Stress–Strain Analysis of Smooth and Ribbed Shells under Local Loads

Static problems for smooth and discretely reinforced cylindrical shells under local loads and complex boundary conditions are solved. The stress–strain states of the casing and ribs are determined by the technical theory of shells and the Kirchhoff–Clebsch theory of rods, respectively. The reinforcing elements are arranged eccentrically. They are of equal or different stiffness, which is also variable along the length. The problems are solved using the finite-difference method. Theoretical results obtained from a refined mesh are compared with experimental data     

Inverse problem of the vibrations of rods and strings of variable mass

We formulate the inverse problem for longitudinal, torsional, and transverse vibrations of rods and transverse vibrations of strings of variable mass. It is shown that if certain assumptions are made concerning the densities of the combining and separating particles, the problem reduces to the integration of independent partial differential equations and a system of ordinary differential equations of first order.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 117–124, September–October, 1974.     

Equations of oscillations of rods of variable composition

We obtain differential equations for the general case of longitudinal, torsional, and transverse oscillations of rods to some parts of which masses are being added or detached. We solve certain special problems concerning the oscillations of such rods of variable composition. In deriving generalized equations of oscillations of rods of variable composition we employ the assumption of planar sections, the assumption of small deformations, and other customary simplifications. We also employ the simplifying assumption of close action; i.e., we assume that the masses being detached and added interact with the rod only at the instant of direct contact. Forces of internal nonelastic resistance are not taken into account. We assume also that in the undeformed state the elastic axis is rectilinear and that the centers of gravity of cross sections are not displaced from their initial positions relative to the cross sections. There may be a change of mass per unit length of the rod both on account of a change in density as well as on account of a change in area of a cross section, the latter being understood to be the union of the initial area of the cross section and the areas of the parts being added and detached. In addition, with the rod there may be associated particles of variable mass distributed continuously or discretely along the length of the rod. We assume that these particles do not interact among themselves but only with the rod.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 103–108, January–February, 1972.     

Postbuckling Behavior of Compressed Rods in an Elastic Medium

The postbuckling of rods loaded by a compressive force P in an elastic medium is considered. The resolving nonlinear equation is obtained, and a method for solving this equation is given. It is shown that, for large lengths, in contrast to the case without elastic medium, the deflection increases as the force P decreases after the loss of stability. Several simple finite-element models, namely, the problems of compression of multilink rods with links connected by springs, are considered to confirm this effect.     

TRIAL FUNCTION METHOD FOR THE LONGITUDINAL VIBRATION OF RODS WITH VARIABLE CROSS-SECTIONS

给出了一种试探函数法,并研究了变截面杆的纵振动问题. 先给出振动控制方程的特殊函数形式的试探解,然后要求此解满足控制方程,反过来确定了控制方程各种可能的系数函数（即截面变化函数）并得到了控制方程的精确解. 作为例子,给出了一种变截面杆在3 种边界条件下的频率方程,计算出了固有频率. 研究表明,试探函数法简单、直接,适合于研究变截面杆的纵振动问题. 对于杆扭转振动、薄膜振动以及管中波传播等问题,该方法同样有推广应用价值.     

Wave propagation and vibration of elastic rods with interfacial frictional slip

Wave propagation and vibration of elastic rods interacting with their environment according to the Coulomb dry friction law, are studied. Exact solutions of the nonlinear problems of impact of semi-infinite or finite rods, with constant stress and velocity or by a rigid body are obtained. Problems of smooth loading and unloading of a semi-infinite rod are solved as well.

A method of exact analytical solution of problems of vibration for steplike loading is developed. The cases of suddenly applied stress which is maintained constant afterwards or changes sign steplike with the period of free vibration, are studied. Continuously distributed and localized friction are considered. It is shown that extension of a zone of disturbances and dissipation in a system with friction, stricly depends on the history of loading.     

A numerical scheme for the study of poynting effect in wave propagation problems with finite boundaries

A second order effect, involving a change of length or an axial force, is present in hyperelastic cylindrical rods subjected to a finite twist. This second order effect leads to a coupling of torsional and longitudinal waves in such rods when they are subjected to finite deformations. In this paper, effects of such a coupling has been studied for cylindrical rods of finite length. The resulting finite deformation elastodynamic problem has been solved by a finite difference method which is a finite deformation elastodynamic problem has been solved by a finite difference method which is a MacCormack two-step variant of Lax-Wendroff second order accurate scheme. The accuracy of the numerical technique has been calibrated by comparing solutions with reported similarity solutions for semi-infinite rods. New results have been presented for finite rods and different loading conditions.     

Analysis of thermal post-buckling of heated elastic rods

ＩｎｔｒｏｄｕｃｔｉｏｎＡｘｉａｌｌｙｃｏｍｐｒｅｓｓｅｄｓｔｒｅｓｓｗｉｌｌｏｃｃｕｒｉｎａｃｏｎｓｔｒａｉｎｅｄｅｌａｓｔｉｃｃｏｌｕｍｎｓｕｂｊｅｃｔｅｄｔｏａｔｅｍｐｅｒａｔｕｒｅｒｉｓｉｎｇ.Ｉｆｔｈｅｖａｌｕｅｏｆｔｈｅｃｏｍｐｒｅｓｓｅｄｓｔｒｅｓｓｅｓｅｘｃｅｅｄａｃｅｒｔａｉｎｌｉｍｉｔ,ｔｈｅｒｍａｌｂｕｃｋｌｉｎｇｏｆｔｈｅｈｅａｔｅｄｒｏｄｓ,ｗｈｉｃｈｉｓｏｕｔｏｆｔｈｅｉｎｉｔｉａｌｓｔｒａｉｇｈｔｌｉｎｅｃｏｎｆｉｇｕｒａｔｉｏｎ,ｗｉｌｌｔａｋｅｐｌａｃｅ.Ｓｏ,ｔｈｅ…     

Spatial Motion of a Rod in a Viscous Fluid Flow

Equations of spatial motion of a curved finitelength rod in a viscous fluid flow are derived. Analytical solutions of problems on the motion of a straight rod under conditions of pure shear, simple shear, and uniaxial extension of the fluid are obtained. Longitudinal stability of the straight rod during its spatial motion is considered. Effective viscosity of a suspension filled by rigid straight rods is evaluated.     

A Material Momentum Balance Law for Rods

A material momentum balance law is presented in this paper where it is also specialized for a variety of rod and string theories. The local form of the law is assumed to be identically satisfied, while the jump condition provides an extra equation which is often needed to solve problems involving the application of rod and string theories. The balance law is also related to several existing conservation laws for strings and rods, including Kelvin’s circulation theorem. A novel identity for the singular sources at a discontinuity is also established. Dedicated to James N. Flavin, my friend and mentor, on the occasion of his 70th Birthday.     

DISPLACEMENT CALCULATION FOR STRAIGHT BAR WITH VARIABLE STIFFNESS BASED ON GRAPH MULTIPLICATION METHOD1)

为解决图乘法计算变截面杆件位移时分割图形数目多,计算容易出错这一问题,提出了一种刚度折算的方法,把长度已知的变截面杆件等价转换为两个一定长度的等截面杆件,通过刚度与位移的关系进行折算,然后叠加即可得到正确结果。可显著减少图乘次数,降低计算量,提高计算效率和正确率,值得推广。     

Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures

Localization phenomena in one-dimensional imperfect continuous structures are analyzed, both in dynamics and buckling. By using simple models, fundamental concepts about localization are introduced and similarities between dynamics and buckling localization are highlighted. In particular, it is shown that strong localization of the normal modes is due to turning points in which purely imaginary characteristic exponents assume a non zero real part; in contrast, if turning points do not occur, only weak localization can exist. The possibility of a disturbance propagating along the structure is also discussed. A perturbation method is then illustrated, which generalizes the classical WKB method; this allows the differential problem to be transformed into a sequence of algebraic problems in which the spatial variable appears as a parameter. Applications of the method are worked out for beams and strings on elastic soil. All these structures are found to have nearly-defective system matrices, so their characteristic exponents are highly sensitive to imperfections.     

Post-buckling of a cantilever rod with variable cross-sections under combined load

IntroductionSinceEuler[1] ,Lagrange[2 ] ,Love[3 ] etal.investigatedtheslenderrod ,asoneofthebasicstructuralstabilityproblems ,manyattentionshavebeenpaidtothepost_bucklingofelasticrodsforalongtime .Today ,flexiblerodshavebeenwidelyusedasspring ,linkages ,robot’sarms,largeantennasandsoon .Hence ,thestudiesofpost_bucklingofelasticrodshavewideengineeringandapplyingbackgroundsinrecentdays .Basedontheassumptionthattheaxiallineoftherodisinextensible ,Timoshenkoetal.[4] examinedthepost_bucklingofco…     

Time- and frequency-domain analysis using lumped mass global collocation

Quite recently, a novel global collocation method for the eigenvalue analysis of freely vibrated elastic structures was proposed (Archive of Applied Mechanics: DOI: ). This paper extends the latter methodology on several levels, in both the time and frequency domain. Firstly the formulation is updated so that it can also deal with rods of variable cross section. Then, the fully populated mass matrices of the previous formulation are properly replaced by lumped masses, thus saving still more computer effort. Subsequently, a new general formulation for the transient response analysis is proposed. Finally, a novel procedure for the coupling of two neighboring collinear rods is presented. The theory is supported by six test cases concerning elastic rods of constant and variable cross sections. Among these, transient analysis refers to the response of a single rod due to a Heaviside-type loading as well as to the impact between two collinear rods of different cross sections.     

Complex-mode Galerkin approach in transverse vibration of an axially accelerating viscoelastic string

Under the consideration of harmonic fluctuations of initial tension and axially velocity, a nonlinear governing equation for transverse vibration of an axially accelerating string is set up by using the equation of motion for a 3-dimensional deformable body with initial stresses. The Kelvin model is used to describe viscoelastic behaviors of the material. The basis function of the complex-mode Galerkin method for axially accelerating nonlinear strings is constructed by using the modal function of linear moving strings with constant axially transport velocity. By the constructed basis functions, the application of the complex-mode Galerkin method in nonlinear vibration analysis of an axially accelerating viscoelastic string is investigated. Numerical results show that the convergence velocity of the complex-mode Galerkin method is higher than that of the real-mode Galerkin method for a variable coefficient gyroscopic system.     

Asymptotic analysis of perforated plates and membranes. Part 1: Static problems for small holes

Static problems for the elastic plates and rods periodically perforated by small holes of different shapes are solved using the asymptotic approach based on the combination of the asymptotic technique and the multi-scale homogenization method. Using the asymptotic homogenization method the original boundary-value problem is reduced to the combination of two types of problems. First one is a recurrent system of unit cell problems with the conditions of periodic continuation. And the second problem is a homogenized boundary-value problem for the entire domain, characterized by the constant effective coefficients obtained from the solution of the unit cell problems. The combination of the perturbation method and the technique of successive approximations is applied for the solution of the unit cell problems. Taking into the account small size of holes the method of perturbation of the shape of the boundary and the Schwarz alternating method are used. The problems of torsion of a rod with perforated cross-section; deflection of the perforated membrane loaded by a normal load; and bending of perforated plates with circular and square holes are considered consecutively. The error of the applied asymptotic techniques is estimated and the high accuracy of the obtained solutions is demonstrated.     

ANALYTICAL SOLUTIONS FOR RESPONSE OF COLLISION OF PARTICLE WITH CONICAL ROD CAUSED BY LONGITUDINAL VIBRATION

The objective is to present exact analytical solutions of longitudinal impact analysis for slender conical rods struck by a particle and a new method is proposed for conical rod-particle impact analysis, in which the superposition method is used and the response of the rod is presented. These analytical results are exact and can be used to validate the numerical methods or other analytical results. The numerical example shows that one of the advantages of the present method is that the analytical form is very simple. The result is that mass ratio and some variables describing the geometrical shape of rods such as taper, length and radius play an important role in impact dynamic system.     

Self-Contact for Rods on Cylinders

We study self-contact phenomena in elastic rods that are constrained to lie on a cylinder. By choosing a particular set of variables to describe the rod centerline the variational setting is made particularly simple: the strain energy is a second-order functional of a single scalar variable, and the self-contact constraint is written as an integral inequality. Using techniques from ordinary differential equation theory (comparison principles) and variational calculus (cut-and-paste arguments) we fully characterize the structure of constrained minimizers. An important auxiliary result states that the set of self-contact points is continuous, a result that contrasts with known examples from contact problems in free rods.     

A novel semi-inverse solution method for elastoplastic torsion of heat treated rods

Torsion rods are a primary component of many power transmission and other mechanical systems. The behavior of these rods under elastoplastic torsion is of major concern for designers. Different methods have so far been proposed which deal with the elastoplastic torsion of rods, most of which assume constant yield stress. This assumption produces rough and inaccurate results when the rods are heat treated, since in the process of heat treatment the form of yield stress distribution across the rod cross section changes. We propose a new method for calculating elastoplastic torsion of rods of simply connected cross section which is based on heat treatment observations. In our method the full plastic stress function is obtained by using the semi-inverse method. Elastoplastic stress function is obtained by generalizing the idea of the membrane analogy and using a piecewise continuous stress function. Since the proposed form of yield stress distribution can not be handled by the current Finite Element packages, we produce a computer package with a 3D graphical interface capable of calculating and displaying the 3D elastoplastic stress function, shear stress contours, and torque-angle of rotation per unit length. We show that our method produces excellent agreement for several known cross sections in comparison to methods which assume constant yield stress.     

Dynamics of strings made of phase-transforming materials

This paper presents a theory to describe the dynamical behavior of a string made of a phase-transforming material like a shape-memory alloy. The study of phase boundaries, the driving force acting on them and the kinetic relation governing their propagation is of central concern. The paper proposes a qualitative experimental test of the notion of a kinetic relation, as well as a simple experimental method for measuring it quantitatively. It presents a numerical method for studying general initial and boundary value problems in strings, and concludes by exploring the use of phase transforming strings to generate motion at very small scales.