Anomalous dependence of the vibration frequencies of a rod in an elastic medium on the rod length

We use numerical-analytic methods to study the influence of the length of a thin inhomogeneous rod on its natural frequencies and the shapes of its plane transverse vibrations. We found that the existence of an external elastic medium described by the Winkler model can lead to an anomalous effect, i.e., to an increase in the natural frequencies of the vibration lower modes as the rod length increases continuously. We discovered rather subtle properties of this phenomenon in the case of variations in the length, the mode number, and the fixation method. We separately studied vibrations for the standard boundary conditions: fixation, hinged fixation, tangential fixation, and free end. We calculated several simple examples illustrating the anomalous dependence of the frequency of the rod natural vibrations in a strongly inhomogeneous elastic medium with different boundary conditions.     

Thermomechanical response of a viscoelastic rod driven by a sinusoidal displacement

The forced vibrations of a rod of thermoviscoelastic material are studied. The rod is considered to be laterally insulated but not constrained, such that a one-dimensional analysis may be employed. Temperature dependence of the material properties and the resulting thermomechanical coupling effects are included. The vibrations are forced by the imposition of a sinusiodal displacement of known amplitude and frequency at one end of the rod. This problem corresponds to a dissipative material bonded to the surface of a relatively rigid, vibrating structure.Initial transient behavior is not considered. A steady-state response is found by means of a finite difference formulation. Material properties of a Lockheed solid propellant are used.The presence of critical frequencies, characterized by high stresses and temperatures, is found for small amplitudes of vibration. Nonlinearities and instabilities lead to a lack of one-to-one correspondence between stress and displacement boundary conditions. No relationship is found between the critical frequencies of the driven rod and the natural frequencies of a rod with an equivalent temperature profile.     

On the extensional and flexural vibrations of rotating bars

The non-linear equations of motion of a slender bar rotating at constant angular velocity about a transverse axis are formulated. Under the assumption that a small perturbed motion occurs about an initially stressed equilibrium configuration, linearized equations of motion for the longitudinal and flexural deformations of a rotating bar carrying a tip mass are derived. Numerical computations for the natural frequencies of the lowest three modes of free vibration reveal that the values of the extensional frequencies increase monotonically, contrary to previously published results, as the angular velocity of rotation increases.     

Free transverse vibration analysis of a toothed gear

Analysis of transverse vibration of the gear found in a high-speed gearbox considered as an annular plate reflecting gear geometry is the subject of this paper. How gear angular velocity affects the deformation of normal modes of transverse vibration of the system under consideration is analysed. Models considered were discretized by the finite elements method. Numerical computations have been performed in the ANSYS environment. The algorithm to identify the proper distorted mode shapes is presented. The Campbell diagram for the system under consideration is elaborated. The problems discussed here can be useful for engineers dealing with dynamics of rotating machine systems.     

Natural Vibrations of a Liquid-Transporting Pipeline on an Elastic Base

Flexural free vibrations of an ideal-liquid-transporting pipeline on an elastic base are studied. A numerical-analytical method for finding the pipeline natural frequencies and vibration modes is developed, which permits one to determine the natural frequencies and modes for the case in which the tension or compression (the longitudinal force acting along the pipeline axis), the pipe diameter, and hence the velocity of the incompressible fluid being transported are arbitrary functions of the longitudinal coordinate measured along the pipeline axis. The least natural frequencies are calculated for the case in which the variable elasticity of the base is given by some test functions.     

On the influence of a transverse magnetic field on the vibration spectra of shallow shells

In the design of electric machines, devices, and plasma generator bearing constructions, it is sometimes necessary to study the influence of magnetic fields on the vibration frequency spectra of thin-walled elements. The main equations of magnetoelastic vibrations of plates and shells are given in [1], where the influence of the magnetic field on the fundamental frequencies and vibration shapes is also studied. When studying the higher frequencies and vibration modes of plates and shells, it is very efficient to use Bolotin’s asymptotic method [2–4]. A survey of studies of its applications to problems of elastic system vibrations and stability can be found in [5, 6]. Bolotin’s asymptotic method was used to obtain estimates for the density of natural frequencies of shallow shell vibrations [3] and to study the influence of the membrane stressed state on the distribution of frequencies of cylindrical and spherical shells vibrations [7, 8]. In a similar way, the influence of the longitudinal magnetic field on the distribution of plate and shell vibration frequencies was studied [9, 10]. It was shown that there is a decrease in the vibration frequencies of cylindrical shells under the action of a longitudinal magnetic field, and the accumulation point of the natural frequencies moves towards the region of lower frequencies [10]. In the present paper, we study the influence of a transverse magnetic field on the distribution of natural frequencies of shallow cylindrical and spherical shells, obtain asymptotic estimates for the density of natural frequencies of shell vibrations, and compare the obtained results with the empirical numerical results.     

旋转运动中弹性梁耦合振动的辛方法

研究旋转梁结构的弹性耦合振动问题。通过引入对偶体系，建立了解决该类问题的辛方法。在辛体系中描述旋转梁纵向和横向耦合振动控制方程，即哈密顿正则方程。进一步求解得到结构的固有振动频率及相应的振动模态，发现固有振动频率随转动角速度先升后降以及模态之间的某种转化规律。     

Stability of forced torsional vibrations of an equipped rod

A model of an equipped elastic rod is considered. In the average sense, this model shows the properties of the one-dimensional Cosserat continuum during longitudinal and torsional motions. Natural and forced torsional vibrations are studied in the case of flow loading. Several conditions for vibration stability and for the end of vibrations are formulated. The following distinctive features of motion are found: each vibration mode has two different shapes and two different frequencies and the onset of the divergence regime is observed when the external loads become more intensive.     

Basic properties of natural vibrations of an extended segment of a pipeline

Natural transverse vibrations of an extended segment of a pipeline containing a uniformly moving fluid are considered. The mechanical model under study takes into account the inertial forces of the pipe and environment and the moment of Coriolis and centrifugal forces arising because of the medium motion. It is proved that all natural frequencies of the pipeline rigidly clamped at both ends are real (and hence no flutter can arise in this model). For the first three modes, the dependence of the eigenvalues on the fluid flow velocity (varying from zero to the buckling velocity) are constructed, and their properties depending on the inertia parameter are studied. Families of vibration mode shapes of the pipeline are constructed and investigated.     

Free flapewise vibration analysis of rotating double-tapered Timoshenko beams

Free vibration analysis of a rotating double-tapered Timoshenko beam undergoing flapwise transverse vibration is presented. Using an assumed mode method, the governing equations of motion are derived from the kinetic and potential energy expressions which are derived from a set of hybrid deformation variables. These equations of motion are then transformed into dimensionless forms using a set of dimensionless parameters, such as the hub radius ratio, the dimensionless angular speed ratio, the slenderness ratio, and the height and width taper ratios, etc. The natural frequencies and mode shapes are then determined from these dimensionless equations of motion. The effects of the dimensionless parameters on the natural frequencies and modal characteristics of a rotating double-tapered Timoshenko beam are numerically studied through numerical examples. The tuned angular speed of the rotating double-tapered Timoshenko beam is then investigated.     

Natural vibrations of a pipeline segment

Transverse natural vibrations of an extended segment of a pipeline conveying a uniformly moving fluid are studied. The mechanical model under study takes into account the pipe and fluid inertia forces and the moment of the Coriolis and centrifugal forces due to the medium motion. It is assumed that both ends are rigidly fixed and the elastic characteristics are constant along the pipe. A mathematical model is developed on the basis of a generalized procedure of separation of variables, and a boundary value problemfor the eigenvalues and eigenfunctions (natural frequencies and vibration shapes) is posed. Ferrari’s formulas are used to solve the fourth-order complex characteristic equation for the wave parameter, and a closed procedure of numerical-analytical determination of roots of the secular equation for the frequencies is obtained. The frequency curves for the firsts two vibration modes against the dimensionless velocity and inertia parameters are constructed. The forms of the observed motions at different times are obtained. Several effects are revealed indicating that there is a dramatic quantitative and qualitative difference between these vibrations and the standard vibrations corresponding to the case of immovablemedium. We discover the absence of a rectilinear configuration of the axis, the variable number and location of nodes, their inconsistency with the mode number, and some other effects.     

A geometrically exact approach to the overall dynamics of elastic rotating blades—part 1: linear modal properties

A geometrically exact mechanical model for the overall dynamics of elastic isotropic rotating blades is proposed. The mechanical formulation is based on the special Cosserat theory of rods which includes all geometric terms in the kinematics and in the balance laws without any restriction on the geometry of deformation besides the enforcement of the local rigidity of the blade cross sections. All apparent forces acting on the blade moving in a rotating frame are accounted for in exact form. The role of internal kinematic constraints such as the unshearability of the slender blades is discussed. The Taylor expansion of the governing equations obtained via an Updated Lagrangian formulation is then employed to obtain the linearized perturbed form about the prestressed configuration under the centrifugal forces. By applying the Galerkin approach to the linearized equations of motion, the linear eigenvalue problem is solved to yield the frequencies and mode shapes. In particular, the natural frequencies of unshearable blades including coupling between flapping, lagging, axial and torsional components are investigated. The angular speeds at which internal resonances may arise due to specific ratios between the frequencies of different modes are determined thus shedding light onto the overall modal couplings in rotating beam structures depending on the angular speed regime. The companion paper (part?2) discusses the nonlinear modes of vibration away from internal resonances.     

Free vibrations of two capillary liquids rotating in a cylindrical vessel

The problem of the small natural vibrations of two coaxially disposed ideal liquids rotating in a cylindrical vessel under conditions of complete weightlessness is considered. The set of normal vibrations of the system consists of internal wave motions and surface waves. Asymptotic formulas are derived for the vibration frequencies of the surface waves. The results of computer calculations are presented in the form of graphs and tables.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 97–104, September–October, 1976.     

Pseudo-nonlinear dynamic analysis of buckled pipes

In this study, the post-divergence behavior of fluid-conveying pipes supported at both ends is investigated using the nonlinear equations of motion. The governing equation exhibits a cubic nonlinearity arising from mid-plane stretching. Exact solutions for post-buckling configurations of pipes with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are investigated. The pipe is stable at its original static equilibrium position until the flow velocity becomes high enough to cause a supercritical pitchfork bifurcation, and the pipe loses stability by static divergence. In the supercritical fluid velocity regime, the equilibrium configuration becomes unstable and bifurcates into multiple equilibrium positions. To investigate the vibrations that occur in the vicinity of a buckled equilibrium position, the pseudo-nonlinear vibration problem around the first buckled configuration is solved precisely using a new solution procedure. By solving the resulting eigenvalue problem, the natural frequencies and the associated mode shapes of the pipe are calculated. The dynamic stability of the post-buckling configurations obtained in this manner is investigated. The first buckled shape is a stable equilibrium position for all boundary conditions. The buckled configurations beyond the first buckling mode are unstable equilibrium positions. The natural frequencies of the lowest vibration modes around each of the first two buckled configurations are presented. Effects of the system parameters on pipe behavior as well as the possibility of a subcritical pitchfork bifurcation are also investigated. The results show that many internal resonances might be activated among the vibration modes around the same or different buckled configurations.     

An image encryption algorithm based on compound homogeneous hyper-chaotic system

A new dynamic model of a rotating flexible beam with a concentrated mass located in arbitrary position is derived based on the absolute nodal coordinate formulation, and its modal characteristics are investigated in this paper. To consider the concentrated mass at an arbitrary location of the beam, a Dirac’s delta function is used to express the mass per unit length of the beam. Based on the proposed dynamic model, the frequency analysis is performed. The nonlinear equation is transformed into the linear one via employing the linear perturbation analysis method. The stiffness matrix of static equilibrium of the system under the deformed condition is obtained, in which the effect of coupling between the longitudinal deformation and transversal deformation is included. This means even if only the chordwise bending equation is solved, the longitudinal vibration effect can be still considered. As we know, once the longitudinal deformation is large, it will significantly affect the chordwise bending vibration. So the proposed model in this paper is more accurate than the traditional dynamic models which are usually lack of the coupling terms between the longitudinal deformation and transversal deformation. In fact, the traditional dynamic models for the chordwise vibration analysis in the existing literature are usually linear due to neglecting the coupling terms, and consequently, they are only suitable for the modal characteristic analysis of a beam under small deformations. In order to get some general conclusions of the natural frequencies and mode shapes, the equation which governs the chordwise bending vibration of the rotating beam is transformed into a dimensionless form. The dynamic model presented in this paper is nonlinear and can be conveniently used to analyze the modal characteristics of a rotating flexible beam with large deformations. To demonstrate the power of the new dynamic model presented in this paper, the dynamic simulations involving the comparisons between the different frequencies obtained using the model proposed in this paper and the models in the existing literature and the investigating in frequency veering and mode shift phenomena are given. The simulation results show that the angular velocity of the flexible beam will give rise to the phenomena of the natural frequency loci veering and the associated mode shift which is verified in the previous studies. In addition, the phenomena of the natural frequency loci veering rather than crossing can be observed due to the changing of the magnitude of the concentrated mass or of the location of the concentrated mass which are found for the first time. Furthermore, there is an interesting phenomenon that the natural frequency loci will veer more than once due to different types of mode coupling between the bending and stretching vibrations of the rotating beam. At the same time, the mode shift phenomenon will occur correspondingly. Additionally, the characteristics of the vibration nodes are also investigated in this paper.     

Real-time response of a rotating disk using image-derotated holographic interferometry

A recently developed technique that shows great promise for studying the structural response of rotating objects is that of image-derotated holographic interferometry. The technique consists of optically subtracting the rotational motion of a disk by passing the image of the rotating disk through a prism that is rotating at half the disk's angular velocity. Heretofore, a pulsed ruby laser had to be used to record the rotating object's out-of-plane modes of vibration. This study reports on the extension of the technique to the real-time analysis of rotating objects by replacing the pulsed ruby laser with an acousto-optically modulated argon laser. Using the strobed argon laser in conjunction with the optical derotator, the rotating disk's normal displacement can be observed as it is being excited in real-time. This technique offers the distinct advantage of being able to observe the rotating disk's structural response over a wide range of pulse frequencies at any point in the disk's cycle of revolution.     

Torsional vibrations of a column of fine-grained material: A gradient elastic approach

The gradient theory of elasticity with damping is successfully employed to explain the experimentally observed shift in resonance frequencies during forced harmonic torsional vibration tests of columns made of fine-grained material from their theoretically computed values on the basis of the classical theory of elasticity with damping. To this end, the governing equation of torsional vibrations of a column with circular cross-section is derived both by the lattice theory and the continuum gradient elasticity theory with damping, with consideration of micro-stiffness and micro-inertia effects. Both cases of a column with two rotating masses attached at its top and bottom, and of a column fixed at its base carrying a rotating mass at its free top, are considered. The presence of both micro-stiffness and micro-inertia effects helps to explain the observed natural frequency shift to the left or to the right of the classical values depending on the nature of interparticle forces (repulsive or attractive) due to particle charge. A method for using resonance column tests to determine not only the shear modulus but also the micro-stiffness and micro-inertia coefficients of gradient elasticity for fine-grained materials is proposed.     

TRANSVERSE NATURAL FREQUENCIES AND FLOW-INDUCED VIBRATIONS OF DOUBLE BELLOWS EXPANSION JOINTS

This paper considers the transverse vibrations of fluid-filled double-bellows expansion joints. The bellows are modelled as a Timoshenko beam, and the fluid added mass includes rotary inertia and bellows convolution distortion effects. The natural frequencies are given in terms of a Rayleigh quotient, and both lateral and rocking modes of the pipe connecting the bellows units are considered. The theoretical predictions for the first six modes are compared with experiments in still air and water and the agreement is found to be very good. The flow-induced vibrations of the double bellows are then studied with the bellows downstream of a straight section of pipe and a 90° elbow. Strouhal numbers are computed for each of the flow-excited mode resonances. The bellows natural frequencies are not affected by the flowing fluid but the presence of an immediate upstream elbow substantially reduces the flow velocity required to excite resonance.     

Modeling of properties and motions of an inhomogeneous one-dimensional continuum of a complicated Cosserat-type microstructure

We consider an approach to modeling the properties of the one-dimensional Cosserat continuum [1] by using the mechanical modeling method proposed by Il’yushin in [2] and applied in [3]. In this method, elements (blocks, cells) of special form are used to develop a discrete model of the structure so that the average properties of the model reproduced the properties of the continuum under study. The rigged rod model, which is an elastic structure in the form of a thin rod with massive inclusions (pulleys) fixed by elastic hinges on its elastic line and connected by elastic belt transmissions, is taken to be the original discrete model of the Cosserat continuum. The complete system of equations describing the mechanical properties and the dynamical equilibrium of the rigged rod in arbitrary plane motions is derived. These equations are averaged in the case of a sufficiently smooth variation in the parameters of motion along the rod (the long-wave approximation). It was found that the average equations exactly coincide with the equations for the one-dimensional Cosserat medium [1] and, in some specific cases, with the classical equations of motion of an elastic rod [4–6]. We study the plane motions of the one-dimensional continuum model thus constructed. The equations characterizing the continuum properties and motions are linearized by using several assumptions that the kinematic parameters are small. We solve the problem of natural vibrations with homogeneous boundary conditions and establish that each value of the parameter distinguishing the natural vibration modes is associated with exactly two distinct vibration mode shapes (in the same mode), each of which has its own frequency value.     

圆截面弹性细杆的平面振动

基于Kirchhoff理论讨论圆截面弹性细杆的平面振动．以杆中心线的Frenet坐标系为参考系建立动力学方程．杆作平面运动时,其扭转振动与弯曲振动解耦．讨论任意形状杆的扭转振动和轴向受压直杆在无扭转条件下的弯曲振动,证明直杆平衡的静态Lyapunov稳定性与欧拉稳定性条件为动态稳定性的必要条件．考虑轴向力和截面转动惯性效应的影响,导出弯曲振动的固有频率．