transverse vibrations of tensioned pipes conveying fluid with time dependent velocity abstract in this study the transverse vibrations of highly tensioned pipes with vanishing flexural stiffness and conveying fluid with time dependent velocity are investigated two different cases the pipes with fixed fixed end and fixed sliding end conditions are considered the time dependent velocity is assumed to be a harmonic function about a mean velocity these systems experience a coriolis acceleration component which renders such systems gyroscopic the equation of motion is derived using hamilton s principle and solved analytically by direct application of the method of multiple scales a perturbation technique the natural frequencies are found increasing the ratio of fluid mass to the total mass per unit length increases the natural frequencies the principal parametric resonance cases are investigated in detail stability boundaries are determined analytically it is found that instabilities occur when the frequency of velocity fluctuations is close to two times the natural frequency of the constant velocity system when the velocity fluctuation frequency is close to zero no instabilities are detected up to the first order of perturbation numerical results are presented for the first two modes

nearly inviscid faraday waves in annular containers of moderately large aspect ratio nearly inviscid parametrically excited surface gravity capillary waves in two dimensional domains of finite depth and large aspect ratio are considered coupled equations describing the evolution of the amplitudes of resonant left and right traveling waves and their interaction with a mean flow in the bulk are derived and the conditions for their validity established under suitable conditions the mean flow consists of an inviscid part together with a viscous mean flow driven by a tangential stress due to an oscillatory viscous boundary layer near the free surface and a tangential velocity due to a bottom boundary layer these forcing mechanisms are important even in the limit of vanishing viscosity and provide boundary conditions for the navier stokes equation satisfied by the mean flow in the bulk for moderately large aspect ratio domains the amplitude equations are nonlocal but decouple from the equations describing the interaction of the slow spatial phase and the viscous mean flow two cases are considered in detail gravity capillary waves and capillary waves in a microgravity environment

non linear vibrations of a beam mass system with both ends clamped abstract a clamped clamped beam mass system is considered the non linear equations of motion including stretching due to immovable end conditions were derived previously 1 ozkaya et al 1997 journal of sound and vibration 199 679 696 in addition to five different end conditions considered in reference 1 the case of clamped clamped edge conditions is treated in this work exact solutions for the mode shapes and frequencies are given for the linear part of the problem for the non linear problem approximate solutions using perturbations are searched alternatively the natural frequencies and non linear corrections are used in training a multi layer feed forward back propagation artificial neural network ann algorithm using the algorithm the numerical calculations are drastically reduced for obtaining the natural frequencies and non linear corrections corresponding to different input parameters

on solving wave equations on fixed bounded intervals involving robin boundary conditions with time dependent coefficients abstract in this paper it is shown how characteristic coordinates or equivalently how the well known formula of d alembert can be used to solve initial boundary value problems for wave equations on fixed bounded intervals involving robin type of boundary conditions with time dependent coefficients a robin boundary condition is a condition that specifies a linear combination of the dependent variable and its first order space derivative on a boundary of the interval analytical methods such as the method of separation of variables sov or the laplace transform method are not applicable to those types of problems the obtained analytical results by applying the proposed method are in complete agreement with those obtained by using the numerical finite difference method for problems with time independent coefficients in the robin boundary condition s the results of the proposed method also completely agree with those as for instance obtained by the method of separation of variables or by the finite difference method

on the transversal vibrations of a conveyor belt with a low and time varying velocity part ii the beam like case in this paper initial boundary value problems for a linear wave string equation are considered these problems can be used as simple models to describe the vertical vibrations of a conveyor belt for which the velocity is small with respect to the wave speed and is assumed to move with a time varying speed formal asymptotic approximations of the solutions are constructed to show the complicated dynamical behavior of the conveyor belt it will also be shown that the truncation method cannot be applied to this problem in order to obtain approximations valid on long time scales

invariant imbedding the box scheme and an equivalence between them we show for general linear two point boundary value problems that the box scheme and invariant imbedding are equivalent in the sense that a specific algorithm for solving the difference equations is valid if and only if an appropriate imbedding is valid a key element in the equivalence is the role played by pivoting to solve the linear equations and switching to get a finite imbedding part of the equivalence proof shows that imbedding is the completion of the box scheme multiple imbeddings are introduced and a lower bound for the critical length problem is obtained

natural frequencies modeshapes and modal interactions for strings vibrating against an obstacle relevance to sitar and veena we study the vibration characteristics of a string with a smooth unilateral obstacle placed at one of the ends similar to the strings in musical instruments like sitar and veena in particular we explore the correlation between the string vibrations and some unique sound characteristics of these instruments like less inharmonicity in the frequencies a large number of overtones and the presence of both frequency and amplitude modulations at the obstacle we have a moving boundary due to the wrapping of the string and an appropriate scaling of the spatial variable leads to a fixed boundary at the cost of introducing nonlinearity in the governing equation reduced order system of equations has been obtained by assuming a functional form for the string displacement which satisfies all the boundary conditions and gives the free length of the string in terms of the modal coordinates to study the natural frequencies and mode shapes the nonlinear governing equation is linearized about the static configuration the natural frequencies have been found to be harmonic and they depend on the shape of the obstacle through the effective free length of the string expressions have been obtained for the time varying mode shapes as well as the variation of the nodal points modal interactions due to coupling have been studied which show the appearance of higher overtones as well as amplitude modulations in our theoretical model akin to the experimental observations all the obtained results have been verified with an alternate formulation based on the assumed mode method with polynomial shape functions

on variable length induced vibrations of a vertical string abstract the purpose of this paper is to study the free lateral responses of vertically translating media with variable length velocity and tension subject to general initial conditions the translating media are modeled as taut strings with fixed boundaries the problem can be used as a simple model to describe the lateral vibrations of an elevator cable for which the length changes linearly in time or for which the length changes harmonically about a constant mean length in this paper an initial boundary value problem for a linear axially moving string equation is formulated in the given model a rigid body is attached to the lower end of the string and the suspension of this rigid body against the guide rails is assumed to be rigid for linearly length variations it is assumed that the axial velocity of the string is small compared to nominal wave velocity and the string mass is small compared to car mass and for the harmonically length variations small oscillation amplitudes are assumed and it is also assumed that the string mass is small compared to the total mass of the string and the car a multiple timescales perturbation method is used to construct formal asymptotic approximations of the solutions to show the complicated dynamical behavior of the string for the linearly varying length analytic approximations of the exact solution are compared with numerical solution for the harmonically varying length it will be shown that galerkin s truncation method cannot be applied in all cases to obtain approximations valid on long timescales

coupled longitudinal transverse dynamics of an axially accelerating beam abstract the coupled longitudinal transverse nonlinear dynamics of an axially accelerating beam is numerically investigated this problem is classified as a parametrically excited gyroscopic system the axial speed is assumed to be comprised of a constant mean value along with harmonic fluctuations hamilton s principle is employed to derive the equations of motion of the system which are in the form of two coupled partial differential equations the equations are discretized using the galerkin method which yields a set of coupled second order nonlinear ordinary differential equations with time dependent coefficients the sub critical dynamics of the system is examined via the pseudo arclength continuation technique while the global dynamics is investigated using direct time integration the mean axial speed and the amplitude of the speed variations are varied so as to construct the bifurcation diagrams of poincare maps the vibration specifications of the system are investigated more detailed via plotting time histories phase plane portraits and fast fourier transforms ffts

on resonances and the applicability of galerkin s truncation method for an axially moving string with time varying velocity abstract in this paper an initial boundary value problem for a linear homogeneous equation describing an axially moving string is investigated the time dependent velocity is assumed to vary harmonically about a constant mean velocity with a small fluctuation amplitude the two timescales perturbation method and the laplace transform method have been employed to the equation of motion in search of infinite mode approximate solutions all resonance cases are investigated in detail it will be shown that galerkin s truncation method cannot be applied to this problem in all cases to obtain approximations valid on long timescales

on the influence of lateral vibrations of supports for an axially moving string in this paper the transverse oscillations in travelling strings due to arbitrary lateral vibrations of the supports will be studied using the method of laplace transforms exact solutions will be constructed for the initial boundary value problems which describe these transverse oscillations

scale disparities in the complex swift hohenberg equation for lasers the complex swift hohenberg csh equation is a generic order parameter equation that applies to many physical systems in the case of class c lasers it can be obtained from the maxwell bloch equations using the assumptions of slow envelope and small detuning we show that the resulting csh equation inevitably contains different asymptotic order terms associated with the dominance of the effect of dispersion over diffusion these scale disparities are usually overlooked or simply not mentioned in the literature assuming that a csh equation with all terms of the same order still provides qualitative information in this paper the asymptotically nonuniform csh equation is carefully deduced using a simpler scaling free procedure and a stability analysis of the simplest solutions together with some numerical simulations are presented in which the mentioned scale disparities are clearly seen

an asymptotic theory for a weakly nonlinear beam equation with a quadratic perturbation in this paper an initial boundary value problem for a weakly nonlinear beam equation with a quadratic nonlinearity will be studied the initial boundary value problem can be regarded as a simple model describing free oscillations of flexible structures like suspension bridges using a two timescales perturbation method an approximation for the solution of this initial boundary problem will be constructed for a class of initial boundary value problems the existence and uniqueness of solutions and the asymptotic validity of approximations on a large timescale are shown it will be shown that for specific values of the beam parameters complicated internal resonances occur

rational approximate solutions and error bounds for the nonsymmetric riccati matrix differential equation in this paper we determine an interval where the exact solution of the nonsymmetric riccati matrix differential equation is defined rational approximate solutions are whose error is smaller than a prefixed number e 0 are constructed in terms of the data

the riccati transformation in the solution of boundary value problems the multiple shooting method and stabilized march have long been advocated for solving boundary value problems even though certain difficulties arise for problems with rapidly increasing and decreasing fundamental solution components we interpret these difficulties in terms of stability of the underlying problem and describe the stabilized march in terms of associated projections onto subspaces relating the riccati transformation to a specific implementation of the stabilized march leads naturally to an alternative method which involves only stable initial value problems and overcomes major difficulties of these other methods the potential of the method is demonstrated in some numerical examples

non linear vibrations of a beam mass system under different boundary conditions an euler bernoulli beam and a concentrated mass on this beam are considered as a beam mass system the beam is supported by immovable end conditions thus leading to stretching during the vibrations this stretching produces cubic non linearities in the equations forcing and damping terms are added into the equations the dimensionless equations are solved for five different set of boundary conditions approximate solutions of the equations are obtained by using the method of multiple scales a perturbation technique the first terms of the perturbation series lead to the linear problem natural frequencies and mode shapes for the linear problem are calculated exactly for different end conditions second order non linear terms of the perturbation series appear as corrections to the linear problem amplitude and phase modulation equations are obtained non linear free and forced vibrations are investigated in detail the effects of the position and magnitude of the mass as well as effects of different end conditions on the vibrations are determined

infinite mode analysis and truncation to resonant modes of axially accelerated beam vibrations abstract the transverse vibrations of simply supported axially moving euler bernoulli beams are investigated the beam has a time varying axial velocity with viscous damping traveling beam eigenfunctions with infinite number of modes are considered approximate analytical solutions are sought using the method of multiple scales a perturbation technique a detailed analysis of the resonances in which upto four modes of vibration involved are performed stability analysis is treated for each type of resonance approximate stability borders are given for the resonances involving only two modes for higher number of modes involved in a resonance sample numerical examples are presented for stabilities

a reflected wave superposition method for vibration and energy of a travelling string this paper considers the analytical free time domain response and energy in an axially translating and laterally vibrating string the domain of the string is either a constant or variable length dependent upon the general initial conditions the translating tensioned strings possess either fixed fixed or fixed free boundaries an alternative analytical solution using a reflected wave superposition method is presented for a finite translating string firstly the cycles of vibration for both constant and variable length strings are provided which for the latter are dependent upon the variable string length each cycle is divided into three time intervals according to the magnitude and the direction of the translating string velocity applying d alembert s method combined with the reflection properties expressions for the reflected waves at the two boundaries are obtained subsequently superposition of all of the incident and reflected waves provides results for the free vibration of the string over the three time intervals the variation in the total mechanical energy of the string system is also shown the accuracy and efficiency of the proposed method are confirmed numerically by comparison to simulations produced using a newmark beta method solution and an existing state space function representation of the string dynamics

universal description of laser dynamics near threshold abstract complex order parameter descriptions of large aspect ratio single longitudinal mode two level lasers with flat end reflectors valid near onset of lasing and for small detunings of the laser from the peak gain are given in terms of a complex swift hohenberg equation for class a and c lasers and by a complex swift hohenberg equation coupled to a mean flow for the case of a class b laser the latter coupled system is a physically consistent generalized rate equation model for wide aperture stiff laser systems these universal order parameter equations provide a connection between spatially homogeneous oscillating states of the complex ginzburg landau equation description of the laser system valid for finite negative detunings and traveling wave states described by coupled newell whitehead segel equations valid for finite positive detunings one of the main conclusions of the present paper is that the usual eckhaus instability boundary associated with a long wavelength phase instability and which delineates the region of the stable traveling wave solutions for class a and c lasers no longer defines the stability boundary for the mathematically stiff class b laser instead a short wavelength phase instability appears causing the stability domain to shrink as a function of increasing stiffness of the system this prediction is consistent with the strong spatiotemporal filamentation instabilities experimentally observed in a borad area semiconductor laser a class b system

computing the dynamic response of an axially moving continuum the aim of this study was to apply an approximate galerkin finite element method to solve the initial boundary value problem of a viscously damped axially moving pre tensioned beam including arbitrary support excitations in order to validate the galerkin finite element method the results were compared with exact solutions when available and with solutions obtained using the finite difference and galerkin methods the energy dissipation of the system was considered in the form of an equivalent viscous damping model it was also shown that for certain values of the parameters especially at high velocities the galerkin method using stationary string eigenfunctions can give a poor prediction of the dynamic response

transverse vibration of an axially accelerating string abstract the transverse vibration of an axially accelerating string is investigated the equation of motion is developed using hamilton s principle the resulting partial differential equations are discretized using galerkin s method assuming the axial velocity to be periodic a stability analysis is performed using floquet theory one two three four six and eight term series approximations are considered in the galerkin s method the one term approximation leads to a mathieu equation the solution of which is well known the numerical results for one term are compared with the analytical solution for the mathieu equation and they are in full agreement the two term approximation leads to gyroscopically coupled equations and the solutions differ significantly from that of the one term approximation whereas the three term approximation solutions look similar to the one term approximation solutions the analysis is carried out for higher order even approximations and the solutions are in qualitative agreement the two and four term approximation solutions are compared with the analytical results from hsu s method and are in reasonable agreement the results show that instabilities occur at much higher amplitudes and frequencies of the periodic axial velocity than that of typical devices such as tape machines and band saws

transition behaviour from string to beam for an axially accelerating material an approximate analytical expression for the natural frequency is given for the problem for velocity profiles harmonically varying about a mean velocity stability borders are determined analytically for fluctuation frequency versus amplitudes beam effects cause the stability boundaries to shift to higher frequency values

vibrations of an axially moving beam with time dependent velocity the dynamic response of an axially accelerating elastic tensioned beam is investigated the time dependent velocity is assumed to vary harmonically about a constant mean velocity these systems experience a coriolis acceleration component which renders such systems gyroscopic the equation of motion is solved by using perturbation analysis principal parametric resonances and combination resonances are investigated in detail stability boundaries are determined analytically it is found that instabilities occur when the frequency of velocity fluctuations is close to two times the natural frequency of the constant velocity system or when the frequency is close to the sum of any two natural frequencies when the velocity variation frequency is close to zero or to the difference of two natural frequencies however no instabilities are detected up to the first order of perturbation numerical results are presented for different flexural stiffness values and for the first two modes

a generalized treatment of the energetics of translating continua part i strings and second order tensioned pipes the energetics of translating one dimensional uniform strings and highly tensioned pipes with vanishing bending stiffness and flowing fluid are analyzed for fixed free and damped boundary conditions the interaction between the translating continua and the boundary supports causes energy transfer at a fixed boundary the transverse component of tension does work and the coriolis forces at a free end cause energy flux into the second order continuum under a symmetric boundary configuration the total energy of free oscillation varies periodically at the fundamental natural frequency asymmetric boundary supports in the pipe fluid system lead to damped or self excited motions at a viscously damped boundary the condition for maximal energy dissipation the destabilizing effect of dissipation and the stabilizing effect of negative damping are examined analytically using travelling wave solutions the energies transferred at the different boundary supports are quantified by energy reflection coefficients which are determined completely by the boundary conditions numerical simulations verify the analytically predicted energy variations

stability and bifurcation of an axially moving beam tuned to three to one internal resonances abstract this study analyzed the nonlinear vibration of an axially moving beam subject to periodic lateral force excitations attention is paid to the fundamental and subharmonic resonances since the excitation frequency is close to the first two natural frequencies of the system the incremental harmonic balance ihb method was used to evaluate the nonlinear dynamic behaviour of the axially moving beam the stability and bifurcations of the periodic solutions for given parameters were determined by the multivariable floquet theory using hsu s method the solutions obtained from the ihb method agreed very well with those obtained from numerical integration furthermore numerical examples are given to illustrate the effects of the three to one internal resonance on the response of the system

multidimensional lindstedt poincare method for nonlinear vibration of axially moving beams abstract the multidimensional lindstedt poincare mdlp method is extended to the nonlinear vibration analysis of axially moving systems galerkin method is used to discretize the governing equations the forced response of an axially moving beam with internal resonance between the first two transverse modes is studied the fundamental harmonic resonance is studied the response curves exhibit the same internal resonance characteristics as that of non transferring thin plates and beams because all these systems possess cubic nonlinearity and similar frequency distribution the examples show that the results of the mdlp method agree reasonably well with that obtained by the incremental harmonic balance ihb method however the former is more straightforward and efficient for obtaining the solution

on initial boundary value problems for weakly semilinear telegraph equations asymptotic theory and application in this paper an asymptotic theory for a class of initial boundary value problems for weakly semilinear telegraph equations is presented the theory implies the well posedness of the problem and the validity of formal approximations on long time scales as an application of the theory an initial boundary value problem for the equation u tt u xx u varepsilon u 3 0 is considered to construct an o varepsilon approximation of the solution of this problem a two time scale perturbation method is applied

on the transversal vibrations of an axially moving continuum with a time varying velocity transient from string to beam behavior in this paper an initial boundary value problem for a linear equation describing an axially moving stretched beam will be considered the velocity of the beam is assumed to be time varying since the order of magnitude of the bending stiffness terms depends on the vibration modes and the frequencies involved a combination of two simplified models that is a string equation for the lower frequencies and a beam with string effect equation for the higher frequencies will be used to describe the transversal vibrations of the system accurately based on the calculations of the natural frequencies the regions of applicability of these sub models will be determined a two time scales perturbation method will be used to construct formal asymptotic approximations of the solutions non resonant and some resonant cases will be studied for four different values of the relative errors an important implication of the earlier results in the literature is that for these types of axially moving continua problems the use of only string like models is not appropriate to describe the dynamics of these types of axially moving continua problems correctly one has to include small bending stiffness in the model in this paper it is explicitly shown how to work with a combined model that is a string model at the low frequencies and a tensioned beam model at the higher frequencies

dynamic stability in parametric resonance of axially accelerating viscoelastic timoshenko beams abstract this paper investigates dynamic stability of an axially accelerating viscoelastic beam undergoing parametric resonance the effects of shear deformation and rotary inertia are taken into account by the timoshenko thick beam theory the beam material obeys the kelvin model in which the material time derivative is used the axial speed is characterized as a simple harmonic variation about the constant mean speed the governing partial differential equations are derived from newton s second law euler s angular momentum principle and the constitutive relation the method of multiple scales is applied to the equations to establish the solvability conditions in summation and principal parametric resonances the sufficient and necessary condition of the stability is derived from the routh hurvitz criterion some numerical examples are presented to demonstrate the effects of related parameters on the stability boundaries

dynamics of a string moving with time varying speed the paper is dedicated to mathematical analysis of the idealized model of a string moving at a variable speed resonance case is studied and it is shown that an infinite system of equations cannot be reduced some exact solutions are obtained

non linear vibrations of a simple simple beam with a non ideal support in between abstract a simply supported euler bernoulli beam with an intermediate support is considered non linear terms due to immovable end conditions leading to stretching of the beam are included in the equations of motion the concept of non ideal boundary conditions is applied to the beam problem in accordance the intermediate support is assumed to allow small deflections an approximate analytical solution of the problem is found using the method of multiple scales a perturbation technique ideal and non ideal frequencies as well as frequency response curves are contrasted

stability analysis of an axially accelerating string the dynamic response of an axially accelerating string is investigated the time dependent velocity is assumed to vary harmonically about a constant mean velocity approximate analytical solutions are sought using two different approaches in the first approach the equations are discretized first and then the method of multiple scales is applied to the resulting equations in the second approach the method of multiple scales is applied directly to the partial differential system principal parametric resonances and combination resonances are investigated in detail stability boundaries are determined analytically it is found that instabilities occur when the frequency of velocity fluctuations is close to two times the natural frequency of the constant velocity system or when the frequency is close to the sum of any two natural frequencies when the velocity variation frequency is close to zero or to the difference of two natural frequencies however no instabilities are detected up to the first order of perturbation numerical results are presented for a band saw and a threadline problem

some aspects of using iv software to solve bvp via a riccati transformation we consider a numerical method for the solution of linear boundary value problems bvps with separated boundary conditions bcs for ordinary differential equations odes by using a particular linear transformation the problem is reformulated as three initial value problems ivps one of them being a riccati differential equation the nonlinearity of the riccati de makes it necessary to deal with the eventuality of modifying the structure of integration so called multiple embedding equations must thus be derived and a marching algorithm is incorporated a stability analysis of the three ivps indicates clearly for what class of problems the method is suited and why stiff iv methods should then frequently be used this leads us to consider the undesirable effect of superstability of implicit schemes which can occur when integrating the riccati de we interpret this occurrence in terms of the original bvp and the resulting analysis suggests some final considerations including the possibility of corrective measures

transient vibration of gyroscopic systems with unsteady superposed motion abstract the equation of motion for a gyroscopic system with unsteady superposed motion is derived for the prototypical problem in which motion of an oscillating particle is measured relative to a non inertial frame the resulting coefficient matrices are time dependent and skew symmetric acceleration terms are present both as coriolis acceleration and as a component of net stiffness such mathematical structure is also demonstrated in the context of other unsteady gyroscopic systems including flexible media that translate with time dependent speed following the asymptotic approach of krylov bogoliubov and mitropolsky a perturbation method is developed for the case in which the superposed motion varies slowly when viewed on the time scale of the system s natural periods of oscillation first order approximations for the modal amplitude and phase are obtained in closed form the method is illustrated through two examples of technical interest a two degree of freedom model of a rotating shaft and a distributed parameter model of a moving tape or web