An inverse eigenvalue method for frequency isolation in spring-mass systems
The action of external vibrating forces on mechanical structures can cause severe damages when resonance occurs. The removal of natural frequencies of the structure from resonance bands is therefore of great importance. This problem is called frequency isolation problem and is the subject of this pa...
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| Published in | Numerical linear algebra with applications Vol. 9; no. 1; pp. 65 - 79 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Chichester, UK
John Wiley & Sons, Ltd
01.01.2002
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| ISSN | 1070-5325 1099-1506 |
| DOI | 10.1002/nla.255 |
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| Abstract | The action of external vibrating forces on mechanical structures can cause severe damages when resonance occurs. The removal of natural frequencies of the structure from resonance bands is therefore of great importance. This problem is called frequency isolation problem and is the subject of this paper. A new inverse eigenvalue method is proposed and applied to spring–mass systems, which have generated much interest in the literature as prototypes of vibrating structures. The novelty of the method lies in using the zeros of the frequency response function at the last mass as control variables in an optimization problem to minimize the impact of redesign. Numerically accurate algorithms for computing the sensitivity with respect to the control variables are presented, which form the basis of an efficient multidimensional search strategy to solve the frequency isolation problem. Copyright © 2001 by John Wiley & Sons, Ltd. |
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| AbstractList | The action of external vibrating forces on mechanical structures can cause severe damages when resonance occurs. The removal of natural frequencies of the structure from resonance bands is therefore of great importance. This problem is called frequency isolation problem and is the subject of this paper. A new inverse eigenvalue method is proposed and applied to spring–mass systems, which have generated much interest in the literature as prototypes of vibrating structures. The novelty of the method lies in using the zeros of the frequency response function at the last mass as control variables in an optimization problem to minimize the impact of redesign. Numerically accurate algorithms for computing the sensitivity with respect to the control variables are presented, which form the basis of an efficient multidimensional search strategy to solve the frequency isolation problem. Copyright © 2001 by John Wiley & Sons, Ltd. |
| Author | Egaña, Juan C. Santos, Luis C. Kuhl, Nelson M. |
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| References | Friedlander A, Martínez JM, Santos SA. New trust region algorithm for bound constrained minimization. Applied Mathematics and Optimization 1994; 30:235-266. Egaña JC, Santos LC, Kuhl N. A strategy for frequencies isolation in discrete dynamical systems. Proceedings VI Pan American Congress of Applied Mechanics, Rio de Janeiro 1999; 6:243-246. Gladwell GM. Inverse Problems in Vibration. Martinus Nijhoff Publishers: Dordrecht, 1989. Hochstadt H. On some inverse problems in matrix theory. Archiv der Mathematik 1974; 8:435-446. Gladwell GM. Inverse Problems in Vibrations. Applied Mechanics Reviews 1986; 39(7). Hald O. Inverse eigenvalue problems for Jacobi matrices. Linear Algebra and Its Applications 1976; 14:63-85. Golub GH, Van Loan CF. Matrix Computations. The Johns Hopkins University Press: Baltimore, 1989. Boley D, Golub GH. A survey of matrix inverse eigenvalue problem. Inverse Problems 1987; 3:595-622. Datta BN. Numerical linear algebra and applications. Brooks/Cole Publishing Company: Pacific Grove, CA, 1995. Gladwell GM. Inverse Problems in Vibrations-2. Applied Mechanics Reviews 1996; 49(10). Gray LJ, Wilson DG. Construction of a Jacobi matrix from spectral data. Linear Algebra and Its Applications 1976; 14:131-134. Dai H, Lancaster P. Newton's method for a generalized inverse eigenvalue problem. Numerical Linear Algebra with Applications 1997; 4(1):1-27. Nylen P, Uhlig F. Inverse eigenvalue problems: existence of special spring-mass systems. Inverse Problems 1997; 13:1071-1081. de Boor C, Golub GH. The numerically stable reconstruction of a Jacobi matrix from spectral data. Linear Algebra and Its Applications 1978; 21:245-260. Ram YM. Inverse eigenvalue problems for a modified vibrating system. SIAM Journal on Applied Mathematics 1993; 53:1762-1775. Joseph KT. Inverse eigenvalue problem in structural design. American Institute of Aeronautics and Astronautics Journal 1992; 30:2890-2896. Golub GH, Boley D. Inverse eigenvalue problems for band matrices. Numerical Analysis 1977; (66):23-31. Inman D. Vibration with Control, Measurement and Stability. Pretince-Hall: Englewood Cliffs, NJ, 1989. Lanczos C. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. Journal Research NBS Section B 1950; 45. 1987; 3 1976; 14 1978; 21 1997; 13 1993; 53 1986; 39 1995 1950; 45. 1996; 49 1999; 6 1974; 8 1997; 4 1994; 30 1989 1992; 30 1977 1999 Hochstadt H (e_1_2_1_10_2) 1974; 8 Gladwell GM (e_1_2_1_6_2) 1989 Gladwell GM (e_1_2_1_5_2) 1996; 49 Datta BN (e_1_2_1_2_2) 1995 Golub GH (e_1_2_1_12_2) 1989 Golub GH (e_1_2_1_19_2) 1977 e_1_2_1_7_2 Inman D (e_1_2_1_3_2) 1989 e_1_2_1_4_2 Lanczos C (e_1_2_1_17_2) 1950; 45 e_1_2_1_11_2 e_1_2_1_22_2 Golub GH (e_1_2_1_18_2) 1977 e_1_2_1_23_2 e_1_2_1_20_2 Egaña JC (e_1_2_1_21_2) 1999; 6 e_1_2_1_15_2 e_1_2_1_16_2 e_1_2_1_13_2 e_1_2_1_14_2 e_1_2_1_8_2 e_1_2_1_9_2 |
| References_xml | – reference: Egaña JC, Santos LC, Kuhl N. A strategy for frequencies isolation in discrete dynamical systems. Proceedings VI Pan American Congress of Applied Mechanics, Rio de Janeiro 1999; 6:243-246. – reference: Gladwell GM. Inverse Problems in Vibrations-2. Applied Mechanics Reviews 1996; 49(10). – reference: Golub GH, Boley D. Inverse eigenvalue problems for band matrices. Numerical Analysis 1977; (66):23-31. – reference: Dai H, Lancaster P. Newton's method for a generalized inverse eigenvalue problem. Numerical Linear Algebra with Applications 1997; 4(1):1-27. – reference: Gladwell GM. Inverse Problems in Vibrations. Applied Mechanics Reviews 1986; 39(7). – reference: Hochstadt H. On some inverse problems in matrix theory. Archiv der Mathematik 1974; 8:435-446. – reference: Golub GH, Van Loan CF. Matrix Computations. The Johns Hopkins University Press: Baltimore, 1989. – reference: Friedlander A, Martínez JM, Santos SA. New trust region algorithm for bound constrained minimization. Applied Mathematics and Optimization 1994; 30:235-266. – reference: Ram YM. Inverse eigenvalue problems for a modified vibrating system. SIAM Journal on Applied Mathematics 1993; 53:1762-1775. – reference: Lanczos C. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. Journal Research NBS Section B 1950; 45. – reference: Joseph KT. Inverse eigenvalue problem in structural design. American Institute of Aeronautics and Astronautics Journal 1992; 30:2890-2896. – reference: Hald O. Inverse eigenvalue problems for Jacobi matrices. Linear Algebra and Its Applications 1976; 14:63-85. – reference: Gray LJ, Wilson DG. Construction of a Jacobi matrix from spectral data. Linear Algebra and Its Applications 1976; 14:131-134. – reference: Datta BN. Numerical linear algebra and applications. Brooks/Cole Publishing Company: Pacific Grove, CA, 1995. – reference: Nylen P, Uhlig F. Inverse eigenvalue problems: existence of special spring-mass systems. Inverse Problems 1997; 13:1071-1081. – reference: Gladwell GM. Inverse Problems in Vibration. Martinus Nijhoff Publishers: Dordrecht, 1989. – reference: Boley D, Golub GH. A survey of matrix inverse eigenvalue problem. Inverse Problems 1987; 3:595-622. – reference: Inman D. Vibration with Control, Measurement and Stability. Pretince-Hall: Englewood Cliffs, NJ, 1989. – reference: de Boor C, Golub GH. The numerically stable reconstruction of a Jacobi matrix from spectral data. Linear Algebra and Its Applications 1978; 21:245-260. – volume: 30 start-page: 2890 year: 1992 end-page: 2896 article-title: Inverse eigenvalue problem in structural design publication-title: American Institute of Aeronautics and Astronautics Journal – volume: 30 start-page: 235 year: 1994 end-page: 266 article-title: New trust region algorithm for bound constrained minimization publication-title: Applied Mathematics and Optimization – volume: 14 start-page: 131 year: 1976 end-page: 134 article-title: Construction of a Jacobi matrix from spectral data publication-title: Linear Algebra and Its Applications – volume: 21 start-page: 245 year: 1978 end-page: 260 article-title: The numerically stable reconstruction of a Jacobi matrix from spectral data publication-title: Linear Algebra and Its Applications – volume: 45. year: 1950 article-title: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators publication-title: Journal Research NBS Section B – volume: 53 start-page: 1762 year: 1993 end-page: 1775 article-title: Inverse eigenvalue problems for a modified vibrating system publication-title: SIAM Journal on Applied Mathematics – year: 1989 – volume: 39 issue: 7 year: 1986 article-title: Inverse Problems in Vibrations publication-title: Applied Mechanics Reviews – volume: 14 start-page: 63 year: 1976 end-page: 85 article-title: Inverse eigenvalue problems for Jacobi matrices publication-title: Linear Algebra and Its Applications – year: 1995 – start-page: 23 issue: 66 year: 1977 end-page: 31 article-title: Inverse eigenvalue problems for band matrices publication-title: Numerical Analysis – volume: 8 start-page: 435 year: 1974 end-page: 446 article-title: On some inverse problems in matrix theory publication-title: Archiv der Mathematik – volume: 3 start-page: 595 year: 1987 end-page: 622 article-title: A survey of matrix inverse eigenvalue problem publication-title: Inverse Problems – year: 1977 – volume: 6 start-page: 243 year: 1999 end-page: 246 article-title: A strategy for frequencies isolation in discrete dynamical systems publication-title: Proceedings VI Pan American Congress of Applied Mechanics – volume: 4 start-page: 1 issue: 1 year: 1997 end-page: 27 article-title: Newton's method for a generalized inverse eigenvalue problem publication-title: Numerical Linear Algebra with Applications – volume: 49 issue: 10 year: 1996 article-title: Inverse Problems in Vibrations–2 publication-title: Applied Mechanics Reviews – volume: 13 start-page: 1071 year: 1997 end-page: 1081 article-title: Inverse eigenvalue problems: existence of special spring–mass systems publication-title: Inverse Problems – year: 1999 – ident: e_1_2_1_15_2 doi: 10.1137/0153082 – ident: e_1_2_1_7_2 – ident: e_1_2_1_14_2 doi: 10.1088/0266-5611/13/4/012 – ident: e_1_2_1_4_2 doi: 10.2514/3.11634 – volume-title: Vibration with Control, Measurement and Stability year: 1989 ident: e_1_2_1_3_2 – volume-title: Inverse Problems in Vibration year: 1989 ident: e_1_2_1_6_2 – volume: 49 issue: 10 year: 1996 ident: e_1_2_1_5_2 article-title: Inverse Problems in Vibrations–2 publication-title: Applied Mechanics Reviews – ident: e_1_2_1_16_2 doi: 10.1016/0024-3795(78)90086-1 – ident: e_1_2_1_9_2 doi: 10.1016/0024-3795(76)90064-1 – volume-title: Lectures Notes in Mathematics, Numerical Analysis year: 1977 ident: e_1_2_1_18_2 – start-page: 23 issue: 66 year: 1977 ident: e_1_2_1_19_2 article-title: Inverse eigenvalue problems for band matrices publication-title: Numerical Analysis – ident: e_1_2_1_22_2 doi: 10.1007/BF01183013 – ident: e_1_2_1_23_2 – volume-title: Matrix Computations year: 1989 ident: e_1_2_1_12_2 – volume: 45 year: 1950 ident: e_1_2_1_17_2 article-title: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators publication-title: Journal Research NBS Section B – volume-title: Numerical linear algebra and applications year: 1995 ident: e_1_2_1_2_2 – ident: e_1_2_1_11_2 doi: 10.1088/0266-5611/3/4/010 – ident: e_1_2_1_20_2 doi: 10.1002/(SICI)1099-1506(199701/02)4:1<1::AID-NLA95>3.0.CO;2-D – ident: e_1_2_1_8_2 doi: 10.1016/0024-3795(76)90020-3 – ident: e_1_2_1_13_2 doi: 10.1115/1.3149517 – volume: 6 start-page: 243 year: 1999 ident: e_1_2_1_21_2 article-title: A strategy for frequencies isolation in discrete dynamical systems publication-title: Proceedings VI Pan American Congress of Applied Mechanics – volume: 8 start-page: 435 year: 1974 ident: e_1_2_1_10_2 article-title: On some inverse problems in matrix theory publication-title: Archiv der Mathematik |
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| SubjectTerms | inverse eigenvalue problem Jacobi matrix spring-mass system δ-Lanczos method |
| Title | An inverse eigenvalue method for frequency isolation in spring-mass systems |
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