Eigenoscillations of mechanical systems with boundary conditions containing the frequency
Authors:
B. P. Belinskiy and J. P. Dauer
Journal:
Quart. Appl. Math. 56 (1998), 521-541
MSC:
Primary 34B24; Secondary 34L10, 34L15, 73D30, 73K05
DOI:
https://doi.org/10.1090/qam/1637056
MathSciNet review:
MR1637056
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Abstract: The problem of eigenoscillations of beam-mass systems is investigated and four examples are developed. For such systems the corresponding Sturm-Liouville problems contain the eigenvalue parameter in the boundary conditions. It is shown that the eigenfunctions for the systems considered form a basis of the appropriate Hilbert space. Rayleigh-Ritz formulas are also developed. Some lower bound estimations for eigenfrequencies are also found.
- James P. Keener, Principles of applied mathematics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1988. Transformation and approximation. MR 943623
E. A. Bambill and P. A. A. Laura, Application of the Rayleigh-Schmidt method when the boundary conditions contain the eigenvalues of the problem, Journal of Sound and Vibration 130, 167–170 (1989)
S.-P. Cheng and N. C. Perkins, Free vibration of a sagged cable supporting a discrete mass, Journal of the Acoustical Society of America 91, 2654–2662 (1992)
M. J. Maurizi and P. M. Bellés, An additional evaluation of free vibrations of beam-mass systems, Journal of Sound and Vibration 154, 182–186 (1992)
H. Abramovich and O. Hamburger, Vibration of a uniform cantilever Timoshenko beam with translational and rotational springs and with a tip mass, Journal of Sound and Vibration 154, 67–80 (1992)
- O. A. Ladyzhenskaya, The boundary value problems of mathematical physics, Applied Mathematical Sciences, vol. 49, Springer-Verlag, New York, 1985. Translated from the Russian by Jack Lohwater [Arthur J. Lohwater]. MR 793735
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
- Eberhard Zeidler, Nonlinear functional analysis and its applications. II/A, Springer-Verlag, New York, 1990. Linear monotone operators; Translated from the German by the author and Leo F. Boron. MR 1033497
- M. V. Keldyš, On the characteristic values and characteristic functions of certain classes of non-self-adjoint equations, Doklady Akad. Nauk SSSR (N.S.) 77 (1951), 11–14 (Russian). MR 0041353
- I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142
- Charles T. Fulton, Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 87 (1980/81), no. 1-2, 1–34. MR 600446, DOI https://doi.org/10.1017/S0308210500012312
- Charles T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), no. 3-4, 293–308. MR 593172, DOI https://doi.org/10.1017/S030821050002521X
- E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, 2nd ed., Clarendon Press, Oxford, 1962. MR 0176151
R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with a fluid, Tôhoku Mathematics Journal 35, 360–375 (1932)
- Bernard Friedman, Principles and techniques of applied mathematics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1956. MR 0079181
- Johann Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary condition, Math. Z. 133 (1973), 301–312. MR 335935, DOI https://doi.org/10.1007/BF01177870
Yu. I. Bobrovnitskii, On oscillations of some mechanical systems with nonorthogonal eigenfunctions, Acoustic Dynamics of Machines and Constructions “Nauk", Moscow, Russia, 1973, pp. 6–9 (in Russian)
- Don B. Hinton, An expansion theorem for an eigenvalue problem with eigenvalue parameter in the boundary condition, Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 117, 33–42. MR 528889, DOI https://doi.org/10.1093/qmath/30.1.33
- Don Hinton, Eigenfunction expansions for a singular eigenvalue problem with eigenparameter in the boundary condition, SIAM J. Math. Anal. 12 (1981), no. 4, 572–584. MR 617716, DOI https://doi.org/10.1137/0512050
- D. B. Hinton and J. K. Shaw, Differential operators with spectral parameter incompletely in the boundary conditions, Funkcial. Ekvac. 33 (1990), no. 3, 363–385. MR 1086767
- D. B. Hinton and J. K. Shaw, Spectrum of a Hamiltonian system with spectral parameter in a boundary condition, Oscillations, bifurcation and chaos (Toronto, Ont., 1986) CMS Conf. Proc., vol. 8, Amer. Math. Soc., Providence, RI, 1987, pp. 171–186. MR 909908
- Herbert Steinrück, Singularly perturbed eigenvalue problems, SIAM J. Appl. Math. 47 (1987), no. 6, 1131–1149. MR 916232, DOI https://doi.org/10.1137/0147076
- Paul Binding and Qiang Ye, Variational principles without definiteness conditions, SIAM J. Math. Anal. 22 (1991), no. 6, 1575–1583. MR 1129400, DOI https://doi.org/10.1137/0522100
- Leiba Rodman, An introduction to operator polynomials, Operator Theory: Advances and Applications, vol. 38, Birkhäuser Verlag, Basel, 1989. MR 997092
- G. V. Radzīēvs′kiĭ, The problem of completeness of root vectors in the spectral theory of operator-valued functions, Uspekhi Mat. Nauk 37 (1982), no. 2(224), 81–145, 280 (Russian). MR 650759
- L. Greenberg and I. Babuška, A continuous analogue of Sturm sequences in the context of Sturm-Liouville equations, SIAM J. Numer. Anal. 26 (1989), no. 4, 920–945. MR 1005517, DOI https://doi.org/10.1137/0726051
- Leon Greenberg, An oscillation method for fourth-order, selfadjoint, two-point boundary value problems with nonlinear eigenvalues, SIAM J. Math. Anal. 22 (1991), no. 4, 1021–1042. MR 1112064, DOI https://doi.org/10.1137/0522067
A. G. Kostyuchenko and A. A. Shkalikov, Self-adjoint quadratic operator pencils and elliptic problems, Functional Analysis and its Applications 17, 109–128 (1983)
A. G. Kostyuchenko and A. A. Shkalikov, On the theory of self-adjoint quadratic operator polynomials, Vestnik Moscov. University Series 1. Mathematics, Mechanics, No. 6, 1983, pp. 40–51 (in Russian)
- G. Chen, S. G. Krantz, D. L. Russell, C. E. Wayne, H. H. West, and M. P. Coleman, Analysis, designs, and behavior of dissipative joints for coupled beams, SIAM J. Appl. Math. 49 (1989), no. 6, 1665–1693. MR 1025953, DOI https://doi.org/10.1137/0149101
- John B. Conway, A course in functional analysis, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985. MR 768926
- Ivar Stakgold, Boundary value problems of mathematical physics. Vol. II, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1968. MR 0243183
- Hyun J. Ahn, Vibrations of a pendulum consisting of a bob suspended from a wire: the method of integral equations, Quart. Appl. Math. 39 (1981/82), no. 1, 109–117. MR 613954, DOI https://doi.org/10.1090/S0033-569X-1981-0613954-4
H. J. Ahn, On random transverse vibrations of a rotating beam with tip mass, Quart. Appl. Math. 39, 97–109 (1981)
L. H. Jones, The transverse vibration of a rotating beam with tip mass, Quart. Appl. Math. 23, 193–203 (1975)
J. A. Burns and E. M. Cliff, An approximation technique for the control and identification of the hybrid system, in Dynamics and Control of Large Flexible Space Structures, Meirovich, Virginia Tech, Blacksburg, VA, 1981, pp. 269–284
F. Riesz and B. St. Nagy, Lectures in Functional Analysis, Ungar, New York, New York, 1955
- Werner Schmeidler, Linear operators in Hilbert space, Academic Press, New York-London, 1965. Translation by Jay Strum; Revised and edited by A. Shenitzer and D. Solitar. MR 0182881
R. D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, Journal of Applied Mechanics 18, A31–A38 (1951)
- Zeev Schuss, Theory and applications of stochastic differential equations, John Wiley & Sons, Inc., New York, 1980. Wiley Series in Probability and Statistics. MR 595164
P. A. Laura, J. A. Reyes, and R. E. Rossi, Dynamic behavior of a cable-payload system suddenly stopped at one end, Journal of Sound and Vibration 34, 81–95 (1989)
J. P. Keener, Principles of Applied Mathematics: Transformation and Approximation, Addison-Wesley Publishing Company, Redwood City, California, 1988
E. A. Bambill and P. A. A. Laura, Application of the Rayleigh-Schmidt method when the boundary conditions contain the eigenvalues of the problem, Journal of Sound and Vibration 130, 167–170 (1989)
S.-P. Cheng and N. C. Perkins, Free vibration of a sagged cable supporting a discrete mass, Journal of the Acoustical Society of America 91, 2654–2662 (1992)
M. J. Maurizi and P. M. Bellés, An additional evaluation of free vibrations of beam-mass systems, Journal of Sound and Vibration 154, 182–186 (1992)
H. Abramovich and O. Hamburger, Vibration of a uniform cantilever Timoshenko beam with translational and rotational springs and with a tip mass, Journal of Sound and Vibration 154, 67–80 (1992)
O. A. Ladyzhenskaia, The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York, New York, 1985
D. Gilbarg and N. S,. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, Berlin, 1983
E. Zeidler, Nonlinear Functional Analysis and its Applications, Part II/A, Linear Monotone Operators, Springer-Verlag, New York, New York, 1990
M. V. Keldysh, On the characteristic values and characteristic functions of a certain class of non-selfadjoint equations, Dokl. Akad. Nauk SSSR 77, 11–14 (1951) (in Russian)
I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Non-Selfadjoint Operators, Translation of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, Rhode Island, 1969
C. T. Fulton, Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proceedings of the Royal Society of Edinburg 87A, 1–34 (1980)
C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proceedings of the Royal Society of Edinburg 77A, 293–308 (1977)
E. C. Titchmarsh, Eigenfunction Expansions Associated with Second Order Differential Equations, I, 2nd edition, University Press, Oxford, London, 1962
R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with a fluid, Tôhoku Mathematics Journal 35, 360–375 (1932)
B. Friedman, Principles and Techniques of Applied Mathematics, John Wiley and Sons, New York, New York, 1956
J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Zeitschrift 133, 301–312 (1973)
Yu. I. Bobrovnitskii, On oscillations of some mechanical systems with nonorthogonal eigenfunctions, Acoustic Dynamics of Machines and Constructions “Nauk", Moscow, Russia, 1973, pp. 6–9 (in Russian)
D. B. Hinton, An expansion theorem for an eigenvalue problem with eigenvalue parameter in the boundary condition, Quarterly Journal of Mathematics, Oxford, Vol. 2, 33–42 (1979)
D. B. Hinton, Eigenfunction expansions for a singular eigenvalue problem with eigenvalue parameter in the boundary condition, SIAM Journal of Mathematical Analysis 12, 572–584 (1981)
D. B. Hinton and J. K. Shaw, Differential Operators with Spectral Parameter Incompletely in the Boundary Conditions, Funkcialaj Ekvacioj (Serio Internacia), Vol. 33, 1990, pp. 363–385
D. B. Hinton and J. K. Shaw, Spectrum of a Hamiltonian System with Spectral Parameter in a Boundary Condition, Canadian Mathematical Society Proceedings 8, 171–186 (1987)
H. Steinrück, Singularly perturbed eigenvalue problems, SIAM Journal of Applied Mathematics 47, 1131–1149 (1987)
P. Binding and Q. Ye, Variational principles without definiteness conditions, SIAM Journal of Mathematical Analysis 22, 1575–1583 (1991)
L. Rodman, An Introduction to Operator Polynomials, Birkhäuser-Verlag, Boston, Massachusetts, 1989
G. V. Radzievskii, A problem of completeness of root vectors in the spectral theory of operator-functions, Uspechi of Mathematical Sciences 37, 81–145 (1982) (in Russian)
L. Greenberg and I. Babuska, A continuous analogue of Sturm sequences in the context of Sturm-Liouville equations, SIAM Journal of Numerical Analysis 26, 920–945 (1989)
L. Greenberg, An oscillation method for fourth-order, self-adjoint, two-point boundary value problems with nonlinear eigenvalues, SIAM Journal of Mathematical Analysis 22, 1021–1042 (1991)
A. G. Kostyuchenko and A. A. Shkalikov, Self-adjoint quadratic operator pencils and elliptic problems, Functional Analysis and its Applications 17, 109–128 (1983)
A. G. Kostyuchenko and A. A. Shkalikov, On the theory of self-adjoint quadratic operator polynomials, Vestnik Moscov. University Series 1. Mathematics, Mechanics, No. 6, 1983, pp. 40–51 (in Russian)
G. Chen, S. G. Krantz, D. L. Russell, C. E. Wayne, H. H. West, and M. P. Coleman, Analysis, design and behavior of dissipative joints for coupled beams, SIAM Journal of Applied Mathematics 49, 1665–1693 (1989)
J. B. Conway, A Course in Functional Analysis, Springer-Verlag, New York, New York, 1985
I. Stakgold, Boundary Value Problems of Mathematical Physics, Volume II, Macmillan Co., New York, New York, 1971
H. J. Ahn, Vibrations of a pendulum consisting of a bob suspended from a wire, Quart. Appl. Math. 39, 109–117 (1981)
H. J. Ahn, On random transverse vibrations of a rotating beam with tip mass, Quart. Appl. Math. 39, 97–109 (1981)
L. H. Jones, The transverse vibration of a rotating beam with tip mass, Quart. Appl. Math. 23, 193–203 (1975)
J. A. Burns and E. M. Cliff, An approximation technique for the control and identification of the hybrid system, in Dynamics and Control of Large Flexible Space Structures, Meirovich, Virginia Tech, Blacksburg, VA, 1981, pp. 269–284
F. Riesz and B. St. Nagy, Lectures in Functional Analysis, Ungar, New York, New York, 1955
W. Schmeidler, Linear Operators in Hilbert Space, Academic Press, New York, New York, 1965
R. D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, Journal of Applied Mechanics 18, A31–A38 (1951)
Z. Schuss, Theory and Applications of Stochastic Differential Equations, 1980
P. A. Laura, J. A. Reyes, and R. E. Rossi, Dynamic behavior of a cable-payload system suddenly stopped at one end, Journal of Sound and Vibration 34, 81–95 (1989)
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