Vibration Based Damage Detection in Smart Non-uniform Thickness Laminated Composite Beams

Hamidreza Ghaffari

School of Science and Engineering, Sharif University of Technology, International Campus

Kish, Iran hamid_r_ghaffari@kish.sharif.edu

Abolghassem Zabihollah

zabihollah@sharif.edu

Esmaeel Saeedi Semnan University

Semnan, Iran es_saeedi@yahoo.com

Roya Ahmadi

Tavanir Organization Tehran, Iran

r.ahmadi@ieee.org

Abstract—Laminated composite beams with non-uniform thickness are being used as primary structural elements in a wide range of advanced engineering applications. Tapered composite structures, formed by terminating some of the plies, create geometry and material discontinuities that act as sources for delamination initiation and propagation. Any small damage or delamination in these structures can progress rapidly without any visible external signs. Due to this reason early detection of damage in these systems during their service life is receiving increasing attention. The presence of a crack in a component or structure leads to changes in its global dynamic characteristics results in decreases in the natural frequencies and modifications of the mode shapes of the component or structure. The present work aims to investigate the influences of delamination on the natural frequencies, frequency and voltage responses of smart non-uniform thickness laminated composite beams. In order to guarantee the continuity of curvature at element interfaces, particularly at ply drop-off locations, a higher-order finite element model by considering the electro-mechanical coupling effect is used which can accurately and efficiently represents the dynamic responses of the structure for forced/random stimuli and provide accurate results using fewer elements.

Keywords- Vibration Based Health Monitoring (VBH); Non-uniform thickness laminated beams; damage detection; FE modeling.

I. INTRODUCTION This Conventional metallic materials for the design of

structural components like wing and fin of an aircraft, helicopter yoke, robot arms, turbine blade and satellite antennas are inhibited by their high densities resulting in a reduction of payload. The laminated composites offer a weight reduction thereby increasing the payload. Tapered composites formed by terminating or dropping off some of the plies in some primary structures have received much attention from researchers. The ply-less areas are filled by resin pockets. Some of the most common types of tapered sections are shown in Fig. 1. The tailoring properties of tapered composites and their potential for creating more significant weight savings than commonly-used laminated components allow an increasing use of them in commercial and military aircraft applications. A typical

example is the helicopter yoke where a progressive variation in the thickness of the yoke is required to provide high stiffness at the hub and relative flexibility at the mid-length of the yoke to accommodate flapping. However, ply drop-off causes a discontinuity within the laminate and therefore, it introduces structural difficulties like stress concentration at the drop station. This leads to failure of the components through delamination and/or failure of resin. The formation of inter-laminar stresses at the drop-off may initiate failure long before the ultimate load carrying capacity of the laminate is reached. Hence, the potential benefits in dropping plies may be compromised through a reduction of the strength of the laminate. Therefore early detection of damage or delamination in these systems during their service life is receiving increasing attention. Changes in the global dynamic characteristics of these structures due to delamination such as impulsive response, frequency response functions, etc. and comparing them with the responses of intact structure leads to the detection, localization and the characterization of the extent of the damage.

Many of researches on laminated beams for investigating their dynamic behaviors and responses were focused on uniform thickness beams [1-2]. Recently higher-order finite element formulations were developed to investigate the free vibration responses of some of the most common tapered configurations [3-4]. In addition to these, at the other investigations [5-6] forced vibration responses of non-uniform thickness laminated beams and vibration responses of axially loaded tapered laminated beams were studied. Significant efforts have already been devoted to developing damage detection algorithms using vibration- based approach [7-8]. But there are some limited studies for damage detection of tapered laminated composite beams based on their vibration responses.

The problems for vibration-based damage detection, quantification and localization are pattern recognition problems since they suggest the discrimination between two or more signal categories, i.e. signals coming from an intact structure and those from a damaged structure. This perspective is used in the present study for non-uniform thickness laminated composite beams.

978-1-4244-3878-5/09/$25.00 ©2009 IEEE

TIC-STH 2009

978-1-4244-3878-5/09/$25.00 ©2009 IEEE 176

Internal taper: model A model B

model C model D

Figure 1. Schematic illustration of taper configurations

II. ANALYSIS OF INTERNALLY-TAPERED BEAMS Due to variety of tapered composite beams and complexity

of the analysis no analytical solution is available at present. Therefore the finite element method is applied for investigating the dynamic behaviors of these structures. Different configuration of laminated tapered beams can be achieved by terminating selected plies at certain planner locations to reach the desired thickness reduction or taper angle. Perhaps, externally tapered section is the simplest way to make a tapered section; however, external tapering is not encouraged since it highly vulnerable to delaminate. In term of manufacturing, internally-tapered sections are the best choice. The most common and yet practical configurations of internally-tapered sections are illustrated schematically in Figure 1, in which; Model A: Basic taper, Model B: Staircase arrangement, Model C: Overlapping dropped plies, Model D: Continuous plies interspersed. The tapered laminates experienced high discontinuity through the thickness at tapered section, thus, a higher-order formulation is required to account for this issue.

A. Governing equation The equation of motion for a laminated composite beam

with non-uniform thickness section is given by [3-4]:

( ) ( ) ( ) ( ) 02

2

2

2

2

2

=∂∂+−⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

twxbxqxb

xwxk

dxd

sρ (1)

where b(x) and q(x) denote the width of the beam and the distributed transverse loading, respectively, with x-axis being the longitudinal axis of the beam. The term, w stand for transverse displacement and t indicates the time. Since the cross-section changes along the length of the beam, the stiffness, k(x), is given as a function of length, x. Further ρs is the mass of the laminated beam per unit length per unit width given by

( ) (2)

11∑

=−−=

n

kkkks hhρρ

where ρk is the density of the material of the kth lamina and further, hk and hk-1 are the distances to the top and bottom surfaces of kth lamina from the mid-plane of the laminate, respectively. When similar materials are used for all the plies, ρs=ρh where h is the total thickness of the laminate and ρ is the density of the composite material.

B. Higher-order interpolation The beam element, considered for the formulation, has two

nodes at the ends and at each end node four degrees of freedom are considered. The transverse displacement w, the slope ∂w/∂x, the curvature ∂2w/∂x2, and the gradient of curvature ∂3w/∂x3 at each node are considered as the four degrees of freedom at that node. Thus there are eight degrees of freedom per element. Correspondingly a seventh-degree polynomial displacement function is required to satisfy boundary conditions. This element represents all the physical situations involved in any combinations of displacement, rotation, bending moment and shear force. The deflection, w(x, t), is approximated as follows:

( ) ( ) [ ]{ } (3) ,, dNtxWtxw e ≡≈ where [N] consists of element shape functions, n is the number of degrees of freedom of the element and {d} is the matrix of nodal degrees of freedom consisting of nodal displacements, rotations, shear forces and bending moments. We(x,t) is given by a seventh-degree polynomial in order to account for the eight degrees of freedom as follows:

( ) (4) , 77

66

55

44

33

2210 xcxcxcxcxcxcxcctxW e +++++++=

Considering this approximation and (1), and taking into account the taper angle at each individual ply, φ, the following matrix equation for the smart non- uniform thickness laminated beam can be obtained:

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡)}({)}({

}{}{

][][][][

}{}{

]0[]0[]0[][

tQtFd

KKKKdM

d

ddddd

ψψ ψψψ

ψ (5)

where [kdd], [kdψ], [kψψ] stand for the elastic, piezoelectric and permittivity stiffness matrices, respectively. Further, [Mdd] is the mass matrix, and {F(t)} and {Q(t)} are the applied mechanical load vector and electrical charge vector, respectively. The nodal displacement matrix {d}= {U, W}T, represents the displacement vector containing axial, {U}, and transverse, {W} , displacements and {ψ} represents the electric potential. The electric potential vector is written as {ψ}= {ψs, ψa}, where ψs and ψa represents, respectively, the voltage output at the sensor and the voltage imposed on the actuator layer. Eq. (5) can also be condensed into the following uncoupled dynamic equations for the structural displacements and sensor voltage, respectively as:

( )

( ) }{][][]][[)}({

}{][]][[][}]{[1

1

asad

sad

ssssdd

ssd

ssssddddd

KKKKtF

dKKKKdM

ψψψψψψ

ψψψψ

−+

=−+−

−

(6)

RESIN POCKET

Mid-plane taper:

177

( )}]{[}]{[][}{ 1 asassd

sss KdKK ψψ ψψψψψ +−= − (7)

The coefficients of the stiffness and mass matrices are given respectively by the following relations [3-4]:

( ) ( ) (8) cos 2

2

2

24

011 dx

dxNd

dxNd

xbDK jil

ij φ∫=

( ) (9)

0

dxNNxhbM j

l

iij ∫= ρ

i , j = 1 – 8

III. FORCED VIBRATION Considering the effect of damping and external loading, (5)

can be expressed as:

(10) )(tFKuuCuM =++

where M is mass, C is damping, K is stiffness matrices, u represents displacement of the physical system and F(t) shows the external excitation. When dealing with vibration problems, usually the effect of the first few modes is dominant. Using the modal expansion {u(t)}=[Ф]{η(t)}, and orthogonal properties of mode shapes, (9) can be cast into the following set of differential equations each representing a single degree of freedom system [9]:

(11) )(tFC Tiiii Φ=Ω++ ηηη

where C = diag[2ξiωi], Ω = diag[ωi2], for i = 1,2,…,n and Ф is

the modal matrix and η denotes the modal coordinates. ωi and ξi are the ith-order natural frequency and damping ratio, respectively.

IV. NUMERICAL EXAMPLES A tapered beam schematically the same as the beam

presented in Fig. 2 with 36 plies at thick and 12 plies at thin section resulting in 24 drop-off plies and taper angle φ = 3° is considered. The configuration of the thick section is (0/90)9s and the length of it (L1) is 50 mm. The stacking sequence for the thin section is (0/90)3s and the length of it (L3) is 50 mm. Length of the tapered section (L2) is 28.6 mm and each ply has the thickness (tk) of 0.125 mm, individually. The height at the thick section (h1) is 4.5 mm and that of the thin section (h2) is 1.5 mm. The mechanical properties of epoxy-resin are: Elastic Modulus (E) is 3.93 GPa, Shear Modulus (G) is 1.034 GPa and Poisson’s Ratio (ν) is 0.37. The mechanical properties of the beam made up of NCT/301 graphite-epoxy composite material are detailed as: E1 is 113.9 GPa, E2 is 7.9856 GPa, ν21 is 0.0178, ν12 is 0.288, G12 is 3.138 GPa, and ρ is 1480 kg/m3. These geometries and mechanical properties are considered for the tapered beams with tapered sections of model A, B, C and D to study their dynamic responses. As it mentioned, tapered composite structures, formed by terminating some of the plies,

create geometry and material discontinuities that act as sources for delamination initiation and propagation. Based on this fact, the dynamic responses of beams with tapered sections made of models A, B, C and D were studied when the delamination with length of 40 mm occurs in the tapered layers in the first interface from the upper surface and extends to the thin section of the beam. To simulate the crack, the nodes at location of the crack were replaced by separated nodes [10-11].

Before discussing about the results its precision should be validated. In order to validate the results, the problem described in this section is solved for a tapered section of model C with a very small angle (0.5˚).

The length of tapered section, thick and thin parts are considered the same as the noted problem but the numbers of plies in thick and thin sections are considered as 36 and 32, respectively. The results for this geometry can be compared with that for a uniform beam with the same length and 36 plies. For fixed-free, simply supported and fixed-fixed conditions, the lowest three natural frequencies for the tapered laminated beam described above with taper angle 0.5˚, are compared with the corresponding natural frequencies of a uniform beam in Tables 1to 3. Considering the fact that there is considerable change in the thin section length, it can be observed that the results for tapered beam and uniform-thickness beam are comparable thus validating the results.

TABLE I. COMPARISON OF NATURAL FREQUENCIES OF TAPERED BEAM (ANGLE = 0.5˚) WITH UNIFORM BEAM.

Mode Number Uniform Fixed-Free Tapered Fixed-Free 1 296.92 290.42 2 1760.3 1729 3 4694.4 4578.7

TABLE II. COMPARISON OF NATURAL FREQUENCIES OF TAPERED BEAM (ANGLE = 0.5˚) WITH UNIFORM BEAM.

Mode Number Uniform- Simply supported

Tapered- Simply supported

1 809.99 788.56 2 3129.3 3052.5 3 6680.3 6536.5

TABLE III. COMPARISON OF NATURAL FREQUENCIES OF TAPERED BEAM (ANGLE = 0.5˚) WITH UNIFORM BEAM.

Mode Number Uniform- Fixed-Fixed Tapered- Fixed-Fixed 1 1755.9 1715.4 2 4553.1 4456.7 3 8328.9 8179.1

Figure 2. Schematic illustration of taper (model D) with internal dropped

plies

178

A. Effect of damage on the natural frequencies Tables 1 to 4 show the three lowest natural frequencies of

the intact and delaminated structures with tapered sections made of models A, B, C and D correspondingly. It can be seen clearly that the resultant reduction in the stiffness of the structure due to the delamination leads to reduction in its natural frequencies. These reductions are present in each of the lower normal frequencies discovered, and become more pronounced at higher frequencies to a degree that corresponding changes between the control and delaminated specimen become considerable. This problem shows the importance of early detection of damage in the components, specifically in aerospace applications which the components work in the frequencies near to higher natural frequencies of the structure. This frequency reduction can be explained by classical structural dynamics. Natural frequencies are in a direct relation with (k/m) 0.5, where k is the stiffness of the structure and m is the mass. When damage is introduced to a specimen, the resulting local loss of stiffness directly affects this ratio, thereby affecting the natural frequencies of the structure.

TABLE IV. NATURAL FREQUENCIES OF BEAM MADE OF MODEL A

Mode Number Control Natural Freq.

(Hz)

Damaged Natural Freq.(Hz)

Reduction (Hz)

1 366.74 332.57 34.17 2 1092.2 958.94 133.26 3 2998.3 1849.9 1148.4

TABLE V. NATURAL FREQUENCIES OF BEAM MADE OF MODEL B

Mode Number Control Natural Freq.

(Hz)

Damaged Natural Freq.(Hz)

Reduction (Hz)

1 370.26 336.05 34.21 2 1076.3 946.29 130.01 3 3033.7 1851.4 1182.3

TABLE VI. NATURAL FREQUENCIES OF BEAM MADE OF MODEL C

Mode Number Control Natural Freq.

(Hz)

Damaged Natural Freq.(Hz)

Reduction (Hz)

1 371.8 338.62 33.18 2 1077.5 947.2 130.3 3 3050.7 1851.8 1198.9

TABLE VII. NATURAL FREQUENCIES OF BEAM MADE OF MODEL D

Mode Number Control Natural Freq.

(Hz)

Damaged Natural Freq.(Hz)

Reduction (Hz)

1 370.45 337.08 33.37 2 1076.7 946.59 130.11 3 3034.3 1851.2 1183.1

B. Effect of damage on the frequency responses There are many advantages of using frequency response method in a SHM system; they can be implemented cheaply, they can be light and conformal, and they can provide good insight as to the global condition of the system. The most attractive implementation of the frequency response method is

one performed passively for low frequencies using ambient vehicle vibrations, caused by the engines or aerodynamic loads for example. Comparing global transfer functions for prescribed frequency ranges at selected positions could provide a good foundation for a first and last line of defense in a SHM system. A passive method such as this could continuously monitor components of a structure without requiring much processing power in order to direct more accurate and energy-intensive active sensor systems where to query for a more detailed survey of potential damage. For studying the frequency responses of the structures made of different tapered sections, the plate is excited using a spread load of 10 N applied at the tip of the beam. Depending on the kind of application that tapered composite beams should be used, each model of tapered sections is suitable for specific applications. In this investigation all the models are considered for studying their frequency responses in the both intact and damaged cases. Due to changes in the cross section of the beam in the tapered location, this part of the beam is highly vulnerable in the operational hazards. Based on this deduction the noted delamination was considered at this location. Figs. 2 to 5 show the comparisons of the frequency responses of intact and damaged tapered beams with different tapered sections.

Figure 2. Comparison of frequency responses for tapered beam model A

Figure 3. Comparison of frequency responses for tapered beam model B

179

Peaks in frequency response correspond to natural vibration modes of the specimen. Shift in the peaks shows the effect of delamination in the reduction of natural frequencies. In this study the frequency range was considered up to 3100 Hz. The reason for this choice is that the lower portion of frequency spectrum appears to be most useful for damage detection due to simple frequency peak correlation. Matching corresponding natural frequency peaks between intact and delaminated specimens become increasingly difficult at higher frequencies due to nonlinear damping due to frictional rubbing of de-bonded surfaces which appears to truncate frequency peaks.

C. Effect of delamination on the voltage responses

Occurrence of delamination in non-uniform thickness laminated composite beams leads to reduction in their stiffness and has considerable effect on their dynamic responses. By investigating the vibration amplitudes of these structures in

Figure 4. Comparison of frequency responses for tapered beam model C

Figure 5. Comparison of frequency responses for tapered beam model D

Figure 6. Comparison of sensor outputs for undelaminated and delaminated

model made of taper section C

Figure 7. Comparison of sensor outputs for undelaminated and delaminated

model made of taper section D

Figure 8. Comparison of sensor outputs for undelaminated and delaminated model made of taper section C

180

both damaged and undamaged conditions for different input excitations, the occurrence of delamination can be identified. For presenting the ability of this method, the voltage responses of PVDF sensor bonded 1 cm away from the tip of the beam for a 25 Hz sinusoidal load of 1 N and random input were studied. Changes in the voltage responses due to the delamination can be observed in Figs. 6 to 9. The responses were obtained for models C and D which had the highest fundamental frequencies in comparison with the other tapered sections. Fig. 6 and Fig. 8 shows the voltage responses of delaminated and undelaminated models made of tapered section of model C for 1 N sinusoidal load and random input correspondingly. Also Fig. 7 and Fig. 9 represent the sensor outputs of delaminated and undelaminated models of tapered sections of model D for the noted loads subsequently.

V. CONCLUSIONS Damage detection in smart non-uniform thickness

laminated composite beams using vibration based health monitoring technique has investigated. Different models of tapered sections including models A, B, C and D were considered for constructing the tapered beams using an accurate and efficient higher order finite element model by considering the electro-mechanical coupling effect. Then the effect of delamination in these structures was studied based on dynamic responses and sensor outputs of each model. Effect of delamination in these structures can be clearly observed on the frequency responses, natural frequencies and voltage outputs of them. Shift in the peaks of frequency response transfer functions correspond to natural vibration modes of the specimen show the effect of delamination in the reduction of natural frequencies. These dynamic responses and sensor

outputs can be considered as suitable criteria and database for detecting the occurrence of delamination and it’s propagation in the real structures.

ACKNOWLEDGMENT The authors would like to thank Sharif University of

Technology, international campus at Kish Island for supporting this research.

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Figure 9. Comparison of sensor outputs for undelaminated and delaminated model made of taper section D

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