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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.

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References

Andronow

A. A.

Chaikin

C. E.

1937 ‘Theory of Oscillations’ (Princeton University Press, Princeton, New Jersey, U.S.A., translated from Russian 1949).

Ayre

R. S.

1952 Jl. Franklin Inst., vol. 253, No. 2, p. 153, ‘Transient Vibration of Linear Multi-degree-of-freedom Systems by the Phase-plane Method’.

Braun

1937 Ingenieur-Archiv, vol. 8, No. 3, p. 198, ‘Über die graphische Lösung der Differentialgleichung der erzwungenen Schwingungen bei beliebigen Gesetz für Dämpfung, Rückstellkraft und Antriebskraft’.

Bruce

1951 Bulletin Seismological Soc. Am., vol. 41, No. 2, p. 101, ‘A Graphical Method for Solving Vibration Problems of a Single Degree of Freedom’.

Coates

R. C.

1950 Engineering, vol. 169, p. 720, ‘Graphical Solution of Axially-loaded Beams’.

Evaldson

R. L.

Ayre

R. S.

Jacobsen

L. S.

1949 Jl. Franklin Inst., vol. 248, No. 6, p. 473, ‘Response of an Elastically Non-linear System to Transient Disturbances’.

Flügge-Lotz

Klotter

1943 See National Advisory Committee Aeronautics Tech. Memoranda 1237, November 1949, ‘On the Motions of an Oscillating System under the Influence of Flip-Flop Controls’.

Fuchs

H. O.

1936 Product Engineering, vol. 7, No. 8, p. 294, ‘Spiral Diagrams to Solve Vibration Damping Problems’.

Howard

H. B.

1928 Aeronautical Research Committee (now Council) Reports and Memoranda 1233, ‘The Graphical and Analytical Determination of Stresses in Single Span and Continuous Beams under End Compression and Lateral Load with Variations in Shear, Distributed Load and Moment of Inertia’.

10.

Jacobsen

L. S.

1951 Structural Dynamics Technical Report, Stanford University, No. 13, ‘On a General Method of Solving Second Order, Ordinary Differential Equations by Phase-Plane Displacements’. (See also 1952 Jl. Applied Mechanics, vol. 19, No. 4, p. 543.).

11.

Jacobsen

L. S.

Ayre

R. S.

Aisawa

1951 Structural Dynamics Technical Report, Stanford University, No. 11.

12.

Kelvin

Lord

1892 Phil. Mag., vol. 34, p. 443, ‘On Graphic Solution of Dynamical Problems’.

13.

Lamoen

1932 Mémoire Académie Royale de Belgique, Classe des Sciences, vol. 12, No. 1437, p. 5, ‘Sur la Sollicitation dynamique des Édifices élancés par le Vent”.

14.

Lamoen

1935 Revue Universelle des Mines, series 8, vol. 11, No. 7, p. 213, ‘Etude graphique des Vibrations de Systèmes à un Seul Degré de Liberté’.

15.

Liénard

1928 Revue Générale de l'Electricite, vol. 23, Nos. 21 and 28, pp. 701 and 946, ‘Etude des Oscillations entrétenues’.

16.

Meissner

1932 ‘Graphische Analysis vermittelst des Linienbildes einer Funktion’ (Verlag der Schweizerischen Bauzeitung, Zurich).

17.

Mindlin

R. D.

Stubner

F. W.

Cooper

H. L.

1948 Proc. Am. Soc. Experimental Stress Analysis, vol. 5, No. 2, p. 69, ‘Response of Damped Elastic Systems to Transient Disturbances’.

18.

Minorsky

1947 ‘Introduction to Non-linear Mechanics’ (Edwards Brothers, Ann Arbor, Michigan, U.S.A.).

19.

Rojansky

1948 Jl. Applied Phys., vol. 19, No. 3, p. 297, ‘Gyrograms for Simple Harmonic Systems Subjected to External Forces’.

20.

Rojansky

Beth

R. A.

1947 Jl. Applied Mechanics, vol. 14, No. 3, p. 202, ‘Camptograms for Beams in Compression’.

21.

Runge

1932 ‘Graphische Methoden’ (Teubner, Leipzig).

22.

Stoker

J. J.

1950 ‘Nonlinear Vibrations in Mechanical and Electrical Systems’ (Interscience Publishers, New York and London).

23.

Timoshenko

S. P.

1937 ‘Vibration Problems in Engineering’, second edition (D. Van Nostrand, Constable, New York, London).

24.

Timoshenko

S. P.

Young

D. H.

1948 ‘Advanced Dynamics’ (McGraw-Hill, New York and London).

25.

Weiss

H. K.

1946 Jl. Aeronautical Sciences, vol. 13, No. 7, p. 364, ‘Analysis of Relay Servomechanisms’.

On the Graphical Solution of Transient Vibration Problems

Abstract

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References