The well-established Dynamic Damage Locating Vector (DDLV) method o ers localization of structural damage through a linear transformation of the null vectors of the transfer matrix change or, in the stochastic case of unknown input, a surrogate hereof. The null vectors are referred to as DDLVs, and it has been previously shown how applying these as loads to the structural domain will result in zero steady-state strain in any subdomain containing sti ness perturbations. The present paper explores the use of DDLVs for localization of mass-related damage and proves that any mass perturbation will, upon application of the DDLVs, be conﬁned to the subdomain exhibiting zero steady-state motion. Since the strain ﬁeld is realized from a surjective linear transformation of the displacement ﬁeld, the DDLVs allow for localization of both mass- and sti ness- related damage without adding computational complexity to the existing procedures. The paper outlines the interrogation scheme for both damage types and demonstrates the basic principles through numerical and experimental examples. Keywords: damage localization, dynamic damage locating vectors, mass and sti ness perturbations, structural health monitoring. 1. Introduction Structural damage localization can be resolved in a setting where damage-induced changes in some set of vibration features are mapped to the structural domain by aid of a model of the undamaged /reference structure [1, 2]. When referring strictly to localization—and not general diagnosis as obtained through, for example, model updating [3, 4, 5]—one of the ﬁrst examples of a model-based approach is the Best Achievable Eigenvector (BAE) method by Lim and Kashangaki [6]. The premise of the BAE method is to interrogate one element at a time in the model and announce damage when the span of a subspace that depends on the element being considered contains the identiﬁed eigenvectors. A requirement here is, as is the case for many of the early model-based localization schemes [2], that the experimental feature is available at all the model coordinates /degrees of freedom (DOF). This requirement will never be satisﬁed in practice, so the missing experimental entries must be estimated through a coordinate expansion procedure, which often degrades the localization performance severely. During the last couple of decades, a vast amount of model-based localization methods have been proposed to cope with the issue of matching experimental and model coordinates [7, 8, 9, 10, 11]. A technique that does not rely on the coordinate match is the Damage Locating Vector (DLV) series by Bernal [12, 13, 14, 15]. The Dynamic Damage Locating Vector (DDLV) method [14] is based on a theorem stating that the null vectors of the transfer matrix change due to sti ness perturbations are dynamic loads inducing zero steady-state strain in the perturbed /damaged subdomain. The stochastic DDLV (SDDLV) method extends to cases where the transfer matrices cannot be computed explicitly because the input is unknown. For this prevailing situation, it is shown by Bernal [15] how null vectors of the transfer matrix change due to sti ness perturbations can be extracted from the null space of a matrix that can be estimated from output-only system realizations. The DDLV and SDDLV methods have demonstrated merit in di erent laboratory and industrial application studies; includ- ing localization of an emulated edge debonding in a residential-sized wind turbine blade [16] and a crack in an onshore wind turbine in idle condition [17]. To increase the robustness of the methods, extensions have been proposed to take into account the statistical uncertainties associated with the embedded output-only realizations from subspace identiﬁcation [18, 19, 20]. The methodological premise is to estimate the uncertainties using well-established techniques [21] and subsequently prop- agate them to the strain /stress estimates using the delta method [22]. The net robustness gain obtained by this propagation technique is exempliﬁed by Marin et al. [18] in the context of a laboratory experiment with a cantilevered beam. *Email: mdu@et.aau.dk. Preprint submitted to Elsevier November 13, 2020 In the literature, the DDLV and SDDLV methods are presented as means to locate sti ness perturbations. The focus of the present paper is to bring attention to the fact that the methods also allow for localization of mass perturbations, which is of practical interest in, for example, o shore applications where marine growth and ice accretion can alter the mass substantially. It will be shown in Section 2 how application of DDLVs as loads to the structural domain in its reference state will conﬁne potential mass perturbations to the structural subdomain exhibiting zero steady-state motion. In this way, mass perturbations can be located directly from the resulting steady-state displacement ﬁelds, while the sti ness perturbations follow from an interrogation of the steady-state strain ﬁelds. Since these strain ﬁelds are computed through a linear transformation of the displacement ﬁelds, the original DDLV and SDDLV methods enable localization of both mass and sti ness perturbations without requiring any methodological extensions. For stochastic systems, it may appear that the DDLVs, which in this instance are extracted from the null space of a surrogate of the damage-induced transfer matrix change, hinge on the assumption of a perturbation-invariant direct transmission term [15, 18, 19]. Since the direct transmission term depends on the mass (but not the sti ness) of the structural system, such a constraint would invalidate the procedure for localization of mass perturbations in stochastic systems. It will, however, be proved in Section 3 that direct transmission invariance is not a requirement for the null space of the aforementioned surrogate to contain DDLVs. As such, mass perturbations can also be located by use of DDLVs in structural systems that are monitored in an output-only setting. The main body of the paper is organized as follows. In Section 2, the DDLV method is outlined and a theorem is presented that generalizes the original DDLV theorem by Bernal [14] to take into account mass perturbations as well. The concept is ex- tended to the stochastic situation in Section 3, while Section 4 contains a brief summary of how the schemes are implemented. Numerical examples with a ﬁnite element truss structure model and an experimental study with a frame structure follow in Sections 5 and 6, and the paper closes with concluding remarks in Section 7. 2. Deterministic load vectors for localization of mass and sti ness perturbations A structural system is referred to as deterministic when both the input and output are known. In this situation, the DDLVs are—as is shown by Bernal [14] for sti ness perturbations and will be shown in Subsection 2.2 for both mass and sti ness perturbations—extracted directly from the null space of the damage-induced shift in transfer matrix. 2.1. Modeling of deterministic structural systems We consider a structural domain, , and assume that it, in its reference state, can be described by the linear, time-invariant (LTI) model M¨x(t)+C˙x(t)+Kx(t)=B2u(t); (1) where ¨ x(t), ˙x(t),x(t)2Rndenote the acceleration, velocity, and displacement vectors, u(t)2Rrcollects the independent inputs that are distributed to byB22Rnr, and M,C,K2Rnnare the mass, damping, and sti ness matrices. System (1) can be rewritten into an N’th order state-space formulation by deﬁning the state vector z(t)2RNand the output vector y(t)2Rm. Hereby, the continuous-time state-space formulation becomes ˙z(t)=Acz(t)+Bcu(t) (2a) y(t)=Ccz(t)+Dcu(t) (2b) where Ac2RNN,Bc2RNr,Cc2RmN, and Dc2Rmrare the state, input, output, and direct transmission matrices. It is noted that in case all the modes of system (1) are identiﬁed, then N=2nandz(t)=[x(t)T˙x(t)T]T. This will, however, seldom be feasible in practice where the typical scenario is to have a modally truncated approximation with N 2n. The receptance transfer matrix of system (2) is given as [14] G(s)=CcAb c(sIAc)1Bc (3) forb=0, 1, 2 depending on whether the output consists of displacements, velocities, or accelerations. The receptance transfer matrix maps the rinputs to mdisplacement outputs. We collect the coordinates of the rinputs in the setRand the coordinates of the moutputs in the setM. From reciprocity, it follows that if R\M ,;andq=#fR[Mg, then the input-output map isCq!Cq. In the derivations made in the present section (the deterministic situation), it will be assumed that this collocated set exists such G(s)=G(s)T2Cqq. 2 2.2. Extracting the DDLVs We now consider the situation where the structural domain is analyzed in its reference state and in a damaged state with mass and sti ness perturbations in 2 . As will be evidenced in Subsection 2.3, 2does not need to be simply connected. Deﬁnition 1. = 1[ 2is a structural domain with 2= K 2[ M 2, where K 2contains any sti ness perturbation K and M 2any mass perturbation M. Thus, in the reference state, 2=;, = 1. Deﬁnition 2. Let G (s)and ˜G(s)denote the transfer matrices of in its reference and damaged states and deﬁne G(s)= ˜G(s)G(s)2Cqq. Then, v (s)2Cqis a DDLV if 0,v(s)2Null (G(s)). Remark 1. A physical interpretation follows from Deﬁnition 2: since G(s)v(s)=˜G(s)v(s)G(s)v(s)=0, the DDLVs are loads that enforce identical state-steady response of in its reference and damaged states. With Null (G(s))2Cq(qp), where p=Rank (G(s)), a necessary constraint for the existence of v(s) is obviously that p<q. Lemma 1 speciﬁes this constraint in terms of the introduced damage. Lemma 1. LetG(s)be given by Deﬁnition 2 with perturbations K andM, then p =Rank (G(s))<q if Rank (M)+ Rank (K)<q. Proof. The proof follows as an extension of that by Bernal [14] and explicit reference to the Laplace variable is skipped for notational convenience. The transfer matrix in the damaged state is deﬁned by ˜G=G+G; (4) where G2Cqqis the reference-state transfer matrix (3). Assuming that Gand ˜Ghave full rank, we can write ˜G1=G1+Ms2+K; (5) thus with X=Ms2+K, ˜G= G1+X1: (6) The Woodbury matrix identity [23] provides ˜G=GGX(I+GX)1G (7) such G=GX(I+GX)1G; (8) and it is seen that Rank (G)=pminn Rank (G);Rank (X);Rank (I+GX)1o : (9) By substituting X=Ms2+K, it follows pRank Ms2+K Rank (M)+Rank (K); (10) hence leading to the constraint q>Rank (M)+Rank (K). 2.3. Damage localization The original DDLV theorem by Bernal [14] states that if G(s) is solely governed by K, then the steady-state strains in 2will be zero when applying v(s) as a load to . In practice, this is realized by applying v(s) to the model transfer matrix of system (1), G(s)= Ms2+Cs+K1; (11) and computing the steady-state displacements x(s)=G(s)B2v(s) (12) from which the steady-state strains (one component per element in the model) are obtained as (s)=Lx(s): (13) We note thatB22Rnqdistributes v(s) to the structural domain, while L2Cnenmaps the displacements to one strain component for each of the neelements in the model. We now extend the DDLV theorem to also consider mass perturbations. 3 Theorem 1. Let be given by Deﬁnition 1 and assume that K andM induce the transfer matrix shift G(s)2Cqqfor which 0,v(s)2Null (G(s)). Then, (12) and(13) yield x M 20and K 20. Proof. The proof follows under the assumption of q>Rank (M)+Rank (K), as developed in Lemma 1. (i) Localization of K: The proof has been provided by Bernal [14], but a summary is included here for clarity. Remark 1 implies that both the steady-state strains and associated strain energy in must be identical in the reference and damaged states, thus Z nf K 2gTEdV+Z K 2T˜EdV=Z nf K 2gTEdV+Z K 2TEdV; (14) where E,˜Eare the constitutive matrices that map strains to stresses in the two structural states. It follows Z K 2T˜EdV=Z K 2TEdV; (15) hence leading to the assertion  K 20 since E;˜E 0 and ˜E,E. (ii) Localization of M: Remark 1 implies that the steady-state motion in must be identical in the reference and damaged states. The same holds for the associated kinetic energy, so when it is assumed, for simplicity but without loss of generality, thatMa ects the density, 2R>0, then Z nf M 2g˙xT˙x dV+Z M 2˜˙xT˙x dV =Z nf M 2g˙xT˙x dV+Z M 2˙xT˙x dV (16) from which Z M 2˜˙xT˙x dV =Z M 2˙xT˙x dV; (17) so ˙x M 2x M 20. 3. Stochastic load vectors for localization of mass and sti ness perturbations A structural system is referred to as stochastic when only the output is known. In this situation, the DDLVs are— as is shown by Bernal [15] for sti ness perturbations and will be shown in Subsection 3.2 for both sti ness and mass perturbations—extracted from the null space of a surrogate of the damage-induced transfer matrix change. When the DDLVs have been extracted from output-only realizations, the localization procedure outlined in Subsection 2.3 applies. 3.1. Modeling of stochastic structural systems The external force in system (1) is unmeasured but assumed to be white noise. In this instance, u(t) will be replaced by a set of ﬁctive inputs that are collocated with the moutputs such Bc2RNmandDc2Rmm. We follow the procedure outlined by Bernal [15] and deﬁne G(s)=Z(s)Bc (18) with Z(s)=CcAb c(sIAc)12CmN. The unknown Bccan be expressed as a function of the direct transmission matrix, Dc=CcA1b cBc, which, of course, is also unknown but is used in the following developments. We have [15] Bc=HyLDc; (19) where superscript ydenotes the Moore-Penrose pseudo-inverse and H="CcA1b c CcAb c# 2R2mN;L="I 0# 2R2mm; (20) with the constraint 2 mNto ensure that (19) is a unique solution. Plugging (19) into (18) and deﬁning R(s)=Z(s)HyL (21) 4 provide G(s)=R(s)Dc; (22) which, due to symmetry in the prescribed collocated set, can be written as G(s)=DcR(s)T: (23) For subsequent developments, we now examine the expression Dc=CcA1b cBc. Let ME;CE;KEbe the mass, damping, and sti ness matrices in the identiﬁed system and ˆCthe matrix that maps the displacement, velocity, or acceleration vector in (1) to the measured output, y(t). For accelerations ( b=2), we get the direct transmission Dc=h ˆCM1 EKEˆCM1 ECEi"0 I M1 EKEM1 ECE#1"0 M1 EˆCT# =h ˆCM1 EKEˆCM1 ECEi"0 0 0C1 EME#"0 M1 EˆCT# =ˆCM1 EˆCT;(24) which is readily found to hold for displacement and velocity measurements ( b=0;1) as well. 3.2. Extracting the DDLVs From the outset of (23), the damage-induced transfer matrix shift can be expressed by G(s)=˜Dc˜R(s)TDcR(s)T(25) or, when deﬁning R(s)=˜R(s)R(s) andDc=˜DcDc, G(s)=h˜DcDci"R(s)T R(s)T# : (26) The literature has explored the situation where M=0 andDc=0 [15, 18, 19]. With these assumptions, it follows that ifv(s)2Null(R(s)T), then v(s)2Null(G(s)). We now examine the situation where M,0 suchDc,0 and prove that 0,v(s)2Null(R(s)T) is still a su cient constraint for v(s) to be a DDLV . Theorem 2. LetG(s)be given by (26), then v (s)is a DDLV if 0,v(s)2Null(R(s)T). Proof. According to Deﬁnition 2, v(s) is a DDLV if 0 ,v(s)2Null(G(s)). With the ﬁnal result in (24) being valid for homogeneous displacement, velocity, and acceleration measurements, it follows from (23) that R(s)T=(ˆCM1 EˆCT)1G(s): (27) Let 0,v(s)2Null(R(s)T) such (ˆC˜M1 EˆCT)1˜G(s)v(s)(ˆCM1 EˆCT)1G(s)v(s)=0; (28) so if we, for notational convenience but without loss of generality, assume that ˆC=IandME=Mwith Mbeing the mass matrix in system (1), then ˜M˜G(s)v(s)MG(s)v(s)=0: (29) Pre-multiplying with M1and plugging in ˜M=M+Mwith M=2666640 0 0M M 2377775 (30) yield (I+M1M)˜G(s)v(s)G(s)v(s)=0,G(s)v(s)=M1M˜G(s)v(s); (31) which can only hold if v(s)2Null(G(s)) suchG(s)v(s)=M1M˜G(s)v(s)=0. That M1M˜G(s)v(s)=0 for v(s)2 Null(G(s)) andMgiven by (30) follows from Theorem 1, as M1M˜G(s)v(s)=M1M[x nf M 2g(s)T0T]T=0. 5 4. Implementation strategy Sections 2 and 3 outline how DDLVs can be used to locate mass and sti ness perturbations in both deterministic and stochastic systems. Since no modiﬁcations of the original DDLV and SDDLV formulations are required, the schemes can be implemented directly as outlined by Bernal [14, 15] and, optionally, with propagation of uncertainties as proposed by Döhler et al. [18] and Marin et al. [19]. In the present paper, the latter is omitted for conciseness. A summary of the DDLV and SDDLV schemes for simultaneous localization of mass and sti ness perturbations is provided below. For more details on each step, the reader is referred to [14, 15]. 1. EstimatefAc;Bc;Cc;Dcg(DDLV) orfAc;Ccg(SDDLV) from the reference and damaged states [24]. Note that consis- tency in the identiﬁed modes is required, thus modes only identiﬁed in one of the two states must be discarded. 2. Select a set of nss-values at which the interrogation is conducted. To avoid undue truncation errors, the s-values must be in the vicinity of the identiﬁed eigenvalues. 3. For each s-value: (a) Compute G(sj) (DDLV) or R(sj)T(SDDLV). (b) Take v(sj) as the right singular vector associated with the lowest singular value of G(sj) orR(sj)T. (c) Compute x(sj) using (12). (d) Compute (sj) using (13). 4. CalculateX=1 nsPns j=1jx(sj)j maxjx(sj)j. 5. CalculateE=1 nsPns j=1j(sj)j maxj(sj)j. 6. The entries inXthat are close to zero point to mass-related damage (node- or DOF-wise). 7. The entries inEthat are close to zero point to sti ness-related damage (element-wise). 5. Numerical examples To exemplify the points made in the previous sections, we examine the ﬁnite element truss model depicted in Fig. 1. The truss, which consists of 21 bars that are assigned identical cross-sections and material models, has a ﬁrst undamped eigenfrequency of 10 rad /s in the reference state. The damping matrix, which is kept ﬁxed in the two structural states, is formed by assuming a classical distribution with 2 % critical damping in each mode in the reference state. In Subsection 5.1, results are presented for simultaneous localization of mass and sti ness perturbations in an idealized setting where noise, modal truncation, and model errors are omitted. Subsection 5.2 outlines localization of a mass perturbation in a stochastic simulation conﬁguration with model errors, noise-contaminated output, and modally truncated approximations of the system matrices. 1 2 3 4 5 6 789 10 11 12 1 2 3 4 5 678 9 10 11 12 13 14 15 16 17 18 19 20 21 L 6L 6L 6L 6L 6L 6L 6 Fig. 1: Finite element truss structure model with node and element numbering. 6 1 3 5 7 9 11 13 15 17 1900.10.20.30.40.5(a) 1 3 5 7 9 11 13 15 17 19 2100.10.20.30.40.5 (b) Fig. 2: Damage localization of mass perturbation in node 9 (DOF 13 and 14) and sti ness perturbation in element 19 using the DDLV scheme; with aggregated steady-state (a) displacement and (b) strain ﬁelds. 5.1. Example 1 To verify the theoretical developments, we analyze the truss model in an idealized context without noise, modal truncation, or model errors. In the damaged state, a mass perturbation in node 9 (DOF 13 and 14) and a sti ness perturbation in element 19 are introduced. Since we operate in the idealized setting, the extents of the perturbations and the s-values at which the interrogation is conducted are immaterial. The localization results using the DDLV scheme are presented in Fig. 2, where Fig. 2a shows the aggregated steady-state displacements of the unconstrained nodes and Fig. 2b shows the aggregated steady- state strains in the elements. Evidently, both damage types are located correctly. Worth of explicit note is that, in agreement with the theoretical developments, the sti ness perturbation is not localizable through the displacement ﬁelds and the mass perturbation is not localizable through the strain ﬁelds. 5.2. Example 2 The truss model is now tested in an output-only simulation setting with a sampling frequency of 200 Hz and a sample length of 400 ;000. The system is subjected to broadband excitation and the output, which is captured in both DOF in each of nodes 2, 4, and 6, is taken as displacements contaminated with 5 % white Gaussian noise. Model errors are introduced such that the simulation models —which replace the physical structure in its two states—are assigned random perturbations to the sti ness of each element, while the interrogation model used to computeG(s) from (11) retains the nominal sti ness. The perturbations are drawn from a uniform distribution with lower and upper limits of 0 :98 and 1:02. In the damaged state, a point mass corresponding to 1 % of the total mass of the truss is added to node 9. The state and output matrices from the two structural states are estimated using stochastic subspace identiﬁcation (SSI) [24] with a ﬁxed model order of N=12 to comply with the constraint N2mestablished in Subsection 3.1. The six ﬁrst modes of the truss are consistently identiﬁed in both structural states and are therefore used in the damage localization analysis. In Table 1, the estimated eigenvalues of the six modes are presented along with the exact ones of the interrogation model. Tb. 1: Estimated eigenvalues of the simulation models and exact eigenvalues of the nominal model. Mode no. Nominal model (exact)Simulation models Reference (estimated) Damaged (estimated) 10:2548 +12:7358i0:2563 +12:7363i0:2475 +12:5059i 20:5697 +28:4816i0:5703 +28:4904i0:5466 +27:9212i 30:6287 +31:4295i0:6294 +31:4502i0:6149 +31:0525i 40:8930 +44:6393i0:8948 +44:7003i0:8908 +44:4694i 51:2336 +61:6693i1:2341 +61:6758i1:1494 +59:9382i 61:2883 +64:4008i1:2942 +64:7070i1:2702 +63:9391i 7 2 4 6 8 10 12 14 16 18 2000.050.10.150.2Fig. 3: Localization of mass perturbation in node 9 (DOF 13 and 14) using the SDDLV scheme. The SDDLV scheme is implemented with four distinct s-values, namely, s2f0:2+10i;0:4+24i;0:7+35i;1+55ig. By aggregating the steady-state displacements obtained for these s-values, the localization results shown in Fig. 3 are obtained for the unconstrained nodes. The two lowest aggregated displacement values are obtained for DOF 13 and 14, which are the DOF in node 9 (where the mass perturbation is added). The third lowest aggregated displacement value—being approximately three times larger than the value for DOF 14—is obtained for DOF 15, which is the horizontal DOF in node 10, see Fig. 1. 6. Experimental study The applicability of DDLVs for locating mass perturbations is tested experimentally in the context of the steel frame depicted in Fig. 4a. The geometrical speciﬁcations of the frame are provided in Fig. 4b, and it is noted that the columns and ﬂoors are welded together. We analyze two structural states; namely, the reference and a “damaged” one in which a mass perturbation of 0 :3 kg (constituting approximately 0 :7 % of the frame’s mass) is added to the second ﬂoor. In each experimental trial, the frame is subjected to a measured impulse load at the third ﬂoor that excites bending around the weak axis, and the response is captured with eight uniaxial accelerometers distributed as shown in Fig. 4b. Here, it is noted that accelerometer 2 is co-located with the input. 6.1. Data processing and DDLV extraction The input transducer and the installed sensors sample with a frequency of 4 ;096 Hz for a duration of 16 s in each trial. Fig. 5a depicts a selected interval of the experimental frequency response functions based on the acceleration output captured in sensor 1 for the two structural states. This interval contains the ﬁrst four bending modes of which the second and fourth exhibit notable sensitivity to the mass perturbation. The transfer matrix for each of the two structural states is computed by applying SSI to the measured temporal input-output realizations. The SSI is set up with a model order of 8, which results in consistent identiﬁcation of the ﬁrst four bending modes in both structural states. The estimated modal parameters are listed in Table 2, where it, in compliance with the frequency response functions in Fig. 5a, is seen how the second and fourth eigenvalues display notable sensitivity to the mass perturbation. It is noted that the mode shapes are evaluated by the modal assurance criterion (MAC) values [25] between “coinciding” modes (that is, the diagonal components in the MAC matrix). The DDLV scheme is implemented with four distinct s-values, namely, si=0:9i,i=1;:::; 4, whereiis the eigenvalue of the ith mode in the reference state. Fig. 5b shows the singular values of G(s), which do not suggest rank deﬁciency of G(s). The lack of rank deﬁciency is expected due to measurement noise, but, as will be evidenced in Subsection 6.3, the right singular vector associated with the smallest singular value may still possess information on the damage location. 6.2. Finite element model formulation A two-dimensional ﬁnite element model of the frame is established by use of 26 two-dimensional Euler-Bernoulli beam elements; each with geometric speciﬁcations in accordance with the physical quantities presented introductory in the present section. The model, which is depicted in Fig. 4c, is classically damped with the damping ratios of the ﬁrst four modes coinciding with the experimental ﬁndings, see Table 2. The mass density of the material model is set to 8 ;085 kg /m3to attain 8 (a) 350 350 350 350 50400 1 2 3 45 6 7 8 (b)123456789 101112131415161718 19202122 k1 k1k2 k2k3 k3k4 k4 (c) Fig. 4: Frame with (a) showing the reference state, (b) a sketch with accelerometer distribution and geometric dimensions in mm, and (c) a node-numbered ﬁnite element beam model representation with rotational springs (with sti ness ki) in the joints. The input is co-located with accelerometer 2. the mass of the real frame structure, and the modulus of elasticity is set to 210 GPa. Rotational springs are implemented in the model to allow for ﬂexibility in the joints between the columns and ﬂoors. It is assumed that the sti nesses associated with springs at a particular ﬂoor are identical, thus a total of four spring sti nesses, ki, are to be determined. This is done through sensitivity-based parameter estimation [26] with the eigenfrequencies of the ﬁrst four modes and the associated MAC values as features. With reference to Fig. 4c, we ﬁnd k1=8:34103,k2=7:48103,k3=5:62103,k4=4:93103N/m, which result in a model with the modal parameters presented in Table 2. 0 50 100 150 200 250 300100101102103104105 (a) 1 2 3 4 5 6 7 810-810-610-410-2100 (b) Fig. 5: Experimental realizations with (a) showing the frequency response functions (with output from sensor 1) from both structural states and (b) the singular values of G(s) for the selected s-values, where iis the eigenvalue of the ith mode in the reference state. 9 Tb. 2: Estimated modal parameters of the frame in its reference (ref.) and damaged (dam.) states and model predictionsain the reference state. Mode numberEigenfrequencies (rad /s) Damping ratios (%) Diagonal MAC values Model Ref. Dam. Ref. Dam. Ref. – Dam Ref. – Model 1 22 :22 22:19 22:13 0:14 0:14 1 :00 1 :00 2 72 :92 72:86 72:61 0:13 0:18 1 :00 1 :00 3 141 :11 141:04 141:00 0:05 0:05 1 :00 0 :99 4 217 :32 217:43 215:33 0:05 0:83 1 :00 0 :99 aThe model is developed in Subsection 6.2. The damping ratios of the ﬁrst four modes are taken as the experimental estimates in the reference state. 6.3. Damage localization results The four DDLVs extracted in Subsection 6.1 are applied to the transfer matrix of the ﬁnite element model developed in Subsection 6.2 evaluated at the selected s-values. The resulting displacement ﬁelds (one per s-value) are aggregated in accor- dance with the implementation strategy outlined in Section 4, which provides the horizontal displacement components shown in Fig. 6. Since the mass perturbation is located in node 20, the DDLV method allows for localization of the perturbation through inspection of the aggregated horizontal displacement ﬁeld. Speciﬁcally, the minimum aggregated horizontal displace- ment is in node 20; with a value that is approximately 1.5 times smaller than the second smallest value, which is attained in the adjacent node 4. 7. Conclusion This paper shows that the DDLV and SDDLV schemes—originally presented as means of locating sti ness perturbations in structural systems—can also be used for mass perturbation localization. It is proved how application of DDLVs as struc- tural loads will conﬁne potential mass perturbations to the structural subdomain exhibiting zero steady-state motion, hence allowing for localization of mass perturbations directly from the displacement ﬁelds. Sti ness perturbations follow from an interrogation of the strain ﬁelds, which are obtained through a linear transformation of the displacement ﬁelds. Therefore, the DDLV and SDDLV schemes enable localization of both damage types without requiring any modiﬁcation of the original formulations by Bernal [14, 15]. In the present paper, this is veriﬁed in the context of numerical examples with a ﬁnite element truss structure model and an experimental study with a frame structure.