Displacement of a rigid rod in a viscous medium

The spatial displacement of an absolutely rigid vertical rod in a viscous medium is considered; the rod is hinged at the upper end to a platform moving on the surface of the viscous medium over a specified curvilinear trajectory. A load is attached to the lower end of the rod. Using a variational Lagrangian equation, a nonlinear system of ordinary differential equations in terms of the angles of rod rotation determining the position of points of this rod at any time is obtained. As an example, the problem is solved for conditions of acceleration, uniform motion, and deceleration of the reference point moving over a trajectory consisting of rectilinear sections and portions of circles. The differential equations obtained may be used in determining the position of rod-type elements suspended in a viscous medium.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 108–112, 1990.     

Three-dimensional non-linear oscillations of a rod with hinged supports

The free and forced flexural oscillations of a rod with hinged supports are investigated analytically and numerically. The geometrical non-linearity due to the change in the length of the central line of the rod accompanying its three-dimensional motion is taken into account. The oscillations of a rod with different natural frequencies in two mutually perpendicular directions as a consequence of the variance in the flexural stiffnesses of the rod or the stiffnesses of the supports in the different directions, are considered. It is shown in the case of natural oscillations that, together with two planar forms of motion, a form exists when a certain threshold value is exceeded, which corresponds to the motion of the cross-sections of the rod in a circle. The amplitude-frequency and phase-frequency characteristics of the system are constructed and qualitatively investigated in the neighbourhood of the principal resonance.     

Modeling and analysis of a contact problem for a viscoelastic rod

We consider a nonlinear viscoelastic rod which is in contact with a foundation along its length and is in contact with an obstacle at its end. The rod is acted up by body forces and, as a result, its mechanical state evolves. Our aim in this paper is twofold. The first one is to construct an appropriate mathematical model which describes the evolution of the rod. The second one is to prove the weak solvability of the problem. To this end, we use arguments on second-order inclusions with multivalued pseudomonotone operators.     

Stability of a growing viscoelastic rod subjected to ageing

The stability of a compressed growing rod of viscoelastic material that possesses the property of ageing /1/ is investigated. In conformity with the Chatayev definition of the stability of dynamic systems and the Lyapunov method described in /2/, stability conditions are obtained for a rod growing during a finite time interval, and in finite and semi-infinite time intervals. Some results of a numerical analysis of the behaviour of such a rod are presented in /3/.     

Existence results for dynamic adhesive contact of a rod

The existence and uniqueness of the weak solution to the model for the dynamics of a viscoelastic rod which is in adhesive contact with an obstacle is established. The model consists of a hyperbolic equation for the vibrations of the rod coupled with a nonlinear ordinary differential equation (ODE) for the evolution of the bonding function. The model allows for failure, i.e., complete debonding, in finite time. The existence of the weak solution is established by using an existence result for ODEs and the Schauder fixed-point theorem. The limit of an elastic rod when the viscosity vanishes is studied, too.     

Inverse mode problems for the finite element model of a vibrating rod

The inverse mode problems for the finite element model of an axially vibrating rod are formulated and solved. It is known that for the finite element model, based on linear shape functions, of the rod, the mass and stiffness matrices are both tridiagonal. It is shown that the finite element model of the rod can be constructed from two eigenvalues, their corresponding eigenvectors and the total mass of the rod. The necessary and sufficient conditions for the construction of a physically realizable rod with positive mass and stiffness elements from two eigenpairs and the total mass of the rod are established. If these conditions are satisfied, then the construction of the model is unique.     

Dynamics of a rod undergoing a longitudinal impact by a body

A longitudinal elastic impact caused by a body on a thin rod is considered. The results of theoretical, finite element, and experimental approaches to solving the problem are compared. The theoretical approach takes into account both the propagation of longitudinal waves in the rod and the local deformations described in the Hertz model. This approach leads to a differential equation with a delayed argument. The three-dimensional dynamic problem is considered in terms of the finite element approach in which the wave propagation and local deformation are automatically taken into account. A benchmark test of these two approaches showed a complete qualitative and satisfactory quantitative agreement of the results concerning the contact force and the impact time. In the experiments, only the impact time was determined. The comparison of the measured impact time with the theoretical and finite element method’s results was satisfactory. Owing to the fact that the tested rod was relatively short, the approximate model with two degrees of freedom was also developed to calculate the force and the impact time. The problem of excitation of transverse oscillation after the rebound of the impactor off the rod is solved. For the parametric resonance, the motion has a character of beats at which the energy of longitudinal oscillation is transferred into the energy of transverse oscillation and vice versa. The estimate for the maximum possible amplitude of transverse oscillation is obtained.     

Buckling of a nonlinear elastic rod

The buckling of a pin-ended slender rod subjected to a horizontal end load is formulated as a nonlinear boundary value problem. The rod material is taken to be governed by constitutive laws which are nonlinear with respect to both bending and compression. The nonlinear boundary value problem is converted to a suitable integral equation to allow the application of bounded operator methods. By treating the integral equation as a bifurcation problem, the branch points (critical values of load) are determined and the existence and form of nontrivial solutions (buckled states) in the neighborhood of the branch points is established. The integral equation also affords a direct attack upon the question of uniqueness of the trivial solution (unbuckled state). It is shown that, under certain conditions on the material properties, only the trivial solution is possible for restricted values of the load. One set of conditions gives uniqueness up to the first branch point.     

Modelling and resonance of a flexible rod using a floating reference frame

A direct variational approach with a floating frame is presented to derive the ordinary differential equations of motion of a flexible rod, constant crank speed slider-crank mechanism. Potential energy terms contained in the derivation include beam-bending energy and energy in foreshortening of the rod tip (which were selected because of the importance of these terms in a pinned-pinned rod parametric resonance). A symbolic manipulator code is used to reduce the constrained equations of motion to unconstrained nonlinear equations. A linearized version of these equations is used to explore parametric resonance stability-instability zones by a monodromy matrix technique.     

Chelomei's problem of the stabilization of a statically unstable rod by means of a vibration

Chelomei's problem of the stabilization of an elastic, statically unstable rod by means of a vibration is considered. Formulae for the upper and lower critical frequencies for the stabilization of the rod are obtained and analysed. It is shown that, unlike the high-frequency stabilization of an inverted pendulum with a vibrating suspension point, a rod is stabilized by frequencies of a periodic force of the order of the fundamental frequency of the transverse oscillations of the uncompressed rod lying in a certain range.     

Flexural vibrations of a moving rod

The problem of the transverse natural vibrations of part of a rod between two coaxially fixed guides, moving with an arbitrary constant velocity, is investigated. The conditions of rigid clamping are taken as the boundary conditions. Additional shear stresses, due to longitudinal tension or compression are taken into account. Relations defining the natural frequencies and forms are constructed in an exact formulation by Fourier method. The dependence of the natural frequencies and forms of the lowest vibration modes on the rate of displacement, unknown in the literature, are constructed, and their features are established. A modelling and animation of unusual wave motions of the rod are presented. The main characteristics for the higher vibration modes are constructed.     

The limiting state of a rigidly fixed nonlinearly elastic multilayer rod

The loss of the load-carrying capacity of a nonlinearly elastic multilayer rod is investigated. The rod, whose layers have various thickness and are made of different materials, is rigidly fixed at both its ends. Rigid contact conditions between the layers are assumed. The problem posed is solved by using the variational method of mixed type in combination with the Rayleigh-Ritz method. The initial analysis is reduced to the solution of the Cauchy problem for a nonlinear ordinary differential equation solved for the first derivative. As the initial condition, the maximum initial eccentricity of the rod is assumed. In the case of zero eccentricity, the Shanley critical force for an axially compressed rod is determined. For a three-layer rod whose outer layers have equal thickness and are made of the same material, numerically, for various degrees of nonlinearity, the effect of physicomechanical and geometric parameters on the critical load of buckling instability is determined. It is found that, by matching the heterogeneity of the rod, it is possible to raise its load-carrying capacity. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 3, pp. 347–360, May–June, 2006.     

Sharp estimate of electric field from a conductive rod and application

We are concerned with the quantitative study of the electric field perturbation due to the presence of an inhomogeneous conductive rod embedded in a homogenous conductivity. We sharply quantify the dependence of the perturbed electric field on the geometry of the conductive rod. In particular, we accurately characterize the localization of the gradient field (i.e., the electric current) near the boundary of the rod where the curvature is sufficiently large. We develop layer‐potential techniques in deriving the quantitative estimates and the major difficulty comes from the anisotropic geometry of the rod. The result complements and sharpens several existing studies in the literature. It also generates an interesting application in EIT (electrical impedance tomography) in determining the conductive rod by a single measurement, which is also known as the Calderón's inverse inclusion problem in the literature.     

The complex three-dimensional elastoplastic bending of a rod with a square cross-section

The complex three-dimensional bending of a long rod (with square cross-section) made of an isotropic ideally plastic and ideally cyclical material is investigated. The bending of the rod occurs due to the action of two moments, applied to its ends in such a way that the longitudinal deformation of the middle fibres of two neighbouring sides is described by a dashed line.     

Derivation of a rod theory for multiphase materials

A rigorous derivation is given of a rod theory for a multiphase material, starting from three-dimensional nonlinear elasticity. The stored energy density is supposed to be nonnegative and to vanish exactly on a set consisting of two copies of the group of rotations SO(3). The two potential wells correspond to the two crystalline configurations preferred by the material. We find the optimal scaling of the energy in terms of the diameter of the rod and we identify the limit, as the diameter goes to zero, in the sense of Γ-convergence.     

A variational characterization of a hyperelastic rod with hard self-contact

We consider an elastic rod, modeled as a curve in space with an impenetrable surrounding tube of radius ρ, subject to a general class of boundary conditions. The impossibility of self-intersection is then imposed as a family of scalar constraints on the physical separation of nonlocal pairs of points on the rod. Thus, the usual variational formulation of energy minimization is considered in a context of nonconvex, nonsmooth optimization. We show the existence of minimizers within suitably defined homotopy classes associated with both the centerline and the frame along the rod. The principle results are then concerned with derivation of first-order necessary conditions for optimality and some consequences of these for the contact forces and for regularity.     

The longitudinal deformation of a rod in friction interaction with a compressing body

The deformation of a rod, confined in a fixed external housing, is considered. The friction forces in the contact surface are related to the deformation of the rod by a power relation. A wide range of variation of the friction parameter and the preliminary clearance parameter with which the rod is inserted into the housing is investigated and the characteristic features of the stress and strain distributions are revealed. The dissipation of energy due to friction and the formation of a hysteresis loop in the dependence of the stresses in the loaded end face on its displacement are considered. The problem is solved in a quasistatic formulation. Analytical relations are found for a number of important cases. Other results are obtained by numerical integration of the initial differential problem.     

The oscillations of a rod in an inhomogeneous elastic medium

The local length-dependence of the natural frequencies and forms of plane transverse oscillations of a thin inhomogeneous rod in an elastic medium with a variable stiffness and arbitrary elastic-fastening boundary conditions is investigated. It is established that the presence of an external elastic medium, described by the Winkler model, can lead to an anomalous effect – an increase in the natural frequencies of lower oscillation modes as the length of the rod increases continuously. The extremely fine properties of this change as a function of the length, the mode number and the method of fastening are revealed. The oscillations in the case of standard methods of fastening are investigated separately. Simple examples, which illustrate the anomalous dependence of the natural oscillation frequencies of the rod in an extremely inhomogeneous elastic medium with different boundary conditions are calculated.     

Propagation of waves along a slightly bent piezoelectric rod

The waveguide modes of a curved piezoelectric rod are considered. The propagation of waves along the rod is studied by the ray method. The transport equations that describe the change in the intensity along the rays are solved. Bibliography:10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 191–202. Translated by V. V. Lukyanov     

A model of a reinforced tyre with a rod protector

A tyre design consisting of a steel-cord-reinforced rigid bond with sides connected to the wheel disc and a protector(tread) in contact with the road is examined. The tread is in the form of a set of rods connected by one end to the band, with the other end either free or in contact with the road. The rod end in contact with the road is acted upon by a force applied from the road, represented by a force normal to the road plane and a shear force due to dry friction. If the modulus of the shear force does not exceed the magnitude of the normal force multiplied by the dry friction coefficient, there is no slip at the contact point. In the opposite case, the rod end will be displaced along the road by an amount sufficient to distribute the normal and shear forces. The dynamics of longitudinal and transverse strains of the rods in contact with the road is analysed using the motion separation method in the quasi-static approximation. The behaviour of the tread rods as a function of the vertical displacement of the wheel centre is investigated, the contact area is found and the conditions are determined under which the contact area is divided into parts in which either slip of the rod ends occurs or does not occur, depending on the magnitude of the longitudinal displacement of the wheel centre or its turning relative to the horizontal axis. An analogue of a continuous model of a rod-like tread is considered, and the magnitudes of the forces and moments are found as a function of the wheel disc displacements. The equations of wheel rolling are obtained, and the conditions under which steady motions exist are found.