On the dynamics of axially moving strings

We investigate an axially moving cable with large sag modelled as string and calculate the steady state solution between two rolls. Alternatively we model the cable as beam with very small bending stiffness and compare this solution with the string solution. The validity of the analytically found boundary layer solution is verified by numerical methods. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Wave-Based Approach to Adaptively Control Self-Excited Vibrations in Drill-Strings

Due to their small diameter-to-length-ratio, drill-strings are vulnerable to torsional vibrations. Moreover, the string is exposed to unknown or uncertain time-variant and nonlinear loads (e.g. friction with falling friction characteristics, contact with the borehole, differential sticking), which can result in severe torsional vibrations and stick-slip. The control law for the boundary controller at the top drive of the string needs to adapt to those unknown loads in order to stabilize the vibrations. The torsional vibrations of a drill-string are governed by the wave equation. Analytical solutions and control laws are often based on a separation of the dynamics into a time- and a space-dependent part (modal representation). Here, we decompose the vibrations into two traveling waves according to the D'Alembert solution, using only few measurements along the string. The wave which travels up the string is then compensated by the actuator at the top drive. With this compensation, the upward-traveling wave is no longer reflected back into the string and vibration energy is absorbed, thus stabilizing the torsional vibrations. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Natural damped vibrations of anisotropic box beams of polymer composite materials. 2. Numerical experiments

The influence of the orientation of reinforcing fibers on the natural frequencies and mechanical loss coefficient of coupled vibrations of unsupported symmetric and asymmetric box beams, as evaluated in numerical experiments, is discussed. The calculations were performed under the assumption that the real parts of the complex moduli and mechanical loss coefficient are frequency-independent. Vibration modes were identified by their surface shapes. The boundaries of the regions of mutual transformation of interacting vibration modes were determined by the joint analysis of the dependences of the coupled and partial eigenfrequencies and the mechanical loss coefficients on the orientation angle of reinforcing fibers. It is established that vibrations of a symmetric box beam give rise to two primary interactions: bending–torsional and longitudinal–shear ones, which are united into a unique longitudinal–bending–torsional–shear interaction by the secondary interaction caused by transverse shear strains. Vibrations of an asymmetric box beam give rise to longitudinal–torsional and bending–bending (in two mutually orthogonal planes) interactions. It is shown that in a number of cases variation in the orientation angle of reinforcing fibers is accompanied with a mutual transformation of coupled vibration modes. If the differential equations for natural vibrations involve odd-order derivatives with respect to the spatial variable (a symmetric beam and the bending–bending interaction of an asymmetric beam), then, with variation in the orientation angle of reinforcing fibers, the mutual transformation of coupled vibration modes proceeds. If the differential equations for natural vibrations involve only even-order derivatives (the longitudinal–torsional interaction of an asymmetric beam), no mutual transformation of coupled vibration modes occurs.     

A contribution to efficient calculation of complex drill string dynamics for deep hole drilling

This paper presents a hybrid model to describe drill string dynamics for deep hole drilling. Generally, a typical rotary drill string has a length of several kilometers, but the diameter is less than half a meter. Due to the large ratio of length to diameter, a drill string is a very flexible system. Consequently, an operating drill string is always affected by axial, torsional and lateral vibrations, which potentially induce serious failures. In order to avoid fatal defects, simulations to forecast vibrations are necessary. The simulation should be capable to exhibit the complex dynamical phenomena, e.g. sick-slip, forward whirl and backward whirl, and interactions between drill string and borehole. Usually, these simulations are very time-consuming. In this work, a hybrid model consisting of lumped masses connected with weightless beam elements representing the drill string is developed. The interaction between the drill string and the borehole is implemented by unilateral constraints to describe the nonlinear contact behavior. It was shown that accuracy and simulating time were improved by this model with respect to classical finite-element models. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamics of an Overhead Line and Pantograph System

The dynamic interaction between pantograph and catenary is studied. The overhead electrification system for high speed trains is modelled by visco‐elastically connected double homogeneous strings system, one finite suspended on rigid concentrated supports and the second of infinite length. The pantograph is represented by one degree of freedom oscillator moving along the string at a constant speed. The model is new in that it gives possibility to consider the influence of the locomotive transverse vibrations on railway overhead contact system. The dynamic state of the investigated system is described by a nonlinear set of coupled partial differential equations with complicated boundary conditions. Dynamic transverse displacements of the connected strings are determined. General results are illustrated by numerical examples.     

Mechanics of Axially Moving and Vibrating in Transverse Direction Orthotropic Thermoelastic Web

We consider deformations and stability of axially moving orthotropic thermoelastic web. The web is modelled by a thin continuous plate moving with a constant velocity with small transverse vibrations and supported by a system of rollers. It is supposed that the plate is subjected to a combined thermomechanical loading including pure mechanical in-plane tension and also centripetal forces. Thermal strains corresponding to thermal tension and bending of the moving plate are taken into account. We formulate and analytically investigate the problem of out-of-plane thermomechanical divergence of orthotropic plate.     

Investigation of Damping of Vibrations in a System With Two‐Disc Inseparable Clutch

This paper presents the results of tests on free and forced harmonic torsional vibrations in a system with a two‐disc inseparable clutch, with structural friction taken into account. Nonlinear histeresis loop describing the frictional‐elastic properties of the system was introduced into the model. The mathematical model of the vibrating system containing two disks inseparable clutch was built. During free vibrations of the system, its damping characteristics were tested by a digital simulation method. The vibration damping decrement as a function of amplitude torsional displacement was determined. When vibrations were harmonically forced, the amplitude ‐ frequency characteristics of the system were determined numerically. The system was used as a nonlinear torsional vibration damper in a linear system with a harmonic force. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An explicit solution for the dynamics of a taut string of finite length carrying a traveling mass: the subsonic case

The authors investigate the linear vibrations induced in an elastic string by a loading point-like mass constrained to moving on it with constant horizontal velocity. Exact solutions are shown in the case of subsonic regime. The displacement is explicitly provided in terms of a power series determined by iteration, which is shown to converge to the solution of the problem. The presence of a discontinuity in the right extremum of the considered space interval is also shown both analytically and numerically.     

Various methods of determining the natural frequencies and damping of composite cantilever plates. 1. Exact solution for the binomial model of deformation

On the basis of the classical theory of thin anisotropic laminated plates the article analyzes the free vibrations of rectangular cantilever plates made of fibrous composites. The application of Kantorovich's method for the binomial representation of the shape of the elastic surface of a plate yielded for two unknown functions a system of two connected differential equations and the corresponding boundary conditions at the place of constraint and at the free edge. The exact solution for the frequencies and forms of the free vibrations was found with the use of Laplace transformation with respect to the space variable. The magnitudes of several first dimensionless frequencies of the bending and torsional vibrations of the plate were calculated for a wide range of change of two dimensionless complexes, with the dimensions of the plate and the anisotropy of the elastic properties of the material taken into account. The article shows that with torsional vibrations the warping constraint at the fixed end explains the apparent dependence of the shear modulus of the composite on the length of the specimen that had been discovered earlier on in experiments with a torsional pendulum. It examines the interaction and transformation of the second bending mode and of the first torsional mode of the vibrations. It analyzes the asymptotics of the dimensionless frequencies when the length of the plate is increased, and it shows that taking into account the bending-torsion interaction in strongly anisotropic materials type unidirectional carbon reinforced plastic can reduce substantially the frequencies of the bending vibrations but has no effect (within the framework of the binomial model) on the frequencies of the torsional vibrations.Institute of Engineering Science Russian Academy of Sciences, St. Petersburg, Russia. St. Petersburg State University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 6, pp. 759–769, November–December, 1996.     

Stochastic field processes in the mathematical modelling of damped transmission line vibrations

Wind-excited vibrations in the frequency range of 10 to 50 Hz due to vortex shedding often cause fatigue failures in the cables of overhead transmission lines. Damping devices, such as the Stockbridge dampers, have been in use for a long time for supressing these vibrations. The dampers are conveniently modelled by means of their driving point impedance, measured in the lab over the frequency range under consideration. The cables can be modelled as strings with additional small bending stiffness. The main problem in modelling the vibrations does however lay in the aerodynamic forces, which usually are approximated by the forces acting on a rigid cylinder in planar flow. In the present paper, the wind forces are represented by stochastic processes with arbitrary crosscorrelation in space; the case of a Kármán vortex street on a rigid cylinder in planar flow is contained as a limit case in this approach. The authors believe that this new view of the problem may yield useful results, particularly also concerning the reliability of the lines and the probability of fatigue damages.     

Forced Vibration Analysis of Euler-Bernoulli Double-Beam Systems by Means of Green’s Functions北大核心CSCD

Double-curved-beam (DCB) systems are usually seen in many engineering fields. Compared to straight double-beam systems, DCB systems are more efficient in noise and vibration control problems. To obtain closed-form solutions of steady-state forced vibrations of DCB systems, the classical Euler-Bernoulli curved beam (ECB) model was employed to model vibration equations for the DCB systems. Green’s functions and the Laplace transform methods were used to get the closed-form solutions to the vibration equations for the DCB systems. These solutions apply to arbitrary boundary conditions. Numerical tests were conducted to verify the present solutions with related results from previous literatures. Effects of some important geometric and physical parameters on vibration responses and the interaction between the elastic layer stiffness and the DCB system, were discussed. The results show that, the DCB system will degenerate to a straight double-beam system when the 2 radii approach infinity, moreover, the DCB system can be simplified as one comprising a straight beam and a curved beam. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

The effect of stiffness on the transverse and longitudinal motions of a musical string: Commemorating the 100th anniversary of the birth of Kh.A. Rakhmatulin

The effect of stiffness on the propagation of longitudinal and transverse waves and vibrations in prestretched strings is considered. The contribution of the longitudinal and transverse components to the dynamic load is of the same order. The longitudinal vibrations occur both at natural frequencies and at frequencies of the transverse vibrations. Resonance phenomena are possible. Low stiffness, which is characteristic for musical strings, leads to a small change in the frequencies of the whole spectrum of transverse and longitudinal vibrations, but to a considerable change in the shape of the string at strike and mounting points and on the transverse wave front.     

Interference between the transverse and longitudinal vibrations in musical strings

Two problems of the vibrations of strings are considered using the approach described previously in [1]: the vibrations of the string of a plucked musical instrument, drawn out at one of the points and at rest at the initial instant of time (Problem 1), and the vibrations of the string of a keyboard musical instrument, the points of which are given an initial velocity at the initial instant of time by a hammer of small width (Problem 2). It is established that forced longitudinal oscillations of the string occur at frequencies of the transverse vibrations, the condition for possible resonance of the longitudinal vibrations is derived, and the nature of the vibrations at the point where the string is fastened due to elasticity and the related shift in the frequency of transverse vibrations is established.     

Green's functions for forced vibration analysis of bending-torsion coupled Timoshenko beam

A method based on Green's functions is proposed for the analysis of the steady-state dynamic response of bending-torsion coupled Timoshenko beam subjected to distributed and/or concentrated loadings. Damping effects on the bending and torsional directions are taken into account in the vibration equations. The elastic boundary conditions with bending-torsion coupling and damping effects are derived and the classical boundary conditions can be obtained by setting the values of specific stiffness parameters of the artificial springs. The Laplace transform technology is employed to work out the Green's functions for the beam with arbitrary boundary conditions. The Green's functions are obtained for the beam subject to external lateral force and external torque, respectively. Coupling effects between bending and torsional vibrations of the beam can be studied conveniently through these analytical Green's functions. The direct expressions of the steady-state responses with various loadings are obtained by using the superposition principle. The present Green's functions for the Timoshenko beam can be reduced to those for Euler–Bernoulli beam by setting the values of shear rigidity and rotational inertia. In order to demonstrate the validity of the Green's functions proposed, results obtained for special cases are given for a comparison with those given in the literature and they agree with each other exactly. The influences of external loading frequency and eccentricity on Green's functions of bending-torsion coupled Timoshenko beam are investigated in terms of the numerical results for both simply supported and cantilever beams. Moreover, the symmetric property of the Green's functions and the damping effects on the amplitude of Green's functions of the beam are discussed particularly.     

Vibrations of Piezoceramic Rods

Presently, most of the research on vibrations of monolithic piezoelectric rods at weak electric fields is restricted to longitudinal oscillations of such structural members where free and forced vibrations have been dealt with and in the case of resonance conditions not only linear but also nonlinear effects within the constitutive relations have been incorporated. On the other hand, bending and torsional vibrations of piezoceramic one‐parametric rods have not been examined yet. The present contribution develops a linear vibration theory of rods with a focus to bending vibrations taking into account rotatory inertia and shear deformation. The governing boundary value problem for beams with both longitudinal and as well transversal polarization is derived, in particular free vibrations are analyzed. Also nonlinear extensions not only of physical nature but also geometrical ones are addressed. A possible technical application is given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Forced vibration analysis of a Timoshenko beam featuring bending-torsion on Pasternak foundation

Free and forced vibration analysis of a Timoshenko beam on viscoelastic Pasternak foundation featuring coupling between flapwise bending and torsional vibrations is studied in this article. The system motion is described through a coupled set of three partial differential equations. The differential transform method, DTM, as an efficient mathematical technique is adopted to obtain the natural frequencies and the mode shapes. The system force response is assessed for a moving concentrated load with a constant velocity. Two different methods are studied and applied in obtaining forced vibration response of the system: (1) the same time functions, STF, by setting out the orthogonality conditions derived in this article and (2) the different time functions, DTF. The difference between the responses of the system is assessed by applying STF and DTF for a constant moving load. The effects of some parameters on the system response are probed. A numerical example is solved to validate the results obtained here with the available ones and a close agreement is found. It is observed that the time functions in DTF and STF are almost identical for transverse displacement and bending angle and are significant for torsion angle, recommending the application of DTF when the bending-torsion coupling is of concern.     

Control of an axially moving viscoelastic Kirchhoff string

The control problem of axially moving strings occurs in a large class of mechanical systems. In addition to the longitudinal displacement, the strings are subject to undesirable transversal vibrations. In this work, in order to suppress these vibrations, we consider a control by a hydraulic touch-roll actuator at the right boundary. We prove uniform stability of the system using a viscoelastic material and an appropriate boundary control force applied to the touch rolls of the actuator. The features of the present work are: taking into account the mass flow entering in and out at the boundaries due to the axial movement of the string and overcoming the difficulty raised by the Kirchhoff coefficient which does not allow us to profit from the dissipativity of the system (as in the existing works so far). We shall make use of an inequality which is new in this theory.     

Application of the resonance method for determining the elastic and dissipative properties of an orthotropic polymer composite

Experimentally determined resonant frequencies and damping of flexural and torsional vibrations of rod-type rectangular test specimens made of an orthotropic GFRP fabric with different ratios of cross-sectional sizes are used for calculating six principal complex elastic and shear moduli. The application of the classical theories of flexural and torsional vibrations, the theory of flexural vibrations of a Timoshenko beam, and a refined theory of torsional vibrations of free-free orthotropic rods is analyzed.     

Modeling of the dynamic phenomena “stick-slip” and “whirl” of a drill string demonstrator

Vibrations of drill strings in oil and gas well bores can reduce drilling performance or lead to damage of drill string components. These vibrations are neither optically observable nor measurable because of the geological formation. Hence usually simulations are used to understand the dynamic behavior. The test stand “OSTrator” adds more insight to this problem. The demonstrator represents a scaled drill rig granting direct access to the complex dynamic behavior of the real drill string. Two phenomena are of special interest: the stick-slip effect, a torsional oscillation, and the whirl effect, a periodic torsional-lateral movement with permanent wall contact. Since the OSTrator was built for showcase purposes, it has to be piloted in and out Stick-Slip and Whirl domains by a control algorithm. Therefore efficient multi-body-systems able to represent these two effects in faster than real time are demanded. In this presentation, the modeling of the stick-slip and the whirl effect are discussed. The models should be able to predict these effects and supply data to avoid or confront them. Key questions are the modeling of the wall contact and the resistance at the drill bit. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Model Order Reduction of a Drill-String-Model with Self-Excited Stick-Slip Vibrations

The dynamical behavior of a drill-string is defined by its small diameter-to-length ratio, which makes the string vulnerable to torsional vibrations. In combination with the nonlinear friction characteristic at the drill bit, this can lead to self-excited stick-slip vibrations which are detrimental to the drilling process. The string can be modeled by the Finite Element Method or as a Multi-Body system to represent the distributed character of the system. The analysis of the resulting high-dimensional model is, however, elaborate and time-consuming. We show that through Galerkin Projection onto the first two Characteristic Functions gained from Karhunen-Loève-Transformation, a reduced system can be obtained which reproduces the essential dynamical properties of the original system, e.g. the stick-slip motion. With the reduced system, the linear stability of the drill-string can be analyzed. We show that by reducing the inertia of the rotary table the system can be stabilized. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)