Brachistochronic motion of mechanical systems with nonideal constraints and applications to technical objects

Analizirano je brahistohrono kretanje materijalne tačke i sistema krutih tela u prisustvu Kulonove sile trenja primenom varijacionog računa. U slučaju brahistohronog kretanja materijalne tačke razmatran je slučaj kada se materijalna tačka kreće u vertikalnoj ravni u homogenom gravitacionom polju duž veze u obliku hrapave linije sa Kulonovim trenjem, pri čemu je početna brzina tačke različita od nule. Analiza brahistohronog kretanja tačke je sprovedena za slučaj kada se veza tretira kao zadržavajuća i za slučaj kada se veza tretira kao nezadržavajuća. Izvršena je generalizacija rezultata iz literature, koji su tretirali ovu problematiku primenim varijacionog računa u slučaju kada je početna brzina tačke jednaka nuli, uvođenjem pretpostavke o znaku normalne reakcije veze kao dodatnog ograničenja u varijacionoj formulaciji problema.The brachistochrone motion of the particle and the rigid multibody systems in the presence of Coulomb friction was analyzed by the application of variational calculus. In case of the brachistochrone motion of the particle, the case when the particle moves in the vertical plane in the homogeneous gravitational field along the coustraint in the form of a rough curve with Coulomb friction was considered, where the initial velocity of the particle was diff'erent fiom zero. The analysis of the brachistocluone ntotion of the particle was performed for the case when the constraint is treated as bilateral and for the case when the constraint is treated as unilateral. By introducing the assumption regarding the sign of the normal reaction of the constraint as an additional constraint in the variational formula of the problem, the results from the references treating these problems by variational calculus with the assumption that the initial velocity of the parlicle is equal to zero were generahzed. The equations of the brachistochroue were obtained in their parameter form, where the slope angle of the tangent on the brachistochrone curve was taken as the parameter. It was shown that the brachistochrone in the general case is a two-segment curve w'ith the initial line segment representing a free-fall parabola in nonresistant medium. The application of obtained results in the problems of optimization in a plant for transportation of granular nraterial was presented. In this technical object, the problem of minimization of losses of rnechanical energy due to the action of Coulomb friction during transporlation of granular material was analytically solved. In the brachistochrone motion of a constrained system of rigid bodies, the case when a certain number of unilateral constraints treated as real constraints with Coulomb friction are fbLrnd among the constraints imposed on the system. A general approach to solution by the application of the ntethodology used in the problem of brachistochrone motion of the particle was given. Within this chapter of the dissertation, a special type of the mechanical system with two degrees of freedom on which the analogy of solving the brachistocluone problem of this system with the brachistochrone problem of the particle considered in the first two chapters of the dissertation was analyzed. The results were obtained in the form which is suitable for illustration on specific technical objects and further numerical analysis. The application of differential evolution, as the optimization method, in solving systems of nonlinear algebraic equations was mentioned

Contribution to the determination of the global minimum time for the brachistochronic motion of a holonomic mechanical system

The problem of the brachistochronic motion of a holonomic scleronomic mechanical system is analyzed. The system moves in an arbitrary field of known potential forces. The problem is formulated as an optimal control task, where generalized speeds are taken as control variables. The problem considered is reduced to solving the corresponding two-point boundary-value problem (TPBVP). In order to determine the global minimal solution of the TPBVP, an appropriate numerical procedure based on the shooting method is presented. The global minimal solution represents the solution with the minimum time of motion. The procedure is illustrated by an example of determining the brachistochronic motion of a disk that performs plane motion in a vertical plane in a homogeneous field of gravity.This is the peer reviewed version of the article: Radulović, R.; Obradović, A.; Šalinić, S. Contribution to the Determination of the Global Minimum Time for the Brachistochronic Motion of a Holonomic Mechanical System. Meccanica 2017, 52 (4–5), 795–805. [https://doi.org/10.1007/s11012-016-0425-z

Free vibrations of planar serial frame structures in the case of axially functionally graded materials

This paper considers the problem of modal analysis and finding the closed-form solution to free vibrations of planar serial frame structures composed of Euler-Bernoulli beams of variable cross-sectional geometric characteristics in the case of axially functionally graded materials. Each of these beams is performing coupled axial and bending vibrations, where coupling occurs due to the boundary conditions at their joints. The numerical procedure for solving the system of partial differential equations, after the separation of variables, is reduced to solving the two-point boundary value problem of ordinary linear differential equations with nonlinear coefficients and linear boundary conditions. In this case, it is possible to transfer the boundary conditions and reduce the problem to the Cauchy initial value problem. Also, it is possible to analyze the influence of different parameters on the structure dynamic behavior. The method is applicable in the case of different boundary conditions at the right and left ends of such structures, as illustrated by an appropriate numerical example

Free vibrations of planar serial frame structures in the case of axially functionally graded materials

This paper considers the problem of modal analysis and finding the closed-form solution to free vibrations of planar serial frame structures composed of Euler-Bernoulli beams of variable cross-sectional geometric characteristics in the case of axially functionally graded materials. Each of these beams is performing coupled axial and bending vibrations, where coupling occurs due to the boundary conditions at their joints. The numerical procedure for solving the system of partial differential equations, after the separation of variables, is reduced to solving the two-point boundary value problem of ordinary linear differential equations with nonlinear coefficients and linear boundary conditions. In this case, it is possible to transfer the boundary conditions and reduce the problem to the Cauchy initial value problem. Also, it is possible to analyze the influence of different parameters on the structure dynamic behavior. The method is applicable in the case of different boundary conditions at the right and left ends of such structures, as illustrated by an appropriate numerical example

The brachistochronic motion of a heavy ball rolling along an imperfect rough surface

The problem of brachistochronic motion of a heavy uniform ball rolling without slip along the upper outside surface of an imperfect rough stationary sphere, is solved. The control forces are located in the tangential plane, and their total power equals zero. In the first part of the paper the determination of the brachistochronic motion is solved as the problem of optimal control using Pontryagin’s maximum principle. This solution corresponds to the motion of the heavy ball along a perfect rough sphere. The second part provides the case when the constraint between the sphere and the ball is imperfectly rough. Here, the problem of optimal control is formulated in such way that the tangential component of the reaction of constraint is taken for the control, with the restriction resulting from Coulomb’s laws of sliding friction. The problem thus formulated belongs to the theory of singular optimal controls, and the solution that satisfies the Maximum principle consists of a singular part and a nonsingular part

Contribution to the problem of in-plane vibration of circular arches with varying cross-sections

Free in-plane vibration analysis of circular arches with varying cross-sections is studied by means of the symbolicnumeric method of initial parameters. The effects of axial extension, transverse shear deformation and rotatory inertia are considered. For various boundary conditions, natural frequencies of free in-plane vibration of circular arches with varying cross-sections are obtained. By comparing obtained results with previous ones available in the literature the effectiveness of application of the symbolic-numeric method of initial parameters to the problem considered is proven

The brachistochronic motion of a heavy ball rolling along an imperfect rough surface

The problem of brachistochronic motion of a heavy uniform ball rolling without slip along the upper outside surface of an imperfect rough stationary sphere, is solved. The control forces are located in the tangential plane, and their total power equals zero. In the first part of the paper the determination of the brachistochronic motion is solved as the problem of optimal control using Pontryagin’s maximum principle. This solution corresponds to the motion of the heavy ball along a perfect rough sphere. The second part provides the case when the constraint between the sphere and the ball is imperfectly rough. Here, the problem of optimal control is formulated in such way that the tangential component of the reaction of constraint is taken for the control, with the restriction resulting from Coulomb’s laws of sliding friction. The problem thus formulated belongs to the theory of singular optimal controls, and the solution that satisfies the Maximum principle consists of a singular part and a nonsingular part

On the brachistochrone of a variable mass particle in general force fields

The problem of the brachistochronic motion of a variable mass particle is considered. The particle moves through a resistant medium in the field of arbitrary active forces. Beginning from these general assumptions, and applying Pontryagin's minimum principle along with singular optimal control theory, a corresponding two-point boundary value problem is obtained and solved. The solution proposed involves an appropriate numerical procedure based upon the shooting method. In this numerical procedure, the evaluation of ranges for unknown values of costate variables is avoided by the choice of a corresponding Cartesian coordinate of the particle as an independent variable. A numerical example assuming the resistance force proportional to the square of the particle speed is presented. A review of existing results for related problems is provided, and it can be shown that these problems may be regarded as special cases of the brachistochrone problem formulated and solved in this paper under very general assumptions by means of optimal control theory

A new approach for the determination of the global minimum time for the Chaplygin sleigh brachistochrone problem

A new approach for the determination of the global minimum time for the case of the brachistochronic motion of the Chaplygin sleigh is presented. The new approach is based on the use of the shooting method in solving the corresponding two-point boundary-value problem and defining either the crossing points of surfaces or the crossing points space of curves in a three-dimensional space of two costate variables and the time of the brachistochronic motion of the sleigh. A number of examples for multiple extremals of the Chaplygin sleigh brachistochrone problem are provided. In these examples, the global minimum is the solution to which the minimum time of motion corresponds.This is the peer reviewed version of the article: Radulović, R.; Šalinić, S.; Obradović, A.; Rusov, S. A New Approach for the Determination of the Global Minimum Time for the Chaplygin Sleigh Brachistochrone Problem. Mathematics and Mechanics of Solids 2017, 22 (6), 1462–1482. [https://doi.org/10.1177/1081286516637234

Contribution to the problem of in-plane vibration of circular arches with varying cross-sections

Free in-plane vibration analysis of circular arches with varying cross-sections is studied by means of the symbolicnumeric method of initial parameters. The effects of axial extension, transverse shear deformation and rotatory inertia are considered. For various boundary conditions, natural frequencies of free in-plane vibration of circular arches with varying cross-sections are obtained. By comparing obtained results with previous ones available in the literature the effectiveness of application of the symbolic-numeric method of initial parameters to the problem considered is proven