Three-dimensional Problems of the Theory of Elasticity. By A. I. Lur’e.1964. (Interscience Publishers)

WE found, on experimental grounds in Article I, that the field of air‐flow past a short body of low resistance shape, such as an aerofoil, comprises two dissimilar parts: (a) a thin boundary layer enveloping the body and dominated by viscous effects, and (b) a motion outside the boundary layer in which viscosity is much less important. It will be remembered that in the external motion occur the large pressure changes, which, transmitted through the boundary layer, account for nearly all the lift and for part of the drag. These pressures we observed to be calculable from the velocities without appreciable error by Bernoulli's equation. In the present Article we confine attention to this external flow, assuming it to be steady, incompressible, and inviscid. Its dependence upon (a), already discussed to some extent, we ignore; the boundary layer is conceived to be everywhere very thin, so that the only role it plays is to allow of relative velocity at the surface of the body. The assumptions made, excepting that of incompressibility, will appear drastic, and it will not be surprising if some of our deductions prove discordant with experimental fact. Nevertheless, they lead to a theory which finds many applications and uses in real fluid motion, and, in particular, gives an intimate view of aerofoil flow that is very close to the truth. It is convenient to develop our reasoning in analytical terms and for simplicity to restrict the flow to two dimensions (Article 1, §5). But the engineer will find special scope in this part of aerodynamics for graphical methods in the solution of particular problems.

Aerodynamics for Engineers

The Super Mechano-System was the research project at the Tokyo Institute of Technology from 1997 to 2002 and the Human Adaptive Mechatronics (HAM) Project is the newly selected COE research project at Tokyo Denki University from 2003 to 2008. Both projects are sponsored by the Ministry of Education, Culture, Sports, Science and Technology in Japan. The author was the SMS project leader until 2000 and Prof. S. Hirose succeeded. Many mechanical systems with functions adapting to varying environment and the relating basic theory have been developed. The concurrent design method of the mechanism and controller has been developed by the control research group in the project. Human adaptive mechatronics is a system, which includes the human in the control loop and changes the functions and structure of the man-machine interface according to the improvement of the human operation skill. In this plenary lecture, the control of the multiple pendulums from the pendant to the upright position is discussed from the viewpoint of varying constraint systems. The swing-up control of pendulums is discussed by considering reachability of an unstable nonlinear system by nonlinear control. A new approach to analyse and to design a nonlinear control based on the fractal is presented. The results show that the map of the control parameters and the initial conditions give interesting results.

Control of pendulum: from Super Mechano-System to Human Adaptive Mechatronics

A unified geometric formulation of the methods used for solving constrained system problems is given. Both holonomic and nonholonomic systems are treated in like manner, and the dynamic equations are expressible in either generalized velocities or quasi-velocities. Moreover, a wide range of ’unconstrained‘ systems are uniformly regarded as generalized particles in the multi-dimensional metric spaces relating to their configuration. The derivation is grounded on the tensor calculus formalism and appropriate geometric interpretations are reported. In its useful matrix form, the formulation turns out short, elementary and general. This unified geometric approach to constrained system dynamics may deserve to become a generally accepted method inacademic and engineering applications.

A Geometric Unification of Constrained System Dynamics

Modelling consists of developing a representation of the properties of an object or a phenomenon with respect to the goals of its analysis. In this paper, the procedure presented is that followed in the CHARM project for human upper limb modelling with respect to the project constraints on the model implementation and simulation. The objective is to develop a Comprehensive Human Animation Resource Model allowing the simulation of human motion, including the finite element simulation of soft tissue deformation and muscular contraction. Generally, models are presented based on assumptions left for a fortiori validation. The a priori considerations which lead to these assumptions are rarely detailed. Here, the aim is to form a basis for the choices and assumptions which are to be made prior to the validation. The analysis is based on the general approach for the modelling of multi-body deformable systems as well as on previous studies on the human upper limb.

A Case Study on Human Upper Limb Modelling for Dynamic Simulation

An alternative technique, called projection method, for solving constrained system problems is presented. This approach can be used to derive equations of motion of both holonomic and nonholonomic systems, and the dynamic equations can be expressed in generalized velocities and/or quasi-velocities. Compared against the other methods of classical mechanics (Lagrange's, Gibbs-Appell, Kane's,...), the present method turns out to be extraordinarily short, elementary and general. As such, it deserves to be promoted as a generally accepted method in academic and engineering applications. Three examples are reported to illustrate advantages of the technique

A unified approach to the modelling of holonomic and nonholonomic mechanical systems

This paper presents two examples of calculations for vehicles with flexible bodies by using mixed multibody and finite element methods. The first example deals with dynamics computations for a bimodal train with a flexible cistern, whereas the second example concerns the dynamics calculations for the PW-6 glider. In the first example, the influence of the chosen friction model on the train dynamics calculations results was discussed. The second example presents several methods of stress calculations and a comparison of results. The achieved conclusions may be used as suggestions towards a modelling method choice for a given problem.Both issues being discussed are of great importance in dynamics of flexible multibody systems modelling practice and durability assessment.

Friction Models and Stress Recovery Methods in Vehicle Dynamics Modelling

The process of determining equations of motion of a system consisting of rigid bodies and springs is often extremely laborious. In this paper, a method is presented whereby all terms of a system of equations for planar rigid body systems are calculated systematically. It is assumed that the system of rigid bodies has a topological tree structure. The method is based on a graph-theoretical approach and is consistent with that for the calculation of kinetic and potential energies presented by Arczewski ( J. Franklin Inst. , Vol. 324, pp. 351–367 and pp. 369–386, 1987). The proof of the method and an example are provided.

Application of graph theory to the mathematical modelling of a class of rigid body systems

The process of determining the kinetic energy as a function of generalized coordinates and velocities of rigid body systems is often extremely laborious. In this paper a method is presented by means of which all the kinetic energy terms of a system possessing a tree structure are calculated in a systematic manner. The paper contains the derivation of a general formula for kinetic energy which is then exemplified in the case of planar motion. A particular example of the six degree-of-freedom rigid body system in planar motion is also provided.

Application of graph theory to the determination of kinetic energy of rigid body systems

The process of determining the generalized forces (as a function of generalized coordinates and system parameters) of a system consisting of mass particles and springs is often extremely laborious. In this paper, a method is presented where the derived formulae have a form of evident functions not only of generalized coordinates and system parameters but also of the topological structure of a system. This paper contains the derivation of general formulae for potential energies of gravity and elastic forces as well as for the generalized forces related to both potential and nonpotential forces acting on a system of particles. It is assumed that the system of particles has a topological tree structure, and that linear springs are situated arbitrarily between the particles. The method is based on a graph-theoretical approach and is consistent with the kinetic energy calculation presented in Arczewski (J. Franklin Inst. , Vol. 329 , pp. 469–481, 1992) .

Krzysztof Arczewski

Papers

A unified approach to the modelling of holonomic and nonholonomic mechanical systems

Friction Models and Stress Recovery Methods in Vehicle Dynamics Modelling

Application of graph theory to the mathematical modelling of a class of rigid body systems

Application of graph theory to the determination of kinetic energy of rigid body systems

Graph theoretical approach—II. determination of generalized forces for a class of systems consisting of particles and springs