On the brachistochronic motion of mechanical systems with unilateral constraints

The problem of brachistochronic motion of a two-degree of freedom mechanical system is considered. The ideal bilateral constraints and one unilateral constraint with Coulomb friction are imposed on the system. It has been assumed that the components of the metric tensor are constant. After introducing the transformation of generalized coordinates, whereby the quadratic form of kinetic energy is reduced to a canonical form, the considered problem is geometrically interpreted as the motion of a representative point in a two-dimensional Euclidean space. Thus, the analogy with the brachistochronic motion of a particle along a curve which is treated as a unilateral constraint is achieved. The results are illustrated via an example.     

The brachistochronic motion of a wheeled vehicle

The paper considers the brachistochronic motion of a wheeled vehicle on a horizontal plane surface. The objective is to transfer the vehicle from the specified initial position with given initial kinetic energy to the specified terminal position in minimum time with conserved total mechanical energy of the vehicle. The problem is solved by applying Pontryagin’s maximum principle and singular optimal control theory. The projection of the reaction force of the horizontal plane applied on the front vehicle wheels onto the axis of the front vehicle axle is taken for a control variable. The cases of unbounded and bounded value of this projection are considered. The shooting method is used to solve the two-point boundary value problem arising from Pontryagin’s maximum principle and singular optimal control theory.     

双面理想完整约束系统的机械能守恒律

研究双面理想完整约束系统在约束不是定常且主动力不是有势时的机械能守恒律. 建立系统的能量变化方程,给出存在机械能守恒律的充分必要条件. 分析有机械能守恒律的12种情况. 最后给出说明性算例.     

Differential equations of motion for a variable-mass holonomic system with a uniformly rotating constraint

In this paper, the differential equations of motion will be established for variable-mass holonomic mechanical systems constrained to the uniform rotation, including the equations in Lagrange form. Nielsen form and Appell form. The application of these new equations is illustrated with an example.     

Constructing a Poincaré map for a holonomic mechanical system with two degrees of freedom subject to impacts

An approach to the construction of Poincaré maps for a nonlinear system with impulsive effect is proposed. The approach is based on linear change of variables that brings the Poincaré map into the simplest form __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 115–122, May 2008.     

Stability in critical cases of a holonomic mechanical system with two degrees of freedom subject to impacts

Stability conditions for a holonomic impulsive mechanical system with two degrees of freedom in the critical case of one pair of complex-conjugate multipliers are established. An analog of the first Lyapunov exponent is calculated __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 105–117, June 2008.     

On Szebehely's problem extended to holonomic systems With a given integral of motion

Summary We consider a family of curves in the n-dimensional configuration space S_n of a holonomic system with n degrees of freedom. We obtain first-order partial differential equations for the potential function U of forces under which any trajectory belonging to the given family of curves can be described by the representative point of the system. We write the potential function U supposing that, in addition to the energy integral, a first integral of motion linear in the lagrangian velocities is assigned. Next we obtain the compatibility conditions between the energy constant E, the parameter which appears in the first integral, and the n–1 geometric constants c₁, c₂,, c_n–1 which characterize the family of trajectories. Finally we discuss two simple examples.

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Sommario Assegnata una famiglia di curve nello spazio S_n delle configurazioni di un sistema olonomo conservativo ad n gradi di libertà, si determinano le equazioni differenziali alle quali deve soddisfare il potenziale delle forze applicate affinchè il punto rappresentativo del sistema possa descrivere una qualunque delle traiettorie appartenenti alla data famiglia. Si scrive poil' espressione del potenziale supponendo che sia assegnato, oltre l'integrale dell'energia, un integrale primo del moto, lineare nelle velocità lagrangiane. Infine si ricavano le condizioni di compatibilità fra la costante dell'energia E, il parametro che interviene nell'integrale primo e le n-1 costanti geometriche c₁, c₂, c₂,, c_n–1 che caratterizzano la famiglia di curve. Si conclude con due semplici esempi che illustrano quanto esposto.

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Work performed under the auspices of G.N.F.M.-C.N.R., Italy.     

Well posed formulations of holonomic mechanical system dynamics and design sensitivity analysis

Abstract

Manifold theoretic ordinary differential equations of motion for holonomic mechanical systems that depend on problem data, or design variables, are shown to be well posed; i.e., they have a unique solution that depends continuously on problem data. It is proved that these differential equations are equivalent to the d’Alembert variational formulation and the index 3 Lagrange multiplier formulation of differential-algebraic equations of motion, which are also shown to be well posed. These results provide a foundation for dynamic system design sensitivity analysis, which requires differentiability of solutions of the equations of motion with respect to design variables.     

The equations of motion of a mechanical system in matrix form

This work recommends methods of construction of equations of motion of mechanical systems in matrix form. The use of a matrix form allows one to write an equation of dynamics in compact form, convenient for the investigation of multidimensional mechanical systems with the help of computers. Use is made of different methods of constructing equations of motion, based on the basic laws of dynamics as well as on the principles of D'Alambert-Lagrange, Hamilton-Ostrogradski and Gauss.     

Stabilization of the motion of an unstable linear mechanical system

Kiev. Translated from Prikladnaya Mekhanika, Vol. 26, No. 4, pp. 79–85, April, 1990.     

Brachistochronic motion of nonholonomic nonlinear mechanical system of the Chaplygin type

In this paper, the differential equations of brachistochronic motion of nonholonomic nonlinear system of the Chaplygin type were determined. The additional forces R_α realizing the corresponding motion of the system together with the given forces were also determined. Lastly, the brachistochrone was solved for a concrete model using mathematical numerical methods.     

RELATIVISTIC VARIATION PRINCIPLES AND EQUATION OF MOTION FOR VARIABLE MASS CONTROLLABLE MECHANICAL SYSTEM

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Application of Radau IIA algorithms to flexible multibody system with holonomic constraints

The paper designs new 2- and 3-stage Radau IIA algorithms to integrate the dynamic responses of flexible multibody system with holonomic constraints. The total translation, the incremental rotation and associated velocities are selected as unknowns to avoid the linearization of angular acceleration which makes it possible to parameterize the finite rotation by using the Wiener–Milenkovi? parameters. The new algorithms release the heavy computational burden through the simplified Newton iterations and are stabilized by the preferable h-scaling technique. Contributions of the paper include: (1) For 2-stage algorithm, the resulting block triangular equations are solved efficiently by an inner iteration scheme. (2) For 3-stage algorithm, the full-size linear system is decoupled into a real and a complex subsystems which reduces the size of the system dramatically. (3) A new scheme is designed to predict the truncation error from the associated deferred correction equations without overestimation. Finally, numerical simulations show the 2- and 3-stage Radau IIA algorithms have excellent stability and convergence properties and behavior in a computationally efficient manner.     

ON THE UNIFICATION OF THE HAMILTON PRINCIPLES IN NONHOLONOMIC SYSTEM AND IN HOLONOMIC SYSTEM

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