journal of solid mechanics and materials engineering - j-stage

Journal of Solid Mechanicsand Materials

Engineering

Vol. 5, No. 11, 2011

610

Free Vibration of Delaminated Composite Shallow Conical Shells*

Sudip DEY**, Tanmoy BANDOPADHYAY**, Amit KARMAKAR**, † and Kikuo KISHIMOTO***, ‡

** Mechanical Engineering Department, Jadavpur University, Kolkata – 700032, India † E-mail: [email protected]

*** Department of Mechanical Sciences and Engineering, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguroku, Tokyo 152-8552, Japan

‡ E-mail: [email protected] Abstract This paper presents a finite element method to investigate the effects of delamination on free vibration characteristics of graphite-epoxy pretwisted shallow angle-ply composite conical shells. The generalized dynamic equilibrium equation is derived from Lagrange’s equation of motion neglecting Coriolis effect for moderate rotational speeds. The theoretical formulation is exercised by using an eight noded isoparametric plate bending element based on Mindlin’s theory. Multi-point constraint algorithm is utilized to ensure the compatibility of deformation and equilibrium of resultant forces and moments at the delamination crack front. The standard eigen value problem is solved by applying the QR iteration algorithm. Finite element codes are developed to obtain numerical results concerning the effect of twist angle, location of delamination and rotational speed on natural frequencies of delaminated angle-ply composite conical shells. The mode shapes are also depicted for a typical laminate configuration. Parametric studies of symmetric and anti-symmetric angle-ply laminates provide the non-dimensional natural frequencies which are the first known results for the type of analyses carried out here.

Key words: Conical Shell, Delamination, Vibration, Angle-Ply, Finite Element, Multi-Point Constraint

1. Introduction

Rotating pretwisted conical shells with low aspect ratio can be idealized as turbo-machinery blades (Fig 1). In weight sensitive applications, composite materials are advantageous because of their light weight, high stiffness and strength. The increasing demand of composite materials is due to their high strength to weight ratio with high compliance to design aspects as well as their cost-effectiveness. Delamination or inter-laminar debonding is the most common feared mode of damage in the composite structures. This damage can cause degradation of strength leading to structural instability and failure. In general, all vibrations of blades are closely related to their natural frequencies. In order to ensure the safety of operation, a profound knowledge of these natural frequencies and the dynamic behaviour of turbomachinery blades are essential for the designers.

*Received 6 July, 2011 (No. 11-0382) [DOI: 10.1299/jmmp.5.610]

Copyright © 2011 by JSME

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(a)

(b)

Fig. 1 Geometry of (a) Twisted plate and (b) Untwisted shallow conical shell model The pioneering work on pretwisted composite plates was carried out by Qatu and

Leissa(1) to determine the natural frequencies of stationary plates using laminated shallow shell theory using Ritz method. Liew et al.(2) investigated on pretwisted conical shell to find out the vibratory characteristics of stationary conical shell by using Ritz procedure and by using the same method, the first known three dimensional continuum vibration analysis including full geometric non-linearities and centrifugal accelerations in composite blades was carried out by McGee and Chu(3). Regarding delamination model, two worth mentioning investigations were carried out. It included analytical and experimental determination of natural frequencies of delaminated composite beam by Shen and Grady(4) and the second one dealt with finite element treatment of the delaminated composite cantilever beam and plate by Krawczuk et al.(5) for free vibration analyses. On the other hand, Lee et al.(6) exhibited the vibration analysis of twisted cantilevered conical composite shell by using finite element method based on the Hellinger–Reissner principle. Later on, Karmakar and Kishimoto(7) analyzed the free vibration characteristics of delaminated composite cylindrical shells. There exists a good number of references on numerical models and experimental investigations of turbomachinery blades idealized according to the Timoshenko theory of beam. It has been found that the application of beam theory is far from straight forward and extracts limited information only. Nabi and Ganesean(8) summarized the quantitative comparison of natural frequencies of metal matrix composite pretwisted blades in stationary condition using beam and plate theories. Although delamination is one of the most feared damage modes in laminated composites, the impact behavior of delaminated structures has been addressed only in two investigations by Sekine et al.(9) and Hu et al.(10) wherein simply supported plates with single and multiple delamination were considered for the analyses. Besides investigation on single delamination, significant work also incurred on multiple delaminations. Considering multiple

Ω'


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delaminations, failure analysis of composite plate due to bending and impact was numerically investigated by Parhi et al.(11) using finite element mentod. Using the same tool, Aymerich et al.(12) simulated impacted cross-ply laminates based on cohesive interface elements. Of late, there are two worth mentioning paper on free vibration characteristics of angle-ply. The first one by Viswanathan and Kim(13) is on free vibration of anti-symmetric angle-ply-laminated plates including transverse shear deformation by Spline method while the second one was carried out by Khare et al.(14) for composite and sandwich laminates with a higher-order facet shell element. Later on, Patel et al.(15) carried out a study to determine the post-buckling characteristics of angle-ply laminated truncated circular conical shells. In contrast, static and vibration analysis of axi-symmetric angle-ply laminated cylindrical shell using state space differential quadrature method was investigated by Alibeigloo(16).

Therefore, considering all the above research findings, it is clear that no attention has been made to the initially stressed delaminated rotating composite plates or shells. It is also observed that only a limited number of investigations has been carried out related to laminated composite cantilever conical shells with initial twist and subsequently the research findings are very limited and scanty. To the best of the authors’ knowledge, there is no literature available, which deals with delaminated pretwisted rotating composite angle-ply conical shells. To fill up this apparent void, the present analyses employed a finite-element based approach to study the free vibration characteristics of pretwisted delminated graphite-epoxy angle-ply composite rotating shallow conical shells. The analyses are carried out using an eight-noded isoparametric plate bending element considering the effects of transverse shear deformation and rotary inertia based Mindlin’s theory. The undelaminated region is modeled by a single layer of plate elements while the delaminated region is modeled using two layers of plate elements whose interface contains the delamination. To ensure the compatibility of deformation and equilibrium of resultant forces and moments at the delamination crack front a multi-point constraint algorithm(17) is incorporated which leads to anti-symmetric element stiffness matrices. The QR iteration algorithm(18) is utilized to solve the standard eigenvalue problem. The first two non-dimensional natural frequencies are obtained considering the effects of triggering parameters like the twist angle, location of delamination, rotational speed and fiber-orientation angle. This paper presents a finite element based numerical approach to determine the non-dimensional natural frequencies of both undelaminated and delaminated composite conical shells neglecting effect of dynamic contact between delaminated layers.

2. Theoretical Formulation

A shallow shell is characterized by its middle surface which is defined by the equation(19),

2 21

2 (1)2 x xy y

x xy yz

R R R= − + +

The radius of twist (Rxy), length (L) of the shell and twist angle (Ψ) are related as,

tan ( 2 )

L

R xyψ = −


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The dynamic equilibrium equation for moderate rotational speeds is derived employing Lagrange’s equation of motion and neglecting Coriolis effect, the equation in global form is expressed as(20),

[M] δ + ([K] + [Kσ]) δ = F(Ω2) (3) where [M], [K], [Kσ] are global mass, elastic stiffness and geometric stiffness matrices, respectively. F(Ω2) is the nodal equivalent centrifugal forces and δ is the global displacement vector. [Kσ] depends on the initial stress distribution and is obtained by the iterative procedure(21) upon solving,

( [K] + [Kσ] ) δ = F (Ω2) (4) Angular velocity matrix components contributing towards acceleration vector is given as(20),(21),

(5)

The element centrifugal force is given by(20),(21),

[Fce] = ρ ∫ [N]T [A] [(hx + x) (hy + y) (hz + z)]T dV (6)

where ρ is the mass density, [N] stands for the shape function matrix and hx, hy, hz are the fixed translational offsets expressed with respect to the plate coordinate system. The element geometric stiffness matrix due to rotation is given by(22),

[Kce] = ∫ [G]T [Mσ] [G] dV (7)

where the matrix [G] consists of derivatives of shape functions and [Mσ] is the matrix of initial in-plane stress resultants caused by rotation. The natural frequencies (ω) are determined from the standard eigen value problem(18) which is represented below and is solved by the QR iteration algorithm,

[A] δ = λ δ (8)

where, [A] = ( [K] + [Kσ] ) - 1 [M]

and λ = 1/ ωn2 (9)

Fig. 2 Plate elements at a delamination crack tip

2 2( ) ( ) ( )

2 2( ) ( ) ( )[ ]

2 2( ) ( ) ( )

y z x y x z

A x y x z y z

x z y z x y

Ω + Ω − Ω Ω − Ω Ω

− Ω Ω Ω + Ω − Ω Ω=

− Ω Ω − Ω Ω Ω + Ω


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2.1 Multi-point Constraints

The cross-sectional view of a typical delamination crack tip is shown in fig. 2 where the nodes of three plate elements meet together to form a common node. The undelaminated region is modeled by plate element 1 of thickness h, and the delaminated region is modeled by plate elements 2 and 3 whose interface contains the delamination (h2 and h3 are the thicknesses of the elements 2 and 3, respectively). The elements 1, 2 and 3 are freely allowed to deform prior to imposition of the constraints conditions. The nodal displacements of elements 2 and 3 at the crack tip are expressed as(17),

Uj = U´j - (Z – Z′ j) θxj (10) Vj = V´j - (Z – Z′ j) θyj (11) Wj = W´j (Where, j = 2, 3) (12)

where U´j , V´j and W´j are the mid-plane displacements, Z′j is the z-coordinate of mid-plane of element j and θx , θy are the rotations about x and y axes, respectively. The above equation also holds good for element 1 and Z′1 equal to zero. The transverse displacements and rotations at a common node have values expressed as(17),

W1 = W2 = W3 = W (13) θx1 = θx2 = θx3 = θx (14)

θy1 = θy2 = θy3 = θy (15) In-plane displacements of all three elements at crack tip are equal and they are related as(17),

U′2 = U′1 - Z′2 θx (16) V′2 = V′1 - Z′2 θy (17) U′3 = U′1 - Z′3 θx (18) V′3 = V′1 - Z′3 θy (19)

where U′1 is the mid-plane displacement of element 1. Equations of (13) to (19) relating the nodal displacements and rotations of elements 1, 2 and 3 at the delamination crack tip, are the multipoint constraint equations used in finite element formulation to satisfy the compatibility of displacements and rotations. Mid-plane strains between elements 2 and 3 are related as(17), ε′ j = ε′j + Z′j k (20)

where ε′ is the strain vector and k is the curvatures vector being identical at the crack tip for elements 1, 2 and 3. This equation can be considered as a special case for element 1 and Z′1 is equal to zero. In-plane stress-resultants, N and moment resultants, M of elements 2 and 3 can be expressed as(17), Nj = [A] j ε′1 + ( Z′j [A]j + [B]j ) k (Where j = 2, 3) (21) Mj = [B] j ε′1 + ( Z′j [B]j + [D]j ) k (Where j = 2, 3) (22) where [A], [B] and [D] are the extension, bending-extension coupling and bending stiffness coefficients of the composite laminate, respectively. The resultant forces and moments at the delamination front for the elements 1, 2 and 3 satisfy the following equilibrium conditions, N = N1 = N2 + N3 (23) M = M1 = M2 + M3 + Z′2 N2 + Z′3 N3 (24)


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Q = Q1 = Q2 + Q3 (25) where Q denotes the transverse shear resultants. An eight noded isoparametric quadratic plate bending element with five degrees of freedom at each node (three translations and two rotations) is employed wherein the shape functions are as follows(18), Ni = (1 + ε εi) (1 + η ηi) (ε εi + η ηi - 1) / 4 (for i = 1, 2, 3, 4) (26) Ni = (1 - ε2) (1 + η ηi) / 2 (for i = 5, 6) (27) Ni = (1 - η2) (1 + ε ε i ) / 2 (for i = 6, 8) (28) where η and ε are the local natural coordinates of the element

3. Results and Discussion Non-dimensional natural frequencies [ω = ωn L2 √(ρh/D)] for conical shells (Rx = ∞)

having a square plan-form (L/bo=1), curvature ratio (bo/Ry) of 0.5 and thickness ratio of (s/h) of 1000 are obtained corresponding to different speeds of rotation Ω = 0.0, 0.5 and 1.0 and relative distance, d/L=0.33, 0.5 and 0.66, considering three different angles of twist, namely ψ=15°, 30° and 45°, in addition to the untwisted one (ψ=0°). Parametric studies are carried out with respect to twist angles, location of delamination and rotational speeds on natural frequencies of angle-ply composite shallow conical shells. The parameters n, Ω´, ωo, ρ, L, bo, h, d, a, θv and θo represent the number of layers, actual angular speed of rotation, fundamental natural frequency of a non-rotating shell, density, length, reference width, thickness, distance of the centerline of delamination from the clamped (fixed) end, crack length, vertex angle and base subtended angle of cone, respectively. The finite element formulation employs eight noded plate bending element with five degrees of freedom at each node. Material properties of graphite-epoxy composite(23) considered as E1=138.0 GPa, E2=8.96 GPa, ν=0.3, G12=7.1 GPa, G13=7.1 GPa, G23=2.84 GPa. Convergence studies are performed using uniform mesh division of (6 x 6) and (8 x 8) and the results are found to be nearly equal, with the difference being well within 1%. In addition to convergence study, a comparative study is also conducted to investigate the differences between the free vibration characteristics of symmetric and anti-symmetric angle-ply conical shells. 3.1 Validation of Results

Based on the present finite element modeling, computer codes are developed and the results obtained are compared and validated with those published in open literature(1),(2) as furnished in Table 1, Table 2 and in Fig 3. The comparative study depicts an excellent agreement with the previously published results and hence it demonstrates the capability of the computer codes developed and insures the accuracy of analyses. Table 1 furnishes non-dimensional fundamental frequencies of graphite-epoxy composite twisted plates with different fibre-orientation angle(1), while Table 2 presents non-dimensional fundamental frequencies of twisted conical shells(2). The span-wise variation of fundamental frequency of composite cantilever beam with relative position of delamination is illustrated in Fig 3. Table 1 Non-dimensional fundamental frequencies [ω=ωn L2 √(ρ/E1h2)] of three layered [θ, -θ, θ] graphite-epoxy twisted plates, L/b=1, b/h=20, ψ=30°

Fibre Orientation Angle, θ Present FEM Qatu and Leissa [1] 15° 0.8618 0.8759 30° 0.6790 0.6923 45° 0.4732 0.4831 60° 0.3234 0.3283


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Fig. 3 Influence of the relative position of delamination on the first natural frequency of the composite cantilever beam (Krawczuk et al. [5]) Table 2 Non-dimensional fundamental frequencies [ω = ωn bo2 √(ρh/D), D=Eh3/12(1- ν2)] for the pretwisted shallow conical shell with ν=0.3, s/h=1000, θv=15° and θo=30°

ψ Aspect Ratio (L/s)

Present FEM (8 x 8)

Present FEM (6 x 6)

Liew et al. [2]

0.6 0.3524 0.3552 0.35990.7 0.2991 0.3013 0.3060

0°

0.8 0.2715 0.2741 0.2783 0.6 0.2805 0.2834 0.2882 0.7 0.2507 0.2528 0.2575

30° 0.8 0.2364 0.2389 0.2417

3.2 Effect of Stacking Sequence

Non-dimensional fundamental and second natural frequencies of six layered both symmetric and anti-symmetric graphite-epoxy composite rotating conical shells for a particular size of the delamination is presented in Tables 3 and 4 for different twist angles. Some aspects of the results are also presented in graphical form in Figures 4 and 5. The shells with 33% mid-plane delamination centered at a relative distance, d/L=0.33 from the fixed end are considered for the analyses. In general, the frequency parameters at stationary conditions are found to decrease with increase in the twist angle for all fibre orientations. At stationary condition, delaminated non-dimensional fundamental frequencies (NDFF) are found lower than the corresponding undelaminated fundamental frequencies irrespective of twist angle and fibre orientation. In all the cases of both twisted and untwisted non-rotating delaminated conical shells, non-dimensional fundamental frequencies are found to be minimum at θ=900. For untwisted delaminated conical shells at stationary condition, the non-dimensional fundamental frequencies are observed to attain the maximum value for θ=150 and gradually decrease to a minimum value for θ=900. This is in agreement with non-rotating shell with no delamination, wherein the non-dimensional fundamental frequency is maximum at θ=150 and gradually decreases to a minimum value for θ=900. For untwisted symmetric rotating delaminated conical shells, the non-dimensional fundamental frequencies are observed to attain a minimum value for θ=750, while the maximum values are attained at θ=450 (Ω=0.5) for both symmetric and anti-symmetric angle-ply wherein the same is identified at θ=00 and 15° (Ω=1.0) for symmetric and anti-symmetric cases respectively. The percentage difference between the maximum and minimum non-dimensional frequencies for in case of untwisted symmetric angle-ply conical shells with the variation of fibre orientation is found to be 71.9%, 90.1%, 79.3% for Ω=0.0, 0.5 and 1.0, respectively.


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For both symmetric and anti-symmetric cases at stationary condition, maximum values of non-dimensional fundamental natural frequencies of undelaminated angle-ply composite conical shells are identified at θ=150 (for ψ= 00, 150) and θ=300 (for ψ=300, 450), while minimum values invariably observed at θ=900 irrespective of twist angles, wherein the differences between maximum and minimum values of non-dimensional fundamental frequencies are found 68.5%, 54.2%, 48.5% and 46.1% (for symmetric case) and 68.4%, 55.6%, 51.8% and 51.1%, corresponding to ψ= 00, 150, 300 and 450 respectively. But in contrast, in both symmetric and anti-symmetric cases at stationary condition, maximum values of non-dimensional fundamental natural frequencies of delaminated angle-ply composite conical shells are identified at θ=150 (for ψ= 00) and θ=300 (for ψ=150, 300 and 450), while minimum values consistently depicted at θ=900 irrespective of twist angles, wherein the differences between maximum and minimum values of non-dimensional fundamental frequencies are found 65.2%, 53.6%, 49.0% and 46.8% (for symmetric case) and 64.8%, 54.4%, 51.5% and 51.2% for ψ= 00, 150, 300 and 450 respectively. Similarly, both symmetric and anti-symmetric cases at stationary condition, maximum values of non-dimensional second natural frequencies (NDSF) of undelaminated angle-ply composite conical shells are identified at θ=00 (for ψ= 00), θ=150 (for ψ= 150) and θ=300 (for ψ=300, 450), while minimum values found to be fixed at θ=900 irrespective of twist angles, wherein the differences between maximum and minimum values of non-dimensional second natural frequencies in both symmetric and anti-symmetric cases are found 48.1%, 56.7%, 53.2% and 51.8%, for ψ= 00, 150, 300 and 450 respectively. In contrast, in both symmetric and anti-symmetric cases at stationary condition, maximum values of non-dimensional second natural frequencies of delaminated angle-ply composite conical shells are identified at θ=00 (for ψ= 00) and θ=300 (for ψ=150, 300 and 450), while minimum values consistently depicted at θ=900 irrespective of twist angles, wherein the differences between maximum and minimum values of non-dimensional second natural frequencies in both symmetric and anti-symmetric cases are found 53.3%, 54.4%, 52.0% and 51.7%, for ψ= 00, 150, 300 and 450 respectively.

From Table 3, it is observed that the effect of centrifugal stiffening (i.e., increase of structural stiffness with increase of rotational speed) is predominantly found with reference to non-dimensional fundamental natural frequency in case of ψ= 00 and 450. For untwisted both symmetric and anti-symmetric laminates, this fact is valid for θ=300, 900. But for ψ= 450, this is only true for symmetric laminates irrespective of twist angle. From Table 4 in contrast to the fundamental frequency values, the centrifugal stiffening effect is very much pronounced in case of non-dimensional second natural frequency for both symmetric and anti-symmetric laminates irrespective of twist angle. For both untwisted and twisted cases, fibre orientation angle, θ=150, always shows that the second natural frequency values increase with with the increase of rotational speed. This fact is corroborated for most values of fibre orientation irrespective of twist angle. In anti-symmetric angle-ply, maximum increase in percentage of rotating frequency with respect to that at stationary condition is found at θ=60° corresponding to Ψ= 0°, 15°, 30°, 45°, respectively. Similarly, maximum reduction in percentage of rotating frequency with respect to that at stationary condition is found at θ=75° corresponding to Ψ=0°, 15°, and at θ=90° corresponding to Ψ=30°, 45°, respectively. For both symmetric and anti-symmetric cases at stationary condition, maximum delaminated non-dimensional fundamental natural frequencies are found at θ=30° corresponding to Ψ= 15°, 30°, 45°, and at θ=15° corresponding to Ψ= 0°.


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(a)

(b)

Fig. 4 Variation of (a) relative frequencies (fundamental) and (b) relative frequencies (second) of six-layered angle-ply conical shells at stationary condition, with respect to fibre orientation, considering h=0.0004, a/L=0.33, s/h=1000, L/s=0.7, θo=45°, θv=20°.

A specific trend against the variation of fibre orientation is depicted in respect of the deviation in non-dimensional fundamental frequency values of delaminated non-rotating shells compared to undelaminated one and maximum deviation for symmetric case is identified at θ=15° irrespective of twist angles. At rotating condition, non-dimensional fundamental and second natural frequencies have a typical trend for both delaminated and undelaminated cases. At lower rotational speed (Ω=0.5), the difference between maximum and minimum values of non-dimensional fundamental frequencies for symmetric laminates are observed 90.1%, 81.2%, 99.8% and 67.9% corresponds to Ψ= 0°, 15°, 30° and 45° respectively, while for anti-symmetric cases, the difference found to be 90.0%, 98%, 97.8% and 88.5%, corresponding to Ψ=0°, 15°, 30° and 45° respectively. In the contrast, at higher rotational speed (Ω=1.0), the difference between maximum and minimum values of non-dimensional fundamental frequencies for symmetric laminates are observed 79.3%, 68.0%, 77.8% and 85.0% corresponds to Ψ= 0°, 15°, 30° and 45° respectively, while for anti-symmetric cases, the difference found to be 81.5%, 39.4%, 54.1% and 97.5%,

Symmetric (θ/-θ/θ/θ/-θ/θ) Anti-symmetric (θ/-θ/θ/-θ/θ/-θ)

Symmetric (θ/-θ/θ/θ/-θ/θ) Anti-symmetric (θ/-θ/θ/-θ/θ/-θ)


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corresponding to Ψ= 0°, 15°, 30° and 45° respectively. On the other hand, at lower rotational speed (Ω=0.5), for both symmetric and anti-symmetric cases, the difference between maximum and minimum values of non-dimensional second natural frequencies are observed 57.3%, 71.6%, 71.1% and 63.6% corresponds to Ψ= 0°, 15°, 30° and 45° respectively, while for higher rotational speed (Ω=1.0), in both symmetric and anti-symmetric cases, the difference found to be 56.8%, 50.4%, 49.3% and 71.6%, corresponds to Ψ= 0°, 15°, 30° and 45° respectively. Table 3 Comparisons of NDFF [ω=ωn L2 √(ρ/E1h2)] between six layered a) Symmetric and b) Anti-symmetric angle-ply conical shells for various twist angles and fibre orientation angle at different rotational speeds (Ω)* considering h=0.0004, a/L=0.33, s/h=1000, L/s=0.7, θo=45°, θv=20°.

# No delamination *Ω=Ω´/ωo (Ω´=Actual angular rotational speed, ωo= Fundamental frequency of non-rotating shell)

Symmetric (θ/-θ/θ/θ/-θ/θ) Anti-symmetric (θ/-θ/θ/-θ/θ/-θ) Mid-plane Delamination Mid-plane Delamination

Ψ

θ

ND# Ω=0.0 Ω=0.5 Ω=1.0

ND Ω=0.0 Ω=0.5 Ω=1.0

0° 0.4007 0.3673 0.2566 0.6537 0.4007 0.3673 0.2667 0.2566

15° 0.4723 0.4278 0.2106 0.6438 0.4712 0.4225 0.1534 0.649630° 0.3677 0.3675 0.3873 0.4464 0.3738 0.3735 0.3937 0.4534

45° 0.2435 0.2434 0.4240 0.2790 0.2452 0.2451 0.4027 0.272260° 0.1813 0.1812 0.1632 0.3966 0.1818 0.1817 0.1615 0.3903

75° 0.1556 0.1555 0.0419 0.1351 0.1558 0.1557 0.0404 0.1202

0°

90° 0.1489 0.1488 0.3003 0.3094 0.1489 0.1488 0.3003 0.3094

0° 0.2498 0.2345 0.2309 0.3748 0.2498 0.2345 0.2309 0.374915° 0.2698 0.2545 0.2852 0.3542 0.2781 0.2608 0.2113 0.3711

30° 0.2666 0.2582 0.1912 0.3386 0.2756 0.2628 0.3840 0.379045° 0.2102 0.2076 0.3867 0.4313 0.2144 0.2095 0.1424 0.3019

60° 0.1631 0.1616 0.1068 0.1378 0.1650 0.1623 0.2123 0.498075° 0.1363 0.1342 0.1192 0.3497 0.1374 0.1344 0.0077 0.2748

15°

90° 0.1235 0.1198 0.0727 0.3041 0.1235 0.1198 0.0729 0.30410° 0.1485 0.1485 0.3241 0.1745 0.1485 0.1404 0.0643 0.1952

15° 0.1563 0.1484 0.0476 0.0705 0.1658 0.1568 0.3151 0.313130° 0.1662 0.1595 0.1420 0.0917 0.1776 0.1676 0.4193 0.2448

45° 0.1504 0.1466 0.5892 0.1302 0.1573 0.1502 0.2018 0.315660° 0.1240 0.1213 0.1637 0.3175 0.1277 0.1227 0.3219 0.2443

75° 0.0998 0.0966 0.1696 0.1336 0.1018 0.0973 0.1188 0.2122

30°

90° 0.0856 0.0813 0.0012 0.1440 0.0856 0.0813 0.0092 0.1448

0° 0.0968 0.0928 0.1399 0.1608 0.0968 0.0928 0.1399 0.160815° 0.0979 0.0937 0.1744 0.7181 0.1046 0.1001 0.1937 0.1839

30° 0.1008 0.0968 0.2461 0.4165 0.1111 0.1055 0.1895 0.111245° 0.0957 0.0927 0.1371 0.1079 0.1031 0.0977 0.3334 0.2964

60° 0.0819 0.0796 0.3379 0.3453 0.0861 0.0818 0.0383 0.331475° 0.0646 0.0622 0.2335 0.2571 0.0667 0.0632 0.1188 0.0571

45°

90° 0.0543 0.0515 0.1083 0.1868 0.0543 0.0515 0.1091 0.0081


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Table 4 Comparisons of NDSF [ω=ωn L2 √(ρ/E1h2)] between six layered a) Symmetric and b) Anti-symmetric angle-ply conical shells for various twist angles and fibre orientation angle at different rotational speeds (Ω) considering h=0.0004, a/L=0.33, s/h=1000, L/s=0.7, θo=45°, θv=20°.

The trend of relative frequencies of fundamental mode (ratio of delaminated frequency and undelaminated frequency) is shown in Fig. 4(a). For ψ= 00, 150 and 450, the trends of relative frequencies are found almost same for both symmetric and anti-symmetric cases. It is noted that there is no variation of relative frequency (fundamental) for higher fibre orientations in case of both symmetric as well as anti-symmetric untwisted laminates. At ψ=300, maximum relative frequency (fundamental) of symmetric laminate is identified at θ=00, on the other hand, the maximum relative frequency (fundamental) of anti-symmetric laminate is found at θ=600. For symmetric twisted laminates, maximum value of relative frequency (fundamental) is found corresponding to θ=600, This is also true for

Symmetric (θ/-θ/θ/θ/-θ/θ) Anti-symmetric (θ/-θ/θ/-θ/θ/-θ) Mid-plane Delamination Mid-plane Delamination

Ψ

θ

ND Ω=0.0 Ω=0.5 Ω=1.0

ND Ω=0.0 Ω=0.5 Ω=1.0

0° 0.5756 0.5750 0.5990 0.6664 0.5756 0.5750 0.4855 0.599015° 0.5269 0.5264 0.5592 0.7517 0.5269 0.5264 0.5592 0.751730° 0.5631 0.5017 0.7356 0.6428 0.5631 0.5017 0.7356 0.642845° 0.4920 0.4904 0.7391 0.5662 0.4920 0.4904 0.7392 0.566260° 0.3658 0.3647 0.3758 0.6234 0.3658 0.3647 0.3758 0.623475° 0.3132 0.3123 0.3155 0.3245 0.3132 0.3123 0.3155 0.3245

0°

90° 0.2988 0.2688 0.4346 0.4354 0.2988 0.2687 0.4346 0.43550° 0.5459 0.5070 0.5151 0.6819 0.5459 0.5070 0.5152 0.6820

15° 0.6292 0.5791 0.5630 0.6931 0.6292 0.5791 0.5631 0.693230° 0.6189 0.5866 0.6043 0.6920 0.6189 0.5866 0.6050 0.692045° 0.4582 0.4517 0.3900 0.5185 0.4582 0.4517 0.3900 0.518560° 0.3486 0.3452 0.3630 0.5221 0.3486 0.3452 0.3630 0.522075° 0.2947 0.2912 0.1717 0.3437 0.2947 0.2912 0.1717 0.3437

15°

90° 0.2725 0.2674 0.3166 0.4837 0.2725 0.2674 0.3165 0.48370° 0.4020 0.3659 0.3338 0.4078 0.4020 0.3659 0.3338 0.4078

15° 0.4565 0.4210 0.6425 0.6483 0.4565 0.4210 0.6425 0.648330° 0.4783 0.4394 0.6939 0.4429 0.4783 0.4394 0.6939 0.442945° 0.3936 0.3729 0.5597 0.6175 0.3936 0.3729 0.5597 0.617560° 0.3091 0.2972 0.4606 0.3865 0.3091 0.2972 0.4607 0.386575° 0.2548 0.2428 0.2578 0.4239 0.2548 0.2428 0.2578 0.3121

30°

90° 0.2238 0.2109 0.2003 0.3288 0.2238 0.2108 0.2004 0.32880° 0.2965 0.2635 0.4293 0.4406 0.2965 0.2635 0.4293 0.4406

15° 0.3261 0.2983 0.3635 0.3716 0.3261 0.2983 0.3635 0.371730° 0.3454 0.3140 0.5124 0.5894 0.3454 0.3140 0.5124 0.589445° 0.3075 0.2803 0.6274 0.6105 0.3075 0.2803 0.6274 0.610560° 0.2503 0.2301 0.2770 0.6582 0.2503 0.2301 0.2770 0.658275° 0.1992 0.1819 0.2283 0.2576 0.1992 0.1819 0.2283 0.2576

45°

90° 0.1666 0.1517 0.2671 0.1872 0.1666 0.1517 0.2671 0.1873


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anti-symmetric twisted laminates for ψ=150, 300, wherein maximum value of relative frequency (fundamental) for ψ=450, is obtained at θ=00. For untwisted angle-ply laminates, the percentage differences between maximum and minimum relative frequencies (fundamental) of symmetric and anti-symmetric laminates found 9.4% and 10.3% respectively. On the other hand for symmetric twisted cases, the same worked out to 5.2%, 5.1% and 2.5% corresponds to ψ= 150, 300 and 450 respectively, while for anti-symmetric twisted cases, the same found 4.6%, 1.7% and 1.2% corresponds to ψ= 150, 300 and 450 respectively. Therefore, the difference between maximum and minimum values of relative frequencies (fundamental) is found to be maximum in case of untwisted shell for both symmetric and anti-symmetric laminates. For both symmetric and anti-symmetric untwisted laminates, there is a striking difference between the values of relative frequencies (fundamental) corresponding to higher values of fibre orientation (θ=300, 450, 600, 750 and 900) and lower values of fibre orientation (θ=00, 150) wherein the minimum value of relative frequency is found at θ=150. The trends of relative frequencies (second) [Fig. 4(b)] are found almost same for both symmetric and anti-symmetric cases irrespective of twist angle. Maximum relative frequencies (second) of both symmetric and anti-symmetric laminate are obtained at θ=600. For untwisted angle-ply, the percentage differences between maximum and minimum relative frequencies (second) of both symmetric and anti-symmetric laminates found 10.3%. In contrast, for twisted both symmetric and anti-symmetric cases, the same worked out to 7.1%, 5.3% and 3.3% corresponds to ψ= 150, 300 and 450 respectively. It is also observed that for both symmetric and anti-symmetric twisted laminates, the value of relative frequency (second) decrease with increase of twist angle for a particular value of fibre orientation. On the other hand, for both symmetric and anti-symmetric untwisted laminates, maximum and minimum values of relative frequencies (second) are obtained corresponding to θ=00 and 300, respectively.

3.3 Effect of Rotational Speed

A delamination of relative length a/L=0.33 is considered and this is centered at a relative distances of d/L=0.33 and 0.66 from the fixed end. The delamination is considered at the interface of 45° and - 45° layers (Table 5). At stationary condition, non-dimensional fundamental natural frequencies are observed to decrease with the increase of twist angle. At stationary condition, non-dimensional fundamental natural frequencies for untwisted conical shell are found to reduce as the delamination moves towards the free end while the same is noted for second natural frequency corresponding to Ψ=30°. It is also noted that non-dimensional fundamental natural frequencies for twisted conical shell at stationary condition are identified to increase as the delamination moves towards the free end while non-dimensional second natural frequencies of stationary twisted conical shells are observed to decrease as the delamination moves towards the free end. It is observed that at d/L=0.33, non-dimensional fundamental natural frequency decreases with increase of twist angle for lower rotational speed (Ω=0.5). In case of fundamental natural frequency, the centrifugal stiffening effect is observed for Ψ=15°, 45° corresponding to d/L=0.66 but for Ψ=30° and at d/L=0.33, this is also observed. The similar trend is found in case of second natural frequency corresponding to Ψ=30°, 45° for location of delamination at d/L=0.33, 0.66 and for Ψ=15° corresponding to d/L=0.66.


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Table 5 Non-dimensional fundamental and second natural frequencies of three layered angle-ply [45°/-45°/45°] composite conical shells by varying relative position of delamination along the span considering d/L= 0.33 and 0.66, h=0.0004, a/L=0.33, s/h=1000, L/s=0.7, θo=45°, θv=20°.

Fig. 5 Variation of non-dimensional fundamental frequencies (NDFF) of six layered angle-ply [45°/-45°/45°]s composite conical shells, by varying relative position of delamination across thickness considering h=0.0004, a/L=0.33, s/h=1000, L/s=0.7, θo=45°, θv=20°.

NDFF NDSF Ψ

Ω d/L = 0.33 d/L = 0.66 d/L = 0.33 d/L = 0.66

0°

0.0 0.5 1.0

0.2427 0.3996 0.2725

0.2426 0.0927 0.2570

0.3453 0.5463 0.3457

0.4144 0.3668 0.5536

15°

0.0 0.5 1.0

0.1705 0.3197 0.1305

0.1856 0.2810 0.5225

0.4082 0.4033 0.4095

0.3997 0.5373 0.7349

30°

0.0 0.5 1.0

0.1038 0.1985 0.7446

0.1175 0.0444 0.3403

0.3109 0.4528 1.1297

0.2914 0.4224 0.5682

45°

0.0 0.5 1.0

0.0612 0.1117 0.0855

0.0688 0.0856 0.4052

0.2108 0.2760 0.4508

0.1965 0.2314 0.7238

Ψ=0° Ψ=15°

Ψ=30° Ψ=45°


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3.4 Effect of Location of Delamination

From the numerical results (Table 6, 7 and Fig. 5), it is observed that non-dimensional fundamental and second natural frequencies attain a minimum value at relative position, h´/h = 0.167 corresponding to n=6 irrespective of twist angle. In other words, minimum value of fundamental frequency is identified when delamination is located between 1st and 2nd layers for any values of twist angle. For a particular relative position of the delamination across the thickness, it is also noted that the non-dimensional fundamental and second natural frequencies are found to decrease with the increase in the angle of twist from 0° to 45°. Variation of non-dimensional fundamental frequencies with respect to relative position across thickness (h′/h) is also depicted in Fig. 5. Irrespective of twist angle, it shows a specific trend i.e., as the location of delamination changes, the frequency value attains the minimum near the top surface and then rises towards the bottom surface. The similar trend is also observed for non-dimensional second natural frequencies with higher values compared to the values of non-dimensional fundamental frequency as furnished in Table 6 and Table 7. Table 6 Non-dimensional fundamental frequencies of six layered angle-ply [45°/-45°/45°]s conical shells by varying relative position of delamination across thickness, considering d/L=0.5, Ω=0.0, h=0.0004, a/L=0.33, s/h=1000, L/s=0.7, θo=45°, θv=20°.

Table 7 Non-dimensional second natural frequencies of six layered angle-ply [45°/-45°/45°]s conical shells by varying relative position of delamination across thickness considering d/L=0.5, Ω=0.0, h=0.0004, a/L=0.33, s/h=1000, L/s=0.7, θo=45°, θv=20°.

Non-dimensional fundamental natural frequency (NDFF) h'/h

Ψ = 0° Ψ = 15° Ψ = 30° Ψ = 45°

0.000 0.245280 0.214466 0.157385 0.103122

0.167 0.245104 0.208174 0.148532 0.096609

0.333 0.245113 0.209008 0.149584 0.097296

0.500 0.245119 0.209500 0.150235 0.097732

0.667 0.245122 0.209733 0.150560 0.097952

0.833 0.245123 0.209742 0.150588 0.097965

1.000 0.245280 0.214466 0.157385 0.103122

Non-dimensional second natural frequency (NDSF) h'/h

Ψ = 0° Ψ = 15° Ψ = 30° Ψ = 45°

0.000 0.492035 0.458235 0.393626 0.307469

0.166 0.490313 0.450049 0.369434 0.278419

0.333 0.490361 0.451111 0.371742 0.279912

0.500 0.490388 0.451659 0.372905 0.280278

0.666 0.490402 0.451863 0.373099 0.279575

0.834 0.490408 0.451784 0.372372 0.277772

1.000 0.492035 0.458235 0.393626 0.307469


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3.5 Mode Shapes The mode shapes corresponding to the fundamental and second natural frequencies are

shown in Fig. 6, for various twist angles (ψ=0°, 15°, 30°, 45°) and rotational speeds (Ω=0.0, 0.5, 1.0), considering three layered angle-ply graphite-epoxy angle-ply composite conical shells of relative crack length a/L=0.33 and by varying the relative position of delamination along the span i.e., the delamination is centered at a relative distance d/L= 0.33, 0.66 from the fixed end. The other parameters considered are as mentioned in Table 5. In this case, the fundamental mode corresponds to the first torsion. It is identified that the symmetry modes are absent when twist angle is non-zero and the nodal lines indicate with zero displacement amplitude. The first span wise bending is observed only for untwisted conical shell at stationary condition corresponding to its fundamental frequency. The effect of first torsional mode increases with rotational speeds both for twisted and untwisted cases in respect of fundamental frequency. The dominance of first torsional mode is found significantly for all the cases corresponding to second natural frequency. 4. Conclusions

In general, the frequency parameters at stationary conditions are found to decrease with increase in the twist angle for fibre orientations. At stationary condition, delaminated non-dimensional fundamental and second natural frequencies are found to be lower than the corresponding undelaminated one irrespective of twist angle and fibre orientation. In all the cases of both twisted and untwisted non-rotating delaminated conical shells, non-dimensional fundamental frequencies are found to be minimum at θ=900. The effect of centrifugal stiffening (i.e., increase of structural stiffness with increase of rotational speed) is predominantly found with reference to non-dimensional fundamental natural frequency in case of ψ= 00 and 450. For untwisted both symmetric and anti-symmetric laminates, this fact is valid for θ=300 and 900. But for ψ=450, this is only true for symmetric laminates irrespective of twist angle. In contrast to the fundamental frequency values, the centrifugal stiffening effect is very much pronounced in case of non-dimensional second natural frequency for both symmetric and anti-symmetric laminates irrespective of twist angle.

At stationary condition, non-dimensional fundamental natural frequencies are observed to decrease with the increase of twist angle corresponding to locations of delaminations near the fixed end as well as near the free end. The non-dimensional fundamental natural frequencies for twisted conical shell at stationary condition increases as the delamination moves towards the free end while non-dimensional second natural frequencies of stationary twisted conical shells decreases as the delamination moves towards the free end. The non-dimensional fundamental and second natural frequencies attain a minimum value when the delamination is located at the interface of first and second layers (i.e., near the top surface) for all values of the twist angle. The non-dimensional fundamental and second natural frequencies decrease with the increase of twist angle for a particular relative position of the delamination across the thickness.

The mode shapes predominantly show the effect of first torsional mode for twisted conical shells, but for untwisted conical shell the first span wise bending mode is observed at stationary condition corresponding to its fundamental frequency at both the locations of delaminations i.e., near the fixed and free ends. The non-dimensional frequencies obtained are the first known results of the type of analyses carried out here and the results could serve as reference solutions for future investigators.


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d/L=0.33 d/L=0.66 Ψ Ω

Mode 1 Mode 2 Mode 1 Mode 2

0.0

0.5

0°

1.0

0.0

0.5

15°

1.0

0.0

0.5

30°

1.0

0.0

0.5

45°

1.0

Fig 6 Effect of twist and rotational speeds on mode shapes of three layered [45°/-45°/45°] graphite-epoxy delaminated composite conical shells by varying relative position of delamination along span with d/L=0.33 and 0.66, h=0.0004, a/L=0.33, s/h=1000, L/s=0.7, θo=45°, θv=20°. References (1) Qatu M. S. and Leissa A. W., “Vibration studies for Laminated Composite Twisted Cantilever Plates”, International Journal of Mechanical Sciences, 33, No. 11, (1991), pp. 927-940.


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(2) Liew K. M., Lim C. M. and Ong L. S., “Vibration of pretwisted cantilever shallow conical shells”, International Journal of Solids Structures, Vol. 31, (1994), pp. 2463-74.

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(8) Nabi, S. M. and Ganesean N., “Comparison of Beam and Plate Theories for Free Vibrations of Metal Matrix Composite Pre-Twisted Blades”, Journal of Sound and

Vibration, Vol. 189, No.2, (1996), pp149-160. (9) Sekine, H., Natsume, T., and Fukunaga H., “Impact Response Analysis of Partially Delaminated Composite Laminates”, JSME International Journal, Vol.63, No. 608A, (1997), pp.787-793. (10) Hu, N., Sekine H., Fukunaga H. and Yao Z.H., “Impact Analysis of Composite Laminates with Multiple Delaminations”, International Journal of Impact Engineering, Vol.22, (1999), pp.633-648. (11) Parhi P. K., Bhattacharyya S. K. and Sinha P. K., “Failure analysis of multiple delaminated due to bending and impact”, Bull. Mater. Sci., Indian Academy of Sciences, Vol. 24, No. 2, (2001), pp. 143–149. (12) Aymerich F., Dore F. and Priolo P., “Simulation of multiple delamination in impacted cross-ply laminates using a finite element model based on cohesive interface elements”, J. Composites Science and Technology, Vol. 69, Issues 11-12, (2009), pp. 1699-1709. (13) Viswanathan K.K., Kim K. S., “Free vibration of antisymmetric angle-ply-laminated plates including transverse shear deformation: Spline method”, International Journal of Mechanical Sciences, Vol. 50, Issues 10-11, (2008), pp.1476-1485. (14) Khare R. K., Kant T., Garg A. K., “Free vibration of composite and sandwich laminates with a higher-order facet shell element”, Journal of Composite Structures, Vol.65, (2004), pp. 405-418. (15) Patel B.P., Singh S., Nath Y., “Postbuckling characteristics of angle-ply laminated truncated circular conical shells”, Journal of Communications in Nonlinear Science and Numerical Simulation, Vol. 13, (2008), pp. 1411-1430. (16) Alibeigloo A., “Static and vibration analysis of axi-symmetric angle-ply laminated cylindrical shell using state space differential quadrature method”, International Journal of Pressure Vessels and Piping, Vol. 86, Issue 11, (2009), pp. 738-747. (17) Gim C.K., “Plate Finite Element Modelling of Laminated Plates”, Journal of Computers and Structures, Vol. 52, (1994), pp.157-168. (18) Bathe K. J., “Finite Element Procedures in Engineering Analysis”, Prentice Hall of India, New Delhi, 1990. (19) Leissa A. W., Lee J. K. and Wang A. J., “Vibrations of Twisted Rotating Blades”, Journal of Vibration, Acoustics, Stress, and Reliability in Design, Trans., ASME, Vol.106, No.2, (1984), pp.251-257. (20) Karmakar A. and Sinha P. K., “Failure Analysis of Laminated Composite Pretwisted Rotating Plates”, JRPC, Vol.20, (2001), pp.1326-1357. (21) Sreenivasamurthy S. and Ramamurthi V., “Coriolis Effect on the Vibration of Flat Rotating Low Aspect Ratio Cantilever Plates”, Journal of Strain Analysis, Vol. 16, No.2, (1981), pp. 97-106. (22) Cook R. D., Malkus D. S. and Plesha M. E., “Concepts and Applications of Finite Element Analysis”, (1989), John Wiley and Sons, New York. (23) Qatu M. S. and Leissa A. W., “Natural Frequencies for Cantilevered Doubly-Curved Laminated Composite Shallow Shells”, Journal of Composite Structures, Vol. 17, (1991), pp. 227-255.

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