Gawronski, W.

Sawicki, Jerzy T.

English

Cleveland-Marshall College of Law

Cleveland State UniversityEngagedScholarship@CSUMechanical Engineering Faculty Publications Mechanical Engineering Department1-6-2000Structural Damage Detection Using Modal NormsW. GawronskiCalifornia Institute of TechnologyJerzy T. SawickiCleveland State University, j.sawicki@csuohio.eduFollow this and additional works at: https://engagedscholarship.csuohio.edu/enme_facpubPart of the Mechanical Engineering CommonsHow does access to this work benefit you? Let us know!Publisher's StatementNOTICE: this is the author’s version of a work that was accepted for publication in Journal of Soundand Vibration. Changes resulting from the publishing process, such as peer review, editing,corrections, structural formatting, and other quality control mechanisms may not be reflected in thisdocument. Changes may have been made to this work since it was submitted for publication. Adefinitive version was subsequently published in Journal of Sound and Vibration, 229, 1,(01-06-2000); 10.1006/jsvi.1999.2179This Article is brought to you for free and open access by the Mechanical Engineering Department at EngagedScholarship@CSU. It has been acceptedfor inclusion in Mechanical Engineering Faculty Publications by an authorized administrator of EngagedScholarship@CSU. For more information,please contact library.es@csuohio.edu.Original CitationGawronski, W., and Sawicki, J.T. (2000) Structural Damage Detection Using Modal Norms. Journal of Sound and Vibration, 229(1),194-198, doi: 10.1006/jsvi.1999.2179.STRUCTURAL DAMAGE DETECTION USING MODAL NORMS W. GAWRONSKI Jet Propulsion ¸aboratory, California Institute of ¹echnology, Pasadena, CA 91109, ;.S.A. AND J. T. SAWICKI Cleveland State ;niversity, Department of Mechanical Engineering, Cleveland, OH 44115, ;.S.A. 1. INTRODUCTION Di!erent approaches and criteria are used to sense and to localize a damage of #exible structures. Most of the methods take advantage of the dynamics behavior of the structural model [1}6]. This paper uses the modal and sensor norms, as de"ned in reference [7], to determine damage locations. The proposed approach allows localization of damaged elements of a structure, and provides the informa­tion about the impact of the damage on the natural modes of the damaged structure. 2. MODAL MODEL A structural model is characterized by its mass, sti!ness, and damping matrices, as well as by the locations of the sensors and actuators. It is represented by the following second MqK #DqR #Kq"Bu, y"Cqq#CvqR . (1a,b) In this equation q is the structural displacement vector of dimension nd, u is the input vector of dimension r, y is the output vector of dimension s, and M, D, K are the mass, damping, and sti!ness matrices, respectively, of dimensions ndxnd. The input matrix B of dimensions ndxr, characterizes the actuator locations, the output displacement and rate matrices C and C of dimensions sxnd characterize the q vdisplacement and rate sensor locations. The mass matrix is positive de"nite, and the sti!ness and damping matrices are positive semide"nite; nd is the number of degrees of freedom, r is the number of actuators, and s is the number of sensors. The above equation is also obtained in the modal co-ordinates using the modal transformation. For a small proportional damping let Wi be the ith natural fre­quency and ci be the ith natural mode, or mode shape. De"ne the matrix of natural frequencies Q"diag (W1, W2,2, W ). and the modal matrix <"[c1 c22 c ], of n ndimensions n xn, consists of n natural modes of a structure. A new variable, qm,dcalled the modal displacement vector is introduced, such that q"<qm. This transformation produces the modal mass matrix Mm "<T M<, and the modal sti!ness matrix Km "<TK<. Both matrices are diagonal. If the damping is propor­tional, the modal damping matrix Dm "<TD< is diagonal as well. Left-multiplying equation (1) by <T, and subsequently by Mm;1 gives the modal model qK #2ZQqR #Q2q "B u, y"C q #C R m, (2a,b)m m m m mq m mv qwhere Z"0)5 M;1D Q;1 is a diagonal matrix of the modal damping, and B ism m mthe modal input matrix, B "M;1<TB, while C "C < and C "C < are the m m mq q mv vmodal displacement and rate matrices, respectively. The modal equations (2a, b) can be re-written as a set of n independent equations for each modal displacement qK #2( W qR #Wi2q "b (3a)mi i i mi mi miu, y"c #c R i"1,2, n, (3b)i mqiqmi mviqmi, where ( is the ith diagonal entry of Z. In the above equations y is the system output i idue to the ith mode dynamics, while b is the ith row of B , and c and c are the mi m mqi mviith columns of C , and C , respectively. De"ne c as the equivalent output matrix mq mv miof the ith mode, c "c /W#c (4)mi mqi i mvi, then  b and  c  2are the input and output gains of the ith mode, see reference mi 2 mi[7], where  x 2denotes the Frobenius norm of x, i.e.,  x 2 "Jtr(xTx). In the case of multiple sensors c consists of s entries cmi , j"1,2, s, that correspond to each mi sensor. 3. MODAL NORMS The H2norm of a system is used to characterize the system itself, along with its modes, and its sensors. Let G(W) be a transfer function of a system. The H2norm of the system is de"ned as" 1 G 22n f+otr(G*(W) G(W) ) dW. (5)2 ;o In modal co-ordinates the equations of a #exible structure are uncoupled, see reference (3). In this case the norms of the ith mode and jth sensor can be approximately determined from the following equations, cf. reference [7]: G  : b   c  /2J( W (6)i .i 2 mi 2 mi 2 i             Figure 1. The beam structure: "xed nodes are "lled, numbers refer to the sensors. De"ne the H norm of the jth mode, G , as the root-mean-square sum over all mi 22 sensors [7]: s" I i"1,2, n.G Gi (7)22 22 ,minorm of the jth sensor, G=1  De"ne also the H , as the root-mean-square sum s 2 2 over all modes, n" I j"1,2, s.Gs 22 Gi 22 , (8) i=1  Figure 2. Sensor and modal indices for the beam structure, damage case 1.                       2Figure 3. Sensor and modal indices for the beam structure, damage case 2. The H norm of the total system is approximately the root-mean-square sum 2222222over either all its modal norms, or all its sensor norms, that is : i=1 =1 The modal and sensor norms are used in the following damage detection proced­ure. 4. DAMAGE DETECTORS damage is de"ned as a weighted di!erence between the jth sensor norm of a healthy n : s  G G G G (9)I 22 I 22 or .mi s  Denote the jth sensor norm of a healthy structure by Gsh. The jth sensor index of the structural , and the jth sensor norm of a damaged structure by Gsd  and damaged structure, i.e.,"  G ! G   / G . (10)22 22 22 sh The sensor index re#ects the impact of the structural damage on the jth sensor. sd  sh s  Similarly, denote the ith mode norm of a healthy structure by Gmhi. The ith mode index of the , and the ith mode norm of a damaged structure by Gmdi 2structural damage is de"ned as a weighted di!erence between the ith mode norm of          a healthy and damaged structure, i.e., " G / G . (11)mi  mdi 22 22 ! Gmhi mhiThe ith mode index re#ects impact of the structural damage on the ith mode. The question arises how to measure the sensor and modal norms. It follows from equation (6) that the norm is determined from the system natural frequencies, modal damping ratios, and the modal input and output gains. The gains, on the other hand, are determined from the natural mode shapes at the actuator and sensor locations. Thus the measurements of natural frequencies, modal damping, and the modal displacements at the actuator and sensor locations of healthy and damaged structures allows for the localization of a structural damage. Note that the natural mode shapes characterize the structure itself since they are determined from the modal matrix <. Thus, they depend neither on the input time history, u(t), nor on the actual system deformations, q(t). 5. EXAMPLE 22A beam structure with "xed corners as in Figure 1 is analyzed. The cross-section area of the steel beams is of 1 cm2. Two damage cases are considered. First, as a 20% reduction of the sti!ness of the beam No. 5, and the second case as a 20% reduction of the sti!ness of the beam No. 17. The structure was more densely divided near the damage locations to better re#ect the stress concentration. Nine­teen strain-gage sensors are placed at the beams 1}19. A vertical force applied at node P excites the structure. For the "rst case the sensor and the modal indices are shown in Figures 2(a,b). The sensor indices in Figure 2(a) indicate that the sensor No. 5, located at the damaged beam su!ered the most changes. The modal indices in Figure 2(b) show that the "rst mode was heavily a!ected by the damage. The sensor and modal indices for the second case are shown in Figure 3(a,b). Figure 3(a) shows the largest sensor index at location No. 17 of the damaged beam. The modal indices in Figure 3(b) show that the tenth and the second modes were mostly a!ected by the damage. REFERENCES 1. M. I. FRISWELL, J. E. T.  PENNY and S. D. GARVEY 1998 Computers and Structures 69, 547}556. A combined genetic and sensitivity algorithm for the location of damage in structures. 2. C.-P. FRITZEN, D.  JENNEWEIN and T. KIEFER 1998 Mechanical Systems and Signal Processing 12, 163}186. Damage detection based on model updating methods. 3. T. MARWALA and P. S. HEYNS 1998 AIAA Journal, 36, 1494}1501. Multiple-criterion method for determining structural damage. 4. A. MESSINA, E.  J.  WILLIAMS and T. CONTURSI 1998 Journal of Sound and <ibration 216, 791}808. Structural damage detection by a sensitivity and statistical-based method. 5. Z. Y. SHI and S. S. LAW 1998 Journal of Sound and <ibration 218, 825}844. Structural damage localization from modal strain energy change. 6. S. K. THYAGARAJAN,  M. J. SCHULTZ and P. F. PAI 1998 Journal of Sound and <ibration 210, 162}170. Detecting structural damage using frequency response function. 7. W. GAWRONSKI 1998 Dynamics and Control of Structures: A Modal Approach, New York Springer. 

Structural Damage Detection Using Modal Norms

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Structural Damage Detection Using Modal Norms

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