“Non-linear normal modes” and the generalized Ritz method in the problems of vibrations of non-linear elastic continuous systems

In the study of natural vibrations of non-linear elastic systems it is shown that the mode shape of the vibration can vary with the amplitude as well as the frequency, and that the amplitude frequency relation is strongly affected by constraints imposed on the mode shape in an approximate solution. A method is developed which assumes the approximate solution in the form of a truncated series in which, instead of the set of coefficients, the set of functions of spatial variables is unknown and then determined by a procedure that can be regarded as a generalization of the Ritz method. The problem of variations of the normal mode shapes and of the associated natural frequencies with the amplitude is illustrated by two examples of beams with non-linear boundary conditions, and the amplitude-frequency relation is compared to that corresponding to the a priori assumed linear normal mode solution. Further possible consequences of the mode shape amplitude variations in forced, resonant motion of nonconservative systems are also indicated.     

Array measurements of the surface pressure beneath a forced axi-symmetric separation bubble

An array of microphones is used to study the space–time characteristics of the wall-pressure field beneath a forced separation bubble downstream of an axi-symmetric backward-facing step. To excite the flow, an externally driven Helmholtz resonator is employed. A unique aspect of the present study is the utilization of an amplitude-modulated forcing scheme in order to avoid contamination of the measured hydrodynamic pressure fluctuations by acoustic radiation from the forcing device. The results lead to the hypothesis that the optimal forcing frequency is achieved when the forced disturbance originates near the center of the unforced separation bubble in the limit of very low levels of forcing. Moreover, a frequency–wavenumber spectrum analysis highlights the possibility for achieving separation control while minimizing potential acoustic radiation due to coupling between the forced disturbance and resonant modes of the underlying surface.     

Non-linear dynamics of the sandwich double circular plate system

Multi-frequency vibrations of a system of two isotropic circular plates interconnected by a visco-elastic layer that has non-linear characteristics are considered. The considered physical system should be of interest to many researches from mechanical and civil engineering. The first asymptotic approximation of the solutions describing stationary and no stationary behavior, in the regions around the two coupled resonances, is the principal result of the authors. A series of the amplitude-frequency and phase-frequency curves of the two frequency like vibration regimes are presented. That curves present the evolution of the first asymptotic approximation of solutions for different non-linear harmonics obtained by changing external excitation frequencies through discrete as well as continuous values. System of the partial differential equations of the transversal oscillations of the sandwich double circular plate system with visco-non-linear elastic layer, excited by external, distributed, along plate surfaces, excitation are derived and approximately solved for various initial conditions and external excitation properties. System of differential equations of the first order with respect to the amplitudes and the corresponding number of the phases in the first asymptotic averaged approximation are derived for different corresponding multi-frequency non-linear vibration regimes. These equations are analytically and numerically considered in the light of the stationary and no stationary resonant regimes, as well as the multi-non-linear free and forced mode mutual interactions, number of the resonant jumps.     

Potential benefits of a non-linear stiffness in an energy harvesting device

The benefits of using a non-linear stiffness in an energy harvesting device comprising a mass–spring–damper system are investigated. Analysis based on the principle of conservation of energy reveals a fundamental limit of the effectiveness of any non-linear device over a tuned linear device for such an application. Two types of non-linear stiffness are considered. The first system has a non-linear bi-stable snap-through mechanism. This mechanism has the effect of steepening the displacement response of the mass as a function of time, resulting in a higher velocity for a given input excitation. Numerical results show that more power is harvested by the mechanism if the excitation frequency is much less than the natural frequency. The other non-linear system studied has a hardening spring, which has the effect of shifting the resonance frequency. Numerical and analytical studies show that the device with a hardening spring has a larger bandwidth over which the power can be harvested due to the shift in the resonance frequency.     

Nonlinear analysis of laser droplet generation by means of 0–1 test for chaos

We apply the recently improved version of the 0–1 test for chaos to real experimental time series of laser droplet generation process. In particular two marginal regimes of dripping are considered: spontaneous and forced dripping. The outcomes of the test reveal that both spontaneous and forced dripping time series can be characterized as chaotic, which coincides with the previous analysis based on nonlinear time series analysis.     

Analyticity and causality of the three-parameter rheological models

In this paper, the basic frequency and time response functions of the three-parameter Poynting–Thomson solid and Jeffreys fluid are revisited. The two rheological models find application in several areas of rheology, structural mechanics, and geophysics. The relation between the analyticity of a frequency response function and the causality of the corresponding time response function is established by identifying all singularities at ω = 0 after applying a partial fraction expansion to the frequency response functions. The strong singularity at ω = 0 in the imaginary part of a frequency response function in association with the causality requirement imposes the addition of a Dirac delta function in the real part in order to make the frequency response function well defined in the complex plane. This external intervention, which was first discovered by PAM Dirac, has not received the attention it deserves in the literature of viscoelasticity and rheology. The addition of the Dirac delta function makes possible the application of time domain techniques that do not suffer from violating the premise of causality.     

Relationship Between Singularities in Classical Mechanics for Complex Initial Conditions and for Complex Time

A relationship is established between the functional forms of two kinds of singularities in dynamical variables that arise in complexified versions classical mechanics: singularities that are treated as a functions of complex initial conditions for real time and those that are treated as a functions of complex time for real initial conditions. The analysis is verified by numerical calculations. The results imply that Kowaleskaya–Painlevé condition for integrability can be phrased in terms of singularities with respect to initial conditions.     

On the “bead,hoop and spring” (BHS) dynamical system

Here, we make the theoretical and numerical analysis of the non-linear equation describing the evolution of the “bead, hoop and spring” (BHS) dynamical system derived by Ochoa and Clavijo in (Eur. J. Phys. 27:1277–1288, 2006). In particular, we solve by standard techniques of non-linear physics an approximation of their equation neglecting the centrifugal effect before giving a more mathematical and exact treatment. The analogy with phase transitions is underlined. We point out the existence of finite-time singularities in the phase-space and we derive a criterion for possible oscillations.     

Isolated Singularities of the 1D Complex Viscous Burgers Equation

The Cauchy problem for the 1D real-valued viscous Burgers equation u _t +uu _x  = u _xx is globally well posed (Hopf in Commun Pure Appl Math 3:201–230, 1950). For complex-valued solutions finite time blow-up is possible from smooth compactly supported initial data, see Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008). It is also proved in Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008) that the singularities for the complex-valued solutions are isolated if they are not present in the initial data. In this paper we study the singularities in more detail. In particular, we classify the possible blow-up rates and blow-up profiles. It turns out that all singularities are of type II and that the blow-up profiles are regular steady state solutions of the equation.     

Broadband passive targeted energy pumping from a linear dispersive rod to a lightweight essentially non-linear end attachment

We examine non-linear resonant interactions between a damped and forced dispersive linear finite rod and a lightweight essentially non-linear end attachment. We show that these interactions may lead to passive, broadband and on-way targeted energy flow from the rod to the attachment, which acts, in essence, as non-linear energy sink (NES). The transient dynamics of this system subject to shock excitation is examined numerically using a finite element (FE) formulation. Parametric studies are performed to examine the regions in parameter space where optimal (maximal) efficiency of targeted energy pumping from the rod to the NES occurs. Signal processing of the transient time series is then performed, employing energy transfer and/or exchange measures, wavelet transforms, empirical mode decomposition and Hilbert transforms. By computing intrinsic mode functions (IMFs) of the transient responses of the NES and the edge of the rod, and examining resonance captures that occur between them, we are able to identify the non-linear resonance mechanisms that govern the (strong or weak) one-way energy transfers from the rod to the NES. The present study demonstrates the efficacy of using local lightweight non-linear attachments (NESs) as passive broadband energy absorbers of unwanted disturbances in continuous elastic structures, and investigates the dynamical mechanisms that govern the resonance interactions influencing this passive non-linear energy absorption.     

The role of buoyancy–frequency oscillations in the generation of mountain gravity waves

There is evidence from balloon measurements that atmospheric buoyancy–frequency profiles, apart from a sharp increase (roughly by a factor of two) at the tropopause, often feature appreciable oscillations (typical wavelength 1–2 km) with altitude. It is argued here that such short-scale oscillatory variations of the background buoyancy frequency, which usually are ignored in theoretical models, can have a profound effect on the generation of mountain waves owing to a resonance mechanism that comes into play at certain wind speeds depending on the dominant oscillation wavelength. A simple linear model assuming small sinusoidal buoyancy–frequency oscillations suggests, and numerical solutions of the Euler equations for more realistic flow conditions confirm, that under resonant conditions the induced gravity-wave activity is significantly increased above and upstream of the mountain, similarly to resonant flow of finite depth over topography.      

The study of self-sustained oscillating plane jet flow impinging upon a small cylinder

 Experimental studies of a plane jet impinging upon a small circular cylinder are conducted by hot-wire measurements. The cylinder is located on the jet centerline within the potential-core region. The jet–cylinder interactions on the instability shear layer frequency, the cylinder wake shedding frequency, and the induced self-sustained oscillation phenomenon are carefully investigated. Test data indicate that the self-sustained flow oscillation is mainly generated by the resonant effect of the flow between the jet exit and the cylinder. Its resonant frequency is found to vary linearly and exhibits jump-stage pattern as a function of the distance between the jet exit and the cylinder. The feedback mechanism and the hydrodynamic instability theorem are proposed to predict correctly the frequency jump position, wave number and the convection speed of the self-sustained oscillating flow for different jet exit velocities. Received: 15 July 1998/Accepted: 9 December 1998     

Second-order hydroelastic analysis of a floating plate in multidirectional irregular waves

Based on an analysis of second-order hydrodynamic forces induced by coupling of first-order wave potentials, second-order hydroelastic equations are established and solved in the frequency domain. The responses of a very large floating structure in multidirectional irregular waves are studied. The characteristics of the difference and sum frequency coordinates are discussed in detail; peaks can be found at the difference and sum frequencies close to the wet resonant frequencies of each mode. We present and analyze the maximum vertical displacement of different points as well as the time history of the vertical displacements of selected points. The differences of the combined (the summation of the linear and non-linear responses) and linear displacements of the selected points are calculated. Our results demonstrate that non-linear fluid forces influence the total responses of the referenced floating structure.     

参数振动系统响应的频谱成分及其分布规律

采用Sylvester理论和Fourier级数展开方法分别研究了参数振动系统自由响应和强迫响应的频谱特性(频谱成分及其分布规律),讨论了系统稳定性和阻尼对于频谱幅值的影响,并给出了系统外激励共振条件. 理论研究结果表明:由于参数激励作用使得系统响应具有多频特点,这些频谱成分与系统固有频率、参数激励频率和外激励频率具有密切联系,而且其在频域分布也呈现出一定的规律. 此外,参数振动系统具有多个外激励共振点,除了外激励频率等于系统固有频率将发生共振外,当外激励频率等于系统固有频率和参数激励频率的组合值时,同样将发生外激励共振现象.      

Measurement of gas transfer across wind-forced wavy air–water interfaces using laser-induced fluorescence

Optical distortions have previously prevented non-intrusive measurements of dissolved oxygen concentration profiles by Laser induced fluorescence (LIF) to within 200 μm of the air–water interface. It is shown that by careful experimental design, reliable measurements can be obtained within 28 μm of moving air–water interfaces. Consideration of previously unidentified optical distortions in LIF imagery due to non-linear effects is presented that is critical for robust LIF data processing and experimental design. Phase resolved gas flux measurements have now been accomplished along wind forced microscale waves and indicate that the highest mean gas fluxes are located in the wave troughs. The local mean oxygen fluxes as determined by LIF techniques can be reconciled to within 40% of those obtained by bulk measurement in the water. These data provide a new perspective on wind-wave enhancement of low solubility gas transfer across the air–water interface.     

Spectral properties for the vibrational response of parametrically excited system

The response spectra are helpful to understand the vibrational characteristics of linear system, and further could be used as important indications for the failure diagnosis and state detecting of the system. Due to the parametric excitation, the vibrational spectra of linear periodic system differ distinctly from that of the linear time-invariant system. Thus, utilizing the Sylvester’s theorem and Fourier series approximations, commonly used in the matrix decomposition, the spectral components for both the free and forced responses of linear parametrically excited system are obtained theoretically in the first part of the paper. The effects of system stability and damping on the response spectra are discussed in detail. Moreover, the external resonant condition for periodic system is found in the analysis. In the following part of the paper, spectral simulations for a spur gear pair under idle load (modeled as the free response) and fluctuating load (modeled as forced response) are done respectively to validate the theoretical results and illustrate the special frequency distributing features of periodic system intuitively.     

The active control of vibrations of composite beams by parametric stiffness modulation

The paper addresses an active control of the resonant vibrations of composite beams performed by a parametric stiffness modulation. A sandwich beam composition with the continuous core is considered. The stiffness modulation is introduced by some fairly small changes in an orientation of elements of the microstructure of a core ply. The controlled vibrations are those of the dominantly flexural type excited by a transverse force acting at a low resonant frequency, whereas the stiffness modulation is performed at a comparatively high frequency identified by the resonance of a mode of the dominantly shear type. This difference in time scales of the controlled vibrations and the input signal facilitates a use of the method of direct partition of motion that predicts an existence of the modal interaction between the low-frequency and the high-frequency motions due to so-called vibrational forces. It is shown that such a parametric control can provide a significant favourable shift of the first eigenfrequency of a controlled beam (the one subjected to the stiffness modulation) from its nominal value for an uncontrolled beam. Heavy fluid loading conditions are accounted for as well as material losses in a structure. Then instead of analysis of eigenfrequencies, a problem of forced vibrations is posed and the forced frequency–amplitude response is analysed. It is demonstrated that although heavy fluid loading reduces resonant frequencies of forced vibrations, the suggested mechanism of control remains valid in these cases.     

On the pressure and stress singularities induced by steady flows of incompressible viscous fluids

Design for structural integrity requires an appreciation of where stress singularities can occur in structural configurations. While there is a rich literature devoted to the identification of such singular behavior in solid mechanics, to date there has been relatively little explicit identification of stress singularities caused by fluid flows. In this study, stress and pressure singularities induced by steady flows of viscous incompressible fluids are asymptotically identified. This is done by taking advantage of an earlier result that the Navier-Stokes equations are locally governed by Stokes flow in angular corners. Findings for power singularities are confirmed by developing and using an analogy with solid mechanics. This analogy also facilitates the identification of flow-induced log singularities. Both types of singularity are further confirmed for two global configurations by applying convergence-divergence checks to numerical results. Even though these flow-induced stress singularities are analogous to singularities in solid mechanics, they nonetheless render a number of structural configurations singular that were not previously appreciated as such from identifications within solid mechanics alone.     

Optimal control of MHD-flow deceleration

A problem of pulsed control for a three-dimensional magnetohydrodynamic (MHD) model is considered. It is demonstrated that singularities of the solution of MHD equations do not develop with time because they are suppressed by a magnetic field. The existence of an optimal control is proved. An optimality system with the solution regular in time as a whole is constructed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 3–10, September–October, 2008.     

SRRs Embedded with MEMS Cantilevers to Enable Electrostatic Tuning of the Resonant Frequency

A microelectromechanical systems (MEMS) cantilever array was monolithically fabricated in the gap region of a split ring resonator (SRR) to enable electrostatic tuning of the resonant frequency. The design consisted of two concentric SRRs each with a set of cantilevers extending across the split region. The cantilever array consisted of five beams that varied in length from 300 to 400 μm, with each beam adding about 2 pF to the capacitance as it actuated. The entire structure was fabricated monolithically to reduce its size and minimize losses from externally wire bonded components. The beams actuate one at a time, longest to shortest with an applied voltage ranging from 30–60 V. The MEMS embedded SRRs displayed dual resonant frequencies at 7.3 and 14.2 GHz or 8.4 and 13.5 GHz depending on the design details. As the beams on the inner SRR actuated the 14.2 GHz resonance displayed tuning, while the cantilevers on the outer SRR tuned the 8.4 GHz resonance. The 14.2 GHz resonant frequency shifts 1.6 GHz to 12.6 GHz as all the cantilevers pulled-in. Only the first two beams on the outer cantilever array pulled-in, tuning the resonant frequency 0.4 GHz from 8.4 to 8.0 GHz.