# Propagation of plane wave in a rotating polygonal cross-sectional panel immersed in fluid

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American Journal of Materials Science 2014, 4(2): 45-55 DOI: 10.5923/j.materials.20140402.01

Plane Wave Propagation in a Rotating Polygonal Cross-sectional Plate Immersed in Fluid

P. Ponnusamy
Department of Mathematics, Government Arts College (Autonomous), Coimbatore, 641 018, Tamil Nadu, India
Abstract The free vibration of homogeneous, isotropic rotating plate of polygonal cross-section immersed in fluid is
studied within the frame work of the linearized, two-dimensional plane theory of elasticity. The equations of motion of the plate are formulated using the constitutive equations of a isotropic material with preferred material direction collinear with the longitudinal axis of the plate. The equations of motion of the fluid are formulated using the constitutive equations of the inviscid fluid. Displacement potentials are used to solve the equations of motion of the plate and the fluid. The frequency equation of the coupled system consisting of the plate and fluid is developed under the assumption of perfect-slip boundary conditions at the solid-fluid interfaces. The non-dimensional frequencies of longitudinal and flexural antisymmetric modes of vibrations are obtained using the Fourier Expansion Collocation Method (FECM), and they are presented in the form of dispersion curves.
Keywords Solid-fluid interface, Wave propagation in plate, Vibration of thermal plate, Piezoelectric plate, Plate
immersed in fluid, Generalized thermo elastic plate, Rotating cylinder/rotating plate

1. Introduction
The plates of circular and plates of polygonal cross-sections are often used as structural components and their vibration characteristics are important for practical design. The frequency responses of rotating arbitrary/ polygonal cross-sectional plates has many applications in various fields of science and technology, namely, submarine structures, pressure vessel, borewells, ship building industries and have many other engineering applications. Nagaya [1,2] studied the simplified method for solving problems of vibrating plates of doubly connected arbitrary shape using the Fourier expansion collocation method, he has detained the frequency equations for a doubly connected polygonal cross-sectional plate and numerical calculation were carried for the above said cross-sections. Following Nagaya, Ponnusamy [3, 4, 5] discussed the frequency responses of generalized thermo elastic cylinder and thermo-elastic plate of arbitrary cross-sections and generalized thermo-elastic plate of polygonal cross-sections by using the Fourier expansion collocation method developed by Nagaya [1,2]. Further the same author [6] investigated the wave propagation in a piezoelectric solid bar of circular cross-section immersed in fluid.
The effect of rotation on elastic waves in solids was

studied by many authors, such as Clarke and Burdess [7], Wren and Burdess [8] and Soderkvist [9]. Ting [10] analyzed the interfacial waves in a rotating anisotropic elastic half space by extending the Stroh [11] formalism. He obtained explicit expression of the polarization vector and the secular equation for monoclinic material half space rotating about the normal to the plane of symmetry. Fang et al [12, 13] studied respectively the effect of rotation on the characteristic of wave propagation in a piezoelectric plate and the effect of rotation on surface acoustic waves in a piezoelectric half space. Sharma and Thakur [14] presented the effect of rotation on Rayleigh-Lamb waves in magneto-thermoelastic plates. Sharma and Othman [15] investigated the effect of rotation on generalized thermo viscoelastic Rayleigh-Lamb waves in plates. Walia et al. [16] studied the propagation characteristics of thermoelastic waves in piezoelectric (6mm class) rotating plate.
For fluid sensor application, vibration modes of an elastic body without normal displacement at its surface are ideal. When the body is vibrating in these modes no pressure waves are generated in the fluid. The fluid produces a drag on the body surface only due to the fluid and the tangential motion of the surface, therby lowering the frequencies of the waves may also cause additional dispersion. Both stationary waves in resonators and propagationg waves in wave guides have been used for fluid sensors. For example, face-shear, thickness-shear and thickness-twist modes in plates are widely used for fluid sensor applications. In these modes, motions of material particles are parallel to the surface of the plates. Understanding the behavior of these waves is

46

P. Ponnusamy: Plane Wave Propagation in a Rotating Polygonal Cross-sectional Plate Immersed in Fluid

fundamentally important to acoustic wave fluid sensors. Sun et al. [17] studied the sheat-horizontal waves in a rotated Y-cut quartz plate in contact with a viscous fluid.
Assaf et al [18] analyzed the vibration and acoustic responses of damped sandwich plates immersed in a light or heavy fluid. Three dimensional vibration analysis of an infinite poroelastic plate immersed in an inviscid elastic fluid was discussed by Ahmed Shah and Tajuddin [19].
Keeping these facts in view, an attempt has been made to investigate the free vibration analysis of rotating polygonal cross-sectional plate immersed in fluid, which has not been discussed by any researchers. The plate is assumed to be rotating with uniform angular velocity about its axis.
2. Formulation of the Problem
We consider a homogeneous, isotropic rotating elastic plate of polygonal cross-section. The system displacements
and stresses are defined by the polar coordinates r and θ in an arbitrary point inside the plate and denote the displacements ur in the direction of r and uθ in the tangential direction θ . The in-plane vibration and
displacements of rotating polygonal cross-sectional plate is obtained by assuming that there is no vibration and a displacement along the z axis in the cylindrical coordinate
( ) system r,θ , z . The two dimensional stress equations of
motion, strain –displacement relations in the absence of body forces for a linearly elastic medium are

( ) σ rr,r + r−1σ rθ ,θ + r−1 σ rr − σθθ + ρΩ2ur =ρur,tt

σ rθ ,r + r −1σθθ ,θ + 2r −1σ rθ = ρuθ ,tt (1)

where

σ rr = λ (err + eθθ ) + 2µerr

σθθ = λ (err + eθθ ) + 2µeθθ

(2)

σ rθ = 2µerθ where σ rr,σθθ ,σ rθ are the stress components,

err, eθθ , erθ are the strain components, ρ is the mass

density, Ω2 is the rotational speed, t is the time, λ and µ are Lame’ constants.
The strain eij related to the displacements are given by
( ) err = ur= ,r , eθθ r−1 ur + uθ ,θ ,

( ) erθ =uθ ,r − r−1 uθ − ur,θ

(3)

in which ur and uθ is the displacement components
along radial and circumferential directions respectively. The comma in the subscripts denotes the partial differentiation with respect to the variables.
Substituting the Eqs. (3) and (2) in Eq. (1), the following displacement equations of motions are obtained as

( ) (λ + 2µ ) ur,rr + r−1ur,r − r−2ur + µr−2ur,θθ + r−1 (λ + µ )uθ ,rθ + r−2 (λ + 3µ )uθ ,θ + ρΩ2ur = ρur,tt ( ) µ uθ ,rr + r−1uθ ,r − r−2uθ + r−2 (λ + 2µ )uθ ,θθ + r−2 (λ + 3µ )ur,θ + r−1 (λ + µ )ur,rθ = ρuθ ,tt (4)

3. Solutions of Solid Medium

The Eq. (4) is a coupled partial differential equation with two displacements components. To uncouple the equation (4), we

follow Mirsky [10] by assuming the vibration and displacements along the axial direction z equal to zero. Hence assuming the

solutions of the Eq. (4) in the form

∑ ( ) ( ) ∞

ur (r,θ= , z,t)

εn

 

φn,r

+

r −1ψ n,θ

+ φ n,r + r −1ψ n,θ eiωt

n=0

∞

∑ ( ) ( ) uθ (r= ,θ , z,t)

ε

n

 

r −1φn,θ

−ψ n,r

+

r −1φ n,θ −ψ n,r

eiωt

(5)

n=0

where

εn

=

1 2

for

n = 0 , ε n = 1 for

n ≥ 1 , i=

−1 , ω is the angular frequency, φn (r,θ ) , ψ n (r,θ ) ,

( ) ( ) φ n r,θ , and ψ n r,θ are the displacement potentials.

American Journal of Materials Science 2014, 4(2): 45-55

47

( ) By introducing the dimensionless quantities λ = λ µ ,

ξ 2 = ρω2a2 µ , c2 = Ω2ξ 2 ω 2 , ω* = ξ a µ ρ ,

T = t µ ρ a , x = r a , c2 is the rotational

velocity, and substituting Eq.(5) in Eq.(4), we obtain

( ) ( ) 


2+λ

∇2 +

ξ 2 + c2

 

φn

=0

(6)

and

( ) ∇2 + ξ 2 + c2 ψ n =0

(7)

where ∇2 ≡ ∂2 ∂x2 + x−1 ∂ ∂x + x−2 ∂2 ∂θ 2

From (6), we obtain

( ) ∇2 + (α1a)2 φn =0

(8)

( ) ( ) where (α1a)2 =ξ 2 + c2 2 + λ . The solution of

Eq.(8) for symmetric mode is

φn = A1Jn (α1ax)cos nθ

(9)

Similarly the solution for the anti symmetric mode φn is obtained by replacing cos nθ by sin nθ in Eq.(9).

φ n = A1Jn (α1ax)sin nθ

(10)

where Jn is the Bessel function of first kind of order n.
Solving Eq.(7), we obtain

ψ n = A2Jn (α2ax)sin nθ

(11)

for symmetric mode and ψ n is obtained from Eq.(11) by replacing sin nθ by cos nθ ,we get

c−f 2urf ,tt = ∆ ,r

(14)

( ) respectly, where urf ,uθf is the displacement vector,

B f is the adiabatic bulk modulus, c f = B f ρ f is the

acoustic phase velocity of the fluid in which ρ f is the

density of the fluid and

( ( )) =∆ urf ,r + r−1 urf + uθf ,θ

(15)

substituting urf = φ,rf and uθf = r −1φ,θf and seeking
the solution of Eq.(14) in the form

∞

∑ φ f (r,θ ,t ) = εnφn f cos nθeiωt

(16)

n=0

where

φn f

=

A3H

(1)
n

(α3ax

)

(17)

where (α3a)2 = ξ 2 ρ f B f in which ρ = ρ ρ f ,

Bf =Bf

µ,

H

(1)
n

is the Hankel function of first kind

( ) of order n. If α3a 2 < 0 , then the Hankel function of

first kind is replaced with Kn , where Kn is the modified

Bessel function of second kind. Substituting the Eq. (16) in

Eq. (13) along with the Eq. (17), we could express the

acoustic pressure of the fluid as

∑ p f

=

∞

ε

n

A3ξ

2

ρ

H

(1)
n

(α3ax

)

cos

nθ

eiωt

(18)

n=0

ψ n = A2Jn (α2ax)cos nθ

(12)

( ) ( ) for the anti symmetric mode. Here α2a=2 ξ 2 + c2 .

If (αia)2 < 0 , i = 1, 2 then the Bessel function Jn of
first kind is to be replaced by the modified Bessel function of
the first kind In .

4. Solution of Fluid Medium

In cylindrical coordinates, the acoustic pressure and radial displacement equation of motion for an in viscid fluid are of the form Achenbach [21]

( ( )) p f = −B f urf ,r + r−1 urf + uθf ,θ

(13)

and

5. Boundary Conditions and Frequency Equations

In this problem, the free vibration of a polygonal (triangle, square, pentagon and hexagon) cross sectional rotating plate immersed in fluid is considered. Since the boundary is irregular, it is difficult to satisfy the boundary conditions of the plate directly. Hence, in the same lines of Nagaya [1, 2], the Fourier expansion collocation method is applied to satisfy the boundary conditions. Thus the boundary conditions are obtained as

( ) ( ) ( ) σ xx + p f

i =σ xy

i =ur − urf

= 0
i

(19)

where x is the coordinate normal to the boundary and y

is the tangential coordinate to the boundary, σ xx is the

( ) normal stress,σ xy is the shearing stress and

i is the

48

P. Ponnusamy: Plane Wave Propagation in a Rotating Polygonal Cross-sectional Plate Immersed in Fluid

value at the i th segment of the boundary. The first and last
conditions in equations (19) are due to the continuity of the stresses and displacements of the plate and fluid on the
curved surface. If the angle γ i between the normal to the

segment and the reference axis is assumed to be constants, thus the transformed expression for the stresses is given by Nagaya [1, 2].

( ( )) ( ) ( ( ) ) σ x=x λ ur,r + r−1 ur + uθ,θ + 2µ[ur,r cos2 (θ − γi ) + r−1 ur + uθ,θ sin2 (θ − γi ) + 0.5 r−1 uθ − ur,θ − uθ,r sin 2(θ − γi )]

( ) ( ( ) ) σ x=y µ[ur,r − r−1 uθ ,θ + ur sin 2(θ − γ i ) + r−1 ur,θ − uθ + uθ ,r sin 2(θ − γ i )]

(20)

The boundary conditions in Eq.(19) are transformed as

( ) (Sxx )i +

S xx

i

 

eiξT

= 0

( ) ( ) 


S xy

+
i

S xy

i

 

eiξT

= 0

( ) (Sr )i +

Sr

i

 

eiξT

= 0

(21)

where

( ) ∑( ) ∞
S=xx 0.5 x10 A10 + x02 A20 + x1n A1n + xn2 A2n + xn3 A3n
n=1

( ) ∑( ) ∞

S=xy 0.5 y10 A10 + y02 A20 +

y1n A1n + yn2 A2n + yn3 A3n

n=1

( ) ∑( ) ∞
S=t 0.5 z10 A10 + z02 A20 + z1n A1n + zn2 A2n + zn3 A3n

(22)

n=1

( ) ∑( ) S=xx

0.5

x30 A30

∞
+

x1n A1n + xn2 A2n + x3n A3n

n=1

( ) ∑( ) S=xy

0.5

x30 A30

∞
+

x1n A1n + xn2 A2n + x3n A3n

n=1

( ) ( ) ∑ = St

0.5

z

3 0

A30

+

∞

z1n

A1n

+

z

2 n

A2n

+

z3n

A3n

(23)

n=1

The coefficients for

xni



z

i n

,

i

=

1, 2,3

are given in the Appendix A.

Performing the Fourier series expansion to Eq.(19) along the boundary, the boundary conditions are expanded in the form

of double Fourier series. For the symmetric mode, the boundary conditions are expressed as follows.

∑ ∑( ) ∞

εm

 X

1 m0

A10

+

X

2 m0

A20

+

∞

X

1 mn

A1n

+

X

2 mn

A2n

+

X

3 mn

A3n

  cos mθ

= 0

m 0=  n 1



∑ ∑( ) ∞ Ym10 A10 + Ym20 A20 + ∞

Ym1n A1n + Ym2n A2n + Ym3n A3n

  sin mθ = 0

=m 1 =n 1



American Journal of Materials Science 2014, 4(2): 45-55

49

∑ ∑( ) ∞

εm

 

Zm1 0

A10

+

Z

2 m0

A20

+

∞

Zm1 n

A1n

+

Z

2 mn

A2n

+

Zm3 n A3n

  cos mθ = 0

(24)

=m 0=  n 1



Similarly, for antisymmetric mode, the boundary conditions are expressed as

∑ ∑( ) ∞

 X

3 m0

A30

+

∞

X

1 mn

A1n

+

X

2 mn

A2n

+

X

3 mn

A3n

  sin mθ = 0

=m 1 =n 1



∑ ∑( ) ∞

εm

 Y

3 m0

A30

+

∞

Y

1 mn

A1n

+

Y

2 mn

A2n

+

Y

3 mn

A3n

  cos mθ = 0

=m 0=  n 1



∑ ∑( ) ∞

 

Z

3 m0

A30

+

∞

Z

1 mn

A1n

+

Z

2 mn

A2n

+

Z

3 mn

A3n

  sin mθ = 0

(25)

=m 1 =n 1



where

I θi

X

j mn

=

( 2ε n

π )∑

∫

xnj ( Ri ,θ )cos mθ dθ

i=1 θi−1

I θl
Ymjn = (2εn π ) ∑ ∫ ynj ( Ri ,θ )sin mθ dθ i=1 θl−1

I θi

Z

j mn

=

( 2ε n

π )∑

∫

znj ( Ri ,θ )cos mθ dθ

(26)

i=1 θi−1

∑ ∫ X

j mn

= (2εn

π)

I

θi xnj ( Ri ,θ )sin mθ dθ

i=1 θi−1

∑ ∫ Y

j mn

= (2εn

π)

I

θl

y

j n

(

Ri

,θ

)

cos

mθ

dθ

i=1 θl−1

∑ ∫ Z

j mn

= (2εn

π)

I

θi

z

j n

(

Ri

,θ

)

sin

mθ

dθ

(27)

i=1 θi−1

where j = 1, 2 , and 3, I is the number of segments, Ri is the coordinate r at the boundary and N is the number of
truncation of the Fourier series. The frequency equations are obtained by truncating the series to N +1 terms, and equating
the determinant of the coefficients of the amplitude Ain = 0 and Ain = 0 , for symmetric and antisymmetric modes of
vibrations. Thus, the frequency equation for the symmetric mode is obtained from Eq. (24), by equating the determinant of
the coefficient matrix of Ain = 0 . Therefore we have

50

P. Ponnusamy: Plane Wave Propagation in a Rotating Polygonal Cross-sectional Plate Immersed in Fluid

 

X

1 00



 

X

1 N

0

 

Y010



 

YN1

0

 

Z

1 00





 

Z

1 N

0

X

2 00



X

2 N

0

Y020

 YN20 Z020



Z

2 N

0

X

1 01



X

1 0N





X

1 N

1



X

1 NN

Y011  Y01N





YN11  YN1N

Z

1 01



Z01N





Z

1 N1



Z

1 NN

X

2 01



X

2 0N





X

2 N1



X

2 NN

Y021  Y02N





YN21  YN2N

Z021  Z02N





Z

2 N1



Z

2 NN

X

3 01



X

3 N1

Y031

 YN31 Z031



Z

3 N

1


 
 


X

3 0N



X

3 NN

Y03N



YN3N

Z

3 0N



Z

3 NN

              

 A10

 

A20

 

A11



 

A1N



 

A31

 A3N

             

=

0

(28)

Similarly, the frequency equation for antisymmetric mode is obtained from the Eq. (25) by equating the determinant of

the coefficient matrix of Ain to zero. Therefore, for the antisymmetric mode, the frequency equation is obtained as

 

X

3 10

X

1 11



X

1 1N

X

2 11



X

2 1N

X

3 11



X

3 1N

  A31 

 



 

X

3 N

0

 

Y

3 10









X

1 N

1



X

1 NN

Y

1 11



Y

1 1N









X

2 N1



X

2 NN

Y

2 11



Y

2 1N







X

3 N1

Y

3 11



 

X


3 NN

  

 A11  

  

Y

3 1N

 

 

A1N

 



  A21 

 

=

0

(29)

 

Y

3 N

0

 

Z

3 10

Y

1 N1



Y

1 NN

Z

1 11



Z

1 1N

Y

2 N1



Y

2 NN

Z

2 11



Z

2 1N

Y

3 N1

Z

3 11

 

Y

3 NN

Z 13N

   



 

A2

N

 A31

   



 









   



Z

3 N

0

Z

1 N1



Z

1 NN

Z

2 N1



Z

2 NN

Z

3 N

1



Z

3 NN



 A3N



6. Numerical Results and Discussions

pentagonal and hexagonal cross sections immersed in fluid.

The numerical analysis of the frequency equation is carried out for rotating polygonal (square, triangle, pentagon and hexagon) cross-sectional plates immersed in fluid, and

The material properties used for the computation are as
follows: For the solid the Poisson ratio ν = 0.3 , density
ρ = 7849kg / m3 and the Young’s modulus

the dimensions of each plate used in the numerical =E

calculation are shown in Figure 1, and its geometric relations for the polygonal cross-sections given by Nagaya [1, 2]

ρf

2.139 ×1011 N = 1000Kg / m3

/ m2 and
and

for the the

fluid: phase

the density velocity

as

= Ri b cos(θ − γ i )−1

(30)

where b is the apothem. The relation given in Eq. (30) is
used directly for the numerical calculation. The axis of
symmetry is denoted by the lines in the figures. where b is

c = 1500m / s . The dimensionless frequencies are
computed using Secant method (applicable for complex
roots 9 ) for polygonal cross sectional rotating plate immersed in fluid. The polygonal cross sectional plate in the
range θ = 0 and θ = π is divided into many segments
for convergence of frequency in such a way that the distance

the apothem.

between any two segments is negligible. Integration is

The frequency equations are obtained in symmetric and performed for each segment numerically by use of Gauss anti symmetric cases given in equations (28) and (29) are Gauss five point formula .The non-dimensional frequencies
analyzed numerically for rotating plate of triangular, square, are computed for 0 < Ω ≤ 1.0 , using the secant method.

American Journal of Materials Science 2014, 4(2): 45-55

51

6.1. Rotating Polygonal cross-sectional Plates

6.1.2. Triangle and Pentagonal Cross-sectional Plate
The triangular and pentagonal cross sectional plate, the vibration displacements are symmetrical about the major axis for the longitudinal mode and anti symmetrical about the minor axis for the flexural mode since the cross section is
symmetric about only one axis. Therefore n and m are
chosen as 0,1, 2,3,... in Eq. (28) for longitudinal mode and
n, m = 1, 2,3,... in Eq. (29) for the flexural anti symmetric
mode.
6.1.3. Dispersion Curves
A graph is drawn between the rotational speeds of Ω =0.5 and non-dimensional frequency of longitudinal modes of a triangular cross-sectional plate in space, immersed in fluid and rotating plate immersed in fluid and is shown in Fig. 2. From Fig. 2, it is observed that as mode increases the dimensionless frequencies increases. Also it is observed that the dimensionless frequencies of plate in space are higher than the plate immersed in fluid and rotating plate immersed in fluid. The dimensionless frequencies of rotating plate immersed in fluid increases and decreases as mode increases.

Plate in space

Plate immersed in fluid

Non-dimensional frequency |ξ|

3.00

Rotating Plate immersed in fluid

2.50

2.00

1.50

1.00

0.50

0.00

Figure 1. Rotating Plate of Polygonal cross-sections. (a) Triangle (b)

0

0.5

1

Square (c) Pentagon (d) Hexagon

Mode

6.1.1. Square and Hexagonal cross-sectional Plate
In the case of longitudinal vibration of square and hexagonal cross sectional plates, the displacements are symmetrical about both major and minor axes since both the cross sections are symmetrical about both the axes. Therefore the frequency equation is obtained by choosing
both terms of n and m are chosen as 0, 2, 4, 6,... in Eq.
(28) . During flexural motion, the displacements are anti symmetrical about the major axis and symmetrical about the minor axis. Hence the frequency equation is obtained by
choosing n, m = 1,3,5,... in Eq. (29).

Figure 2. Comparison between the frequency response of longitudinal modes of triangular cross-sectional plate in space, immersed in fluid and rotating with a speed of Ω=0.5
A graph is drawn between the rotational speed and
non-dimensional frequency ξ of longitudinal modes of
triangular cross-sectional plate immersed in fluid and is shown in Fig. 3. From Fig. 3, it is observed that as mode increases the non-dimensional frequency increases. Further it is observed that as rotational speed increases the dimensional frequencies increases.

52

P. Ponnusamy: Plane Wave Propagation in a Rotating Polygonal Cross-sectional Plate Immersed in Fluid

Non-dimensional frequency|ξ|

Ω=0.01

1.4

Ω=0.5

Ω=1.0

1.2

Ω=2.0

1

Ω=3.0

0.8

0.6

0.4

0.2

0

0

0.2 0.4 0.6 0.8

1

Mode

Figure 3. Rotational speed versus non-dimensional frequency |ξ| of longitudinal modes of triangular cross-sectional plate immersed in fluid
Graph is drawn between the mode and non-dimensional
frequency ξ for flexural antisymmetric modes of
triangular cross-sectional rotating plate immersed in fluid and is shown in Fig. 4. From Fig. 4, it is observed that as mode increases both the real and imaginary part of non-dimensional frequencies increases. It is interesting to note that the real part of dimensional frequencies is higher than that of the imaginary part as rotation increases.

and are is shown in Figs. 5 and 6. From Figs. 5 and 6, it is observed that the dimensional frequencies increases as rotating speed increases. Also it is observed that the cross over points in the trend lines denote the transfer of energy between the modes of vibration.

Rotation Ω=0.1

Rotation Ω=0.5

2.5

Rotation Ω=1.0

Rotation Ω=1.5

2

Rotation Ω=2.0

Rotation Ω=3.0

1.5

Non-dimensional frequency |ξ|

1

0.5

0

0

0.2 0.4 0.6 0.8

1

Mode
Figure 5. Rotational speed Ω=0.1, 0.5, 1.0, 2.0 and 3.0versus dimensionaless frequency |ξ| for longitudinal modes of square cross-sectional rotating plate immersed in fluid

Non-dimensional frequency |ξ|

Non-dimensional freequency |ξ|

Rotation Ω=0.1, Re

Rotation Ω=0.1, Im

1.4

Rotation Ω=0.5, Re

Rotation Ω=0.5, Im

1.2

Rotation Ω=1.0, Re

1

Rotation Ω=1.0, Im

0.8

Ω=0.1

Ω=0.5

1.80

Ω=1.0

1.60

Ω=1.5

1.40

Ω=3.0

1.20

1.00

0.80

0.6

0.60

0.40

0.4

0.20

0.2

0.00

0

0.5

1

0

0

0.2

0.4

0.6

0.8

1

Mode

Rotational speed Ω

Figure 6. Rotational speed versus non-dimensional frequency |ξ| of longitudinal modes of hexagonal cross-sectional plate immersed in fluid

Figure 4. Rotational speed Ω=0.1, 0.5, 1.0 versus dimensionaless frequency |ξ| for a flexural antisymmetric modes of triangular cross-sectional rotating plate immersed in fluid
Graphs are drawn between mode and non-dimensional
frequency ξ for longitudinal modes of square and
hexagonal cross-sectional rotating plate immersed in plate

A graph is drawn between the mode and non-dimensional frequency for different longitudinal modes of pentagonal cross-sectional plate immersed in fluid with fixed rotational speed Ω =0.1 and is shown in Fig. 7. From Fig. 7, it is observed that as mode increases the non-dimensional frequency increases. The notation Lm denotes for longitudinal modes of vibration. Also it is noted that as

American Journal of Materials Science 2014, 4(2): 45-55

53

longitudinal modes of vibration non-dimensional frequency increases.

Non-dimensional frequency |ξ|

1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00
0

Lm1 Lm2 Lm3 Lm4
0.2 0.4 0.6

increases 0.8

the 5. Conclusions
In this paper, a method for solving the wave propagation problem of rotating polygonal cross sectional plate immersed in fluid has been presented. The frequency equation has obtained using Fourier expansion collocation method. Numerical calculations have been carried out for triangular, square, pentagonal and hexagonal cross sectional rotating plate immersed in fluid. This method is straightforward and the numerical results for any other polygonal cross section can be obtained directly for the same frequency equation by substituting geometric values of the boundary of any cross section analytically or numerically with satisfactory convergence.
1

Mode
Figure 7. Rotation Ω=0.1, the frequency response for different longitudinal modes of pentagonal cross-sectional plate immersed in fluid

Appendix A
e1n = 2{n(n −1) Jn (α1ax) + (α1ax) Jn+1 (α1ax)}cos 2(θ − γ l )cos nθ
{ ( )} − x2 (α1a)2 λ + 2cos2 (θ − γl ) Jn (α1ax)cos nθ

e1n = 2{n(n −1) Jn (α1ax) + (α1ax) Jn+1 (α1ax)}cos 2(θ − γ l )sin nθ
{ ( )} − x2 (α1a)2 λ + 2cos2 (θ − γl ) Jn (αax)sin nθ

e2n = 2n{(n −1) Jn (α2ax) + (α2ax) Jn+1 (α2ax)}cos 2(θ − γ l )cos nθ
{( ) } + 2 n(n −1) − (α2ax)2 Jn (α2ax) + (α2ax) Jn+1 (α2ax) sin 2(θ − γl )sin nθ

en2 = 2n{(n −1) Jn (α2ax) + (α2ax) Jn+1 (α2ax)}cos 2(θ − γ l )sin nθ
{( ) } − 2 n(n −1) − (α2ax)2 Jn (α2ax) + (α2ax) Jn+1 (α2ax) sin 2(θ − γl )cos nθ

en3

=

ξ

2

ρ

H

(1)
n

(α3ax ) cos

nθ

e3n

=

ξ

2

ρH

(1)
n

(α 3ax ) sin

nθ

{ } f= 1n 2 n(n −1) − (α1ax)2 Jn (α1ax) + (α1ax) Jn+1 (α1ax) sin 2(θ − γ l )cos nθ

+ 2n{(α1ax) Jn+1 (α1ax) − (n −1) Jn (α1ax)}cos 2(θ − γ l )sin nθ

54

P. Ponnusamy: Plane Wave Propagation in a Rotating Polygonal Cross-sectional Plate Immersed in Fluid

{ } = f 1n 2 n(n −1) − (α1ax)2 Jn (α1ax) + (α1ax) Jn+1 (α1ax) sin 2(θ − γ l )sin nθ

− 2n{(α1ax) Jn+1 (α1ax) − (n −1) Jn (α1ax)}cos 2(θ − γ l )cos nθ

f

3 n

= 2n {( n

−1) Jn

(α 2ax )

− (α2ax) Jn+1 (α2ax)}sin 2(θ

− γl

)cos nθ

{ ( ) } + 2 (β ax) Jn+1 (α2ax) − n(n −1) − (α2ax)2 Jn (α2ax) cos 2(θ − γl )sin nθ

f

3 n

= 2n {( n

−1)

Jn

(α 2ax )

−

(α 2 ax )

J n+1

(α 2ax )} sin

2 (θ

−

γl

) sin

nθ

{ ( ) } − 2 (α2ax) Jn+1 (α2ax) − n(n −1) − (α2ax)2 Jn (α2ax) cos 2(θ − γl )cos nθ

= g1n {nJn (α1ax) − (α1ax) Jn+1 (α1ax)}cos nθ

= g1n {nJn (α1ax) − (α1ax) Jn+1 (α1ax)}sin nθ

g

2 n

=

nJ n

(α2ax)cos nθ

g

2 n

=

nJ n

(α 2ax ) sin

nθ

g

3 n

= − nHn(1) (α3ax)

−

(α 3ax )

H

(1)
n+1

(α 3ax )

cos

nθ

g

3 n

= − nHn(1) (α3ax)

−

(α 3ax )

H

(1)
n+!

(α3ax

)

sin

nθ

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