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American Institute of Aeronautics and Astronautics

1

Vibration Characteristics of Axially Loaded Rotating Functionally Graded Cylindrical Shell

Rohit Saha1 Banaras Hindu University, Varanasi, Uttar Pradesh, 221005, India

and

Pabitra R Maiti2 Banaras Hindu University, Varanasi, Uttar Pradesh, 221005, India

This paper presents the free vibration analysis of axially loaded rotating functionally graded (FG) cylindrical shells made up of stainless steel and nickel. The properties are graded in the thickness direction according to a volume fraction power-law distribution. Taking account, the effects of centrifugal and Coriolis forces as well as the initial hoop tension, the equations of motion for the rotating FG shell are formulated using Love’s first approximation theory. The frequency characteristics for FG cylinders are investigated using closed form solutions for simply supported cylindrical shell. The effects of length to radius ratio, thickness to radius ratio, material gradient index, and axial load are investigated on the natural frequencies of rotating cylinder.

Nomenclature N = Material gradient index L = Cylinder length R = Cylinder radius Ω = External rotational velocity ρ = Density of cylinder Ncr = Critical buckling load Na = Applied axial load

I. Introduction unctionally gradient materials (FGMs) are obtained by combining two or more materials. Most of the functionally gradient materials are employed in high-temperature environments and many of the constituent

materials may possess temperature-dependent properties. FGMs are also considered as a potential structural material for the future high-speed spacecrafts. The rotating cylindrical shell has a wide range of engineering application such as gas turbines, locomotives engines, electric motors and rotor systems. This paper is thus, based on investigation of vibration characteristics of rotating FG shells , as to the best of author’s knowledge, there has been no work done on rotating FG shell.

II. Functionally Graded Materials Functionally graded materials (FGMs) are obtained by combining two or more materials. Most of the

FGMs are employed in high temperature environments and many of the constituent materials may possess temperature dependent properties P can be expressed as a function of temperature 6 as

1 2 3

0 1 1 2 31P P P T PT PT PT

(1)

1 Senior Undergraduate, Department of Civil Engineering, Institute of Technology, email: [email protected]. 2 Lecturer, Department of Civil Engineering, Institute of Technology, email: [email protected].

F

49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference <br> 16t7 - 10 April 2008, Schaumburg, IL

AIAA 2008-1876

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


2

where P0, P-1, P1, P2 and P3 are the coefficients of temperature T(K) expressed in Kelvin and are unique to the constituents materials.

The material properties P of FGMs are a functional of material properties and volume fractions of the constituent materials and are expressed as

1

k

j fjj

P PV

(2)

where Pj and Vfj are respectively, the material property and volume fraction of the constituent material j. The volume fractions of all constituents materials should add up to one, i.e.

1

1k

fij

V

(3)

For a cylindrical shell with a uniform thickness h and a reference surface at its middle surface, the volume fraction can be written as

/ 2 N

f

z hV

h

(4)

where N is the material gradient index , 0 N . For a functionally gradient material with two constituent materials, the Young’s modulus E, poisson’s ratio and the mass density can be expressed as

1 2 2

/ 2 Nz hE E E E

h

(5)

1 2 2

/ 2 Nz h

h

(6)

1 2 2

/ 2 Nz h

h

(7)

III. Formulations Consider a cylindrical shell with radius R, length L and thickness h, (fig.1). The deformations defined with reference to a coordinate to a coordinate system (x,θ,z) taken at the middle surface are u, v, and w in the x, θ and z directions respectively rotating along its axis with angular velocity of .

For a thin cylindrical shell, plane stress condition is assumed and the constituent relation is given by

Q e (8)

where , given by T

x x is the stress vector

in x, θ and xθ directions respectively and e is the strain vector

given by, T

x xe e e e

. The reduced stiffness matrix is defined as

Figure 2. Geometry of thin rotating cylinder

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

4

z/h

E1

E2

n=100n=10

n=5

n=2

n=0.2

n=0.1

n=0.5

n=1

Youn

g's

Mod

ulus

Figure 1. Variation of Young’s Modulus with depth


3

11 12

12 22

66

0

0

0 0

Q Q

Q Q Q

Q

(9)

For isotropic materials the reduced stiffness Qij(i,j=1,2, and 6) are defined as

11 22 21

EQ Q

, 12 21

EQ

,

66 2 1

EQ

(10)

From Love’s shell theory 7, the components in the strain vector {e} are defined as

1 1xe e zk , 2 2e e zk , 2xe z (11)

where e1, e2 and are the reference surface strains, and k1, k2 and are the surface curvatures. These surface strains and curvatures are defined as

1 2

1 1u v v ue e w

x R x R

(12)

2 2 2

1 2 2 2 2

1 1w w v w vk k

x R R x x

(13)

For a thin cylindrical shell, the force and moment resultants are defined as

/ 2

/ 2

h

x x x x

h

N N N dz

(14)

/ 2

/ 2

h

x x x x

h

M M M zdz

(15)

The constitutive equations is given by

[ ]{ }FN S (16)

where TF

x x x xN N N N M M M and 1 2 1 2 2T e e k k and [S] is defined as

11 12 11 12

12 22 12 22

66 66

11 12 11 12

12 22 12 22

66 66

0 0

0 0

0 0 0 0[ ]

0 0

0 0

0 0 0 0

A A B B

A A B B

A BS

B B D D

B B D D

B D

(17)

where Aij, Bij and Dij (i,j=1,2 and 6 ) are the extensional, coupling and bending stiffness defined as

/ 2

2

/ 2

(1, , )h

ij ij ij ij

h

A B D Q z z dz

(18)

The equations of motions of rotating cylinder 2can be written as 2 2

2 2 2

1 1 1x xt

N N u w uN

x R R R x t

(19)


4

2 22

2 2

1 1 12x x

t

N N M M N u v wv

R x R x R R x t t

(20)

2 2 2 2 2 22

02 2 2 2 2 2 2

2 12x x

t

M M M N N w v w w vN w

x R x R R R x t t

(21)

where, and t are the density and density per unit length, respectively. 2 2

tN R is defined as the initial hoop tension due to centrifugal force and No is the applied axial load.

Substituting equation 17 in equ19-21 then the resulting equation can be written as in matrix form

11 12 13

21 22 23

31 32 33

0

0

0

L L L u

L L L v

L L L w

(22)

Where Lij (i,j=1,2,3) are the differential operator with respect to x and θ. In the parametric studies, only the simply supported cylindrical shell is considered. The simply supported boundary conditions at the two ends of the shell, x=0 and L, are

0x xv w N M (23) For SS boundary conditions, the solution which satisfies the boundary conditions can be expressed in closed form as

cos cosm x

u U n tL

sin sinm x

v V n tL

(24)

sin cosm x

w W n tL

Substituting eqn (24) into Eqn (22), the resulting equations can be written in matrix form as

11 12 13

21 22 23

31 32 33

0

0

0

C C C U

C C C V

C C C W

(25)

To solve for the eigen values ω, the non-trivial solutions condition is imposed by setting the determinant of the characteristics matrix in eqn(25) to zero; the eigen values ω can be obtained by using the Newton Raphson procedure.

IV. Results and Discussions In this paper, studies are presented on the vibration of the simply supported rotating functionally graded (FG) cylindrical shells. The FG shell is composed of stainless steel on its outer side and nickel on its inner surface. The material properties are graded in the thickness direction according to the power law distribution. Material properties of the FG shell taken from Loy et al.1. The program developed for numerical computations was validated by comparing the results with those of Loy et al.1 for non rotating FG cylinder listed in Table. 1. To further verify the present analysis, our results are compared to those presented by Lam and Loy2 and Liew et al.3 for the cross-ply laminated cylindrical shells of lamination scheme [0°/90°/0°]. The comparisons for the rotating shell with L/R= 1are presented in Table 2 in which *

f and *b are the forward-wave and backward-wave non-dimensional frequency

parameters, respectively. Again it is evident that very good agreement is achieved, thus further verifying the validity and accuracy of the present formulation as well as the program. In succeeding figures, Y and Ω denote the normalized natural frequency and the rotational speed, respectively, where Y= ωmn/ωmno , Ω is the rotating speed in rad/s, ωmn and ωmno respectively correspond to the rotational and non-rotational natural frequency of each respective mode. B and F are backward-wave and forwardwave non-


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dimensional frequency parameters, respectively. The effects of constant axial load on the shells with different length ratios, thickness ratio and material gradient indices are examined. The axial loads considered are only a fraction of buckling load. For isotropic shell of intermediate length the buckling load is given by Timoshenko and Gere4 as

2

23(1 )cr

EhNR

(26)

Table 1. Comparison of frequency for simply supported (SS) FG shell (m=1, R=1,L/R=20, h/R=0.002) NSS (Hz) N=0.5(Hz) N=1(Hz) N=5(Hz) n

Present Loy et al (1999)




1 13.5477 13.548 13.265 13.31 13.208 13.211 12.954 12.998 2 4.5919 4.592 4.4681 4.5168 4.431 4.48 4.4053 4.4068 3 4.2632 4.2633 4.0918 4.1911 4.1389 4.1569 4.0654 4.0891 4 7.2249 7.2250 7.0612 7.0972 7.0284 7.0384 6.8942 6.9251 5 11.5419 11.542 11.216 11.336 11.241 11.241 11.053 11.061 6 16.897 16.897 16.234 16.594 16.455 16.455 16.065 16.192 10 48.1676 48.168 47.181 47.301 46.605 46.905 46.096 46.155

Table.2 Non-dimensional frequency parameter 222* R E for [0/90/0] SS rotating laminated

cylindrical shell (h/R=0.002, L/R=1, E11/E22=40, G12/E22=2.5 , ν=0.25) Ω Lam and Loy2 Liew et al .3 Present (rev/s) n *

b *f *

b *f *

b *f

0.1 1 1.061429 1.06114 1.061428 1.061139 1.061425 1.061137 2 0.804214 0.803894 0.804212 0.803892 0.804208 0.803886 3 0.598476 0.598157 0.598472 0.598183 0.59847 0.598179 4 0.45027 0.450021 0.450263 0.450015 0.450266 0.450011 5 0.345363 0.345149 0.345355 0.34514 0.345352 0.34511 6 0.270852 0.270667 0.27084 0.270654 0.27082 0.270657 0.4 1 1.061862 1.060706 1.061862 1.060705 1.061858 1.060702 2 0.804696 0.803415 0.804694 0.803413 0.804692 0.803411 3 0.598915 0.597762 0.598911 0.597758 0.598908 0.597754 4 0.450662 0.449667 0.450654 0.44966 0.450653 0.44963 5 0.345724 0.34487 0.345714 0.34486 0.345712 0.34483 6 0.271207 0.270468 0.271193 0.270454 0.271186 0.270457 1.0 1 1.062728 1.059836 1.062728 1.059837 1.062723 1.059832 2 0.805667 0.802464 0.805664 0.802461 0.805661 0.802458 3 0.59982 0.596937 0.599813 0.59693 0.599806 0.59694 4 0.451513 0.449027 0.451502 0.449015 0.451501 0.449011 5 0.346593 0.344459 0.346577 0.344442 0.346575 0.344446 6 0.272197 0.270349 0.272174 0.270326 0.272171 0.270322

Fig. 3 and 4 shows the effects of variations of the material gradient index on the forward and backward-traveling waves for the rotating cylindrical shells compressive load and tensile loads. The normalizing frequency ωmno , is for mode (m,n)=(1,4) of a shell of length ratio L/R=5.0, thickness ratio h/R=0.005, and under compressive and tensile loading. It can be inferred that the shell with intermediate length becomes stiffer with increase in material gradient


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index. However, some variation in frequencies only takes place in range of N between 0.1 and 10. Beyond this range the variations are negligible for engineering considerations. Fig. 5 and 6 illustrate the effects of variations of the length ratio on the forward and backward-traveling waves for the rotating cylindrical shells compressive load and tensile load respectively. The value of the circumferential wave number n at which the minimum frequency occurs decreases with the L/R ratio. From the present results, it is observed that the frequencies of the shells with larger length ratio are lower than that of the shells with smaller length ratio. This is expected, as an increase in the length of the shell will cause the shell to become less stiff.

0.0 0.5 1.0 1.5 2.0

1.06

1.08

1.10

1.12

1.14

1.16

Na= -0.1N

cr (1,4)

L/R=5, h/R=1/200

Backward waveForward wave

(rev/s)

Y

N=0.1 N=1 N=10

Figure 3. Variation of bifurcation of natural frequencies with external rotational velocity for mode (m,n)=(1,4) of rotating FG cylindrical shell of h/R=0.005,L/R=5 and under compressive loading of N0= -0.1Ncr for different material gradient index

0.0 0.5 1.0 1.5 2.00.92

0.94

0.96

0.98

1.00

1.02

Na= 0.1Ncr (1,4)

L/R=5, h/R=1/200


(rev/s)

Y

N=0.1 N=1 N=10

Figure 4. Variation of bifurcation of natural frequencies with external rotational velocity for mode (m,n)=(1,4) of rotating FG cylindrical shell of h/R=0.005,L/R=5 and under tensile loading of N0= 0.1Ncr for different material gradient index

0.0 0.5 1.0 1.5 2.01.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

1.18

1.20

1.22

1.24

1.26 (1,3)

(1,6)

Na= -0.1Ncr

(1,4)

N=1, h/R=1/200


(rev/s)

Y

L/R=2.5 L/R=5 L/R=10

Figure 5. Variation of bifurcation of natural frequencies with external rotational velocity of rotating FG cylindrical shell of h/R=0.005, N=1 and under compressive loading of N0= -0.1Ncr for different L/R ratio

0.0 0.5 1.0 1.5 2.00.90

0.92

0.94

0.96

0.98

1.00

1.02

1.04

1.06

1.08

1.10

1.12

(1,3)

(1,3)

(1,6)

(1,6)(1,4)

Na= 0.1N

cr

(1,4)

N=1, h/R=1/200


(rev/s)

Y

L/R=2.5 L/R=5 L/R=10

Figure 6. Variation of bifurcation of natural frequencies with external rotational velocity of rotating FG cylindrical shell of h/R=0.005, N=1 and under tensile loading of N0=0.1Ncr for different L/R ratio


7

Fig. 7 and 8 show the effect of variation of the thickness ratio on the forward and backward traveling wave for the rotating FG cylindrical shell under compressive and tensile axial load. It is observed that the frequencies of the shells with higher-thickness ratio are lower than that of the shells with smaller-thickness ratio. This is expected, as a decrease in the thickness of the shell will cause it to become less stiff. Similar characteristics are observed for the shells under compressive axial loading with those described in Fig. 5 & 6. Fig. 9 and 10 depict the effects of the magnitude of the axial compressive and tensile loads on the forward- and backward-traveling waves for the rotating cylindrical shell. The normalizing frequency mno ω , is given for mode (m,n)=(1,4) of a shell of length to radius ratio L/R=5.0, thickness ratio h/R=0.005 . Results are presented for the mode (m,n)=(1,4). It is observed that the frequencies of the shell will become lower when the magnitude of the compressive axial loading increases and conversely the magnitude of tensile load increases results in reduction in natural frequencies. This is reasonable as an increase in the magnitude of the compressive axial loading causes a enhancement in the stiffness whereas increase in magnitude of tensile load causes reduction in the stiffness.

0.0 0.5 1.0 1.5 2.01.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

1.18

1.20

1.22

1.24

1.26 (1,3)

(1,6)

Na= -0.1N

cr

(1,4)

N=1, L/R=5


(rev/s)

Y

h/R=1/100 h/R=1/200 h/R=1/500

Figure 7. Variation of bifurcation of natural frequencies with external rotational velocity of rotating FG cylindrical shell of L/R=5, N=1 and under compressive loading of N0=-0.1Ncr for different h/R ratio

0.0 0.5 1.0 1.5 2.00.92

0.94

0.96

0.98

1.00

1.02

1.04

1.06

1.08

1.10

1.12 (1,3)

(1,5)

Na= 0.1Ncr

(1,4)

N=1, L/R=5


(rev/s)

Y

h/R=1/100 h/R=1/200 h/R=1/500

Figure 8. Variation of bifurcation of natural frequencies with external rotational velocity of rotating FG cylindrical shell of L/R=5, N=1 and under tensile loading of N0=0.1Ncr for different h/R ratio

0.0 0.5 1.0 1.5 2.01.0

1.1

1.2

1.3

1.4

1.5 (1,4)

(1,4)

(1,4)

N=1, L/R=5, h/R=1/200


(rev/s)

Y

Na= -0.1N

cr Na= -0.2Ncr Na= -0.4Ncr

Figure 9. Variation of bifurcation of natural frequencies with external rotational velocity of rotating FG cylindrical shell of L/R=5, h/R=0.005, N=1 under different compressive loading

0.0 0.5 1.0 1.5 2.00.78

0.80

0.82

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

1.02 (1,4)

(1,4)

(1,4)

N=1, L/R=5, h/R=1/200


(rev/s)

Y

Na= 0.1N

cr N

a= 0.2N

cr N

a= 0.4N

cr

Figure 10. Variation of bifurcation of natural frequencies with external rotational velocity of rotating FG cylindrical shell of L/R=5, h/R=0.005, N=1 under different tensile loading


8

V. Conclusion The vibration analysis of the rotating FG cylindrical shell under compressive and tensile axial loading is investigated considering the effects of the centrifugal, Coriolis and initial hoop tension due to rotation. The effects of the length ratio, thickness ratio, and magnitude of the axial loading, material gradient index and rotating speed on the frequency characteristics of the rotating cylinder have been examined. It was observed that the circumferential wave number at which the fundamental frequencies occurred is dependent on the linear parameters of the shell.

References 1Loy, C. T., Lam, K. Y., and Reddy J. N., “Vibration of functionally graded cylindrical shell”, Int J Mech Sci, Vol. 41,1999, pp.309-324. 2Lam, K. Y., and Loy, C. T. , ‘‘Analysis of rotating laminated cylindrical shells by different thin shell theories.’’ J. Sound Vib., 186, 1995, pp. 23–35. 3Liew, K. M., Ng, T. G., and Zhao, X., “Vibration of axially loaded rotating cross ply laminated cylindrical shells via ritz method.” ASCE J Engg. Mech, Vol. 128, No. 9, 2002, pp. 1001-1007. 4Timoshenko, S. P., and Gere, J. M., Theory of elastic stability, 2nd ed., McGraw-Hill, New York, 1961.

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