exponential stabilization of an axially moving string with geometrical nonlinearity by linear boundary feedback this paper addresses the vibration suppression problem of an axially moving string using boundary control taking into account large transverse motion of the string via the second order component of lagrangian strain a nonlinear oscillation equation is derived for the problem by employing the hamilton s principle direct lyapunov method is then adopted to demonstrate that a linear boundary control of negative speed feedback ensures the vibration response of the moving string exponentially stable the main contributions are a new stability condition in terms of controller gain parameter and a detailed demonstration on the equilibrium condition at the boundary where the control is applied

stabilization of an axially moving web via regulation of axial velocity in this paper a novel control algorithm for suppression of the transverse vibration of an axially moving web system is presented the principle of the proposed control algorithm is the regulation of the axial transport velocity of an axially moving beam so as to track a profile according to which the vibration energy decays most quickly the optimal control problem that generates the proposed profile of the axial transport velocity is solved by the conjugate gradient method the galerkin method is applied in order to reduce the partial differential equation describing the dynamics of the axially moving beam into a set of ordinary differential equations odes for control design purposes these odes are rewritten into state space equations the vibration energy of the axially moving beam is represented by the quadratic form of the state variables in the optimal control problem the cost function modified from the vibration energy function is subjected to the constraints on the state variables and the axial transport velocity is considered as a control input numerical simulations are performed to confirm the effectiveness of the proposed control algorithm

the numerical study of the nonlinear dynamics of a light axially moving string abstract the vibration characteristics of a light axially moving band are investigated by a numerical study in the subcritical and supercritical speed ranges equations of motion for the geometrically nonlinear axially moving string are formulated using hamilton s principle and discretized by the finite element method the periodic nonlinear problem for the string is solved by the fourier galerkin newton fgn method the nonlinear dynamic behaviour of an axially moving band is examined through the dependences between the fundamental frequency axial velocity and vibration amplitude resulting from nonlinear free vibration in general the behaviour of the nonlinear axially moving string is similar to that of a nonlinear beam the most notable differences being the fact that the string does not undergo bifurcation from a straight configuration to curved equilibrium states in a supercritical transport speed regime nor does the equilibrium state remain straight

the transient amplitude of the viscoelastic travelling string an integral constitutive law abstract the problem considered is an initially stressed viscoelastic string subjected to steady state and harmonic variation of axial motion the string material is assumed to be constituted by the hereditary integral type the partial differential integral equation of motion is derived first then by applying galerkin s method the governing equation is reduced to a set of second order non linear differential integral equations which are solved by finite difference numerical integration procedures the viscoelastic string with a linear spring dashpot model is considered as the voigt element in series with a spring three parameter model the qualitative aspect of parametric excitation due to the non uniform travelling velocity of a viscoelastic string is investigated finally the results are compared with those obtained by a constitutive law of a differential type method the effects of elastic and viscoelastic parameters constant and non constant transport speeds the wave propagation speed ratio and the non linear term on the transient amplitudes are also investigated

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

symmetries of distributional domain wall geometries generalizing the lie derivative of smooth tensor fields to distribution valued tensors we examine the killing symmetries and the collineations of the curvature tensors of some distributional domain wall geometries the chosen geometries are rigorously the distributional thin wall limit of self gravitating scalar field configurations representing thick domain walls and the permanence and or the rising of symmetries in the limit process is studied we show that for all the thin wall space times considered the symmetries of the distributional curvature tensors turns out to be the killing symmetries of the pullback of the metric tensor to the surface where the singular part of these tensors is supported remarkably enough for the nonreflection symmetric domain wall studied these killing symmetries are not necessarily symmetries of the ambient space time on both sides of the wall

products and compositions with the dirac delta function the need to define pointwise products and compositions with distributions is pointed out in the context of the problems of renormalisation junction conditions and curved shock waves earlier definitions are briefly reviewed and new definitions are proposed using non standard analysis basic properties are established and some products and compositions with the delta distribution are explicitly evaluated with these definitions the domain of validity of the nonlinear differential equations of classical field theory can be extended to include rankine hugoniot equations are derived from the euler equations an immediate application to quantum field theory is pointed out

a generalized treatment of the energetics of translating continua part ii beams and fluid conveying pipes abstract the energetics of translating tensioned beams and fluid transporting pipes under fixed simply supported and free boundary conditions are analyzed in a generalized manner the conservative and non conservative forces acting at the boundaries lead to energy transfer between the translating continua and the boundary supports the forces and associated convective velocities are identified from the one dimensional transport theorem the group velocity and the wavenumber of propagating and evanescent waves in the dispersive continua are defined the time variation of total energy is represented in terms of the impedances and the reflection coefficients of the propagating waves in the continuum and the dynamic stability of the translating continua is discussed based on the energy expressions the critical fluid speed in a cantilevered pipe at which the resultant energy flux into the pipe at the free end vanishes is determined by the use of travelling wave solutions

robust adaptive boundary control of a flexible marine riser with vessel dynamics in this paper robust adaptive boundary control for a flexible marine riser with vessel dynamics is developed to suppress the riser s vibration to provide an accurate and concise representation of the riser s dynamic behavior the flexible marine riser with vessel dynamics is described by a distributed parameter system with a partial differential equation pde and four ordinary differential equations odes boundary control is proposed at the top boundary of the riser based on lyapunov s direct method to regulate the riser s vibration adaptive control is designed when the system parametric uncertainty exists with the proposed robust adaptive boundary control uniform boundedness under the ocean current disturbance can be achieved the proposed control is implementable with actual instrumentation since all the required signals in the control can be measured by sensors or calculated by a backward difference algorithm the state of the system is proven to converge to a small neighborhood of zero by appropriately choosing design parameters simulations are provided to illustrate the applicability and effectiveness of the proposed control

asymptotic stabilization of a nonlinear axially moving string by adaptive boundary control in this paper a robust adaptive boundary control for an axially moving string that shows nonlinear behavior resulting from spatially varying tension is investigated a hydraulic actuator equipped with a damper is used as the control actuator at the right boundary of the string the lyapunov redesign method is employed to derive a robust control algorithm employing adaptation laws that estimate three unknown system parameters mass per unit length of string lumped mass of hydraulic actuator and damping coefficient of damper and an unknown boundary disturbance the uniform asymptotic stability when the three parameters are all unknown the exponential stability when they are known and the uniform ultimate boundedness with a bounded boundary disturbance of the closed loop system are investigated the convergence of the parameter estimates to the true values is shown numerical simulations are performed to demonstrate the effectiveness of the proposed robust adaptive boundary control

optimal investments for robust utility functionals in complete market models this paper introduces a systematic approach to the problem of maximizing the robust utility of the terminal wealth of an admissible strategy in a general complete market model where the robust utility functional is defined by a set q of probability measures our main result shows that this problem can often be reduced to determining a least favorable measure q0 q which is universal in the sense that it does not depend on the particular utility function the robust problem is thus equivalent to a standard utility maximization problem with respect to the subjective probability measure q0 by using the huber strassen theorem from robust statistics it is shown that q0 always exists if q is the i core of a 2 alternating capacity besides other examples we also discuss the problem of robust utility maximization with uncertain drift in a black scholes market and the case of weak information

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

vibrations and stability of axially traveling laminated beams abstract in this paper vibrations and stability of an axially traveling laminated composite beam are investigated analytically via the method of multiple scales based on classical laminated beam theory the governing equations of motion for a time variant axial speed are obtained using newton s second law of motion and constitutive relations the method of multiple scales an approximate analytical method is applied directly to the gyroscopic governing equations of motion and complex eigenfunctions and natural frequencies of the system are obtained the stability boundaries of the system near resonance are determined via the routh hurwitz criterion finally a parametric study is conducted which considers the effects of laminate type and configuration as well as the mean speed and amplitude of speed fluctuations on the vibration response natural frequencies and stability boundaries of the system

vibration suppression of a string through cyclic application and removal of constraints abstract a new method for the control of string vibration is presented the method relies on application and removal of a constraint at one point on the string close to one of its boundaries application of the constraint temporarily results in two vibrating strings one of which is significantly shorter in length than the other the vibration of the shorter length string decays rapidly due to high damping and results in reduction in the overall energy of the system removal of the constraint does not change the energy of the system but allows the constraint to be applied repeatedly for vibration suppression a mathematical model of the cycle of constraint application and removal is presented and multiple cycles of constraint application and removal are simulated experiments were performed with a coiled extension spring which behaves like a lightly damped string simulation and experimental results match well with each other and demonstrate the efficacy of the simple control strategy

boundary control design for extensible marine risers in three dimensional space abstract a design of boundary controllers is proposed for practical exponential stabilization of extensible marine risers in three dimensional 3d space under sea loads the design removes flaws in existing works two lyapunov type theorems are developed for study of existence and uniqueness and stability of nonlinear evolution systems in hilbert space these theorems have their potential use in control design and stability analysis for flexible systems including marine risers

mean stochastic comparison of diffusions stochastic bounds are derived for one dimensional diffusions and somewhat more general random processes by dominating one process pathwise by a convex combination of other processes the method permits comparison of diffusions with different diffusion coefficients one interpretation of the bounds is that an optimal control is identified for certain diffusions with controlled drift and diffusion coefficients when the reward function is convex an example is given to show how the bounds and the liapunov function technique can be applied to yield bounds for multidimensional diffusions

vibration suppression of a non linear axially moving string by boundary control the string is pulled at a constant speed through two eyelets which are distanced from each other by one unit of length one of the eyelets is fixed and the other one can move freely in the direction of the y axis a control input force can be applied to the free to move eyelet transversally in the direction of y

vibration of an axially moving string with geometric non linearity and translating acceleration the vibration of an axially moving string is studied when the string has geometric non linearity and translating acceleration based upon the von karman strain theory the equations of motion are derived considering the longitudinal and transverse deflections the equation for the longitudinal vibration is linear and uncoupled while the equation for the transverse vibration is non linear and coupled between the longitudinal and transverse deflections these equations are discretized by the galerkin method after they are transformed into the variational equations i e the weak forms so that the admissible and comparison functions can be used for the bases of the longitudinal and transverse deflections respectively with the discretized equations the natural frequencies the time histories of the deflections and the distributions of the deflection and stress are investigated in addition comparisons between the results of linear and non linear theories are provided

technical communique exponential stabilization of an axially moving string by linear boundary feedback the system modeled by an axially moving string and a mass damper spring mds controller applied at the right hand side rhs boundary of the string is considered the feature of the controller is not restricted at all and is determined by the system implicitly the mathematical model of this system is composed of an ordinary differential equation ode describing the mds and a partial differential equation pde describing the string the c 0 semigroup theory provides an elegant state space representation for the analysis of the coupled system the exponential stability is verified through the total mechanical energy dissipation and the semigroup theory the lower bound of the feedback gain is obtained

vibration control of an euler bernoulli beam under unknown spatiotemporally varying disturbance in this article the vibration suppression of an euler bernoulli beam system is considered by using the adaptive boundary control technique the dynamics of the beam are represented by a partial differential equation and four ordinary differential equations involving functions of both space and time by using lyapunov synthesis the robust boundary control with a disturbance observer is first proposed to suppress the vibration and attenuate the effect of the external disturbances to compensate for both the system parametric uncertainties and the disturbances uncertainties the adaptive boundary control is developed with the proposed boundary control the state of the beam system is proven to be uniformly ultimately bounded and converge to a small neighbourhood of zero by appropriately choosing the design parameters the effectiveness of the proposed control is successfully verified by numerical simulations

nonlinear transversal vibration and stability of an axially moving viscoelastic string supported by a partial viscoelastic guide abstract in this paper transversal nonlinear vibration of an axially moving viscoelastic string supported by a partial viscoelastic guide is analytically investigated the string is traveling under time variant velocity which includes a mean velocity along with small harmonic fluctuations the model of the viscoelastic guide is also a parallel combination of springs and viscous dampers the governing partial differential equation is derived from hamilton s principle and geometrical relations the method of multiple scales is applied to the governing partial differential equation to obtain solvability conditions for both non resonance and principal parametric resonance cases additionally in the case of principal parametric resonance the stability and bifurcation of trivial and non trivial steady state responses are analyzed through the routh hurwitz criterion eventually numerical simulations are presented to highlight the effects of mean velocity guide length stiffness and damping coefficient of the guide and viscosity coefficient of the string on the natural frequencies stability frequency response curves and bifurcation points of the system

dynamic characteristics of the out of plane vibration for an axially moving membrane the dynamic characteristics of the out of plane vibration for an axially moving membrane are investigated in this study the equations of in plane and out of plane motions are derived by using the extended hamilton principle two different types of boundary conditions are considered at the rollers one is the condition of no friction and the other is the condition of no slipping after discretizing the equations by the galerkin method the natural frequencies and mode shapes are computed from the computation results it is found that the translating speed boundary conditions and aspect ratio of the membrane have effects on the natural frequencies mode shapes and stability for the out of plane vibration of the moving membrane

stability of an axially accelerating string subjected to frictional guiding forces abstract the dynamic response of an axially translating continuum subjected to the combined effects of a pair of spring supported frictional guides and axial acceleration is investigated such systems are both non conservative and gyroscopic the continuum is modeled as a tensioned string translating between two rigid supports with a time dependent velocity profile the equations of motion are derived with the extended hamilton s principle and discretized in the space domain with the finite element method the stability of the system is analyzed with the floquet theory for cases where the transport velocity is a periodic function of time direct time integration using an adaptive step runge kutta algorithm is used to verify the results of the floquet theory this approach can also be employed in the general case of arbitrary time varying velocity results are given in the form of time history diagrams and instability point grids for different sets of parameters such as the location of the stationary load the stiffness of the elastic support and the values of initial tension this work showed that presence of friction adversely affects stability but using non zero spring stiffness on the guiding force has a stabilizing effect this work also showed that the use of the finite element method and floquet theory is an effective combination to analyze stability in gyroscopic systems with stationary friction loads

vibrations of an axially accelerating beam with small flexural stiffness transverse vibrations of an axially moving beam are considered the axial velocity is harmonically varying about a mean velocity the equation of motion is expressed in terms of dimensionless quantities the beam effects are assumed to be small since in this case the fourth order spatial derivative multiplies a small parameter the mathematical model becomes a boundary layer type of problem approximate solutions are searched using the method of multiple scales and the method of matched asymptotic expansions results of both methods are contrasted with the outer solution

robust boundary control of an axially moving string by using a pr transfer function in this note a scheme for the vibration suppression of a translating string using a positive real pr transfer function is investigated the transverse vibration of the string is controlled by hydraulic touch rolls located at the right end of the string the mathematical model of the system which consists of a hyperbolic partial differential equation pde describing the dynamics of the moving string and an ordinary differential equation ode for the actuator dynamics is derived by using hamilton s principle for translating continua the transfer function of the proposed boundary controller is a nonproper but pr function the asymptotic stability of the closed loop system in the presence of output disturbance is proved

non linear parametric vibration and stability of axially moving visco elastic rayleigh beams abstract an axially moving visco elastic rayleigh beam with cubic non linearity is considered and the governing partial differential equation of motion for large amplitude vibration is derived through geometrical constitutive and dynamical relations by directly applying the method of multiple scales to the governing equations of motion and considering the solvability condition the linear and non linear frequencies and mode shapes of the system are analytically formulated in the presence of damping terms it can be seen that the amplitude is exponentially time dependent and as a result the non linear natural frequencies of the system will be time dependent for the resonance case through considering the solvability condition and routh hurwitz criterion the stability conditions are developed analytically eventually the effects of system parameters on the vibrational behavior stability and bifurcation points of the system are investigated through parametric studies

the equivalence of darmois israel and distributional method for thin shells in general relativity a distributional method to solve the einstein s field equations for thin shells is formulated the familiar field equations and jump conditions of darmois israel formalism are derived a careful analysis of the bianchi identities shows that for cases under consideration they make sense as distributions and lead to jump conditions of darmois israel formalism

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

steady state response of axially moving viscoelastic beams with pulsating speed comparison of two nonlinear models abstract principal parametric resonance in transverse vibration is investigated for viscoelastic beams moving with axial pulsating speed a nonlinear partial differential equation governing the transverse vibration is derived from the dynamical constitutive and geometrical relations under certain assumption the partial differential reduces to an integro partial differential equation for transverse vibration of axially accelerating viscoelastic nonlinear beams the method of multiple scales is applied to two equations to calculate the steady state response closed form solutions for the amplitude of the vibration are derived from the solvability condition of eliminating secular terms the stability of straight equilibrium and nontrivial steady state response are analyzed by use of the lyapunov linearized stability theory numerical examples are presented to highlight the effects of speed pulsation viscoelascity and nonlinearity and to compare results obtained from two equations

the dual theory of choice under risk in this essay a new theory of choice under risk is being proposed it is a theory which in a sense that will become clear is dual to expected utility theory hence the title dual theory risky prospects are evaluated in this theory by a cardinal numerical scale which resembles an expected utility except that the roles of payments and probabilities are reversed this theme the reversal of the roles of probabilities and payments will recur throughout the paper i should emphasize that playing games with probabilities masquerading as payments and payments masquerading as probabilities is not my object rather i hope to convince the reader that the dual theory has intrinsic economic significance and that in some areas its predictions are superior to those of expected utility theory while in other areas the reverse will be the case two reasons have prompted me to look for an alternative to expected utility theory the first reason is methodological in expected utility theory the agent s attitude towards risk and the agent s attitude towards wealth are forever bonded together at the level of fundamental principles risk aversion and diminishing marginal utility of wealth which are synonymous under expected utility theory are horses of different colors the former expresses an attitute towards risk increased uncertainty hurts while the latter expresses an attitude towards wealth the loss of a sheep hurts more when the agent is poor than when the agent is rich a question arises therefore as to whether these two notions can be kept separate from each other in a full fledged theory of cardinal utility the dual theory will have this property the second reason that leads me to look for an alternative to expected utility theory is empirical behavior patterns which are systematic yet inconsistent with expected utility theory have often been observed two prominent references among many others are allais 1953 and kahneman tversky 1979 so deeply

transverse vibrations of elastically connected double string complex system part i free vibrations this paper analyzes the forced transverse vibrations of an elastically connected double string complex continuous system the general solutions of forced vibrations of strings subjected to arbitrarily distributed continuous loads are found by using the method of expansion in a series of the mode shape functions different cases of exciting loadings are analyzed the action of stationary harmonic loads and moving concentrated forces is considered vibrations caused by the harmonic exciting forces are discussed and conditions of resonance and dynamic vibration absorption are formulated thus the string type dynamic absorber can be used to suppress the excessive vibrations of corresponding string systems the dynamic string type damper is a new concept of a continuous dynamic vibration absorber cdva it is shown that a corresponding two degree of freedom discrete system is an analogue of an elastically connected double string complex system theoretical analysis is illustrated by a numerical example

comparison theorems for neutral stochastic functional differential equations abstract the comparison theorems under wu and freedman s order are proved for neutral stochastic functional differential equations with finite or infinite delay whose drift terms satisfy the quasimonotone condition and diffusion term is the same

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

bounded input bounded output stability of a damped nonlinear string in this paper we prove that the displacement of a damped nonlinear string under bounded distributed forces is bounded the proof is achieved by showing that an energy like lyapunov function corresponding to the string is bounded

boundary control of two dimensional marine risers with bending couplings abstract the aim of this paper is to design a boundary controller for global stabilization of two dimensional marine risers under environmental disturbances based on the energy approach equations of motion including bending couplings for the risers are derived due to the couplings the riser dynamics exhibit mutual effects between transverse motions the lyapunov direct method is used to design the boundary controller proof of the existence and uniqueness of the solutions of the closed loop system is provided based on the galerkin approximation method stability analysis of the closed loop system is carried out using the lyapunov stability theory numerical simulations illustrate the results

technical communique boundary control of the axially moving kirchhoff string in this note the axially moving kirchhoff string is considered it is proved that this non linear string can be stabilized by the linear boundary control which is the negative feedback of the transversal velocity of the string at one end furthermore it is concluded that the axially moving kirchhoff string is stable without an active control when one of its ends is free to move transversally

nonlinear dynamics of axially moving plates abstract the nonlinear dynamics for forced motions of an axially moving plate is numerically investigated using von karman plate theory and retaining in plane displacements and inertia the equations of motion are obtained via an energy method based on lagrange equations this yields a set of second order nonlinear ordinary differential equations with coupled terms the equations are transformed into a set of first order nonlinear ordinary differential equations and are solved via the pseudo arclength continuation technique the near resonance nonlinear dynamics is examined via plotting the frequency response curves of the system results are shown through frequency response curves time histories and phase plane diagrams the effect of system parameters such as the axial speed and the pretension on the resonant responses is also highlighted

free and forced response of an axially moving string transporting a damped linear oscillator abstract the transverse response of a cable transport system which is modelled as an ideal constant tension string travelling at constant speed between two supports with a damped linear oscillator attached to it is predicted for arbitrary initial conditions external forces and boundary excitations the exact formulation of the coupled system reduces to a single integral equation of volterra type governing the interaction force between the string and the payload oscillator the time history of the interaction force is discontinuous for non vanishing damping of the oscillator these discontinuities occur at the instants when transverse wavefronts propagating along the string interact with the oscillator the discontinuities are treated using the theory of distributions numerical algorithms for computing the integrals involving generalized functions and for solution of the delay integro differential equation are developed the response analysis shows a discontinuous velocity history of the payload attachment point special conditions leading to absence of the above discontinuities are given

adaptive vibration isolation for axially moving strings theory and experiment high speed transport of continuous materials such as belts webs filaments or bands can cause unwanted vibration vibration control for these systems often focuses on restricting the response resulting from external disturbances e g support roller eccentricity or aerodynamic excitation to areas not requiring high precision positioning this paper introduces vibration controllers for an axially moving string system consisting of a controlled span coupled to a disturbed span via an actuator the system model includes a partial differential equation for the two spans and an ordinary differential equation for the actuator exact model knowledge and adaptive isolation controllers based on lyapunov theory regulate the controlled span from bounded disturbances in the adjacent uncontrolled span assuming distributed damping in the uncontrolled span the exact model knowledge and adaptive controllers exponentially and asymptotically drive the controlled span displacement to zero respectively while ensuring bounded uncontrolled span displacement and control force experiments demonstrate the effectiveness of the proposed controller in isolating the controlled span from disturbances and damping the controlled span displacement

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

decay of solutions of wave equations in a bounded region with boundary dissipation abstract an energy decay rate is obtained for solutions of wave type equations in a bounded region in rn whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface

robust adaptive boundary control of an axially moving string under a spatiotemporally varying tension in this paper a vibration suppression scheme for an axially moving string under a spatiotemporally varying tension and an unknown boundary disturbance is investigated the lower bound of the tension variation is assumed to be sufficiently larger than the derivatives of the tension the axially moving string system is divided into two spans i e a controlled span and an uncontrolled span by a hydraulic touch roll actuator which is located in the middle section of the string the transverse vibration of the controlled span part of the string is controlled by the hydraulic touch roll actuator and the position of the actuator is considered as the right boundary of the controlled span part the mathematical model of the system which consists of a hyperbolic partial differential equation describing the dynamics of the moving string and an ordinary differential equation describing the actuator dynamics is derived by using the hamilton s principle the lyapunov method is employed to design a robust boundary control law and adaptation laws for ensuring the vibration reduction of the controlled span part the asymptotic stability of the closed loop system under the robust adaptive boundary control scheme is proved through the use of semigroup theory simulation results verify the effectiveness of the robust adaptive boundary controller proposed

lie group theory and analytical solutions for the axially accelerating string problem abstract transverse vibrations of a string moving with time dependent velocity v t have been investigated analytical solutions of the problem are found using the systematic approach of lie group theory group classification with respect to the arbitrary velocity function has been performed using a newly developed technique of equivalence transformations from the symmetries of the partial differential equation the method for deriving exact solutions for the arbitrary velocity case is shown special cases of interest such as constant velocity constant acceleration harmonically varying velocity and exponentially decaying velocity are investigated in detail finally for a simply supported strip approximate solutions are presented for the exponentially decaying and harmonically varying cases

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

absolute stability of the kirchhoff string with sector boundary control this paper addresses the stability problem of the nonlinear kirchhoff string with nonlinear boundary control the nonlinear boundary control is the negative feedback of the transverse velocity of the string at one end which satisfies a sector constraint employing the integral type multiplier method we establish explicit absolute exponential stability of the kirchhoff string system

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

principal parametric resonance of axially accelerating viscoelastic strings with an integral constitutive law the steady state transverse responses and the stability of an axially accelerating viscoelastic string are investigated the governing equation is derived from the eulerian equation of motion of a continuum which leads to the mote model for transverse motion the kirchhoff model is derived from the mote model by replacing the tension with the averaged tension over the string the method of multiple scales is applied to the two models in the case of principal parametric resonance closed form expressions of the amplitudes and the existence conditions of steady state periodical responses are presented the lyapunov linearized stability theory is employed to demonstrate that the first second non trivial steady state response is always stable unstable numerical calculations show that the two models are qualitatively the same but quantitatively different numerical results are also presented to highlight the effects of the mean axial speed the axial speed fluctuation amplitude and the viscoelastic parameters

note on boundary stabilization of wave equations an energy decay rate is obtained for solutions of wave type equations in a bounded region in mathbb r n whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface unlike most previous results on this subject the results presented here are potentially valid for regions having connected boundaries

brief paper stabilization analysis of a generalized nonlinear axially moving string by boundary velocity feedback this paper addresses vibration suppression of an axially moving nonlinear string subject to boundary control at either the downstream or the upstream end the equation of motion and the boundary conditions are derived from hamilton s principle using the exact expression of the strain to describe the geometrical nonlinearity due to the finite transverse deformation we can approximate this model by various simpler models found in the literature we show that the oscillations of the moving string can be stabilized exponentially via linear negative velocity feedback acting at one end of the string our proofs rely on lyapunov s direct method using a carefully constructed energy functional

the incremental harmonic balance method for nonlinear vibration of axially moving beams in this paper the incremental harmonic balance ihb method is formulated for the nonlinear vibration analysis of axially moving beams the galerkin method is used to discretize the governing equations a high dimensional model that can take nonlinear model coupling into account is derived the forced response of an axially moving strip with internal resonance between the first two transverse modes is studied particular attention is paid to the fundamental superharmonic and subharmonic resonance as the excitation frequency is close to the first second or one third of the first natural frequency of the system numerical results reveal the rich and interesting nonlinear phenomena that have not been presented in the existent literature on the nonlinear vibration of axially moving media

boundary control of transverse motion of marine risers with actuator dynamics this paper presents a boundary controller to reduce transverse motion of flexible marine risers driven by a hydraulic system at the top end of the risers under environmental disturbances induced by waves wind and ocean currents the boundary controller is designed based on lyapunov s direct method and the backstepping technique proof of existence and uniqueness of the solutions of the closed loop control system is carried out by using the galerkin approximation method simulation results illustrate the effectiveness of the proposed boundary controller

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

dynamic stability of a three dimensional string subjected to both magnetic and tensioned excitations this paper presents the dynamic stability of an axially moving string subjected to both actions of an alternating uniform transverse magnetic field and a periodic tension force owing to time varying actions terms with time dependent coefficients appear in the equations of motion which resutls in the existence of parametric instability

delay induced dynamics of an axially moving string with direct time delayed velocity feedback abstract the local dynamics of an axially moving string under aerodynamic forces is investigated with a time delayed velocity feedback controller the retarded differential difference governing equation is obtained in modal coordinates of a two degree of freedom system through galerkin s discretization procedure the stability of trivial equilibrium is examined with the change of counting multiplicity of eigenvalue with positive real part the hopf bifurcation curves are determined in the controlling parameter spaces with the aid of the center manifold reduction a functional analysis is carried out to reduce the modal equation to a single ordinary differential equation of one complex variable on the center manifold the approximate analytical solutions in the vicinity of hopf bifurcations are derived in the case of primary resonance the curves of excitation response and frequency response curves are shown with the effect of time delay the stability analysis for steady state periodic solutions of the reduced system indicates the onset of local control parameter for vibration control and response suppression moreover the poincare bendixson theorem and energy considerations are used to investigate the existences and characteristics of quasi periodic solutions when stability of the periodic solution is lost numerical results demonstrate the validity of the analytical prediction two different kinds of quasi periodic solutions are found