Modeling And Simulation Of Complex Mechanical Systems Based On The Natural-Coordinate Approach

The method of natural coordinates, a creative idea, uses purely Cartesian variables to describe the motion of multibody systems, with the result that the dynamics models, Differential Algebraic Equations (DAE), are particularly simple. Although the method has the potential for meeting the real-time simulation challenge, it has not been applied in industry successfully, for the conventional approaches to achieving such a natural-coordinate DAE are involved. In this dissertation, the following creative researches on basic theories of natural coordinates are carried out, in order to improve the natural-coordinate method.1) A novel idea, natural-coordinate system (NCS), is put forward to uniformly describe different types of traditional selection of base points and unit vectors. With this concept, to choose natural coordinates for each body of a multibody system is to build a general NCS consisting of one origin and three nonzero underlying vectors, not to select points or vectors traditionally. Based on the idea, several modeling methods are proposed as follows.a) A new selection method, based on the idea of NCS, is presented to choose all the natural coordinates for a multibody system in a rational way. The method includes two steps:First, construct a so-called standard natural-coordinate system for every rigid element; second, adapt it to the joints attached to the element, resulting in an actual natural-coordinate system which includes all the rational natural coordinates for the element. Without most of the conventional shortcomings, the method serves for all kinds of three-dimensional elements and is compatible with all traditional methods.b) A self-adaptation method for natural-coordinate systems is proposed, in order to automate the selection of natural coordinates for multibody system. The four-step method includes:First, find out all empty positions, which come from the feature points or vectors of the joints attached to the element, and give equal weight to them; second, delete redundant empty positions and add their weight to the unique one; third, select at most four empty positions which have a maximum total weight and can be occupied by a natural-coordinate system at the same time; fourth, the standard natural-coordinate system on the element can adapt itself to the selected empty positions, leading to an actual natural-coordinate system, which contains twelve rational natural coordinates for the element.c) It is somewhat complicated and inflexible to use the classical method to calculate mass matrices with natural coordinates. Based on the novel idea (natural-coordinate system), a new method is presented, aimed at simplifying and unifying the calculation of mass matrix for all kinds of three-dimensional elements which have constant natural-coordinate mass matrices. In this method, calculating the mass matrix for each element involves three steps:First, carry out a so-called standard mass matrix, which is diagonal and only related to its own inertia characteristics; second, construct a so-called adaptation matrix, which is closely related to the distinguishing features of the joints attached to the element; third, work out the new actual mass matrix, which is usually intended for the requirement of simplified joint constraint equations. The existence, uniqueness, and invertibility of adaptation matrices for all kinds of natural-coordinate systems are validated mathematically.2) A new strategy "Automatically Reduce Normal Natural-Coordinate DAEs Assembled Automatically" is developed to obtain DAEs of moderate size. It differs from traditional strategies in that natural coordinates are always chosen on rigid bodies of a multibody system, not on joints. To implement the new strategy, two reduction methods are presented as follows.a) With the strategy in mind, the authors have developed a reduction method on natural-coordinate constant constraint (NCCC). A NCCC constrains a state variable to a constant, leading to a simplest constraint equation which often appears in the joints which connect elements defined with natural coordinates to the ground. The reduction method can be performed once for every NCCC, with the result that one column and one row are removed from the system mass matrix. The reduction operations can be done independently and repeatedly, a characteristic suitable for computer implementation.b) Another reduction method is also presented, which is on natural-coordinate equivalent constraint (NCEC); an NCEC equalizes two state variables from two different bodies. This reduction method is an enhanced version of the reduction method on NCCC. The works in the thesis make an effective approach to address natural-coordinate DAEs. With those modeling methods, it is easy to assemble normal natural-coordinate DAEs, where each body is defined with 12 natural coordinates. With the reduction methods, it is convenient to reduce the normal natural-coordinate DAEs, according to those simplified joint constraints, with the result that nearly all the advantages of natural coordinates remain. All the methods should be performed by computer so that the approach of natural coordinates could be applied in industry.