Stability of columns and strings under periodically varying forces
Authors:
S. Lubkin and J. J. Stoker
Journal:
Quart. Appl. Math. 1 (1943), 215-236
MSC:
Primary 73.2X
DOI:
https://doi.org/10.1090/qam/8982
MathSciNet review:
8982
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R. Einaudi, Sulle configurazioni di equilibrio instabile di una piastra sollecitata da sforzi tangentiali pulsante, Atti. Accad. Gioenia Catania, in two parts: I) mem. 20, 1–5 (1936); II) mem. 5, 1–20 (1937). Part I) deals with the rectangular plate having simply supported edges, II) with the rectangular plate assuming other boundary conditions. The theory is applied only to cases in which the steady part of the applied load is less than the lowest Euler load and the harmonic component is small in amplitude.
S. Goldstein, Mathieu functions, Trans. Cambridge Phil. Soc., 23, 303–336 (1937).
G. Gorelik, Resonanzerscheinungen in linearen Systemen mit periodisch veränderlichen Parametern, (in Russian), Zeit. tech. Phys., Leningrad, in three parts: I) 4, 1783–1817 (1934); II) 5, 195–215 (1935); III) 5, 489–517 (1935). Treats the nonhomogeneous “Mathieu equation” with a viscous damping term added. Among other things it is proved that the solutions of the homogeneous Mathieu equation with a damping term all tend to zero with increase of the independent variable if the solutions without damping are stable. (These observations were taken from the Zentralblatt f. Mech., 3, 302 (1935); 4, 18 (1936).)
G. Hamel, Über lineare homogene Differentialgleichungen zweiter Ordnung mit periodischen Koeffizienten, Math. Ann., 73, 371 (1912).
P. Humbert, Fonctions de Lamé et fonctions de Mathieu, Mém. des Sci. Math. X, Gauthier-Villars, Paris, 1926.
E. L. Ince, Researches into the characteristic numbers of the Mathieu equation, Proc. Royal Soc. of Edinburgh, 46, 20–29 (1925); 46, 316–322 (1926); 47, 294–301 (1927).
E. L. Ince, Ordinary differential equations, Longmans, Green and Co., London, 1927.
S. Lubkin, Stability of columns under periodically varying loads, 1939, manuscript in the library of New York University. A thesis for the degree of Doctor of Philosophy. Outlines a treatment of the problem for any boundary conditions.
S. Lubkin, Stability of columns under periodically varying loads, 1943, privately printed. 11 pages. A brief summary of the above thesis.
C. V. Raman, Experimental investigations on the maintenance of vibrations, Proc. Indian Assoc. for the Cultivation of Sci. Bulletin 6, 1912. A lengthy experimental investigation, beautifully and profusely illustrated with photographs of vibrating strings. Some anomalies are explained by considering a nonlinearity in the phenomena; for this purpose the differential equation \[ \ddot u + k\dot u + \left ( {{\eta ^2} - 2\alpha \sin 2pt + \beta {u^2}} \right )u = 0\] is introduced.
Lord Rayleigh, On the maintenance of vibrations by forces of double frequency and on the propagation of waves through a medium endowed with a periodic structure, Phil. Mag. (5) 24, 145–159 (1887). Considers only the possibility of excitation of the lowest mode of vibration of the string. The effect of damping is considered.
A. Stephenson, On a class of forced oscillations, Quart. Journ. of Math. 37, 353–360 (1906). Amplifies the results of Rayleigh noted above on the vibration of strings. Observes the possibility of exciting vibrations when the frequency of the applied force is a rational multiple of the fundamental frequency of lateral oscillation of the string.
A. Stephenson, On a new type of dynamical stability, Mem. and Proc. Manchester Literary and Phil. Soc. 52, No. 8 (1908). The theoretical possibility of converting the instable equilibrium position of a rigid rod standing on end by applying a vertical periodic force at the bottom seems to have been pointed out for the first time in this paper.
- M. J. O. Strutt, Der charakteristische Exponent der Hillschen Differentialgleichung, Math. Ann. 101 (1929), no. 1, 559–569 (German). MR 1512551, DOI https://doi.org/10.1007/BF01454859
M. J. O. Strutt, Lamésche, Mathieusche u. verwandte Funktionen in Physik u. Technik, Ergeb. der Math., Springer, Berlin, 1932. A very complete summary of both theory and applications, with an extensive bibliography.
I. Utida and K. Sezawa, Dynamical stability of a column under periodic longitudinal forces, Report of the Aero. Res. Inst., Tokyo Imp. Univ., 15, 193 (1940). It is assumed that the constant part of the applied load is zero (that is, $P = 0$ in our notation). The paper is both experimental and theoretical in character. In the experiments in some cases jump phenomena similar to those observed in working with forced oscillations of systems with one degree of freedom and a nonlinear restoring force were noted when the amplitude of the oscillations was large. The boundary conditions in the experiments were nearly those for clamped ends rather than pinned ends. In this paper a reference is given to a paper by K. Nisino: Journ. Aero. Res. Inst., Tokyo, No. 176, May 1939, which apparently deals with the same problem; the writers were not able to find this paper.
- E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469
N. Krylov and N. Bogoliuboff, Investigation of the influence of resonance in transverse vibration of rods caused by periodic normal forces at one end, (in Russian), Ukrainian Scientific Research Institute of Armament, Recueil, Kiev, 1935. The writers were not able to locate this paper.
R. Einaudi, Sulle configurazioni di equilibrio instabile di una piastra sollecitata da sforzi tangentiali pulsante, Atti. Accad. Gioenia Catania, in two parts: I) mem. 20, 1–5 (1936); II) mem. 5, 1–20 (1937). Part I) deals with the rectangular plate having simply supported edges, II) with the rectangular plate assuming other boundary conditions. The theory is applied only to cases in which the steady part of the applied load is less than the lowest Euler load and the harmonic component is small in amplitude.
S. Goldstein, Mathieu functions, Trans. Cambridge Phil. Soc., 23, 303–336 (1937).
G. Gorelik, Resonanzerscheinungen in linearen Systemen mit periodisch veränderlichen Parametern, (in Russian), Zeit. tech. Phys., Leningrad, in three parts: I) 4, 1783–1817 (1934); II) 5, 195–215 (1935); III) 5, 489–517 (1935). Treats the nonhomogeneous “Mathieu equation” with a viscous damping term added. Among other things it is proved that the solutions of the homogeneous Mathieu equation with a damping term all tend to zero with increase of the independent variable if the solutions without damping are stable. (These observations were taken from the Zentralblatt f. Mech., 3, 302 (1935); 4, 18 (1936).)
G. Hamel, Über lineare homogene Differentialgleichungen zweiter Ordnung mit periodischen Koeffizienten, Math. Ann., 73, 371 (1912).
P. Humbert, Fonctions de Lamé et fonctions de Mathieu, Mém. des Sci. Math. X, Gauthier-Villars, Paris, 1926.
E. L. Ince, Researches into the characteristic numbers of the Mathieu equation, Proc. Royal Soc. of Edinburgh, 46, 20–29 (1925); 46, 316–322 (1926); 47, 294–301 (1927).
E. L. Ince, Ordinary differential equations, Longmans, Green and Co., London, 1927.
S. Lubkin, Stability of columns under periodically varying loads, 1939, manuscript in the library of New York University. A thesis for the degree of Doctor of Philosophy. Outlines a treatment of the problem for any boundary conditions.
S. Lubkin, Stability of columns under periodically varying loads, 1943, privately printed. 11 pages. A brief summary of the above thesis.
C. V. Raman, Experimental investigations on the maintenance of vibrations, Proc. Indian Assoc. for the Cultivation of Sci. Bulletin 6, 1912. A lengthy experimental investigation, beautifully and profusely illustrated with photographs of vibrating strings. Some anomalies are explained by considering a nonlinearity in the phenomena; for this purpose the differential equation \[ \ddot u + k\dot u + \left ( {{\eta ^2} - 2\alpha \sin 2pt + \beta {u^2}} \right )u = 0\] is introduced.
Lord Rayleigh, On the maintenance of vibrations by forces of double frequency and on the propagation of waves through a medium endowed with a periodic structure, Phil. Mag. (5) 24, 145–159 (1887). Considers only the possibility of excitation of the lowest mode of vibration of the string. The effect of damping is considered.
A. Stephenson, On a class of forced oscillations, Quart. Journ. of Math. 37, 353–360 (1906). Amplifies the results of Rayleigh noted above on the vibration of strings. Observes the possibility of exciting vibrations when the frequency of the applied force is a rational multiple of the fundamental frequency of lateral oscillation of the string.
A. Stephenson, On a new type of dynamical stability, Mem. and Proc. Manchester Literary and Phil. Soc. 52, No. 8 (1908). The theoretical possibility of converting the instable equilibrium position of a rigid rod standing on end by applying a vertical periodic force at the bottom seems to have been pointed out for the first time in this paper.
M. J. O. Strutt, Der characteristische Exponent der Hillschen Differentialgleichung, Math. Ann., 101, 559–569 (1929). A study of the asymptotic character of the stable regions of the general Hill’s equation.
M. J. O. Strutt, Lamésche, Mathieusche u. verwandte Funktionen in Physik u. Technik, Ergeb. der Math., Springer, Berlin, 1932. A very complete summary of both theory and applications, with an extensive bibliography.
I. Utida and K. Sezawa, Dynamical stability of a column under periodic longitudinal forces, Report of the Aero. Res. Inst., Tokyo Imp. Univ., 15, 193 (1940). It is assumed that the constant part of the applied load is zero (that is, $P = 0$ in our notation). The paper is both experimental and theoretical in character. In the experiments in some cases jump phenomena similar to those observed in working with forced oscillations of systems with one degree of freedom and a nonlinear restoring force were noted when the amplitude of the oscillations was large. The boundary conditions in the experiments were nearly those for clamped ends rather than pinned ends. In this paper a reference is given to a paper by K. Nisino: Journ. Aero. Res. Inst., Tokyo, No. 176, May 1939, which apparently deals with the same problem; the writers were not able to find this paper.
E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th Ed., Cambridge University Press, 1927.
N. Krylov and N. Bogoliuboff, Investigation of the influence of resonance in transverse vibration of rods caused by periodic normal forces at one end, (in Russian), Ukrainian Scientific Research Institute of Armament, Recueil, Kiev, 1935. The writers were not able to locate this paper.
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Article copyright:
© Copyright 1943
American Mathematical Society
