-structure in solids. The typical nonlinear phenomenon is generation of second harmonics. Second harmonic generation of ultrasonic waves propagation has been vigorously studied for tracking material micro-damages in unbounded media and plate-like waveguides. However, there are few studies of launching second harmonic guided wave propagation in tube-like structures. Considering that second harmonics could provide useful information sensitive for material degradation condition, this research aims at developing a procedure for detecting second harmonics of ultrasonic guided wave in an isotropic pipe. The second harmonics generation of guided wave propagation in an isotropic and stress-free elastic pipe is investigated. Flexible polyvinylidene fluoride (PVDF) comb transducers are used to measure fundamental wave and second harmonic one. Experimental results show that nonlinear parameters increase monotonically with propagation distance. This work experimentally verifies that the second harmonics of guided waves in pipe have the cumulative effect with propagation distance. The proposed procedure is applied to assessing thermal fatigue damage indicated by nonlinearity in an aluminum pipe. The experimental observation verifies that nonlinear guided waves can be used to assess damage levels in early thermal fatigue state by correlating them with the acoustic nonlinearity.     

多尺度法的推广及在非线性黏弹性系统的应用

黏弹性材料在航空、机械、土木等领域具有广阔的应用前景,而具有1.5自由度的非线性Zener模型能更好地描述其特性.因此,研究多尺度法的推广和应用具有重要意义.在传统多尺度法的基础上,推广并利用多尺度法对非线性奇数阶微分方程进行研究,解决非线性奇数阶系统的动力学求解问题.以非线性Zener模型为例,首先通过推广的多尺度法...     

Interesting effects in harmonic generation by plane elastic waves

The harmonics of plane longitudinal and trans-verse waves in nonlinear elastic solids with up to cubic nonlinearity in a one-dimensional setting are investigated in this paper. It is shown that due to quadratic nonlinearity, a transverse wave generates a second longitudinal harmonic. This propagates with the velocity of transverse waves, as well as resonant transverse first and third harmonics due to the cubic and quadratic nonlinearities. A longitudinal wave generates a resonant longitudinal second harmonic, as well as first and third harmonics with amplitudes that increase linearly and quadratically with distance propagated. In a second investigation, incidence from the linear side of a pri-mary wave on an interface between a linear and a nonlinear elastic solid is considered. The incident wave crosses the interface and generates a harmonic with interface conditions that are equilibrated by compensatory waves propagating in two directions away from the interface. The back-propagated compensatory wave provides information on the nonlinear elastic constants of the material behind the interface. It is shown that the amplitudes of the compensatory waves can be increased by mixing two incident longitudinal waves of appropriate frequencies.     

Resonant excitation op long waves in a two-layer medium by variable pressure on the free surface

The amplitudes of the stationary internal waves are estimated for exact resonance. The dependence of the amplitudes on the densities and depths of the layers is investigated. It is shown that dispersion considerably reduces the amplitude of the stationary waves. In this case higher harmonics appear in the solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 90–98, March–April. 1990.     

Study for ball bearing outer race characteristic defect frequency based on nonlinear dynamics analysis

The characteristic defect frequencies are widely used for diagnosing the local defect of the ball bearing. The varying compliance (VC) frequency of a fault-free rotor–bearing system equals to the BPFO (ball bearing outer race defect frequency) due to the internal kinematic relationship of a bearing assembly. In order to indicate this issue, a semi-analytical method—the harmonic balance method with alternating frequency/time domain technique—is exploited to obtain the solutions of rotor–ball bearing systems with /without an outer race defect. The solutions and the features of a rotor–ball bearing system with essentially nonlinear parametric excitation are analyzed. We prove the VC frequency equals the BPFO and explain the reasons that the harmonics of the characteristic defect frequency generally appear in the frequency domain. The VC, BPFO as well as their harmonics affected by the primary and super-harmonic resonance of the system are found out. Finally, a test rig of a rigid rotor–bearing system is established to verify the theoretical analysis qualitatively by presenting the performance of VC, BPFO and their harmonics in the frequency domain. In addition, the tests are accomplished in a cycle of running up and down to reveal the primary and super-harmonic resonance characteristics. On the basis of the theoretical and experimental results, the basic BPFO is not enough to judge an outer race defect. The discussion on frequency spectrum, the primary and super-harmonic resonance provides a more reliable way to elucidate the characteristic defect frequencies.     

Discriminating linear from nonlinear elastic damage using a nonlinear time reversal DORT method

The DORT method is a selective detection and focusing technique originally developed to detect defects and damages which induce linear changes of the elastic moduli. It is based on the time reversal (TR) where a signal collected from an array of transducers is time reversed and then back-propagated into the medium to obtain focusing on selected targets. TR is based on the principle of spatial reciprocity. Attenuation, dispersion, multiple scattering, mode conversion, etc. do not break spatial reciprocity. The presence of defects or damage, may cause materials to show nonlinear elastic wave propagation behavior that will break spacial reciprocity. Therefore the DORT method will not allow focusing on nonlinear elastic scatterers. This paper presents a new method for the detection and identification of multiple linear and nonlinear scatterers by combining nonlinear elastic wave spectroscopy, time reversal and DORT method. In the presence of nonlinear hysteretic elastic scatterers, forcing the solid with a harmonic excitation, the time reversal operator can be obtained not only at the fundamental frequency of excitation, but also at the odd harmonics. At the fundamental harmonic, either inhomogeneities and linear damages can be individually selected but only at odd harmonics nonlinear hysteretic elastic damages exist. A procedure was developed where by decomposing the operator at the odd harmonics, it was possible to focus on nonlinear scatterers and to differentiate them from the linear inhomogeneities. A complete mathematical nonlinear DORT formulation for 1 and 2D structures is presented. To model the presence of nonlinear elastic hysteretic scatterers a Preisach–Mayergoyz (PM) material constitutive model was used. Results relative to 1 and 2 dimensional structures are reported showing the capability of the method to focus and discern selectively linear and nonlinear scatterers. Furthermore, an analysis was conducted to study the influence of the number of sources and their location on the imaging process showing that using a higher numbers of sensors does not automatically bring to a minor uncoupled behaviour between the nonlinear targets.     

Uniform Expansions of Periodic Solutions for the Third Superharmonic Resonance

A new method of uniform expansions of periodic solutions to ordinary differential equations has recently been proposed to study quasi-harmonic processes in non-linear dynamical systems, in particular, when a small parameter of non-linearity is absent. The main idea of the method consists in using the ratio of the amplitudes of higher harmonics to the amplitude of the first harmonic of a periodic solution as a small formal parameter that appears due to descending the amplitudes of harmonics monotonically with increasing their number (this is the condition that the term quasi-harmonic implies). In this paper, the method is generalized for the third superharmonic resonance (when the first and the third harmonics become of the same magnitude) in a harmonically forced oscillator with arbitrary odd polynomial non-linearity.     

On the local stability analysis of the approximate harmonic balance solutions

Periodic response of nonlinear oscillators is usually determined by approximate methods. In the "steady state" type methods, first an approximate solution for the steady state periodic response is determined, and then the local stability of this solution is determined by analyzing the equation of motion linearized about this predicted "solution". An exact stability analysis of this linear variational equation can provide erroneous stability type information about the approximate solutions. It is shown that a consistent stability type information about these solutions can be obtained only when the linearized variational equation is analyzed by approximate methods, and the level of accuracy of this analysis is consistent with that of the approximate solutions. It is demonstrated that these consistent stability results do not imply that the approximate solution is qualitatively correct. It is also shown that the difference between an approximate and the next higher order stability analysis can be used to "guess" the role of higher harmonics in the periodic response. This trial and error procedure can be used to ensure the qualitatively correct and numerically accurate nature of the approximate solutions and the corresponding stability analysis.     

Three-wave resonance interaction of disturbances in a boundary layer

The three-frequency resonance of Tolman-Schlichting waves, one of which propagates along the stream while the other two propagate at adjacent angles to it, is investigated as a function of the spectrum and initial intensity in incompressible flows of the boundary-layer type within the framework of a weakly nonlinear theory. In the parallel-flow approximation such an interaction leads to the formation of unstable self-oscillations. The spatial evolution of the associated disturbances is studied with allowance for the self-similar deformation of the velocity profile of the main flow. It is shown that such development can lead to a sharp amplification of the oscillations, primarily of those propagating at an angle to the flow. The role of the effects under consideration in the transitional process and the connection with experimental data are discussed. As experiments [1, 2] show, in the process of a transition from a laminar boundary layer to a turbulent region, well described by the linear theory of hydrodynamic stability, there first comes a section of the excitation of harmonics of a Tolman-Schlichting wave, the appearance of three-dimensional structures, and a rapid growth in the intensity of low-frequency oscillations. There is no doubt that in this section the phenomena are dependent on the nonlinear character of the development with disturbances. The resonance interaction of wave triads can play an important role in this. For small enough amplitudes such an interaction is described by a first-order theory [3, 4], and in the general case the nonlinear effects associated with them should occur sooner than others. The importance of resonance triads for the explanation of the development of three-dimensional structures in a layer and the generation of intense pulsations has already been emphasized in [5, 6]. The clarification of the properties of the evolution of resonantly interacting disturbances therefore is important for an understanding of this transitional process.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 78–84, September–October, 1978.The authors thank V. Ya. Levchenko for a discussion of the work.