# Showing papers in "Journal of Sound and Vibration in 1986"

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TL;DR: In this paper, a set of beam characteristic orthogonal polynomials in the Rayleigh-Ritz method was used to obtain the natural frequencies of rectangular plates. But the results for lower modes were not as good as those obtained by other methods.

Abstract: Natural frequencies of rectangular plates are obtained by employing a set of beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. The orthogonal polynomials are generated by using a Gram-Schmidt process, after the first member is constructed so as to satisfy all the boundary conditions of the corresponding beam problems accompanying the plate problems. Natural frequencies obtained by using the orthogonal polynomial functions are compared with those obtained by other methods. The method yields superior results for lower modes, particularly when plates have some of the edges free.

450 citations

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TL;DR: In this paper, a piecewise linear displacement field which allows the contact conditions for the displacements and the transverse shearing stresses at the interfaces to be satisfied simultaneously, the nonlinear (in the von Karman sense) equations of motion for thick multilayered orthotropic plates are developed.

Abstract: Based upon a piecewise linear displacement field which allows the contact conditions for the displacements and the transverse shearing stresses at the interfaces to be satisfied simultaneously, the non-linear (in the von Karman sense) equations of motion for thick multilayered orthotropic plates are developed. Successively, the equations are specified to the linear boundary value problem of the bending and to the linear eigenvalue problems of the undamped vibration and buckling of rectangular plates. In order to assess the accuracy of the proposed theory, the sample problem of the bending, free undamped vibration and buckling of a three-layered, symmetric cross-ply, square plate simply supported on all edges is investigated. For purposes of comparison, numerical results from the exact elasticity theory, the classical lamination (Kirchhoff) theory and the shear deformation theory (Timoshenko and Mindlin) with three different values of the shear correction factor are also presented. It is found that the proposed approach is very efficient in predicting the global responses (deflection, natural frequencies and buckling loads) of thick multilayered plates and models effects, such as the distortion of the deformed normals, not attainable from the classical lamination theory, as well as the shear deformation theory.

353 citations

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TL;DR: Criteria are established to distinguish veerings from crossings in both continuous and discretized models and the existence of curve veering in continuous models is illustrated by presenting the exact solution of an elementary eigenvalue problem.

Abstract: The dependence of eigenvalues on a system parameter is frequently illustrated by a family of loci. When two loci approach each other, they often cross or abruptly diverge. The latter case, called “curve veering”, has been observed in approximate solutions associated with discretized models. The influence of discretization in producing curve veering has raised doubt on the validity of many approximate solutions. The existence of curve veering in continuous models is illustrated by presenting the exact solution of an elementary eigenvalue problem. Veering is then examined in a general eigenvalue problem. Criteria are established to distinguish veerings from crossings in both continuous and discretized models. The application of the criteria is illustrated by examples.

327 citations

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TL;DR: In this paper, the relationship between broadband shock associated noise and screech tones is examined and it is shown that from the spectral characteristics point of view, the screech tone may be regarded as a special case of broadband wave associated noise.

Abstract: It is recognized that the three principal components of the noise of shock containg supersonic jets, namely, the dominant part of turbulent mixing nose, the boradband shock associated noise and the screech tones are all generated directly by the large scale turbulence structures/instability waves of the jet flow. In this paper the relationship between broadband shock associated noise and screech tones is examined. It is shown that from the spectral characteristics point of view, the screech tones may be regarded as special cases of broadband shock associated noise. Theoretical calculations for the peak frequencies of broadband shock associated noise and the fundamental frequencies of screech tones are carried out. These calculations are free from empirical constants. The calculated results are found to agree favorably with experimental measurements.

263 citations

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TL;DR: In this paper, a mathematical model of an unloaded symmetric rotor supported by one rigid and one fluid lubricated bearing is proposed, where the rotor model is represented by generalized (modal) parameters of its first bending mode.

Abstract: A mathematical model of an unloaded symmetric rotor supported by one rigid and one fluid lubricated bearing is proposed. The rotor model is represented by generalized (modal) parameters of its first bending mode. The rotational character of the bearing fluid force is taken into account. The model yields synchronous vibrations due to rotor unbalance as a particular solution of the equations of motion, rotor/bearing system natural frequencies and corresponding self-excited vibrations known as “oil whirl” and “oil whip”. The stability analysis yields rotative speed threshold of stability. The model also gives the evaluation of stability of the rotor synchronous vibrations. In the first balance resonance speed region two more thresholds of stability are encountered. The width of this stability region is directly related to the amount of rotor unbalance. The results of the analysis based on this model stand with very good agreement with field observations of rotor dynamic behavior and the experimental data.

257 citations

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TL;DR: In this paper, the response of a single degree of freedom system with dry friction is considered and the asymptotic stability of the steady state motions and some transient behaviors are also determined.

Abstract: The response of a single degree of freedom system with dry friction is considered. Den Hartog [1] gave an exact solution for the symmetric steady state motions of such a system in 1930. In this paper these results are extended to include a static coefficient of friction different from the dynamic one. More importantly, the asymptotic stability of the steady state motions and some transient behaviors are also determined. It is shown that for positive viscous damping the non-sticking steady state solutions of the same period as the forcing are nearly always asymptotically stable, but that for negative viscous damping, which may arise from aerodynamic forces [2], such motions can become unstable. By using bifurcation theory it is shown that new behaviors, such as aperiodic motions containing two distinct frequency components, can result from such dynamic instabilities. It is also shown that the symmetric motions with two stops per period can be unstable and that pairs of unsymmetric motions are born at the bifurcation points.

246 citations

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TL;DR: In this paper, a physically motivated, continuous microslip model of friction is developed for analyzing the dynamic response of frictionally damped structures in which the friction interface is subjected to high normal loads.

Abstract: A new, physically motivated, continuous microslip model of friction is developed for analyzing the dynamic response of frictionally damped structures in which the friction interface is subjected to high normal loads. By using this model with a single-degree-of-freedom oscillator it is found that incorporating the effects of partial slip of the friction interface can result in significant reductions of the resonant response with respect to that of a system in which only gross slip is allowed. Comparisons with experimental results are presented in a companion paper [1].

173 citations

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TL;DR: In this article, a response function is derived for an infinite, uniform, one-dimensional structure which is subjected to an array of harmonic forces or moments, spaced equidistantly, and which have a constant phase or ratio between any adjacent pair.

Abstract: A response function is found for an infinite, uniform, one-dimensional structure which is subjected to an array of harmonic forces or moments, spaced equidistantly, and which have a constant phase or ratio between any adjacent pair. Receptance functions are derived for these “phased arrays”. They are used to set up a general determinantal equation for the propagation constants of the infinite structure when it is made periodic by the addition of an infinite set of regular constraints. They are also used to set up equations for the response of the structure to a convected harmonic pressure field. The method enables the equations for the propagation constants and for the response to convected loading to be set up with much greater facility than by earlier methods. It only requires a knowledge of the response function of the infinite uninterrupted structure under a single-point harmonic force or moment. The general equation for the propagation constants is used to study (a) a simply supported periodic Timoshenko beam, and (b) a parallel plate with periodic beam-type stiffeners. Some calculated propagation constants are presented and discussed. The periodic plate results are relevant to integrally stiffened skins of the type used in aeroplanes.

172 citations

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Virginia Tech

^{1}TL;DR: In this paper, a mixed shear flexible finite element, with relaxed continuity, is developed for the geometrically linear and nonlinear analysis of layered anisotropic plates, based on a refined higher order theory which satisfies the zero transverse shear stress boundary conditions on the top and bottom faces of the plate and requires no shear correction coefficients.

Abstract: A mixed shear flexible finite element, with relaxed continuity, is developed for the geometrically linear and nonlinear analysis of layered anisotropic plates. The element formulation is based on a refined higher order theory which satisfies the zero transverse shear stress boundary conditions on the top and bottom faces of the plate and requires no shear correction coefficients. The mixed finite element developed herein consists of eleven degrees of freedom per node which include three displacements, two rotations and six moment resultants. The element is evaluated for its accuracy in the analysis of the stability and vibration of anisotropic rectangular plates with different lamination schemes and boundary conditions. The mixed finite element described here for the higher order theory gives very accurate results for buckling loads and natural frequencies.

168 citations

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TL;DR: In this article, orthogonal polynomial functions are used in the Rayleigh-Ritz method to generate results for a number of flexural vibration and buckling problems for rectangular isotropic and orthotropic plates.

Abstract: Series composed of recently proposed, orthogonal polynomial functions are used in the Rayleigh-Ritz method to generate results for a number of flexural vibration and buckling problems for rectangular isotropic and orthotropic plates. It is demonstrated that the approach yields excellent results not only for natural frequencies and buckling loads but also for the more sensitive mode shape, bending moment and shear force distribution problem.

165 citations

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TL;DR: In this article, Schlieren photographs of the flow field along with hot-wire measurements in the jet were obtained at different pressure ratios and it was shown that the self-excitation helps to induce large scale vortical motions in a converging rectangular nozzle.

Abstract: An experimental investigation has been carried out on an underexpanded jet of air issuing from a converging rectangular nozzle of moderate aspect ratio. Schlieren pictures of the flow field along with hot-wire measurements in the jet were obtained at different pressure ratios. At the pressure ratio corresponding to the maximum screeching sound, schlieren photographs show a very strong organized cylindrical wave pattern on either side of the jet, with their respective sources being located at the end of the third shock cell. Associated with this wave pattern is a large increase in the angle of spread of the jet. It is shown that the self-excitation helps to induce large scale vortical motions in the jet both in the plane containing the small dimension of the nozzle and in the plane containing the long dimension of the nozzle. However, the locations of these structures are different in the two planes. Nevertheless, the characteristic Strouhal number corresponding to these large scale structures in both planes is the same and equal to 0·12. The influence of the self-excitation on the mean velocities and rms intensities has also been investigated. For the full range of pressure ratios studied, similarity was found both in the mean velocity and rms intensity profiles in the two central planes beyond 80 widths downstream of the nozzle exit. However, the shapes of the similarity profiles are different in the two planes. For the downstream distances studied, complete axisymmetry (identical mean velocity profiles in both planes) was not found, which suggests it may persis for a large distance downstream of the nozzle exit.

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TL;DR: In this paper, a procedure for generalizing harmonic balance to obtain higher-order approximations to the periodic solutions of differential equations for two systems, one a conservative system and the other non-conservative, is presented.

Abstract: A procedure is provided for generalizing the method of harmonic balance to obtain higher-order approximations to the periodic solutions of differential equations for two systems, one a conservative system and the other nonconservative. The procedure is currently being applied to investigate the possible solution behaviors of singular, nonlinear oscillators, where the usual perturbation methods do not work.

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TL;DR: The theoretical and experimental status of helicopter impulsive noise is reviewed in this article, where two major sources of helicopter noise are addressed: high-speed impulsive noises and blade- vortex interaction noise.

Abstract: The theoretical and experimental status of helicopter impulsive noise is reviewed The two major source mechanisms of helicopter impulsive noise are addressed: high-speed impulsive noise and blade- vortex interaction impulsive noise A thorough physical explanation of both generating mechanisms is presented together with model and full-scale measurements of the phenomena Current theoretical prediction methods are compared with experimental findings of isolated rotor tests The noise generating mechanisms of high speed impulsive noise are fairly well understood—theory and experiment compare nicely over Mach number ranges typical of today’s helicopters For the case of blade-vortex interaction noise, understanding of noise generating mechanisms and theoretical comparison with experiment are less satisfactory Several methods for improving theory/experiment are suggested

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TL;DR: In this article, a 2.54 cm diameter air jet at a Mach number of 0.50 and a Reynolds number of 300,000 was used to calibrate subsonic jet noise theories, showing that large scale coherent structures of the jet govern the source locations by controlling the turbulence and also strongly influence the near field pressure fluctuations.

Abstract: Flow and sound field data are presented for a 2.54 cm diameter air jet at a Mach number of 0.50 and a Reynolds number of 300,000. Distributions of mean velocity, turbulence intensities, Reynolds stress, spectral components of turbulence as well as of the near field pressure, together with essential characteristics of the far field sound are reported. This detailed set of data for one particular flow, erstwhile unavailable in the literature, is expected to help promoote and calibrate subsonic jet noise theories. 'Source locations' in terms of the turbulence maxima, coupling between the entrainment dynamics and the near pressure field, the sound radiation paths, and the balance in mass, momentum and sound energy fluxes are discussed. The results suggest that the large scale coherent structures of the jet govern the 'source locations' by controlling the turbulence and also strongly influence the near field pressure fluctuations.

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TL;DR: In this article, the influence of microslip on the resonant response of structures that are restrained by a friction contact is discussed, and a new model of micro-slip described in the companion paper is used in simulating the vibratory response of three sets of experiments.

Abstract: The influence of microslip on the resonant response of structures that are restrained by a friction contact is discussed. Microslip or partial slip of the friction interface becomes important and needs to be taken into account when the friction contact pressure is large. A new model of microslip described in the companion paper [1] is used in simulating the vibratory response of three sets of experiments; in each case partial slip of the friction interface resolves anomalies that could not be explained by the simple point contact model of friction that has been used in the past to analyze these types of problems.

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TL;DR: In this paper, a rotating uniform cantilever beam with a tip mass was studied by using beam characteristic orthogonal polynomials in the Rayleigh-Ritz method, and the results were compared with those obtained by the Myklestad method, the extended Galerkin method and finite element methods.

Abstract: Natural frequencies and mode shapes of a rotating uniform cantilever beam with a tip mass are studied by using beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. The set of orthogonal polynomials which satisfy the geometrical boundary conditions are generated by using the Gram-Schmidt process. The results are compared with those obtained by the Myklestad method, the extended Galerkin method and finite element methods. The variation of natural frequencies with the speed of rotation is plotted for several parameter combinations such as setting angle, tip mass, moment of inertia of tip mass, etc. Mode shapes at different rotational speeds are also plotted. Use of orthogonal polynomials for the deflection shapes enables the computation of higher natural frequencies of any order to be accomplished without facing any numerical difficulties, which is not the case when arbitrary polynomial expressions are used.

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TL;DR: In this paper, the effects of tip clearance on both noise and performance of four commercially representative fans (three low pressure axial flow and one mixed flow) in a circular duct, both with and without coincidence of the fan and duct axes (fan/duct eccentricity).

Abstract: Results are presented of a systematic experimental investigation of the effects of tip clearance on both noise and performance of four commercially representative fans (three low pressure axial flow and one mixed flow) in a circular duct, both with and without coincidence of the fan and duct axes (fan/duct eccentricity). It is shown that with eccentricity significant pure tone noise is generated due to blade tip/duct wall interaction, with a plane wave mode component. Reducing tip clearance both improves performance and reduces noise, not only at the maximum efficiency operating point but also in an appreciable low flow rate operating region. With tip clearances small enough to optimize performance and minimize noise fan/duct eccentricity must be kept to a very small value, implying strict precautions in fan and duct manufacture, installation and service conditions.

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TL;DR: In this article, the effects of the thickness of the plate and the diameter of the holes on the impedance of the perforate have been included, and it is found from the results that resistance increases with the grazing flow velocity, but is more or less independent of thickness and hole diameter.

Abstract: This paper is concerned with measurement of grazing flow impedance of a perforated plate. The measurements have been made for various grazing flow velocities with an impedance tube facility, at different frequencies. The effects of the thickness of the plate and the diameter of the holes on the impedance of the perforate have been included. It is found from the results that resistance increases with the grazing flow velocity, but is more or less independent of thickness and hole diameter, whereas reactance increases with thickness and diameter of the hole. Finally an empirical formula for computing the normal incidence impedance of an orifice or cluster of orifices (perforated plate) is given, and verified for use in prediction of noise reduction across concentric tube resonators.

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TL;DR: In this article, a simple theoretical model has been developed from first principles for cross flow induced fluid-elastic instabilities in heat exchanger tube bundles, which includes the effects of tube array pattern and pitch.

Abstract: A simple theoretical model has been developed from first principles for cross flow induced fluid-elastic instabilities in heat exchanger tube bundles. The model is an extension of the authors' tube-in-channel physical representation used for the case of transverse dynamic stability of tube arrays. The theory has been modified and extended to treat static divergence as well as fluid-elastic stability in both streamwise and transverse directions to the flow. While the mechanism responsible for transverse dynamic instability was found to be one of flow redistribution, no flow redistribution takes place for symmetric streamwide tube motion nor during transverse divergence. The model includes the effects of tube array pattern and pitch. Theoretical descriptions of heat exchanger instability mechanisms developed by other authors are compared and contrasted with the present model.

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TL;DR: In this article, the interaction between flow and flow-induced acoustic resonances near rigid plates with square leading edges located in a hard-walled duct is described, and a potential flow model for a plate with a square leading edge is developed and the acoustic power generated by vortices, which grow and shed from the leading edge, is calculated as they move through the resonant acoustic field.

Abstract: The interaction between flow and flow-induced acoustic resonances near rigid plates with square leading edges located in a hard-walled duct is described. These plates generate β-mode acoustic resonances over several discrete ranges of flow velocity. A potential flow model for a plate with a square leading edge is developed and the acoustic power generated by vortices, which grow and shed from the leading edge, is calculated as they move through the resonant acoustic field. Vortices grow rapidly during the first half cycle and the vortex growth rate is quantified by using a Kutta condition. This model indicates that a major phase-dependent sound soccurs when a vortex passes the trailing edge of the plate. It is the remoteness of this source from the site at the leading edge where the sound affects the flow which makes possible the discrete resonance ranges.

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TL;DR: In this paper, a single microphone output is de-Dopplerized, and results from a Lockheed TriStar graphically illustrate the capability of the de-dopplerization for the analysis of noise from counterrotating propeller driven aircraft.

Abstract: The technique described in this paper eliminates the Doppler effect from aircraft flyover noise measurements and generates narrow band spectra at required angles. Such a capability allows more accurate interpretation of flight data, and is necessary for a detailed comparison with predictions and static measurements, since 1 3 octave or narrow band levels, before de-Dopplerization, yield limited information on tonal content. The paper first explains how a single microphone output is de-Dopplerized, and includes details of aircraft tracking and computer simulation of flyover measurements. The technique is especially relevant to the analysis of noise from counter-rotating propeller driven aircraft, and results are shown for an Avro Shackleton. It is also applied to a Boeing 757, with high bypass ratio turbofan engines. Narrow band spectra at selected angles, density plots of complete flyovers, and field shapes at constant frequencies are all presented. Acoustic imaging, achieved by focussing the de-Dopplerized signals from an array of microphones, is also described, and results from a Lockheed TriStar graphically illustrate its capability.

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TL;DR: In this article, an exact conservation equation is developed which generalizes the familiar acoustic energy equation for arbitrary homentropic disturbances to a homentropy steady flow, and a perturbation energy density and flux are defined and identified as the exact physical quantities whose leading order representations are the usual acoustic energy densities and flux.

Abstract: An exact conservation equation is developed which generalizes the familiar acoustic energy equation. The new relation is valid for arbitrary homentropic disturbances to a homentropic steady flow. It is suggested by a derivation of the acoustic energy equation by means of a regular perturbation expansion of the general energy equation for an ideal compressible fluid. A perturbation energy density and flux are defined and are identified as the exact physical quantities whose leading order representations are the usual acoustic energy density and flux.

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TL;DR: Theoretical stability threshold curves are determined for standard heat exchanger tube bundle geometries based on the first principles model developed in Part I. as mentioned in this paper, and an investigation into the sensitivity of the predicted threshold curves on variations in model parameters is also presented.

Abstract: Theoretical stability threshold curves are determined for standard heat exchanger tube bundle geometries based on the first principles model developed in Part I. Over the range of mass-damping parameter relevant to practical heat exchangers, transverse dynamic instability is predicted to occur at lower reduced velocity than either streamwise or transverse static instabilities. No streamwise dynamic instability is predicted for any array by the current model. Quite good agreement is found with available experimental data, particularly for parallel triangular and square arrays. No empirical fluid force coefficients are required. An investigation into the sensitivity of the predicted threshold curves on variations in model parameters is also presented.

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TL;DR: In this paper, an approximate solution of the title problem is obtained by using the Rayleigh-Schmidt approach, which yields accurate engineering results which in the case of a uniform cantilever beam are in very good agreement with To's eigenvalues.

Abstract: An approximate solution of the title problem is obtained by using the Rayleigh-Schmidt approach. It is shown that a simple polynomial co-ordinate function yields accurate, engineering results which in the case of a uniform cantilever beam are in very good agreement with To's eigenvalues. Only the fundamental mode of vibration is considered in the present investigation.

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TL;DR: In this article, a cylindrical shell which is reinforced at regular intervals by flexible stiffeners parallel to the shell generator is considered, and the structure effectively constitutes a one-dimensional periodic system and is analyzed as such.

Abstract: Wave propagation around a cylindrical shell which is reinforced at regular intervals by flexible stiffeners parallel to the shell generator is considered. The shell itself is restricted to a section between two circumferential frames on to which the shell is simply supported. The structure effectively constitutes a one-dimensional periodic system and is analyzed as such. Four degrees of freedom are allowed between each periodic element and equations are set up for the four pairs of propagation constants which characterize the possible wave motions. Symmetric or asymmetric stiffener sections may be accommodated in the analysis together with structural damping. Computed propagation constants are presented for two different stiffener cross-sections, each pitched at two different intervals around the shell. Natural frequencies are calculated for one of these.

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TL;DR: In this paper, the forced vibration of a system of coupled beams is examined and it is shown that violent non-synchronous torsion and bending vibration observed in such a system may be explained by the existence of quadratic nonlinear coupling terms and internal resonance effects between three and four modes.

Abstract: Small non-linear interactions of the type referred to as autoparametric may have a considerable effect on the forced oscillatory behaviour of structures. Under conditions of internal resonance these effects may bring about complex forms of response where the normal linear resonance of a directly excited mode is absorbed, and indirectly excited modes show simultaneous large non-synchronous responses. In this paper the forced vibration of a system of coupled beams is examined and it is shown that violent non-synchronous torsion and bending vibration observed in such a system may be explained by the existence of quadratic non-linear coupling terms and internal resonance effects between three and four modes. A finite degree of freedom model was formulated and the method of multiple scales was applied. For the case of three-mode interactions stationary solutions were obtained which gave very satisfactory agreement with measured responses, and showed the effects of detuning of the internal resonance. The analysis also gave a clear indication of the cause of strong four-mode interactions but did not yield stationary solutions in this case. Experimental results are included to illustrate the four-mode interactive response.

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TL;DR: In this article, it was shown that plane rotors with one or two particles free to move, subject to viscous damping, exhibit autobalancing, a property attributed to non-autonomous systems possessing an hyperbolic stable fixed point in the (non-extended) phase space for an open domain in parameter space.

Abstract: It is shown that plane rotors with one or two particles free to move, subject to viscous damping,, in a groove on the rotor exhibit autobalancing, a property attributed to non-autonomous systems possessing an hyperbolic stable fixed point in the (non-extended) phase space for an open domain in parameter space. The analysis is also extended to non-plane rotors. Dynamical systems of this kind may be represented by perturbed Hamiltonian systems and averaging methods may be applied. Here the driving is considered in the form of a constraint and the particular structure emerging is found to be most easily analyzed by using the method of multiple scales. With this method it is also shown that non-plane rotors are autobalancing. The first order solutions showing how this physically six-dimensional problem approaches the fixed point are also given.

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TL;DR: The dynamic stability of supported cylindrical pipes converying fluid, when the flow velocity is harmonically perturbed about a constant mean value, is considered in this paper, and explicit stability conditions for perturbations of small intensity are obtained by using the method of averaging.

Abstract: The dynamic stability of supported cylindrical pipes converying fluid, when the flow velocity is harmonically perturbed about a constant mean value, is considered in this paper. Explicit stability conditions for perturbations of small intensity are obtained by using the method of averaging. For large periodic excitation a numerical method based on the Floquet theory is used to extend the stability boundaries. The effects of the mean flow velocity, dissipative forces, boundary conditions, and virtual mass on the extent of the parametric instability regions are then discussed.

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TL;DR: In this article, a 12-d.o.f, geometrically non-linear, doubly curved, thin shell finite element of revolution with laminate composite materials was used for free vibration analysis of a pneumatic tire without suspension.

Abstract: Natural frequencies and mode shapes of a pneumatic tire without suspension are investigated using a 12-d.o.f., geometrically non-linear, doubly curved, thin shell finite element of revolution with laminate composite materials. The wheel is assumed to be free to move within its own plane. The results of the free vibration analysis indicate that only the radial modes of n = 1 are affected by the wheel's freedom to move. To evaluate the finite element modeling, a simplified elastic ring-spring model is studied. The tire is modeled as a circular, elastic ring supported by distributed spring in both radial and circumferential directions. The wheel is modeled as a rigid mass to which the disturbed spring is attached. The two models are found to agree and complement each other. While the simplified ring-spring model is easy and practical to use to obtain preliminary results, the complex finite element model can give more detailed and accurate results for both free vibration and dynamic response analyses.