determination of the mutual radiation resistances of a rectangular plate and their impact on the radiated sound power abstract the mutual radiation resistances resulting from cross modal coupling have been mostly excluded from sound power calculations on the belief or premise that either for some reasons they have an insignificant impact on the power radiation or consideration of them can be an enormous computational burden in this paper the characteristics of the mutual radiation resistances are investigated of a simply supported rectangular plate it is shown that once the self radiation resistances or the so called modal radiation efficiencies have been calculated the mutual radiation resistances can be readily obtained at virtually no cost of equal importance because no approximation is made the present formulation is fully accurate in the entire frequency range an asymptotic expression is also derived for large modal wavenumbers numerical results are presented to illustrate the spectral characteristics of the mutual radiation resistances and the corresponding impact on the radiated sound power

natural extensions for the rosen fractions the rosen fractions form an infinite family which generalizes the nearest integer continued fractions we find planar natural extensions for the associated interval maps this allows us to easily prove that the interval maps are weak bernoulli as well as to unify and generalize results of diophantine approximation from the literature

vibration analysis of high speed rotating conical shell with arbitrary boundary conditions based on the energy method an analytical model for the vibration characteristics of high speed rotating conical shell with arbitrary boundary is constructed effects of centrifugal and coriolis accelerations as well as initial hoop tension due to rotating are all taken into account a set of translational and rotational artificial springs to simulate the elastic constraints imposed on the conical shell edges under the framework each of the displacement components of the conical shell is invariantly described by a modified fourier series which is composed of a standard fourier series and closed form auxiliary functions all the expansion coefficients are determined by using the familiar rayleigh ritz procedure as the generalized coordinates to validate the present analysis a series of comparisons between the present results and those available in the open literature are carried out and very good agreements are obtained on this basis some numerical results are given to illustrate the effects of rota

linear estimate for the number of limit cycles of a perturbed cubic polynomial differential system abstract perturbing the cubic polynomial differential systems x y a 1 x a 0 b 1 y b 0 y x a 1 x a 0 b 1 y b 0 having a center at the origin inside the class of all polynomial differential systems of degree n we obtain using the averaging theory of second order that at most 17 n 15 limit cycles can bifurcate from the periodic orbits of the center

vibroacoustic analysis of rectangular plates with elastic rotational edge restraints this investigation deals with the vibration of and the acoustic radiation from a simply supported rectangular plate with elastic restraints against edge rotations the displacement of the plate is first sought as a series expansion in terms of the beam functions each of the beam functions is then expressed as the linear combination of a fourier sine series and a complementary sufficiently smooth function that is introduced to ensure and improve the convergence of the fourier series expansion to facilitate the acoustic analysis the plate displacement is eventually simplified to a standard fourier sine series the modal and acoustic characteristics of square plates are studied for different restraining stiffnesses and configurations it is shown that the modes of the restrained plates can be considerably different from those in the simply supported case so are the corresponding modal radiation efficiencies the proposed method is generally applicable to rectangular plates elastically restrained along any edge s and the acoustic calculations are valid for an arbitrary acoustic or structural modal wave number

a simple formula for predicting resonant frequencies of a rectangular plate with uniformly restrained edges abstract this paper provides empirical formulas for rapidly calculating the resonant frequencies of an orthotropic rectangular plate with its edges constrained by elastic supports in particular the classical boundary condition with guided supports at its edges is considered other boundary conditions such as the guided free edges guided clamped edges and guided free support edges have also been included in the present study simple and closed form empirical formulas have been derived to allow straightforward computations of the modal resonant frequencies the empirical formulas are based on the analytical results obtained from the rayleigh ritz method however the coefficient is determined empirically from the results obtained by other more accurate computational scheme e g finite element method the method is further generalized to predict the resonant frequencies for general boundary conditions of a square orthotropic plate

the number of limit cycles from a cubic center by the melnikov function of any order abstract in this paper we consider the system x y 1 x 2 p x y y x 1 x 2 q x y where p x y and q x y are arbitrary quadratic polynomials we study the maximum number of limit cycles bifurcating from the periodic orbits by using the melnikov function of any order we prove that the upper bound for the number of limit cycles is 3 and reached

the number of limit cycles of a quintic polynomial system in this paper we consider the bifurcation of limit cycles of the system email protected y x 2 a 2 y 2 b 2 email protected x y email protected x x 2 a 2 y 2 b 2 email protected x y for e sufficiently small where a email protected r 0 and p q are polynomials of degree n we obtain that up to first order in e the upper bounds for the number of limit cycles that bifurcate from the period annulus of the quintic center given by e 0 are 3 2 n sin 2 email protected 2 1 if a b and n 1 if a b moreover there are systems with at least 3 2 n sin 2 email protected 2 1 if a b and n 1 limit cycles if a b

on the short wave acoustic radiation from planar panels or beams of rectangular shape an exact representation which involves single as opposed to double integrals is established for the time average power output generated by individual modes of normal surface velocity on a simply supported and baffled rectangular panel fluid loading is not taken into account the representation enables analytical estimates to be found in the short wavelength range and provides details hitherto unresolved by numerical analysis

the radiation efficiency of baffled plates and strips the average radiation efficiency of point excited rectangular plates including those with a very large aspect ratio strips is investigated by using a modal summation method based on the farfield sound intensity by taking an average over all possible forcing positions on the plate the cross modal contributions average out to zero the numerical results from the modal summation are compared with established formulae for rectangular plates for wavenumbers where acoustic circulation takes place it is shown that the previously published formulae are not applicable for predicting the average radiation efficiency for a strip the analysis for the strip reveals that the radiation efficiency at frequencies below the first structural natural frequency of the strip is proportional to the square of the shortest edge length at frequencies between the fundamental natural frequency and that of the second order mode of bending across the strip the average radiation efficiency is found to be proportional to the structural damping loss factor but differs from the results obtained from the usual model of the nearfield radiation from the forcing point of a rectangular plate approximate expressions for calculating the average radiation efficiency of the strip are derived the maximum radiation efficiency around the critical frequency is found to vary less with helmholtz number kca where kc is the wavenumber at the critical frequency and a is the width than previously published models suggest

limit cycles of cubic polynomial vector fields via the averaging theory abstract in this paper we study the maximum number of limit cycles that can bifurcate from the period annulus surrounding the origin of a class of cubic polynomial differential systems using the averaging method more precisely we prove that the perturbations of the period annulus of the center located at the origin of the cubic polynomial differential system x y f x y y x f x y where f x y 0 is a conic such that f 0 0 0 by arbitrary cubic polynomial differential systems provide at least six limit cycles bifurcating from the periodic orbits of the period annulus using only the first order averaging method

bifurcation of periodic orbits of periodic equations with multiple parameters by averaging method abstract we investigate a class of n dimensional periodic differential equations with multiple small parameters by averaging method and obtain sufficient conditions for the existence of multiple periodic orbits as applications of the main results a cubic system is studied

a note on vibrations of generally restrained beams exact frequency and normal mode shape expressions are derived in this note for generally restrained bernoulli euler beams with unsymmetrical translations and rotations at either end the eigenfrequencies and mode shape parameters are presented for a wide range of restraint parameters several degenerate cases have been studied and some of these have been compared with those available in the published literature it is believed that the results presented in this paper will be of use in design of beams shafts and piping under dynamic considerations

free vibration of two elastically coupled rectangular plates with uniform elastic boundary restraints abstract an analytical method is derived for determining the vibrations of two plates which are generally supported along the boundary edges and elastically coupled together at an arbitrary angle the interactions of all four wave groups bending waves out of plane shearing waves in plane longitudinal waves and in plane shearing waves have been taken into account at the junction via four types of coupling springs of arbitrary stiffnesses each of the transverse and in plane displacement functions is expressed as the superposition of a two dimensional 2 d fourier cosine series and several supplementary functions which are introduced to ensure and improve the convergence of the series representation by removing the discontinuities that the original displacement and its derivatives will potentially exhibit at the edges when they are periodically expanded onto the entire x y plane as mathematically implied by a 2 d fourier series the unknown expansions coefficients are calculated using the rayleigh ritz procedure which is actually equivalent to solving the governing equation and the boundary and coupling conditions directly when the assumed solutions are sufficiently smooth over the solution domains numerical examples are presented for several different coupling configurations a good comparison is observed between the current results and the fea models although this study is specifically focused on the coupling of two plates the proposed method can be directly extended to structures consisting of any number of plates

on the efficacy of the wavelet decomposition for high frequency vibration analyses abstract this paper reports the extraordinary ability of the wavelet decomposition for vibration analyses under the framework of rayleigh ritz method using a beam as an example daubechies wavelet scale functions are used as admissible functions for decomposing the flexural displacement of the structure along with the artificial springs at the boundary to predict vibration of an euler bernoulli beam in an extremely large frequency range it is shown that the use of wavelet basis allows reaching very high frequencies typically covering more than 1000 modes using conventional computational facility within the available numerical dynamics of the computers with no particular care needed for round off errors as a side benefit the use of spring boundary also allows handling any elastic boundary conditions through a dynamic contribution in the hamiltonian of the beam the wavelet decomposed approach combines the flexibility of the global methods and the accuracy of local methods by inheriting the versatility of the rayleigh ritz approach and the superior fitting ability of the wavelets numerical results on both free and forced vibrations are given in excellent agreement with predictions of classical methods

application of chebyshev series to solution of non prismatic beam vibration problems the problem of the vibration of a non prismatic beam resting on a two parameter elastic foundation has been solved by applying the approximation by chebyshev series as a result closed analytical formulas defining the coefficients of the sought solutions were obtained the method was used to solve the eigenproblem for a simply supported beam and a cantilever beam the obtained results were compared with the results reported by other authors

a series expansion of the acoustic power radiated from planar sources a series expansion in ascending powers of the wavenumber k is derived for the acoustic power delivered by baffled or unbaffled planar sources this series provides a relatively simple means of deriving expressions for the power radiated by a baffled source with a known velocity distribution and can be used for unbaffled plates when the velocity field outside the plate is also known the terms in the series are calculated from the moments of this velocity distribution in the plane containing the source if these moments are written as derivatives in wavenumber space it is shown that a maclaurin expansion of the fourier transformed velocity provides an easy technique for computing the first few terms of the acoustic power examples are provided for baffled rectangular plates with various boundary conditions the arbitrarily shaped plate with free boundaries is particularly interesting it is proven that the volume flow across its surface must be zero and as a result corner and edge mode radiation cannot e

free vibration analysis of circular cylindrical shells abstract a general analytical method is presented for evaluating the free vibration characteristics of a circular cylindrical shell with classical boundary conditions of any type the solution is obtained through a direct solution procedure in which sanders shell equations are used with the axial modal displacements represented as simple fourier series expressions stokes transformation is exploited to obtain correct series expressions for the derivatives of the fourier series an explicit expression of the exact frequency equation can be obtained for any kind of boundary conditions the accuracy of the method is checked against available data the method is used to find the modal characteristics of the thermal liner model of the u s fast test reactor ftp the numerical results obtained are compared with finite element method solutions

an exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports abstract an analytical method is developed for the vibration analysis of rectangular plates with elastically restrained edges the displacement solution is expressed as a two dimensional fourier series supplemented with several one dimensional fourier series mathematically such a series expansion is capable of representing any function including the exact displacement solution whose third order partial derivatives are required to be continuous over the area of the plate since the discontinuities or jumps potentially related to the partial derivatives at the edges when they are periodically extended onto the entire x y plane as implied by a two dimensional fourier series expansion have been explicitly absorbed by the supplementary terms all the series expansions for up to the fourth order derivatives can be directly obtained through term by term differentiations of the displacement series thus an exact solution can be obtained by letting the series simultaneously satisfy the governing differential equation and the boundary conditions on a point wise basis because the series solution has to be truncated numerically the exact solution should be understood as a solution with arbitrary precision several numerical examples are presented to illustrate the excellent accuracy of the current solution the proposed method can be directly extended to other more complicated boundary conditions involving non uniform elastic restraints point supports partial supports and their combinations

free transverse vibrations of nano to micron scale beams in the present work the integral equation approach and the non local elasticity theory are employed to investigate the free transverse vibrations of nano to micron scale beams the frequency equation is analytically formulated into an eigenvalue problem of a matrix with an infinite order the numerical calculation is implemented by truncating this matrix to a finite order one it is found that the impact of the non local effect on the natural frequencies and vibrating modes is negligible for the beams with micrometre scale length but when the length of beams reaches the nanoscale the non local effect becomes important especially for the high order natural frequencies and vibrating modes

metrical diophantine approximation for continued fraction like maps of the interval we study the metrical properties of a class of continued fraction like mappings of the unit interval each of which is defined as the fractional part of a mobius transformation taking the endpoints of the interval to zero and infinity

free vibrations of beams with general boundary conditions a simple and unified approach is presented for the vibration analysis of a generally supported beam the flexural displacement of the beam is sought as the linear combination of a fourier series and an auxiliary polynomial function the polynomial function is introduced to take all the relevant discontinuities with the original displacement and its derivatives at the boundaries and the fourier series now simply represents a residual or conditioned displacement that has at least three continuous derivatives as a result not only is it always possible to expand the displacement in a fourier series for beams with any boundary conditions but also the solution converges at a much faster speed the reliability and robustness of the proposed technique are demonstrated through numerical examples

bifurcation of limit cycles by perturbing a piecewise linear hamiltonian system with a homoclinic loop abstract in this paper we study limit cycle bifurcations for a kind of non smooth polynomial differential systems by perturbing a piecewise linear hamiltonian system with the center at the origin and a homoclinic loop around the origin by using the first melnikov function of piecewise near hamiltonian systems we give lower bounds of the maximal number of limit cycles in hopf and homoclinic bifurcations and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in e in the case when the degree of perturbing terms is low we obtain a precise result on the number of zeros of the first melnikov function

bifurcation of a piecewise smooth cubic system via expansion of melnikov function abstract asymptotic expansion formulae of the first order melnikov function near the perturbed center of a cubic system are investigated studied through a suitable transformation by expanding the formulae we obtain that the given cubic system can have at least ten limit cycles

acoustic radiation from a rectangular plate reinforced by springs at arbitrary locations abstract the vibration of a simply supported rectangular plate which is reinforced by a finite number of springs at arbitrary locations is sought as a linear combination of the flexural modes of the corresponding unloaded or simple plate although these modes cannot be directly used to decouple the dynamic equations of the motion of the reinforced plate it is still advantageous to use them as a complete set of basis functions to represent the interested quantities such as the deflection of the plate a linear transformation is established between the two modal spaces spanned by the modes respectively for the reinforced and the unloaded plates the radiation characteristics of the spring reinforced plate are investigated by using a near field formulation in order to reduce computational burden a frequency interpolation technique is employed so that numerical integration now needs to be carried out only at some key frequencies the specific radiation resistances of the basis modes are first determined as a set of invariants from which the modal radiation efficiency of the reinforced plate can be readily extracted the effects of cross modal coupling between the basis modes have been considered which is necessary for an accurate determination of the modal radiation efficiency for the reinforced plate a qualitative procedure is also described for locating the coincidence frequencies where the modal radiation efficiencies of the reinforced plate tend to reach their peak values

a unified approach for predicting sound radiation from baffled rectangular plates with arbitrary boundary conditions this paper discusses sound radiation from a baffled rectangular plate with each of its edges arbitrarily supported in the form of elastic restraints the plate displacement function is universally expressed as a 2 d fourier cosine series supplemented by several 1 d series the unknown fourier expansion coefficients are then determined by using the rayleigh ritz procedure once the vibration field is solved the displacement function is further simplified to a single standard 2 d fourier cosine series in the subsequent acoustic analysis thus the sound radiation from a rectangular plate can always be obtained from the radiation resistance matrix for an invariant set of cosine functions regardless of its actual dimensions and boundary conditions further this radiation resistance matrix unlike the traditional ones for modal functions only needs to be calculated once for all plates with the same aspect ratio in order to determine the radiation resistance matrix effectively an analytical formula is derived in the form of a power series of the non dimensional acoustic wavenumber the formula is mathematically valid and accurate for any wavenumber several numerical examples are presented to validate the formulations and show the effect of the boundary conditions on the radiation behavior of planar sources

an analytical solution for the self and mutual radiation resistances of a rectangular plate it is widely accepted that for an arbitrary acoustic wavenumber the radiation resistance of a simply supported rectangular plate has to be calculated through numerical integration in this study an analytical solution for the self and mutual radiation resistances is obtained in the form of the power series of the non dimensional acoustic wavenumber unlike the previous analytical or asymptotic solutions it is not subject to any of the restrictions usually imposed upon the acoustic wavenumber a few numerical examples are given simply to verify the solution and elucidate the reliance of the self and mutual radiation resistances on the acoustic wavenumber and the plate aspect ratio it is shown that the present formulae are extremely efficient by cutting the cpu time by orders of magnitude in comparison with the traditional numerical integration scheme this investigation successfully fills the long existing gap in solutions for the moderate wavenumbers that are often of primary concern in an acoustic analysis

dynamic analysis of generally supported beams using fourier series the applicability of fourier series to the dynamic analysis of beams having arbitrary boundary conditions is explored for the simplicity of presenting the analysis procedure beams having zero end displacement are used in the discussion numerical results for free vibration and dynamic response based on the present analysis procedure are compared with existing results for natural frequencies and dynamic behavior obtained by the conventional modal analysis these results indicate that the present analysis is accurate and provides a unified and systematic procedure which is seemingly simpler and more straightforward than the conventional modal analysis