20
Appl. Math. Mech. -Engl. Ed., 34(4), 437–456 (2013) DOI 10.1007/s10483-013-1682-8 c Shanghai University and Springer-Verlag Berlin Heidelberg 2013 Applied Mathematics and Mechanics (English Edition) Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method K. DANESHJOU 1 , M. TALEBITOOTI 1 , R. TALEBITOOTI 2 (1. School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16846-13114, Iran; 2. School of Automotive Engineering, Iran University of Science and Technology, Tehran 16846-13114, Iran) Abstract The generalized differential quadrature method (GDQM) is employed to con- sider the free vibration and critical speed of moderately thick rotating laminated compos- ite conical shells with different boundary conditions developed from the first-order shear deformation theory (FSDT). The equations of motion are obtained applying Hamilton’s concept, which contain the influence of the centrifugal force, the Coriolis acceleration, and the preliminary hoop stress. In addition, the axial load is applied to the conical shell as a ratio of the global critical buckling load. The governing partial differential equations are given in the expressions of five components of displacement related to the points ly- ing on the reference surface of the shell. Afterward, the governing differential equations are converted into a group of algebraic equations by using the GDQM. The outcomes are achieved considering the effects of stacking sequences, thickness of the shell, rotating velocities, half-vertex cone angle, and boundary conditions. Furthermore, the outcomes indicate that the rate of the convergence of frequencies is swift, and the numerical tech- nique is superior stable. Three comparisons between the selected outcomes and those of other research are accomplished, and excellent agreement is achieved. Key words generalized differential quadrature method (GDQM), natural frequency, rotating conical shell, first-order shear deformation theory (FSDT), critical speed Chinese Library Classification O343.8 2010 Mathematics Subject Classification 74B05 1 Introduction Rotating conical shells are applied in various engineering applications including centrifugal separators, gas turbines, rotary kilns, and rotary dryers. Vibrations of such structures are also widely studied [1–3] . Meanwhile, most researches are constrained to the vibration study of rotating cylindrical and conical shells developed from the classical laminated shell theory (CLST). These literatures include the researches done by Lam ang Loy [4] and Hua [5] on the multilayered cylindrical and conical shells with a constant angular velocity and also the studies on different CLSTs as well as an investigation on the effects of boundary conditions (BCs) on Received Feb. 1, 2012 / Revised Nov. 29, 2012 Corresponding author M. TALEBITOOTI, Ph. D., E-mail: [email protected]

Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

  • Upload
    r

  • View
    214

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

Appl. Math. Mech. -Engl. Ed., 34(4), 437–456 (2013)DOI 10.1007/s10483-013-1682-8c©Shanghai University and Springer-Verlag

Berlin Heidelberg 2013

Applied Mathematicsand Mechanics(English Edition)

Free vibration and critical speed of moderately thick rotatinglaminated composite conical shell using generalized

differential quadrature method∗

K. DANESHJOU1, M. TALEBITOOTI1, R. TALEBITOOTI2

(1. School of Mechanical Engineering, Iran University of Science and Technology,

Tehran 16846-13114, Iran;

2. School of Automotive Engineering, Iran University of Science and Technology,

Tehran 16846-13114, Iran)

Abstract The generalized differential quadrature method (GDQM) is employed to con-

sider the free vibration and critical speed of moderately thick rotating laminated compos-

ite conical shells with different boundary conditions developed from the first-order shear

deformation theory (FSDT). The equations of motion are obtained applying Hamilton’s

concept, which contain the influence of the centrifugal force, the Coriolis acceleration,

and the preliminary hoop stress. In addition, the axial load is applied to the conical shell

as a ratio of the global critical buckling load. The governing partial differential equations

are given in the expressions of five components of displacement related to the points ly-

ing on the reference surface of the shell. Afterward, the governing differential equations

are converted into a group of algebraic equations by using the GDQM. The outcomes

are achieved considering the effects of stacking sequences, thickness of the shell, rotating

velocities, half-vertex cone angle, and boundary conditions. Furthermore, the outcomes

indicate that the rate of the convergence of frequencies is swift, and the numerical tech-

nique is superior stable. Three comparisons between the selected outcomes and those of

other research are accomplished, and excellent agreement is achieved.

Key words generalized differential quadrature method (GDQM), natural frequency,

rotating conical shell, first-order shear deformation theory (FSDT), critical speed

Chinese Library Classification O343.8

2010 Mathematics Subject Classification 74B05

1 Introduction

Rotating conical shells are applied in various engineering applications including centrifugalseparators, gas turbines, rotary kilns, and rotary dryers. Vibrations of such structures arealso widely studied[1–3]. Meanwhile, most researches are constrained to the vibration studyof rotating cylindrical and conical shells developed from the classical laminated shell theory(CLST). These literatures include the researches done by Lam ang Loy[4] and Hua[5] on themultilayered cylindrical and conical shells with a constant angular velocity and also the studieson different CLSTs as well as an investigation on the effects of boundary conditions (BCs) on

∗ Received Feb. 1, 2012 / Revised Nov. 29, 2012Corresponding author M. TALEBITOOTI, Ph.D., E-mail: [email protected]

Page 2: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

438 K. DANESHJOU, M. TALEBITOOTI, and R. TALEBITOOTI

cylindrical shells with a high rotary speed[6–7]. A nine-node curvilinear super-parametric finiteelement is used by Chen et al.[8] to solve problems of vibrations in shells rotating at high speedsabout their longitudinal axes. Lam and Hua[9] investigated the vibration analysis of rotatingconical shells under simply supported constraints by using the most applicable classical shellcalled as Love theory. Lim and Liew[10] established a new approach developed from energyconcept to find out the free vibration of shallow conical shells.

Most of the literatures noted above are based on CLST. Therefore, they are not able to modelthe thick shell in broad band frequencies or even the thin shell in high frequency domains as aconsequence of neglecting the shear waves addition to rotary inertia in their model. This paperconsiders the first-order shear deformation theory (FSDT)[11] to analyze relatively thick shells.It takes account of the influences of rotation of the shell and also the transverse shear stresses.

The researches point to the fact that the problems devoting to the vibration of rotatingconical structures is too complicated to be solved with conventional routines, such as thoseused for cylindrical structures. The generalized differential quadrature method (GDQM) isa well-organized numerical procedure which comes from the differential quadrature method(DQM)[12]. The main structures of GDQM are explained in Section 2.5, while the more detailedstudies on this technique as well as some engineering advanced topics are discussed by Shu[13].In fact, more researches in this context have been presented[14–21] due to excessive simplicityand adaptability.

The works mentioned above indicate that, although the critical speed and vibration ofconical shells are important in many industrial usages, no study is devoted in the field ofvibration of axially loaded rotating laminated conical shells with arbitrary BCs. Noting thatthe CLST cannot include the favorable modeling for moderately thick shells as it assume thatthe transverse normal to the plane is remained normal after deformation, the FSDT whichdiminishes these assumptions with considering the thickness field variables as linear functionsis applied in this study.

In this paper, the governing equations included five 2D partial differential equations (PDEs)can be extracted by the aid of Hamilton’s concept into the energy function. Having substitutedthe displacements and rotations of the reference surface into equations of motion, the second-order PDEs will be transformed into five ordinary differential equations (ODEs). With the aidof GDQM, the ODEs are converted into algebraic equations. Having imposed the assumed BCs,the eigenvalue equation of the problem is obtained. Developed from eigenvalues, the results areattained to interpret the effects of stacking sequence, thickness of the shell, boundary condition,rotating speed, and half-vertex cone angle on the frequency characteristics. The results arecompared with those other researches, especially for rotating cylindrical shells where the coneangle approaches into zero, and also the non-rotating composite conical shells where the rotatingspeed goes into zero. These comparisons indicate the excellent precision of the present study.

2 Problem specification

2.1 Geometrical outlineFigure 1 depicts a moderately thick laminated conical shell which rotates with a constant

angular velocity Ω about its longitudinal symmetric axis. As illustrated, Na is the axial load,α is a half-vertex cone angle, h is the thickness, L is the length, a is the radius at small end,and b is the radius at the large end of the conical shell. The reference surface of the shellis considered at its mid-plane, where a coordinate spot (x, θ, z) is fixed. The radius at anycoordinate position (x, θ, z) is followed as

r(x, z) = a+ x sinα+ z cosα. (1)

It is noteworthy that if the parameter α goes to zero, the formulation of the conical shell isreduced to that of the cylindrical shells.

Page 3: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

Free vibration and critical speed of moderately thick rotating laminated composite conical shell 439

Fig. 1 Geometry of composite conical shell structure

2.2 Stress-strain formulaThe stress-strain relation of a lamina is formulated in terms of the fiber direction of the

layers as follows:⎛⎜⎜⎜⎜⎝

σ1

σ2

τ12τ13τ23

⎞⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎝

Q11 Q12 0 0 0Q12 Q22 0 0 00 0 Q66 0 00 0 0 Q55 00 0 0 0 Q44

⎞⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝

ε1ε2ε12ε13ε23

⎞⎟⎟⎟⎟⎠. (2)

The coordinates of the laminates are described with subscripts of 1 and 2, in which thesubscript 1 represents the fiber direction, and the subscript 2 denotes the orthogonal one. Inaddition, the stiffness arrays Qij are introduced in expressions of the mechanical properties ofthe lamina as follows:

⎧⎪⎪⎨⎪⎪⎩

Q11 =E11

1 − υ12υ21, Q12 = Q21 =

υ12E22

1 − υ12υ21,

Q22 =E22

1 − υ12υ21, Q44 = KG23, Q55 = KG13, Q66 = G12,

(3)

where Eii are the elastic moduli, Gij are the shear moduli, υij are Poisson’s ratios, and K isthe shear correction coefficient equaled as 5/6 in the present implementation of the FSDT. Thetransformation matrix T is used to convert the stresses and strains from the 1, 2 coordinatesto x, θ coordinates, which is defined as

T =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

C2β S2

β 0 0 0 2CβSβ

S2β C2

β 0 0 0 −2CβSβ

0 0 1 0 0 00 0 0 Cβ −Sβ 00 0 0 Sβ Cβ 0

−CβSβ CβSβ 0 0 0 C2βS

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠, Cβ = cosβ, Sβ = sinβ, (4)

where β is the orientation of the fibers. Moreover, the stress-strain relationship of a typical kthlayer of a laminated composite shell becomes

σ = Qijε, σT = (σx, σθ, τxθ, τθz, τxz), εT = (εx, εθ, εxθ, εθz, εxz), (5)

Page 4: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

440 K. DANESHJOU, M. TALEBITOOTI, and R. TALEBITOOTI

where the constants Qij are the elastic stiffness coefficients, which are deduced from

Q = T−1QT. (6)

2.3 First-order shear deformation theoryIn the FSDT for laminated shells, the Kirchhoff assumptions are relaxed, which consider

that the normal to the plane is not remained normal after deformation. In addition, some otherassumptions are made in deriving a relatively thick shell theory. They are expressed as follows:

(i) The transverse normal strain is assumed to be small and negligible, i.e., εz = 0.(ii) The displacements are assumed to be small. Thus, strains seem to be infinitesimal.(iii) The shell is relatively thick. Thus, the normal stress in the thickness direction is assumed

to be negligible, i.e., σz = 0.(iv) The laminated shell behaves as a linear elastic material.(v) The rotary inertia of the shell is considered.According to the mentioned hypothesis, the displacement components based on the FSDT

are of the form⎧⎪⎨⎪⎩

u(x, θ, z, t) = u0(x, θ, t) + zψx(x, θ, t),

v(x, θ, z, t) = v0(x, θ, t) + zψθ(x, θ, t),

w(x, θ, z, t) = w0(x, θ, t),

(7)

where u0, v0, w0, ψx, and ψθ will be predicted in the paper. The kinematics principle formulatedby Eq. (7) will be accompanied with the expression that the displacements are assumed to besmall. In other words, w(x, θ, t) � h. A relationship between strains and displacements of theshell can be written as[22]

⎛⎜⎜⎜⎜⎝

εx

εθ

γxθ

γxz

γθz

⎞⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

∂x0 0

sinαr

1r

∂θ

cosαr

1r

∂θ

∂x− sinα

r0

∂z0

∂x

0∂

∂z− cosα

R

1r

∂θ

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎝uvw

⎞⎠ , (8)

where r represents the radius of the conical shell at the desired node toward the x-directionrepresented by Eq. (1), and (u, v, w)T is expressed in Eq. (7).2.4 Governing equations

The PDEs corresponding to the problem can be developed applying Hamilton’s concept as∫ t2

t1

(δT − δUε − δUh − δUNa)dt = 0, (9)

where δT , δUε, δUh, and δUNa are the variations of the kinetic energy, the strain energy, thestrain energy through hoop stress, and the work of external axial forces, respectively, and tdenotes the time. The shell strain energy is formulated as

Uε =12

∫ h2

−h2

∫ L

0

∫ 2π

0

(Q11ε2x +Q22ε

2θ +Q66γ

2xθ +Q55γ

2xz +Q44γ

2θz

+ 2(Q12εxεθ +Q16εxγxθ +Q26εθγxθ +Q45γxzγθz))r(x, z)dθdxdz. (10)

Page 5: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

Free vibration and critical speed of moderately thick rotating laminated composite conical shell 441

The work generated by the rotation of the shell due to the centrifugal force is represented as

Uh =12

∫ h2

−h2

∫ L

0

∫ 2π

0

( 1r(x, z)

∂v

∂θ+u sinα+ w cosα

r(x, z)

)2

r(x, z)dθdxdz, (11)

where Nθ is the preliminary hoop stress as a consequence of the centrifugal force given by

Nθ = ρ hΩ2r2(x, z). (12)

Besides, the work done on the shell as a consequence of the axial load is described as[23]

UNa = Nxa

∫ h2

−h2

∫ 2π

0

∫ L

0

12

(∂w∂x

)2

r(x, z) dxdθdz, (13)

where Nxa is the axial load on edge of shell in the x-direction. The effects of Nθ

a and Nza are

null for clamped and simply supported BCs.The kinetic energy of the shell is represented as

T =12ρ

∫ h2

−h2

∫ 2π

0

∫ L

0

V · V r(x, z)dxdθdz, (14)

where the velocity at the desired node is formulated as

V = r + (Ω cosα i − Ω sinαk) × r, r =∂r

∂t, (15)

where r is represented with its components in the x-, θ-, and z-directions followed as

r = ui + vj + wk. (16)

Substituting Eqs. (15) and (16) into Eq. (14), the kinetic energy can be extended into

T =12ρ

∫ h2

−h2

∫ 2π

0

∫ L

0

((u20 + v2

0 + w20 + z2ψ2

x + z2ψ2θ) + 2zu0ψx + Ω2(v2

0 + z2ψ2θ + zvψθ

+ (z2ψ2x + zu0ψx + u2

0) sin2 α+ w20Ω

2 cos2 α+ (u0w0 + zw0ψx) sinα cosα)

+ Ω sinα(v0u0 − u0v0 + z(u0ψθ − u0ψθ + vψx − vψx) + z2(ψθψx − ψθψx))

+ Ω cosα(v0w0 − v0w0 + z(ψθw0 − ψθw0)))r(x, z)dxdθdz. (17)

Using Hamilton’s concept with substituting Eqs. (17), (13), (10), and (11) into Eq. (9), thefollowing equations are derived:

⎛⎜⎜⎜⎜⎝

L11 L12 L13 L14 L15

L21 L22 L23 L24 L25

L31 L32 L33 L34 L35

L41 L42 L43 L44 L45

L51 L52 L53 L54 L55

⎞⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝

u0

v0w0

ψx

ψθ

⎞⎟⎟⎟⎟⎠

= 0, (18)

where the coefficients Lij (i, j = 1, 2, · · · , 5) are differential operators of (u0, v0, w0, ψx, ψθ)T.

Page 6: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

442 K. DANESHJOU, M. TALEBITOOTI, and R. TALEBITOOTI

2.5 GDQM solution of governing equationsThe GDQM is a numerical technique for estimating derivatives of an adequately smooth

function. The derivatives of a smooth function are estimated with weighted sums of the functionvalues at a set of so-called nodes. It should be noted that the node is also an entitled grid pointor a mesh point. Therefore, with applying the GDQM, the derivatives of a function f(xi, θi) ata node (xi, θi) are written as[12]

∂pg(x, θ)∂xp

∣∣∣x=xj

=NGP∑k=1

Cpjkg(xj , θk), j = 1, 2, · · · , NGP, (19)

where NGP is the number of grid points, g can be taken as u, v, or w, and parameters of Cpjk

are the corresponding weighting constants associated to the pth-order derivatives which areachieved as below.

If p = 1, then

C1jk =

M (1)(xj)(xj − xk)M (1)(xk)

for j �= k and j, k = 1, 2, · · · , NGP, (20)

and

C1jj = −

NGP∑k=1(k �=j)

C1jk for j = 1, 2, · · · , NGP, (21)

where M (1)(x) is the first derivative of M(x), and they can be defined as

M(x) =NGP∏k=1

(x− xk), M (1)(xk) =NGP∏

j=1(j �=k)

(xk − xj). (22)

If p > 1, namely for the second- and higher-order derivatives, the weighting constants areacquired by means of the subsequent recurrence formulation, i.e.,

Cpjk =r

(C1

jk · Cp−1jj − Cp−1

jk

xj−xk

)for j �= k and j, k = 1, 2, · · ·, NGP, p = 2, 3, · · ·, NGP−1, (23)

and

Cpjj = −

NGP∑k=1(k �=j)

Cpjk for j = 1, 2, · · · , NGP. (24)

Since no restriction is obligatory on the coordinate discretization and NGP is in the imple-mentation of the GDQM, the following discretization of the mesh points toward the x-directioncan be presumed in this relation:

xj =L

2

(1 − cos

( j − 1NGP − 1

π)), j = 1, 2, · · · , NGP. (25)

The vibration modes of the conical shell are described by the circumferential wave number nand the natural frequency ω. The statements for the displacement components are presumedto have the form of a product with indefinite continuous functions varying smoothly toward the

Page 7: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

Free vibration and critical speed of moderately thick rotating laminated composite conical shell 443

x-direction and the trigonometric functions toward the θ-direction, i.e.,

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

u0 = U cos(nθ + ω t),

v0 = V sin(nθ + ω t),

w0 = W cos(nθ + ω t),

ψx = ζx cos(nθ + ω t),

ψθ = ζθ sin(nθ + ω t),

(26)

where U, V, W, ζx, and ζθ are the indefinite functions of the x-direction. Having substitutedthe displacement components defined in Eq. (26) into the set of PDE of Eq. (18), a group ofODEs with the coefficients varying toward the x-direction is produced as

L∗X∗ = 0, (27)

where X∗T = (U, V, W, ζx, ζθ) is the null space basis which express the mode shape, andL∗ = (L∗

ij) (i, j = 1, 2, · · · , 5) is a 5 × 5 matrix which includes the differential operative of U∗

and is defined as

L∗ =

⎛⎜⎜⎜⎜⎜⎜⎝

M11 M12 M13 M14 M15

M21 M22 M23 M24 M25

M31 M32 M33 M34 M35

M41 M42 M43 M44 M45

M51 M52 M53 M54 M55

⎞⎟⎟⎟⎟⎟⎟⎠ω2 +

⎛⎜⎜⎜⎜⎜⎜⎝

G11 G12 G13 G14 G15

G21 G22 G23 G24 G25

G31 G32 G33 G34 G35

G41 G42 G43 G44 G45

G51 G52 G53 G54 G55

⎞⎟⎟⎟⎟⎟⎟⎠ω

+

⎛⎜⎜⎜⎜⎜⎜⎝

K11 K12 K13 K14 K15

K21 K22 K23 K24 K25

K31 K32 K33 K34 K35

K41 K42 K43 K44 K45

K51 K52 K53 K54 K55

⎞⎟⎟⎟⎟⎟⎟⎠, (28)

where the coefficients of Mij , Gij , and Kij (i, j = 1, 2, · · · , 5) are expressed in Appendix A.

With imposing Eq. (19) to Eq. (27) and rearranging Eq. (27) with respect to the orders ofderivatives, the linear discrete algebraic equations are achieved as

L∗X∗∣∣∣x=xj

= L∗∗5×15X

∗∗15×1

∣∣∣x=xj

= 0, j = 1, 2, · · · , NGP, (29)

where NGP is the entire discrete grid point number toward the x-direction, and X∗∗ is given by

X∗∗T∣∣∣x=xj

=(U(xj), U (1)(xj), U (2)(xj), V (xj), V (1)(xj), V (2)(xj),W (xj),W (1)(xj),W (2)(xj),

ζx(xj), ζ(1)x (xj), ζ(2)

x (xj), ζθ(xj), ζ(1)θ (xj), ζ

(2)θ (xj)), (30)

Page 8: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

444 K. DANESHJOU, M. TALEBITOOTI, and R. TALEBITOOTI

where⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

U (p)(xj) =NGP∑k=1

CpjkU(xk),

V (p)(xj) =NGP∑k=1

CpjkV (xk),

W (p)(xj) =NGP∑k=1

CpjkW (xk),

ζ(p)x (xj) =

NGP∑k=1

Cpjkζx(xk),

ζ(p)θ (xj) =

NGP∑k=1

Cpjkζθ(xk), p = 1, 2.

(31)

Hence, the entire system of PDE is discretized. Therefore, a group of algebraic equations canbe produced from the general combination of these equations,

(Mω2)d+ (Gω)d +Kddd+Kdbb = 0. (32)

In the above equations, the dimensions of M , Kdd, and G are 5(NGP − 2) × 5(NGP − 2) andthe dimension of Kdb is 5(NGP − 2) × 10. In addition, the dimensions of vectors d and bare 5(NGP − 2) and 10, respectively, which represent the indefinite degrees of freedom at thesampling points within the inner territory, and those on the boundary are as follows:

dT = (U(x2), V (x2),W (x2), ζx(x2), ζθ(x2), · · · , U(xNGP−1),

V (xNGP−1),W (xNGP−1), ζx(xNGP−1), ζθ(xNGP−1)),

bT = (U(x1), V (x1),W (x1), ζx(x1), ζθ(x1),

U(xNGP), V (xNGP),W (xNGP), ζx(xNGP), ζθ(xNGP)).

(33)

Similarly, the discretized form of the BCs becomes

Kbdd+Kbbb = 0. (34)

In the current practical use of GDQM, four BCs considered for shells are as follows[16]:(i) Clamped at both edges (Cs-Cl):

u0 = 0, v0 = 0, w0 = 0, ψx = 0, Mxθ = 0 at x = 0 and L. (35)

(ii) Simply supported at both edges (Ss-Sl):

v0 = 0, w0 = 0, Nx = 0, Mx = 0, Mxθ = 0 at x = 0 and L. (36)

(iii) Simply supported at the small edge-clamped at the large edge (Ss-Cl):{v0 = 0, w0 = 0, Nx = 0, Mx = 0 , Mxθ = 0 at x = 0,u0 = 0, v0 = 0, w0 = 0, ψx = 0, Mxθ = 0 at x = L.

(37)

(iv) Clamped at the small edge-simply supported at the large edge (Cs-Sl):{u0 = 0, v0 = 0, w0 = 0, ψx = 0, Mxθ = 0 at x = 0,v0 = 0, w0 = 0, Nx = 0 , Mx = 0 , Mxθ = 0 at x = L.

(38)

Page 9: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

Free vibration and critical speed of moderately thick rotating laminated composite conical shell 445

By means of Eq. (34) to remove vector (b) from Eq. (32), it can be recognized that

(Mω2)d+ (Gω)d+ (Kdd −KdbKbb−1Kbd)d = 0. (39)

Equation (39) is a non-standard eigenvalue equation that equivalently converted into a standardformula for a certain frequency as[17]

( ( 0 I−K −G

)

︸ ︷︷ ︸A∗

−( I 0

0 M

)

︸ ︷︷ ︸B∗

ω

)(dωd

)= 0, (40)

where I is an identity matrix, and its dimension is 5(NGP − 2) × 5(NGP − 2). Applying amainstream method, the eigenvalue equation (40) can be resolved, and 10(NGP − 2) eigenvaluesare achieved. The two eigenvalues among these outcomes are chosen, for which the absolutesof real values are the minimum amount. One of these eigenvalues is negative and matches tobackward, and the other one is positive and agrees to the forward wave. In one instance, for anon-rotating conical shell, these smallest eigenvalues are the same in values, and the vibrationof the shell behaves as the stationary motion.

3 Results and discussion

Because of proper demonstration of the results, the backward and forward waves are depictedby solid lines and dashed lines, respectively, with the unit of the angular velocity Ω being inr/s (revolution per second). Besides, four BCs, including Cs-Cl, Ss-Sl, Ss-Cl, and Cs-Sl, areregarded here to be examined. Mechanical constants of the structure used in current work arelisted in Table 1.

Table 1 Mechanical properties of material

Modulus of elasticity/Poisson’s ratio

Modulus of rigidity/ Density/

(GN·m−2) (GN·m−2) (kg·m−3)

Isotropic E=7.6 υ = 0.3 G= 2.9 ρ = 1600Orthotropic E11=125 υ12 = 0.4 G11=5.9 ρ = 1600

Carbon- E22=10 υ13 = 0.2 G13=3.0 ρ = 1600Epoxy E33=10 υ23 = 0.2 G23=5.9 ρ = 1600

The GDQM is particularly efficient for the comprehensive analysis such as buckling insta-bility or vibration. The accuracy of the GDQM is strongly trustworthy, and its application isboth uncomplicated and effective. In order to exhibit the adaptability and effectiveness of thecurrent work, three comparisons are accomplished with the accessible results in the existingliterature. The first, as presented in Table 2, is a stationary conical shell with Ss-Sl by takingΩ=0 into the current relations[24–25]. The second comparison, as listed in Table 3, is a station-ary isotropic fully clamped conical shell. The third, as displayed in Fig. 2, is a cross-ply conicalshell with a fully simply supported BC[17]. The outcomes listed in Tables 2–3 and also thoseillustrated in Fig. 2 show the excellent accurateness of the current study.

As noted, NGP is arbitrary and should be chosen properly. Therefore, a convergence checkingis done for the conical shell with different lengths in Figs. 3(a)–3(b). It is apparent that at least11 grid points should be chosen in GDQM to obtain accurate numerical results. Therefore, allpresented results followed in this study employ 11 grid points in GDQM. Noteworthy, sinceNGP is small, the computational needed time is very short in comparison with other numericalmethods such as finite element method (FEM).

Page 10: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

446 K. DANESHJOU, M. TALEBITOOTI, and R. TALEBITOOTI

Table 2 Comparison of parameter f = ωbp

(1 − υ2)ρ/E for stationary conical shell with Ss-Sl (m=1,υ=0.3, h/b=0.01, L sin (α/b)=0.25)

nα = 30◦ α = 45◦ α = 60◦

Ref.[24] Ref.[25] Present Ref.[24] Ref.[25] Present Ref.[24] Ref.[25] Present

2 0.791 0 0.842 0 0.800 3 0.687 9 0.765 5 0.705 1 0.572 2 0.634 8 0.585 43 0.724 8 0.737 6 0.728 8 0.697 3 0.721 2 0.701 5 0.600 1 0.623 8 0.604 34 0.635 2 0.636 2 0.634 8 0.666 4 0.673 9 0.666 4 0.605 4 0.614 5 0.605 35 0.553 1 0.552 8 0.552 7 0.630 4 0.632 3 0.629 1 0.607 7 0.611 1 0.605 56 0.494 9 0.495 0 0.494 3 0.603 2 0.603 5 0.601 3 0.615 9 0.617 1 0.612 47 0.465 3 0.466 1 0.464 1 0.591 8 0.592 1 0.589 2 0.634 3 0.635 0 0.629 78 0.465 4 0.466 0 0.462 6 0.599 2 0.600 1 0.595 9 0.665 0 0.666 0 0.659 39 0.489 2 0.491 6 0.486 5 0.625 7 0.627 3 0.621 4 0.708 4 0.710 1 0.701 6

Table 3 Comparison of parameter f = ωbp

(1 − υ2)ρ/E for stationary conical shell with Cs-Cl(m=1, υ=0.3, h/b=0.01, L sin (α/b)=0.5)

nα = 45◦ α = 60◦

Ref.[24] Ref.[25] Present Ref.[24] Ref.[25] Present

1 0.812 0 0.845 2 0.812 8 0.631 6 0.644 9 0.632 42 0.669 6 0.680 3 0.671 3 0.552 3 0.556 8 0.553 53 0.543 0 0.555 3 0.544 9 0.478 5 0.481 8 0.479 84 0.457 0 0.477 8 0.458 8 0.429 8 0.436 1 0.430 85 0.409 5 0.439 5 0.410 8 0.409 3 0.420 2 0.409 8

Fig. 2 Frequencies of rotating cross-ply conical shell versus rotating speed in comparison betweenresults of current work and those reported in Ref. [17] (L=2m, a=0.5 m, m=1, n=1, h=3mm,α=15◦, [0◦/90◦/0◦], Ss-Sl)

Fig. 3 Convergence of frequency for stationary cross-ply conical shell (a=0.5m, h=3mm, m=1,α=15◦, [0◦/90◦/0◦], Ss-Sl)

Page 11: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

Free vibration and critical speed of moderately thick rotating laminated composite conical shell 447

Figures 4(a)–4(b) show the variations of the natural frequency with n by using both FSDTand CLST for cylindrical and conical shells, respectively. It is exciting to note that the differencebetween FSDT and CLST becomes significant as n is increased as a consequence of shorteningthe wave length and then considerable effects of the shear waves. This discrepancy will be moresignificant for a conical shell where it compares with the cylindrical one. In other word, theresults confirm that the FSDT should be preferred in analyzing the conical shells where theCLST is not able to predict the accurate results.

Fig. 4 Natural frequencies versus parameter n for isotropic shells with CLST and FSDT (L=5m,a=0.5m, h=10 mm, Ω=0 r·s−1, Ss-Sl)

The variation of the ratio fCLST/fFSDT is plotted with the ratio h/b in Fig. 5 for the conicalshell with different cone angles in circumferential wave number, where fCLST and fFSDT arequoted as frequencies of conical shell calculated by means of CLST and FSDT, respectively. Itis expected that as the thickness of cylindrical shell increases, the influence of transverse shearforce becomes more significant on the frequency. Therefore, as seen in Fig. 5, it is detected thatthe response to increase thickness is mostly expected for fCLST/fFSDT with an increase for allthe curves. Moreover, an interesting result should be paid attention is that a sudden changeis made on the curves where the cylinder is compared with the cones, and also this increasingmanner is continued as the cone angles goes up.

Fig. 5 Frequency ratio versus thickness for isotropic conical shell at various cone angles in mode (1,10) (L sin (α/b)=0.5, Ω=0 r·s−1, Ss-Sl)

The influence of stacking sequence on the natural frequency of conical shell is displayedin Fig. 6. Five configurations with different stacking sequences of plies are considered. Themaximum natural frequencies at low values of the parameter n belong to configuration [45◦/−45◦/45◦]. Though, two other configurations, i.e., [90◦/45◦/90◦] and [90◦/0◦/90◦] provide thehigher natural frequencies in a broad band of high parameter n. It is due to the fact that

Page 12: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

448 K. DANESHJOU, M. TALEBITOOTI, and R. TALEBITOOTI

they contain more 90◦ layers, and make the structure stiffer in circumference at high parametern wherever the length of the wave descends. Meanwhile, the natural frequencies of two otherconfigurations [0◦/90◦/0◦] and [0◦/45◦/0◦] are less than other ones in high circumferential modes.

Fig. 6 Natural frequencies of rotating conical shell versus parameter n at various stacking sequencelayers (L=2m, a=0.5m, h=3mm, α=15◦, Ω=10 r·s−1, Ss-Sl)

Figure 7 presents the influence of parameter α on the variation of the frequency versusparameter n. In this figure, it can be observed that, at small parameter n, frequencies descendmonotonically, while at great n, frequencies of the shells generally ascend, and the rising rate ofgradient comes to be great with the reduction in the parameter α. It is because of an increasein the bending stiffness of the conical shell as a consequence of ascending the curvature of theshell. Meanwhile, the equivalent mass of the shell will be decreased. Therefore, as depicted inthis figure, the fundamental frequency (f) of the shell will be enhanced and the correspondingcircumferential wave number will be decreased.

Fig. 7 Natural frequencies of cross-ply conical shell versus parameter n at various cone angles (L=2m,a=0.5m, h=3mm, Ω=0 r·s−1, [0◦/90◦/0◦], Ss-Sl)

Figures 8(a)–8(b) illustrate the influences of variety of BCs on the frequency of the backwardtraveling wave at both different length shells. As depicted, it decreases rapidly at first, and thenraised monotonically by ascending the parameter n. The shell with Cs-Cl has the maximumfrequencies, followed by the Ss-Cl, Cs-Sl, and S-S shells. This behavior was simply expectedbefore, as Cs-Cl is a fully limited boundary. At lower parameter n, moderately great discrep-ancies between the frequencies of these boundary conditions are observed, indicating that theeffect of BC is noteworthy. At higher parameter n, both boundary conditions Ss-Cl, Cs-Cl, andthe both ones Cs-Sl and Ss-Sl come together as a consequence of shortening the wavelengths.It has a duty to be finally mentioned that the differences in the frequencies between four BCsdiminish with ascending the shell length.

Page 13: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

Free vibration and critical speed of moderately thick rotating laminated composite conical shell 449

Fig. 8 Natural frequencies of rotating cross-ply conical shell versus parameter n for different bound-ary conditions (a=0.5m, h=3mm, m=1, Ω=20 r·s−1, α=15◦, [0◦/90◦/0◦])

The phenomenon of the critical speed of the shell is illustrated in Fig. 9, in which the for-ward travelling wave crosses longitudinal coordinate. In this crossover, an unsteady occurrencepossibly appears as this wave turns into stationary in connection with the travelling coordinate,and thus will be prepared to shift to the backward mode. At this moment in time, any unbal-anced mass can be synchronized with the whirl and amplitude of the displacement resonate[26].For various cone angles α at two cases of Ss-Sl and Cs-Cl, the variations of frequency versusangular velocity are revealed in Figs. 9(a)–9(b). In particular, it can be comprehended that thisvariation behaves non-linearly in the conical shell in mode (1, 1), contrary to the cylindricalone which behaves linearly. Also, with increasing the parameter α, the non-linearity in thefrequencies of the forward wave increases. With comparing Fig. 9(a) and Fig. 9(b), it can be de-termined that the effect of BC on the critical speed of cylindrical shell is increasingly noticeablethan that of conical one.

Fig. 9 Natural frequencies for cross-ply conical shell with rotating speed (L=5m, a=0.5m, m=1,n=1, h=3 mm, [0◦/90◦/0◦])

In Figs. 10(a)–10(d), the effect of the constant axial load on the frequencies and the criticalspeeds of conical shell for different cone angles α is revealed. The parameter Ncr is the criticalglobal buckling of the conical shell in this figure and the axial compressive and tensile loadis represented by negative and positive sign, respectively. It should be mentioned here thatthe compressive axial load should be a proper fraction of the static global instability load. Toachieve the static critical global buckling load, the global buckling differential equations canbe obtained by neglecting the terms involving Ω and ω in Eq. (39). Four configurations of the

Page 14: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

450 K. DANESHJOU, M. TALEBITOOTI, and R. TALEBITOOTI

conical shell, α=0◦, 15◦, 30◦, and 45◦, are used to investigate the sensitivity of the critical speedto the axial load. From Figs. 10(a)–10(d), it is observed that the critical speed phenomena inconical shell exhibits in mode n = 1 as the cylindrical one. Furthermore, it is detected that forconical case at various cone angles, the response to compressive and tensile loads is generally likethe cylindrical one with a downward and a upward shifts for all rotating speeds, respectively. Inaddition, for all configurations, it is noteworthy that, the level of the sensitivity of critical speeddecreases by raising the tensile axial load, whereas by rising the compressive load, the level ofsensitivity of critical speed increases. As illustrated in this figure, the increasing percentagerate for the whole configurations is almost the same. It is an interesting result for designersas lets them to guess the critical speed of an axially loaded conical shell with estimating theone for a unloaded conical shell. Figure 11 represents the same results for the conical shell withdifferent geometry specifications. It confirms that the effects of the axial load on critical speedare predictable and definite, independent of geometry specifications.

Fig. 10 Influence of axial load on critical speed for cross-ply conical shell (L=6m, a=1m, h=2mm,m=1, n=1, Ss-Sl)

4 Conclusions

The vibration analysis and the critical speed of the rotating laminated conical shells areinvestigated by using GDQM. The FSDT is employed to consider the effects of transverse sheardeformation. The effects of the centrifugal force, the Coriolis acceleration, and the preliminaryhoop stress caused by angular velocity are considered. Results are derived to investigate theinfluences of the thickness of shell, boundary conditions, stacking sequence, cone angle, andangular velocity. Particularly, the main conclusions are followed as below:

Page 15: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

Free vibration and critical speed of moderately thick rotating laminated composite conical shell 451

Fig. 11 Influence of axial load on critical speed for cross-ply conical shell (L=5m, a=0.5 m, h=3mm,m=1, n=1, Ss-Sl, [0◦/90◦/0◦])

(i) The accuracy of GDQM is strongly trustworthy, and its application is both uncomplicatedand effective. In addition, the performance is efficient to consider the vibration of conical shells.

(ii) The CLST in comparison with FSDT leads into significant errors in the vibration ofconical shell even for those of thin-walled shell.

(iii) Increasing the parameter α seems to impose reductions in the fundamental frequencyof the cone whereas the corresponding parameter n increases.

(iv) The frequency of the travelling wave at first is quickly decreased. Then, it ascendsmonotonically by raising the parameter n in different boundary conditions.

(v) Contrary to cylindrical shell, the variation of the frequency with angular velocity behavesnon-linearly in conical shell in mode (1, 1). Moreover, by increasing the cone angle, the non-linearity in the frequency of the forward wave increases.

(vi) The effect of BC on the critical speed of conical shell increases by decreasing the pa-rameter α.

(vii) For all configurations of conical shells, by raising the tensile axial load, the level of thesensitivity of critical speed decreases, whereas with an increase in compressive load, it increases.

(viii) The effects of the axial load on critical speed are predictable and definite, independentof geometry specifications.

References

[1] Leissa, A. W. Vibration of Shells, NASA, Washington, D. C., SP-288 (1973)

[2] Sivadas, K. R. Vibration analysis of prestressed rotating thick circular conical shell. J. SoundVibr., 148, 477–491 (1995)

Page 16: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

452 K. DANESHJOU, M. TALEBITOOTI, and R. TALEBITOOTI

[3] Hua, L., Lam, K. Y., and Ng, T. Y. Rotating Shell Dynamics, Elsevier, London (2005)

[4] Lam, K. Y. and Loy, C. T. Influence of boundary conditions for a thin laminated rotating cylin-drical shell. Compos. Struct., 41, 215–228 (1998)

[5] Hua, L. Influence of boundary conditions on the free vibrations of rotating truncated circularmulti-layered conical shells. Composites: Part B, 31, 265–275 (2000)

[6] Lam, K. Y. and Loy, C. T. Analysis of rotating laminated cylindrical shells by different thin shelltheories. J. Sound Vibr., 186, 23–25 (1995)

[7] Hua, L. Frequency analysis of rotating truncated circular orthotropic conical shells with differentboundary conditions. Composites Science and Technology, 60, 2945–2955 (2000)

[8] Chen, Y., Zhao, H. B., and Shea, Z. P. Vibrations of high speed rotating shells with calculationsfor cylindrical shells. J. Sound Vibr., 160, 137–160 (1993)

[9] Lam, K. Y. and Hua, L. On free vibration of a rotating truncated circular orthotropic conicalshell. Composites: Part B, 30, 135–144 (1999)

[10] Lim, C. W. and Liew, K. M. Vibratory behavior of shallow conical shells by a global Ritz formu-lation. Eng. Struct., 17(1), 63–70 (1995)

[11] Qatu, M. S. Vibration of Laminated Shells and Plates, Elsevier, The Netherlands (2004)

[12] Wu, T. Y., Wang, Y. Y., and Liu, G. R. A generalized differential quadrature rule for bendinganalyses of cylindrical barrel shells. Comput. Meth. Appl. Mech. Eng., 192, 1629–1647 (2003)

[13] Shu, C. Differential Quadrature and Its Application in Engineering, Springer, Berlin (2000)

[14] Wang, Y., Liu, R., and Wang, X. On free vibration analysis of nonlinear piezoelectric circularshallow spherical shells by the differential quadrature element method. J. Sound Vibr., 245, 179–185 (2001)

[15] Liew, K. M., Ng, T. Y., and Zhang, J. Z. Differential quadrature-layerwise modeling technique forthree dimensional analysis of cross-ply laminated plates of various edge supports. Comput. Meth.Appl. Mech. Eng., 191, 3811–3832 (2002)

[16] Karami, G. and Malekzadeh, P. A new differential quadrature methodology for beam analysisand the associated differential quadrature element method. Comput. Meth. Appl. Mech. Eng.,191, 3509–3526 (2002)

[17] Ng, T. Y., Hua, L., and Lam, K. Y. Generalized differential quadrature for free vibration ofrotating composite laminated conical shell with various boundary conditions. Int. J. Mech. Sci.,45, 567–587 (2003)

[18] Huang, Y. Q. and Li, Q. S. Bending and buckling analysis of antisymmetric laminates usingthe moving least square differential quadrature method. Comput. Meth. Appl. Mech. Eng., 193,3471–3492 (2004)

[19] Wang, X. and Wang, Y. Free vibration analyses of thin sector plates by the new version ofdifferential quadrature method. Comput. Meth. Appl. Mech. Eng., 193, 3957–3971 (2004)

[20] Wang, X. Nonlinear stability analysis of thin doubly curved orthotropic shallow shells by thedifferential quadrature method. Comput. Meth. Appl. Mech. Eng., 196, 2242–2251 (2007)

[21] Haftchenari, H., Darvizeh, M., Darvizeh, A., Ansari, R., and Sharama, C. B. Dynamic analysisof composite cylindrical shells using differential quadrature method (DQM). Compos. Struct., 78,292–298 (2007)

[22] Tornabene, F. Free vibration analysis of functionally graded conical, cylindrical shell and annularplate structures with a four-parameter power-law distribution. Comput. Meth. Appl. Mech. Eng.,198, 2911–2935 (2009)

[23] Nedelcu, M. GBT formulation to analyze the buckling behaviour of isotropic conical shells. Thin-Walled Struct., 49, 812–818 (2011)

[24] Irie, T., Yamada, G., and Tanaka, K. Natural frequencies of truncated conical shells. J. SoundVibr., 92, 447–453 (1984)

[25] Lam, K. Y. and Hua, L. Influence of boundary conditions on the frequency characteristic of arotating truncated circular conical shell. J. Sound Vibr., 223, 171–195 (1999)

[26] Ghayour, M., Rad, S. Z., Talebitooti, R., and Talebitooti, M. Dynamic analysis and critical speedof pressurized rotating composite laminated conical shells using generalized differential quadraturemethod. Journal of Mechanics, 26(1), 61–70 (2010)

Page 17: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

Free vibration and critical speed of moderately thick rotating laminated composite conical shell 453

Appendix A

M11 = M22 = M33 =ρπ

2

Z h2

− h2

Z L

0

(a + x sin α + z cos α)dxdz, (A1)

M14 = M41 = M25 = M52 =ρπ

2

Z h2

− h2

Z L

0

(a + x sin α + z cos α)zdxdz, (A2)

M44 = M55 =ρπ

2

Z h2

− h2

Z L

0

(a + x sin α + z cos α) z2 dxdz, (A3)

(M12 = M13 = M15 = M21 = M23 = M24 = M31 = M32 = 0,

M34 = M35 = M42 = M43 = M45 = M51 = M53 = M54 = 0,(A4)

G12 = G21 = −2ρπΩsin α

Z h2

− h2

Z L

0

(a + x sin α + z cos α)dxdz, (A5)

G15 = G51 = −2ρπΩsin α

Z h2

− h2

Z L

0

(a + x sin α + z cos α)zdxdz, (A6)

G23 = G32 = −2ρπΩcos α

Z h2

− h2

Z L

0

(a + x sin α + z cos α) dxdz, (A7)

G24 = G42 = −2ρπΩsin α

Z h2

− h2

Z L

0

(a + x sin α + z cos α) zdxdz, (A8)

G35 = G53 = −2ρπΩcos α

Z h2

− h2

Z L

0

(a + x sin α + z cos α) zdxdz, (A9)

G45 = G54 = −2ρπΩsin α

Z h2

− h2

Z L

0

(a + x sin α + z cos α) z2 dxdz, (A10)

(G11 = G13 = G14 = G22 = G25 = G31 = G33 = 0,

G34 = G41 = G43 = G44 = G52 = G55 = 0,(A11)

K11 = π

Z h2

− h2

Z L

0

“−Q11

∂2

∂x2+

Q22 sin2 α

r2+

Q66n2

r2+ 2

Q16n

r

∂x

”rdxdz, (A12)

K12 = π

Z h2

− h2

Z L

0

“Q22 + Q66

r2n sin α − Q66 + Q12

rn

∂x− Q16

∂2

∂x2+

Q16 + Q26

rsin α

∂x

+Q26

r2(n2 − sin2 α)

”rdxdz + ρΩ2πn sin2 α

Z h2

− h2

Z L

0

(a + x sin α + z cos α)dxdz, (A13)

K13 = π

Z h2

− h2

Z L

0

“Q22 sin α cos α

r2− Q12 cos α

r

∂x+

Q26n cos α

r2

”rdxdz, (A14)

K14 = π

Z h2

− h2

Z L

0

“−Q11

∂2

∂x2+

Q22 sin2 α

r2+

Q66n2

r2+ 2

Q16n

r

∂x

”zrdxdz, (A15)

Page 18: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

454 K. DANESHJOU, M. TALEBITOOTI, and R. TALEBITOOTI

K15 = π

Z h2

− h2

Z L

0

“Q22 + Q66

r2n sin α − Q66 + Q12

rn

∂x− Q16

∂2

∂x2+

Q16 + Q26

rsin α

∂x

+Q26

r2(n2 − sin2 α)

”zrdxdz + ρΩ2πn sin α

Z h2

− h2

Z L

0

(a + x sin α + z cos α)zdxdz, (A16)

K21 = π

Z h2

− h2

Z L

0

“Q22 + Q66

r2n sin α +

Q66 + Q12

rn

∂x− Q16

∂2

∂x2− Q16 + Q26

rsin α

∂x

+Q26

r2(n2 − sin2 α)

”rdxdz + ρΩ2πn sin2 α

Z h2

− h2

Z L

0

(a + x sin α + z cos α)dxdz, (A17)

K22 = π

Z h2

− h2

Z L

0

“Q22n2 + Q66 sin2 α + Q55 cos2 α

r2− Q66

∂2

∂x2− 2

Q26n

r

∂x

”rdxdz

+ ρΩ2π(1 + n2)

Z h2

− h2

Z L

0

(a + x sin α + z cos α)dxdz, (A18)

K23 = π

Z h2

− h2

Z L

0

“ (Q22 + Q55) cos α n

r2− (Q26 + Q45) cos α

r

∂x− Q26 cos α sin α

r2

”rdxdz

+ ρΩ2πn cos α

Z h2

− h2

Z L

0

(a + x sin α + z cos α)dxdz, (A19)

K24 = π

Z h2

− h2

Z L

0

“Q22 + Q66

r2n sin α +

Q66 + Q12

rn − Q16

∂2

∂x2−Q16+Q26

rsin α

∂x+

Q26

r2(n2−sin2 α)

− Q45 cos α

zr

”zrdxdz + ρΩ2πn sin α

Z h2

− h2

Z L

0

(a + x sin α + z cos α)zdxdz, (A20)

K25 = π

Z h2

− h2

Z L

0

“Q22n2 + Q66 sin2 α + Q55 cos2 α

r2− Q55 cos α

zr− Q66

∂2

∂x2− 2

Q26n

r

∂x

”zrdxdz

+ ρΩ2π(1 + n2)

Z h2

− h2

Z L

0

(a + x sin α + z cos α)zdxdz, (A21)

K31 = π

Z h2

− h2

Z L

0

“Q22 sin α cos α

r2+

Q12 cos α

r

∂x− Q26n cos α

r2

”rdxdz, (A22)

K32 = π

Z h2

− h2

Z L

0

“ (Q22 + Q55) cos α n

r2+

(Q26 + Q45) cos α

r

∂x− Q26 cos α sin α

r2

”rdxdz

+ ρΩ2πn cos α

Z h2

− h2

Z L

0

(a + x sin α + z cos α)dxdz, (A23)

K33 = π

Z h2

− h2

Z L

0

“Q22 cos2 α + Q55n2

r2− Q44

∂2

∂x2+

Q45

rn

∂x

”rdxdz, (A24)

Page 19: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

Free vibration and critical speed of moderately thick rotating laminated composite conical shell 455

K34 = π

Z h2

− h2

Z L

0

“Q22 sin α cos α

r2+

“Q12 cos α

r− Q44

z

” ∂

∂x+

“Q45n

zr− Q26n cos α

r2

””zrdxdz, (A25)

K35 = π

Z h2

− h2

Z L

0

““Q22 + Q55

r2cos α n − Q55n

zr

”+

“Q26 + Q45

rcos α − Q45

z

” ∂

∂x

− Q26 cos α sin α

r2

”zrdxdz + ρΩ2πn cos α

Z h2

− h2

Z L

0

(a + x sin α + z cos α)zdxdz, (A26)

K41 = π

Z h2

− h2

Z L

0

“− Q11

∂2

∂x2+

Q22 sin2 α

r2+

Q66n2

r2+ 2

Q16n

r

∂x

”zrdxdz, (A27)

K42 = π

Z h2

− h2

Z L

0

“Q22 + Q66

r2n sin α − Q66 + Q12

rn

∂x− Q16

∂2

∂x2+

Q16 + Q26

rsin α

∂x

+Q26

r2(n2−sin2 α)−Q45 cos α

zr

”zrdxdz+ρΩ2πn sin α

Z h2

−h2

Z L

0

(a+x sin α+z cos α)zdxdz, (A28)

K43 = π

Z h2

− h2

Z L

0

“Q22 sin α cos α

r2−

“Q12 cos α

r− Q44

z

” ∂

∂x−

“Q45n

zr− Q26n cos α

r2

””zrdxdz, (A29)

K44 = π

Z h2

− h2

Z L

0

“− Q11

∂2

∂x2+ 2

Q16n

r

∂x+

“Q22 sin2 α

r2+

Q66n2

r2+

Q44

z2

””z2rdxdz, (A30)

K45 = π

Z h2

− h2

Z L

0

“Q22+Q66

r2n sin α−Q66+Q12

rn

∂x−Q16

∂2

∂x2+

Q16+Q26

rsin α

∂x+

Q26

r2(n2−sin2 α)

−“Q45 cos α

zr− Q45

z2

””z2rdxdz + ρΩ2πn sin α

Z h2

− h2

Z L

0

(a + x sin α + z cos α)z2 dxdz, (A31)

K51 = π

Z h2

− h2

Z L

0

“Q22 + Q66

r2n sin α +

Q66 + Q12

rn

∂x− Q16

∂2

∂x2− Q16 + Q26

rsin α

∂x

+Q26

r2(n2 − sin2 α)

”zrdxdz + ρΩ2πn sin α

Z h2

− h2

Z L

0

(a + x sin α + z cos α)zdxdz, (A32)

K52 = π

Z h2

− h2

Z L

0

“Q22n2 + Q66 sin2 α + Q55 cos2 α

r2− Q55 cos α

zr− Q66

∂2

∂x2− 2

Q26n

r

∂x

”zrdxdz

+ ρΩ2π(1 + n2)

Z h2

− h2

Z L

0

(a + x sin α + z cos α)z2dxdz, (A33)

K53 = π

Z h2

− h2

Z L

0

““Q22+Q55

r2cos α n−Q55n

zr

”−

“Q26+Q45

rcos α−Q45

z

” ∂

∂x−Q26 cos α sin α

r2

”zrdxdz

+ ρΩ2πn cos α

Z h2

− h2

Z L

0

(a + x sin α + z cos α)zdxdz, (A34)

Page 20: Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

456 K. DANESHJOU, M. TALEBITOOTI, and R. TALEBITOOTI

K54 = π

Z h2

− h2

Z L

0

“Q22+Q66

r2n sin α+

Q66+Q12

rn

∂x−Q16

∂2

∂x2−Q16+Q26

rsin α

∂x+

Q26

r2(n2− sin2 α)

−“Q45 cos α

zr− Q45

z2

””z2rdxdz + ρΩ2πn sin α

Z h2

− h2

Z L

0

(a + x sin α + z cos α)z2 dxdz, (A35)

K55 =π

Z h2

− h2

Z L

0

““Q22n2+Q66 sin2 α+Q55 cos2 α

r2−2

Q55 cos α

zr+

Q55

z2

”−Q66

∂2

∂x2−2

Q26n

r

∂x

”z2rdxdz

+ ρΩ2π(1 + n2)

Z h2

− h2

Z L

0

(a + x sin α + z cos α)z2 dxdz. (A36)