# Scaling Mode Shapes in Output-Only Structure by a Mass-Change-Based Method

Authors: Liangliang Yu; Hanwen Song
Journal: Shock and Vibration
 (2017)

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## Abstract

A mass-change-based method based on output-only data for the rescaling of mode shapes in operational modal analysis (OMA) is introduced. The mass distribution matrix, which is defined as a diagonal matrix whose diagonal elements represent the ratios among the diagonal elements of the mass matrix, is calculated using the unscaled mode shapes. Based on the theory of null space, the mass distribution vector or mass distribution matrix is obtained. A small mass with calibrated weight is added to a certain location of the structure, and then the mass distribution vector of the modified structure is estimated. The mass matrix is identified according to the difference of the mass distribution vectors between the original and modified structures. Additionally, the universal set of modes is unnecessary when calculating the mass distribution matrix, indicating that modal truncation is allowed in the proposed method. The mass-scaled mode shapes estimated in OMA according to the proposed method are compared with those obtained by experimental modal analysis. A simulation is employed to validate the feasibility of the method. Finally, the method is tested on output-only data from an experiment on a five-storey structure, and the results confirm the effectiveness of the method.


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## Body

## 1. Introduction

Operational modal analysis (OMA) is an engineering field in which the modal properties of structures under ambient vibrations are studied [1]. Its development can be traced to 1965, when the method using the cross-correlation function to estimate the frequency response function (FRF) was proposed by Clarkson and Mercer [2], which was also the origin of using the cross-correlation function instead of the impulse response function (IRF) to identify the mode parameters in cases where the excitation is unknown. After approximately 50 years of development, especially due to studies of autoregressive moving average (ARMA) [3], random decrement technique (RDT) [4], operational deflection shapes (ODS) [5], natural excitation technique (NExT) [6], and stochastic subspace identification (SSI) [7], OMA has matured and has been extended to a wider range of applications in engineering.Furthermore, if the identified mode parameters are further used to predict structural response, structural modification, or health monitoring, the FRF or IRF should be identified. In fact, the FRF is obtained using experimental modal analysis (EMA) in the laboratory. Since the boundary conditions of the structure during a laboratory test are not entirely identical with real in-operation working conditions of the structure, the FRF of the system in the laboratory is different from that in an operating state. Therefore, reconstructing the FRF based on the mode parameters (natural frequencies, damping ratios, and mass-scaled mode shapes) identified using the OMA method is necessary. Because the excitation cannot be measured in OMA, the mode shapes cannot be normalized or scaled by mass. As a consequence, an additional procedure to obtain the mass-scaled modal shapes is needed. In the past few years, many methods have been proposed to solve this problem. Doebling and Farrar [8] used a finite-element model (FEM) mass matrix that is reduced to the measurement degree of freedom (DOF) to normalize the mode shapes, according to the work of Guyan [9]. However, their method is only valid for the lower-frequency modes. Another method used the mode parameters identified using the OMA technology to upgrade the FEM, from which the mass matrix that scales the mode shapes was obtained [10–15]. Randall et al. [16] strictly limited the form of excitation and developed cepstral methods for updating mode parameters identified using the OMA method when the logarithmic spectrum of the excitation is smooth and flat enough. Using the mass perturbation as a mode shape scaling method originated from Parloo et al. [17, 18]. By adding one or more masses whose weights are calibrated to the structure, the operational mode shapes can be mass-scaled by the shift in mode frequencies between the original and modified structures. On the basis of this idea, much work has been done. The approximate formula for determination of the scaling factors based on the frequency shift when introducing mass perturbation on the structure was derived directly from the governing equation of motion [19]. It explained how testing should be performed in order to significantly reduce approximation errors. The uncertainties of the estimated scaling factors were also investigated [20]. An exact formulation computing the square of the scaling factors was given by Bernal [21, 22]. This formulation can be used in structures with both non-closely spaced modes and closely spaced modes. López-Aenlle et al. [23] provided two new exact formulations for scaling factors, illustrated the large difference in accuracy between the approximate and exact formulations, and confirmed that the results are similar only when the mass perturbation is relatively small. Taking into account a drawback that the first mode of a structure is less sensitive to the mass perturbation, a new mass-stiffness perturbation method was suggested to scale the mode shapes [24]. A procedure to optimize the mass perturbation strategy was proposed by López-Aenlle et al. [25, 26], which used the mode parameters (natural frequencies and shapes) of the original structure as basic information. Different mass perturbation methods were applied to estimate the scaling factors of a 15-T concrete slab by Fernández et al. [27]. The results showed that all of the methods estimate the scaling factors with good accuracy when an appropriate mass change strategy is adopted. Poozesh et al. employed the drive point scaling method to scale optically measured operating deflection shapes (ODS) [28]. Additionally, they demonstrated the benefits and drawbacks of the mass sensitivity technique. However, the accuracy of the mass matrix obtained by the finite-element method commonly depends on the accuracy of the modeling of the structure. Meanwhile, the mass change in the aforementioned method should not be too high in order to minimize the difference of the mode shapes between the original and modified structures.In this paper, the mass distribution matrix is defined and calculated using unscaled mode shapes given that the structure is discrete and the mass matrix is diagonal [8, 9, 29]. The mass matrix is the product of the mass distribution matrix and a coefficient. Then, a small mass with calibrated weight is added to a certain location of the structure, and the mass distribution vector of the structure is estimated. It is noteworthy that removing a small mass with calibrated weight from a certain location of the structure is also feasible. Based on the difference of the mass distribution vectors between the original and modified structures, the mass matrix is identified. Mode shapes are scaled by the mass matrix. It is also worth noting that a universal set of modes is unnecessary when calculating the mass distribution matrix. This means that modal truncation is allowed in the proposed method.The rest of this paper is organized as follows. In Section2, the mass distribution matrix of the structure is defined and calculated. The mass matrix is obtained based on the method that a small mass with calibrated weight is added to a certain DOF of the structure. Then, the unscaled mode shapes are scaled by the mass matrix. Finally, a simulation and an experiment are presented in Sections 3 and 4, respectively.


## 2. Scaling Mode Shapes

In this section, the mass distribution matrix is defined and calculated by unscaled mode shapes. Then, the mass matrix is obtained by the mass change method and the mode shapes are scaled by the mass matrix. Finally, modal truncation is discussed.


### 2.1. Mass Distribution Matrix

The equation of motion of a linear time-invariant discrete system withN DOFs can be written as(1)Mx¨t+Cx˙t+Kxt=ft,where M, C, and K represent the mass, damping, and stiffness matrices, respectively. M is a real, symmetric, positive definite matrix. C and K are real, symmetric, and positive or semipositive definite matrices. t denotes time. ft is the excitation, and xt represents the displacement response vector.According to the property that mode shapes are weighted orthogonal with respect to the mass matrix, one has(2)ΨTMΨ=I,where Ψ is the mass-scaled mode shape matrix and I is the identity matrix.The relation between the unscaled or arbitrary scaled mode shape matrix and the mass-scaled mode shape matrix can be expressed as(3)Φ=Ψα=α1ψ11ψ21⋮ψN1α2ψ12ψ22⋮ψN2⋯αNψ1Nψ2N⋮ψNN,where Φ is the unscaled or arbitrarily scaled mode shape matrix. The scaling factor α is a diagonal matrix with unknown but positive elements.Substituting (3) into (2) yields(4)ΦTMΦ=α2.On the basis of the assumption that the structure is a discrete system whose mass matrix can be assumed to be diagonal [8, 9, 29], the expanded form of (4) can be written as(5)ϕ11ϕ12⋯ϕ1i⋯ϕ1Nϕ21ϕ22⋯ϕ2i⋯ϕ2N⋮⋮⋯⋮⋯⋮ϕN1ϕN2⋯ϕNi⋯ϕNNTdiag⁡mϕ11ϕ12⋯ϕ1j⋯ϕ1Nϕ21ϕ22⋯ϕ2j⋯ϕ2N⋮⋮⋯⋮⋯⋮ϕN1ϕN2⋯ϕNj⋯ϕNN=diag⁡α12α22⋮αN2,where(6)m=m1m2⋯mNT,and diag⁡m returns a square diagonal matrix with the elements of vector m on the main diagonal.Considering the symmetry on the left-hand side of (5), NN+1/2 equations can be obtained from (5), in which mii=1,2,…,N is the independent variable.Wheni=j, N equations can be extracted from (5) as(7)ϕ11ϕ11ϕ21ϕ21⋯ϕN1ϕN1ϕ12ϕ12ϕ22ϕ22⋯ϕN2ϕN2⋮⋮⋱⋮ϕ1Nϕ1Nϕ2Nϕ2N⋯ϕNNϕNNm1m2⋮mN=α12α22⋮αN2.Equation (7) cannot be solved since the scaling factor matrix α is unknown.Wheni<j, NN-1/2 equations can be extracted from (5) as(8)Am=0,where(9)A=ϕ11ϕ12ϕ21ϕ22⋯ϕN1ϕN2ϕ11ϕ13ϕ21ϕ23⋯ϕN1ϕN3⋮⋮⋱⋮ϕ11ϕ1Nϕ21ϕ2N⋯ϕN1ϕNNϕ12ϕ13ϕ22ϕ23⋯ϕN2ϕN3ϕ12ϕ14ϕ22ϕ24⋯ϕN2ϕN4⋮⋮⋱⋮ϕ12ϕ1Nϕ22ϕ2N⋯ϕN2ϕNN⋮⋮⋮⋮ϕ1,N-1ϕ1Nϕ2,N-1ϕ2N⋯ϕN,N-1ϕNNNN-1/2×N.Owing to the symmetry on the left-hand side of (5), the case when i>j is similar to that when i<j and is omitted here. In (8), there are NN-1/2 equations. m is the null space of A. Given that m exists and is unique, the dimension of the null space of A is 1. Therefore, the rank of A is N-1. The mass distribution vector M~, which denotes the ratios among the elements of m, is the unique basis of the null space of A. Since singular value decomposition (SVD) is an efficient method of calculating the null space of a matrix, it is adopted to obtain M~:(10)A=USVT.Because the rank ofA is N-1 and the singular values in S are sorted in descending order, M~ can be obtained as(11)M~=VN,where M~ is scaled by its 2-norm because V is a unit orthogonal matrix. Consequently, the mass distribution matrix is expressed as(12)M^=diag⁡M~.


### 2.2. Mass Matrix

A small mass with calibrated weightΔmk is added to the DOF k of the structure. According to the method similar to that used in Section 2.1, the mass distribution vector of the structure modified by Δmk, denoted as M~δ, can be obtained.Owing to the fact thatM~ is scaled by its 2-norm, M~ can be rewritten as(13)M~=m1m⋯mkm⋯mNmT,where m is the 2-norm of vector m.Similarly,M~δ can be rewritten as(14)M~δ=m1Δmk2+2mkΔmk+m21/2⋯mk+ΔmkΔmk2+2mkΔmk+m21/2⋯mNΔmk2+2mkΔmk+m21/2T.Combining thekth equation in (13) and thekth equation in (14), the following equations can be obtained:(15)mkm=M~k,mk+ΔmkΔmk2+2mkΔmk+m21/2=M~kδ.Solving the equations in (15), one obtains(16)m=M~k1-M~kδ2+M~k2M~kδ4-M~k2M~kδ2-M~kδ4+M~kδ2M~kδ2-M~k2Δmk.Substituting (16) into (6) and (13) yields(17)m=M~k1-M~kδ2+M~k2M~kδ4-M~k2M~kδ2-M~kδ4+M~kδ2M~kδ2-M~k2ΔmkM~.Then, the mass matrix of the structure is obtained as(18)M=diag⁡m=M~k1-M~kδ2+M~k2M~kδ4-M~k2M~kδ2-M~kδ4+M~kδ2M~kδ2-M~k2Δmk·M^.


### 2.3. Scaling Factor

Since the mass matrix has been obtained in Section2.2, the mode shape matrix can be scaled. Substituting (18) into (4), one has(19)α2=M~k1-M~kδ2+M~k2M~kδ4-M~k2M~kδ2-M~kδ4+M~kδ2M~kδ2-M~k2ΔmkΦTM^Φ.Since the matrixΦTM^Φ is diagonal and all diagonal elements are positive, α can thus be written as(20)α=M~k1-M~kδ2+M~k2M~kδ4-M~k2M~kδ2-M~kδ4+M~kδ2M~kδ2-M~k2Δmk1/2·ΦTM^Φ,where ΦTM^Φ represents the square root of the diagonal element of matrix ΦTM^Φ.Substituting (20) into (3), one can scale the mode shapes by the mass matrix:(21)Ψ=M~k1-M~kδ2+M~k2M~kδ4-M~k2M~kδ2-M~kδ4+M~kδ2M~kδ2-M~k2Δmk-1/2Φ·ΦTM^Φ-1.


### 2.4. Modal Truncation

Generally, a universal set of modes cannot be obtained due to modal truncation. When the order of the modes is truncated asT (T≤N), (8) becomes (22)A~TT-1/2×NmN=0N.Considering the fact that the rank ofA~ is N-1, both T and N should satisfy the following inequality:(23)12TT-1≥N-1.Solving the inequality in (23), one has(24)T≥ceil⁡8N-7+12=N0,where ceil⁡8N-7+1/2 means rounding 8N-7+1/2 to the nearest integer greater than or equal to 8N-7+1/2 and N0 is the smallest positive integer that satisfies the inequality in (24). The details of the relationship between N and N0 are presented in Figure 1.Figure 1
Relationship betweenN and N0.WhenN≥4, as is shown in Figure 1, N0<N. This means that the universal set of modes is unnecessary when calculating the mass distribution matrix. In other words, the modal truncation is allowed. The growth rate of N0 is much smaller than the growth rate of N. When N is sufficiently large, it follows that N0≪N.It is worth noting that (24) is the premise of calculating the mass distribution matrix in this paper. If the total number of natural modes identified is smaller than N0, the mass matrix or mass distribution matrix cannot be obtained by the aforementioned method.Considering the modal truncation, when arbitraryN0 mode shapes are taken from all T mode shapes, (22) can be written as(25)A~iN0N0-1/2×Nmi=0,where i=1,2,…,CTN0 (CTN0 is the total number of combinations of taking N0 items at a time from T). According to (11), CTN0 mass distribution vectors can be obtained.Similarly, when arbitraryN0+1 mode shapes are taken from all T ones, (22) can be written as(26)A~iN0N0+1/2×Nmi=0,where i=CTN0+1,CTN0+2,…,CTN0+CTN0+1. Thus, CTN0+1 mass distribution vectors can be extracted.Inductively, whenT mode shapes are taken, CTT mass distribution vector can be obtained.Using the enumerating method,ST mass distribution vectors can be obtained from T identified modes; namely,(27)ST=CTN0+CTN0+1+CTN0+2+⋯+CTT.Theoretically, the mass distribution vectors calculated by different mode shape combinations should be identical.


## 2.1. Mass Distribution Matrix

The equation of motion of a linear time-invariant discrete system withN DOFs can be written as(1)Mx¨t+Cx˙t+Kxt=ft,where M, C, and K represent the mass, damping, and stiffness matrices, respectively. M is a real, symmetric, positive definite matrix. C and K are real, symmetric, and positive or semipositive definite matrices. t denotes time. ft is the excitation, and xt represents the displacement response vector.According to the property that mode shapes are weighted orthogonal with respect to the mass matrix, one has(2)ΨTMΨ=I,where Ψ is the mass-scaled mode shape matrix and I is the identity matrix.The relation between the unscaled or arbitrary scaled mode shape matrix and the mass-scaled mode shape matrix can be expressed as(3)Φ=Ψα=α1ψ11ψ21⋮ψN1α2ψ12ψ22⋮ψN2⋯αNψ1Nψ2N⋮ψNN,where Φ is the unscaled or arbitrarily scaled mode shape matrix. The scaling factor α is a diagonal matrix with unknown but positive elements.Substituting (3) into (2) yields(4)ΦTMΦ=α2.On the basis of the assumption that the structure is a discrete system whose mass matrix can be assumed to be diagonal [8, 9, 29], the expanded form of (4) can be written as(5)ϕ11ϕ12⋯ϕ1i⋯ϕ1Nϕ21ϕ22⋯ϕ2i⋯ϕ2N⋮⋮⋯⋮⋯⋮ϕN1ϕN2⋯ϕNi⋯ϕNNTdiag⁡mϕ11ϕ12⋯ϕ1j⋯ϕ1Nϕ21ϕ22⋯ϕ2j⋯ϕ2N⋮⋮⋯⋮⋯⋮ϕN1ϕN2⋯ϕNj⋯ϕNN=diag⁡α12α22⋮αN2,where(6)m=m1m2⋯mNT,and diag⁡m returns a square diagonal matrix with the elements of vector m on the main diagonal.Considering the symmetry on the left-hand side of (5), NN+1/2 equations can be obtained from (5), in which mii=1,2,…,N is the independent variable.Wheni=j, N equations can be extracted from (5) as(7)ϕ11ϕ11ϕ21ϕ21⋯ϕN1ϕN1ϕ12ϕ12ϕ22ϕ22⋯ϕN2ϕN2⋮⋮⋱⋮ϕ1Nϕ1Nϕ2Nϕ2N⋯ϕNNϕNNm1m2⋮mN=α12α22⋮αN2.Equation (7) cannot be solved since the scaling factor matrix α is unknown.Wheni<j, NN-1/2 equations can be extracted from (5) as(8)Am=0,where(9)A=ϕ11ϕ12ϕ21ϕ22⋯ϕN1ϕN2ϕ11ϕ13ϕ21ϕ23⋯ϕN1ϕN3⋮⋮⋱⋮ϕ11ϕ1Nϕ21ϕ2N⋯ϕN1ϕNNϕ12ϕ13ϕ22ϕ23⋯ϕN2ϕN3ϕ12ϕ14ϕ22ϕ24⋯ϕN2ϕN4⋮⋮⋱⋮ϕ12ϕ1Nϕ22ϕ2N⋯ϕN2ϕNN⋮⋮⋮⋮ϕ1,N-1ϕ1Nϕ2,N-1ϕ2N⋯ϕN,N-1ϕNNNN-1/2×N.Owing to the symmetry on the left-hand side of (5), the case when i>j is similar to that when i<j and is omitted here. In (8), there are NN-1/2 equations. m is the null space of A. Given that m exists and is unique, the dimension of the null space of A is 1. Therefore, the rank of A is N-1. The mass distribution vector M~, which denotes the ratios among the elements of m, is the unique basis of the null space of A. Since singular value decomposition (SVD) is an efficient method of calculating the null space of a matrix, it is adopted to obtain M~:(10)A=USVT.Because the rank ofA is N-1 and the singular values in S are sorted in descending order, M~ can be obtained as(11)M~=VN,where M~ is scaled by its 2-norm because V is a unit orthogonal matrix. Consequently, the mass distribution matrix is expressed as(12)M^=diag⁡M~.


## 2.2. Mass Matrix

A small mass with calibrated weightΔmk is added to the DOF k of the structure. According to the method similar to that used in Section 2.1, the mass distribution vector of the structure modified by Δmk, denoted as M~δ, can be obtained.Owing to the fact thatM~ is scaled by its 2-norm, M~ can be rewritten as(13)M~=m1m⋯mkm⋯mNmT,where m is the 2-norm of vector m.Similarly,M~δ can be rewritten as(14)M~δ=m1Δmk2+2mkΔmk+m21/2⋯mk+ΔmkΔmk2+2mkΔmk+m21/2⋯mNΔmk2+2mkΔmk+m21/2T.Combining thekth equation in (13) and thekth equation in (14), the following equations can be obtained:(15)mkm=M~k,mk+ΔmkΔmk2+2mkΔmk+m21/2=M~kδ.Solving the equations in (15), one obtains(16)m=M~k1-M~kδ2+M~k2M~kδ4-M~k2M~kδ2-M~kδ4+M~kδ2M~kδ2-M~k2Δmk.Substituting (16) into (6) and (13) yields(17)m=M~k1-M~kδ2+M~k2M~kδ4-M~k2M~kδ2-M~kδ4+M~kδ2M~kδ2-M~k2ΔmkM~.Then, the mass matrix of the structure is obtained as(18)M=diag⁡m=M~k1-M~kδ2+M~k2M~kδ4-M~k2M~kδ2-M~kδ4+M~kδ2M~kδ2-M~k2Δmk·M^.


## 2.3. Scaling Factor

Since the mass matrix has been obtained in Section2.2, the mode shape matrix can be scaled. Substituting (18) into (4), one has(19)α2=M~k1-M~kδ2+M~k2M~kδ4-M~k2M~kδ2-M~kδ4+M~kδ2M~kδ2-M~k2ΔmkΦTM^Φ.Since the matrixΦTM^Φ is diagonal and all diagonal elements are positive, α can thus be written as(20)α=M~k1-M~kδ2+M~k2M~kδ4-M~k2M~kδ2-M~kδ4+M~kδ2M~kδ2-M~k2Δmk1/2·ΦTM^Φ,where ΦTM^Φ represents the square root of the diagonal element of matrix ΦTM^Φ.Substituting (20) into (3), one can scale the mode shapes by the mass matrix:(21)Ψ=M~k1-M~kδ2+M~k2M~kδ4-M~k2M~kδ2-M~kδ4+M~kδ2M~kδ2-M~k2Δmk-1/2Φ·ΦTM^Φ-1.


## 2.4. Modal Truncation

Generally, a universal set of modes cannot be obtained due to modal truncation. When the order of the modes is truncated asT (T≤N), (8) becomes (22)A~TT-1/2×NmN=0N.Considering the fact that the rank ofA~ is N-1, both T and N should satisfy the following inequality:(23)12TT-1≥N-1.Solving the inequality in (23), one has(24)T≥ceil⁡8N-7+12=N0,where ceil⁡8N-7+1/2 means rounding 8N-7+1/2 to the nearest integer greater than or equal to 8N-7+1/2 and N0 is the smallest positive integer that satisfies the inequality in (24). The details of the relationship between N and N0 are presented in Figure 1.Figure 1
Relationship betweenN and N0.WhenN≥4, as is shown in Figure 1, N0<N. This means that the universal set of modes is unnecessary when calculating the mass distribution matrix. In other words, the modal truncation is allowed. The growth rate of N0 is much smaller than the growth rate of N. When N is sufficiently large, it follows that N0≪N.It is worth noting that (24) is the premise of calculating the mass distribution matrix in this paper. If the total number of natural modes identified is smaller than N0, the mass matrix or mass distribution matrix cannot be obtained by the aforementioned method.Considering the modal truncation, when arbitraryN0 mode shapes are taken from all T mode shapes, (22) can be written as(25)A~iN0N0-1/2×Nmi=0,where i=1,2,…,CTN0 (CTN0 is the total number of combinations of taking N0 items at a time from T). According to (11), CTN0 mass distribution vectors can be obtained.Similarly, when arbitraryN0+1 mode shapes are taken from all T ones, (22) can be written as(26)A~iN0N0+1/2×Nmi=0,where i=CTN0+1,CTN0+2,…,CTN0+CTN0+1. Thus, CTN0+1 mass distribution vectors can be extracted.Inductively, whenT mode shapes are taken, CTT mass distribution vector can be obtained.Using the enumerating method,ST mass distribution vectors can be obtained from T identified modes; namely,(27)ST=CTN0+CTN0+1+CTN0+2+⋯+CTT.Theoretically, the mass distribution vectors calculated by different mode shape combinations should be identical.


## 3. Simulation

In this section, a 10-DOF structure is designed to verify the method of scaling mode shapes.The mass matrix is(28)Mgiven=diag3.813.04.63.80.61.80.618.83.22.0Tkg.The stiffness matrix is(29)Kgiven=317800-88000-16300-63100-7070000000-88000312600-19700-16800-4600-969000000-16300-19700279600-32800-65200-35200-65200000-63100-16800-32800280200-33000-65800-30400-2560000-70700-4600-65200-33000342000-73500-19100-29800-4610000-96900-35200-65800-73500368100-67100-20200-3000-640000-65200-30400-19100-67100311500-44000-41100-44600000-25600-29800-20200-44000163500-5400-385000000-46100-3000-41100-5400192800-9720000000-6400-44600-38500-97200186700N/m.Consequently, the natural mode frequencies are 8.34, 18.92, 26.67, 33.74, 42.02, 48.69, 54.57, 70.05, 116.68, and 122.89 Hz. Taking the damping ratio for every order of the structure to beζ1=0.008, ζ2=0.008, ζ3=0.008, ζ4=0.007, ζ5=0.007, ζ6=0.007, ζ7=0.006, ζ8=0.006, ζ9=0.006, and ζ10=0.006, the damping matrix of the system is computed by(30)Cgiven=Φ-Tdiagc~1⋯c~r⋯c~NΦ-1,where c~r=2ζrm~rωr, m~r=ΦrTMgivenΦ, and r=1,2,…,10.


### 3.1. Mass Distribution Matrix

The loads on the structure are uncorrelated random signals subjected to Gaussian distribution. In this simulation, they fitN0,10000, N0,10000, N0,10000, N0,8100, N0,8100, N0,8100, N0,6400, N0,6400, N0,6400, and N0,6400, where Nμ,σ2 denotes a Gaussian distribution with μ being a mean value and σ2 the variance. In order to illustrate the advantage of modal truncation, all of the random signals are filtered by Chebyshev Type I filters, and the passband edge frequency fp=60 Hz. The sampling frequency fs=512 Hz. All responses of the structure are recorded.The modal parametersfr, ζr, and ϕr are identified by the SSI method. Owing to the band-limited white noise, only the first seven modes can be obtained. According to (11) and (27), the 29 mass distribution vectors or the mass distribution matrices of the structure can be calculated by the unscaled mode shapes Φ. In practice, according to (24), N0=5. Therefore, at least five mode shapes of the structure are required to calculate the mass distribution vector or mass distribution matrix. A different total number of identified mode shapes can be used to calculate the mass distribution vectors, the details of which are shown in Table 1.Table 1
Total number of mass distribution vectors calculated by (27).
T
5
6
7
S
T
1
7
29The mass distribution vectors calculated by the first seven modes are illustrated in Figure2. Apparently, the given mass distribution vector is(31)M~given=0.15640.53490.18930.15640.02470.07410.02470.77410.13170.0823T.Figure 2
Mass distribution vectors calculated by the first seven modes.In Figure2, the dark, thin solid lines coincide well with the thick dashed lines. Calculating the average of the 29 results, the identified mass distribution vector is(32)M~=0.15580.53390.18850.15600.02560.07520.02480.77440.13170.0825T.The consistency of the lines means that the method used to calculate the mass distribution matrix is reliable.


### 3.2. Mass Matrix and Mass-Scaled Mode Shapes

To identify the mass matrix of the structure, a small mass,Δm9=1.0 kg, is added to DOF 9 of the structure. According to a method similar to that used in Section 3.1, M~δ can be obtained as(33)M~δ=0.15500.53200.18860.15530.02410.07340.02440.76860.17150.0822T.According to (17) and (18), m and M can be acquired. The relative error between the given mass matrix Mgiven and the identified mass matrix M is shown in Table 2.Table 2
Relative error between the given mass matrixMgiven and the identified mass matrix M.
DOF
1
2
3
4
5
6
7
8
9
10
M
given (kg)
3.8
13.0
4.6
3.8
0.6
1.8
0.6
18.8
3.2
2.0
M (kg)
3.820
13.094
4.622
3.826
0.628
1.843
0.609
18.991
3.231
2.022
Error (%)
0.52
0.73
0.48
0.69
4.69
2.41
1.56
1.02
0.96
1.10As can be seen from Table2, the maximum relative error of the identified mass is 4.69% (DOF 5). Therefore, the relative error between the given mass matrix Mgiven and the identified mass matrix M is acceptable.Since the mass matrix was obtained in Section3.2, the mode scaling factor can be computed according to (20) and is given by(34)α=diag12.62414.1549.1519.82110.1109.5507.4796.6093.9673.963T.According to (21) or (3), the mass-scaled mode shapes can be obtained.


## 3.1. Mass Distribution Matrix

The loads on the structure are uncorrelated random signals subjected to Gaussian distribution. In this simulation, they fitN0,10000, N0,10000, N0,10000, N0,8100, N0,8100, N0,8100, N0,6400, N0,6400, N0,6400, and N0,6400, where Nμ,σ2 denotes a Gaussian distribution with μ being a mean value and σ2 the variance. In order to illustrate the advantage of modal truncation, all of the random signals are filtered by Chebyshev Type I filters, and the passband edge frequency fp=60 Hz. The sampling frequency fs=512 Hz. All responses of the structure are recorded.The modal parametersfr, ζr, and ϕr are identified by the SSI method. Owing to the band-limited white noise, only the first seven modes can be obtained. According to (11) and (27), the 29 mass distribution vectors or the mass distribution matrices of the structure can be calculated by the unscaled mode shapes Φ. In practice, according to (24), N0=5. Therefore, at least five mode shapes of the structure are required to calculate the mass distribution vector or mass distribution matrix. A different total number of identified mode shapes can be used to calculate the mass distribution vectors, the details of which are shown in Table 1.Table 1
Total number of mass distribution vectors calculated by (27).
T
5
6
7
S
T
1
7
29The mass distribution vectors calculated by the first seven modes are illustrated in Figure2. Apparently, the given mass distribution vector is(31)M~given=0.15640.53490.18930.15640.02470.07410.02470.77410.13170.0823T.Figure 2
Mass distribution vectors calculated by the first seven modes.In Figure2, the dark, thin solid lines coincide well with the thick dashed lines. Calculating the average of the 29 results, the identified mass distribution vector is(32)M~=0.15580.53390.18850.15600.02560.07520.02480.77440.13170.0825T.The consistency of the lines means that the method used to calculate the mass distribution matrix is reliable.


## 3.2. Mass Matrix and Mass-Scaled Mode Shapes

To identify the mass matrix of the structure, a small mass,Δm9=1.0 kg, is added to DOF 9 of the structure. According to a method similar to that used in Section 3.1, M~δ can be obtained as(33)M~δ=0.15500.53200.18860.15530.02410.07340.02440.76860.17150.0822T.According to (17) and (18), m and M can be acquired. The relative error between the given mass matrix Mgiven and the identified mass matrix M is shown in Table 2.Table 2
Relative error between the given mass matrixMgiven and the identified mass matrix M.
DOF
1
2
3
4
5
6
7
8
9
10
M
given (kg)
3.8
13.0
4.6
3.8
0.6
1.8
0.6
18.8
3.2
2.0
M (kg)
3.820
13.094
4.622
3.826
0.628
1.843
0.609
18.991
3.231
2.022
Error (%)
0.52
0.73
0.48
0.69
4.69
2.41
1.56
1.02
0.96
1.10As can be seen from Table2, the maximum relative error of the identified mass is 4.69% (DOF 5). Therefore, the relative error between the given mass matrix Mgiven and the identified mass matrix M is acceptable.Since the mass matrix was obtained in Section3.2, the mode scaling factor can be computed according to (20) and is given by(34)α=diag12.62414.1549.1519.82110.1109.5507.4796.6093.9673.963T.According to (21) or (3), the mass-scaled mode shapes can be obtained.


## 4. Experimental Verification

To verify the aforementioned theory, a five-storey prototype structure was constructed for experimentation and is shown in Figure3.Figure 3
Experimental structure.As can be seen from Figure3, in this experiment, ft is white noise. The acceleration responses x¨t=x¨1tx¨2tx¨3tx¨4tx¨5tT are measured and stored by the SO Analyze© system (m+p international, Germany). The sampling frequency fs=512 Hz.The mode parameters are identified by the SSI method, and the stability diagram is reproduced in Figure4. The natural frequency fr and the damping ratio ζr are shown in Table 3.Table 3
Identified mode frequencies and damping ratios.
Order
1
2
3
4
5
Frequency (Hz)
7.965
24.833
36.758
43.678
49.593
Damping ratio (%)
0.66
0.83
0.79
1.03
0.85Figure 4
Stability diagram (∗, frequency and damping stability poles; ×, frequency stability poles; +, new poles).Five modes can be identified. However, in Figure4, the poles at the first modes are not as stable as the others. Meanwhile, the coherence functions at the first modes are not as satisfied as at the other modes. One reason for this is the fact that the responses of the first mode are larger than that of the others. This is also because the shaker in our laboratory is not good enough in the low frequency band. According to (24), N0=4. Therefore, at least four modes shapes of the structure are required to calculate the mass distribution vector or mass distribution matrix. According to the stability diagram in Figure 4, the second, third, fourth, and fifth mode shapes are employed to calculate the mass distribution vector that is shown in Figure 5(a).Figure 5
Mass distribution vector calculated by four modes.
(a)
Original structure
(b)
Modified structureIn Figure5(a), the mass distribution vector can be identified as(35)M~=0.61000.37830.37110.50990.2951T.To identify the mass matrix of the structure, a small mass,Δm3=0.311 kg, is added to DOF 3 of the structure. Repeated modal testing after changing the mass is undertaken. Similarly, the mass distribution vector of the modified structure can be computed, with the results shown in Figure 5(b). Then, M~δ can be obtained as(36)M~δ=0.59610.36900.42500.49540.2874T.According to (17) and (18), m and M can be acquired, where(37)M=diag⁡m=diag⁡2.92311.81301.77812.44531.4143T.According to (21) or (3), mass-scaled mode shapes can be obtained, which are shown in Figure 6. For comparison, the mass-scaled mode shapes estimated by EMA serve as the control group.Figure 6
Mass-scaled mode shapes obtained using EMA (light, thick, and dashed lines) and the proposed method (dark, thin, and solid lines).
(a)
f
1
=
7.97 Hz
(b)
f
2
=
24.83 Hz
(c)
f
3
=
36.76 Hz
(d)
f
4
=
43.68 Hz
(e)
f
5
=
49.59 HzIn Figure6, the mass-scaled mode shapes obtained using EMA are represented by the light, thick, and dashed lines, while those identified using the proposed method are represented by the dark, thin, and solid lines. The dark, thin, and solid lines coincide with the light, thick, and dashed lines. The consistency of the dashed and solid lines means that the relative error between the mass-scaled mode shapes identified using EMA and those identified using the method proposed in this paper is acceptable.


## 5. Conclusions

In this paper, a mass-change-based method for the rescaling of mode shapes in OMA is introduced, which only requires response data. The features of the proposed method are listed as follows.(
1
) The mass distribution matrix, which is defined as a diagonal matrix whose diagonal elements represent the ratios among the diagonal elements of the mass matrix, was calculated using the unscaled mode shapes given that the structure is discrete and the mass matrix is diagonal. Based on the theory of null space, the mass distribution vector or matrix was obtained.(
2
) A small mass with calibrated weight was added to a certain DOF of the structure, and then the mass distribution vector of the modified structure was estimated. Based on the difference of the mass distribution vectors between the original and modified structures, the mass matrix was identified. In this method, the mass with calibrated weight was not necessary to be of light weight in order to minimize the difference between the original and modified structures.(
3
) A universal set of modes was unnecessary when calculating the mass distribution matrix, indicating that modal truncation is allowed in the proposed method.(
4
) A simulation was employed to validate the feasibility of the method. Furthermore, the method was tested on the output-only data from a five-storey structure under laboratory conditions. The mass-scaled mode shapes estimated by OMA according to the method proposed in this paper were compared with those obtained using EMA. The effectiveness of the method was confirmed and validated.This method is not only enforceable in the simulation and experimental cases proposed in this paper but is also applicable for other engineering problems.



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*Source: 2617534-2017-04-26.xml*