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20082008 27: 447 originally published online 31 JanuaryJournal of Reinforced Plastics and Composites

H.J. Lin and J.F. TsaiAnalysis of Underwater Free Vibrations of a Composite Propeller Blade

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Analysis of Underwater Free Vibrations of

a Composite Propeller Blade

H. J. LIN*

Department of Electrical Engineering, National Penghu University, Taiwan

J. F. TSAI

Department of Engineering Science and Ocean Engineering

National Taiwan University, Taiwan

ABSTRACT: The free vibration characteristics of a composite marine propeller blade were studied.MAB and composite materials were used. Composites with symmetric, balanced and unbalancedstacking sequences were analyzed. Rotational effects and added mass were considered using thefinite element method. The natural frequency of the blade in water was much lower than in air.The mode shapes of the blade are almost the same in air as in water. The anisotropy of thecomposites shift the contours of the mode shape. Generally, greater anisotropy corresponds toa lower natural frequency. The rotational effects can be ignored because the marine propellerrotates slowly.

KEY WORDS: underwater, free vibration, composite, propeller blade.

INTRODUCTION

COMPOSITE MATERIALS ARE used in numerous structural applications. They are used

not only in industry, such as in the mobile, aerospace and shipbuilding industries,

but also in daily life, since they are strong, rigid, lightweight and inexpensive. Composite

materials may now be effectively applied to produce propeller blades. The applica-

tion of composite materials technology to marine architecture has increased with

particular benefits of weight and special characteristics of the materials. The application

of composites may achieve weight, noise and pressure fluctuation reduction and increase

the fuel efficiency. The performance of a composite marine propeller blade involves its

structural, fluid, acoustic, vibrational, material, and other characteristics. Now, manysmall or middle size composite marine propellers were commercialized or tested. Thus, the

structural analysis including vibration of the composite marine propeller may be necessary

in the future. The composite marine propeller may be the alternative to metal.

Performing a structural analysis of a propeller blade is difficult because it has complex

geometry and loading. Classical curved beam, plate and shell theories have applied to

analyze the structural analysis of a propeller during the early age [13]. The propeller blade

is considered to be a cantilever rigidly attached to a boss. These approaches have been

*Author to whom correspondence should be addressed. E-mail: [email protected]

Journal of REINFORCED PLASTICS AND COMPOSITES, Vol. 27, No. 5/2008 447

0731-6844/08/05 044712 $10.00/0 DOI: 10.1177/0731684407082539 SAGE Publications 2008

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shown to be successful, but particular assumptions restrict their application. The finite

element method is so popular that various varieties of elements have been used [48]. Plate

elements, thin and thick-shell elements and solid elements have been extensively adopted.

Surface loading has been calculated based on assumed loadings or on analytical results

concerning for example, the lifting line, the lifting surface and panel analysis. Coupled

fluid-structure analysis [9] has also been performed to evaluate the strength of composite

propeller blades. An iterative procedure was applied herein to calculate new blade

geometry and modify the surface loading. This process is repeated until stability is once

again achieved. Numerous researchers have performed the free vibration analysis of a

blade in air or in vacuum. However, such a blade is really operated in water, so the

added mass effects due to the water must be considered in the free vibration analysis.

The added mass effect comes from the transmission of pressure to the hull due to the

inertia of the water. Various methods of calculating the added mass are available. They

include, for example, the finite element method, the strip method and ideal flow theory.

This study considers the potential flow of an ideal fluid. A so-called panel method

is adopted, in which the source elements are distributed on the surface of the bodyto simulate the flow field. The added mass can be calculated from the surface pressure

induced by the fluid.

GEOMETRY AND STACKING SEQUENCE OF BLADE

The basic blade geometric data of a typical propeller blade are the number of blades (N),

the diameter of the propeller (D or 2R), the radius of the section (r), the length of the chord

(C), the pitch (P), the pitch angle (), the skew angle (), the rake (Z), the camber (f), the

maximum thickness of the section (t) and the offset of the blade surface. Figure 1 shows

the notation used herein: the Z-axis is the direction of advance; S is a non-dimensional

t(s)

f(s)

s

0

C/2

C/2

1r

Zm

f

X

Z

Y

qm

Figure 1. Coordinate system and definitions of variables.

448 H.J. LIN AND J.F. TSAI



4/13

curvilinear coordinate along the nose-tail helix, which are zero at the leading edge and one

at the trailing edge. The coordinates of a point on the face or back surface of the blade

can be written as:

m

C

S

1=2

cos=r

f

t=2

sin

=r

1

X r cos 2

Y r sin 3

Z Zm C S 1=2 sin f t=2 cos: 4

The propeller selected in the example is the MAU3-60 propeller designed for a single-

screw fishing boat. The propeller blade has a fixed pitch with diameter of 140 cm, a pitch

ratio of 0.77, an expansion ratio of 0.6 and three blades. Table 1 presents the geometric

parameters of the blade. Figure 2 shows the finite element mesh of the MAU3-60

0 10 20 30 40 50 60 70 80

X-axis (spanwise, cm)

40

30

20

10

0

10

20

30

40

Y-axis(chordwis

e,cm)

Figure 2. Projected view and finite element mesh of MAU3-60 propeller blade.

Table 1. Basic geometric parameters of blade of MAU 3-60 propeller.

r/R C/D P/D hm (8) Zm/D t/D f/D

0.2 0.3108 0.77 9.00 0.0176 0.0430 0.13850.3 0.3629 0.77 7.40 0.0265 0.0383 0.10550.4 0.4067 0.77 5.50 0.0353 0.0335 0.08240.5 0.4406 0.77 3.50 0.0441 0.0288 0.06530.6 0.4629 0.77 1.30 0.0529 0.0240 0.05190.7 0.4654 0.77 1.08 0.0617 0.0193 0.0414

0.8 0.4340 0.77 3.75 0.0705 0.0146 0.03370.9 0.3439 0.77 6.68 0.0794 0.0098 0.0284

1.0 0.0000 0.77 9.10 0.0882 0.0000 0.0000

Analysis of Underwater Free Vibrations of a Composite Propeller Blade 449



5/13

propeller blade. The rotational speed n of the propeller is 12 rps. The advancing speed, Va,

of the propeller is 1.68 m/s. A uniform inflow condition is considered. The mesh size of

the propeller blade is 8 16 in the structural analysis and 8 8 in the fluid analysis.The root is assumed to be fixed ended boundary condition.

The symmetric stacking sequence of graphite/epoxy of T300/1076E was adopted.

Table 2 presents the material properties of MAB (manganese aluminum bronze) and

T300/1076E. The balanced and unbalanced stacking sequences [. . .//90/0]s and[. . ./2/90/0]s, were considered. The subscript s denotes symmetric with respect to themiddle surface, i.e., camber surface. The first stacking layer with 08 is on the camber

surface. The number of layers is counted from the middle surface to the upper and lower

surfaces of the propeller blades. The number of layers varies with the thickness. For

example, a stacking sequence of [. . .2/90/0]S means 08, 908, , , 08, 908, , , . . . starting

from the camble surface to the suction and pressure sides, respectively. Figure 3 shows the

stacking sequence [. . .//90/0]S in the propeller blade. The generation line (X-axis) ofthe propeller blade is taken as the reference of fiber direction of the composites.

The positive fiber orientation is from the root to the leading edge.

XY

... ...

q

[...q/q/90

/0

/0

/90

/q/q...]

Figure 3. Stacking sequence [. . .//90/0]S of composite in propeller blade.

Table 2. Material properties of MAB and T300/1076E.

Properties: MAB Data

Longitudinal modulus, E 100 GPa

poisson ratio, 0.3

Density, 8200 Kg/m3

Properties: T300/1076E Data

Longitudinal modulus, E11 145 GPaTransverse modulus, E22 8.91 GPa

Shear modulus, G12 4.35 GPa

In-plane poisson ratio, 12 0.3

Density, 1577 Kg/m3

Layer thickness 0.001 m




6/13

FINITE ELEMENT METHOD

A geometrically nonlinear degenerate shell element [9] was used. There are five degrees

of freedom at each node, three translations and two rotations. The static finite-element

equation for a geometrically non-linear three-dimensional degenerate shell element is

expressed as [1013]:

Ko KG q Fext 5

in which Ko and KG are the linear stiffness matrix and the geometric matrix; q is the nodal

displacement, and Fext is the external forces. The matrices are defined as:

Ko Z

BTo DBo dV 6

KG Z

GTx xy

xy y

!G dV: 7

In these equations, Bo represents the linear strain-displacement transformation matrix; D

is the material property matrix, and G is a matrix defined by the derivative of the

coordinates. x, y and xy are stresses on the tangent plane of the shell surface.

The centrifugal forces are considered even though the marine propeller rotates relatively

slowly. They may affect the stiffness of the structure and so much be taken into account.

The Lagrange equation and kinetic energy of the propeller blade are used to formulate

the centrifugal force and rotational stiffness [1416]. The kinetic energy of a propeller

blade is given by:

T 12

Z V2 dV 8

where is the density of the material, and V*

is the velocity (V2 V* V* ). Thevelocity of a point includes the translational and rotational speeds, _u

*

and !*

, and is

defined as:

V*

_u*

!*

X*

u*

9

where X

* x,y, z T is the position coordinate, !* !x,!y,!z T, and _u* _u, _v, _wf gT.Equation (9) can be expressed in detail as:

V*

_u

_v

_w

8>>>:

9>>=>>;

!y z w !z y v !z x u !x z w !x y v !y x u

8>>>:

9>>=>>;: 10

Computing V2

and canceling the terms proportional to X

*

X*

, which do nocontribute to the Lagrangian, and substituting Equation (10) into Equation (8)




7/13

yields the kinetic energy,

T

1

2 Z_u

_v

_w

8>:

9>=>;

T_u

_v

_w

8>:

9>=>;

dV

1

2 Z_u

_v

_w

8>:

9>=>;

T

A1

u

v

w

8>:

9>=>;

dV

12

Z

u

v

w

8>:

9>=>;

T

A2 u

v

w

8>:

9>=>; dV

Z

x

y

z

8>:

9>=>;

T

A2 u

v

w

8>:

9>=>; dV 11

where A1 and A2 are expressed as:

A1

0 2!z 2!y2!z 0

2!x

2!y 2!x 0

264

375

12

A2 !2y !2z !x !y !x !z!x !y !2x !2z !z !y!x !z !z !y !2y !2x

264

375 13

Applying the finite element method yields the Lagrangian equation of motion,

d

dt

@T

@ _u

@T

@u Mq C_q KRq FR 14

in which q, _q and q are the nodal displacement, the velocity and the acceleration,

respectively; M is the mass matrix; KR is the rotational stiffness matrix; and FR is the

centrifugal force, as shown below.

MZ

N T N dV 15

C Z N T A1 N dV 16

KR Z

N T A2 N dV 17

FR Z

N T A2 x

y

z

8>:

9>=>; dV: 18

Based on the static finite element analysis, the governing equation of motion, neglecting

damping, can be expressed as:

Mq Ko KG KR q Fext FR: 19




8/13

ADDED MASS FORMULATION

The propeller blade operates in the water. The additional pressure transmitted to the

blade surface, due to the inertia of the water, is represented as added mass. Assume a node

on the propeller blade generating a movement of displacement, the generated velocity can

be transfer to the normal velocity, _qn, of the node [9], by

_qn T _q 20in which [T] is the transformation matrix between _q and _qn.

The vibration of the structure submerged in the water exerts a pressure normal to the

shell surface. The ideal flow of the source is assumed to be distributed on the surface of

the structure. The velocity potential of a source is expressed as [1719]:

ZZ

14

1

R*

p

R*

q

dS P f g 21

in which is the source intensity, and R*p and R*q are the position vectors of two points,

p and q. Thus, the normal velocity can be obtained by the dot product of the derivative of

velocity potential and the surface normal vector.

_qn r n* ZZ

r 14

1

R*p R*q

0B@

1CAdS C f g: 22

The pressure can be derived using the unsteady Bernoulli equation as

p @@t

23

where is the density of the fluid. Substituting Equations (20)(22) into Equation (23)

yields the pressure in terms of q, as,

p P _f g P C 1 qn P C 1 T q: 24

Integrating the surface pressure yields the force induced by the fluid

FA ZZ

p dS A q 25

in which A is the so-called added mass matrix. Substituting Equation (25) into

Equation (19) yields the equation of motion in the form,

M A q Ko KG KR q Fext FR: 26For harmonic motion of the structure, the structural displacement q can be expressed

in the harmonic form,

q qo ei!t 27




9/13

in which qo is the amplitude of harmonic motion. According to the free vibration analysis,

Equation (26) can be simplified and rewritten as:

Ko KG KR !2 M A

qo 0: 28

The solution to the eigenvalue and eigenfunction problem, Equation (28) is determined

using by

Det Ko KG KR !2 M A 0: 29

The subspace method was used to solve the eigen-value problem of Equation (29).

However, matrix A in Equation (29) is asymmetric. Therefore, A was replaced with

1/2(A AT) [18].

NUMERICAL RESULTS AND DISCUSSION

MAB propeller blade

Manganesealuminumbronze (MAB) is commonly used in the manufacture of marine

propeller blades. Table 3 presents its first three natural frequencies and vibration modes.

The natural frequency of the blade in water, !w, is lower than that in air, !a, by 23% to

30%. Since ! / ffiffiffiffiffiffiffiffiffiffi1=Mp , it means that the added mass is almost equal to 0.7 to 1.0 timesof the mass of the blade itself. Thus, the added mass must be considered in evaluating

the dynamic characteristics of the propeller blades. Figures 4 and 5 show the modes

of vibration in air and water. The mode shapes are almost the same in air and water,

although they differ slightly. The first and second modes are pure bending and torsion

modes. The third mode is the secondary bending mode in the direction of span.

Table 3 also presents the effects of rotation on the free vibration characteristics of

marine propeller blades. Marine propeller blades rotate more slowly than fans, jets,

turbines or other types of blade. The rotational speed of the marine blade is 12 rps.

The value of!w when still is almost the same as that when running. Thus, the rotational

effects can be ignored since the marine propeller blade rotates slowly.

Composite propeller blade

Symmetrical blades with balanced and unbalanced stacking sequences were consideredin the analysis. Table 4 shows the first three natural frequencies of the blade with [. . ./152/

90/0]S. The natural frequency of the blade in water is 4363% lower than that in air.

The difference between the natural frequencies is much larger than that of the blade made

Table 3. Natural frequency (Hz) of propeller blade made of MAB.

MAB MAB

Mode xa xw Mode (xw)still/(xw)running

1 15.2 10.8 1 0.996

2 31.1 24.5 2 0.973

3 36.9 28.6 3 0.991




10/13

of MAB. Since ! / ffiffiffiffiffiffiffiffiffiffi1=Mp , then the added mass is almost six times the mass of thecomposite blade. The effects of the added mass on the vibration characteristics of the blade

with [. . ./152/90/0]S exceeds that on those of the MAB blade. Figures 6 and 7 plot the

vibration modes in air and water. The mode shapes are almost the same. The first and

third modes are the first pure bending and torsion modes. The second mode is the second

(a)

(c)

(b)

Figure 4. Vibration modes of blade made of MAB in air. (a) mode I; (b) mode II and (c) mode III.

(a) (b)

(c)

Figure 5. Vibration modes of blade made of MAB in water. (a) mode I; (b) mode II and (c) mode III.




11/13

bending mode in the direction of span. The second and third modes shape of the blade

made of [. . ./152/90/0]S are the same as those of the third and second modes of the MAB

blade, respectively. Also, the contour is shifted toward the trailing edge of the blade with

[. . ./152/90/0]S because the 158 carbon fibers stiffen the leading edge. The anisotropy of the

composite distorts its mode shapes.

Table 4 also presents the first three natural frequencies of the blade with [. . .15/15/90/0]S. The natural frequency of the blade in water is 50% lower than that in air.

The added mass is also approximately six times that of the mass of the composite blade.

The natural frequencies of the blade with [. . .15/15/90/0]S exceed those of the bladewith [. . ./152/90/0]S, perhaps because the former is less anisotropic.

Table 5 shows the natural frequencies of the blades with [ . . ./2/90/0]S, [. . ./2/90/0]Sand [. . .//90/0]S in water. A typical airfoil section of the propeller blade is thickerat its leading edge than at its trailing edge. The leading edge is stiffer than the trailing

edge. When fibers are negatively stacked, the stiffness of the trailing edge is increased.

Accordingly, negative fiber stacking increases the natural frequency of the blade.

Therefore many of the data obtained for blades with [. . ./2/90/0]S exceed thoseobtained for blades with [. . ./2/90/0]S.

(a) (b)

(c)

Figure 6. Vibration mode of blade with [. . ./152/90/0]S in air. (a) mode I; (b) mode II and (c) mode III.

Table 4. Natural frequency (Hz) of composite propeller blade.

[. . .152/90/0]S [. . .15/15/90/0/]SMode xa xw Mode xa xW

1 23.18 8.71 1 26.82 9.92

2 42.10 22.80 2 41.60 23.06

3 51.86 29.58 3 55.32 29.83




12/13

CONCLUSIONS

The free vibration characteristics of metal and composite propeller blades

were analyzed. The effects of rotation and added mass were noted, using the finite

element method. Symmetric composite blades with balanced and unbalanced stacking

sequences were analyzed. The first three natural frequencies and vibration modes are

presented and discussed. Numerical results reveal that the added mass is almost equal to

that of the blade made of MAB, and seven times that of the composite blade. The natural

frequencies of the blade made of MAB and the [. . ./152/90/0]S blade in water are 2330%

and 4363%, respectively, lower than that of the blade in air. Since ! / ffiffiffiffiffiffiffiffiffiffi1=Mp , thea dded m ass is al most d oub le an d six ti mes t he m asses of t he MAB a nd

(a) (b)

(c)

Figure 7. Vibration mode of blade with [. . ./152/90/0]S in water. (a) mode I; (b) mode II and (c) mode III.

Table 5. Nature frequency (Hz) of composite propeller blade in water.

[. . .152/90/0]S [. . .302/90/0]S [. . .452/90/0]S

Mode !W Mode !W Mode !W

1 8.71 1 8.45 1 8.282 22.80 2 23.62 2 23.91

3 29.58 3 29.55 3 27.49

1 10.22 1 9.90 1 9.34

2 20.46 2 23.14 2 24.67

3 32.96 3 33.00 3 31.70

1 9.92 1 10.35 1 9.82

2 23.06 2 25.92 2 26.17

3 29.83 3 30.68 3 29.72




13/13

composite blades, respectively. Submergence in water greatly reduces the natural

frequency of a blade. The mode shapes are almost the same in air and water. The first,

second and third modes are the first bending, torsion and second bending modes in the

direction of the span of the MAB blade, respectively. The second and third modes of the

blade with [. . ./152/90/0]S are the second bending mode in the direction of the span and

torsion mode. The contours of the mode shape are shifted because of the anisotropy of the

composites. A blade with [. . ./2/90/0]S has a higher natural frequency than a blade with[. . ./2/90/0]S. The rotational effects can be ignored because a marine propeller blade

rotates slowly.

ACKNOWLEDGMENT

The authors would like to thank the National Science Council of the Republic of China

for financially supporting of this research.

REFERENCES

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2. Cohen, J. W. (1955). On Stress Calculations in Helicoidal Shells and Propeller Blades, Netherlands ResearchCenter, T. N. O. Shipbuilding and Navigation, Delft Report 21S.

3. Conolly, J. E. (1974). Strength of Propeller, Trans RINA, 103: 139204.

4. Genalis, P. (1970). Elastic Strength of Propellers An Analysis by Matrix Methods. Ph.D. Thesis, Universityof Michigan, USA.

5. Atkinson, P. (1968). On the Choice of Method for the Calculation of Stress in Marine Propeller, TransRINA, 110: 447463.

6. Ma, J. H. (1974). Stresses in Marine Propellers, Journal of Ship Research, 18: 252264.

7. Sontvedt, T. (1974). Propeller Blade Stresses Application of Finite Element Methods, Computers &Structures, 4(1): 193204.

8. Atkinson, P. and Glover, E. J. (1988). Propeller Hydroelastic Effects, SNAME on the Propeller88Symposium No.21, Jersey, NJ.

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12. Zienkiewicz, O. C. (1991). The Finite Element Method, 4th edn, McGraw-Hill, London.

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15. Dokainish, M. A. and Rawtani, S. (1971). Vibration Analysis of Rotating Cantilever Plates, InternationalJournal for Numerical Methods in Engineering, 3(2): 233248.

16. Bossak, M. A. J. and Zienkiewicz, O. C. (1973). Free Vibration of Initially Stressed Solid with ParticularReference to Centrifugal Force Effects in Rotating Machinery, Journal of Strain Analysis, 8(4): 245252.

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