Forces associated with non-linear non-holonomic constraint equations

A concise method has been formulated for identifying a set of forces needed to constrain the behavior of a mechanical system, modeled as a set of particles and rigid bodies, when it is subject to motion constraints described by non-holonomic equations that are inherently non-linear in velocity. An expression in vector form is obtained for each force; a direction is determined, together with the point of application. This result is a consequence of expressing constraint equations in terms of dot products of vectors rather than in the usual way, which is entirely in terms of scalars and matrices. The constraint forces in vector form are used together with two new analytical approaches for deriving equations governing motion of a system subject to such constraints. If constraint forces are of interest they can be brought into evidence in explicit dynamical equations by employing the well-known non-holonomic partial velocities associated with Kane's method; if they are not of interest, equations can be formed instead with the aid of vectors introduced here as non-holonomic partial accelerations. When the analyst requires only the latter, smaller set of equations, they can be formed directly; it is not necessary to expend the labor first to form the former, larger set and subsequently perform matrix multiplications.     

Non-holonomic mechanics: A geometrical treatment of general coupled rolling motion

In the presented paper, a problem of non-holonomic constrained mechanical systems is treated. New methods in non-holonomic mechanics are applied to a problem of a general coupled rolling motion. Two goals are stressed.The first of them lies in the solution of an originally formulated problem of rolling motion of two rigid cylindrical bodies in the homogeneous gravitational field leading typically to non-linear equations of motion. A solid cylinder can roll inside a ring under the static frictional force assuring rolling without slipping, the ring rolls again without slipping along a generally shaped terrain formed by hills and valleys. “Surprising behaviour” of the mechanical system which permits interesting applications is studied and discussed.The second purpose of the paper is to show that the geometrical theory of non-holonomic constrained systems on fibered manifolds proposed and developed in the last decade by Krupková and others is an effective tool for solving non-holonomic mechanical problems. A comparison of this method to alternative methods is given and the benefits of coordinate-free formulation are mentioned.In this paper, the geometrical theory is applied to the abovementioned mechanical problem. Both types of equations of motion resulting from the theory—deformed equations with the so-called Chetaev-type constraint forces containing Lagrange multipliers, and reduced equations free from multipliers—are found and discussed. Numerical solutions for two particular cases of the motion of the cylindrical system along a cylindrical surface are presented.     

A field method for integrating the equations of motion of nonholonomic controllable systems

This paper presents a field method for integrating the equations of motion of nonholonomic controllable systems. An example is given to illustrate the application of the method.     

多狗追击问题的力学解法

对于多狗依次追击问题,运用相对运动的思想,可方便地求出追击的时间、路程及相对运动方程.     

Analysis of a weakly non-linear autonomous oscillator by means of the field method

In this paper the field method is extended to the study of oscillatory systems with two degrees of freedom and weak quadratic non-linearity. The basic field method concept is combined with the technique of multiple time scales and the solution for both non-resonant case and the case out of the first resonance are found. The qualitative analysis of behavior in the resonant area is done by determining the values of the “adelphic” integral.     

THE METHOD OF DERIVATION OF MAC－MILLAN＇S EQUATION FOR THE NON－LINEAR NON－HOLONOMIC SYSTEM

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A field method for solving the equations of motion of nonholonomic systems

In this paper, the field method^[1] for solving the equations of motion of holonomic nonconservative systems is extended to nonholonomic systems with constant mass and with variable mass. Two examples are given to illustrate its application. The project supported by the National Natural Science Foundation of China.     

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

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On some important problems in analytical dynamics of non-holonomic systems

By using deductive method,Chetaev condition is derived in this paper.We pointout that the processes of variation and differentiation are not permutable innon-holonomic dynamics is a misunderstanding.The paper gives two classical relationsof non-holonomic systems and points out integral variational principles can be appliedin non-holonomic systems.     

Bifurcation and buckling analysis of a unilaterally confined self-rotating cantilever beam

A nonlinear dynamic model of a simple non-holonomic system comprising a self-rotating cantilever beam subjected to a unilateral locked or unlocked constraint is established by employing the general Hamilton's Variational Principle. The critical values, at which the trivial equilibrium loses its stability or the unilateral constraint is activated or a saddle-node bifurcation occurs, and the equilibria are investigated by approximately analytical and numerical methods. The results indicate that both the buckled equilibria and the bifurcation mode of the beam are different depending on whether the distance of the clearance of unilateral constraint equals zero or not and whether the unilateral constraint is locked or not. The unidirectional snap-through phenomenon (i.e. catastrophe phenomenon) is destined to occur in the system no matter whether the constraint is lockable or not. The saddle-node bifurcation can occur only on the condition that the unilateral constraint is lockable and its clearance is non-zero. The results obtained by two methods are consistent. The project supported by the National Natural Science Foundation of China (10272002) and the Doctoral Program from the Ministry of Education of China (20020001032) The English text was polished by Yunming Chen.     

概念力学分析中的延拓方法

在概念力学分析基本方法如动因素法、主响应法和极端形态界定法等基础上发展了延拓分析的方法, 亦即以分析对象的一个概念力学基本分析模型为出发点, 通过不断延拓的分析过程来达到解决一系列具有工程实际意义问题的目的. 给出了管道有暖流通过时管壁的内力与变形分析和多层框架在均匀温降或楼面载荷作用下的弯矩分析等两个实际分析案例. 揭示了如何在延拓分析过程中结合运用概念力学分析的其他基本方法以取得突破, 并进而确定结构受力响应基本形态的过程.     

Travelling wave solutions for a quasilinear model of field dislocation mechanics

We consider an exact reduction of a model of Field Dislocation Mechanics to a scalar problem in one spatial dimension and investigate the existence of static and slow, rigidly moving single or collections of planar screw dislocation walls in this setting. Two classes of drag coefficient functions are considered, namely those with linear growth near the origin and those with constant or more generally sublinear growth there. A mathematical characterisation of all possible equilibria of these screw wall microstructures is given. We also prove the existence of travelling wave solutions for linear drag coefficient functions at low wave speeds and rule out the existence of nonconstant bounded travelling wave solutions for sublinear drag coefficients functions. It turns out that the appropriate concept of a solution in this scalar case is that of a viscosity solution. The governing equation in the static case is not proper and it is shown that no comparison principle holds. The findings indicate a short-range nature of the stress field of the individual dislocation walls, which indicates that the nonlinearity present in the model may have a stabilising effect. We predict idealised dislocation-free cells of almost arbitrary size interspersed with dipolar dislocation wall microstructures as admissible equilibria of our model, a feature in sharp contrast with predictions of the possible non-monotone equilibria of the corresponding Ginzburg-Landau phase field type gradient flow model.     

Semi-analytical finite element method for fictitious crack model in fracture mechanics of concrete

Based on the Hamiltonian governing equations of plane elasticity for sectorial domain, the variable separation and eigenfunction expansion techniques were employed to develop a novel analytical finite element for the fictitious crack model in fracture mechanics of concrete. The new analytical element can be implemented into FEM program systems to solve fictitious crack propagation problems for concrete cracked plates with arbitrary shapes and loads. Numerical results indicate that the method is more efficient and accurate than ordinary finite element method.     

Generalized Whittaker equations and field method in generalized classical mechanics

This paper presents the energy integral in generalized classical mechanics. The integral enables us to reduce a given canonical system with 2n order to another system with only (2n-2) order and to obtain generalized Whittaker equations. And then, this paper extends a field method integrating the equations of motion for classical mechanics to generalized classical mechanics. Finally, this paper gives an example to illustrate the application of these equations and the field method.The Project Supported by the National Natural Science Foundation of China.     

Finite and impulsive motion of constrained mechanical systems via Jourdain's principle: discrete and hybrid parameter models

The main purpose of this paper is to present a unified analytical dynamics framework for the analysis of finite and impulsive motion of mechanical systems using Jourdain's principle. Emphasis is given to the general case when a mechanical system is described by a hybrid (discrete-distributed) parameter model. A large group of finite and impulsive, generally non-holonomic, constraints are analysed in detail and a so-called extended Appellian classification is presented for these constrained motion problems. The fundamental dynamic equation of constrained systems is developed in terms of velocity variations (Jourdain's principle). Based on this equation and the constraints, the methods of quasivelocities and Lagrangian multipliers are adopted and interpreted for the finite motion of hybrid parameter models of mechanical systems; and the methods of independent quasivelocity variations and Lagrangian multipliers are introduced for the analysis of impulsive motion of such models. To illustrate the proposed material, an example of a one-link flexible arm intercepting and capturing a moving target is considered.     

Dynamics modelling of the four-wheeled mobile platform

The model of dynamics of the four-wheeled mobile platform has been presented. Model of construction of prototype of has also been presented. The proposed model is useful to examine different configurations of the drive wheels and to analyze the relations between causes and effects of the motion parameters. The solution presented in the work allows to study the behavior of the platform also under slippage and in the circumstances to refrain the platform from falling into the skid. The problem of the forced motion and free motion of the platform with the possibility of modification the drive modulus positions has been considered. Analysis of the active forces with the resistive forces is also included. The formulated initial problem has been solved numerically with use of the Runge-Kutta method of the fourth order. The sample simulation results for the solution and conclusions are in the final sections.     

On the local average of random field in stochastic finite element analysis

The distinguishing feature of stochastic finite element analysis is that it involves the discretization of the parameter space of random fields of material properties, the geometry of structure and / or the loads. It is shown in earlier investigations that a reasonable procedure of discretization is to take the local averages of the random fields on each element. In the present paper the formulae for the covariance of the local averages of a homogeneous random vector field on rectangular elements are generalized by relaxing the condition. For an inhomogeneous random field and /or non-rectangular elements, a procedure of using Gaussian quadrature to evaluate the means and covariances of the local averages is proposed. Thus, the stochastic finite element method (SFEM) based on the local averages of random fields is adapted to a structure with irregular shape and / or inhomogeneous random fields. The effects of the mesh geometry, the ratio of element size to the correlation scale as well as the number of Gaussian quadrature points on the convergence of SFEM are discussed. It is found that even better results could be obtained by utilizing appropriate Gaussian quadrature instead of exact local average.Project supported by National Natural Science Foundation of China.     

Discrete time transfer matrix method for dynamics of multibody system with flexible beams moving in space

In this paper,by defining new state vectors and developing new transfer matrices of various elements moving in space,the discrete time transfer matrix method of multi-rigid-flexible-body system is expanded to study the dynamics of multibody system with flexible beams moving in space.Formulations and numerical example of a rigidflexible-body three pendulums system moving in space are given to validate the method.Using the new method to study the dynamics of multi-rigid-flexible-body system moving in space,the global dynamics equations of system are not needed,the orders of involved matrices of the system are very low and the computational speed is high,irrespective of the size of the system.The new method is simple,straightforward,practical,and provides a powerful tool for multi-rigid-flexible-body system dynamics.     

On the representation of finite rotation in nonlinear field theory of continuum mechanics

One of the important basic theoretical problems in the development of continuum mechanics is the separation of finite strain and finite rotation at a point in the displacement field. Now it is certain that S-R decomposition theorem provides a rational theoretical solution for this problem. The purpose of this paper is to clarify some misleading basic concepts of finite rotation of deformed body in current literature, and to promote further progress.The detailed abstract of this paper was published in 1985 Shanghai International Conference of Nonlinear Mechanics. In the present paper some new results are included.     

Two-dimensional numerical calculation of the flow field about a scoop inlet by the piecewise linear method

The piecewise linear method (PLM) based on time operator splitting is used to solve the unsteady compressible Euler equations describing the two-dimensional flow around and through a straight wall inlet placed stationary in a rapidly rotating supersonic flow. The PLM scheme is formulated as a Lagrangian step followed by an Eulerian remap. The inhomogeneous terms in the Euler equations written in cylindrical coordinates are first removed by Sod's method and the resulting set of equations is further reduced to two sets of one-dimensional Lagrangian equations, using time operator splitting. The numerically generated flow fields are presented for different values of the back pressure imposed at the downstream exit of the inlet nozzle. An oblique shock wave is formed in front of the almost whole portion of the inlet entrance, the incoming streamlines being deflected towards the higher pressure side after passing through the oblique shock wave and then bending down to the lower pressure side. A reverse flow appears inside the inlet nozzle owing to the recovery pressure of the incoming streams being lower than the back pressure of the inlet nozzle.