Abstract
In this paper, we propose an algorithm for constructing a pentadiagonal matrix with given prescribed three spectra. Sufficient conditions for solvability of the problem are given. We generate an algorithmic procedure to construct the solution matrices and we given a numerical example illustrating the construction algorithm.
Introduction
The free undamped infinitesimal vibrations, of frequency , of a thin straight beam of length are governed by the Euler–Bernoulli equation(1.1) (1.1) Here is Young’s modulus, is the density, is the cross sectional area at section and is the second moment of this area about the axis through the centroid at right angles to the plane of vibration. The problem can be discretized by assuming that the interval is divided into intervals with length , and that , , and has the values and on the subintervals , respectively. Figure shows a simple discrete model for the transversely vibrating Euler–Bernoulli beam. The parameters of discrete model can be defined as(1.2) (1.2) Figure shows the configuration around the mass and the discrete system is obtained. For small displacement, the rotations areThe moment needed to produce a relative rotation of the two rigid rods on either side of isEquilibrium of the rod linking and yields the shearing forcesBalance equation for mass isHere and denote external shearing forces and bending moments, respectively, applied to the ends. Suppose that the left hand end is clamped so thatThe governing equations in the matrix form can be written as
(1.3) (1.3) (1.4) (1.4) (1.5) (1.5) (1.6) (1.6) where , andEquations (Equation1.3(1.3) (1.3) ), (Equation1.4(1.4) (1.4) ), (Equation1.5(1.5) (1.5) ), (Equation1.6(1.6) (1.6) ) lead to(1.7) (1.7) where(1.8) (1.8) If we seek the solution of the formthenso that for free vibration Equation (Equation1.7(1.7) (1.7) ) leads to(1.9) (1.9) where is a pentadiagonal matrix and . We let and then we obtain the following eigenvalue problem(1.10) (1.10) where(1.11) (1.11) is a symmetric pentadiagonal matrix as follows(1.12) (1.12) where(1.13) (1.13) See [Citation1–Citation4]. Thus, the inverse problem for a discrete beam corresponds to an inverse problem for a symmetric pentadiagonal matrix in which the the elements of the first off-diagonals are negative, all other elements being positive. In this paper we consider the problem of constructing a symmetric pentadiagonal matrix of the form (Equation1.12(1.12) (1.12) ) and by procedure in Appendix 2 we construct masses , stiffnesses and lengths of a discrete beam. We propose an algorithm for constructing a symmetric pentadiagonal matrix of the form (Equation1.12(1.12) (1.12) ) with given three spectra. The classical problem of constructing pentadiagonal matrix by spectra was solved by Boley and Golub,[Citation3, Citation5] Singh [Citation4, Citation6] and Chu et al. [Citation7]. The set of eigenvalues of matrix are called the spectrum of and is denoted by . The matrix obtained by knocking the th row and column of is denoted by and the matrix obtained from by knocking off the th row and th column and th row and th column is denoted by , respectively. and denote the spectra of matrix and respectively. Boley and Golub [Citation5] proposed an inverse eigenvalue problem for a general symmetric matrix with bands, the case , is a pentadiagonal matrix. They construct, a pentadiagonal matrix with given spectral data , and . Singh [Citation4] derived a discrete model of beam equation that leads to a pentadiagonal matrix and solved direct and inverse problem of the beam equation. Chu et al. [Citation7] proposed a numerical method for constructing a symmetric pentadiagonal Toeplitz matrix from three largest eigenvalues. Caddemi and Calio [Citation8] presented the exact closed-form solution for the vibration modes and the eigen-value equation of the Euler–Bernoulli beam-column in the presence of an arbitrary number of concentrated open cracks, and they presented the exact closed-form expressions for the vibration modes of the Euler–Bernoulli beam in the presence of multiple concentrated cracks.[Citation9] Gladwell [Citation10] derive a closed form procedure to construct the in-line system of masses and springs with a minimal mass for given overall stiffness from the one spectrum, also he studied the problem of constructing a discrete model of a contilever beam in flexural vibration having a specified two spectrum, and formulate the problem of finding a minimal mass solution for the given length and stiffness, and obtain explicit solutions in simple cases. Using the idea of discrete model [Citation10] Ghanbari et al [Citation11] construct a Rod using one spectrum and minimal mass condition. In this paper, we construct a pentadiagonal matrix with given spectral data , and , for some . Physically are the eigenvalues corresponding to the end conditions such as fixed-free, are the eigenvalues of discrete system which the mass th is fixed and are the eigenvalues of discrete system which the masses th and th are fixed. Clearly the eigenvalue must interlace; and for simplicity we assume that the interlacing is strict.(1.14) (1.14)Statement of results
The matrices and are of the form(2.1) (2.1) where and are pentadiagonal matrix of order and , respectively that , and . Express the eigenvalue problem for in terms of the normalized eigenvectors and of and , respectively. Thus if is eigenvector of then we can write as followsThe eigenvalue problem becomesPremultiplying this equation byand using orthogonality of matrices and and we obtainwhere and are eigenvalues of matrices and , in respectively and(2.2) (2.2) Thusso that Theorem 2.1. If all and are distinct then . Proof
Suppose that there exists an such that is eigenvalue of and , th component of eigenvector corresponding to , is zero; thusand(2.3) (2.3) Since is eigenvector of corresponding to , thus and from (2.3) is eigenvalue of which is a contradiction, since eigenvalues of matrices and are distinct.
By Theorem 2.1 we havewhere and are characteristic polynomials of the matrices and , respectively. This implies that(2.4) (2.4) where Interlace condition (Equation1.14(1.14) (1.14) ) implies that and in (Equation2.4(2.4) (2.4) ) are positive. Now and may be computed from(2.5) (2.5) where . From trace formula we can compute and as follows(2.6) (2.6) Now we consider eigenvalue problem for matrix . Thus if is eigenvector of then we can write as follows(2.7) (2.7) The eigenvalue problem becomes as followswhere(2.8) (2.8) (2.9) (2.9) Thusso that for we have(2.10) (2.10) Theorem 2.2. If all and are distinct then . Proof
The proof is similar to Theorem 2.1. If we denote the coefficient matrix of the system (Equation2.10(2.10) (2.10) ) by , then by Theorem 2.2, for . Thus(2.11) (2.11) where denote characteristic polynomial of matrix . Thus(2.12) (2.12) The system of nonlinear equations (Equation2.12(2.12) (2.12) ) can be solved by Newton’s method [Citation12] for , and . With known and Equation (Equation2.9(2.9) (2.9) ) implies that(2.13) (2.13) where . Since vectors and are orthonormal, multiplying the Equation (Equation2.8(2.8) (2.8) ) by we find(2.14) (2.14) where and denote Euclidean norm of a vector. Also vectors and are orthonormal, multiplying the Equation (Equation2.2(2.2) (2.2) ) by we obtain(2.15) (2.15) where . In this procedure, a Gram-Schmidt process for finding orthonormal vectors and elements is necessary. Having and , matrices and in can be computed by using the block Lanczos algorithm (see the Appendix 1). The following algorithm lists the steps for constructing the matrix :
Construction Algorithm:
Compute and from (Equation2.6(2.6) (2.6) ).
Compute and from (Equation2.5(2.5) (2.5) ).
Compute from (Equation2.13(2.13) (2.13) ) and from (Equation2.14(2.14) (2.14) ).
Compute from (Equation2.15(2.15) (2.15) ).
Construct matrices and from Lanczos algorithm (see Appendix 1).
Numerical examples
Here we illustrate the construction procedure by an example. We consider the Euler–Bernoulli beam equation as followsSuppose that then the prescribed three spectra of discrete beam are as followsThese eigenvalues satisfy the conditions of Theorem 2.1, Theorem 2.2 and condition (Equation1.14(1.14) (1.14) ). Step 1 of algorithm givesfrom (Equation2.4(2.4) (2.4) ) we obtain and by step 2 we findSolving the system (Equation2.12(2.12) (2.12) ) by using Newton’s method, we findand step 3 givesUsing step 4, we findUsing the block Lanczos algorithm (Appendix 1), we findThus we constructed a symmetric pentadiagonal matrix of the form (Equation1.12(1.12) (1.12) ) as followsComputing the spectrum of the solution by MATLAB confirms the spectral data of the matrix . With this and by Appendix 2 we find masses , stiffnesses and lengths of discrete beam as followsThe pentadiagonal matrices in (Equation1.8(1.8) (1.8) ) and in (Equation1.12(1.12) (1.12) ) are independent of . But for different values of we construct different discrete beam which have the same spectrum.
For we obtainThis pentadiagonal matrix have the same spectrum but for this we cannot construct a discrete beam since by (Equation1.13(1.13) (1.13) ) all must be positive and all must be negative.
Conclusion
In this paper, we formulated and solved the inverse problem of constructing a pentadiagonal matrix using three given spectra with interlacing property (Equation1.14(1.14) (1.14) ). This problem will have multiple solutions because in the constructing process we have different choices for and vectors and . Also for pentadiagonal matrix we construct masses , stiffnesses and lengths of corresponding discrete beam. Numerical example shows the accuracy of this method and details of the algorithm confirms the simplicity of the construction algorithm.
References
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- Barcilon V. Inverse problems for a vibrating beam. J. Appl. Math. Phys. 1976;27:346–358.
- Gladwell GML. Inverse problems in vibration. New York (NY): Kluwer Academic; 2004.
- Singh KV. The transcendental eigenvalue problem and its application in system identification [Phd thesis]. Baton Rouge (LA): Department of Mechnical Engineering, Louisiana State University; 2003.
- Boley D, Golub GH. A survey of matrix inverse eigenvalue problems. Inverse Probl. 1987;3:595–622.
- Singh KV. Transcendental eigenvalue problem and its application. AIAA Journal. 2002;40:1402–1407.
- Chu MT, Diele F, Ragni S. On the inverse problem of constructing symmetric pentadiagonal Toeplitz matrices from their three largest eigenvalues. Inverse Probl. 2005;21:1879–1894.
- Caddemi S, Calia I. The influence of the axial force on the vibration of the Euler–Bernoulli beam with an arbitrary number of cracks. Arch. Appl. Mech. 2012;82:827–839.
- Caddemi S, Calio I. Exact closed-form solution for the vibration modes of the Euler–Bernoulli beam with multiple open cracks. J. Sound Vib. 2009;327:473–489.
- Gladwell GML. Minimal mass solutions to inverse eigenvalue problems. Inverse Probl. 2006;22:539–551.
- Ghanbari K, Mirzaei H, Gladwell GML. Reconstruction of a rod using one spectrum and minimal mass condition. Inverse Probl. Sci. Eng. 2013. doi:10.1080/17415977.2013.782543.
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The block Lanczos algorithm [Citation3]
Suppose that and two vectors and are given. Let , this algorithm constructs a pentadiagonal matrix of the formand orthogonal matrix such that(5.1) (5.1) Here for some integer , are symmetric matrices of order and are upper triangular matrix of order 2. The first block of Equation (EquationAppendix 1.1(1.1) (1.1) ) gives(5.2) (5.2) since matrix is orthogonal, premultiplying (EquationAppendix 1.2(1.2) (1.2) ) by we obtain(5.3) (5.3) Equation (EquationAppendix 1.2(1.2) (1.2) ) can be written as(5.4) (5.4) Suppose andThe first column of Equation (EquationAppendix 1.4(1.4) (1.4) ) givesand the second column of (EquationAppendix 1.4(1.4) (1.4) ) giveswhich leads toso thatIn this process, a Gram–Schmidt process for finding orthonormal vectors and matrix is necessary. By repeating the above process we can find matrices and . This algorithm can be applied for the similarly. We have used this case in our numerical example.
Reconstruction of masses , stiffnesses and lengths [Citation3]
Suppose that we have computedwhere and is the pentadiagonal matrix in (Equation1.8(1.8) (1.8) ). We now untangle to give masses , stiffnesses and lengths .
First, we apply external static forces to masses 1 and 2, and deform the system as shown in Figure . For this configuration the static equation is
i.e.orConsider this equation. We know and the last two components . But is pentadiagonal so that, knowing , we can compute , and find and hence .Having found the masses , we find the lengths. We apply a single force at , and deform the system as shown in Figure . Now the equation
yieldsThis means that if we invert the equationwe will findthis yields the as followsThe last step is to find the . By Equations (Equation1.8(1.8) (1.8) ) and (Equation1.11(1.11) (1.11) ) we findwhich gives and the stiffnesses . From (Equation1.8(1.8) (1.8) ) and (Equation1.11(1.11) (1.11) ) we note that the matrices are independent of constant .