Vibrations of rod-rigid element systems with a local non-linearity

The paper deals with vibrations of systems consisting of non-coaxial rods connected by rigid bodies and of a local non-linearity. The motion of the rods is described by classical wave equation and the solution of the d’Alembert type is applied in the study. This leads to solving ordinary differential equations with a retarded argument. The local non-linearity is described through irrational functions and in a special case it includes the polynomial of the third degree. Detailed considerations are given for a system consisting of three rods and two rigid bodies. In numerical analysis non-linear effects are discussed. The results concerning harmonic vibrations are presented for the local non-linearities having characteristics of a soft type as well as of a hard type.     

Dynamics Investigation of Three Coupled Rods with a Horizontal Barrier

This report is a part of the larger project of non-linear dynamics investigation of three coupled physical pendulums with damping and with arbitrary situated barriers, and externally driven. The set of differential equations and the set of algebraic inequalities (representing a barrier) governing the motion of three coupled rods are presented in the non-dimensional form. The system of governing equations is integrated between two successive impacts, and the discontinuity points are detected (by halving time step until a required precision is obtained). In each impact time, the state of the system is transformed using the extended restitution coefficient rule. The theory of Aizerman and Gantmakher is used to calculate the fundamental solution matrices in the analyzed system exhibiting discontinuities. The fundamental matrices are used during calculation of Lyapunov exponents, during stability analysis of periodic solutions (Floquet multipliers) and in shooting method applied to detect and trace periodic orbits. Some examples for three coupled identical rods with horizontal barrier are reported.     

Korn Inequalities for a Reinforced Plate

Asymptotically optimal Korn inequalities are derived for a composite material that consists of two families of stiff rods surrounded by a homogeneous soft material. The composite plate is fixed through the protruding stiff rods only. The asymptotic behaviour is shown to be crucially different for families of connected rods and for those where the rods are isolated.     

Magnetic torque attitude control of a satellite using the state-dependent Riccati equation technique

A non-linear attitude control method for a satellite with magnetic torque rods using the state-dependent Riccati equation (SDRE) technique has been developed. The magnetic torque caused by the interaction with the Earth's magnetic field and the magnetic moment of torque rods plays a role of the control torque. The detailed equations of motion for this system are presented using angular velocity and quaternions. The SDRE controller is developed for the non-linear systems which can be formed in pseudo-linear representations using the state-dependent coefficient (SDC) method without linearization procedure. The aim of this control system is to achieve a stable attitude within 5°, and minimize the control effort. The stability regions for the SDRE controlled satellite system are estimated through the investigation of the stability conditions developed for pseudo-linear systems and the application of Lyapunov's theorem. For comparisons, the Linear Quadratic Regulator (LQR) method using the solution of the algebraic Riccati equation (ARE) is also applied to this non-linear system. The performance of the SDRE non-linear control system demonstrates more robustness and stability than the LQR control system when subjected to a wide range of initial conditions.     

Calculation of the critical velocity of a stepwise rod system under a longitudinal impact

A longitudinal elastic central impact of a system of a constant-section and stepwise rods on a rigid target is modeled by an exact analytical solution of the differential wave equation by the D’Alembert method with imposing appropriate initial and boundary conditions. Based on the Euler formula, an analytical expression is derived for calculating the critical velocity at which buckling of the rod system occurs.     

Efficiency of a corrugated shell as a radial spring

This paper deals with the problem of radial support of prismatical rods. The radial expansion of the rods and the limits of radial pressure acting on them are prescribed, so that an upper limit of the spring rate is imposed. The weight of the radial support must be kept as low as possible; hence, its efficiency of material utilization should be high. Specifically, the efficiency of a corrugated cylindrical shell employed as a radial-spring system is investigated. As a preliminary convenience, general formulas are set up to describe the efficiency of material utilization in springs. Then, the elastic properties and the stresses of corrugated shells with various shapes of convolutes are established. Finally, from the formulas for the spring efficiency, the elastic properties, and the stress of the corrugated shells, their efficiency of material utilization as springs is established.     

Waves, normal modes and frequencies in periodic and near-periodic rods. Part II

A systematic approach to the study of normal modes and frequencies of disordered periodic rods is presented within a new transfer matrix framework proposed earlier by the authors. The normal frequency structure and mode localization of multiply-disorder periodic rods are investigated. The Monte Carlo and the perturbation method are applied to study the effects of material parameter uncertainties on normal modes and frequencies of randomly-disordered periodic rods. Some intricate aspects are investigated statistically, and it is shown that for this strongly-coupled elastic system, multiple and/or random disorders lead to more localized modes in or near stop-bands in a more complex way. In addition, high frequency wave localization is a typical feature of such a strongly-coupled but randomly-disordered periodic rod system.     

Input shaping control of a nuclear power plant’s fuel transport system

In this paper, the residual vibration control problem of a nuclear power plant’s fuel-transport system is discussed. The purpose of the system is to transport fuel rods to the target position within the minimum time. But according to observations, the rods oscillate at the end of the maneuver, causing an undesirable delay in the operation and affecting the system’s performance in terms both of productivity and of safety. In the present study, a mathematical model of the system was developed to simulate the under-water sway response of the rod while keeping in view the effects of the hydrodynamic forces imposed by the surrounding water. Experiments were performed to validate the model’s correctness. Further, simulation results were used to design the input shaping control that generates shaped velocity commands for transport of the fuel rods to the target position with the minimum residual vibration. It was observed that due to the under-water maneuvering, the fuel-handling system behaves as a highly damped process and that the generated shaped velocity commands fail to effect the desired suppression of the residual vibration. Therefore, keeping in view the highly damped nature of the system, a modified shaped command was generated that transported the fuel rods to the target position with the minimum residual vibration.     

Thermomechanics of a heterogeneous fluctuating chain

In this paper we present a theory to efficiently calculate the thermo-mechanical properties of fluctuating heterogeneous rods and chains. The central problem is to evaluate the partition function and free energy of a general heterogeneous chain under the assumption that its energy can be expressed as a quadratic function in the kinematic variables that characterize the configurations of the chain. We analyze the effects of various types of boundary conditions on the fluctuations of the rods and chains and show that our results are in agreement with recent work on homogeneous rods. The results for the heterogeneous chains are verified through Monte Carlo simulations. Finally, we consider a special heterogeneous chain with only two bending moduli and use it as a model to interpret experiments on partially unfolded protein oligomers.     

On stability of relative equilibria of a double pendulum with vibrating suspension point

We consider the motions of a double pendulum consisting of two hinged identical rods. The pendulum suspension point is assumed to perform harmonic vibrations of arbitrary frequency and arbitrary amplitude in the vertical direction. We carry out a complete nonlinear analysis of the stability of the four pendulum relative equilibria on the vertical. The problem on the stability of the relative equilibria of the mathematical pendulum in the case where the suspension point performs vertical harmonic vibrations of arbitrary frequency and arbitrary amplitude was considered in a linear setting [1–3] and a nonlinear setting [4, 5]. In the case of small-amplitude rapid vertical vibrations of the suspension point, linear and (mathematically not fully rigorous) nonlinear stability analysis of the relative equilibria was carried out for an ordinary pendulum [6–9] and a double pendulum [10, 11]. In [12], for the same case of rapid vibrations, stability conditions in the linear approximation were obtained for the four relative equilibria of a system consisting of two physical pendulums. In the special case of a system consisting of two identical rods, the problem was solved in the nonlinear setting.     

Large elastic deformations in thin cantilever rods due to concentrated loadings

In this paper the problem of large elastic deformations in cantilever thin rods subjected to concentrated loads is considered. Taking into account the incompressibility assumption of the center line and the equations relating the internal moments with the curvatures and torsion of the rod before and after the deformation, the non-linear equilibrium system, composed of six coupled differential equations of first order, is transformed to a new system of higher order. The cases of geometries of initially curved rods and their cross-sections were investigated, for which the higher order system of equations may be decoupled and solved in a closed form. Several applications of thin curved cantilever rods were made and the potentialities of the method were shown with these examples.     

The thermoelastic Aldo contact model with frictional heating

In the study of the essential features of thermoelastic contact, Comninou and Dundurs (J. Therm. Stresses 3 (1980) 427) devised a simplified model, the so-called “Aldo model”, where the full 3D body is replaced by a large number of thin rods normal to the interface and insulated between each other, and the system was further reduced to 2 rods by Barber's Conjecture (ASME J. Appl. Mech. 48 (1981) 555). They studied in particular the case of heat flux at the interface driven by temperature differences of the bodies, and opposed by a contact resistance, finding possible multiple and history dependent solutions, depending on the imposed temperature differences.The Aldo model is here extended to include the presence of frictional heating. It is found that the number of solutions of the problem is still always odd, and Barber's graphical construction and the stability analysis of the previous case with no frictional heating can be extended. For any given imposed temperature difference, a critical speed is found for which the uniform pressure solution becomes non-unique and/or unstable. For one direction of the temperature difference, the uniform pressure solution is non-unique before it becomes unstable. When multiple solutions occur, outermost solutions (those involving only one rod in contact) are always stable.A full numerical analysis has been performed to explore the transient behaviour of the system, in the case of two rods of different size. In the general case of N rods, Barber's conjecture is shown to hold since there can only be two stable states for all the rods, and the reduction to two rods is always possible, a posteriori.     

Modeling MEMS resonators with rod-like components accounting for anisotropy, temperature, and strain dependencies

Numerous systems can be described using masses and rods in transverse vibration. Motivated by the desire to model the effects of axial strain and temperature on systems of this type, we develop a procedure to determine the frequencies of transverse vibration of devices composed of rods and rigid masses as functions of these effects. Our models allow for the rods to be composed of anisotropic materials with material symmetry contained in the cubic system. The goal of the present paper is to demonstrate the modeling procedure using the example of a double-ended tuning fork (DETF). Following this, the results of a slightly more complicated model are compared with the experimental results found for a prototype MEMS DETF sensor composed of polycrystalline silicon.     

Mild hydrothermal preparation of a layered metal hydroxide salt with microtube/rod morphology

Microtubes/rods of the layered metal hydroxide salt compound Cd2(OH)3(DS).nH2O, where DS stands for dodecyl sulfate sandwiched between two adjacent inorganic sheets, have been synthesized for the first time through a mild hydrothermal reaction route. The microtubes/rods have a diameter of about 1 μm and a length ranging from several microns to 20 μm. The growth process of microtubes/rods under the experimental conditions employed follows a dissolution-recrystallization route.     

Approximate calculation of the added mass in flow around a multirow array of rods

A method of calculating effective masses of multirow clusters of elastic-cylindrical rods is presented. The flow of liquid caused by lateral vibrations of the rods is described approximately using a model of cells.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 50–54, September–October, 1970.     

Vibration of rods with a concentrated mass in the presence of non-conservative forces

Summary Separable and non-selfadjoint boundary-value problems representing the vibration of linearly elastic unidimensional systems are considered. The elastic system is modelled as a continuous distributed-parameter system where singularities in the mass distribution function can be neatly taken into account. Specifically, extending Green's function approach, the free vibration, stability and forced vibration of fixed-free rods with a tip mass and under the action of uniformly distributed non-conservative loads have been investigated analytically.Support of this research through the NSERC Grant No. OGPIN013 is greatly appreciated     

Reaction Forces under Static Loading of a Wheel Pair with Camber

The static loading of a wheel pair with a deformable periphery is considered when the wheels are mounted on a common axle with a non-zero camber angle. A tire tread (protector) is modeled by a set of elastic rods interacting with the motion plane according to the dry friction law. The wheel camber effect on the slip in the contact zone and on the magnitude of reaction forces from the road is studied. An analog of the continuous model for a rod tread is discussed. The normal and tangential reaction forces are found depending on the vertical displacement of the center of the mechanical system under discussion.     

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.     

Homogenization of the signorini boundary-value problem in a thick plane junction

We consider a mixed boundary-value problem for the Poisson equation in a plane thick junction Ω_ε that is the union of a domain Ω₀ and a large number of ε-periodically located thin rods. The nonuniform Signorini conditions are given on the vertical sides of the thin rods. The asymptotic analysis of this problem is made as ε → 0, i.e., in the case where the number of thin rods infinitely increases and their thickness tends to zero. With the help of the integral identity method, we prove the convergence theorem and show that the nonuniform Signorini conditions are transformed (as ε → 0) into the limiting variational inequalities in the domain that is filled up with thin rods when passing to the limit. The existence and uniqueness of a solution to this nonstandard limit problem are established. The convergence of the energy integrals is proved as well. Published in Neliniini Kolyvannya, Vol. 12, No. 1, pp. 44–58, January–March, 2009.     

The Slip Velocity at the Edge of a Porous Medium: Effects of Interior Resistance and Interface Curvature

Singularity methods are used to analyze creeping planar flow in the annulus between concentric cylinders, when a portion of the annulus is filled with an array of regularly spaced rods adjacent to the inner cylinder. The rods are evenly spaced on concentric circles, and the circles are spaced such that the array resembles a square lattice bent into a circle. The rods and inner cylinder are stationary, and steady rotation of the outer cylinder generates the flow. The quantity of interest is the slip velocity, the mean velocity at the interface between the array and the unfilled portion of the annulus. The primary part of the study concerns the influence of the interior rods on the interfacial velocity, and to this end the velocity is found as successive circles of rods are removed, starting with the circle closest to the inner cylinder. The calculations are carried out for solid volume fractions from 0.0001 to 0.1, and these show that the slip velocity is virtually unchanged as the interior circles of rods are removed, until only one circle remains and then the velocity is of order 10% larger than that for the full array. Hence the velocity at the edge of a sparse porous medium depends minimally on the hydrodynamic resistance of the obstacles in the interior. In the secondary part of the study, it is found that curvature of the interface does not influence the velocity there.