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.     

弹性细杆弯曲的Kirchhoff方程的违约校正求解

采用Euler四元数表示的Kirchhoff方程来研究受力挤压作用下的弹性细杆的拓扑构形，进一 步研究弹性细杆的力学性质；将得到的微分方程与约束条件组成微分代数方程后再转化为微 分方程规范形式以便求解；为满足边界条件，应用数值打靶法求解边值条件，并将弹性细杆 在力作用下的拉压过程用Matlab仿真出来.同时对由于误差导致的违约现象进行处理，并针 对欧拉参数的特征，选取合适的修正系数以保持方程的稳定性. 关键词： DNA Euler四元数 Kirchhoff方程 弹性细杆 违约修正     

Supersonic flows with small perturbations in the presence of external effects in the flow. Part 1: “Thin body of revolution”

Axisymmetric supersonic flow about a thin body of revolution with an external energy supply and an external force localized near the body surface is considered in the linear approximation. An analytic theory is constructed for calculating spatial fields of flow parameters in this case for an arbitrary dependence of external effects on the longitudinal coordinate. Formulas are derived for the pressure ratio on the surface of the thin body of revolution. The results of calculations based on the analytic theory are in good agreement with numerical data obtained from the solution of hydrodynamic equations in the Euler approximation.     

Kirchhoff弹性直杆在力螺旋作用下的稳定性

研究受力螺旋作用的圆截面Kirchhoff弹性直杆在各种边界条件下的稳定性问题. 用直角坐标和Cardano角表示截面的形心位置和姿态. 由Kirchhoff方程得到弹性细杆的直线平衡特解,导出线性化扰动方程及其通解. 根据边界条件确定积分常数的非零解存在条件,讨论了各种边界条件,如两端铰支、两端固定、一端铰支一端固定以及一端固定一端自由的弹性细杆直线平衡状态的稳定性,导出了临界载荷的表达式,绘制了稳定域,将Greenhill公式推广到其他边界条件,并且使压杆的Euler 公式成为其特例. 关键词： Kirchhoff弹性杆 稳定性 力螺旋 Greenhill公式     

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.     

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.     

Solution to the problem of Nicolai

We consider the problem of Nicolai on dynamic stability of an elastic cantilever rod loaded by an axial compressive force P and a twisting tangential torque L in continuous formulation. The problem is to find the stability region for non-equal principal moments of inertia of the rod in the space of three parameters: P, L and the parameter α for the ratio of principal moments of inertia. New governing equations and boundary conditions, which form the basis for analytical and numerical studies, are derived. An important detail of this formulation is that the pre-twisting of the rod due to the torque L is taken into account. The singular point on the stability boundary at the critical Euler force P_E is recognized and investigated in detail. For an elliptic cross-section of a uniform rod the stability region is found numerically with the use of the Galerkin method and the exact numerical approach. The obtained numerical results are compared with the analytical formulas of the asymptotic analysis.     

Shape transition and propulsive force of an elastic rod rotating in a viscous fluid

The deformation of an elastic rod rotating in a viscous fluid is considered, with applications related to flagellar motility. The rod is tilted relative to the rotation axis, and experiments and theory are used to study the shape transition when driven either at constant torque or at constant speed. At low applied torque, the rod bends gently and generates small propulsive force. At a critical torque, the rotation speed increases abruptly, and the rod forms a helical shape with increased propulsive force. We find good agreement between theory and experiment. A simple physical model is presented to capture and explain the essential behavior.     

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.     

ASYMPTOTIC APPROACH FOR NON-LINEAR PERIODICAL VIBRATIONS OF CONTINUOUS STRUCTURES

An asymptotic approach for determining periodic solutions of non-linear vibration problems of continuous structures (such as rods, beams, plates, etc.) is proposed. Starting with the well-known perturbation technique, the independent displacement and frequency is expanded in a power series of a natural small parameter. It leads to infinite systems of interconnected non-linear algebraic equations governing the relationships between modes, amplitudes and frequencies. A non-trivial asymptotic technique, based on the introduction of an artificial small parameter is used to solve the equations. An advantage of the procedure is the possibility to take into account a number of vibration modes. As examples, free longitudinal vibrations of a rod and lateral vibrations of a beam under cubically non-linear restoring force are considered. Resonance interactions between different modes are investigated and asymptotic formulae for corresponding backbone curves are derived.     

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...     

Bulk Nonlinear Elastic Strain Waves in a Bilayer Coaxial Cylindrical Rod

The problem of the propagation of long nonlinear elastic strain waves in a bilayer coaxial cylindrical rod with an ideal contact between the layers has been considered. Expressions for transverse displacements through longitudinal displacements have been derived. The former satisfies free boundary conditions and continuity conditions for displacements and stresses at the interlayer interface with the desired accuracy. It has been shown how these expressions generalize the well-known plane-section and Love hypotheses for an isotropic homogeneous rod. An equation for the propagation of a nonlinearly elastic strain longitudinal wave has been derived, and its particular solution in the form of a solitary traveling wave has been studied.     

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.     

双层杆结构中纵向模态的超声导波

通过分析应力和位移边界条件得到了双层杆结构中纵向模态的频散方程，并进行数值求解，得到相应的频散曲线。对该结构中纵向模态的主要传播特性进行了分析。然后，利用长度伸缩型压电陶瓷片在内层为冰外层为钢的双层杆结构中进行了激励接收纵向模态的实验。实验结果与理论分析较为吻合。结果表明，双层杆结构的频散曲线可用作模态选取的理论指导，用于该结构的缺陷检测。     

Impact of a linearly elastic rod on a thin linearly viscoelastic target

The impact of a linearly elastic rod on a thin linearly viscoelastic target which rests on a rigid foundation is considered. The behaviour of the target is quasi-static. The special case of a cylindrical linearly elastic rod impacting on a thin Kelvin target is studied in detail and the maximum impact force, the coefficient of restitution, and the energy absorption of the target are determined. The type of results obtained is useful, e.g., in the design of impact protection devices.     

颗粒物质中圆棒受到的静摩擦力

通过实验研究了圆棒在颗粒物质中受到的摩擦力与颗粒填充高度及棒径的关系.观测到摩擦力随颗粒高度和棒径而增大.用连续介质模型推导了摩擦力公式，与实验观测基本一致.结果表明，摩擦力和棒径成正比；当颗粒高度很小时，摩擦力与颗粒高度平方成正比；而当颗粒堆积很高时，摩擦力与高度成正比，即静摩擦力与接触面积成正比. 关键词： 颗粒物质 摩擦力     

The reconstruction of density and Young’s modulus of an inhomogeneous rod

The problems of reconstructing the Young modulus and the density of an inhomogeneous elastic rod by analyzing its longitudinal and bending vibrations are considered. For solving these problems, an iteration algorithm based on the apparatus of the Fredholm integral equations is proposed. Specific examples of reconstruction are presented.     

Generation of elastic stress waves at a T-junction of square rods

Fourier techniques have been used to predict transmitted and reflected waves at a T-joint in rods of square cross-section for an arbitrary longitudinal impulse approaching the joint in the terminating rod. Elementary and Timoshenko beam theory were used for longitudinal and flexural wave motions respectively. Good agreement is obtained between the predicted and experimental results for an input pulse with large wavelengths compared with the lateral dimensions of the rod. The results demonstrate the similarity to transverse impact of an infinite rod and the unidirectional translation of the joint as a valid model.     

Some features of bending of a rod under a strong longitudinal compression

Typical processes of rod bending under strong longitudinal compression are studied. The corresponding dynamic equation of bending is considered as a perturbation of the two-dimensional Laplace equation. It is established that, for these processes, the expantion of domains of rapid increase of bending begins in small neighborhoods of singularity points of solutions of the limiting Laplace equation. The initial stages of these increases are described using the Hardy integral.     

Longitudinal wave propagation in a rod with variable cross-section

Vibration data are required for condition monitoring in machinery, and can only be collected indirectly after transferring through rods, shells, rotating shafts or other components in many engineering applications. Investigation on the transfer characteristics of vibration in these components is very helpful to guarantee the efficiency of the data collected indirectly. Here, the longitudinal wave propagation in a rod with variable cross-section is investigated. First, the equations of motion are established for the rod based upon the elementary wave theory, the Love theory and the Mindlin–Herrmann theory. Second, the transfer matrix method is employed to explore the propagation characteristics of the rod from the derived equations of motion. Finally, two kinds of rods with the cross-sections varying in the exponential and the polynomial forms are used to illustrate the analytical predictions of the propagation characteristics of the longitudinal wave, which are compared with the results from the finite element analysis (FEA) method. It is shown that Poisson's effect or the shear deformation plays a very important role in the longitudinal wave propagation in the rod and can widen the rod's stop band moderately. Moreover, the cut-off frequency of the rod is unconcerned with the variation form of the cross-section, but dependent on the area ratio between both the ends of the rod, even though Poisson's effect or shear deformation is included.