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Journal of Sound and Vibration

Journal of Sound and Vibration 333 (2014) 99–113

0022-46http://d

n CorrE-m

journal homepage: www.elsevier.com/locate/jsvi

Inverse vibration problem for un-damped 3-dimensionalmulti-story shear building models

Kiarash M. Dolatshahi a, Fayaz R. Rofooei b,n

a Department of Civil, Structural and Environmental Engineering, University at Buffalo, The State University of New York,Buffalo, NY 14260, USAb Civil Engineering Department, Sharif University of Technology, Tehran, Iran

a r t i c l e i n f o

Article history:Received 29 March 2013Received in revised form25 August 2013Accepted 30 August 2013

Handling Editor: I. Trendafilova

However, in case of three dimensional (3-D) structural models, those methods are no

Available online 3 October 2013

0X/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.jsv.2013.08.045

esponding author. Tel.: þ98 21 66164233; fail addresses: km256@buffalo.edu (K.M. Dola

a b s t r a c t

Various researchers have contributed to the identification of the mass and stiffnessmatrices of two dimensional (2-D) shear building structural models for a given set ofvibratory frequencies. The suggested methods are based on the specific characteristics ofthe Jacobi matrices, i.e., symmetric, tri-diagonal and semi-positive definite matrices.

longer applicable, since their stiffness matrices are not tri-diagonal. In this paper theinverse problem for a special class of vibratory structural systems, i.e., 3-D shear buildingmodels, is investigated. A practical algorithm is proposed for solving the inverseeigenvalue problem for un-damped, 3-D shear buildings. The problem is addressed intwo steps. First, using the target frequencies, a so-called normalized eigenvector matrix,which is a banded matrix containing the information related to the frequencies and modeshapes of the target structural system, is determined. In this regard, similar to the solutionof inverse problem for 2-D shear building structural models in which an auxiliarystructure is constructed by adding constraints (or springs) to the original system, threeauxiliary structures are proposed to solve the problem for 3-D cases. In the second step,the normalized eigenvector matrix is utilized to obtain the normalized stiffness matrix; inturn, this matrix is decomposed into the stiffness and mass matrices of the system. Finally,a numerical example is presented to demonstrate the efficiency of the proposed algorithmin determining the mass and stiffness matrices of a 3-D structural model for a given set oftarget vibrational frequencies.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The inverse eigenvalue problem (IEP) has attracted attention of many mathematicians and researchers over time. Theinverse vibration problem is a particular case of the inverse eigenvalue problem that deals with the dynamic characteristicsof structures. For the distributed parameter systems such as beams or plates, partial differential equations govern thedynamic behavior. However, for the lumped mass structural models, the structural properties are defined in terms of mass,stiffness and damping matrices. Depending on the type of these matrices, different methods have been introduced forsolving inverse vibration problems [1,2].

In case of a 2-D multi-story shear building, since each story needs to be described by just one degree of freedom (itsrelative displacement), the stiffness matrix becomes a symmetric, tri-diagonal positive definite matrix which by definition is

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ax: þ98 21 66014828.tshahi), rofooei@sharif.edu (F.R. Rofooei).

Fig. 1. Mass-spring system [2].

K.M. Dolatshahi, F.R. Rofooei / Journal of Sound and Vibration 333 (2014) 99–113100

called a Jacobi matrix. This is the simplest mathematical model of a building that could be represented by a mass-springsystem shown in Fig. 1. In the characteristic equation of a lumped mass, 2-D shear building model, i.e., ð½K��λ½M�Þx¼ 0, orð½M��1=2½K�½M��1=2�λ½I�Þx¼ 0; the ½M��1=2½K�½M��1=2 is by definition a Jacobi matrix. This matrix could be used to identifythe mass and stiffness matrices of the dynamic system once its target eigenvalues, λi's (the square of the vibrationalfrequencies of the system, ω2

i 's) are known in advance [1].To solve the inverse eigenvalue problem of Jacobi matrices, an extra set of eigenvalues is used to recover the mass and

stiffness matrices of the original system [3–6]. The extra set of eigenvalues can be determined using auxiliary structures thatare constructed by adding extra masses or springs to the original system in different ways, e.g., adding a spring to the lastmass of the structural system [1].

The inverse vibration problem of 2-D structures has been handled in two steps. First, the normalized eigenvector matrixand subsequently the normalized stiffness matrix (with respect to the mass matrix) are computed. This normalized stiffnessmatrix contains the information of both mass and stiffness matrices of the original system. In the next step, the mass andstiffness matrices are extracted from the normalized stiffness matrix. The first step is a general mathematical problem, andhas mostly been considered by mathematicians. However, the second step is more interesting for structural engineers.

As was mentioned before, the main purpose of solving the inverse vibration problem is to identify the mass and stiffnessmatrices of a dynamic system, knowing its eigenvalues (frequencies squared) or eigenvectors (mode shapes). This problemwas first addressed by Gladwell [7,8] for a continuous beam in which he discretized the beam into a number of masses andrigid links that were connected to each other using rotational springs. Later on, Gladwell generalized the problem for finite-element models with tri-diagonal stiffness matrix and diagonal mass matrix [9]. The hybrid inverse problemwas studied byTakewaki [10–14], in which the objective is to find the system properties (stiffness and mass matrices), having some dataabout its spectral characteristics (eigenvalues or eigenvectors). With regard to the seismic design, some researchers havetried to determine the stiffness distribution of the structural system to satisfy specific drift limits [15].

Boley and Golub [16] generalized the mathematical solution of inverse eigenvalue problem of a Jacobi matrix to matriceswith any bandwidth. Using the block Lanczos method, they developed a procedure for calculating the components of anybanded matrix. In the case of tri-diagonal matrices, by having two sets of eigenvalues, it can be proved that the inverseproblem always has a unique solution [1]. However, Boley and Golub [16] have shown that, contrary to the tri-diagonalmatrices, this conclusion cannot be extended to the banded matrices with larger bandwidths. For example, it is not possibleto find a unique matrix with a bandwidth of five by exclusively using three sets of eigenvalues.

A number of researchers have tried to solve the inverse vibration problem for damped dynamic systems [17–21]. Ramand Elhay [19] used the early steps of Halt [5] to solve the inverse vibration problem for a 2-D damped system withtri-diagonal stiffness matrix considering two sets of eigenvalues. The problem in this case does not have a unique solution.Later on, Gladwell and Foltete [20,21] extended their work to include the damped structural systems. They tried todetermine the system parameters by using a given set of complex eigenvalue data.

Huanget et al. [22] and Wang [23] worked on physical parameter reconstruction of fixed–fixed mass-spring systemsusing their characteristic data (frequencies and mode shapes). Dong et al. [24] investigated the parameter reconstruction ofvibratory systems from their partial eigen-information.

The inverse vibration problem can also be considered as an important part of system identification problem, in which theresponse of a structural system under a known excitation is used to identify its dynamic characteristics [25–28]. In this case,utilizing inverse vibration methods can be beneficial in reconstructing the mass and stiffness matrices of the structuralsystem.

Not much literature is available on the inverse vibration problem for 3-D structural models. In this work, a new approachfor solving the inverse vibration problem for 3-D lumped mass and un-damped shear building structures is presented. First,the typical stiffness and mass matrices of 3-D structural models are presented. Then, a mathematical procedure is proposedto reconstruct the normalized eigenvector matrix from the target vibratory frequencies considered for the structural system,assuming the last three rows of the related normalized eigenvector matrix are known. These rows are determined using thethree proposed auxiliary structures. Finally, the calculated normalized stiffness matrix is decomposed into the stiffness andmass matrices of the dynamic system. A numerical example is used to demonstrate the applicability of the proposedprocedure in determining the structural characteristics of 3-D building models.

2. The stiffness and mass matrices of a 3-D shear building model

As was mentioned before, the current work is limited to lumped mass shear buildings where, in contrast to flexuralbuildings, axial deformations of columns are neglected. Moreover, it is assumed that the floors behave as rigid diaphragms[26,27,29–32]. For 2-D shear buildings, the stiffness matrix is similar to that of the mass-spring system shown in Fig. 1.

K.M. Dolatshahi, F.R. Rofooei / Journal of Sound and Vibration 333 (2014) 99–113 101

However, for 3-D building models, it is not common to represent the problem with a series of mass-spring system. That isdue to the existence of eccentricity caused by the unsymmetrical placement of lateral load-carrying elements, or non-uniform distribution of floor masses.

In a typical 3-D shear building model, assuming the floors as rigid diaphragms, only three degrees of freedom (dof) arerequired to characterize the displacement of each floor. Depending on the numbering of the degrees of freedom, as shown inFigs. 2a and b, different stiffness matrices can be developed with various bandwidths. Fig. 2a shows a common type of dofnumbering that is easy to use while leading to a sparse stiffness matrix. Eq. (1) presents the stiffness matrix for a typicalthree-story building model using the numbering approach shown in Fig. 2a [29,30],

(1)

In Eq. (1), kx, ky, kθ, ex and ey represent the system's lateral shear stiffnesses along x and y directions, torsional stiffnessand the coordinates of the center of rigidity of floors, respectively. In the considered lateral-torsional coupled structuralmodel, the center of coordinates system is located on the mass center of each floor, which is presumably its centroid.

As shown in Eq. (1), the stiffness matrix can be partitioned into 3�3 sub-matrices. The diagonal sub-matrices provideinformation about the x, y and θ components of the stiffness matrix of the system, whereas the off-diagonal sub-matricesindicate the lateral-torsional coupling of the system. The main difficulty of this numbering approach is that the stiffnessmatrix bandwidth increases by growing the number of stories.

Eq. (2) shows the stiffness matrix of the system according to an alternative dof numbering method shown in Fig. 2b[29,30]. Evidently, this approach leads to a stiffness matrix with a smaller bandwidth

(2)

The obtained stiffness matrix can also be divided into a number of (3�3) sub-matrices, in which each sub-matrixincludes the stiffness properties of one or two stories. The advantage of this numbering approach, making it appropriatefor the current study, is that the maximum number of sub-matrices in each row or column remains constant andequal to three for any number of stories. The bandwidth of a one story building in this case is three and increases to 11 forbuilding models with three stories or more. Hence, the latter approach is considered in this paper for assembling thestiffness matrix.

By ignoring the torsional stiffness of individual columns, the overall torsional stiffness of any typical floor, i.e., kθ, can beconsidered as a function of the lateral stiffness of its load carrying elements, and the eccentricities along the x and y

Fig. 2. Two different dofs numbering methods for a typical three story building model.

K.M. Dolatshahi, F.R. Rofooei / Journal of Sound and Vibration 333 (2014) 99–113102

directions. However, in this study the torsional stiffness of the floors is considered as an independent parameter togeneralize the proposed approach.

The stiffness matrix of a one story 3-D building is a tri-diagonal matrix (Jacobi matrix) and is similar to the stiffnessmatrix of 2-D multi-story structures. The inverse problem for this case has been thoroughly examined by Gladwell [1].As was mentioned before, the bandwidth of the stiffness matrix increases to 11 for 3-D shear buildings with three stories ormore. Since the problem formulation and the obtained results for the three story model are similar to those with morenumber of stories, only the three story building models are considered in this study.

The mass matrix of a 3-D lumped mass shear building model would be a diagonal matrix with two unknowns for eachfloor, i.e., equal mass components along the x and y translational dofs and the mass moment of inertia for the floor'srotational dof:

M9�9 ¼

m1 0 0 0 0 0 0 0 00 m1 0 0 0 0 0 0 00 0 m3 0 0 0 0 0 00 0 0 m4 0 0 0 0 00 0 0 0 m4 0 0 0 00 0 0 0 0 m6 0 0 00 0 0 0 0 0 m7 0 00 0 0 0 0 0 0 m7 00 0 0 0 0 0 0 0 m9

266666666666666664

377777777777777775

(3)

The mass matrix shown in Eq. (3) can be easily divided into a number of (3�3) diagonal sub-matrices, which contain themass information of each story.

3. Problem formulation

A new procedure is introduced here to identify the stiffness and mass matrices of the 3-D shear buildings with threestories or more. Fig. 3 shows the flowchart indicating different steps of the proposed procedure for calculating the elementsof the mass and stiffness matrices. The solution starts by considering the characteristic equation of the original 3-D, N-storyshear building structural system [27,28],

ð½K��λi½M�Þxi ¼ 0 (4)

where K and M are 3N�3N matrices, respectively. The parameter λi represents the ith eigenvalue (the square of the ithnatural frequency), whereas xi is the ith eigenvector (mode shape) of the original system such that xTi Mxi ¼ 1:0. Bydecomposing the mass matrix into [M]¼[L][L]T where

½L�3N�3N ¼ ½L�T3N�3N ¼ ½M�1=2 ¼ diag½m1=21 ; m1=2

1 ;m1=23 ;m1=2

4 ; m1=24 ;m1=2

6 ; …;m1=23N �3N�3N (5)

and using the following linear transformation:

fug3N�1 ¼ ½L�fxg3N�1 ) fxg ¼ ½L��1fug (6)

Fig. 3. Proposed procedure for inverse vibration problem for 3-D shear buildings.

K.M. Dolatshahi, F.R. Rofooei / Journal of Sound and Vibration 333 (2014) 99–113 103

The characteristic equation given by Eq. (4) changes into

½B�fug ¼ λfug; fug ¼ ½M�1=2fxg (7)

in which

½B�3N�3N ¼ ½L�½M��1½K�½L��1 ¼ ½M��1=2½K�½M��1=2 (8)

Subsequently [1],

ð½B��λ½I�Þfug ¼ 0 (9)

where [B] is the symmetric normalized stiffness matrix, λ is the eigenvalue and {u} is the normalized eigenvector of thestructural model. For each normalized eigenvector one can write

½B�fugi ¼ λifugi (10)

Considering the normalized eigenvectors matrix as [U]¼[{u}1, {u}2,…,{u}N], it can be easily shown that these eigenvectorsare orthonormal, i.e.,

½U�T ½U� ¼ ½U�½U�T ¼ ½I�; ½U�T ¼ ½U��1 (11)

Generalization of Eq. (10) for the whole system gives

½B�½U� ¼ ½U�½Λ� (12)

in which [Λ]3N�3N is the diagonal matrix of the eigenvalues (λ). Assuming [U]T¼[Z], Eq. (11) could be written as

½Z�½Z�T ¼ ½Z�T ½Z� ¼ I (13)

Multiplying Eq. (12) by [U]T from both sides leads to

½U�T ½B�½U�� �½U�T ¼ ½U�T ½U�½Λ�� �½U�T ) ½Z�½B� ¼ ½Λ�½Z� (14)

The transposed normalized eigenvector matrix (Z) can be decomposed into a number of Zi3N�3 sub-matrices. Knowing thatthe matrix [B] can also be partitioned into 3�3 sub-matrices, the solution can be written as follows:

½Z1 Z2…Zn�

A1 B1 0 0 0 0B1T A2 B2 0 0 00 B2T : : 0 00 0 : : : 00 0 0 : : Bðn�1Þ0 0 0 0 Bðn�1ÞT An

26666666664

37777777775¼ ½Λ�½Z1 Z2…Zn� (15)

In Eq. (15), [Ai] and [Bi] are 3�3 matrices. For a three story building model it leads to

½Z1 Z2 Z3�9�9

A1 B1 0B1T A2 B20 B2T A3

264

3759�9

¼ ½Λ�9�9½Z1 Z2 Z3�9�9 (16)

Following Eq. (8), matrix B for three story buildings becomes

(17)

K.M. Dolatshahi, F.R. Rofooei / Journal of Sound and Vibration 333 (2014) 99–113104

Using the sub-matrices [ai](3�3), and [mi](3�3) as the stiffness and mass matrices of the ith story of the building model,Eqs. (2) and (3) can be re-written as follows:

½K� ¼½a1þa2� �½a2� ½0��½a2� ½a2þa3� � a3½ �½0� �½a3� ½a3�

264

375 (18)

½M� ¼½m1� ½0� ½0�½0� ½m2� ½0�½0� ½0� ½m3�

264

375 (19)

Now, one can revise matrix [B] given by Eq. (17) in terms of [ai] and [mi] matrices:

½B� ¼½m1�� 1=2½a1þa2�½m1��1=2 �½m1��1=2½a2�½m2�� 1=2 ½0��½m2�� 1=2½a2�½m1�� 1=2 ½m2�� 1=2½a2þa3�½m2��1=2 �½m2�� 1=2½a3�½m3�� 1=2

½0� �½m3�� 1=2½a3�½m2�� 1=2 ½m3��1=2½a3�½m3��1=2

264

375 (20)

or

½B� ¼A1 B1 0B1T A2 B20 B2T A3

264

3759�9

(21)

Expanding the last column of matrix [B] in Eq. (16) yields

½Z2�½B2�þ½Z3�½A3� ¼ ½Λ�½Z3�: (22)

Multiplying Eq. (22) by [Z3]T from the left-hand side (LHS) and using Eq. (13) gives

½A3� ¼ ½Z3�T ½Λ�½Z3�: (23)

Therefore, Eq. (22) becomes

½Z2�½B2� ¼ ½Λ�½Z3��½Z3�½A3� ¼ ½R3� (24)

If the last three rows of the normalized eigenvector matrix U, or the last three columns of the matrix Z, i.e., Z3, are givenas part of the input data, the right-hand side (RHS) of Eq. (24) could be determined. Later on in Section 4, the matrix Z3 willbe determined using a number of auxiliary systems. It should be observed that in the prescribed matrix decomposition, thesub-matrices Zi are all orthogonal (Eq. (13)). The next step is to decompose the [R3] matrix into two matrices ([Z2][B2])using any of the available mathematical algorithms such as the QRmethod [2]. However, in that method, Rmust be an uppertriangular matrix in order to have a unique answer. In this problem, the equivalent R matrix ([B2]) is not an upper triangularmatrix. Thus, based on the given configuration of matrix [B] for 3-D shear buildings, certain information regardingcomponents of [B] can be used to calculate the remaining unknown terms. Using Eq. (17), one can find the followingrelations between matrices [B2] and [A3]:

½B2� ¼ �½m2��1=2½m3�1=2½A3� (25)

Substituting Eq. (25) into Eq. (24) yields

½Z2�½m2�� 1=2½m3�1=2 ¼ �½R3�½A3��1 (26)

According to Eq. (3), m2½ ��1=2 m3½ �1=2 is a diagonal, (3�3) matrix of the following form:

½m2�� 1=2½m3�1=2 ¼

ffiffiffiffiffim7m4

q0 0

0ffiffiffiffiffim7m4

q0

0 0ffiffiffiffiffim9m6

q

2666664

3777775 (27)

Thus, knowing that (½Z2�9�3 ¼ ½ fZ21g9�1 ; fZ22g9�1 ; fZ23g9�1 �), Eq. (26) becomesffiffiffiffiffim7m4

qZ21f g ;

ffiffiffiffiffiffiffim7

m4

rZ22f g;

ffiffiffiffiffiffiffim9

m6

rZ23f g

� �¼ �½R3�½A3��1 ¼ ½N1�9�3 (28)

Note that the vectors Z21 ; Z22 and Z23 are unit vectors, i.e., jfZ21gj ; jfZ22gj and jfZ23gj are all equal to 1. Therefore,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim7=m4

pand

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim9=m6

pare equal to the norm of the first and third columns of matrix [N1], respectively. So, the vectors

fZ21g; fZ22g and fZ23g can be easily calculated.Having identified the matrices [A3], [B2] and [Z2] in terms of matrix [Z3], as well as determining the second column of [B]

that has three sub-matrices, the rest of matrix [B] for a multi-story shear building can be similarly calculated. Expanding

K.M. Dolatshahi, F.R. Rofooei / Journal of Sound and Vibration 333 (2014) 99–113 105

Eq. (16) from the second column of matrix [B] leads to

½Z1�½B1�þ½Z2�½A2�þ½Z3�½B2�T ¼ ½Λ�½Z2� (29)

A2½ � ¼ Z2½ �T Λ½ � Z2½ � (30)

or

½Z1�½B1� ¼ ½Λ�½Z2��½Z2�½A2��½Z3�½B2�T ¼ ½R2� (31)

According to Eqs. (20) and (21), we can find a relation between matrices ½B2�T , ½B1� and ½A2� as follows:

� ½m2��1=2½m3�1=2½B2�T � ½m2��1=2½m1�1=2½B1� ¼ ½A2� (32)

Rearranging Eq. (32) and using Eq. (31) leads to

½Z1�ð½m1�� 1=2½m2�1=2Þ ¼ �½R2�½P��1 (33)

where P is a known matrix defined as

P ¼ ½A2�þ½m2�� 1=2½m3�1=2½B2�T (34)

Again, by expanding matrix Z1½ � as½Z1�9�3 ¼ ½ fZ11g 9�1 fZ12g 9�1 fZ13g 9�1 �;

Eq. (33) changes to ffiffiffiffiffim4m1

qZ11f g

ffiffiffiffiffiffiffim4

m1

rZ12f g

ffiffiffiffiffiffiffim6

m3

rZ13f g

� �¼ �½R2�½P��1 ¼ ½N2�9�3 (35)

Similar to ½Z2�, jfZ11gj ¼ jfZ12gj ¼ jfZ13gj ¼ 1. Therefore,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim4=m1

pand

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim6=m3

pare equal to the norm of the first and third

columns of matrix [N2], respectively, and vectors fZ11g; fZ12g and fZ13g can now be determined.A similar procedure can be adopted to find the rest of matrix [B]. Note that the equations in this section are very similar

to the formulations of 2-D structures developed in Ref. [1]. Clearly, the formulations of 3-D structural models are moredifficult to derive, since one has to deal with sub-matrices instead of scalar values in Eq. (17).

4. Recovering the last three rows of the normalized eigenvector matrix [U]

The purpose of this section is to determine the last three rows of the normalized eigenvector matrix [U], or the last threecolumns of its transpose matrix [Z], that has already been used in constructing the normalized stiffness matrix in theprevious section (Eq. (23)). It can be shown that for 3-D multi-story shear buildings, this matrix can be determined using thefrequencies of three auxiliary structures. The proposed auxiliary structures are constructed by adding three additionalsprings (or constraints) to the mass center of the last floor, i.e., one for each dof of the floor, as shown in Fig. 4. Adding theseextra springs will change the last three diagonal terms of the stiffness matrix, i.e., K3N�2;3N�2, K3N�1;3N�1, and K3N;3N terms.

Fig. 4. The proposed auxiliary structures.

K.M. Dolatshahi, F.R. Rofooei / Journal of Sound and Vibration 333 (2014) 99–113106

For example, in a three story building, adding a spring along the x-direction of the last story will only change the K7;7 term ofthe stiffness matrix (Fig. 2a).

Following Gladwell's procedure [1], and modifying it for the 3N�1 and 3N�2 dofs of the 3-D structural system, the lastthree rows of the normalized eigenvector matrix [U] can be determined. Considering the eigenvalue equation of the thirdauxiliary structure

ðK0�λMÞy¼ 0 (36)

and substituting M1=2 y¼ u into Eq. (36) gives

½B0�fug i ¼ λifugi (37)

in which ½B0� is the normalized stiffness matrix for the third auxiliary structure. ½B0� differs from ½B� only in the last diagonalterm, K9;9 for three story buildings. Thus,

½B0�fugi ¼ ½B�fugiþða0n�anÞun;ifeg n (38)

fugi ¼ ∑n

i ¼ 1p ifug i; ½B�fugi ¼ ∑

n

i ¼ 1λip ifug i (39)

where n¼3N for an N-story building. The parameters fugi and λi represent the eigenvectors and eigenvalues of the auxiliarystructure, whereas uf gi and λi are the respective values for the original structural system. Moreover, an and a0n show the (n,n)terms in [B] and ½B0� matrices, respectively. Thus, Eq. (38) changes to

∑n

k ¼ 1λkpkfug kþða0N�aNÞun;ifegn ¼ λi ∑

n

k ¼ 1pkfug k (40)

�uTj ) λjp jþða0n�anÞun;iun;j ¼ λip j (41)

So,

ðλi�λjÞp j ¼ ða0n�anÞun;iun;j

p j ¼ ða0n�anÞun;iun;j

ðλi�λjÞ(42)

Substituting into Eq. (39) yields

fugi ¼ ða0n�anÞun;i ∑n

j ¼ 1

un;juj

ðλi�λjÞ(43)

Equating the last components in each side provides

1¼ ða0n�anÞ ∑n

j ¼ 1

u2nj

ðλi�λjÞ(44)

According to the definition, one can write unj ¼ zjn. The answers of Eq. (44) are the eigenvalues of the third auxiliarystructure ðλiÞn1: So,

1�ða0n�anÞ ∑n

j ¼ 1

z2jnðλ�λjÞ

¼ ∏n

j ¼ 1

λ�λjλ�λj

� �(45)

Multiplying Eq. (45) by λi�λð Þ and letting λ¼ λi leads to

ða0n�anÞz2in ¼ ðλi�λiÞ ∏n

j ¼ 1

λi�λjλi�λj

� �(46)

The difference in sum of the eigenvalues of ½B0� and B½ � matrices will yield the difference in the sum of their diagonalelements, which is equal to the difference of their last diagonal terms. The prime sign in Eq. (46) denotes the index “j”cannot be equal to “i”. Therefore,

ða0n�anÞ ¼ ∑n

j ¼ 1ðλj�λjÞ40 (47)

The same procedure can be used when the (3N�1) and (3N�2) diagonal terms are changed in the second and thirdauxiliary structures. Thus, it may be concluded that

ða0S �aSÞz2is ¼ ðλi�λiÞ ∏3N

j ¼ 1

λi�λjλi�λj

� �(48)

where s¼3N, 3N�1 and 3N�2, and in each case the related a0s and λi will be substituted for the corresponding auxiliarystructure. This equation is very similar to the one for 2-D buildings, which has been described in detail in Ref. [1].

K.M. Dolatshahi, F.R. Rofooei / Journal of Sound and Vibration 333 (2014) 99–113 107

Note that, based on Eq. (48), it is only possible to calculate the absolute value of each component at each step. Afterfinding the components of each vector, iteration is required to satisfy the orthogonality of the calculated vectors.Consequently, 2� 2n cases should be checked to determine the sign of different elements of the second and third vectors.Clearly, this iteration will not necessarily lead to a unique answer.

5. Decomposition of matrix B into K and M matrices

Having found the components of matrix B, it remains to decompose it into theM and K matrices. If a unit displacement isapplied to the dofs of the first story (1st, 2nd and 3rd dofs), all the remaining masses displace by unit amount (the other twofloors would have the same displacements) leading to

fXg ¼ f1;1;1;1;1;1;1;1;1gTfug ¼ ½M�1=2fXg ¼ fm1=2

1 ;m1=21 ;m1=2

3 ;m1=24 ;m1=2

4 ;m1=26 ;m1=2

7 ;m1=27 ;m1=2

9 gT (49)

Based on this displacement vector, the induced elastic forces in the structural model become

½K�fXg ¼ fkx1�ey1kx1; ky1þex1ky1; kθ1þex1ky1�ey1kx1;0;0;0;0;0;0gT (50)

From Eqs. (7) and (8) it is deduced that

½B�fug ¼ ð½M��1=2½K�½M��1=2Þð½M�1=2fXgÞ ¼ ½M��1=2½K�fXg (51)

Partitioning the RHS and LHS of Eq. (51) into vectors with dimension of (1�3), one obtains

½B�fM1;M2;M3gT ¼ fC;0;0gT (52)

where Mi includes the three components of the mass matrix for each story (for example M11�3 ¼ fm1=21 ;m1=2

1 ;m1=23 g), and

C1�3 is an unknown vector.

A1 B1 0B1T A2 B20 B2T A3

264

375

M1M2M3

8><>:

9>=>;¼

C

00

8><>:

9>=>; (53)

By expanding the third row

½B2�T fM2gþ½A3 �fM3g ¼ 0 (54)

This matrix system of equations can be decomposed into three equations with four unknowns. Knowing that equally scalingthe mass and stiffness matrices of any structural system would not change its frequencies, Eq. (54) could be expanded interms of the last floor mass (M3) as a given data. Expanding the second row of Eq. (53) yields

½B1�T fM1gþ½A2�fM2gþ½B2�fM3g ¼ 0 (55)

in Eq. (55), [M1] is the only unknown that can be easily calculated. Following the same procedure, the vector of mass foreach story can be determined for a multi-story shear building model. An easier approach can also be used to find thecomponents of mass matrix. In Section 3 we found the relation between the masses of different floors, for exampleffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim7= m4

pand

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim9=m6

p. Therefore, by knowing the mass and mass moment of inertia of the last floor, all components of

mass for the rest of the building can be determined.Finally, based on the proposed procedure, any similar inverse problem for identification of the mass and stiffness

matrices of 3-D shear building models can be solved in two phases. In the first phase, the normalized stiffness matrix ([B]) isgenerated using a target set of vibrational frequencies for the structural system (Sections 3 and 4 of the paper). In the secondphase, the obtained normalized stiffness matrix is decomposed into the stiffness ([K]) and mass ([M]) matrices of the targetstructural system (see Fig. 3).

6. Required conditions for having a realistic structural model

The main purpose of this study is to reconstruct the mass and stiffness matrices when the frequencies of the main andauxiliary structures are given. In this paper, it is assumed that the input data are provided according to an experimentalinvestigation. However, it is possible to make some comments about the required conditions for input data (frequencies ofthe real and auxiliary structures), which will lead to a realistic structural system. For 2-D structural models, it can be easilyproved that in cases where Eq. (56) holds, not only a realistic solution exists, but also the solution is unique [1].

0rλ1oλ1oλ2o⋯oλnoλn (56)

In Eq. (56) λi and λi are the eigenvalues of the main and auxiliary structures, respectively. However, in order to have arealistic solution for the 3-D structures, the required conditions are not as easy as the conditions for 2-D structures. In otherwords, these conditions for 3-D building models can be expressed in the following questions. First, for which values of theinput data a realistic eigenvector matrix can be constructed (zis in Eq. (48))? Second, for which conditions the constructedmatrix [B] matches with the matrix [B] for 3-D building models (according to Eq. (17))? In particular, using the proposed

K.M. Dolatshahi, F.R. Rofooei / Journal of Sound and Vibration 333 (2014) 99–113108

algorithm, a matrix should be constructed with the bandwidth of 11, positive diagonal terms and zero value for some of thematrix elements. According to Eq. (48) the first condition will be met if Eq. (56) holds for all three auxiliary structuresindependently. In other words, Eq. (56) guarantees the formation of positive sign for the RHS of Eq. (48). However, thecalculated eigenvector matrix ([Z3]) should be checked to be orthogonal. Regarding the second question, if the properconstruction of the matrix [A3] (in Eqs. (16) and (17)) is guaranteed, the rest of the sub-matrices will be constructedcorrectly according to the proposed algorithm (see for example Eqs. (25) and (32)). According to Eq. (23), the diagonal termsof matrix [A3] will be equal to

½A3�ð1;1Þ ¼ fZ31gT ½Λ�fZ31g½A3�ð2;2Þ ¼ fZ32gT ½Λ�fZ32g½A3�ð3;3Þ ¼ fZ33gT ½Λ�fZ33g (57)

Eq. (57) indicates that without any requirement the diagonal terms of matrix [A3] are positive and consequently realistic.The sign of [A3] (1,3), [A3] (2,3), [A3] (3,1) and [A3] (3,2) is a function of each floor's stiffness eccentricity, which can be eitherpositive or negative; thus, it is automatically satisfied. Finally, the most important condition that should be maintained is thevalue of [A3] (1,2) and [A3] (2,1), which must be equal to zero. Therefore,

½A3�ð1;2Þ ¼ fZ31gT ½Λ�fZ32g ¼ 0

½A3�ð2;1Þ ¼ fZ32gT ½Λ�fZ31g ¼ 0 (58)

Substituting Eq. (48) into Eq. (58) results in

7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðλ1�λ1Þ ∏

3N

j ¼ 1

0λ1�λjλ1�λj

� �sλ1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðλ1�λ1Þ ∏

3N

j ¼ 1

0λ1�λjλ1�λj

!vuut

7⋯7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðλ3N�λ3NÞ ∏

3N

j ¼ 1

0λ3N�λjλ3N�λj

� �vuut λ3N

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðλ3N�λ3NÞ ∏

3N

j ¼ 1

0λ3N�λjλ3N�λj

!vuut ¼ 0 (59)

In Eq. (59), the parameters λi, λi and λi are referred to the eigenvalues of the main, first and second auxiliary structuralmodels, respectively. The sign of different terms in Eq. (59) should be determined similar to Eq. (48). In conclusion, Eqs. (56)and (59) guarantee the correct formation of matrix [A3] in Eqs. (16) and (17), and consequently correct formation of matrix [B](Eq. (17)). However, the orthogonality of matrix [Z3] should be checked. This would lead to a realistic structural model.

7. Numerical example

In this section, a 3-D, three-story shear building structure is considered. First, based on the given information (stiffnessand mass of the structure), the frequencies of the structural model are calculated (direct solution). In the second phase, theproblem is solved in an inverse manner, i.e., based on the given frequencies as the target frequencies, the mass and stiffnessmatrices of the original structure are determined and the obtained data are compared with the given data in phase I (directsolution). Assuming the following information for the structural model:

kx1 ¼ 12ky1 ¼ 10kθ1 ¼ 13

8><>: ;

ex1 ¼ 0:15ey1 ¼ 0:1

(;

m1 ¼ 5m1 ¼ 5m3 ¼ 10

8><>: ;

kx2 ¼ 8ky2 ¼ 7kθ2 ¼ 9

8><>: ;

ex2 ¼ 0:1ey2 ¼ 0:2

(;

m4 ¼ 8m4 ¼ 8m6 ¼ 12

8><>:

kx3 ¼ 5ky3 ¼ 6kθ3 ¼ 7

8><>: ;

ex3 ¼ 0:2ey3 ¼ 0:3

(;

m7 ¼ 6m7 ¼ 6m9 ¼ 11

8><>:

the stiffness and mass matrices of the system are assembled as (according to Eqs. (2) and (3))

K ¼

20 0 �2:8 �8 0 1:6 0 0 00 17 2:2 0 �7 �0:7 0 0 0

�2:8 2:2 22 1:6 �0:7 �9 0 0 0�8 0 1:6 13 0 �3:1 �5 0 1:50 �7 �0:7 0 13 1:9 0 �6 �1:21:6 �0:7 �9 �3:1 1:9 16 1:5 �1:2 �70 0 0 �5 0 1:5 5 0 �1:50 0 0 0 �6 �1:2 0 6 1:20 0 0 1:5 �1:2 �7 �1:5 1:2 7

K.M. Dolatshahi, F.R. Rofooei / Journal of Sound and Vibration 333 (2014) 99–113 109

M¼

5 0 0 0 0 0 0 0 00 5 0 0 0 0 0 0 00 0 10 0 0 0 0 0 00 0 0 8 0 0 0 0 00 0 0 0 8 0 0 0 00 0 0 0 0 12 0 0 00 0 0 0 0 0 6 0 00 0 0 0 0 0 0 6 00 0 0 0 0 0 0 0 11

Matrix [B] now can be constructed as

B¼

4 0 �0:396 �1:2649 0 0:2066 0 0 00 3:4 0:3111 0 �1:1068 �0:0904 0 0 0

�0:396 0:3111 2:2 0:1789 �0:0783 �0:8216 0 0 0�1:2649 0 0:1789 1:625 0 �0:3164 �0:7217 0 0:1599

0 �1:1068 �0:0783 0 1:625 0:1939 0 �0:866 �0:12790:2066 �0:0904 �1 �0:3164 0:1939 1:3333 0:1768 �0:1414 �0:6093

0 0 0 �0:7217 0 0:1768 0:8333 0 �0:18460 0 0 0 �0:866 �0:1414 0 1 0:14770 0 0 0:1599 �0:1279 �0:6093 �0:1846 0:1477 0:6364

The eigenvalues of the structural model can be determined as (direct solution)

fωi2g ¼ ð4:7383 4:0545 2:5373 1:9338 1:6888 1:0063 0:16070:2844 0:2489Þ

Also, the eigenvectors of the system (normalized mode shapes) are

φ¼

�0:8546 �0:1870 0:2908 �0:1021 �0:2757 0:1526 �0:0511 �0:1843 �0:06170:1212 �0:8414 �0:2404 0:3160 �0:2367 �0:1142 0:0626 0:0496 �0:21030:2320 �0:1567 0:7287 0:3578 0:2150 0:4009 �0:2056 0:1117 �0:02900:3992 0:0732 0:0214 �0:2490 �0:5571 0:3260 �0:1751 �0:5378 �0:1907�0:0678 0:4454 0:0609 0:5043 �0:3125 �0:1795 0:1675 0:1518 �0:5983�0:1657 0:0987 �0:5315 0:1826 0:0855 0:5539 �0:5160 0:2366 �0:1032�0:0835 �0:0089 �0:0836 0:2364 0:5175 �0:2453 �0:2499 �0:6913 �0:24840:0238 �0:1326 0:0318 �0:5367 0:3457 0:1815 0:2126 0:1740 �0:68190:0469 �0:0361 0:1786 �0:2609 �0:1384 �0:5144 �0:7201 0:2734 �0:1396

For solving the inverse vibration problem, three auxiliary structural systems are generated by adding extra springs,Δkx3 ¼ 0:1; Δky3 ¼ 0:2; and Δkθ3 ¼ 0:30 to the three dofs of the last floor, respectively (the values are selected arbitrarily).Consequently, the squared frequencies of the auxiliary structures become

fωi2g1 ¼ ð4:7384 4:0545 2:5382 1:9357 1:6893 1:0136 0:1744 0:2868 0:2495Þ

fωi2g2 ¼ ð4:7384 4:0550 2:5374 1:9437 1:6926 1:0074 0:1620 0:2862 0:2638Þ

fωi2g3 ¼ ð4:7385 4:0545 2:5374 1:9348 1:6932 1:0073 0:1617 0:2926 0:2498Þ

Note that all these calculations are performed just to generate real sets of frequencies to be used as the input data for theinverse problem. Generally, the sets of frequencies can be obtained in different ways such as experimental data, directsolutions, etc. In the next step, the inverse problem is solved.

8. Recovering the mass and stiffness matrices

According to Fig. 3, once the frequencies of the auxiliary structures are known, the last three rows of the normalizedeigenvector matrix are calculated using Eq. (48). As mentioned before, to determine the sign of the elements of the

K.M. Dolatshahi, F.R. Rofooei / Journal of Sound and Vibration 333 (2014) 99–113110

eigenvectors in the second and first columns of matrix [Z3], 2�29 iterations are required.

½Z3� ¼

�0:0835 0:0238 0:04690:0089 0:1326 0:0361�0:0836 0:0318 0:1786�0:2364 0:5367 0:2609�0:5175 �0:3457 0:13840:2453 �0:1815 0:51440:2499 �0:2126 0:7201�0:6913 0:1740 0:27340:2484 0:6819 0:1396

Using Eq. (23) leads to

A3½ � ¼ Z3½ �T Λ½ � Z3½ � ¼0:8333 0 �0:1846

0 1 0:1477�0:1846 0:1477 0:6364

The matrix [A3] that is determined numerically, accurately estimates the last diagonal component of the given matrix [B].Using Eq. (24), one could write

Z2½ � B2½ � ¼ Λ½ � Z3½ �� Z3½ � A3½ � ¼ R3½ � ¼

�0:3174 0:0821 0:17350:0354 0:3997 0:1054�0:1094 0:0224 0:3195�0:2120 0:4626 0:2156�0:4171 �0:2585 0:10120:1374 �0:0771 0:2624�0:0352 0:0721 �0:26500:4299 �0:1649 �0:2495�0:1194 �0:5328 �0:1089

Also, according to Eq. (28)

N1½ � ¼

0:3457 �0:0586 �0:1587�0:0634 �0:3857 �0:09440:0185 0:0527 �0:50890:2156 �0:4367 �0:17480:4823 0:2705 �0:0818�0:2823 0:1554 �0:53030:1517 �0:1450 0:4941�0:4657 0:1314 0:22640:1651 0:5182 0:0987

Norms of the columns of matrix [N1] are as follows:Norms¼(0.8660, 0.8660, 0.9574)¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim7=m4

p;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim7=m4

p;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim9=m6

p �.

Therefore, based on Eq. (28), [Z2] can be calculated as

Z2½ � ¼

0:3992 �0:0677 �0:1657�0:0732 �0:4454 �0:09860:0213 0:0609 �0:531540:2490 �0:5043 �0:18260:5570 0:3124 �0:0855�0:3260 0:1794 �0:55380:17519 �0:1675 0:5160�0:5377 0:1517 0:23640:1907 0:5983 0:1031

Following Eq. (30),

A2½ � ¼ Z2½ �T Λ½ � Z2½ � ¼1:6251 0 �0:3164

0 1:6250 0:1938�0:3164 0:1938 1:3334

K.M. Dolatshahi, F.R. Rofooei / Journal of Sound and Vibration 333 (2014) 99–113 111

Also, using Eq. (25) leads to

½B2� ¼ � ½m2�� ð1=2Þ½m3�1=2½A3� ¼�0:7217 0 0:1599

0 �0:866 �0:12790:1768 �0:1414 �0:6093

Application of Eq. (31) gives

½R2� ¼

1:1225 �0:1523 �0:3781�0:2084 �0:9436 �0:1662�0:2375 0:209 �0:5169�0:1932 0:3778 0:3436�0:3872 �0:2451 0:21220:1213 �0:095 0:2876�0:0279 0:0532 �0:15270:2531 �0:0637 �0:1343�0:0729 �0:235 �0:0301

Matrix [N2] is computed using Eq. (35)

½N2� ¼

�1:081 0:1533 0:25420:2365 1:0643 0:17170:3679 �0:304 0:79830:1292 �0:3998 �0:39190:3487 0:2994 �0:2355�0:193 0:1444 �0:43920:0647 �0:0792 0:2252�0:2331 0:0628 0:12240:078 0:266 0:0317

Norms of the columns of ½N2� are as follows:Norms¼(1.2649, 1.2649, 1.0954)¼ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim4=m1

p;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim4=m1

p;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim6=m3

pÞ.

Thus

½Z1� ¼

�0:8546 0:1212 0:2320:187 0:8414 0:15670:2908 �0:2404 0:72870:1021 �0:316 �0:35780:2757 0:2367 �0:215�0:1526 0:1142 �0:40090:0511 �0:0626 0:2056�0:1843 0:0496 0:11170:0617 0:2103 0:029

Using Eq. (31), [B1] can be calculated as

½B1� ¼ ½Z1�T ½R2� ¼�1:2649 0 0:2066

0 �1:1068 �0:09040:1789 �0:0783 �0:8216

Finally,

½A1� ¼ ½Z1�T ½Λ�½Z1� ¼4 0 �0:3960 3:4 0:3111

�0:396 0:3111 2:2

The numerically calculated B matrix in the inverse solution is equal to the calculated B matrix in the direct solution, with avery good level of accuracy. Employing Eqs. (54) and (55) and using the calculated B matrix, matrix M can be determined as

m1 ¼ 5:0001; m3 ¼ 10:0001; m4 ¼ 7:9998; m6 ¼ 12:0000

K.M. Dolatshahi, F.R. Rofooei / Journal of Sound and Vibration 333 (2014) 99–113112

Eq. (8) yields

K ¼

20:0004 0:0000 �2:8000 �8:0000 0:0000 1:6000 0:0000 0:0000 0:00000:0000 17:0003 2:2000 0:0000 7:0000 �0:7000 0:0000 0:0000 0:0000�2:8000 2:2000 22:0002 1:6000 �0:7000 �9:0000 0:0000 0:0000 0:0000�7:9999 0:0000 1:6001 12:9997 0:0000 �3:1000 �4:9999 0:0000 1:50000:0000 �7:0000 �0:7003 0:0000 12:9997 1:9000 0:0000 �5:9999 �1:20001:6003 �0:7002 �9:0002 �3:1000 1:9000 16:0000 1:5000 �1:2000 �7:00000:0000 0:0000 0:0000 �5:0000 0:0000 1:5002 5:0000 0:0000 �1:50000:0000 0:0000 0:0000 0:0000 �5:9997 �1:1998 0:0000 6:0000 1:20000:0000 0:0000 0:0000 1:5000 �1:1998 �7:0003 �1:5000 1:2000 7:0000

In case there is no initial data available for constructing the auxiliary structures, it is possible to generate auxiliary structuresby assuming very flexible springs connected to the dofs of the last floor (Δkx;y;z3). In this case, the frequencies of the auxiliarystructures would be almost equal to the original structure with minor differences similar to the case of the example.Consequently, as an option for choosing the frequency sets of auxiliary structures, it is possible to use frequencies that areslightly larger than the frequencies of the original structure. It should be noted that all the conditions of Section 6 should besatisfied.

9. Conclusions

In this paper, the inverse vibration problem formulations for 3-D multi-story shear building structures are developed.First, the mathematical procedure is proposed to reconstruct the normalized eigenvector matrix [U] from the target sets offrequencies considered for the structural system, assuming that the last three rows of the related eigenvector matrix areknown. Then, using three proposed auxiliary structures, following Gladwell's procedure for 2-D problems, the last threerows of the eigenvector matrix are determined. The relations between the components of mass, stiffness and [B] matricesare used in developing the solution for reconstruction of matrix [B]. Once the normalized stiffness matrix [B] is determinedusing the normalized eigenvector matrix, a method is presented to decompose it into the stiffness and mass matrices of thetarget structural model.

The presented procedure only calculates the absolute values of eigenvector components; therefore, iterations are neededto determine the sign of different components in order to satisfy the orthogonality condition among the eigenvectors. Thisin turn may lead to several answers for the given problem. However, some of the mathematical solutions to the problemmay not be applicable for setting up the stiffness and/or mass matrices of real shear building structures. In general, thediversity of the answers could be interpreted as a benefit due to more options that it provides in recovering the structuralproperties. Finally, a numerical example is presented to demonstrate the capability of the proposed procedure in determiningthe target structural mass and stiffness properties.

Acknowledgment

The authors would like to thank Professor Andrei M. Reinhorn (Clifford C. Furnas Eminent Professor of StructuralEngineering, University at Buffalo) for reading the manuscript and making the necessary suggestions.

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