stabilization of elastic plates with dynamical boundary control we consider a hybrid system composed of a plate equation and two ordinary differential equations we prove that the system is strongly but not uniformly stable by a new approach we show that the smooth solution has a rational decay rate finally we establish the uniform energy decay rate for a simplified hybrid system

exact boundary controllability of vibrating non classical euler bernoulli micro scale beams abstract this study investigates the exact controllability problem for a vibrating non classical euler bernoulli micro beam whose governing partial differential equation pde of motion is derived based on the non classical continuum mechanics in this paper it is proved that via boundary controls it is possible to obtain exact controllability which consists of driving the vibrating system to rest in finite time this control objective is achieved based on the pde model of the system which causes that spillover instabilities do not occur

boundary value problems for nth order lipschitz equations on considere l unicite des solutions de problemes aux valeurs limites pour une equation scalaire d ordre n y n f t y y y n 1 ou f est continue sur a b r n et satisfait une condition de lipschitz

non linear vibrations of viscoelastic moving belts part i free vibration analysis abstract the non linear free vibration of viscoelastic moving belts is studied based on the linear viscoelastic differential constitutive law the generalized equation of motion is derived for a moving belt with geometric non linearities the method of multiple scales is applied directly to the governing equation which is in the form of continuous autonomous gyroscopic systems with weak non linearity this direct treatment does not involve a prior assumption regarding the spatial solutions the non linear natural frequencies and free response amplitude for autonomous systems are predicted by the perturbation method the results obtained with the quasi static assumption and those without this assumption are compared the effects of elastic and viscoelastic parameters axial moving speed and the geometric non linearity on natural frequencies and amplitudes of the free response are investigated from numerical examples

explicit lie poisson integration and the euler equations we give a wide class of lie poisson systems for which explicit lie poisson integrators preserving all casimirs can be constructed the integrators are extremely simple and have widespread applicability examples are the free rigid body a moment truncation and a new fast algorithm for the sine bracket truncation of the 2d euler equations

optimality for boundary value problems for lipschitz equations on considere les solutions de problemes aux valeurs limites pour une equation scalaire d ordre n x n f t x x x n 1 ou f est continue sur a b r n et satisfait une condition de lipschitz on determine les sous intervalles de longueur optimale de a b en fonctions des coefficients de lipschitz sur lesquels des solutions sont uniques

two point boundary value problems for second order ordinary differential equations across many resonant points abstract several optimality results on the existence and uniqueness of solutions to two point boundary value problems across many resonant points y x y 0 y 0 y0 y 1 y1 are proved by the optimal control theory method

efficient time symmetric simulation of torqued rigid bodies using jacobi elliptic functions if the three moments of inertia are distinct the solution to the euler equations for the free rigid body is given in terms of jacobi elliptic functions using the arithmetic geometric mean algorithm abramowitz and stegun 1992 handbook of mathematical functions with formulas graphs and mathematical tables new york dover these functions can be calculated efficiently and accurately compared to standard numerical ode and lie poisson solvers the overall approach yields a faster and more accurate numerical solution to the euler equations this approach is designed for mass asymmetric rigid bodies in the case of symmetric bodies the exact solution is available in terms of trigonometric functions see dullweber et al 1997 j chem phys 107 5840 51 reich 1996 fields inst commun 10 181 91 and benettin et al 2001 siam j sci comp 23 1189 203 for details in this paper we consider the case of asymmetric rigid bodies subject to external forces we consider a strategy similar to the symplectic splitting method proposed in reich 1996 fields inst commun 10 181 91 and dullweber et al 1997 j chem phys 107 5840 51 the method proposed here is time symmetric we decompose the vector field of our problem into a free rigid body frb problem and another completely integrable vector field the frb problem consists of the euler equations and a differential equation for the 3 3 orientation matrix the euler equations are integrated exactly while the matrix equation is approximated using a truncated magnus series in our experiments we observe that the overall numerical solution benefits greatly from the very accurate solution of the euler equations we apply the method to the heavy top and the simulation of artificial satellite attitude dynamics

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

an introduction to lie group integrators basics new developments and applications we give a short and elementary introduction to lie group methods a selection of applications of lie group integrators are discussed finally a family of symplectic integrators on cotangent bundles of lie groups is presented and the notion of discrete gradient methods is generalised to lie groups

asymptotic stabilization of vibrating composite plates this paper presents a solution to the boundary stabilization problem of free vibration of composite plates it is assumed that an object such as a flange which has mass and mass moment of inertia can be attached to the free portion of the boundary of the plate both cases of with and without boundary object have been analyzed for other types of plates by many researches in the presence of boundary mass and mass moment of inertia two ordinary differential equations govern their motion for both cases the plate dynamics is presented by a linear fourth order partial differential equation pde linear boundary control laws are constructed to stabilize the composite plates asymptotically the control forces and moments consist of feedbacks of the velocity and normal derivative of the velocity at the boundaries of the plate asymptotic stabilization of free vibration of composite plates in both cases i e with and without boundary object is achieved via boundary actions finally it is shown that exponentially stability of the hybrid system cannot be fulfilled in the other words the asymptotical stability is the best our main tools in this article are semigroup techniques spectral theory of the linear operators and lyapunov stability method

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

a changing chart symplectic algorithm for rigid bodies and other hamiltonian systems on manifolds we revive the elementary idea of constructing symplectic integrators for hamiltonian flows on manifolds by covering the manifold with the charts of an atlas implementing the algorithm in each chart thus using coordinates and switching among the charts whenever a coordinate singularity is approached we show that this program can be implemented successfully by using a splitting algorithm if the hamiltonian is the sum h1 h2 of two or more integrable hamiltonians profiting from integrability we compute exactly the flows of h1 and h2 in each chart and thus compute the splitting algorithm on the manifold by means of its representative in any chart this produces a symplectic algorithm on the manifold which possesses an interpolating hamiltonian and hence it has excellent properties of conservation of energy we exemplify the method for a point constrained to the sphere and for a symmetric rigid body under the influence of positional potential forces

comparison of splitting algorithms for the rigid body we compare several different second order splitting algorithms for the asymmetric rigid body with the aim of determining which one produces the smallest energy error for a given rigid body namely for given moments of inertia the investigation is based on the analysis of the dominant term of the modified hamiltonian and indicates that different algorithms can produce energy errors which differ by several orders of magnitude as a byproduct of this analysis we remark that for the special case of a flat rigid body with moments of inertia proportional to 1 0 75 0 25 one of the considered algorithms is in fact of order four

the vibration and stability of a friction guided translating string abstract eyelets capstans and cylindrical surfaces are often used in thread fiber and paper handling machinery to guide the axially moving element in addition to providing positional control these guides introduce dry friction forces that alter the vibration and stability characteristics of the system this paper examines the lateral response of a string that slides through an elastically supported dry friction guide exact expressions are derived for the linear response under free and forced conditions solutions for the eigenvalue spectrum exhibit unusual features including multiple divergence instabilities regions of flutter instability and regions of curve veering associated with mode localization a second order perturbation solution is derived to examine the behavior of the eigenvalue spectrum in regions of flutter instability and curve veering the analysis highlights the common features of these two phenomena and suggests ways to minimize vibration by adjusting various design variables the analysis also demonstrates that the eigenvalue loci in regions of flutter instability and curve veering are naturally described by a local hyperbolic approximation

wave synthesis in a non linear stretched string with time varying length or tension abstract this paper is concerned with the numerical tone synthesis of a non linear stretched string the length or tension of which varies with time 1 at a constant rate 2 at a cosine function rate and 3 at a rate determined by a handwritten input function listening to the results for the analytical waveforms transformed by a digital to analog converter indicates that the case of the constant rate time varying length corresponds closely to a guitar string positive or negative feedback compensation for the analytically obtained frequency brings the analytical waveforms close to the numerically calculated waveforms a simple method is proposed with lower order polynomials for smoothly connecting two discontinuous waveforms into continuous waveforms a fourier series synthesizer procedure is described to aid rapid real time synthesis of tones of desired timbre

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

precise and fast computation of a general incomplete elliptic integral of second kind by half and double argument transformations this is a continuation of our works to compute the incomplete elliptic integrals of the first and second kind fukushima 2010 2011 21 23 we developed a method to compute an associate incomplete elliptic integral of the third kind j f n m p f n m f f m n by the half argument formulas of the sine and cosine amplitude functions and the double argument transformation of the integral the relative errors of j f n m computed by the new method are sufficiently small as less than 20 machine epsilons meanwhile the simplicity of the adopted algorithm makes the new method run 1 5 to 3 7 times faster than carlson s duplication method the combination of the new method and that to compute simultaneously two other associate incomplete elliptic integrals of the second kind b f m e f m 1 m f f m m and d f m f f m e f m m which we established recently 23 enables a precise and fast computation of arbitrary linear combination of legendre s incomplete elliptic integrals of all three kinds f f m e f m and p f n m these new procedures share the same device the half argument transformations while the double argument transformation of j f n m includes those of b f m and d f m as its sub component as a result the simultaneous computation of the three associate integrals is significantly faster than computing them separately in fact our combined procedure is 2 7 to 5 9 times faster than the combination of carlson s duplication method to compute r d and r j

numerical implementation of the exact dynamics of free rigid bodies in this paper the exact analytical solution of the motion of a rigid body with arbitrary mass distribution is derived in the absence of forces or torques the resulting expressions are cast into a form where the dependence of the motion on initial conditions is explicit and the equations governing the orientation of the body involve only real numbers based on these results an efficient method to calculate the location and orientation of the rigid body at arbitrary times is presented this implementation can be used to verify the accuracy of numerical integration schemes for rigid bodies to serve as a building block for event driven discontinuous molecular dynamics simulations of general rigid bodies and for constructing symplectic integrators for rigid body dynamics

boundary controllability of a coupled wave kirchoff system abstract we consider two problems in boundary controllability of coupled wave kirchoff systems let be a bounded region in r n n 2 with lipschitz continuous boundary in the motivating structural acoustics application represents an acoustic cavity let 0 be a flat subset of which represents a flexible wall of the cavity let z denote the acoustic velocity potential which satisfies a wave equation in and let v denote the displacement on 0 which satisfies a kirchoff plate equation on 0 these equations are coupled via z v t on 0 where is the exterior unit normal to 0 and the backpressure z t appears in the kirchoff equation in the first problem we consider a control u 0 in the kirchoff equation on 0 and an additional control u 1 in the neumann conditions on a subset 1 of where 1 satisfies geometric conditions using both controls we obtain exact controllability of the wave and plate components in the natural state space in the second problem we consider only the control u 0 without geometric conditions exact controllability is not possible but we show that for any initial data we can steer the plate component exactly and the wave component approximately

vibration and stability analysis of a multi guided rotating string abstract the small displacement vibrations of a rotating circular string restrained by point and distributed springs are studied in the present paper the model used includes the translating string as a special case the governing equations and boundary conditions are derived using hamilton s principle analytical solutions are obtained for free vibrations and some specific cases are discussed in addition approximate solutions with trigonometric shape functions are constructed by the rayleigh ritz method numerical results are given showing the effectiveness of the rayleigh ritz formulation constructed and some features of the string vibration and stability are discussed

exponential stabilization of a transversely vibrating beam via boundary control abstract this paper presents a solution to the boundary stabilization of a beam in free transverse vibration the dynamics of the beam are presented by a non linear partial differential equation pde a linear control law is constructed to stabilize the beam the control force consists of feedback from the slope and velocity at the boundary of the beam the novelty of this article is that it has been possible to stabilize exponentially a free transversely vibrating beam via boundary control without resorting to truncation of the model

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

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

the exact computation of the free rigid body motion and its use in splitting methods this article investigates the use of the computation of the exact free rigid body motion as a component of splitting methods for rigid bodies subject to external forces we review various matrix and quaternion representations of the solution of the free rigid body equation which involve jacobi ellipic functions and elliptic integrals and are amenable to numerical computations we consider implementations which are exact i e computed to machine precision and semiexact i e approximated via quadrature formulas we perform a set of extensive numerical comparisons with state of the art geometrical integrators for rigid bodies such as the preprocessed discrete moser veselov method our numerical simulations indicate that these techniques combined with splitting methods can be favorably applied to the numerical integration of torqued rigid bodies

semidiscrete approximations of optimal robin boundary control problems constrained by semilinear parabolic pde an optimal robin boundary control problem associated with semilinear parabolic partial differential equations is considered existence of an optimal solution is proved and an optimality system of equations is derived semidiscrete finite element approximations of the optimality system are defined and error estimates are obtained

uniqueness existence and optimality for fourth order lipschitz equations for each t y y y y4 t zi z2 z3 z e a 6 x r4 in particular we will characterize optimal length subintervals of a b in terms of the lipschitz coefficients k i 1 2 3 4 on which certain two three and four point boundary value problems for 1 have unique solutions the techniques we employ here involve applications of the pontryagin maximum principle 14 p 3141 in conjunction with uniqueness implies existence results for solutions of boundary value problems for 1 these techniques are motivated by works of melentsova 15 and melentsova and mil shtein 16 171 and most notably by the two papers by jackson ll 121 furthermore the results contained herein can be considered as extensions of a recent paper by henderson 6 dealing with boundary value problems for third order equations in relating the results of this paper to previous works we will formulate the boundary value problems in terms of the nth order differential equation

preprocessed discrete moser veselov algorithm for the full dynamics of a rigid body the discrete moser veselov algorithm is an integrable discretization of the equations of motion for a free rigid body it is symplectic and time reversible and it conserves all first integrals of the system the only drawback is its low order we present a modification of this algorithm to arbitrarily high order which has negligible overhead but considerably improves the accuracy

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

lie group methods for rigid body dynamics and time integration on manifolds recently there has been an increasing interest in time integrators for ordinary differential equations which use lie group actions as a primitive in the design of the methods these methods are usually phrased in an abstract sense for arbitrary lie groups and actions we show here how the methods look when applied to the rigid body equations in particular and indicate how the methods work in general an important part of the lie group methods involves the computation of a coordinate map and its derivative various options are available and they vary in cost accuracy and ability to approximately conserve invariants we discuss how the computation of these maps can be optimized for the rigid body case and we provide numerical experiments which give an idea of the performance of lie group methods compared to other known integration schemes

a parametrically driven harmonic analysis of a non linear stretched string with time varying length abstract this paper is concerned with the first second and third harmonic lateral vibrations of a string having a forced periodic small length variation account is taken of the non linear term arising from the correction to the tension of the string due to local elongation it is found that increasing the amplitude of the length variation of the string results in a narrowing of the frequency response range of each of the harmonic oscillation components both numerical and analytical calculations of the harmonic frequency responses have been carried out the relationships for initial values between the original differential equation and the approximate differential equations for the harmonic responses are discussed in detail

reducing round off errors in rigid body dynamics in several recent publications numerical integrators based on jacobi elliptic functions are proposed for solving the equations of motion of the rigid body although this approach yields theoretically the exact solution a standard implementation shows an unexpected linear propagation of round off errors we explain how deterministic error contribution can be avoided so that round off behaves like a random walk