Stability of periodic solutions of the nonlinear string
Author:
R. W. Dickey
Journal:
Quart. Appl. Math. 38 (1980), 253-259
MSC:
Primary 70K20; Secondary 35L10, 73C02
DOI:
https://doi.org/10.1090/qam/580884
MathSciNet review:
580884
Full-text PDF Free Access
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Additional Information
- G. F. Carrier, On the non-linear vibration problem of the elastic string, Quart. Appl. Math. 3 (1945), 157–165. MR 12351, DOI https://doi.org/10.1090/S0033-569X-1945-12351-4
R. Narasimha, Non-linear vibration of an elastic string, J. Sound Vib. 8, 134 (1968)
- R. W. Dickey, Infinite systems of nonlinear oscillation equations related to the string, Proc. Amer. Math. Soc. 23 (1969), 459–468. MR 247189, DOI https://doi.org/10.1090/S0002-9939-1969-0247189-8
- Takaaki Nishida, A note on an existence of conditionally periodic oscillation in a one-dimensional anharmonic lattice, Mem. Fac. Engrg. Kyoto Univ. 33 (1971), 27–34. MR 0279387
- R. W. Dickey, The initial value problem for a nonlinear semi-infinite string, Proc. Roy. Soc. Edinburgh Sect. A 82 (1978/79), no. 1-2, 19–26. MR 524669, DOI https://doi.org/10.1017/S0308210500011008
- Shou-Chen Hu, THE INITIAL VALUE PROBLEM FOR A NONLINEAR INFINITE STRING, ProQuest LLC, Ann Arbor, MI, 1979. Thesis (Ph.D.)–State University of New York at Buffalo. MR 2630307
G. S. S. Murthy and B. S. Ramakrishna, Non-linear character of resonance in stretched strings, J. Acoust. Soc. Amer. 38, 461 (1965)
- John W. Miles, Stability of forced oscillations of a vibrating string, J. Acoust. Soc. Amer. 38 (1965), 855–861. MR 191153, DOI https://doi.org/10.1121/1.1909816
G. V. Anand, Non-linear resonance in stretched strings with viscous damping, J. Acoust. Soc. Amer. 40, 1517 (1966)
E. W. Lee, Non-linear forced vibration of a stretched string, Br. J. Appl. Phys. 8, 411 (1957)
- Donald W. Oplinger, Frequency response of a nonlinear stretched string, J. Acoust. Soc. Amer. 32 (1960), 1529–1538. MR 120900, DOI https://doi.org/10.1121/1.1907948
- R. W. Dickey, Infinite systems of nonlinear oscillation equations with linear damping, SIAM J. Appl. Math. 19 (1970), 208–214. MR 265654, DOI https://doi.org/10.1137/0119019
- Edward L. Reiss and Bernard J. Matkowsky, Nonlinear dynamic buckling of a compressed elastic column, Quart. Appl. Math. 29 (1971), 245–260. MR 286344, DOI https://doi.org/10.1090/S0033-569X-1971-0286344-8
- J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publishers, Inc., New York, N.Y., 1950. MR 0034932
L. Brillouin, Wave propagation in periodic structures, Dover, New York, 1946
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR 0069338
G. F. Carrier, On the nonlinear vibration problem of the elastic string,Quart. Appl. Math. 3, 157 (1945)
R. Narasimha, Non-linear vibration of an elastic string, J. Sound Vib. 8, 134 (1968)
R. W. Dickey, Infinite systems of nonlinear oscillation equations related to the string, Proc. Amer. Math. Soc. 23, 459 (1969)
T. Nishida, A note on the nonlinear vibrations of an elastic string, Mem. Fac. Eng., Kyoto Univ., 34 (1971)
R. W. Dickey, The initial value problem for a nonlinear semi-infinite string, Proc. Roy. Soc. Edin. 82A, 19 (1978)
S. C. Hu, The initial value problem for a non-linear infinite string, Ph.D. Thesis, State University of New York at Buffalo, 1979
G. S. S. Murthy and B. S. Ramakrishna, Non-linear character of resonance in stretched strings, J. Acoust. Soc. Amer. 38, 461 (1965)
J. W. Miles, Stability of forced oscillations of a vibrating string, J. Acoust. Soc. Amer. 38, 1855 (1965)
G. V. Anand, Non-linear resonance in stretched strings with viscous damping, J. Acoust. Soc. Amer. 40, 1517 (1966)
E. W. Lee, Non-linear forced vibration of a stretched string, Br. J. Appl. Phys. 8, 411 (1957)
D. W. Oplinger, Frequency response of a non-linear stretched string, J. Acoust. Soc. Amer. 32, 1529 (1960)
R. W. Dickey, Infinite systems of nonlinear oscillation equations with linear damping, SIAM J. Appl. Math. 19, 208 (1970)
E. L. Reiss and B. J. Matkowsky, Nonlinear dynamic buckling of a compressed elastic column, Quart. Appl. Math. 29, 245 (1971)
J. J. Stoker, Nonlinear vibrations, Interscience, New York, 1950
L. Brillouin, Wave propagation in periodic structures, Dover, New York, 1946
E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955
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Article copyright:
© Copyright 1980
American Mathematical Society