Numerical Reconstruction of Spring-Mass System from Two Nondisjoint Spectra

The inverse problems in structures vibration look for determining or estimating the physical properties of a system in vibration (mass density, elastic constants, etc.) from a known dynamic behavior (natural frequencies, electric flux, tension, etc.) (see [1–4]). The model used, which has generated much interest in the literature, as a prototype of structure, is a nonuniform thin rod with one end fixed to a surface (see [2–5]), whose discrete model is a spring-mass system, which consists of n mass m i > 0, associated with the masses of each element of the rod and connected by n springs with rigidity constants


Introduction
The inverse problems in structures vibration look for determining or estimating the physical properties of a system in vibration (mass density, elastic constants, etc.) from a known dynamic behavior (natural frequencies, electric flux, tension, etc.) (see [1][2][3][4]).
The model used, which has generated much interest in the literature, as a prototype of structure, is a nonuniform thin rod with one end fixed to a surface (see [2][3][4][5]), whose discrete model is a spring-mass system, which consists of  mass   > 0, associated with the masses of each element of the rod and connected by  springs with rigidity constants   > 0 corresponding to the rigidity of each one of these elements (Figure 1).
It is known (see [4]) that the matrices  and  can be uniquely reconstructed if the following information is given: the eigenvalues (  )   1 of the original system (, ), the eigenvalues (  ) −1 1 of the auxiliary system (, ), corresponding to the original system whose last mass is fixed (Figure 2), and an additional factor, for example, the total mass of the system   = ∑  =1   .The structural properties of the matrices  and  allow us to reduce the generalized eigenvalue equation (1) to the standard form (see [1][2][3][4]) where the Jacobi matrix  is tridiagonal symmetric positive definite, with the same eigenvalues (  )  1 of the system (, ), Figure 1: Fixed-free spring-mass system.
which are real, positive, and distinct.Therefore, a fundamental step to determine the system (, ) is to reconstruct the matrix .Without loss of generality, we assume that  is of the following form: In [6], stable numerical procedures to reconstruct the Jacobi matrix  are discussed.This reconstruction uses as initial spectral information the eigenvalues (  )  1 of  and the eigenvalues (  ) −1 1 of the matrix , which is obtained by deleting the last row and last column of .A fundamental property in these procedures is the interlacing property (see [1,4,6]) which is a necessary and sufficient condition for the existence of a physically real system and for constructing  as well.In [7,8] the authors generalize the reconstruction of the system (, ) by using the interlaced spectrum corresponding to an auxiliary system that consists in fixing any mass of the system (, ), other than the extreme masses (Figure 3).
Clearly, if the auxiliary system (, ) is the system with its ( + 1)th mass, 1 ≤  ≤  − 2, being fixed, then (, ) is uncoupled in two auxiliary spring-mass systems, (  ,   ) and (  ,   ), with natural frequencies (  )   1 and (  )  1 , respectively, where  =  −  − 1.The structural properties of the matrices   ,   ,   , and   allow us to partition  as where the submatrices reconstruct the matrix  in (6), such that In this problem two cases arise: in the first one, all natural frequencies (  )   1 and (  ) In terms of the  matrix, the meaning is that no eigenvector V () of  has a node in its coordinate V ()  +1 ; that is, V () +1 ̸ = 0,  = 1, . . ., .In this case, the reconstruction is unique.In the second case, one or more natural frequencies (  )   1 and (  )  1 , are identical.The meaning of this situation is that some eigenvector of , let us say V (ℓ) , has a node in V (ℓ)  +1 ; that is, , and a family of isospectral matrices is obtained.
In [7], the authors study the first case; that is, they reconstruct the system (, ), using a modification of the fast orthogonal reduction method, when the auxiliary spectra are separated.In the next section we study the second case, using the same method, and thus the problem is completely solved.This method is less computationally expensive than others in literature [8].

Reconstructing the System from Nondisjoint Spectra
We denote by   (),   (), and   () the characteristic polynomials of the matrices ,   , and   , respectively; that is, We define the vectors corresponding, respectively, to the last and the first row of the matrices of eigenvectors of   and   .We also define the diagonal matrices ,  = −−1 be given, satisfying the interlacing property (7); that is, Then there exists an isospectral family of  ×  matrices (),  ∈ (0, /2), of the form (6) Proof.We suppose that there is a pair   ,   of frequencies such that   =   =   , where  =  + .From the expansion of det(  −   ), throughout its ( + 1)th row, we find that where  \1 () and  \ () are the characteristic polynomials of   and   after we delete its first row and column, and th row and column, respectively.On the other hand, if we denote by  +1 () the characteristic polynomial of the principal submatrix obtained from   by adding ( + 1)th row and column, we have that Thus, (11) is Analogously, (11) can be written as Now, if we denote by  +1 () the characteristic polynomial of the principal submatrix obtained from   by adding a row and column above   , we have Then, From ( 12) we have and from (15) we have Since the polynomials   (),   (), and   () in ( 9) have common factors  −   ≡  −   ≡  −   , (13) and ( 16) are respectively, where Replacing ( 17) and (18) in these last two equations, we obtain and dividing (22) by    (  )   (  ), we get It is known that if   = [ (1)    (2)   ⋅ ⋅ ⋅  ()  ] is the orthogonal matrix of eigenvectors of   , then    (  −   )  =   − Λ  , and we have The left side in ( 24) is while the right side is Comparing the entries in position (, ) in both sides in (24) we find that Taking the limit when  tends to   , we obtain Analogously, we can obtain Then, by replacing (28) and ( 29) in (23), we get For  ∈ (0, /2) we can define Thus, given that (31) allow us to know  () , () and  () ,1 ().
Subsequently, once the vectors   () and   () in ( 10) are known, we can form the ( + 1) × ( + 1) matrices: where the entries   0 and   0 are arbitrary real numbers.Then, we apply the Modified Fast Orthogonal Reduction Algorithm (see [9]) to orthogonally reduce the matrices  +1 () and  +1 () to their tridiagonal form, obtaining in this way the desired matrices   () and   ().To do this, we first permute the arrowhead matrix  +1 () by applying  = [ ].We point out that similar relationships are analyzed by Jessup in [10].
Finally, considering that the diagonal entry  +1 of () can be computed as and the codiagonal entries   () and  +1 () can be computed from (32) and (33), respectively, the matrix () of the form ( 6) is obtained completely, having a common eigenvalue with   () and   ().
If we have more common eigenvalues, we can repeat the previous procedure for each pair of common eigenvalues.That is, if  is the number of identical pairs    ≡    ≡    ,  = 1, . . ., , then we have ) sin 2 ,  = 1, 2, . . ., .

An Optimization Procedure to Find an Objective Jacobi Matrix
In this section we want to find an objective matrix within a family of matrices.First, we observe that the construction procedure depends continuously on the parameter .Then, by means of an optimization procedure, we find an appropriate , so that the procedure reconstructs a matrix with a desired structure.
Theorem 3. Let J be a given symmetric tridiagonal matrix partitioned in the following form: where ( J) = (  )  1 , ( J ) = (  )  1 , and where the matrix () is obtained by using the Modified Fast Orthogonal Reduction process, has a minimum in [0 + , /2 − ].
Example 1.In Table 1, we show the results associated with the reconstructed matrix Ĵ, of the form (44), with  = 11,  = 3, and  = 7 for  = 0.5.The given eigenvalues are shown in the second, third, and fourth column.In the fifth and sixth column we list the diagonal and codiagonal entries of Ĵ.In the last column, we show the relative error    = ‖  − λ ‖ 2 /‖  ‖ 2 with respect to the exact eigenvalues   of the matrix  and the eigenvalues λ of Ĵ.
Example 2. In Table 2, we show the results associated with the reconstructed matrix Ĵ, by considering appropriate orders of Ĵ and for arbitrary values of , listed from the first to fourth  Example 3. In the reconstructed matrix Ĵ, by considering the same orders of Example 2, we add an optimization process of Golden Section Search [11], of parameter , obtaining an optimal , denoted as  opt and listed in the fourth column of Table 3.In the last three columns the relative errors   ,   , and   are shown.The results of our numerical experiments confirm that our method works quite well.

Figure 3 :
Figure 3: Spring-mass system with a fixed interior mass.
}, are related to the systems (  ,   ) and (  ,   ), respectively.As the system (, ) can be reconstructed from the matrix , it is enough to reconstruct  from the sets (  )1 and (  ) −1