Engineering Structures 30 (2008) 3497–3504

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Engineering Structures

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

Damage detection by means of structural damping identificationR.O. Curadelli a,∗, J.D. Riera b, D. Ambrosini a, M.G. Amani aa Laboratory of Experimental Dynamic, National University of Cuyo, CONICET, Mendoza, Argentinab Laboratory of Structural Dynamic and Reliability, Federal University of Rio Grande do Sul Porto Alegre, Brazil

a r t i c l e i n f o

Article history:Received 12 November 2007Received in revised form18 May 2008Accepted 19 May 2008Available online 2 July 2008

Keywords:Structural damageDynamic properties identificationInstantaneous frequencyInstantaneous damping coefficientWavelet transform

a b s t r a c t

Vibration measurements in structures under ambient or controlled excitation open a way to damageassessment by correlating changes in the dynamic system parameters – natural frequencies, modalshapes or damping – with damage. Most methods proposed in the literature are based on themeasurement of natural frequencies or modal shapes, associating damage to local reduction of thestructural stiffness while completely ignoring damping, both in the identification procedure as wellas in the correlation of changes in the spectral properties of the structure with damage. In numerousapplications thesemethods, however, present some practical limitations on account of the low sensitivityof natural frequencies or modal shapes to damage. The main objective of this paper is to present a novelscheme to detect structural damage by means of the instantaneous damping coefficient identificationusing a wavelet transform. Laboratory tests as well as numerical simulations showed that in manycommonly used structural systems damage causes important changes in damping. Thus, parameters thatcharacterize structural damping can be used as damage-sensitive system properties. It is further shownthat the wavelet transform can be conveniently employed in a procedure for instantaneous frequenciesand damping identification from free vibration structural response.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Structural monitoring and damage detection are areas of cur-rent interest in civil, mechanical and aerospace engineering. Mostdamage detection methods are visual or localized experimentalprocedures such as acoustic or ultrasonic methods, magnetic fieldmethods, radiographs, eddy-current and thermal fieldmethods. Allthese experimental techniques require that the location of dam-age be known a priori and that the portion of the structure underinspection be easily accessible. These limitations led to the devel-opment of global monitoring techniques based on changes in thevibration characteristics of the structure.Damage or fault detection by monitoring changes in the

dynamic properties or response of the structure has receivedconsiderable attention in recent literature. The basic idea springsfrom the notion that spectral properties, described in terms ofthe so-called modal parameters (frequencies, mode shapes, andmodal damping), are functions of the physical properties of thestructure (mass, energy dissipation mechanisms and stiffness).

∗ Corresponding author.E-mail addresses: ocuradelli@fing.uncu.edu.ar (R.O. Curadelli),

riera@cpgec.ufrgs.br (J.D. Riera).URLs: http://www.fing.uncu.edu.ar (R.O. Curadelli), http://www.ufrgs.br

(J.D. Riera).

0141-0296/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2008.05.024

Therefore, changes in the physical properties will cause changes inthe modal properties. Literature reviews on damage identificationand health monitoring of structures based on changes in theirmeasured dynamic properties were published, for example, byDoebling et al. [1,2], Zou et al. [3], Sohn et al. [4], Salawu [5] andChang et al. [6]. Moreover, in large civil engineering structuresit is usually impossible to measure all components of the inputexcitation. Experimental modal identification through ambientvibration response records, in which only the structural output ismonitored, has consequently become a very attractive alternative.Structural damage usually results in a decrease in mass,

stiffness and strength of structural elements. In consequence, itsdetection and localization have great practical importance and ledto the rapid growth of a research field rather incorrectly known asStructural Health Monitoring (SHM) [7], although what is actuallymonitored is the structural integrity and it would sound awkwardto qualify any structure as healthy. At any rate, four SHM categoricallevelswere proposed to quantify damage in engineering structures[8]: Level 1: damage detection; Level 2: damage location; Level 3:quantification of the degree of damage; Level 4: estimationof the remaining service life. For SHM purposes, perhaps themost frequently used techniques are vibration-based methods.Vibrationmethods are based on the fact that damage in a structureusually implies a local stiffness reduction or increase in flexibility,which induce changes in its dynamic properties. These changes canbe used for damage identification.

3498 R.O. Curadelli et al. / Engineering Structures 30 (2008) 3497–3504

The amount of literature describing damage detection schemesbased on shifts in the natural frequencies is quite large. Theforward problem, which usually falls into the category of Level 1damage identification, consists of calculating frequency shifts for aknown type of damage. Typically, damage is modelled numericallyby a reduction in the stiffness and then the measured frequenciesare compared to the frequencies predicted by themodel.Moreover,this technique is used extensively in damage diagnosis and healthmonitoring of existing highway bridges, seismic behaviour andresidual capacity of structures.It should be noted that frequency shifts have practical

limitations in several applications, especially in case of largestructures. Farrar et al. [9,10] conducted tests on the I-40 Bridgeand verified that for a reduction in the bending stiffness of theoverall bridge cross section equal to 21%, no significant reduction inthe modal frequencies was observed. Chen et al. [11] investigatedchanges in natural frequencies on steel space structures subjectedto atmospheric corrosion showing that atmospheric corrosion doesnot perceptibly affect the natural frequencies of the structure.Several references cited in literature reviews by Doebling et al.[1], Salawu [5] and Sohn et al. [4] stress that the usually smallfrequency shifts caused by damage require either very precisemeasurements or high levels of damage. Ongoing research may,however, provide ways to circumvent these difficulties.Damping properties, on the other hand, have seldom been

used for damage diagnosis. Crack detection in a structure basedon damping, however, has advantages over detection schemesbased on frequencies and modal shapes. In fact, when cracksresult in small or no frequency variations, damping changes mayrender the detection of the nonlinear, dissipative effects thatcracks produce. Modena et al. [12] show that visually undetectablecracks cause very little change in resonant frequencies andrequire higher mode shapes to be detected, while these samecracks cause larger changes in damping. In some cases, dampingchanges of around 50% were observed. Their study focuses onidentifying manufacturing defects causing structural damage inprecast reinforced concrete elements. The particular dynamicresponse of reinforced concrete justifies the use of damping asa damage-sensitive property, proposing two new methods basedon changes in damping to detect cracking. These techniques areemployed to locate cracks in a 1.20× 5.80 m precast hollow floorpanel excited with stepped sins and shocks, and the diagnosisresults are comparedwith those of frequency and curvature-basedapproaches.Kawiecki [13] measures damping in a 90× 20× 1 mm metal

beam and metal blanks used to fabricate 3.5-inch computer harddisks, noting that damping can be a useful damage-sensitiveindicator. Modal damping is determined using the half-powerbandwidth method applied to the FRF within the frequency rangeof 5–9 kHz obtained from measured data. The author claims thatthe approach should be particularly suitable for structural healthmonitoring of lightweight structures and microstructures.After conducting vibration tests on prestressed reinforced

concrete hollow panels, Zonta et al. [14] observe that crackingproduces a frequency splitting in the frequency domain and thebeat phenomenon of the free decay signals in the time domain. Itis argued that crack formation in prestressed reinforced concretecauses a no viscous dissipative mechanism, making dampingmoresensitive to damage, proposing thereby to resort to this dispersivephenomenon for damage diagnosis. Zonta et al. [14] point outthat the dispersive phenomenon cannot be represented by thestandard linear model of a single degree of freedom system andemphasize that the oscillator has a variable stiffness, and thisvariability is proportional to the frequency. They recognize theneed for further research using the dispersion phenomenon fordamage detection because additional effects such as hysteresis and

other non-classical dissipative mechanisms must be considered. Itis important to note that these difficulties to accurately identifymodal damping led to a scattering higher than the frequency, as itis showed below.Recent advances in signal processing based on the Hilbert–

Huang transform [15] and wavelet analysis opened a new perspec-tive on the development of the system identificationmethodologies[16–19] and on SHM [20–22].In this paper a new scheme to detect structural damage em-

ploying the instantaneous damping coefficient identification usingthewavelet transform is presented. Experimental and simulated ex-amples confirm that, in the structures analyzed, damage causesimportant variation of the coefficient that characterizes damp-ing, which proves therefore to be a generally damage-sensitivesystem property. Because the wavelet estimation technique re-quires free decay response observations, which is rarely feasibleon large structures, the modal identification herein proposed is in-deed an appealing technique. Ambient excitations such as trafficor wind have the advantage that no equipment is needed to excitethe structure. To convert ambient random response to free decayresponses the well established Random Decrement Technique wasused [38,39,18].

2. Identification of instantaneous undamped natural fre-quency and damping coefficient

Although various complex phenomena occur when differentmaterials/structural elements are damaged, principally underseismic excitation, only two of those phenomena are commonlyassociatedwith damage: stiffness and strength degradation. Hencemost models describing the hysteretic behaviour of structuralmembers have been derived from the shape of experimentalforce–displacement relationship (hysteretic curves). Examples ofmacro models proposed to represent the complex behaviour ofreinforced concrete beams or beam-columns are Takeda-type [23]and their variations [24,25], which simulate phenomena such asstiffness and strength degradation and pinching. An oscillatorwith such behaviour can be modelled by second-order systemswith nonlinear restoring force k(x)x and nonlinear damping forceho (x) x and represented by the equation x+2ho (x) x+k (x) x = 0,which leads to a free response of the form:

x(t) = A(t) cos(ϕ(t)). (1)

Note that functions ho and k represent apparent (viscous)damping and apparent (elastic) stiffness coefficients, respectively.In the following they will be referred to simply as damping andstiffness coefficients.Feldman [26,27] showed, by applying the multiplication

property of the Hilbert Transform for overlapping functions tothe equation of motion for viscously damped systems, that theinstantaneous undamped natural frequency and the instantaneousdamping coefficient may be calculated according to the formulae:

ω2o (t) = ω2−AA+2A2

A2+AωAω,

ho (t) = −AA−ω

2ω

(2)

in which ωo(t) is the instantaneous undamped natural frequencyand ho(t) the instantaneous damping coefficient of the system.A(t) and ω(t) are the envelope (instantaneous amplitude) and theinstantaneous damped angular frequency of the vibratory systemsolution with their first and second time derivatives

(ω, A, A

).

The envelope A(t) and the instantaneous phase ϕ(t) can beextracted from the vibration signals employing the continuouswavelet transform [28,29], as follows:

R.O. Curadelli et al. / Engineering Structures 30 (2008) 3497–3504 3499

According to analytic signal theory, the analytic signal xa(t) =x(t) + jH(x(t)) of a large number of processes including thevibration of a under-critically-damped system described by Eq. (1),where it is assumed that the phase varies much faster than theamplitude (asymptotic signal), can be approximated by a complexsignal (Torresani [30]):

xa(t) = A(t)ejϕ(t) (3)

in which A(t) and ϕ(t) were defined before and H(x(t)) is theHilbert transform of x(t):

H [x (t)] =1π

∫+∞

−∞

x (τ )1t − τ

dτ . (4)

The continuous wavelet transform of an asymptotic signal x(t) isdefined as (Chui [31]):

Wg [x] (a, b) =1√a

∫+∞

−∞

x (t) g∗(t − ba

)dt (5)

where g(t) is the analyzing or mother wavelet and g∗(t) thecomplex conjugate of g(t), any function g(t) can be used asan analyzing wavelet if it satisfies the admissibility condition(Chui [31]); b is the parameter localizing the wavelet function inthe time domain: Wg [x](a, b) shows the local information aboutx(t) at the time t = b (translation) and a is the dilatationor scale parameter defining the analyzing window stretching.The idea of the continuous wavelet transform is to decompose afunction x(t) into wavelet coefficientsWg [x](a, b) using the basisof wavelet functions. The wavelet coefficients represent a measureof the similitude between the dilated and shifted wavelet and thefunction x(t) at time b and scale a.In terms of the analytic signal the continuouswavelet transform

of an asymptotic signal x(t) is:

Wg [x] (a, b) =12√a

∫+∞

−∞

A (t) ejωtg∗(t − ba

)dt. (6)

Developing, the Eq. (6) by Taylor’s formula for t = b, using theFourier transform G∗(aω) of g∗[(t − b)/a] and neglecting terms ofsuperior order:

Wg [x] (a, b) =√a2A(b)ejωbG∗ (aω) . (7)

From Eq. (3) the concept of instantaneous angular frequency asthe time varying derivative of the phase can be derived: ω (t) =ϕ (t).Using the complex Morlet wavelet in the frequency domain,

G (aω) = e−12 (aω−ωc )

2as analyzing function; Eq. (7) can be written

as:

Wg [x] (a, b) =√a2A(b)ejϕ(b)e−

12 (aϕ(b)−ωc )

2(8)

in which ωc is the wavelet central frequency.The square of themodulus of the continuouswavelet transform

can be interpreted as an energy density distribution overthe time–scale plane. The energy of the signal is essentiallyconcentrated on the time–scale plane around a region called theridge of the continuouswavelet transform. In otherwords, the ridgeof the continuous wavelet transform is the region containing thepoints defined by a = a(b), where the amplitude of the continuouswavelet transform is maximum. The ridges are identified byseeking out the points where the continuous wavelet transformcoefficients take on local maximum values: for each value of b,weobtain a value as |Wg [x] (a(b), b) | = maxaWg [x] (a, b). To obtainthe ridge, the dilatation parameter a = a(b) has to be calculated inorder to maximize the analyzing wavelet G∗ (aϕ(b)), that is, using

the complex Morlet wavelet, for a = a(b) = ωc/ϕ(b). It can beobtained:

Wg [x] (a(b), b) =√a (b)2A(b)ejϕ(b). (9)

The values of the continuous wavelet transform that arerestricted to the ridge are the skeleton of the continuous wavelettransform.It is important to note that the real components of the continu-

ous wavelet transform along the ridge are directly proportional tothe signal given by Eq. (1) and from Eq. (9) we obtain

A(b) = 2|Wg [x](a(b), b)|√a(b)

(10)

ϕ(b) = Arg(Wg [x] (a(b), b)

). (11)

Finally, using the Eqs. (10) and (11) it is possible to estimatethe instantaneous amplitude A(t) and the phase ϕ(t). Being thederivative ofA(t), A(t) and the derivative ofϕ(t), the instantaneousdamped angular frequency ω (t) = ϕ(t), in conjunction withthe Eq. (2), the instantaneous undamped natural frequency anddamping of the system can be determined.This procedure establishes an identification technique to deter-

mine the undamped natural frequency and damping coefficient ofthe system as instantaneous functions of time at every point of thevibration process.Employing thismethod, the evolution of the systemparameters

with increasing damage will be analyzed in numerical andexperimental examples.

3. Changes in frequency and damping with damage

In nonlinear systems (nonlinear dissipative and elastic forces)the damping coefficient and the natural frequency become func-tions of the vibration amplitude. With the aim of determining theevolution of the dynamic system parameters (undamped naturalfrequency and damping coefficient) with vibration amplitude andincreasing damage, three examples were analyzed.

3.1. Numerical simulation of the response of 2D reinforced concreteframe with strength and stiffness degradation

In order to assess changes in the system parameters causedby damage, in this section, a 2D reinforced concrete frame andmodelled with inelastic elements described by Takeda-based [23–25] constitutive rules capable of representing hysteretic stiffnessdegradation and pinching was analyzed using the Drain 2D [32]software. The example is a three bays, six-stories high reinforcedconcrete frame, designed for a peak ground acceleration (PGA)of 0.17 g (value that presents a 10% probability of exceedancein 50 years, in accordance with the provisions of the ArgentineCode INPRES-CIRSOC 103 [33] or ACI, which in this example resultin similar designs (Curadelli and Riera [34]). The total mass perfloor is 100 t, Young’s modulus of concrete E = 24.8 GPa whichlead to a fundamental period T1 = 1 s. The internal dampingratio was assumed equal to 5% of critical. In the nonlinearconstitutive relations, the yield strength of reinforcing steel andthe compressive strength of concrete were assumed equal to fy =413 MPa and fc = 27.6 MPa, respectively. Fig. 1 shows the basicdimensions of the frame.The structure was subjected to the Caucete, San Juan, Argentina

1997 seismic acceleration time history. From the last 10 s of theroof response, which is actually a free vibrations record, changesin the system parameters were determined by the proceduredescribed above. In order to assess different degrees of structural

3500 R.O. Curadelli et al. / Engineering Structures 30 (2008) 3497–3504

Fig. 1. Reinforced concrete 2D frame structure.

damage, the record was normalized to four intensity levels [1, 2, 3and 4 m/s2 (collapse)], defined in terms of their PGA.Fig. 2(a) and (b) show the free response and the corresponding

envelopes, determined by the identification technique, for the firstand ultimate degrees of damage.Figs. 3 and 4 illustrate the dependence of the instantaneous

fundamental frequency and instantaneous damping coefficient onthe amplitude of vibration, respectively, for each damage level.Figs. 3 and 4 provide valuable evidence for the influence of dam-

age on the dynamic properties of the reinforced concrete frame.The maximum undamped fundamental frequency reduction, ob-served for a PGA = 0.4 g, reaches nearly 25%, increasing linearlywith the PGA. Damping, on the other hand, presents more pro-nounced and typical variations.

3.2. Experimental study of a reinforced concrete beam

The performance of the identification procedure was alsoassessed using experimental data, for which purpose a reinforcedconcrete beam tested by Palazzo [35] was analyzed. The specimenwas a 0.20 m × 0.10 m × 5.60 m reinforced concrete beam inflexure under two points loading (Fig. 5(a) and (b)). The yield

Fig. 3. Instantaneous undamped fundamental frequency. Lowest excita-tion level (PGA = 1.0 m/s2 , undamaged), highest excitation level (PGA =4 m/s2 , severe damage).

Fig. 4. Instantaneous damping coefficient. Lowest excitation level (PGA =1.0 m/s2 , undamaged), highest excitation level (PGA = 4 m/s2 , severedamage).

strength of reinforcing steel bars and the compressive strength ofconcrete were fy = 420 MPa and fc = 17 MPa, respectively.Four KYOWA AS-GB accelerometers (accuracy 100 mV/g) wereused to measure the dynamic response (vertical acceleration atdifferent points) of the structure, sampled at 500 Hz each channel.The signals generated by the accelerometers were amplified bya dynamic strain amplifier KYOWA DPM-612B and recorded by

Fig. 2. Free vibration response and envelope. (a) PGA = 1 m/s2 and (b) PGA = 4 m/s2 .

R.O. Curadelli et al. / Engineering Structures 30 (2008) 3497–3504 3501

Fig. 5. Sketch of reinforced concrete simply supported beam tested (b) load location.

Fig. 6. Reinforced concrete beam test setup.

a Computerboars PCM-DAS16D/16 data acquisition board. Fordetails, see Palazzo [35]. Loads were increasedmonotonically untilthe intensities indicated in Table 1 were reached, as schematicallyshown in Fig. 6. At that point loadswere removed and free vibrationtests conducted to determine dynamic properties. The procedurewas then repeated for the next loading step. Fig. 7(a–b) shows theobserved free vibration response of the undamaged state and afterthe last loading step, respectively.The correlations between the instantaneous fundamental fre-

quency and instantaneous damping coefficient and the amplitudeof vibration, for the healthy and damaged condition (after LoadStep 5), are presented in Figs. 8 and 9, respectively. The scatter-ing of parameters for ten samples in each state represented by onestandard deviation to both sides of mean values are indicated indashed line.

Table 1Load step values (from Palazzo [35])

Loadstep

Load (kN) Fundamental frequency measured after each load step

0 (undamaged) 4.521 2× 1.46 4.342 2× 2.25 4.293 2× 3.23 4.254 2× 5.19 4.015 2× 7.25a 3.63a Maximum load considered (correspond to 3.50/00 , maximum compress strain,INPRES-CIRSOC 201 [41]).

Figs. 8 and 9 show the instantaneous modal parameters alongtime for two different levels of damage. It is clear that dampinghas an important variation.

3.3. Experimental 3D frame model

As a second experimental example, the one bay, six-storeyshigh aluminium 3D frame model tested by Amani et al. [36] wasanalyzed (Fig. 10). Geometric properties are indicated in Table 2(more details are reported in Amani [37]. Structural damage wascaused by subjecting the model to a horizontal unidirectional basemotion in a shaking table. The excitation consisted of a series ofnine simulated acceleration time histories (Gaussian white noise)with increasing intensity. The standard deviations of the baseacceleration in the tests were σa = 1.8; 2.1; 6.1; 12; 20; 32;38; 44; and 48 m/s2. As it was mentioned before, the wavelettransform procedure for modal parameters identification operateson the free vibration. Thus, in order to obtain free decay response,

Fig. 7. Free vibration response and his envelope: (a) for healthy condition and (b) after load step 5 test. (Palazzo [26]).

3502 R.O. Curadelli et al. / Engineering Structures 30 (2008) 3497–3504

Fig. 8. Instantaneous undamped fundamental frequency.

Fig. 9. Instantaneous damping coefficient.

Table 2Geometrical and mechanical properties of the model

Total height 0.50 mStory height 0.083 mSpan length 0.10 mColumns cross section Stories 1&2 0.6× 18 (mm)

Stories 3&4 0.6× 12 (mm)Stories 5&6 0.6× 6 (mm)

Total mass 0.665 kgMass density (columns) 2698 kg/m3Young’s modulus 67 GPa

themeasured horizontal acceleration response at the top floor wasprocessed by random decrement technique [38,39,18].Fig. 11(a–b) show free vibration normalized acceleration

response at the top floor obtained from random decrementtechnique for the lowest and highest excitation level, respectively.Figs. 11 and 12 show for both, lowest excitation level (σ a =

1.8 m/s2, undamaged) and highest excitation level (σ a =

48 m/s2, severe damage), the dependence of instantaneousfundamental frequency and instantaneous damping coefficienton the instantaneous amplitude of vibrations, respectively. Thescattering is indicated by one standard deviation to both sides ofthe mean values.Similarly, Figs. 12 and 13 show that increasing excitation yield

to a damage level reducing the undamped fundamental frequencyby about 9%, while damping reveals more marked variations.It is important to point out that the relationship between

instantaneous damping coefficient and vibration amplitude for

Fig. 10. One bay, six-storeys high aluminum 3D framemodel tested by Amani et al.[37].

every example is similar to those obtained by harmonic lineariza-tion of a bilinear oscillator [40]; after reaching a maximum valuefor a certain value of amplitude, a smooth decrease takes place.

4. Conclusions

For the kind of structures analyzed, a damage detection tech-nique using the instantaneous damping coefficient as sensitive-damage feature, has shown that it is potentially useful. It seemsquite clear that, in systems as analyzed, damping is a promisingdamage indicator in structural health monitoring because it hasmore sensibility to damage than the natural frequency. As shownon Figs. 8 and 9 for reinforced concrete beam and 12 and 13 forthe experimental model, the statistical variations of damping arehigher than the natural frequencies, however, the results are notmasked. The paper describes an approach, based on awavelet trans-form, to detect structural damage by means of the instantaneousdamping coefficient identification.With examples using experimental results and numerical

simulation on structures subjected to seismic base excitation,it was demonstrated that the identification technique based onwavelets is useful to assess changes in the vibration characteristicsdue to incremental damage of nonlinear systems.Since thewavelet transform is a signal decomposition procedure

working as a filter in the time–frequency domain defined by thelocalizing parameter and dilatation parameter, it is possible toanalyse a signal only locally. Thus, a multidegrees of freedomsystem can be decoupled into single degrees of freedom. For a fixedvalue of the dilatation parameter (a = ai), which maximises theamplitude of the continuous wavelet transform, only the modeassociated with ai gives a relevant contribution, while all the otherterms are negligible. Hence, it is possible to choose the value of thedilatation parameter ai corresponding to the analysed mode andby means of Eqs. (10) and (11) to determine the amplitude and thephase variations in the time domain of this mode.It is also stressed that this work is a start point for damage

identification based on damping evaluation, and further researchis needed in the subject.

Acknowledgements

The authors wish to thank the financial aid of CONICET andSECyT, Argentina and CNPq and CAPES, Brazil.

R.O. Curadelli et al. / Engineering Structures 30 (2008) 3497–3504 3503

Fig. 11. Free vibration normalized acceleration response and the envelope for: (a) lowest and (b) highest excitation level. One bay, six-storey high aluminium 3D framemodel tested by Amani et al. [37].

Fig. 12. Instantaneous undamped fundamental frequency. Lowestexcitation level (σ a = 1.8 m/s2 , undamaged), highest excitation level(σ a = 48 m/s2 , severe damage).

Fig. 13. Instantaneous damping coefficient. Lowest excitation level (σ a =1.8 m/s2 , undamaged), highest excitation level (σ a = 48 m/s2 , severedamage).

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