Szebehely's problem for nonholonomic systems with a given first integral

I.IntroductionTheinverseproblemofdynamicsisoneoftheimportantsubjectsinmechanics.In1977,Szebehelysetforthaninverseproblemforthedeterminationofthet'orcefunctiontoamaterialpointintheplanefromparametricfamilyoftrajectories,andobtainedalinearfirstorderpartialdifferentialequationfortheforcefunction.Later,Erdil'l,MellsandPirast=l,MellsandBorgherol'l,BoilsandMertnsl4]extendedSzebehely'sproblemtoboththreeandndimensionalholonomicsystem.Recently,theauthorandProfessorMetFengxiangl'1studiedtheSzebehe…     

About the basic integral variants of holonomic nonconservative dynamical systems

In this paper, we prove that for holonomic nonconservative dynamical system the Poincaré and Poincaré-Cartan integral invariants do not exist. Instead of them, we introduce the integral variants of Poincaré-Cartan's type and of Poincare's type for holonomic nonconservative dynamical systems, and use these variants to solve the problem of nonlinear vibration. We also prove that the integral invariants introduced in references [1] and [2] are merely the basic integral variants given by this paper. Project supported by National Natural Science Foundation of China     

On quadruple integral equations related to a certain crack problem

An exact solution of a four part mixed boundary value problem representing a three colinear crack system connected with specified crack opening displacements between the cracks is obtained. The three cracks thus become one with pressure and/or opening displacement prescribed on the crack face. From considerations of dual symmetry and a formulation based on Papkovich-Neuber harmonic functions, the boundary value problem is reduced to solving a quadruple set of integral equations. An exact solution of these equations is derived using a modified finite Hilbert transform technique. The closed form results for the stress distributions and the crack-tip stress intensity factors are presented. Limiting cases of the solution yield results which agree with well known solutions.     

On the problem of stabilization of motion for a parametric family of nonlinear singularly perturbed systems

For nonlinear uncertain singularly perturbed systems, we construct a control guaranteeing their absolute parametric stability and estimate the set of values of the parameters for which this property of the system is preserved.     

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.     

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.     

On the stability of motion of uncertain systems

The basic theorem of the direct Lyapunov method on the stability of solutions of uncertain systems with respect to a moving invariant set is generalized. A corollary of general assertion is cited in terms of scalar Lyapunov functions. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 2, pp. 105–109, February, 1999.     

On the formulation of the problem of the motion of a body in water

The selection of solutions describing steady irrotational flow of an ideal incompressible fluid over bodies is considered. The selection is based on restrictions that follow from the physical properties of a real fluid and from the presence of a boundary layer on the body. In particular, for any body one can specify a minimal Euler number below which flow without cavitation becomes physically impossible. In the limiting case of an Euler number equal to zero, only the Kirchhoff scheme is physical admissible, and the cavity section tends to a circle. An equation is derived for the limiting shapes of cavities at small cavitation numbers, and a comparison is made with known results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 19–26, September–October, 1980.     

Singular integral equations for a patch repair problem

Within the scope of linear elasticity, an in-plane problem related to the repair of an infinite thin elastic plate with a hole by a patch is considered. The patch and the plate are joined together only along their boundaries. The plate is subjected to stresses applied at infinity. The problem is reduced to a system of four singular integral equations. Existence and uniqueness of the solution of the system is proved. The proposed solution allows one to evaluate the efficiency of a patch repair with little computational effort.     

Lie symmetries and conserved quantities of holonomic variable mass systems

In this paper, the Lie symmetries and the conserved quantities of the holonomic variable mass systems are studied. By using the invariance of the ordinary differential equations under the infinitesimal transformations, the determining equations and the conserved quantities are given. And an example is given to illustrate the application of the result. Foundation item: the National Natural Science Foundation of China (19572038)     

On a path-independent integral approach to fast tearing of elastomers

Summary The feasibility of a path-independent integral approach to fracture mechanics of elastomers and the objectivity of the pertinent crack stability criterion are discussed with reference to tearing with negligible mechanical dissipations.

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Sommario In questo articolo si discutono le possibilità offerte dall'impiego dell'integrale J e del relativo criterio di stabilità della cricca nello studio della frattura veloce di materiali elastomerici.

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List of symbols and abbreviations F Deformation gradient. - Feled Finite Element for Large Elastic Deformations. - T Strain energy release per unit crack growth (termedG in fracture mechanics of metals). - T _l First Piola-Kirchhoff stress tensor. - u Displacement vector. - W Stored energy function per unit of volume in the undeformed configuration. - x Spatial co-ordinates. - X Material co-ordinates.