On the Graphical Solution of Transient Vibration Problems

Abstract

Phase-plane methods have been used with great success in the theory of non-linear mechanics. They have also been applied to a different class of problems; viz. to dynamical systems with transient loading. The paper concerns this second use of the plane and some recent extensions of it. A survey is made of the literature covering the twenty years since these constructions were introduced. This relates to: (1) the transient disturbance of single-degree-of-freedom systems (which may include damping, non-linearity, hysteresis, etc.) and of multi-degree-of-freedom systems; (2) the deflexions of beam columns in the theory of elastic stability; and (3) Jacobsen's general method of graphical integration. Some new additions to the theory are given. These are: (1) a method of graphical differentiation; (2) a simplified approach to multi-degree-of-freedom systems leading to the treatment of transient loading of beams by moving and moving-and-varying loads (damping may be contemplated in these beam problems and the method that is devised easily carries over to certain plate problems); and (3) a demonstration that the graphical method can be used for deflexions—as well as bending moments —in the theory of beam columns. It is suggested that these constructions are of real importance, not only for practical engineering problems, but also for teaching purposes.