DAMAGE DETECTIONS IN NONLINEAR VIBRATING THERMALLY LOADED STRUCTURES 1

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1 11 th National Congress on Theoretical an Applie Mechanics, 2-5 Sept. 2009, Borovets, Bulgaria DAMAGE DETECTIONS IN NONLINEAR VIBRATING THERMALLY LOADED STRUCTURES 1 E. MANOACH Institute of Mechanics, Bulgarian Acaemy of Sciences, Aca. G. Bonchev Street, Bl. 4, 1113 Sofia, Bulgaria e.manoach@imbm.bas.bg ABSTRACT. In this work, geometrically nonlinear vibrations of fully clampe beams an plates subjecte to thermal changes are use to stuy the sensitivity of some vibration response parameters to the presence of amage an elevate temperature. The geometrically nonlinear version of the Minlin plate theory an Timoshenko beam theory are use to moel the structure behaviour. Damage is represente as a thickness reuction in a small area of the structure. The structures are subjecte to harmonic loaing leaing to large amplitue vibrations an temperature changes. The main results are focusse on establishing the influence of amage on the vibration response of the heate an the unheate structures an the change in the time-history iagrams an the Poincaré maps cause by amage an elevate temperature. The amage etection criterion formulate earlier for non-heate plates, base on analyzing the points in the Poincaré sections of the amage an healthy plate, is moifie an teste for the case of plates an beams aitionally subjecte to elevate temperatures. The importance of taking into account the actual temperature in the process of amage etection is shown. KEYWORDS: Plates, beams, amage etection, Poincaré maps, thermal loas 1. Introuction The main objective of structural health monitoring is to ascertain whether amage is present or not in a structure. Vibration base structural health monitoring methos (VSHM) are wiely use for structural health monitoring an amage assessment purposes. They are base on the fact that amage will alter the stiffness, mass or energy issipation properties of a structure which in turn will alter its measure vibration response. Most of the previous efforts of researchers in the area of VSHM were irecte towars methos base on linear moal analysis [1-5]. One of the main problems with these methos comes from the fact that in general amage starts as a local phenomenon an 1 The author wishes to thank the Bulgarian Research Fun for the support through grant TN-1518/2005.

2 E. Manoach oes not necessarily affect significantly the moal characteristics of the structure. In many cases the lower orer resonance frequencies an moe shapes are not very sensitive to amage, except in cases of very large amage [3]. Another problem with a number of VSHM methos is that they rely on a linear moel of the structure. As the theoretical moel itself can only approximate the actual behaviour of the vibrating structure, it will introuce computational errors [3]. These errors will be greater if the non-linearities of the system are substantial. Since they are not taken into account in the moel such methos might give false alarms ue to a iscrepancy between the measure an the moelle/expecte response. Temperature changes can an o affect substantially the vibration response of a structure. Thermal loas introuce stresses ue to thermal expansion, which lea to changes in the moal properties. Thermal loas can also cause buckling an in some cases even lea to chaotic behaviour [6-10]. Thus, on a lot of occasions the presence of a temperature fiel can either mask the effect of amage or increase it, which will rener a VSHM metho ineffective - it might give no alarm when a fault is present or give a false alarm. This is why it is vital to be able to take into account the temperature changes when eveloping VSHM proceures. To aress some of the above mentione problems, new concepts in vibrationbase monitoring have been emerging recently. These employ measure time series of the structural vibration response, or, often concomitantly, non-linear systems theory. Most of the stuies in this fiel are evote to the extraction of features from the structural vibration response, which can inicate the presence of amage an its location. In [11] the authors use the beating phenomenon for amage etection purposes. In [12] an [13 ] new attractor-base metrics are introuce as amage sensitive features. The results are promising. In our previous works [14] an [15] a numerical approach to stuy the geometrically non-linear vibrations of rectangular plates with an without amage is evelope. A amage inex an a metho for amage etection an location, base on the Poincaré map of the response, have been propose. The suggeste amage assessment metho shows goo capability to etect an localize amage in plates. Although the approach seems to hol a lot of potential, there is limite research aressing VSHM methos base on time series analysis an non-linear ynamics. The main objectives of this stuy are twofol: (i) to stuy the influence of efects, elevate temperatures an their combination on the ynamic characteristics of the plates an beams an on its geometrically nonlinear ynamic response; (ii) to test the criteria for ientification of irregularities (efects) in structures propose in [14, 15] taking into account the elevate temperature by analyzing the Poincaré map of the structural vibration response. The application of the propose approach is emonstrate on rectangular plates an beams with efects at elevate temperatures. 2. Theoretical moel The objects of the investigation are beams an rectangular plates. Their geometry, imensions an the coorinate systems are shown on Fig.1. The geometrically

3 Damage etections in structures nonlinear version of the Minlin plate theory an the geometrically nonlinear version of the Timoshenko beam theory are use to moel the structure behaviour, so that the shear eformation an rotatory inertia are taken into account. At each point of the mile surface of the plate, the isplacements in the x, y, z irections are enote by u, v, w, respectively. ψx( x, yt, ), ψ y( xyt,, ) are the angles of the rotation of the normal of the cross section to the plate mi-plane (see Fig. 1 c). The enotations in the case of beams are similar. The presence of a efect can be moelle as a reuction of the structure thickness or a stiffness reuction an therefore a variation of the flexural rigiity in the governing equations is use. Only the basic equations of the plate motion are escribe below because the beam can be consiere as a particular case of a plate. z y h x l a) b O(x,y,z) x p(x,y,t) h x y w x,u y a b b) c) Fig. 1. Structures geometry an coorinate system. a, b) Beam an plate imensions imensions an loaing, c) Mi-plane of the plate an the components of the generalize isplacement vector. 2.1 Constitutive equations Assuming that the material of the plate is linear elastic an isotropic the relations for the stress an strain components are given by: σ E( x, y) ε νε E( x, y) = + ν 1 ν α Δ σ E( x, y) = 1 ν ε + νε E( x, y) x x y T T, y y x 1 ν α TΔT, 2 2 (2.1 a-) σ xz = ngεxz, σ yz = ng εyz In terms of generalize stresses the above equations take the form : y,v z,w a ψ x ψy b

4 E. Manoach 0 0 T 0 0 T 1 0 N x = A( ε x + νε y) Aα Tγ, Ny = A( ε y + νε x) Aα Tγ, N v xy = Aε xy 2 o o T 0 0 T Mx = D( κx + νκy) AαTκ, M y = D( κy + νκx) AαTκ, (2.2a-h) Mxy = (1 ν ) Dκxy, Qx = (1 ν) n Aεxz, Qy = (1 ν) n Aεyz where (2.3 a-) h /2 T T γ ( x, y) = ΔT( x, y, z) z, κ ( x, y) = ΔT( x, y, z) zz, h /2 h /2 h /2 E( xyhxy, ) (, ) Axyhxy (, ) (, ) Axy (, ) =, Dxy (, ) = 2 1 v 12 In Eqns ( ) E is the Young moulus, ν is the Poison ratio, N x, N y an N xy are the stress resultants in the mi-plane of the plate, M x, M y an M xy are the stress couples an Q x an Q y are the transverse shear stress resultants, α T is the coefficient of thermal expansion an ΔT (Kelvin) is the temperature variation ( in general it can be assume non-uniform along the plate length an thickness) with respect to a reference temperature. n 2 is a shear correction factor which is assume equal to 5/6 throughout the paper. 2.2 Equations of motion The equilibrium equations may be eucte by consiering the conitions for translational equilibrium in the x, y an z irections an for rotational equilibrium about x an y. They are as follows: N N x xy Ny Nxy + + ρhu&& x = ρhu&& y = 0 x y y x 3 3 M M x xy ψ x ρh y xy y x 2 x 0 M M ψ ρh + Q + c + && ψ = + Qy + c2 + && ψ y = 0( x y t 12 y x t a-e) Q Q x y w w w w + + Nx + N 2 2 y + N 2 xy + c1 + ρhw&& = p x y x y x y t Here an throughout in the paper ots over variables represents erivation with respect to time, c 1 an c 2 enote the amping coefficients, an ρ is the ensity of the plate material Bounary an initial conitions In the present work fully clampe beams an plates, i.e. plates for which all their four eges are clampe an in-plane fixe, are consiere. This means that all isplacements u, v an w an angular rotations ψ x anψ y are zero along the bounaries. The influence of the temperature variation is more essential for such plates ue to the thermal expansion. 2

5 Damage etections in structures The initial conitions are accepte in the following general form: 0 0 w( x, y, 0) = w ( x, y), w& ( x, y, 0) = w& ( x, y), 0 0 ψ xy,, 0 = ψ ( x, y), ψ& x, y, 0 = ψ& ( x, y,), x 0, a, y 0, b (2.5 a-) ( ) ( ) [ ] [ ] x x y y 3. Damage ientification technique There are a lot of techniques to treat the nonlinear structural vibration response in the time omain. The state-space representation of the structural vibration response is a suitable an powerful tool for stuying the ynamic behaviour of a structure. A wwt, &, of perioically riven stanar technique for ealing with phase space ( ) oscillators is to stuy the projection of ( ww&, ) at moments in time t, where t is a multiple of the perio T=2π/ω. Here ω can be the frequency of the excitation of the mechanical system, an eigen frequency of the structure, or its multiple, an T is a perio of the forcing function, an eigen perio of the system, or its multiple. The result of inspecting the phase projection ( ww&, ) only at specific times t=kt is a sequence of ots, representing the so-calle Poincaré map. In papers [14, 15] the following amage inex base on the analysis of the Poincaré map was introuce:. u Si Si (3.6) Ii =, u Si where N p u u u u u i = ( i, j+ 1 i, j) + ( & i, j+ 1 & i, j) j= 1 N p i = i, j+ 1 i, j + & i, j+ 1 & i, j j= 1 S w w w w (3.7 a,b) S ( w w ) ( w w ) In these equations i=1,2 N noes, N noe is the number of noes, N p is the number of u u points in the Poincaré map an ( wij, w& ij ) an ( wij, w& ij ) enote the j th point on the Poincaré maps of the unamage an the amage states, respectively. A small (close to 0) amage inex will inicate no amage, while a big amage inex will inicate the presence of a fault at the corresponing location. The above amage inex epens on the location of the point on the plate, an consequently it is a function of the plate coorinates x an y. One can expect that the maxima of the surface I ( x, y ) (3.7 a) will represent the locations of the amage, i.e. max (, ) max { i } I x y = I. i The amage criterion base on this inex presumes setting a threshol value T for the amage inex an if (3.8) I ( xy, ) > T

6 E. Manoach then one can conclue that the plate is amage an the areas of points (x,y) for which Eqn (3.8) is fulfille, form the amage area (areas). In the present work we shall use the same amage inex an amage criterion but taking into account the temperature changes as well, I = I ( x, y, Δ T). This suggestion presumes that the amage inex efine by Eqns (3.6) an (3.7) is calculate for equal values of Δ T for the healthy an amage plate. 5. Results an iscussions. Numerical calculation of the vibrational isplacements of the healthy an the amage rectangular plates subjecte to mechanical an thermal loaing were performe. The amage was moelle as a reuction (up to 50%) of the plate thickness in small parts of the plate. The first example concerns the same plate as the one consiere in [10 ]. The plate has the following imensions an material properties: a = 0.25 m, b = 0.24 m, h = m, E = Pa, ρ = 7850 kg/m 3, ν=0.3 an α T = 17.3 x 10-6 K -1. This very thin plate is subjecte to harmonic loaing with frequency of excitation ω h = 172 ra/s (0.7ω 1,1 ) an amplitue p = 0.3N. The amplitues of oscillations obtaine here are very close to the ones shown in Fig. 9 in Reference [10], so the verification of the present results is satisfactory (not shown here). Then the same plate but with increase thickness h = m (case B from [10]) was subjecte to thermal an ynamic loaing. For this plate two cases were consiere: 1) unamage plate an 2) plate with reuce thickness in a small part of the plate - the white area from the plate shown in Fig 2. It was shown in [10] that the buckling temperature for this plate is ΔT = 0.9 K. It is clear that the attempt to inspect such a plate for amage without consiering the temperature changes is conemne to fail. The plate is subjecte to a harmonic loaing p=0.9 N applie in the plate centre with frequency of excitation ω h = 319 ra/s. (ω 1,1 =455,6 ra/s) The introuce small efect oesn t influence essentially the response of the plate but small changes in the eigen frequencies an moes lea to phase shift an the ifferences between the two responses increase with time. (not shown here). The Poincaré maps of the responses of the healthy an the amage plate in the centre of the efect are shown in Fig 3. The Poincaré plots shown are obtaine as a projection of ( ww&, ) at moments t, where t is a multiple of the perio T=2π/ω h. The amage oesn t change essentially the form of the Poincaré plot. As can be expecte the ifference between the two responses is larger at the points with reuce thickness. A contour plot of the amage inex obtaine by using Eqns (3.7) is plotte in Fig 6 where a threshol value T =0.06 is use.

7 Damage etections in structures 0.12 velocity, m/s Damage Fig. 2. FE iscretization an amage area (white colour) of the plate isplacements, m Fig. 3. Poincaré map at the centre of the efect. Unamage- black ots; Damage grey ots Fig. 4. Contour plot of the Damage inex As can be seen the amage criterion in this case works quite well an preicts rather precisely the amage location espite of the fact that the amage inexes have low values. Then the same plates were consiere at elevate temperature namely ΔT=0.7 K. This temperature leas to increase amplitues of vibrations of the plates. Again, the ifferences in the plate history iagrams are visible but they are not very large in the beginning of the time histories (not shown here). However the Poincaré plots for the amage an the unamage plate have very ifferent shapes, as can be seen from Fig. 5. This phenomenon may inicate that for these loaing parameters the ynamic

8 E. Manoach system changes its position in the basin of attractions moving from one region to another. This observation agrees with the fact that the plate buckles at ΔT = 0.9K [10] velocity, m isplacements, m Fig. 5 Poincaré map of the response of the plate centre of heate unamage (black ots) an amage (grey ots) plates. ΔT = 0.7 K The shapes of the Poincaré plots at the amage noes are similar. Obviously, in such case the amage criterion (8) is not appropriate an oesn t give satisfactory results for the amage location (not shown here). As can be expecte neglecting the temperature influence is impossible for the amage etection purpose an leas to wrong results. The secon numerical example concerns a thicker rectangular plate with the following geometrical an material properties: a=10 m, b=2.5 m, h=0.05 m, Young moulus E = N/m 2, Poison ratio ν=0.34, ensity ρ= 2778 kg/m 3. The amping 12 Ns coefficient c1 = c2 was chosen to be The finite element iscretization h 2 3 m an the amage area are shown in Fig. 6. Again, the amage area has a thickness h amage =h/2. The plate is fully clampe an the applie harmonic loa p=500 N is uniformly istribute over the whole plate surface. The excitation frequency is 260 ra/s, which is only 7 % less than the first eigen frequency of the healthy plate. A strong beating can be observe in the responses of the healthy an amage plates (not shown here). The phase of the response of the amage plate shifts an the ifference between the responses increases with the time. The same conclusion applies in the case of the rectangular plate at elevate temperature. The elevate temperature leas to larger values of the vibration amplitue. Again, the ifferences between the Poincaré plots of the heate an unheate plates are largest for the points from the amage areas (see Fig. 7 a-c). Accoringly, the amage inexes corresponing to the amage area have the biggest values, which give the possibility to locate the amage. The contour plots of I i corresponing to three ifferent temperatures are shown in Fig. 8 a-c. It can be seen that the amage location is preicte very precisely in the case of the unheate plate as well as in the cases of the heate plate with two ifferent

9 Damage etections in structures temperatures ΔT = 50 K an ΔT = 100 K. The threshol value T is set to 0.28 for all cases an the maximal value of I is almost the same (I =0.4 for ΔT=0, ΔT = 50 K an I =0.42 for ΔT = 100 K). 0 velocity, m/s -10 Fig. 6. FE moel of the plate with amage isplacements, m 0 (a) velocity isplacements 10 (b) 0 velocity isplacements Fig. 7. Poincaré map at the centre of the efect for: (a) unheate plate, (b) - heate plate ΔT= 50 K, (c) heate plate -ΔT = 100 K. Unamage plate - black ots; Damage plate grey ots. Fig. 8. Contour maps of the amage inex for unheate an heate rectangular plate with amage.

10 E. Manoach If, however one calculates, for example the amage inex of the healthy unheate plate an the one for the amage but heate plate then the amage location cannot be preicte precisely. This is ue to the temperature change which is not taken into account for the healthy plate. The vibration responses of the healthy an the amage plates shoul be compare for the same temperatures. The thir example concerns thin clampe beam. It has the following geometrical an material characteristics: L=0.5 m. h=0.002 m, b=0.02m, E=7. 10 N/m2, ν = 0.34, ρ=2778 kg/m3, ΔT = 5 K, α T =23.9 x The FE moel consist of 41 Noes an 40 beam elements. The beam is subjecte to harmonic loaing istribute in five noes aroun the beam center with magnitue 100 N/m unheate heate w/l 0.00 Damage Inex imensionles time Fig. 9. Time history iagram of unheate an heate amage beam at the beam centre beam lenght Fig. 10. Damage inex versus the imensionless beam length. The cases when beam has reuce thickness h 1 =h/2 for x [0.1125, ] an when the beam is subjecte to elevate temperature ΔT= 5 K. The elevate temperature lea beam to buckle an its vibrations continue aroun a new equilibrium position. This can be seen on the time history iagrams for the amage beam shown at Fig. 9 (for the healthy beam the figure is very similar). The amage inex iagram for the unheate beam at Fig. 10 shows that the applie metho preicts precisely the beam presence an its location. For heate beam however the results are not so clear an there are two peaks of the amage inex - at the place of amage an at the beam en (not shown here). Again in the case of a ramatically change of the structure behaviour ue to the temperature changes the suggeste amage criterion oesn t work so well.

11 Damage etections in structures 6. Conclusins In this paper the compute time omain vibration responses are use to analyse the ynamic behaviour of plates in the intact conition an in the case when efects are present taking into account the temperature changes. A amage assessment metho is suggeste which is base on the phase space representation of the time omain nonlinear vibration response of the plate an uses the analysis of its Poincaré map. It has been emonstrate that amage as well as elevate temperatures can influence substantially the time omain response of the plate an its Poincaré maps. It can be conclue that: 1) The influence of the temperature changes is essential an can change substantially the nonlinear ynamic response of the plate an this is why temperature changes shoul be taken into account when eveloping a amage assessment proceure; 2) Temperature loaings which lea to either buckling or chaotic behaviour of the plate, might rener the amage criterion suggeste by Eqns ( ) inappropriate. This is because even small amage, resulting in stiffness reuction of the plate, coul lea to ramatic changes in the Poincaré maps of the response an consequently to unreliable results. The potential, the sensitivity an the applicability of the evelope metho still have to be teste for real measurements an for more structures, efects an loaing conitions. R E F E R E N C E S [1] ZOU, Y, TONG, L., STEVEN, G. P. Vibration base moel- epenent amage (elamination) ientification an health monitoring for composite structures a review. J. Soun & Vibrations 230(2) (2000) [2] RIZOS, P.F., ASPRAGATHOS N.,. DIMAROGONAS, A. D. Ientification of crack location an magnitue in a cantilevere beam from the vibration moes. J. Soun & Vibrations 138 (1990) [3] BANKS H.T., INMAN, LEO, D.J., WANG, Y. An experimentally valiate amage etection theory in smart structures. J. Soun & Vibrations 191 (1996) [4] VERBOVEN, P.,. PARLOO, E, GUILLAUME, P., OVERMEIRE, M. VAN. Autonomous structural health monitoring part I: moal parameter estimation an tracking. Mech. Systems an Signal Processing 16 (2002) [5] PARLOO, E., VERBOVEN, P., GUILLAUME, P., OVERMEIRE, M. VAN. Autonomous structural health monitoring part II: vibration-base inoperation amage assessment. Mech. Systems an Signal Processing 16 (2002) [6] RIBEIRO P., MANOACH, E. The effect of temperature on the large amplitue vibrations of curve beams. J. Soun & Vibrations 285 pp (2005) [7] MANOACH, E., RIBEIRO, P. Couple, thermoelastic, large amplitue vibrations of Timoshenko beams. Int. J. Mechanical Sciences, 46, pp (2004) [8] RIBEIRO, P. Thermally inuce transitions to chaos in plate vibrations. J of Soun & Vibration 299 (1-2) (2007)

12 E. Manoach [9] THORTON, E.A Thermal structures for aerospace applications, AIAA Eucation Series, Reston, [10] AMABILI, M., CARRA, S. Thermal effects on geometrically nonlinear vibrations of rectangular plates with fixe eges. J. Soun & Vibrations 321(2009) [11] CATTARIUS, J., INMAN, D.J. Time omain analysis for amage etection in smart structures. Mech. Systems an Signal Processing 11 (1997) [12] TODD, M., NICHOLS J.M, PECORA,L. M. VIRGIN L., Vibration-base Damage Assessment Utilizing State Space Geometry Changes: local Attractor Variance Ratio. Smart Materials an Structures 10 (2001) [13] MONIZ, L., NICHOLS, J.M,NICHOLS,C.J., SEAVER,M., TRICKEY, S.T., TODD, M.D., PECORA, L.M., VIRGIN, L.N., A multivariate, attractor-base approach to structural health monitoring. J. Soun & Vibrations 283 (2005) [14] MANOACH, E. TRENDAFILOVA,I Large Amplitue Vibrations an amage etection of rectangular plates J. Soun & Vibrations, 315 (3), pp (2008) [15] TRENDAFILOVA, I., MANOACH, E. Vibration Base Damage Detection in Plates by Using Time Series Analysis. Mech. Systems an Signal Processing 22 (2008)