This textbook was initially developed for the introductory course in finite element methods at the Department of Mechanical Engineering and Applied Mechanics, the University of Michigan, Ann Arbor, Michigan. It is based on four years of teaching experience of first-year graduate students and some senior undergraduate students in the engineering college. Because of the mechanical engineering environment, heat conduction problems are covered, as are standard stress analysis of solids and structures. Many small-size BASIC and FORTRAN programs are given so that readers can apply them to solve exercises using either microcomputers such as IBM PCs and compatibles or mainframes of computer networks that support FORTRAN iv. BASIC programs are available in separate volumes with a diskette for the IBM PC and compatibles by reader's request (see back of book for ordering instructions). FORTRAN programs are also available to readers who specially request them from the author. At present, BASIC compilers are available for some microcomputers in order to improve speed of computation. In the author's opinion, they are very impressive and encourage the use of microcomputers even for finite element methods. It is noted, however, that these programs are primarily designed for educational purposes – to teach the theory of finite element methods.

Fundamental considerations

Many non-linear systems of engineering interest possess more than one stationary solution stable against small disturbances, that is, more than one attractor. Such solutions are termed locally stable. In practical applications usually only one of these is desirable; the others, for the most part, are unwelcome because they signal danger to safe and reliable operation of the device whose model is being analyzed. The problem is to establish the conditions which lead to a particular steady state, or alternatively, to examine the disturbances which are apt to cause one stationary state to change to another (for example, small-amplitude stationary vibration to large-amplitude motion, a non-oscillatory state to an oscillatory condition, etc.).

The solution to the first aspect of the problem, i.e. establishing the domains of the initial conditions which lead to different stationary solutions, is well known. These domains are called the domains of attraction of a particular solution. For systems which are directly described by a set of two first-order differential equations of the type (4.1, 1) or whose set of such equations is identical with the original second-order differential equation the problem is solved by means of phase portrait analysis outlined in Chapter 4. The differential equations of motion of one-degree-of-freedom systems excited harmonically by an external force or parametrically can be converted, by application of known procedures (van der Pol or Krylov and Bogoljubov methods) to a system of two first-order differential equations of the type (4.1, 1) which, however, are not identical with the original equation of motion.