Aleksandar Obradović

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

Abstract 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.

Appropriate Modeling of Dynamic Behavior of Quayside Container Cranes Boom Under a Moving Trolley

The paper deals with the analysis of moving trolley effects on the dynamic behavior of flexible structure of a mega high-performance quayside container crane (QCC) boom. The boom is modeled as a system with distributed parameters, comprising reduced stiffnesses and lumped masses from other parts of the upper structure. This paper looks both at the “moving force” and “moving mass” trolley modeling approaches to achieve the required performance of the QCC boom structure. Deterministic simulation for both considered approaches gives dynamic structural response of the boom for container transfer from quay-to-ship. The obtained results for “moving force” and “moving mass” models are compared in the scope of real values of parameters and future expectations in design of QCC. The conclusions lead to an appropriate way of model selection that can be used by engineers in practice.

On the brachistochronic motion of a variable-mass mechanical system in general force fields

The problem of the brachistochronic motion of a mechanical system composed of rigid bodies and variable-mass particles is solved. The laws of the time-rate of mass variation of the particles as well as relative velocities of the expelled (or gained) masses are assumed to be known. The system moves in an arbitrary field of known potential and nonpotential forces. Applying Pontryagin’s minimum principle along with singular optimal control theory, a corresponding two-point boundary value problem is obtained. The appropriate numerical procedure based on the shooting method to solve the obtained two-point boundary value problem is presented. The considerations in the paper are illustrated by an example of determining the brachistochronic motion of a system composed of a rigid rod and two variable-mass particles attached to the rod.

On the instability of equilibrium of nonholonomic systems with nonhomogeneous constraints

The first Lyapunov method, extended by V. Kozlov to nonlinear mechanical systems, is applied to the study of the instability of the equilibrium position of a mechanical system moving in the field of potential and dissipative forces. The motion of the system is subject to the action of the ideal linear nonholonomic nonhomogeneous constraints. Five theorems on the instability of the equilibrium position of the above mentioned system are formulated. The theorem formulated in [V.V. Kozlov, On the asymptotic motions of systems with dissipation, J. Appl. Math. Mech. 58 (5) (1994) 787-792], which refers to the instability of the equilibrium position of the holonomic scleronomic mechanical system in the field of potential and dissipative forces, is generalized to the case of nonholonomic systems with linear nonhomogeneous constraints. In other theorems the algebraic criteria of the Kozlov type are transformed into a group of equations required only to have real solutions. The existence of such solutions enables the fulfillment of all conditions related to the initial algebraic criteria. Lastly, a theorem on instability has also been formulated in the case where the matrix of the dissipative function coefficients is singular in the equilibrium position. The results are illustrated by an example.

The brachistochronic motion of a vertical disk rolling on a horizontal plane without slip

This paper deals with the brachistochronic motion of a thin uniform disk rolling on a horizontal plane without slip. The problem is formulated and solved within the frame of the optimal control theory. The brachistochronic motion of the disk is controlled by three torques. The possibility of the realization of the brachistochronic motion found in presence of Coulomb dry friction forces is inspected. Also, the influence of values of the coefficient of dry friction on the structure of the extremal trajectory is analyzed. Two illustrative numerical examples are provided.

Brachistochronic motion of a nonholonomic variable-mass mechanical system in general force fields:

In this paper, the brachistochronic motion of a mechanical system composed of variable-mass particles is analysed. Workless (ideal) holonomic and linear nonholonomic constraints are imposed on the system. It is assumed that the system moves in an arbitrary field of known potential and nonpotential forces with prescribed both laws of the time-rate of mass variation of the particles and relative velocities of the expelled (or gained) masses. The first time-derivatives of quasi-velocities are taken as control variables. Using Pontryagin’s maximum principle and singular optimal control theory, the problem of brachistochronic motion of the nonholonomic variable-mass mechanical system is solved as a two-point boundary value problem. In addition, a discussion about the realization of control forces is given. The results are illustrated via an example.

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.

Determination of the equivalent torsional vibration system

The procedure of replacing the elements of a complex torsional vibration system of ship diesel engine propulsion with simplified equivalent ones with the same dynamic characteristics is shown in this work. The given procedure comprises the methods for the determination of equivalent lengths, stifnesses and moments of inertia based on the equality between kinetic and potential energy of real and equivalent elements of the system. Additionally, the exciting moments which excite forced torsional vibrations of the considered system are analysed.

Biologically inspired optimal control of robotic system:synergy approach

This paper suggests a new optimal control of robotic system based on a biologically inspired control principle-synergy, which allows resolving actuator redundancy. The redundancy control problem has been discussed in the framework of optimal control problem which is solved by Pontryagins maximum principle. Joint actuator synergy approach is suggested which is established by optimization law at coordination level, where is introduced a central control as suggested Bernstein in [1]. In that way, one may obtain a specific constraint(s) on the control variables. Finally, the effectiveness of suggested biologically inspired optimal control is demonstrated with a suitable robot with three degrees of freedom and four control variables as the illustrative example.