CORRECTED SOLVABILITY CONDITIONS FOR NON-LINEAR ASYMMETRIC VIBRATIONS OF A CIRCULAR PLATE

An investigation into non-linear asymmetric vibrations of a clamped circular plate under a harmonic excitation is made. We re-examined a primary resonance studied by Sridhar, Mook and Nayfeh, in which the frequency of excitation is near the natural frequency of an asymmetric mode of the plate. We corrected their solvability conditions and found that in the absence of internal resonance, the steady state response can have not only the form of standing wave but also the form of travelling wave, which is a remarkable contrast to their conclusion.     

Electric impedance and amplitude-frequency response of a one-dimensional ultrasonic liquid-filled resonator with flat piezoelectric plates

The impedance method is used to determine the electric impedance of a resonator. The amplitude-frequency response of a one-dimensional liquid-filled ultrasonic resonator is calculated by directly solving the wave equations and piezoelectric effect equations under the corresponding boundary conditions. An analysis of the amplitude-frequency response shows that the simple analytical expression obtained from the aforementioned solution is in good agreement with experimental data. An anomalous variation of the electric current in the radiating piezoelectric plate versus the excitation frequency is theoretically revealed near the high-Q resonance peaks. This effect is confirmed experimentally. It gives rise to errors in the measured absorption coefficient and multiply broadens the resonance peaks when the measurements are performed near the resonance frequencies of the piezoelectric plates.     

Parametric excitation of two internally resonant oscillators

The response of two-degree-of-freedom systems with quadratic non-linearities to a principal parametric resonance in the presence of two-to-one internal resonances is investigated. The method of multiple scales is used to construct a first-order uniform expansion yielding four first-order non-linear ordinary differential (averaged) equations governing the modulation of the amplitudes and the phases of the two modes. These equations are used to determine steady state responses and their stability. When the higher mode is excited by a principal parametric resonance, the non-trivial steady state value of its amplitude is a constant that is independent of the excitation amplitude, whereas the amplitude of the lower mode, which is indirectly excited through the internal resonance, increases with the amplitude of the excitation. However, in addition to Poincaré-type bifurcations, this response exhibits a Hopf bifurcation leading to amplitude- and phase-modulated motions. When the lower mode is excited by a principal parametric resonance, the averaged equations exhibit both Poincaré and Hopf bifurcations. In some intervals of the parameters, the periodic solutions of the averaged equations, in the latter case, experience period-doubling bifurcations, leading to chaos.     

Improvement of the semi-analytical method, based on Hamilton's principle and spectral analysis, for determination of the geometrically non-linear response of thin straight structures. Part III: steady state periodic forced response of rectangular plates

In Parts I and II of this series of papers, a practical simple “multi-mode theory”, based on the linearization of the non-linear algebraic equations, written on the modal basis, in the neighbourhood of each resonance, has been developed for beams and fully clamped rectangular plates.¹ Simple explicit formulae have been derived, which allowed, via the so-called first formulation, direct calculation of the basic function contributions to the first three non-linear mode shapes of clamped-clamped and clamped-simply supported beams, and the two first non-linear mode shapes of FCRP. Also, in Part I of this series of papers, this approach has been successively extended, in order to determine the amplitude-dependent deflection shapes associated with the non-linear steady state periodic forced response² of clamped-clamped beams, excited by a concentrated or a distributed harmonic force in the neighbourhood of the first resonance.This new approach has been applied in the present work to obtain the NLSSPFR formulation for FCRP, SSRP, and CCCSSRP, leading in each case to a non-linear system of coupled differential equations, which may be considered as a multi-dimensional form of the well-known Duffing equation. The single-mode assumption, and the harmonic balance method, have been used for both harmonic concentrated and distributed excitation forces, leading to one-dimensional non-linear frequency response functions of the plates considered. Comparisons have been made between the curves based on these functions, and the results available in the literature, showing a reasonable agreement, for finite but relatively small vibration amplitudes. A more accurate estimation of the FCRP non-linear frequency response functions has been obtained by the extension of the improved version of the semi-analytical model developed in Part I for the NLSSPFR of beams, to the case of FCRP, leading to explicit analytical expressions for the “multi-dimensional non-linear frequency response function”, depending on the forcing level, and the amplitude of the response induced in the range considered for the excitation frequency.     

Transition scenario to turbulence in thin vibrating plates

A thin plate, excited by a harmonic external forcing of increasing amplitude, shows transitions from a periodic response to a chaotic state of wave turbulence. By analogy with the transition to turbulence observed in fluid mechanics as the Reynolds number is increased, a generic transition scenario for thin vibrating plates, first experimentally observed, is here numerically studied. The von Kármán equations for thin plates, which include geometric non-linear effects, are used to model large amplitude vibrations, and an energy-conserving finite difference scheme is employed for discretisation. The transition scenario involves two bifurcations separating three distinct regimes. The first regime is the periodic, weakly non-linear response. The second is a quasiperiodic state where energy is exchanged between internally resonant modes. It is observed only when specific internal resonance relationships are fulfilled between the eigenfrequencies of the structure and the forcing frequency; otherwise a direct transition to the last turbulent state is observed. This third, or turbulent, regime is characterized by a broadband Fourier spectrum and a cascade of energy from large to small wavelengths. For perfect plates including cubic non-linearity, only third-order internal resonances are likely to exist. For imperfect plates displaying quadratic nonlinearity, the energy exchanges and the quasiperiodic states are favored and thus are more easily obtained. Finally, the turbulent regime is characterized in the light of available theoretical results from wave turbulence theory.     

The response of two-degree-of-freedom systems with quadratic non-linearities to a combination parametric resonance

The response of two-degree-of-freedom systems with quadratic non-linearities to a combination parametric resonance in the presence of two-to-one internal resonances is investigated. The method of multiple scales is used to construct a first order uniform expansion yielding four first order non-linear ordinary differential equations governing the modulation of the amplitudes and the phases of the two modes. Steady state responses and their stability are computed for selected values of the system parameters. The effects of detuning the internal resonance, detuning the parametric resonance, the phase and magnitude of the second mode parametric excitation, and the initial conditions are investigated. The first order perturbation solution predicts qualitatively the trivial and non-trivial stable steady state solutions and illustrates both the quenching and saturation phenomena. In addition to the steady state solutions, other periodic solutions are predicted by the perturbation amplitude and phase modulation equations. These equations predict a transition from constant steady state non-trivial responses to limit cycle responses (Hopf bifurcation). Some limit cycles are also shown to experience period doubling bifurcations. The perturbation solutions are verified by numerically integrating the governing differential equations.     

Theory and experiments for large-amplitude vibrations of empty and fluid-filled circular cylindrical shells with imperfections

The large-amplitude response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of some of the lowest natural frequencies is investigated. Donnell's non-linear shallow-shell theory is used and the solution is obtained by the Galerkin method. Several expansions involving 16 or more natural modes of the shell are used. The boundary conditions on the radial displacement and the continuity of circumferential displacement are exactly satisfied. The effect of internal quiescent, incompressible and inviscid fluid is investigated. The non-linear equations of motion are studied by using a code based on the arclength continuation method. A series of accurate experiments on forced vibrations of an empty and water-filled stainless-steel shell have been performed. Several modes have been intensively investigated for different vibration amplitudes. A closed loop control of the force excitation has been used. The actual geometry of the test shell has been measured and the geometric imperfections have been introduced in the theoretical model. Several interesting non-linear phenomena have been experimentally observed and numerically reproduced, such as softening-type non-linearity, different types of travelling wave response in the proximity of resonances, interaction among modes with different numbers of circumferential waves and amplitude-modulated response. For all the modes investigated, the theoretical and experimental results are in strong agreement.     

Non-linear gas oscillations in a pipe

The results of an experimental study of non-linear longitudinal gas oscillations in a closed tube are presented. Forced oscillations greater than in other experiments to date are obtained. A brief representation of the theory holding for excitation frequencies near and equal to the natural frequencies of a gas column is given. The experimental amplitude and the periodic shock wave form are compared with the calculated values. Oscillations whose excitation frequencies are twice as small as the first natural frequency are also studied.     

PARAMETRIC STABILIZATION OF A GYROSCOPIC SYSTEM

This paper studies the stabilization of a gyroscopic system using parametric stabilization near a combination resonance. The gyroscopic system is near its primary instability, i.e., the bifurcation parameter is such that the system possesses a double zero eigenvalue. The stability of the system is studied for the linear Hamiltonian system, the damped linear system, the forced linear Hamiltonian system, and finally the damped and forced linear system. The addition of the periodic excitation near the critical combination resonance provides the system with an extended stability region when the excitation frequency is slightly above the combination resonance. A non-linear numerical example shows that these results may persist for the non-linear problem. The results of this work, are then discussed in relation to an example gyroscopic problem, a rotating shaft with periodically perturbed rotation rate.     

On the dynamic response at the wheel axle of a pneumatic tire

A method for calculating the steady state displacement response and force transmission at the wheel axle of a pneumatic tire-suspension system due to a steady state force or displacement excitation at the tire to ground contact point is developed. The method requires the frequency responses (or receptances)_of both tire-wheel and suspension units. The frequency response of the tire-wheel unit is obtained by using the modal expansion method. The natural frequencies and mode shapes of the tire-wheel unit are obtained by using a geometrically non-linear, ring type, thin shell finite element of laminate composite. The frequency response of the suspension unit is obtained analytically. These frequency responses are used to calculate the force-input and the displacement-input responses at the wheel axle. This method allows the freedom of designing a vehicle and its tires independently and still achieving optimum dynamic performance.     

Non-linear structural vibrations under combined parametric and external excitations

A set of second order equations with weak quadratic and cubic non-linearities is considered. Simultaneous parametric and external (forcing) excitations are included. The frequency of the parametric excitation is near a natural frequency of the system, and three cases are analyzed: (i) the external excitation is absent; (ii) the external excitation is present but is not involved in a resonance; and (iii) the external frequency is the same as the parametric frequency. Results are obtained by the method of multiple scales. Frequency-response curves are presented for various combinations of excitation amplitudes, damping coefficients, and phase shift between the excitations. It is found that stable multi-modal responses may exist in the first-order asymptotic solution, even though only one mode is involved in the resonance and no internal resonance condition is present.     

Dynamic response of composite plates with cut-outs,part II: Clamped-clamped plates

The forced and free dynamic response of plates with cut-outs formulated in Part I [1] is used to investigate the effect of cut-outs on the natural frequencies of clamped-clamped plates. The size, shape and location of the cut-out is expressed as a displacement dependent external loading. The plates considered are homogeneous and anisotropic. Lagrange's equations of motion lead to an infinite system of differential equations in time-dependent generalized co-ordinates with generalized forces which include the effects of the cut-outs. There is an infinite system of frequency equations for free vibrations. The infinite system is truncated to a finite system of equations depending upon the accuracy desired in frequency values. Results are given for square, clamped-clamped plates with centrally located square cut-outs for different modulus ratios. Good agreement is obtained when results for isotropic plates with cut-outs are compared with available theoretical and experimental results.     

Finite-amplitude waves in isotropic elastic plates

The propagation of finite-amplitude waves in a homogeneous, isotropic, stress-free elastic plate is investigated theoretically. Geometric and weak material non-linearities are included, and perturbation is used to obtain solutions of the non-linear equations of motion for harmonic generation in the waveguide. Solutions for the second-harmonic, sum, and difference-frequency components are obtained via modal decomposition. Ordinary differential equations for the modal amplitudes in the expansion of the second-order solution are obtained using a reciprocity relation. There are no restrictions on the modes or frequencies of the primary waves. Two conditions for internal resonance are quantified: phase matching, and transfer of power from the primary to the secondary wave.     

Wave Resonance in Media with Modular,Quadratic and Quadratically-Cubic Nonlinearities Described by Inhomogeneous Burgers-Type Equations

The phenomenon of “wave resonance” which occurs at excitation of traveling waves in dissipative media possessing modular, quadratic and quadratically-cubic nonlinearities is studied. The mathematical model of this phenomenon is the inhomogeneous (or “forced”) equation of Burgers type. Such nonlinearities are of interest because the corresponding equations admit exact linearization and describe real physical objects. The presence of “accompanying sources” (traveling with the wave) on the right-hand side of the inhomogeneous equations ensures the inflow of energy into the wave, which thereafter spreads throughout the wave profile, flows to emerging shock fronts, and then dissipates due to linear and nonlinear losses. As an introduction, the phenomenon of wave resonance in ideal and dissipative media is described and physical examples are given. Exact expressions for nonlinear steady-state wave profiles are derived. Non-stationary processes of wave generation, spatial “beating” of amplitudes with different relationship between the speed of motion of the sources and the natural wave velocity in the medium are studied. Resonance curves are constructed that contain a nonlinear shift of the absolute maxima to the “supersonic” region. The features of the resonance in each of the three types of nonlinearity are discussed.     

The effect of the number of nodal diameters on non-linear interactions in two asymmetric vibration modes of a circular plate

In order to investigate the effect of the number of nodal diameters on non-linear interactions in asymmetric vibrations of a circular plate, a primary resonance of the plate is considered. The plate is assumed to have an internal resonance in which the ratio of the natural frequencies of two asymmetric modes is three to one. The response of the plate is expressed as an expansion in terms of the linear, free oscillation modes, and its amplitude is considered to be small but finite, and the method of multiple scales is used. In view of the corrected solvability conditions for the responses, it has been found that in order for the modes to interact, the ratio of the numbers of nodal diameters of two modes must be either three to one or one to one. In this study the one-to-one case, in which the modes have the same number of nodal diameters, is examined. The non-linear governing equations are reduced to a system of autonomous ordinary differential equations for amplitude and phase variables by means of the corrected solvability conditions. The steady state responses and their stability are determined by using this system. The result shows very complicated interactions between two modes by telling existence of non-vanishing amplitudes of the mode not directly excited.     

Asymmetric non-linear forced vibrations of free-edge circular plates. Part II: experiments

This article is devoted to an experimental validation of a theoretical model presented in an earlier contribution by the same authors. The non-linear forced vibrations of circular plates, with the excitation frequency close to the natural frequency of an asymmetric mode, are investigated. The experimental set-up, which allows one to perform precise measurements of the vibration amplitudes of the two preferential configurations, is presented. Experimental resonance curves showing the amplitude and the phase of each configuration as functions of the driving frequency are compared to the theoretical ones, leading to a quantitative validation of the predictions given by the model. Finally, all the approximations used are systematically discussed, in order to show the scope and relevance of the approach.     

有界大洋对纬向风强迫的响应及共振

为揭示北太平洋主、次要气候模态即太平洋年代际振荡(PDO)和北太平洋流涡振荡(NPGO)的形成机理及其振荡周期与大洋水平尺度之间的联系,采用中纬β通道中的约化重力准平衡线性大洋模型,解析求解了纬向风强迫下的大洋流场响应,讨论了其中的共振问题.1)有界大洋的响应形态分别类似于冬季PDO和NPGO的流场模.2)响应形态分别表现为在大洋西海岸以东,前者有一个椭圆状流涡,后者有南北两个旋转方向相反的流涡并构成流涡偶;在整个大洋,前者有一个洋盆尺度环流,后者在大洋南北分别有两个旋转方向相反的洋盆尺度环流;在中纬度西风急流异常位置偏北和偏南,则能分别强迫出以上的两种情况.3)大洋流场对纬向风场强迫的响应频率(周期)与纬向风强迫频率(周期)相同,但大洋响应要滞后于纬向风的强迫;而响应流场即流函数的强度则与纬向风强迫的大小成正比.当纬向风强迫频率(周期)与该大洋固有频率(周期)相同时,二者会有共振发生,此时大洋响应最为强烈;而二者频率(周期)相差较远时,响应则不大.摩擦越小共振就越强,共振的个数也越多.有界大洋东西向的长度对其固有频率(周期)即共振频率(周期)有明显影响,并起着决定作用;当该长度减小时,相邻两个共振周期的间隔会增大.海洋大气间的两两非线性相互作用,使得随机风场的振荡包含了从极低频到高频的各种成分;通过该共振,可从中挑选出与大洋固有频率相同或相近的共振频率,在该频率上流场对风场的响应最为强烈,从而也就锁定了PDO和NPGO的周期.最终结论为:非线性相互作用、风场对流场的强迫、共振是造成PDO和NPGO的三个关键因子;该解析解的性质为时变的共振Rossby波.     

DYNAMIC RESPONSE OF TWO ROTORS CONNECTED BY RIGID MECHANICAL COUPLING WITH PARALLEL MISALIGNMENT

The effect of parallel misalignment on the lateral and torsional responses of two rotating shafts (Jeffcott rotors) is examined with theoretical and numerical analysis. The general equations of motion are derived and given in dimensionless form to represent the general case. The equations of motion revealed that parallel misalignment couples the translation and angular deflections through the stiffness matrix and the force vector. The non-linear equations are solved numerically using a combination of Newmark and Newton-Raphson methods to determine the dimensionless frequency and transient responses in terms of misalignment magnitude. The numerical results show that the system natural frequencies are excited at transient condition due to the presence of pure parallel misalignment. At steady state condition, the 1×-rotational speed excitation is present in the translation and angular directions, which indicates that parallel misalignment can be a source of both torsional and lateral excitations.     

Improved frequency resolution from transient tests with short record lengths

A method is described for improving the accuracy of the natural frequencies obtained from the Fourier transform of the structural response to an impulse. Results are presented from tests in which the input was at a single frequency and from impulse tests on an aluminium plate. It is shown to be possible to obtain frequency resolution of one-tenth of the spacing between the frequency points produced by the Fourier transform, at a low cost in terms of computer time and store. The natural frequencies of the aluminium plate obtained by this method are also compared with those measured when using steady state excitation. Excellent agreement is shown between the results obtained by using the two techniques.