Rods Of Variable Cross-section Research Articles

Stability of a compressed-tensioned rod of variable cross-section with combined loading

Objective. The purpose of the study is to determine the stability of a straight rod of variable cross-section under combined axial loading. Method. The longitudinal bending of the rod is described by the classical theory using Bernoulli’s hypothesis, and the critical forces are determined from the Euler problem with appropriate assumptions. Result. An algorithm for a numerical method for solving the problem of determining the eigenvalues of the differential equation for longitudinal bending of a rod is proposed. External loads are considered “dead”. The functions of changing the variable cross-sectional area, variable stiffness and distributed load are considered given. The curved axis of the rod after bifurcation is described using a linear ordinary differential equation. Conclusion. The implementation of the numerical method was carried out by the finite difference method using numerical methods and modern computer software.

Method of calculation for dynamic strength of complex configuration frame structures of locomotives for transport engineering

The article presents a numerical and analytical method for studying the bending-longitudinal vibrations of the load-bearing frame of the locomotive frame of a complex configuration in the form of an elastic rod of variable cross-section with variable mass, bending and longitudinal stiffness, based on methods: piecemeal linear approximation, iterations and the Boundary Element Technology. Numerical studies are performed in C # and in MATHCAD 15.

Developing a method of accounting for the existence of local surface heat exchange in rods of variable cross-section

The load-bearing elements of a number of strategic equipment are of limited length and variable cross-section. Most of them are exposed to certain types of heat sources. In order to ensure the reliable operation of this equipment, it is necessary to know the temperature field along the length of the variable cross-section rod. In this paper, a computational algorithm and a method for determining the temperature field along the length of a rod with a limited length and variable cross-section are proposed. They are based on the fundamental laws of conservation of energy. The nonlinearity of the process is due to nonlinear dependencies of the areas of the variable cross-section on the coordinate. The radius of the cross-section of the rod decreases linearly along the entire length, starting from the left end. The side surface of the first and third discrete elements of the rod is heat-insulated. Convective heat exchange with the environment takes place on the side surface of the second discrete element of the rod. The cross-sectional area of the left end of the rod is under the heat flow with a constant intensity, and a heat flow with different intensities is supplied to the right end, wherein the heat transfer coefficient is considered constant. For this task, you must first determine the law of temperature distribution along the length of the rod. In addition, if one end is rigidly fixed and the other end is free, the elongation must be calculated depending on the available heat sources, the physical and geometric characteristics of the rod, taking into account the presence of insulation. In case of pinching of both ends of the investigated rod, the value of axial compressive force is calculated taking into account the addition of real factors. The distribution of all components of the strain, stress, and displacement field is also defined

Уточнение результатов численных расчетов частот поперечных колебаний стержня переменного сечения с упругим закреплением экстраполяционным методом

Thearticle describes a numerical experiment with the results of calculating the natural frequencies of transverse vibrations of a rod of variable cross-section with elastic fixation at one of the ends. The possibility of refining by extrapolation the calculated values of natural frequencies obtained by the method of algebraic polynomials of the fifth degree is analyzed. It is shown that ex-trapolation made it possible to increase the accuracy of calculations by the numerical method by 2-3 orders of magnitude

Nonlinear stochastic vibration of a variable cross-section rod with a fractional derivative element

The article deals with the problem of computing efficiently the nonlinear response of a rod involving a fractional constitutive model, and exposed to random excitation. The constitutive model is a three-parameter model comprising an instant elasticity modulus, a prolonged elasticity modulus, and a relaxation parameter. The nonlinear term is a linear-plus-cubic force of the Winkler kind. The resulting nonlinear fractional partial differential equation governing the rod displacement has no known exact solution. Thus, the article proposes an approximate analytical solution by relying on the statistical linearization technique. Further, it develops a Boundary Element Method (BEM)-based approach to estimate numerically the rod response statistics. The statistical linearization solution is obtained by representing the rod displacement as the superposition of linear modes of vibration having time-dependent coefficients. In this context, it is shown that the equation governing the time variation of the mode coefficients is a nonlinear fractional ordinary differential equation, whose solution is computed by a surrogate linear system identified by minimizing the response error between the linear system and the nonlinear one in a mean square sense. Relevant Monte Carlo studies pertaining to rods with fixed-fixed, and fixed-free ends show that the proposed analytical solution is in a good agreement with data obtained by the numerical (BEM) approach.

On the tension of a cylindrical rod of variable cross-section

The paper considers the solution of linearized equations of the theory of small elastic-plastic deformations in the case of an axisymmetric problem. It is assumed that in the initial state there is a simple stretching. In the first approximation, the relations for the components of displacements, deformations and stresses are obtained. Solutions are expressed in terms of zero-and first-order Bessel functions. The paper deals with the stretching of an infinitely long cylindrical rod of variable cross-section.

The research of discrete and continuous models of impact devices by numerical methods

The article gives the research results of discrete and continuous models of an impact device by numerical methods. The discrete model makes it possible to evaluate the process of impulsive force of the striker with the tool taking into account the aftereffect of the resistance of the working medium. For the rod model, the initial boundary conditions are formulated, which take into account the impulsive force with the determination of the impact force by the Hertz method. The characteristics of nonlinear rigid and dissipative constraints depend on the difference in the displacements of the contact rod ends. The interaction of the tool with the working medium, in particular with the rock, is modeled by similar constraints. These processes are described by initial-boundary value problems with wave equations, which are different in initial and boundary conditions. Mixed difference schemes are used to solve problems. For the rods of variable cross-section, the voltage distribution along their length is determined. The results obtained are compared with the solutions by the Fourier method of model problems.

The inverse problem of the oscillation of a rod with a variable cross section

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in percussion mechanisms. The strength of such parts directly depends on the natural frequencies of vibrations. Oscillations and vibration are also used in diagnostics, for example in acoustic diagnostics of defects. The paper presents a method that allows numerically finding a variable cross section of an elastic rod from the natural frequencies of longitudinal vibrations. It is assumed that the cross-sectional area varies along the axis and is described by an exponential function of a polynomial of degree n. The boundary condition on the left end is hard, on the right-elastic. It is shown that to determine n unknown coefficients of the cross section function, n natural frequencies are required.

Algorithm of applying the energy conservation law to study the thermal stress-strain state of a rod of variable cross section with the simultaneous presence of local temperatures, heat exchanges and thermal insulations

The bearing elements of power plants, internal combustion engines of jet engines, nuclear thermal power plants, gas-generating power plants, processing lines, and spacecraft antennas are rods of variable cross-section. They work in a difficult temperature environment. In this paper, we propose an algorithm for applying the fundamental law of energy conservation for a deep study of the resulting thermal-stress-strain state of a rod of limited length and variable cross-section in the simultaneous presence of local temperatures, heat exchange and thermal insulation. At the same time, the obtained solutions in the form of determining the temperature field, displacement, all the components of deformations and stresses, as well as the calculated values of the rod elongation and the resulting axial force are highly accurate at the level of satisfying the energy conservation law.

The Dynamics of a Rotor System of a 1100 MW Turbine Unit

The calculations data of the critical rotational speeds of the shaft line on yielding supports with the support yield in the vertical and horizontal directions of the modernized turbine unit K-1000-60/1500-2 (after modernization, the turbine unit type is designated as K-1100-60/1500-2M) have been presented. 1000 MW turbine units comprising a high-pressure cylinder and three low-pressure cylinders have been in operation at nuclear power plants since 1984. The design models of shafting are based on technical and improved theories of variable cross-section rods on elastic-damper supports, taking into account the coupling of vibrations in two planes, inertia of rotation, gyroscopic moments and aerodynamic forces on the disks. Additionally, the stress condition of the most dangerous sections of welded rotors has been investigated in case of a sudden short circuit in the generator. The conditions of the shafting for the cases of two- and three-phase short circuits were considered. The calculations of the critical rotational speeds of the shaft line and of the strength of the rotors in case of a short circuit were performed using software packages developed in the IPMash of the NAS of Ukraine, Joint-Stock Corporation “Turboatom”. The calculation data show that no dangerous resonance vibrations of the shaft line of the upgraded К-1100-60/1500-2М turbine unit are detected and the strength of the shaft line in case of a sudden short circuit in the generator is in conformity with the requirements of regulatory and technical documents.

Mathematical modeling of unsteady elastic stress waves in a console with a base (half-plane) under fundamental seismic action

The aim of the work is to consider the problems of numerical modeling of seismic safety of the console with the base in the form of an elastic half-plane under unsteady wave influences. Stress waves of different nature, propagating in the deformed body interact with each other. After three or four times the passage and reflection of stress waves in the body, the process of propagation of disturbances becomes steady, the body is in oscillatory motion. The problem of modeling problems of the transition period is an actual fundamental and applied scientific problem. Methods. The finite element method in displacements is used to solve the two-dimensional plane dynamic problem of elasticity theory with initial and boundary conditions. On the basis of the finite element method in displacements, an algorithm and a set of programs for solving linear plane two-dimensional problems have been developed, which allow solving problems with non-stationary wave effects on complex systems. The algorithmic language “Fortran-90” was used in the development of the complex of programs. The study area is divided by spatial variables into finite elements of the first order. According to the time variable, the study area is also divided into finite elements of the first order. Results. The problem of the influence of a plane longitudinal elastic wave in the form of a Heaviside function on a console with a base (the ratio of width to height is one to ten) is considered. The initial conditions are taken as zero. The system of equations from 16 016 084 unknowns is solved. Contour stresses and stress tensor components are obtained in characteristic areas of the problem. On the basis of the conducted researches it is possible to draw the following conclusions: the console (the ratio of width to height one to ten) is modeled with the elastic basis in the form of an elastic half-plane; the elastic contour stresses on the faces of the console are almost a mirror image of one another, that is, antisymmetric; the console under seismic action works as a rod of variable cross-section, that is, if there are tensile stresses on one face, then compressive stresses on the other; on the contours of the console under seismic action, bending waves mainly prevail.

Influence of force load on the stability of ultrasonic longitudinal–torsional composite drilling system

This study aimed to investigate the problems of system detuning, amplitude attenuation, and even vibration stop after the load was applied to the acoustic system during ultrasonic-assisted vibration processing. In the theoretical part, the overall theoretical model of the ultrasonic longitudinal–torsional composite drilling system (ULTCDS), including the longitudinal–torsional composite horn, transducer, and load was established based on the four-terminal network, combined with the longitudinal and torsional vibration equations of the variable cross-section rod. Also, the influence of the load on the input impedance and output amplitude of the acoustic system was preliminarily analyzed. In the simulation part, the static and dynamic drilling force was simulated by the pre-stressed modal and transient dynamic analyses, and the influence of the force load on the resonant frequency and output amplitude of the system was further explored. In the experiment part, the static force loading test and the ultrasonic drilling test of titanium alloy were designed. The micro-morphology of drilling holes under different processing parameters was observed, and the influence of force load on the system characteristics and stability was explored. The results showed that when the system was subjected to a force load greater than 400 N, the system current and frequency were greatly offset, the electromechanical conversion efficiency reduced, and the amplitude was attenuated to below the amplitude at empty load. In the ultrasonic-assisted drilling of titanium alloy processing, the acoustic system undergoes an obvious detuning phenomenon and the drilling quality is attenuated when the current increased by more than 60% and the frequency changed by more than 350 Hz, affecting the stability of the system and the ultrasonic-assisted processing effect.

Transverse Oscillations of a Cantilever Rod of Rectangular Cross Section and Variable Thickness

A numerical-analytical algorithm for calculating (a type of accelerated convergence method) of eigenvalues and modes of transverse vibrations of a rod of variable cross-section is proposed. The algorithm is implemented as a software package in the Maple symbolic computations language. The use of the package was demonstrated by the example of solving problems of determining the eigen forms and natural frequencies of rods with different dependences of the cross section on the longitudinal distance.

FREE LONGITUDINAL VIBRATIONS OF A VERTICAL ROD WITH DISCRETE MASSES WITH DAMPING FORCES

Objectives. The longitudinal oscillations of a vertical rod of a continually discrete system with kinematic seismic disturbances in the form of a stationary random process are considered.Method. A method for determining the variance of the output process of displacements, using the representation of the input random process as a sum of harmonic deterministic perturbations, is proposed and implemented.Result. The dependence function of the dispersion of displacements on the longitudinal coordinate is determined. Longitudinal vibrations of vertical rods near the epicenter of earthquakes are dangerous for their strength and stability. The methods of finite differences and coordinate descent allow you to create universal algorithms and computer programs that easily solve complex spectral problems.Conclusion. To date, research on random vibrations of buildings and structures, as well as regulatory documents, has been devoted to horizontal seismic effects and transverse bending vibrations caused by them. Examples indicate the need to expand the scope of research with the inclusion of other types of vibrations: combinations of longitudinal with transverse, angular, torsional, parametric, etc. This design can be easily adapted to vibrations of rods of variable cross section, to vibrations of continually discrete rods.

DESIGN CALCULATION TECHNIQUE FOR THICK-WALLED COMPOSITE CONSTRUCTIONS OPERATING UNDER HIGH-SPEED LOADING

The scientific paper presents a technique for calculating a thick-walled axially symmetrical construction consisting of a steel rod (reinforcing component) and an external thick-walled shell made of carbon fiber. The construction is loaded with linear load distributed along its length, associated with the operating external pressure, and inertial forces. The technique is based on a one-dimensional model of a composite rod of variable cross-section, roughly taking into account transverse deformations, which is necessary when analyzing a thickwalled construction operating under pressure. In the proposed approach, the geometry of the product is divided into sections and approximated by fragments in the form of truncated cones and, in the particular case, in the form of cylinders. As a result of calculations, the distribution of normal tensile/compression stresses in the shell and in the reinforcing rod, as well as tangential stresses at the boundary of their contact, are determined. As a result of test calculations, a fairly good consistency of the considered one-dimensional model with refined finiteelement modeling is shown. It has been established that the concentration of shear stresses is realised due to the presence of substantially inhomogeneous deformations in the areas of sharp changes in geometry, a reduction in the radius of the shell (the appearance of edge effects), and in the area of a sharp increase in the radius of the bevel of the shell where external pressure is in place.

An accurate and efficient series solution for the longitudinal vibration of elastically restrained rods with arbitrarily variable cross sections

Longitudinal vibration of non-uniform rod has been of great significance in various engineering occasions. The existing works are usually limited to the certain area variation and/or classical boundary condition. Motivated by this limitation, an efficient accurate solution is developed for the longitudinal vibration of a general variable cross-section rod with arbitrary boundary condition. Displacement function is invariantly expressed as the summation of standard Fourier series and supplementary polynomials, with an aim to make the calculation of all derivatives more straightforwardly in the whole solving region [0, L]. Energy principle is employed for system dynamics formulation, with the elastic boundaries considered as potential energy stored in the restraining spring. Arbitrary cross-section area variation is uniformly expanded into Fourier series. Numerical examples are presented for the natural frequency and mode shapes of non-uniform rod of free and clamped boundary conditions and compared with literature data. Results show good agreement with the previous analytical solutions. Influence of cross section area variation on vibration characteristics of non-uniform rods is then studied and discussed. One of the most advantages of the proposed model is that there is no need to reformulate the problem or rewrite the codes when the cross-section area distribution and/or boundary conditions change arbitrarily.

Longitudinal oscillation of a rod with a variable cross section

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in percussion mechanisms. The strength of such parts directly depends on the natural frequencies of longitudinal vibrations. The paper presents a method that allows numerically finding the natural frequencies of longitudinal vibrations of an elastic rod with a variable cross section. This method is based on representing the cross-sectional area as an exponential function of a polynomial of degree n. Based on this idea, it was possible to formulate the Sturm-Liouville problem with boundary conditions of the third kind. The linearly independent functions of the general solution have the form of a power series in the variables x and λ, as a result of which the order of the characteristic equation depends on the choice of the number of terms in the series. The presented approach differs from the works of other authors both in the formulation and in the solution method. In the work, a rod with a rigidly fixed left end is considered, fixing on the right end can be either free, or elastic or rigid. The first three natural frequencies for various cross-sectional profiles are given. From the analysis of the numerical results it follows that in a rigidly fixed rod with thinning in the middle part, the first natural frequency is noticeably higher than that of a conical rod. It is shown that with an increase in the rigidity of fixation at the right end, the natural frequencies increase for all cross section profiles. The results of the study can be used to solve inverse problems of restoring the cross-sectional profile from a finite set of natural frequencies.

Natural Transverse Oscillations of a Rotating Rod of Variable Cross Section

The problem of natural oscillations of a rotating rod of variable cross section is considered. It is believed that the rod is rigidly attached at one end to a rotor with a constant speed orthogonal to the axis of rotation, the other end is assumed to be free. Flexural movements occur in the plane of rotation or perpendicular to it. To determine the natural oscillations, the Euler rod oscillation model is used, taking into account, in addition to the tensile force, the bond reaction force caused by the movements of the neutral axis. Using an original numerical-analytical method, the lower frequencies were calculated for the power and exponential laws of the cross section change.

A computational algorithm and the method of determining the temperature field along the length of the rod of variable cross section

Abstract Bearing elements of a number of strategic equipment are limited length rods with a variable cross-section. Most of them are exposed to certain types of heat sources. To ensure reliable operation of these equipment it is necessary to know the temperature field along the length of the rod with a variable cross section. This paper proposes a computational algorithm and method to determine the temperature field along the length of the rod with limited length and variable cross-section. They are based on fundamental laws of energy conservation. Also obtained is an approximate analytical solution of the problem.

A Practical Estimation of Frequency Response Functions for System Decoupling Indirectly Using a Variable Cross Section Rod

It is of high importance to be able to decouple a system to obtain the dynamic characteristics of its substructures; however, the necessary frequency response functions (FRFs) of the coupling interface are usually challenging to measure due to the limited accessible space and complex geometries. In this paper, a measurement technique in the decoupling process of a coupled system is proposed in order to obtain the FRFs at coupling interface. Specifically, a variable cross section rod is adopted to transmit the dynamic behavior of coupling interface. The proposed technique has three advantages: (a) the thick end with large cross section can provide enough area for applying excitation force like using impact hammer and/or setting up sensors; (b) the slender end with small cross section can break through the spatial limitation more easily; and (c) the convenience that no additional experimental setup is required but just using an available variable cross section rod. Vibrational equation of the variable cross section probe method is derived and then combined with the existing decoupling theories. Finally, the proposed probe method and the new decoupling theory combining probe theory are validated through numerical simulations (FEM) and laboratory experiments, respectively. The results show its great practicability in decoupling process especially in low frequency range.