Waves and vibrations in strings with windings: Commemorating the 100th anniversary of the birth of Kh.A. Rakhmatulin

The equations of the propagation of transverse, twisting and longitudinal waves and vibrations are obtained, taking into account their interactions in musical strings with windings. Their solutions are obtained. The occurrence of transverse and twisting motions leads to the appearance of longitudinal motions, while the transverse and longitudinal components play the role of inducing forces for the twisting components. The contributions of the transverse, twisting and longitudinal components to the dynamic loading of the string are of the same order. The longitudinal-twisting vibrations occur both at natural frequencies and at frequencies of the transverse vibrations. Resonance phenomena between the individual modes of these vibrations are possible.     

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.     

Dynamic motion of whirling rods with Coriolis effect

Longitudinal vibrations coupled with transverse vibrations of whirling rods are investigated. It is known that longitudinal and transverse vibrations are governed by second and fourth order differential equations, respectively. Due to the Coriolis effect, a system of equations that governs the longitudinal and transverse displacements will be constructed by coupling these two equations together. Solutions of the equations assume small oscillations of vibration being superimposed on the steady state of the whirling rod. Exact and approximate solutions are obtained from the proposed governing equations, where the approximate solutions on displacements and natural frequencies are acquired by neglecting the Coriolis effect. A proposed numerical scheme known as complete function collocation method is implemented to solve the governing equations coupled with longitudinal and transverse displacements. The approximate results on both longitudinal and transverse natural frequencies show that natural frequencies are decreasing while the angular velocity of the rod is increasing. Exact and numerical results on both longitudinal and transverse natural frequencies show that there are no predictable trends whether natural frequencies are increasing or decreasing while the angular velocity of the rod is increasing.     

Various methods of determining the natural frequencies and damping of composite cantilever plates. 2. Approximate solution by Galerkin's method for the trinomial model of damping

The application of Kantorovich's method to a trinomial model of deformation taking into account transverse bending of a plate leads to a connected system of three ordinary differential equations of fourth order with respect to three unknown functions of the longitudinal coordinate and to the coresponding boundary conditions for them at the fixed end and on the free edge. For the approximate calculation of the frequencies and forms of natural vibrations Galerkin's method is used, and as coordinate functions we chose orthogonal Jacobi polynomials with weight function. The dimensionless frequencies depend on the magnitude of the four dimensionless complexes, three of which characterize the anisotropy of the elastic properties of the composite. For the fibrous composites used at present we determined the possible range of change of the dimensionless complexes d₁₆ and d₂₆ attained by oblique placement. The article examines the influence of the angle of reinforcement on some first dimensionless frequencies of a plate made of unidirectional carbon reinforced plastic. It also analyzes the asymptotics of the frequencies when the length of the plate is increased, and it shows that for strongly anisotropic material with the structure []_T the frequencies of the flexural as well as of the torsional vibrations may be substantially lower when flexural-torsional interaction is taken into account.For Communication 1 see [4].Institute of Engineering Science of the Russian Academy of Sciences, St. Petersburg, Russia. St. Petersburg State University, Russia. Translated from Mekhanika Kompozitnykh Materialov, No. 1. pp. 23–33, January–February, 1997.     

井下钻柱纵向横向耦合振动模型建立与数值分析

针对井下钻柱运动的复杂性,基于动力学理论,建立了井下钻柱纵向和横向耦合振动的数学模型,并进行数值求解及分析.根据井下钻柱的实际工况,以整个井下钻柱为研究对象,提出了钻柱纵向和横向耦合振动的动力方程,并利用解析法和无量纲法分别求解出其动刚度和动阻尼的表达式,以及钻柱前两阶振动的固有频率.分析结果表明:当井下钻柱振动频率增大时,其动刚度呈幅值衰减的周期性变化,而其动阻尼呈幅值增强的周期性变化;井下钻柱长度和横截面面积越大,其动刚度和动阻尼的幅值越小;井下钻柱的Poisson(泊松)比对其振动的动刚度、动阻尼和前两阶固有频率没有影响;同时,井下钻柱的第二阶固有频率始终大于第一阶固有频率.该文的研究方法和模型为井下钻柱钻具分析和结果优化提供了理论参考和实际意义.     

Parametric resonances in the problem of longitudinal impact on a thin rod

The longitudinal impact on a thin elastic rod, which generates a periodic system of longitudinal waves in it, is considered. At definite values of the parameters of the problem in the linear approximation, these waves induce parametric resonances, which are accompanied by an unlimited increase in the amplitude of the transverse vibrations. To obtain finite values of the amplitudes, a quasilinear system is considered in which the effect of the transverse vibrations on the longitudinal vibrations is taken into account. This system was previously solved using the Bubnov–Galerkin method and beats accompanied by energy transfer between the transverse and longitudinal vibrations were obtained. In this work, an approximate analytical solution of the system has been derived that is based on double-scale expansions. A qualitative analysis of this solution has been carried out. An estimate of the maximum transverse bending has been obtained for various methods of loading. Both shortand long-term pulses have been considered. It has been shown that, in the case of a spontaneously applied long-term pulse that is lower than the Euler critical load, intensive transverse vibrations can occur.     

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.     

On the determination of natural frequencies and mode shapes of cable-stayed bridges

An accurate analysis of the natural frequencies and mode shapes of a cable-stayed bridge is fundamental to the solution of its dynamic responses due to seismic, wind and traffic loads. In most previous studies, the stay cables have been modelled as single truss elements in conventional finite element analysis. This method is simple but it is inadequate for the accurate dynamic analysis of a cable-stayed bridge because it essentially precludes the transverse cable vibrations. This paper presents a comprehensive study of various modelling schemes for the dynamic analysis of cable-stayed bridges. The modelling schemes studied include the finite element method and the dynamic stiffness method. Both the mesh options of modelling each stay cable as a single truss element with an equivalent modulus and modelling each stay cable by a number of cable elements with the original modulus are studied. Their capability to account for transverse cable vibrations in the overall dynamic analysis as well as their accuracy and efficiency are investigated.     

基于非约束模态的中心刚体-Timoshenko梁动力学建模与分析北大核心CSCD

梁的横向变形会导致梁纵向缩短,建模过程中考虑梁横纵变形二次耦合项则存在动力刚化现象,这说明梁的纵向变形会对模型的广义刚度造成影响.对于做旋转运动的梁结构,旋转运动时还会受到离心力的作用而产生轴向拉力,轴向拉力同样也会引起梁的轴向变形,这种影响对粗短梁更加明显.以大范围运动中心刚体-Timoshenko梁模型为研究对象:首先,运用Timoshenko梁理论以及Hamilton原理建立含离心力的动力学模型;其次,引入非约束模态概念,采用Frobenius方法求解非约束模态振型函数以及固有频率;最后,通过数值仿真探究不同恒定转速时非约束模态与约束模态广义刚度的差异和非约束模态条件下离心力对模型的影响.     

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.     

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.     

Effects of a small length scale on vibrations of an embedded double-walled carbon nanotube

Extensive continuum analyses are carried out to estimate the influence of matrix stiffness, a small length scale, and intertubular radial displacements on free vibrations of an individual double-walled carbon nanotybe. The analyses are based on both local and classical Euler–Bernoulli and Timoshenko elasticity theories with concentricity and nonconcentricity assumptions. The effect of a small length scale is incorporated in the formulations. New intertubular resonant frequencies are calculated based on these theories. Detailed results are demonstrated for the resonant frequencies as functions of matrix stiffness and the small length scale. The results indicate that the internal radial displacement and the stiffness of the surrounding matrix can greatly affect the resonant frequencies, especially at higher frequencies, and thus the latter does not keep the otherwise concentric structure at ultrahigh frequencies. More over, at high frequencies and small aspect ratios, the effect of the small length scale be comes more significant.     

Flexural vibrations of a moving rod

The problem of the transverse natural vibrations of part of a rod between two coaxially fixed guides, moving with an arbitrary constant velocity, is investigated. The conditions of rigid clamping are taken as the boundary conditions. Additional shear stresses, due to longitudinal tension or compression are taken into account. Relations defining the natural frequencies and forms are constructed in an exact formulation by Fourier method. The dependence of the natural frequencies and forms of the lowest vibration modes on the rate of displacement, unknown in the literature, are constructed, and their features are established. A modelling and animation of unusual wave motions of the rod are presented. The main characteristics for the higher vibration modes are constructed.     

Rigorous van der Waals effect on vibration characteristics of multi-walled carbon nanotubes under a transverse magnetic field

An analytical method is presented to investigate rigorous van der Waals interaction effect on vibration characteristics of multi-walled carbon nanotubes embedded in matrix under a transverse magnetic field. Each of the concentric tubes of multiwall carbon nanotubes is considered as an individual elastic shell and coupled with any two walls through a rigorous van der Waals interaction being dependent on the change of interlayer spacing and the radii of tubes. Results show that the rigorous van der Waals interaction effect makes the lowest magneto-vibration frequency of multi-walled carbon nanotubes decrease and the highest magneto-vibration frequency increase. The effect of rigorous van der Waals interaction on magneto-elastic vibrations of multi-walled carbon nanotubes is dependent on the transverse magnetic strength and the matrix constrained stiffness.     

Asymptotic solutions of non-classical boundary-value problems of the natural vibrations of orthotropic shells

The natural vibrations of orthotropic shells are considered in a three-dimensional formulation for different versions of the boundary conditions on the faces: rigid clamping rigid clamping, rigid clamping free surface, and mixed conditions. Asymptotic solutions of the corresponding dynamic equations of the three-dimensional problem of the theory of elasticity are obtained. The principal values of the frequencies of natural vibrations are determined. It is shown that three types of natural vibrations occur in the shell: two shear vibrations and a longitudinal vibration, which are due solely to the boundary conditions on the faces. It is proved that each boundary layer has its own natural frequency. The boundary-layer functions are determined and the rates at which they decrease with distance from the faces inside the shell are established.     

The effect of high-frequency vibration on the onset of Marangoni convection in a horizontal liquid layer

The effect of high-frequency translational harmonic vibrations on the onset of thermocapillary convection in a horizontal liquid layer, bounded above by a free surface and below by a solid wall, is investigated by averaging the convection equations. It is shown that longitudinal vibrations have no effect on convective instability. If the direction of the vibration contains a transverse component, and the action of the vibration has a stabilizing effect: the free boundary of a uniform liquid is smoothed and thermal convection of a non-uniform liquid may be suppressed. The maximum stabilizing effect is obtained for vertical vibrations.     

Solution of some dynamic problems of viscoelasticity theory by the averaging method

Damping of longitudinal vibrations of a viscoelastic bar and transverse vibrations of a freely supported orthotropic plate is investigated by the averaging method. The problem of the forced movement of a viscoelastic cylinder enclosed in a thin elastic shell under the effect of an arbitrarily time-varying internal pressure is studied.Institute of Cybernetics at the Computer Center Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 2, pp. 246–252, March–April, 1970.     

Vibrations of the surface of an elastic double-layered half-space with a periodic system of cracks

The two-dimensional problem of the normal incidence of a plane transverse wave from the far field on to the free surface of an elastic double-layered half-space, comprising a homogeneous layer attached to a semi-infinite base of a different elastic material, is considered. At the boundary between the two media there is a system of plane cracks, arranged periodically along the separation line, which models the fracture zone at the interface between dense solid rock and soft sedimentary rock. The effect of the fractures on the transmission of a transverse seismic wave generated by a deep-focus earthquake, and of the type of vibrations of the free surface of the ground that result, is studied. It is difficult to predict whether the seismic wave is strengthened or weakened by the fracture zone. The effect of the system of cracks on vibrations of the free surface largely depends on the physical and geometrical parameters and, primarily, on the vibration frequencies.     

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.     

On the Effect of Contact Compliance on Vibrational Smoothing of Dry Friction

High-frequency vibrations may be utilized in order to smooth the characteristics of dry friction at low sliding velocities and, consequently, quench undesired friction induced phenomena. Many studies have been published so far, most of them using classical Coulomb friction models and yielding compact results. Unfortunately, the agreement with related experimental results is insufficient. As the Coulomb model overestimates the smoothing effect, improved modelling seems to be necessary. Based on Dahl's friction model, the effect of longitudinal and transverse high-frequency vibrations on a 1-DoF-friction oscillator is investigated here. Accounting for contact compliance, a reduction of the smoothing effect is observed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)