Transverse rod vibrations under a short-term longitudinal impact

Transverse vibrations of a thin rod caused by a short-term longitudinal impact are considered. After the impact, a system of compression-tension waves, which causes transverse vibrations, appears in the rod. Parametric resonance, which leads to an unbounded rise in the amplitude of transverse vibrations, is investigated in the linear approximation. With a nonlinear approach, beats appear in the resonance vicinity, in which the mutual exchange of the energy of longitudinal and transverse vibrations occurs. The influence of viscoelastic resistance forces is investigated.     

Dynamic blocking of the influence of surface point defects on the glide of edge dislocations

The influence of image forces on an edge dislocation gliding parallel to the free crystal surface having point defects is studied theoretically. It is shown that image forces block the drag mechanism that involves the excitation of dislocation vibrations by surface defects.     

Numerical modelling of longitudinal vibrations of a sucker rod string

A new technique for analyzing the dynamic behavior of a sucker rod string used in the oil well industry is presented. The main difficulty in the numerical calculation of the examined structure is a multivalued velocity—force relation determined by Coulomb's friction and by loads generated during operation of pump valves. Both the monotonic and nonmonotonic velocity—force relations are considered. A quasi-variational inequality formulation of the problem is proposed. The solution of the inequality amounts to finding the minimum of a convex nonsmooth functional at each time step by means of the Newmark difference time scheme, successive iterations and finite element discretization. The problem of functional minimization is reduced to construction of a sequence of smooth nonlinear programming problems by introducing the auxiliary variables and applying the augmented Lagrangian method. The proposed approach is used to study the longitudinal vibrations of sucker rod strings under near-real conditions. In such systems the most commonly occurring vibration modes are the stick-slip vibrations and the vibrations with natural force excited twice a cycle. The nonmonotonic character of the friction law leads to intensification of these vibrations. In the case of nonmonotonic friction law the stick-slip vibrations can occur even under the action of constant external forces.     

Rod Vibrations Caused by Axial Impact

The axial impact by an elastic body on an elastic-rod end with a fixed opposite end is considered. The propagation of elastic waves in the rod and the local deformations in the contact zone are taken into account. After recoil of the body, the rod performs free longitudinal vibrations, which can under certain conditions cause parametric transverse vibrations having the character of beats. Depending on the parameters of the problem, the collision time, the shock-pulse shape, and the greatest amplitude of the transverse vibrations under parametric resonance are determined.     

Pulsed Sounding of Cracks with the Use of the Modulation of Ultrasound by Vibrations

The possibility of using the effect of the modulation of ultrasound by vibrations due to the presence of cracks for the nonlinear acoustic detection of cracks is demonstrated. The method is based on a pulsed ultrasonic sounding with gating the received signal and simultaneously exciting low-frequency vibrations in the sample. The presence of a crack is characterized by the modulation of the ultrasonic wave reflected from the crack. The visualization of the crack position in a model object (a metal rod) is performed. The possibility of selecting a crack on the background of an intense signal reflected from a cavity is experimentally demonstrated. The manifestation of the nonlinear properties of a crack is studied as a function of the polarization of the flexural vibrations of the rod.     

Singular inextensible limit in the vibrations of post-buckled rods: Analytical derivation and role of boundary conditions

In-plane vibrations of an elastic rod clamped at both extremities are studied. The rod is modeled as an extensible planar Kirchhoff elastic rod under large displacements and rotations. Equilibrium configurations and vibrations around these configurations are computed analytically in the incipient post-buckling regime. Of particular interest is the variation of the first mode frequency as the load is increased through the buckling threshold. The loading type is found to have a crucial importance as the first mode frequency is shown to behave singularly in the zero thickness limit in the case of prescribed axial displacement, whereas a regular behavior is found in the case of prescribed axial load.     

Electric fields in polymers during dynamic deformation

The transverse vibrations of a beam and the propagation of tension, compression, and shear deformation waves along the axis of a rod are studied by recording the electric field that appears under these conditions near the rod. Experiments are performed on samples made of various plastic materials in order to compare the effect of the properties of a material on the electric response during dynamic deformation, all other things being equal. Dynamic Young’s moduli are determined during bending vibrations and the propagation of longitudinal waves. It is shown that the location and type of antenna should be taken into account to adequately interpret a recorded signal and a dynamic mechanical process.     

Forced transverse vibrations of an elastically connected complex simply supported double-beam system

The present paper is devoted to analyzing undamped forced transverse vibrations of an elastically connected complex double-beam system. The problem is formulated and solved in the case of simply supported beams. The classical modal expansion method is applied to ascertain dynamic responses of beams due to arbitrarily distributed continuous loads. Several cases of particularly interesting excitation loadings are investigated. The action of stationary harmonic loads and moving forces is considered. In discussing vibrations caused by exciting harmonic forces, conditions of resonance and dynamic vibration absorption are determined. The beam-type dynamic absorber is a new concept of a continuous dynamic vibration absorber (CDVA), which can be applied to suppress excessive vibrations of corresponding beam systems. A numerical example is presented to illustrate the theoretical analysis.     

Forced torsional vibrations of a cylindrical rod connected to an elastic half-space

An analysis is presented of the forced torsional vibrations of a cylindrical rod connected to an elastic half-space under the condition that the circumferential displacement at the free end of the rod, where the disturbing moment is applied, varies proportionally with the distance from the rod axis. Both the Pochhammer-Chree and elementary theory are utilized. The response curve of the rotational amplitude at the free end of the rod, obtained from the Pochhammer-Chree theory, and that obtained from the elementary theory almost coincide with each other except in the case where the rod and the half-space are of the same material. The amplitude attenuation is largest when the rod and the half-space are of the same material. The maximum values of half-space displacement distribution at the interface lie within the rod edge.     

黏性介质中圆截面弹性细杆的平面振动

讨论圆截面弹性细杆在黏性介质中的平面振动. 基于Kirchhoff理论，以杆中心线的Frenet坐标系为参考系，建立其动力学方程，杆中心线为任意平面曲线时，其扭转振动与弯曲振动解耦. 讨论两端固定条件下任意形状杆的平面扭转振动，以及无扭转的轴向受压直杆和圆环杆的平面弯曲振动，导出其自由振动频率和阻尼系数. 证明空间域内压杆的Lyapunov稳定性和欧拉稳定性条件为时域内渐近稳定性的充分必要条件，或无阻尼压杆的稳定性必要条件. 圆环杆平衡恒满足渐近稳定性条件. 关键词： 弹性细杆 黏性介质 扭转振动 弯曲振动     

Stability and vibration of a helical rod with circular cross section in a viscous medium

The stability and vibration of a thin elastic helical rod with circular cross section in a viscous medium are discussed. The dynamical equations of the rod in the viscous medium are established in the Frenet coordinates of the centreline with the Euler angles describing the attitudes of the cross section as variables. We have proved that the Lyapunov and Euler conditions of stability of a helical rod in the space domain are the necessary conditions for the asymptotic stability of the rod in the time domain. The free frequencies and damping coefficients of torsional and flexural vibrations of the helical rod in the viscous medium are calculated.     

金属管(杆)横驻波演示实验

对两端自由金属管(杆)弯曲振动实验进行了分析,对观察到的现象给出了相应的解释.     

Dynamic Interaction of Longitudinal and Transverse Laser-Induced Oscillations of a Rod

Doklady Physics - The problem of the interaction of longitudinal and transverse vibrations of a rod under thermal actions using laser radiation is considered. Simultaneously taking into account the...     

Identification of the type and parameters of fastening from the natural frequencies of a fastened rod

A rod can be fastened in a number of ways. If the rod ends are inaccessible to visual observation, the question arises of whether it is possible to determine the type and parameters of their fastening from sound vibrations induced by an impact on the rod. In this connection, a possible statement of the problem is as follows: to identify the type and parameters of fastening on both ends of a homogeneous rod from its natural frequencies. In other words, it is necessary to determine the type of fastening of each of the ends of a homogeneous rod (clamping, free support, elastic support, elastic fixing, floating fixing, or free end) and its parameters (the stiffness coefficients of springs for elastic fixing). The duality and stability of the solution to this problem is demonstrated. A method of solving this problem with the use of 9 natural frequencies is proposed. The results of numerical experiments on reconstructing various types of fastening from 9 natural frequencies are presented.     

On dynamics of elastic rod based on exact Cosserat model

The analysis of kinematics and dynamics of an elastic rod with circular cross section is studied on the basis of exact Cosserat model under consideration of the tension and shear deformation of the rod. The dynamical equations of a rod with arbitrary initial shape are established in general form. The dynamics of a straight rod under axial tension and torsion is discussed as an example. In discussion of static stability in the space domain the Greenhill criteria of stability and the Euler load are corrected by the influence of tension and shear strain. In analysis of dynamical stability in the time domain it is shown that the Lyapunov and Euler stability conditions of the rod in space domain are the necessary conditions of Lyapunov's stability in the time domain. The longitudinal, torsional and lateral vibrations of a straight rod based on exact model are discussed, and an exact formula of free frequency of lateral vibration is obtained. The free frequency formulas of various simplified models, such as the Rayleigh beam, the Kirchhoff rod, and the Timoshenko beam, can be seen as special cases of the exact formula under different conditions of simplification.     

Again on the Ishlinskii-Lavrentyev problem

The problem of deformation and transverse vibrations of a thin rectilinear rod under a longitudinal force is considered. It is established in the classic Ishlinskii and Lavrentyev paper in the linear statement that with the longitudinal force essentially exceeding the Euler critical force, the stability loss generates one of the upper buckling modes. Below, the evolution of post-critical rod deformations is considered for long-term force excitation in the nonlinear statement and the relation of the deformation pattern is noted both with the Ishlinskii-Lavrentyev effect and with the Euler elasticas.     

Vibrations of post-buckled rods: The singular inextensible limit

The small-amplitude in-plane vibrations of an elastic rod clamped at both extremities are studied. The rod is modeled as an extensible, shearable, planar Kirchhoff elastic rod under large displacements and rotations, and the vibration frequencies are computed both analytically and numerically as a function of the loading. Of particular interest is the variation of mode frequencies as the load is increased through the buckling threshold. While for some modes there are no qualitative changes in the mode frequencies, other frequencies experience rapid variations after the buckling threshold, the thinner the rod, the more abrupt the variations. Eventually, a mismatch for half of the frequencies at buckling arises between the zero thickness limit of the extensible model and the inextensible model.     

Study on the prestressed sandwich piezoelectric ceramic ultrasonic transducer of torsional-flexural composite vibrational mode

Based on the classical torsional and flexural vibrational theory of a slender rod, the prestressed sandwich torsional-flexural composite mode piezoelectric ceramic ultrasonic transducer is studied. This type of transducer consists of the slender metal rods and the longitudinally and tangentially polarized piezoelectric ceramic rings. The resonance frequency equations for the torsional and flexural vibrations in the transducers are derived. The simultaneous resonance of the torsional and flexural vibrations in the transducer is acquired by correcting the length of the metal slender rods resulting from the piezoelectric ceramic elements. The experimental results show that the measured resonance frequencies of the transducers are in good agreement with the computed ones, and the measured resonance frequencies of the torsional and the flexural vibrations in the composite transducers are also in good agreement with each other.     

Numerical investigation of the flow in the cathode melt droplet of electric arc

The role of electromagnetic forces and the forces of viscous friction with the arc plasma flow in the flow formation within the cathode melt is considered within the framework of numerical modelling; a comparative estimation of the separate influence of each of the above forces is carried out. The melt flow pattern is found to form mainly by electromagnetic forces. The character of the effect of electromagnetic forces is determined to a considerable extent by the ratio of the radius of cathode attachment of the arc on the melt droplet and the rod cathode radius.     

A theorem on the representation of the field of forced vibrations of a composite elastic system

A theorem on the representation of a vibration field of an elastic system comprising two subsystems is proved in a general form. It is assumed that the system is linear and the subsystems are rigidly connected and interact along a continuous surface S. According to the theorem, forced vibrations of the system can be represented in the form of the sum of two components, which are the solutions of two simpler auxiliary boundary-value problems. The first component is the field of vibrations of the isolated (separated or blocked along S) subsystems under the effect of preset external forces. The second component represents the forced vibrations of the junction of the subsystems, where the external forces are taken equal to zero and only the reaction forces obtained in solving the first auxiliary problem act at the surface S. The theorem is applied to the problem on the reflection and transmission of elastic waves through a junction of two media. It is demonstrated that the theorem utilization reduces the amount of calculations. Other applications are discussed.