Shock Waves in Anisotropic Cylinders

We study small-amplitude longitudinal and torsional shock waves in circular cylinders consisting of an anisotropic medium such that the velocities of the longitudinal and torsional waves are close to each other. Previously, simple waves were considered in the same situation and conditions were found for these waves to overturn and for the corresponding shock waves to form. Here we present the study of shock waves: the shock adiabat and the evolutionary conditions. The results obtained can also be related to shock waves in unbounded media with quadratic nonlinearity.     

Torsional rigidity of reinforced concrete bars with arbitrary sectional shape

In this paper, the torsional rigidity of arbitrarily shape bar made of different materials is studied on the basis of theory of elasticity and finite element approach. With additional boundary conditions for the common boundaries of different materials from the continuous conditions of deformation and traction across the interior boundary, the torsion function can be solved numerically from the second boundary-value problem of potential theory. The traction jump boundary conditions across the interior surfaces are enforced in the alternate finite element approach. Several examples are shown to check the computational approach proposed, and the approach, at last, is applied to calculate the torsional rigidity of reinforced concrete bar and some multiply connected cross sections such as tower leg section of the Tsing Ma Bridge and other engineering structures.     

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.     

缫丝工程中不同茧层的茧粒数分布模型及其应用

在定粒缫丝工程中采用粗细茧丝搭配的方法进行配茧缫丝时,需要根据落绪茧及茧丝粗细搭配状况选择适当的接绪茧接绪,以保持厚薄茧型的搭配,减小生丝纤度方差.本文对该缫丝动态过程进行数学建模,给出差微分方程组及其平衡状态下不同茧层茧粒数的分布.通过对定粒缫丝动态过程的计算机模拟,对获得的理论结果及其在缫丝工程管理中的应用问题进行了讨论.     

考虑屋盖整体空间工作时的单层偏心厂房的平扭耦联地震反应分析

本文研究了单层偏心厂房的平动-扭转耦联地震反应.建立了考虑屋盖整体空间工作时的地震反应运动方程,并进行了分析,讨论.指出在包含有地震动转动分量作用时,地震动各运动分量独立作用的迭加原理和动力反应分析的振型分解法已不再适用.本文在不计地震转动分量的前提下,对单层偏心厂房的平扭耦联地震反应进行了大量的计算及对比分析,结论是:考虑屋盖变形后的计算结果能更好地解释地震破坏现象,并指出了不考虑屋盖变形的适用范围.     

The effects of lateral–torsional coupling on the nonlinear dynamic behavior of a rotating continuous flexible shaft–disk system with rub–impact

This study investigates the lateral–torsional coupling effects on the nonlinear dynamic behavior of a rotating flexible shaft–disk system. The system is modeled as a continuous shaft with a rigid disk in its mid span. Coriolis and centrifugal effects due to shaft flexibility are also included. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed mode method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work include time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The main objective of the present study is to investigate the torsional coupling effects on the chaotic vibration behavior of a system. Periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for cases with and without torsional effects. As demonstrated, inclusion of the torsional–lateral coupling effects can primarily change the speed ratios at which rub–impact occurs. Also, substantial differences are shown to exist in the nonlinear dynamic behavior of the system in the two cases.     

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.     

Torsional oscillations in structures subject to ground motion

A multiple-scale technique has been employed to study nonlinear torsional oscillations in single-storey structures with cubic softening stiffness members subject to a single frequency ground excitation. The structures are studied under free oscillation conditions, primary resonance, and combination resonance. Time history analyses are employed to quantify the dynamic behaviour of a single-storey structure. The instabilities are related to the jump in response which exists in single degree-of-freedom softening oscillators subjected to harmonic excitation. It is shown that this jump can be quantified analytically, and appears as a cusp catastrophe in the bending mode. Time history analyses show that the bending mode can exhibit a limit cycle behaviour of constant amplitude.     

Efectos torsionales en estructuras sobre suelo blando

In this work the torsional vibrations of structures on soft soil caused by the combined effects of structural asymmetry and base rotation are evaluated. A simplified soil-structure system with five degrees of freedom is analyzed. The foundation is considered to be embedded into a uniform layer resting on an elastic half-space, under the incidence of inclined SH waves. Instead of matching the maximum torsional moments due to the independent effects of structural asymmetry and base rotation, a new approach for computing the design eccentricity is applied. Specifically, torsional eccentricity coefficients compatible with the peak coupled lateral-torsional response are obtained, using an ensemble of narrow-band earthquake motions typical of soft-soil sites in Mexico City. It is shown that these coefficients are not constant along the principal axes, but they depend on the position of the resistant element. Hence, it is necessary to compute them without assuming a linear variation, as is done nowadays. The torsional eccentricity coefficients can significantly differ from the currently codified values, particularly for torsionally flexible structures for which both coefficients may be negative. This implies that the lateral displacement is reduced by the effects of torsion.     

Propagation of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space

In the present paper, the dispersion equation which determines the velocity of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space has been obtained. The dispersion equation obtained is in agreement with the classical result of Love wave when the initial stresses and inhomogeneity parameters are neglected. Numerical results analyzing the dispersion equation are discussed and presented graphically. The result shows that the initial stresses have a pronounced influence on the propagation of torsional surface waves. It has also been shown that the effect of density, directional rigidities and non-homogeneity parameter on the propagation of torsional surface waves is prominent.     

横观各向同性饱和地基上刚性圆板的扭转振动

通过解析方法研究了横观各向同性饱和半空间上刚性圆板在简谐扭转荷载作用下的振动问题．运用Hankel变换求解了横观各向同性饱和土的动力控制方程,结合混合边界条件得出了刚性基础的扭转对偶积分方程,并将对偶积分方程转化为第二类Fredholm积分方程求解了基础的扭转振动问题,同时给出了动力柔度系数,基础的角位移幅值和基底接触剪应力的表达式．通过数值算例研究了地基的各向异性程度对基础扭转振动的影响．     

A Note on a Longitudinal and Torsional Type of Ultrasonic Motor with two Stators

This paper deals with an ultrasonic motor, which consist of one rotor and two stators. The main parts of each stator are one longitudinal and one torsional piezoceramic actuator, which are used to generate the desired elliptical orbit on the stator surface. For the purpose of a high efficiency, the longitudinal and the torsional resonant frequency should ideally coincide with the operational frequency. To match the resonant frequencies, two adjustable rings are added to the stators. The influence of the mass and the position of the rings on the first longitudinal and the first torsional eigenfrequency and their mode shapes is investigated. The experimental results fit very well with the theoretical predictions and show that the adjusting rings can be used to match the eigenfrequencies of the two mode shapes. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytical study on dynamic responses of a curved beam subjected to three-directional moving loads

Considering the warping resistance, inertia force and moving three-directional loads, a more comprehensive set of governing equations for vertical, torsional, radial and axial motions of the curved beam are derived. The analytical solutions for vertical, torsional, radial and axial responses of the curved beam subjected to three-directional moving loads are obtained, using the Galerkin method to discretize the partial differential equations and the modal superposition method to decouple the ordinary differential equations. The analytical results are compared with the numerical integration and a published work to verify the validity of the proposed solutions. Effects of Galerkin truncation terms and damping ratio on solution convergence are also discussed. Considering first-mode and higher-mode truncation respectively, the conditions of resonance and cancellation are analyzed for vertical, torsional, radial and axial motions of the curved beam. Taking a curved bridge under passage of a vehicle as an example, the influences of system parameters, such as vehicle speed, braking acceleration, bridge curve radius, bridge span and bridge deck elastic modulus, on bridge midpoint vibration are explored. The proposed approach and results may be beneficial to enhance understanding the three-directional vehicle-induced dynamic responses of curved bridges. It is shown that when the axial motion, or the multiple moving loads are involved, the first-order truncation are not accurate enough and one should use higher-mode truncation to study the responses of curved beams. In addition, it is necessary to consider damping in the vibration study of curved beams.     

A stochastic dynamic model of train-track-bridge coupled system based on probability density evolution method

An innovative stochastic dynamic model of a 3D train-track-bridge coupled system (TTBS) with refined wheel/rail interaction is established for a high-speed railway based on the random theory of probability density evolution method (PDEM). The multi-coupling effect of excitations can be simultaneously input into the new model, e.g. random track irregularity, random vehicle loads, stochastic system parameters, et al. Moreover, a new approach, named “Number theoretic method of multi-target probability functions” (NTM-mp), is developed to obtain the discrete point sets of multidimensional random parameters in hypercube space, aims to solve the point design of system uncertainty. The stochastic harmonic function (SHF) is applied to generate representative random track irregularity samples. The results of TTBS got by PDEM are verified with several typical case studies for its efficiency and reliability, which are the deterministic results in the representative publication, the Monte Carlo method (MCM) results, and the field testing results on the high-speed railway. At last, a typical case study of TTBS on a high-speed railway is presented for numerical analysis. Discussions and significant conclusions on the random dynamic responses are presented.     

Investigation of critical influential speed for moving mass problems on beams

A traveling mass due to its mass inertia has significant effects on the dynamic response of the structures. According to recent developments in structural materials and constructional technologies, the structures are likely to be affected by sudden changes of masses and substructure elements, in which the inertia effect of a moving mass is not negligible. The transverse inertia effects have been a topic of interest in bridge dynamics, design of railway tracks, guide way systems and other engineering applications such as modern high-speed precision machinery process. In this study an analytical–numerical method is presented which can be used to determine the dynamic response of beams carrying a moving mass, with various boundary conditions. It has been shown that the Coriolis acceleration, associated with the moving mass as it traverses along the vibrating beam shall be considered as well. Influences regarding the speed of the moving mass on the dynamic response of beams with various boundary conditions were also investigated. Results illustrated that the speed of a moving mass has direct influence on the entire structural dynamic response, depending on its boundary conditions. Critical influential speeds in the moving mass problems were introduced and obtained in numerical examples for various BC’s.     

Regularization of optimal design problems for bars and plates,part 2

For the problems of optimal design considered in Ref. 1, contradictions arising in the necessary conditions of optimality are eliminated by suitable extension of the initially given class of admissible materials. The extended class includes composites of some special (layered) microstructure. Elastic properties of such composites are described, and alternative (regularized) formulations of the optimal design problems are given. Necessary conditions of Weierstrass are shown to be satisfied, both for the case in which the strip of variations is small compared with the width of the layers and for the opposite case. Numerical results are given for the regularized problem of a bar of extremal torsional rigidity.The authors are indebted to Dr. N. A. Lavrov for performing numerical calculations.     

具有弹性边拱及拉杆支承的双曲扁壳(一)

本文利用变分原理建立了具有弹性边拱及拉杆支承的双曲扁壳的平衡方程式及相应的边界条件和角点条件,这里假定边拱只在其本身平面内有刚度,边拱的扭转刚度和垂直于其平面的弯曲刚度都略去不计,本文研究了不许自由外伸的角点铰支条件,以及能够自由外伸的角点简支条件,前者相当于周边有拉杆限制角点外伸位移的情况,后者相当于周边无拉杆的情况.对于前者而言,本文近似地假定边拱沿弧方向的抗拉伸刚度为无穷大,亦即假定扁壳的边界切向位移为零,边拱只通过其垂直于扁壳平面的弯曲来产生弹性支承的作用.这些支承条件是近似地符合当前双曲扁壳屋盖的设计条件的.本文利用双三角级数解法求得具有弹性边拱及拉杆支承的方形底球面扁壳在自重载荷下的正确解.其特点在于先将边界条件积分处理使先满足角点条件,然后求解平面应力微分方程使满足积分后的边界条件.本文的结果直接给出拉杆中的拉力,对于具体设计问题是有用的.本文提出的积分形式的边界条件方法,对于弹性支承的边界问题在板壳方面的应用中是有它的普遍实用意义的.本文还给出了具有弹性边拱支承的方形底扁球壳的数值结果,角点为铰支或简支的,选取的参数值为λ=11.5936.计算结果表明级数收敛很快,并得出了边拱的弹性变形对壳体内力、内力矩及挠度分布规律的影响.     

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)     

Nonlinear dynamics of statically displaced tendons with non-conventional end conditions

This study deals with the nonlinear dynamic behavior of TLP tendons subjected to heaving motions imposed at the top. It investigates a realistic configuration that is determined by static displacements and non-conventional boundary conditions defined by torsional springs attached at the end connections. Both considerations alter the characteristics of the tendon dynamics, both linear and nonlinear. The study is dedicated to the investigation of the dynamic impacts that could greatly affect tendon's structural integrity under severe loading. Two major issues are identified and examined: (i) possible tension cancellation and (ii) identification of the nonlinear internal resonances due to parametric-like excitation and analytical processing of the nonlinear system at relevant events. The whole process makes use of the complicated static configuration that takes into account the initial tendon displacement(s) and the boundary conditions determined by the existence of the torsional springs. The tension cancellation occurrences due to heavy vertical loading imposed at the top, are assessed by means of the numerical solution of the underlying boundary value problem. It is shown that the use of springs reduces the possibility for the nullification of the total tension and the destabilization of the floater which could be encountered under “ringing motions”. The identification of the internal resonances is implemented through the analytic processing of the reduced governing set, using perturbation analysis.     

L形和十形截面杆的抗扭刚度

本文是作者前三篇论文的一个继续[1-3].利用调和函数延拓方法,L形和+形截面杆的抗扭刚度很容易求得.数值结果如表1-3所示.