Damage detection using finite element and laser operational deflection shapes

Finite Elements in Analysis and Design 38 (2002) 193–226 www.elsevier.com/locate/!nel

Damage detection using !nite element and laser operational de$ection shapes K. Waldrona , A. Ghoshala , M.J. Schulza; ∗ , M.J. Sundaresana , F. Fergusona , P.F. Paib , J.H. Chungc a Intelligent Structures and Mechanisms Laboratory, Department of Mechanical Engineering, 1601 E Market Street, North Carolina A& T State University, Greensboro, NC 27411, USA b Structural Mechanics and Control Laboratory, Department of Mechanical and Aeronautical Engineering, University of Missouri—Columbia, Columbia, MO 65211, USA c L E Contour and Co., 1145 Ridge Rd. West, Rockwall, TX 75087, USA

Abstract Damage detection can increase safety, extend serviceability, reduce maintenance costs, and de!ne reduced operating limits for structures. Vibration-based methods are a new approach for damage detection and are more globally sensitive to damage than localized methods such as ultrasonic and eddy current methods. The interpretation of vibration responses to identify damage is dependent on the numbers and types of sensors and actuators used. One technique that is promising is the use of operational de$ection shapes (ODS) for structural damage detection. The ODS are the actual vibration displacement or velocity patterns of a structure that is vibrating in the steady-state condition due to a speci!c structural loading. In this paper, the ODS are represented as summations of scaled mode shapes and it is demonstrated through !nite-element simulations how the vibration excitation parameters can be chosen to aid in identifying structural damages. Generation of experimental ODS is performed using a scanning laser Doppler vibrometer and piezoceramic actuators, and examples of damage detection on a beam are presented. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Laser; Vibrometry; Smart structures; Operational de$ection shapes; Finite element; Mode shapes; Damage detection

1. Introduction Damage detection using vibration measurements is being investigated as an alternative to labor-intensive conventional non-destructive evaluation (NDE) methods [1–3] in which a sensor ∗

Corresponding author. Tel.: +1-336-334-7620, Ext. 313; fax: +1-336-334-7417. E-mail address: [email protected] (M.J. Schulz). 0168-874X/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 8 7 4 X ( 0 1 ) 0 0 0 6 1 - 0

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Nomenclature FRF ODS Oh (!; ) dof SLDV x(t); v(t) fm !m Re H j M; C; K a h, d

frequency response function operational de$ection shape ODS vector degree of freedom scanning laser Doppler vibrometer displacement and velocity vectors excitation vector excitation frequency real part receptance FRF matrix √ −1 mass, damping, stiHness matrices phase angle healthy and damaged structure

must be mapped over the surface of the structure, and because hidden damage inside the structure is diIcult to detect using conventional methods. Relatively larger damages located away from a sensor can be detected by changes in the vibration response of the structure. However, interpretation of the vibration responses is often diIcult, and it is desirable to detect damage such as cracking in the initial stage, which has a small eHect on the vibration response. Some approaches store pre-damage or healthy vibration responses, and detect damages by changes in the response [4 –11]. These techniques require storage of large data sets and the response can change due to changes in the environment such as temperature, and also changes in the boundary conditions of a structure. It is also diIcult to discriminate between changes due to the environment and damage. Using a model of the structure can aid in damage detection, but it is diIcult to correlate the healthy model with the healthy structure and this limits the accuracy of damage detection on complex structures [12–19]. Using an input–output model or identifying the dynamics of the healthy structure is another approach used, but again the environmental changes must be removed from the problem [20 –25]. The input–output model approach solves the inverse or reverse problem of detecting and locating damage using vibration measurements. The approaches found in the literature provide a theoretical basis for approximating the damage force magnitude and bounding the damage location between the closest sensors. The down side of the techniques is that the full FRF matrix is needed for all measurement points. This is often not practical because the excitation force cannot be put individually at each point. The use of ODS to detect damage is an approach that is being increasingly investigated. The literature shows that ODS have been used for damage detection on bridges, wind turbines, and other structures [26 –39]. The use of ODS for structural damage detection is promising because the ODS provide a visual interpretation of the vibration patterns of the structure and anomalies can be detected in terms of the geometric features of the structure. A scanning laser Doppler vibrometer (SLDV) can be used to measure the vibration response in a dense spatial pattern

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Fig. 1. Structure with p actuators and the laser sensor.

to generate ODS and detect small damages. For this purpose, the structure can be excited at the high frequencies necessary to detect small damages by using piezoceramic PZT patches or a PZT inertial actuator bonded to the structure. The advantages of using ODS for damage detection, and the diHerences between ODS and mode shapes are discussed in this paper. The derivation of the ODS is presented, FE simulations are performed, and an experiment is presented. Suggestions to improve the technique are also given. 2. Operational deection shape theory This section discusses the measurement and use of ODS. A technique to measure ODS using piezoceramic PZT patches or inertial actuators and a SLDV sensor is presented. The ODS are derived and described in terms of modal data, and use of ODS for damage detection is discussed. The structure with p actuators and a laser sensor is shown in Fig. 1. The excitation measured by the voltage supplied to the individual PZT actuators, is proportional to the force applied to the structure. The laser velocity measurement at a point on the structure is vr (t). The equations of motion for a linear structure with p sinusoidal excitation forces acting on the structure are
p   m M xN + C x˙ + Kx = Re fm ej! t ; (1) m=1

√

where j = −1; t is time, M; C; K are the mass, damping and stiHness matrices, x is the displacement vector, and Re is the real part of the forces. The sinusoidal forces are applied with possibly complex magnitude vectors fm and at the excitation frequencies !m . Herein, the superscript m denotes the mth forcing term. The subscripts used later denote elements of matrices or vectors. To solve (1) for the steady-state response, the displacement response is assumed to be
p   m j!m t x(t) = Re X e : (2a) m=1

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Thus the velocity response is
p   m m j!m t v(t) = Re j! X e ;

(2b)

m=1

X m = X m (j!m )

where is the mth response vector containing complex constants due to the forcing function m. It is important to note in (2) that the steady-state response occurs only at the driving frequencies, and that the total response is a superposition of the responses due to the individual forcing functions. Substituting (2a) into (1) gives
p 
p 
p     m m m Re − (!m )2 MX m ej! t + Re j!m CX m ej! t + Re KX m ej! t m=1




= Re

p 

m=1



fm ej!

m

t

m=1

:

(3)

m=1

Using Re(a + b) = Re(a) + Re(b), (3) becomes
p 
p    m m Re Am X m ej! t = Re fm ej! t ; m=1

(4)

m=1

where Am = (K − (!m )2 M + j!m C) is the system matrix evaluated at frequency !m . The damping matrix C is frequency dependent due to a constant multiplier and it is proportional to stiHness matrix K. The K and C matrices are well conditioned and invertible. The system matrix Am contains the consistent mass matrix M , stiHness matrix K, and damping matrix C. Hence it is a positive de!nite symmetric matrix that is well conditioned and invertible. For gyroscopic systems, the stiHness matrix is not symmetric, and we did not consider gyroscopic systems. To satisfy (4), it is required that X m = H m fm ;

(5)

where H m = (Am )−1 is the receptance (displacement=force) FRF matrix of the system evaluated at frequency !m . Gaussian elimination has been used to invert matrix Am using the identity matrix method to get the FRF matrix H m in Matlab. By considering (5) and (2b), we can de!ne the velocity response in the frequency domain due to the mth forcing function as V m = j!m H m fm :

(6)

From (6), the velocity frequency response at measurement point r due to force m is Vrm = j!m

N 

Hrkm fkm ;

(7)

k=1

where N is the number of measurement points, fkm is the kth row element of fm , and Hrkm is the rth row and kth column element of H m . The mobility FRF is de!ned as the velocity response as a function of frequency measured at point r, divided by a sinusoidal input force of

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frequency ! applied only at point s, which is Vr =fs = j!Hrs :

(8)

The receptance and mobility FRF’s of the structure can be computed from measurements using (8). In this procedure, the sinusoidal force at one location is varied over the frequency range of interest. The force must be individually put at each measurement point to obtain the full FRF matrix. The mode shapes of the structure can then be computed using the amplitude and damping of the FRF’s at resonance. By assuming that damping is proportional or modal, the modes can be calculated using the force at only one measuring point. The ODS is de!ned by evaluating (2) at diHerent angles or times during a steady-state periodic response. The angles are de!ned to be a = !ref ta ;

(9)

where !ref is one speci!c reference or excitation frequency. The ODS can be evaluated at speci!ed angles, a = 2a=b, where b is the number of points in one cycle of vibration to evaluate the ODS, and a = 0; 1; 2; : : : ; b − 1. Therefore, the times to evaluate the ODS are given by ta (!ref ) =

a 2a = : !ref b!ref

(10)

Using (2), (5), (6), and (10), the displacement and velocity ODS are then given by
p     a m x = Re H m fm ej !ref

(11)

m=1

and



a v !ref






= Re

p 



j!m H m fm ej

m

;

where m = !m (a =!ref ). Eq. (12) can also be written as
p    N   a m jm m m = Re j! e hk fk ; v !ref m=1

hm k

(12)

m=1

(13)

k=1

H m.

is the kth column of The ODS can be plotted as mesh plots or fringe contours where by the software supplied with the laser measurement system, or by MATLAB algorithms. Eq. (12) can be used to study the ODS in terms of the force vector, the excitation frequency, and the damping ratio. Another way to understand the ODS is to write H m in terms of modal parameters, i.e., mode shapes and damping ratios. To achieve this, the damping must be modal or proportional as discussed in [1], so that the modes are real and the equations can decouple. ˜ T , where This approach has the modal decomposition: K = RT ; M = T , and C = C 2 ˜  = [1 2 : : : N ]; i are the mode shapes, R = diag(!i ), and C is the diagonal damping matrix in modal coordinates with damping ratios &i . Using the modal substitutions, the FRF

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matrix H m is



1 H =  diag 2 m 2 !i − (! ) + j!m !i 2&i m



T :

(14)

Another way to write H m is Hm =

N 

i Ti : (!i2 − (!m )2 + j!m !i 2&i ) i=1

Thus, from (12)
p    N T   a   m i i v fm ej : = Re j!m !ref (!i2 − (!m )2 + j!m !i 2&i ) m=1

(16)

i=1

Writing (16) as a real quantity gives    p N  a i (Ti fm )((!m !i 2&i ) cos m − (!i2 − (!m )2 ) sin m ) v : = !m !ref ((!i2 − (!m )2 )2 + (!m !i 2&i )2 ) m=1

(15)

(17)

i=1

Eq. (17) can be used to study the ODS in terms of the mode shapes, force vectors, the excitation frequencies, and the damping. From (17), it is obvious that there are almost an in!nite number of possible ODS that can be generated and the ODS are complicated because they depend on forces at diHerent frequencies. To study the ODS in the simplest way, it can be assumed that only one forcing term is present and the excitation frequency of this one term is !ref . In this case (17) simpli!es to   N  a i (Ti f)((!ref !i 2&i ) cos a + (!i2 − (!ref )2 ) sin a ) v : (18) = !ref !ref ((!i2 − (!ref )2 )2 + (!ref !i 2&i )2 ) i=1 Note that the velocity ODS in (18) depends on all of the mode shapes of the structure. If the damping ratios are small and the structure is excited at its ith natural frequency, that is !ref = !i , then the denominator of (18) will be small and the ith term in the series, i.e., the ith modal response, may be the largest contribution to the response. The response in each mode, however, also depends on the degree of orthogonality between that mode and the forcing vector. If (Ti f) = 0, then there will be no response in the ith mode. The eHect of the force is the major diHerence between mode shapes and operational de$ection shapes. The mode shapes are computed from the FRF matrix and do not depend on the forcing function magnitude or location. Conversely, the ODS depends on the location and relative magnitudes of the forces acting at diHerent locations on the structure. In practice, if a single input is used and the structure is excited at a resonance, and damping is small, then the mode shapes and ODS are similar. However, other cases occur when the ODS have non-stationary nodal points, and complicated ODS do not reach their maximum displacements and do not pass through equilibrium at the same time. This “non-modal” behavior has been observed when two excitations are used, when the modal density is high and when a crack is present in a structure.

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Fig. 2. First mode ODS for a healthy !xed–!xed beam and an equivalent beam damaged with a crack.

Operational de$ection shapes computed by the SLDV system for the structure in the healthy and damaged condition can be compared to detect damage. Changes in shape or curvature of the ODSs locate the damage. The ODS are the actual structural response from all modes combined and damping is included. Thus the ODS may be more sensitive in detecting damage because no assumptions or approximation other than the FFT operation are used. The expected !rst mode ODS for a healthy !xed–!xed beam (clamped–clamped) and an equivalent beam damaged with a crack, are shown in Fig. 2. Note the higher amplitude expected in the damaged beam and that can be compared later with the similar results with obtained the FE simulation. A damage indicator is needed to quantify the damage level, based upon the measured ODS. One procedure suggested is similar to the methods of checking the orthogonality of mode shapes. The ODS can be represented as a vector that is dependent on the excitation frequency and the phase angle at which the ODS is evaluated. The ODS vectors for the healthy and damaged structures are de!ned as Oh (!; ) and Od (!; ), where !;  are the excitation frequency and phase angle, and the superscripts h and d represent the healthy and damaged structures, respectively. The damage can be quanti!ed by considering the angle between the healthy and damaged normalized ODS vector. This is written as   h O ◦ Od −1 $(!; ) = cos ; (19) |Oh | |Od | where ◦ denotes the dot product and | | denotes the magnitude of the vector. This indicator is based on the diHerence in the shapes, not the magnitudes of the ODS. 3. Damage detection FE simulation using a beam model A !nite-element model of an Euler–Bernoulli beam with diHerent boundary conditions, damage characteristics, and force combinations is used to study the characteristics of ODS and detecting damage using ODS. Consistent mass and stiHness matrices are used in the analysis [37]. One must determine the minimum number of degrees of freedom (DOFs) in the model required to represent a particular ODS to detect damage accurately. Generally, it can be stated, that a given set of elements can accurately depict the operating de$ecting shape up to a

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Fig. 3. Fixed–!xed beam !nite-element model for damage detection simulation, edge view.

mode number (at the corresponding frequency) that is at most half the number of elements. It follows that at least one beam element is required to depict a half sine wavelength between two consecutive nodes. Accuracy, however, increases when a greater number of elements are used to represent the mode shapes. The parameters that need to be investigated when using ODS for damage detection are: (1) the eHect of boundary conditions (!xed–!xed, !xed–free, pinned–pinned) on damage detection, (2) the eHect of loading on damage detection (one load, symmetry of loading, multiple loads, the load is a mode shape, location of the loading, loads of diHerent frequency applied simultaneously), (3) the frequency range of the excitation, (4) the sensitivity of translational and rotational ODS to damage, (5) the eHect of damage magnitude and location on detection, and (6) the eHect of diHerent phases of the ODS on detecting damage. Also, two hypotheses regarding damage detection are investigated using this FE simulation model. These are: (1) a high frequency range is more sensitive for detecting damage, and (2) when the excitation is close to the damage, the damage will be easier to detect. A FE simulation example is presented in which the eHect of damage on the ODS of a vibrating !xed–!xed uniform beam is determined. The !nite element model of the beam is shown in Fig. 3. The modeled beam is 12 in long, 0:5 in wide, and 1 in deep. It is made of aluminum 6061 T6 alloy material with a weight density of 0:098 lb=in3 and a modulus of elasticity of 1:0e07 ksi. In this example FE simulation, the beam consists of 100 elements unless otherwise noted. The damping ratio for proportional damping (i.e., the damping matrix is proportional to the stiHness matrix) is speci!ed to be 0.03. Damage is modeled as a 10% reduction in all of the values of the elemental stiHness matrix in element number 48 unless otherwise speci!ed. Once the natural frequencies of the healthy and the damaged beam are determined, several FEM simulations are conducted and the ODS are computed for diHerent sets of frequency ranges of a given bandwidth. The frequency ranges generally encompass the natural frequencies of the beam. The healthy and damaged translational ODS have been plotted for the !rst and 10th modes. Case 1 (E=ect of di=erent boundary conditions): The eHect of the boundary condition on detection of damage using ODS has been studied for three diHerent boundary conditions: (a) pinned–pinned (b) !xed–free and (c) !xed–!xed. The FEM model of the beam consists of 100 elements. The excitation is introduced by a unit sinusoidal load at element number 40 in translational DOF 79. The damage is seeded as a 10% reduction in all of the stiHness terms in the 48th element. For the pinned–pinned condition, the !rst natural frequency of the healthy beam is 625:28 Hz and for the damaged beam it is 624:59 kHz. The 10th natural frequency of

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the healthy beam is 75:659 kHz and for the damaged beam is 75:624 kHz. Figs. 4(a–f) show the ODS for the pinned–pinned healthy and the damaged beams encompassing the 1st natural frequency, thus the !rst $exural mode of both the healthy and damaged beams. Figs. 4(a–c) represent the translational ODS while 4(d–f) represent the rotational ODS. Figs. 5(a–f) show the ODS for the pinned–pinned healthy and the damaged beams encompassing the 10th natural frequency, thus the 10th $exural mode of both the healthy and damaged beams. Figs. 5(a–c) represent the translational ODS while Figs. 5(d–f) represent the rotational ODS. The diHerence in amplitude of ODS between the healthy and the damaged beam is distinctly observable in both the translational healthy–damaged ODS (Figs. 4(c) and 5(c)) and rotational healthy–damaged ODS (Figs. 4(f) and 5(f)). The localization of the damage near the 48th element is diIcult to ascertain in the !rst mode for the translational healthy–damaged ODS (Fig. 4(c)). However, the rotational healthy–damaged ODS (Fig. 4(f)) does show a kink=discontinuity and an abrupt change caused by the damaged element. For the 10th mode, localization of the damage near the 48th element is clearer in the rotational healthy–damaged Beam ODS (Fig. 5(f)) than in the translational healthy–damaged ODS (Fig. 5(c)) where a distinct abrupt change in the amplitude causes a signi!cant discontinuity near the 95th and 96th DOF, i.e., the 48th element (Fig. 5(f)). This discontinuity and abrupt change in the sinusoidal waveform of the ODS is caused by the damage which acts like an internal hinge. Clearly, the rotational healthy–damaged ODS gives more information in detection and localization of damage than the translational healthy– damaged ODS. Also the higher frequency modes are a better indicator of damage detection and localization. Similar observations are also seen for other boundary conditions (Figs. 6 –9). The !xed–free condition is shown in Figs. 6 and 7 and the !xed–!xed condition is shown in Figs. 8 and 9. The !rst natural frequency of the healthy beam is 222:75 Hz and for the damaged beam it is 222:69 Hz for the !xed–free boundary condition. The 10th natural frequency of the healthy beam is 68:937 kHz and for the damaged beam it is 68:896 kHz for the same boundary condition. For the !xed–!xed boundary condition, the !rst natural frequency of the healthy beam is 1417:4 Hz and for the damaged beam it is 1416:3 Hz. The 10th natural frequency of the healthy beam is 82:694 kHz and for the damaged beam, it is 82:659 kHz. Figs. 6(a–f) and 7(a–f) show the ODS for the !xed–free healthy and damaged beam encompassing the !rst and the 10th $exural mode, respectively. Figs. 8(a–f) and 9(a–f) show the ODS for the !xed–!xed healthy and damaged beam encompassing the !rst and the 10th $exural mode, respectively. The diHerence in amplitude of ODS between the healthy and the damaged beam is distinctly observable in all the cases of the translational healthy–damaged ODS (Figs. 6(c), 7(c), 8(c) and 9(c)) and rotational healthy–damaged ODS (Figs. 6(f), 7(f), 8(f), 9(f)). The damage cannot be localized at all in the !rst mode translational healthy–damaged ODS (Fig. 6(c)) for the !xed–free boundary condition. In Fig. 6(f), a ridge is observable near the damaged element 48 but localization is not accurate for the !rst mode rotational healthy–damaged ODS for the !xed–free boundary condition. However, in case of the !xed–!xed beam, the !rst mode rotational healthy–damaged ODS (Fig. 8(f)) do clearly indicate the discontinuity near the damaged element. Fig. 8(f) also gives a better localization than the !rst mode translational healthy–damaged ODS (Fig. 8(c)) for the same case. In case of the 10th mode, the rotational healthy–damaged ODS (Figs. 7(f) and 9(f)) give a more accurate localization of the damage than their translational healthy– damaged ODS counterpart (Figs. 7(c) and 9(c)), respectively, because of the distinct change

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Fig. 4. ODS for the pinned–pinned beam in the range of 620 –629 Hz encompassing the !rst mode; 100 elements, 10% damage at EL 48, excitation at DOF 79 at EL40. Translational ODS: (a) healthy (b) damaged (c) healthy–damaged. Rotational ODS: (d) healthy (e) damaged (f) healthy–damaged.

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Fig. 5. ODS for the pinned–pinned beam in the range of 75.6 –75:67 kHz encompassing the 10th mode; 100 elements, 10% damage at EL 48, excitation at DOF 79 at EL40. Translational ODS: (a) healthy (b) damaged (c) healthy–damaged. Rotational ODS: (d) healthy (e) damaged (f) healthy–damaged.

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Fig. 6. ODS for the !xed–free beam in the range of 215 –225 Hz encompassing the !rst mode; 100 elements, 10% damage at EL 48, excitation at DOF 79 at EL40. Translational ODS: (a) healthy (b) damaged (c) healthy–damaged. Rotational ODS: (d) healthy (e) damaged (f) healthy–damaged.

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Fig. 7. ODS for the !xed–free beam in the range of 68.88–68:95 kHz encompassing the 25th mode; 100 elements, 10% damage at EL 48, excitation at DOF 79 at EL40. Translational ODS: (a) healthy (b) damaged (c) healthy–damaged. Rotational ODS: (d) healthy (e) damaged (f) healthy–damaged.

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Fig. 8. ODS for !xed–!xed beam in the range of 1410 –1419 Hz encompassing the !rst mode; 100 elements, 10% damage at EL 48, excitation at DOF 79 at EL 40. Translational ODS: (a) healthy (b) damaged (c) healthy–damaged. Rotational ODS: (d) healthy (e) damaged (f) healthy–damaged.

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Fig. 9. ODS for the !xed–!xed beam in the range of 82.64 –82:71 kHz encompassing the 25th mode; 100 elements, 10% damage at EL 48, excitation at DOF 79 at EL 40. Translational ODS: (a) healthy (b) damaged (c) healthy–damaged. Rotational ODS: (d) healthy (e) damaged (f) healthy–damaged.

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in the amplitude causing a discontinuity near the damaged element (i.e. the 48th element). However, Figs. 7(f), 9(c) and (f) show a similarity in healthy–damaged ODS patterns. The huge diHerence in the ODS amplitude between the two halves separated by the damage and the sudden change in the ODS at the damage (discontinuity) clearly localizes the damage. Fig. 9(f) is also much more de!nitive in damage localization than Fig. 8(f), once again showing the importance of higher modes in damage detection and localization. Also both Figs. 8(f) and 9(f) localize the damage far more accurately than Figs. 6(f) and 7(f), respectively, which means that for !xed–!xed boundary condition it is clearer to detect and localize damage than for the !xed–free boundary condition. This may be because the load transference from the actuator do not take place at the free-end of the cantilever beam. The translational ODS normally give a higher amplitude de$ection than the rotational ODS yet the !gures show demonstratively that rotational healthy–damaged ODS give better accuracy in damage detection and localization for all the cases in both lower and higher frequency ODS. In the translational lower frequency ODS, when the amplitude of the ODS is high, the damage may be too small to make any signi!cant eHect in terms of a discontinuity that would be discernible in the translational ODS. In comparison, the rotational ODS in the lower and higher frequencies and the translational ODS in the higher frequencies have signi!cantly smaller de$ection ODS. Under such conditions, the eHect of the smaller damage has a greater probability to show up as a discontinuity=kink in the structural ODS caused by the internal hinge like behavior. Under these circumstances, the visibility of the damage eHect on the ODS increases considerably when the damage is near or at an anti-nodal point for the high frequency modes. In summation, the eHect of the damage on the ODS has to be large enough in comparison to the amplitude of the ODS at a particular frequency to be observable. The boundary conditions also aHect the detection capability of the present analysis in the lower modes. It is distinctly easier to localize damage for the clamped–clamped beam than for the cantilever beam. Comparing Figs. 4 –7, it is also shown that the pinned–pinned boundary condition (simply supported beam) for the beam is more favorable for damage detection using the present method than the cantilever beam. Clearly, the boundary conditions make a diHerence in the ability to detect damage especially in the lower frequencies. Case 2 (E=ect of the position and type of load): The !xed–!xed healthy and damaged beams from Figs. 8 and 9 are actuated by a unit sinusoidal load applied at the translational DOF 79 at the element number 40. Figs. 10 and 11 show the same beams being excited by a unit sinusoidal load applied at the rotational DOF 10 of element number 5. Figs. 12 and 13 show the same beams being excited by a unit sinusoidal load applied at the translational DOF 9 of element number 5. In all the cases here, the damaged element is element number 48 with 10% damage. It should be noted that the load at the translational DOF actually simulates an inertial mass actuator, while the load at the rotational DOF simulates a PZT patch. Figs. 8, 10, and 12 depict the !rst mode and nearby frequencies, while Figs. 9, 11 and 13 depict the 10th frequency mode and the nearby frequencies. Clearly, Figs. 8 and 9 display higher amplitude displacement followed by Figs. 10 and 11 and then by Figs. 12 and 13, respectively, in the translational and rotational ODS. This indicates how the position and the type of the actuator make a diHerence in the ODS amplitudes, thus aHecting the damage detection sensitivity of the present method. The diHerence in amplitude of ODS between the healthy and the damaged beam is distinctly observable for all the cases of the translational healthy–damaged ODS (Figs. 8(c), 9(c), 10(c),

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Fig. 10. ODS for !xed–!xed beam in the range of 1410 –1419 Hz encompassing the !rst mode; 100 elements, 10% damage at EL 48, excitation at DOF 10 at EL 5. Excitation is introduced at the rotational DOF 10 of the !fth element. Translational ODS: (a) healthy (b) damaged (c) healthy–damaged. Rotational ODS: (d) healthy (e) damaged (f) healthy–damaged.

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Fig. 11. ODS for the !xed–!xed beam in the range of 82.64 –82:71 kHz encompassing the 10th mode; 100 elements, 10% damage at EL 48, Excitation at DOF 10 at EL 5. Translational ODS: (a) healthy (b) damaged (c) healthy–damaged. Rotational ODS: (d) healthy (e) damaged (f) healthy–damaged.

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Fig. 12. ODS for !xed–!xed beam in the range of 1410 –1419 Hz encompassing the !rst mode; 100 elements, 10% damage at EL 48, excitation at DOF 9 at EL 5. Translational ODS: (a) healthy (b) damaged (c) healthy–damaged. Rotational ODS: (d) healthy (e) damaged (f) healthy–damaged.

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11(c), 12(c) and 13(c)) and the rotational healthy–damaged ODS (Figs. 8(f), 9(f), 10(f), 11(f), 12(f) and 13(f)). In all the cases, the rotational healthy–damaged ODS for both the lower and the higher mode (1st and the 10th mode in this case) gives an accurate localization of the damage. The order of change in the discontinuity and the sensitivity of the present method varies depending upon the mode number and the placement of the actuator (ODS (Figs. 8(f), 9(f), 10(f), 11(f), 12(f) and 13(f)). In the translational healthy–damaged ODS, while all of the lower mode !gures do indicate a discontinuity near the damage (Figs. 8(c), 10(c), and 12(c)), all the higher modes (Figs. 9(c), 11(c), and 13(c)) show the sudden and distinct change in the amplitude near the damage element and the considerable diHerence in ODS amplitude between the two halves separated by the damage. However, the localization is better discernible in the rotational healthy–damaged ODS than in the translational healthy–damaged ODS even in the higher modes. When the excitation load is applied near the damage, the translational and the rotational healthy–damaged ODS amplitudes are much higher than when the excitation load is applied away from the damage, thereby actually making the kink discontinuity magni!ed near the damaged element. Similarly, the simulated PZT load gave a higher ODS amplitudes for healthy–damaged cases than the simulated inertial actuator load having a similar eHect on the magni!cation of the discontinuity. In conclusion, the position of the actuating load and the type of actuator do make a diHerence in damage detection sensitivity for this method. A PZT patch near the damage is the optimal con!guration for this damage detection system for the cases presented. Case 3 (E=ect of mixed mode ODS on damage detection): Figs. 14(a–f) show the ODS for the !xed–!xed healthy and the damaged beams encompassing the frequencies around 50 kHz which means the frequency range lies between the eighth and the ninth natural frequencies. Away from the resonant frequencies this ODS is comprised of two or more modes. In this case, the FE beam model consists of 100 elements. The damage is seeded as 10% reduction in all of the stiHness terms in the 48th element. The excitation is introduced by a unit sinusoidal load at element number 40th at its translational DOF 79. Figs. 14(a–c) represent the translational ODS while Figs. 14(d–f) represent the rotational ODS. The diHerence in amplitude of ODS in between the healthy and the damaged beam is shown in the translational healthy–damaged ODS (Fig. 14(c)) and rotational healthy–damaged ODS (Fig. 14(f)). However, the localization of the damage near the 48th element due to the sudden change in the sinusoidal amplitude is diIcult to ascertain because in this case the ODS is an irregular sinusoidal waveform comprised of mixed modes. Case 4 (The e=ect of higher levels of damage): Figs. 15(a–f) and 16(a–f) show the ODS for the !xed–!xed healthy and the damaged beams encompassing the !rst mode and the 10th $exural mode, respectively. In this case, the excitation is introduced by a unit sinusoidal load at element number 40 and translational DOF 79. The damage is seeded as a 75% reduction in all of the stiHness terms in the 48th element. Figs. (15(a–c) and 16(a–c)) represent the translational ODS while the Figs. (15(d–f) and 16(d–f)) represent the rotational ODS. The !rst ←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Fig. 13. ODS for the !xed–!xed beam in the range of 82.64 –82:71 kHz encompassing the 10th mode; 100 elements, 10% damage at EL 48, excitation at DOF 9 at EL 5. Translational ODS: (a) healthy (b) damaged (c) healthy– damaged. Rotational ODS: (d) healthy (e) damaged (f) healthy–damaged.

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Fig. 14. ODS for !xed–!xed beam in the range of 50 –50:020 kHz between the 74th and the 75th modes. The FE beam model consists of 150 elements. Ten per cent damage is seeded in the 74th element for the damaged FE beam model. Excitation is introduced at the translational DOF 119 of the 60th element. Translational ODS: (a) healthy (b) damaged (c) healthy–damaged. Rotational ODS: (d) healthy (e) damaged (f) healthy–damaged.

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Fig. 15. ODS for !xed–!xed beam in the range of 1370 –1430 Hz encompassing the !rst mode; 100 elements, 75% damage at EL 48, Excitation at DOF 79 at EL 40. Translational ODS: (a) healthy (b) damaged (c) healthy–damaged. Rotational ODS: (d) healthy (e) damaged (f) healthy–damaged.

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Fig. 16. ODS for the !xed–!xed beam in the range of 82.18–82:9 kHz encompassing the 10th mode; 100 elements, 75% damage at EL 48, excitation at DOF 79 at EL 40. Translational ODS: (a) healthy (b) damaged (c) healthy–damaged. Rotational ODS: (d) healthy (e) damaged (f) healthy–damaged.

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natural frequency for the healthy beam is 1416:7 Hz and for the damaged beam it is 1388:3 Hz. The 10th natural frequency of the healthy beam is 82:694 kHz and for the damaged beam, it is 81:945 kHz. The healthy and the damaged ODS in both cases of rotational and translational ODS show the distinct phase changes that occur due to the large frequency shifts between the healthy and the damaged beam. This, however, aHects the accuracy as the FE simulation has to cover a wider bandwidth to discern the diHerence between the healthy and the damaged body especially in higher frequency ODS. The diHerence in amplitude of ODS between the healthy and the damaged beam is shown in both the translational healthy–damaged ODS (Figs. 15(c) and 16(c)) and rotational healthy–damaged ODS (Figs. 15(f) and 16(f)). However, in contrast to Figs. 8(c and f) and 9(c and f) where the damage is 10%, in this case the amplitude of healthy–damaged ODS for both the translational and the rotational is greater. There is a distinct amplitude change observable in the healthy and the damaged beam translational ODS in the !rst mode (Figs. 15(a and b)). The localization of the damage near the 48th element due to the sudden change in the sinusoidal amplitude is ascertained in Fig. 15(e) by observing the signi!cant kink=discontinuity near the damage. Fig. 15(e) is the ODS of the damaged beam in its !rst mode. Unlike other cases (Fig. 8(e)) where the damage is 10%, in this case the kink=discontinuity is easily visible in the damaged beam rotational ODS due to higher levels of seeded damage. In Fig. 15(f), the kink is also distinctly observable identifying the damage and the damage location. The kink=discontinuity is probably a manifestation of an internal hinge created due to the damage. In the higher mode (Figs. 16(a–f)), the sudden discontinuity is observable in both of the translational healthy–damaged ODS (Fig. 16(c)) and rotational healthy–damaged ODS (Fig. 16(f)), accurately localizing and identifying the damage. However, the internal hinge like appearance was diIcult to ascertain in the damaged ODS (Fig. 16(e)). Thus, in comparison to the lower level of damage (Figs. 8(a–f) and 9(a–f), a higher level of damage (Figs. 15(a–f) and 16(a–f)) is generally much easier to locate even in the lower modes. Case 5 (E=ect of damage on ODS when close to a ?xed end): In this case Figs. 17(a–f) and 18(a–f) show the ODS for the !xed–!xed healthy and the damaged beams encompassing the !rst mode and the 10th $exural mode, respectively. In this case, the excitation is introduced by a unit sinusoidal load at element number 40 and translational DOF 79. The damage is seeded as 10% reduction in all of the stiHness terms in the !fth element, which is near one of the !xed ends. While the !rst and the 10th natural frequency for the healthy beam as in the earlier cases are 1416:7 Hz and 82:694 kHz, respectively, the frequencies change for the damaged case to 1415:5 Hz and 82:660 kHz, respectively. In comparison to the earlier case of Figs. 8(a–f) and 9(a–f), where the damaged element is the 48th element and far away from the !xed ends, the amplitude of the ODS is considerably less in the present case for both the modes (Figs. 17(a–f) and 18(a–f)). However, the change in the healthy–damaged ODS (Figs. 17(c) and (f), 18(c) and (f)) is noticeable. Also the localization of the damage is actually distinctly noticeable as the kink discontinuity and sudden change in amplitude in the healthy–damage ODS near the !fth element in Figs. (17(f), 18(c), and (f)). Thus, the rotational healthy–damage ODS indicates the damage in both the lower and higher modes while the translational healthy–damage ODS indicates it in the higher mode. Fig. 19 shows the damage indicator for the frequency range 100 Hz–20 kHz. In this case study, 10% damage has been seeded to element number 40. The peaks in this !gure indicate

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Fig. 17. ODS for !xed–!xed beam in the range of 1410 –1419 Hz encompassing the !rst mode; 100 elements, 10% damage at EL 5, excitation at DOF 79 at EL 40. Translational ODS: (a) healthy (b) damaged (c) healthy–damaged. Rotational ODS: (d) healthy (e) damaged (f) healthy–damaged.

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Fig. 18. ODS for the !xed–!xed beam in the range of 82.65 –82:71 kHz encompassing the 10th mode; 100 elements, 10% damage at EL 5, excitation at DOF 79 at EL 40. Translational ODS: (a) healthy (b) damaged (c) healthy–damaged. Rotational ODS: (d) healthy (e) damaged (f) healthy–damaged.

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Fig. 19. Damage indicator for the frequency range of 100 Hz–20 kHz with 10% damage to element 40 of the FE beam model.

the frequencies around which damage localization and detection would be most obvious. The accuracy of the FE simulations depends on the damage model and the number of elements. The FE damage model could be !ne-tuned depending on the type of damage including crack, indentation, distributed fatigue and others. The damage like crack edges can be far more accurately represented by isoparametric elements and denser meshes. 4. Experimentation Two Aluminum (6061 T6) beams having dimensions of length 12 in, width 0:5 in, depth 1 in, are tested in the healthy and damage cases. A crack 0:157 in long was grown into one of the beam by tension testing. By growing a fatigue crack in the beam, the damage is realistic. The crack was grown using a tension machine (Instron Universal Tensile Testing Machine). The beam is placed in the tension machine and a tension test is conducted using 80% of the maximum load (15:2 kips), a frequency of 0:5 Hz, and a sample rate of 0:2 kHz. In 1915 cycles the crack is grown. This process requires that the crack be closely watched to make sure that the beam is not completely fatigued. Once the crack begins to propagate, the machine is suddenly shut oH when the crack reaches the desired length. Each beam is coated with re$ective paint to improve laser re$ectivity. The mass actuator is mounted in the center, on the backside of the beams (Fig. 21). The actuator is driven at various frequency bandwidths or at a speci!ed frequency depending on whether a periodic chirp excitation or sine excitation, respectively, is needed for the testing. The laser scan is done for a rectangular mesh of 200 points on the beams for the healthy and damaged (beam with the fatigue crack) conditions using a SLDV as shown in Figs. 20 and 22. Fig. 20 shows the laboratory instrumentation setup including the SLDV and Fig. 21 shows the front and the side view of the aluminum test beams mounted with the

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Fig. 20. Experimental setup includes the scanning laser Doppler vibrometer, vibration controller, function generator, focus instrument, and the main CPU.

!xed end on a bracket. The scanning was done on the front (outer side which is painted with the re$ective paint) edge. The damaged beam has the crack on the interior side, which is on the opposite side of the scanned edge (Fig. 22). First, the beams are excited with a periodic chirp of various frequency bandwidths to !nd the resonant frequencies. The resonant frequencies were determined by observing the peaks in the FRF measurements. Fig. 23 shows the FRF measurements for both the healthy and the damaged beams, from 0:5 Hz to 10 kHz. To determine the !rst frequency, the beams are excited with the periodic chirp of the frequency bandwidth of 600 –800 Hz. Once the !rst natural frequency of the fundamental mode was determined, the excitation is changed to a sine wave at that frequency to get the maximum de$ection at the !rst mode. The frequency of the !rst mode determined for the healthy case is 728 Hz and for the damaged case is 710 Hz, respectively (Fig. 23). The discrepancies between the experimental results and the computational results are attributed to attachment of the mass-actuator, not perfectly clamped ends, and the frame on which the beams are mounted. The test shows that at the resonant frequencies there are diHerences in de$ection amplitude between the healthy and damaged ODS. The ODS comparison for the healthy and damaged cases at their !rst mode is shown in Fig. 24. The ODS shows that, in the damaged case, the beam experienced some twisting. The twist indicates that there is damage in the beam. The twist is probably due to the non-uniform crack present in the damaged beam and=or the eHect of surface heat-treating. Scans are also made at higher frequencies. ODS at 35; 920 Hz

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Fig. 21. Front view and side view of the aluminum test beams mounted with !xed ends. The beams are mounted with a mass inertial actuator

Fig. 22. Detail of the non-uniform crack pattern.

for the healthy case and 34; 940 Hz for the damaged case are shown in Fig. 25. There is a distinct diHerence in the operational de$ection shapes for the healthy and the damaged cases. Interestingly, in this case, the ODS of the damaged beam shows that the sinusoidal pattern distinctly changes just beyond the crack and to the end of the beam. The amplitude for the

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Fig. 23. ODS of the healthy (red) at 728 Hz and damaged (green) 710 Hz beams at the !rst frequency modes.

Fig. 24. ODS of the healthy (red) at 35; 920 Hz and damaged (green) 34; 940 Hz beams.

Fig. 25. FRF comparison of healthy and damage cases from 0.5 to 10 kHz (red—damaged, green—healthy).

healthy case is also observed to be higher than the amplitude for the damaged case. Clearly, the ODS at higher frequencies are far more sensitive to the damage than the ODS at lower frequencies. In the same test, a FRF comparison is used to describe the frequency shift of the beams. The velocity FRF’s of the beams are computed for the healthy and damaged cases for various

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frequency ranges from 0.5 to 100 kHz. Fig. 23 shows the FRF comparison of healthy and damaged beams from 0.5 to 10 kHz. It shows that the FRF’s for the damaged case shift to lower frequencies but the amplitude of vibration for the damaged case is lower than the response of the healthy structure at some frequencies and higher at other frequencies. 5. Conclusions Detecting damage due to crack propagation using the SLDV and ODS is shown to be an eHective technique. The ODS method was able to detect a crack hardly visible to the eye. It should be noted that the crack existed on the interior side while the scanning was done on the outer side where the crack had not propagated. Also the fatigue crack was a realistic damage to the structure. The FEM simulations and the experimental results show that at high frequency ranges, i.e., at higher natural modes, damage is easier to detect. Also, when the excitation is closest to the damage area the damage is more accurately found. The comparison of the !nite element computer simulation and the laser scan is a good way of optimizing a method of damage detection. The FE model showed that when the beam is modeled using a large number of elements, damage could be predicted accurately for higher modes. In both the FEM simulations and the laser, determination of the natural frequencies is important because excitation at the proper natural frequencies can eliminate the possibility of mixed mode shapes and harmonics. These lead to irregular ODS which make ODS changes due to the damage diIcult to detect and interpret. The FEM simulations also indicate that the ODS method for the detection of damage is sensitive enough to detect at least 10% damage. For the higher frequencies, there is a much higher probability of detecting damage when the damage is located near the anti-nodal point. This is due to the fact that the de$ection is at its highest amplitude at the anti-nodal point for a half wavelength. This is true both for the FEM simulations and the laser experiment. The FEM simulations indicate that the rotational healthy–damaged ODS is more sensitive for damage detection than the translational healthy–damaged ODS. It is however diIcult to measure rotational ODS. The damage detection capability of this method is aHected by the diHerent boundary conditions, placement of the excitation, placement of the damage near or away from the ends, and the level of damage. Acknowledgements This material was supported by NASA Marshall Space Flight Center under grant number NAG8-1646 and the National Renewable Energy Laboratory under grant number XCX-7-16469-01. This support is gratefully acknowledged. References [1] Nondestructive Testing Handbook, 2nd Edition, Vol. 7, Ultrasonic Testing, Paul McIntire, ASM International. [2] P. Cawley, R.D. Adams, The location of defects in structures from measurements of natural frequencies, J. Strain Anal. 14(2) (1979).

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