Dynamic response of rotating flexible cantilever pipe conveying fluid with tip mass

ARTICLE IN PRESS

International Journal of Mechanical Sciences 49 (2007) 878–887 www.elsevier.com/locate/ijmecsci

Dynamic response of rotating flexible cantilever pipe conveying fluid with tip mass Han-Ik Yoona,, In-Soo Sonb a

Division of Mechanical Engineering, Dong-eui University, 995 Eomgwangno, Busanjin-Gu, Busan, 614-714, Korea b The Center for Industrial Technology, Dong-eui University, 995 Eomgwangno, Busanjin-Gu, 614-714, Korea Received 8 November 2006; accepted 17 November 2006 Available online 28 December 2006

Abstract In this study the vibration system is consisted of a rotating cantilever pipe conveying fluid and a tip mass. The equation of motion is derived by using the Lagrange’s equation. Also, the equation of motion is derived applying a modeling method that employs hybrid deformation variables. The influences of the rotating angular velocity and the velocity of fluid flow on the dynamic behavior of a cantilever pipe are studied by the numerical method. The effects of a tip mass on the dynamic behavior of a rotating cantilever pipe are also studied. The influences of a tip mass, the velocity of fluid, the angular velocity of a cantilever pipe and the coupling of these factors on the dynamic behavior of a cantilever pipe are analytically clarified. The natural frequencies of a cantilever pipe conveying fluid are proportional to the angular velocity of the pipe and a tip mass in both axial direction and lateral direction. r 2006 Elsevier Ltd. All rights reserved. Keywords: Dynamic response; Rotating flexible cantilever pipe; Cantilever pipe conveying fluid; Tip mass

1. Introduction Rotating cantilever beams are found in several practical engineering examples such as turbine blades and aircraft rotary wings. For reliable and practical design of the structures, it is necessary to estimate the modal characteristics of those structures accurately. Since significant variations of modal characteristics result from rotational motion of the structures, they have been investigated by many researchers. In addition, the fluid flowing inside the pipe acts as the concentrated tangential follower force at the tip of the pipe, and exerts a lot of influences on the dynamic behavior of a pipe. Therefore, a large number of papers have presented about the dynamic behavior of fluidconveying pipes since the early 1960s. The transfer of energy between the flowing fluid and the pipe was discussed by Benjamin [1,2]. The problem of flutter induced by a pure rocket thrust, which has applications to missiles, spacecraft and space structure, is also closely related to a stability [3]. Corresponding author. Tel.: +82 51 890 1645; fax: +82 51 890 2232.

E-mail addresses: [email protected] (H.-I. Yoon), [email protected] (I.-S. Son). 0020-7403/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2006.11.006

Langthjem and Sugiyama [4] studied the dynamic stability of a cantilevered two-pipe system conveying different fluids. Lim et al. [5] executed the nonlinear dynamic analysis of a cantilever tube conveying fluid with system identification. Recently, Yoon and Son [6] investigated the effects of the open crack and the moving mass on the dynamic behavior of simply supported pipe conveying fluid. They studied about the influences of the crack, the moving mass and its velocity, the velocity of fluid, and the coupling of these factors on the Timoshenko beam. The first modeling approach for beams, which hereafter will be referred to as the classical linear Cartesian (CLC) approach, was introduced [7,8] in the 1970s when the speed of computer and numerical methods was progressing rapidly. This approach is based on the classical linear elastic modeling, where geometric as well as material linearity is assumed. Later, a modeling method for straight beams undergoing large overall motions as well as small elastic deformation was presented [9,10]. This modeling method involves a stretch deformation instead of three Cartesian deformation variables used conventionally; it is named hybrid deformation variable (HDV) modeling. An early analytical model to calculate natural frequencies of

ARTICLE IN PRESS H.-I. Yoon, I.-S. Son / International Journal of Mechanical Sciences 49 (2007) 878–887

those structures was suggested in Ref. [11]. Based on the Rayleigh energy theorem, a simple equation that related the natural frequency to the rotating frequency of a beam was suggested. Also, the effects of a tip mass, elastic foundation and cross-section variation, shear deformation and the gyroscopic damping effect on the modal characteristics of rotating beams were studied, respectively [12–16]. Kane [17] investigated a rotating flexible cantilever beam using the traditional zero-order approximation coupling (ZOAC) model, and showed that this model failed to describe the dynamic behavior of the beam when the beam is in high rotational velocity. Cai et al. [18] studied the dynamic characteristics of the rotating flexible beam system by using the traditional ZOAC model and the first-order approximation coupling (FOAC) model. They considered two cases of low and high rotational speeds of the system. An exact solution [19] and an approximation formula [20] for the natural frequencies and mode shapes for out-of-plane lateral vibrations of the rotating uniform linear Euler–Bernoulli beam have been derived. Panussis and Dimarogonas [21] investigated the case of simultaneous in-plane and out-of-plane lateral vibrations of small amplitude of a horizontally rotating fluid-tube cantilever conveying fluid. In this study, the effects of a tip mass and fluid flow on the dynamic behavior of a rotating cantilever pipe conveying fluid are investigated. Other’s researches did not study about the coupling effect of a tip mass, fluid flow and rotation. Therefore, the influences of a tip mass, the velocity of fluid, a rotating angular velocity and the coupling of these factors on the natural frequencies and tip-response of a rotating cantilever pipe conveying fluid are depicted. The cantilever pipe is modeled by the Euler–Bernoulli beam theory. 2. Mathematical model Fig. 1 shows the schematic diagram of a rotating cantilever pipe conveying fluid with a tip mass. Consider a cantilever pipe of a length L, which is fixed at point O of a rigid hub with a radius r. The hub is rotating about the axis of symmetry with a rotating angular velocity o. The orthogonal unit vectors i and j are rotating with the hub. P0 and P are the positions of a generic point before and after j i Tm x+s



U

r

Po

O hub A

U

w w1

P w2

x

Fig. 1. Schematic diagram of a rotating cantilever pipe conveying fluid with a tip mass.

879

deformation, respectively. When the point P0 moves to the point P, using the Cartesian co-ordinates, the deformations of the pipe in the directions of i and j are generally described by the axial deformation w1 and the lateral deformation w2 , respectively. In addition, T m is a tip mass, U is the velocity of fluid flow and s is the arc length stretch. Using the assumed mode method, the tip-responses of a rotating cantilever pipe can be assumed to be sðx; tÞ ¼

n X

f1i ðxÞq1i ðtÞ,

(1)

i¼1

w2 ðx; tÞ ¼

n X

f2i ðxÞq2i ðtÞ,

(2)

i¼1

where f1i ðxÞ and f2i ðxÞ are the spatial mode functions for s and w2 , respectively. f1i ðxÞ are the modal function of the longitudinal vibration, and f2i ðxÞ are the modal functions of the bending vibration. qi ðtÞ are generalized co-ordinates which is time dependent, and n is the total number of the generalized coordinates. The functions f1i ðxÞ and f2i ðxÞ can be obtained for a cantilever beam with a tip mass as follows: p x f1i ðxÞ ¼ sin ai , (3) L f2i ðxÞ ¼ cosðpi xÞ  coshðpi xÞ þ bi ½sinðpi xÞ  sinhðpi xÞ,

ð4Þ

where L is the un-deformed length of the pipe, and bi can be expressed as follows: bi ¼

sinðpi LÞ  sinhðpi LÞ . cosðpi LÞ þ coshðpi LÞ

(5)

In Eqs. (3)–(5), pai and pi are the frequency parameters, which are easily calculated using the frequency equations of a cantilever beam. 2.1. Energy of rotating cantilever pipe The velocity vector of a particle at a generic point P can be obtained by using the following equation: vP ¼ vO þ vP=A þ x  ðx þ wÞ,

(6)

where O is a reference point identifying a point fixed in the rigid hub A. vO is the inertial velocity of point O, vP=A is relative velocity of point P with respect to the rigid hub A, and x is the angular velocity of the rigid hub. These vectors of Eq. (6) can be expressed as follows: vO ¼ roj, vP=A ¼ w_ 1 i þ w_ 2 j, x ¼ ok, x ¼ xi, w ¼ w1 i þ w2 j,

ð7Þ

where () denotes q=qt, i and j are the orthogonal unit vectors. By using Eqs. (6) and (7),, the velocity of point P

ARTICLE IN PRESS H.-I. Yoon, I.-S. Son / International Journal of Mechanical Sciences 49 (2007) 878–887

880

can be obtained as follows: vP ¼ ðw_ 1  ow2 Þi þ ðro þ w_ 2 þ ox þ ow1 Þj.

(8)

In the present modeling method, a non-Cartesian variable s denoting the arc length stretch of the neutral axis is employed. There is a geometric relation between the arc length stretch and the Cartesian variable. In order to use the arc length stretch s instead of w1 , the geometric relation between the arc length stretch s and the Cartesian variables is given by  Z  1 x qw2 2 s  w1 þ dx. (9) 2 0 qx Using the arc length stretch and Eq. (8), the energy of rotating cantilever pipe with a tip mass can be written as Z L 1 1 Tp ¼ m ðvP vP Þ dx þ T m ðvP jx¼L Þ2 , (10) 2 2 0 1 Vp ¼ 2

Z L" EAp 0

  2 2 # qs 2 q u2 þ EI dx, qx qx2



(11)

where E is the modulus of elasticity of the pipe, I is the moment of inertia of the pipe cross-section, m is the mass per unit length of a pipe, and Ap is the cross-sectional area of the pipe, respectively. 2.2. Work and energy due to fluid flow The relative velocity of fluid flow with respect to the pipe is assumed U, the velocity of fluid have to include the motion of a pipe. Therefore, the flow velocity components are (  ) 1 qw2 2 U x ¼ w_ 1  w2 o þ U 1  , (12) 2 qx qw2 . qx The sum of the flow velocity is given by 2   qw2 2 6 vf ¼ 4 ðro þ w_ 2 þ ox þ ow1 Þ þ U qx

U y ¼ ðro þ w_ 2 þ ox þ ow1 Þ þ U

3   ))2 1=2 1 qw2 2 5 . þ w_ 1  w2 o þ U 1  2 qx (

(13)

(

ð14Þ

The kinetic energy of fluid flow inside the pipe can be expressed as Z L 1 Tf ¼ M ðvf vf Þ dx, (15) 2 0 where M is the fluid mass per unit length of a pipe. The work of a follower force due to the fluid discharge is divided into two kinds of work, one is the work done by a conservative force component, and the other is the work

done by a non-conservative force component. The work W c due to the conservative component of a tangential follower force is  Z L 1 qw2 2 2 W c ¼ MU dx. (16) 2 qx 0 The work dW nc due to the non-conservative component of a follower force is as follows:     qw2 þ w_ 2 dw2  . (17) dW nc ¼ MU U qx x¼L 2.3. Equation of motion 2.3.1. Dimensionless equation of motion The equation of motion of the system is obtained by substituting the above energy functions into the Lagrange’s equation     d qLa qLa  ¼ 0, (18) dt qq_ i qqi where La can be defined as follows: La  ðT p þ T f  V p Þ þ W c þ dW nc .

(19)

For simplicity, the following dimensionless quantities are introduced: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x t EI q ; d1 ¼ 1 , x¼ ; t¼ 2 L m þ M L L rffiffiffiffiffiffi q M M d2 ¼ 2 ; b ¼ ; u ¼ UL , mþM EI L rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi Tm M 2 mþM 2 ; O ¼ oL ; Of ¼ oL , m¼ EI EI ðm þ MÞL rffiffiffiffiffiffiffiffiffiffi Ap L2 T mL r . ð20Þ ; r¯ ¼ ; I¯ ¼ Om ¼ oL EI L I Therefore, the dimensionless equation of motion is obtained as follows: " # " # " # M11 0 C11 C12 K11 K12 d€ þ d_ þ 0 M22 C21 C22 K21 K22 " # 0 P11 d¼ , ð21Þ 0 P22 where () denotes the q=qt, and d ¼ ½d 1i ðtÞ d 2i ðtÞT . The matrices of Eq. (21) can be written as M11 ¼

n X

½ðM1 Þij þ mðM2 Þij ,

(22)

½ðM3 Þij þ mðM4 Þij ,

(23)

i¼1

M22 ¼

n X i¼1

C11 ¼

n X i¼1

2u

pffiffiffi bðC1 Þij ,

(24)

ARTICLE IN PRESS H.-I. Yoon, I.-S. Son / International Journal of Mechanical Sciences 49 (2007) 878–887

C12 ¼ 2

n X

pffiffiffi ½OðC2 Þij þ Om mðC3 Þij ,

(25)

C21 ¼ C12 , n X

(26)

2u

pffiffiffi bðC4 Þij ,

K11 ¼

½I¯ ðK1 Þij  O2 ðM1 Þij  O2m ðM2 Þij ,

n

X dðOÞ

dt

i¼1

(28)

ðC2 Þij þ

 dðOm Þ ðK3 Þij þ uOf ð2K4 þ K5 Þij , dt (29)

K21 ¼

n

X dðOÞ

 dðOm Þ ðC2 Þij þ ðK3 Þij þ uOf ðK5 Þij , dt dt

i¼1

K22 ¼

n

X

(30)

ðK6 Þij þ u2 fðK7 Þij þ ðK8 Þij g  O2 ðM3 Þij  O2m ðM2 Þij

i¼1

þ¯rO2 ðK9 Þij þ

P11 ¼ 

n

X

Z

2

   1 2 1 O ðK10 Þij þ O2m r¯ þ ðK11 Þij , 2 2 ð¯r þ xÞf1i ðxÞ dx þ

0

O2m ð¯r

ð31Þ 

1

O

i¼1

þ 1Þf1i ð1Þ , (32)

P22 ¼

n  X dO

Z

1

þ Of u

dt

i¼1

ð¯r þ xÞf2i ðxÞ dx 0

 dOm pffiffiffi þ mð¯r þ 1Þf2i ð1Þ . dt

ð33Þ

These matrices in the above equations are defined in Appendix A. 2.3.2. Modal formulation Eq. (21) can be transformed into the following equation for the free vibration: M g_ þ K g ¼ 0,

(34)

where 2"

M11

0

0

M22

6 M ¼ 6 4 2"

where l is the eigenvalue, and H is the corresponding mode shape. From the eigenvalues in Eqs. (34)–(36), the frequencies can be obtained. 3. Numerical results and discussion

i¼1

K12 ¼ 

(36)

(27)

i¼1 n X

expressed as g ¼ elt H,

i¼1

C22 ¼

881

½0 C11

C12

6 C21 C22 K ¼ 6 4 ½I g ¼ ½d_ dT ,

#

# "

3 ½0 7 7, 5 ½I K11 K21

K12

In this study, the dynamic behavior of a rotating cantilever pipe conveying fluid influenced by the tip mass, the rotating angular velocity and the velocity of fluid. Those are computed by the fourth order Runge–Kutta method. The properties of the rotating cantilever pipe conveying fluid are given in Table 1. In this study, the base performs a planar rotational motion around the vertical axis which is given as Ref. [17], that is 8

  > < O t  1 sin 2pt if 0ptpts ; s ts 2p ts o¼ (37) > : Os if tXts ; where Os and ts are the steady-state angular velocity and the time to reach the angular velocity, respectively. This motion smoothly increases the angular velocity until it reaches the steady state and the steady-state angular velocity is sustained. It is so smooth and slow that the lateral oscillation after reaching the steady state remains quite small. Fig. 2(a) and (b) are shows the simulation results of the angular velocity and angular acceleration of pipe and fluid by Eq. (37), respectively. First of all, the accuracy of the present numerical results needs to be confirmed. Fig. 3 shows a comparison between our results and the results of Ref. [18] for the bending tip-displacement of a rotating cantilever beam with o ¼ 4 rad=s, where the length of beam is L ¼ 1:8 m and the cross-section area is 2:5  104 m2 . Fig. 4 shows a comparison between our results and Ref. [22] of the lowest two natural frequencies of a rotating cantilever beam without fluid flow. In Fig. 4, the horizontal axis is the scale of the dimensionless angular velocity of beam ðOm ¼ pffiffiffiffiffiffiffiffiffiffiffiffi oL2 m=EI Þ. These results show that our results are a good agreement with the result of references. In Table 2, the lowest three natural frequencies of a cantilever beam with no rotation (O ¼ 0) are given. Also, Table 2 represents Table 1 Specifications of a rotating cantilever pipe conveying fluid

#3

7 K22 7, 5 ½0 ð35Þ

where I represents a unit matrix. For the complex modal analysis, it is assumed that g is a harmonic function of t

Property

Values

Length of pipe Out-radius of pipe In-radius of pipe Bending stiffness Density of pipe Hub radius

1m 0.025 m 0.02 m 8.9782 Nm2 2:766  103 kg=m3 0.05 m

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Fig. 2. Spin-up motion of a rotating cantilever pipe and fluid. (a) Angular velocity, (b) angular acceleration.

Fig. 3. Comparison between present numerical results and results of Ref. [18] ½o ¼ 4 rad=s; ts ¼ 15 s.

a comparison between the present results and others (Refs. [22,23]) for natural frequency of a cantilever beam with a tip mass (m ¼ 1) and without a tip mass. As observed in the table, the three results are found to be identical. Figs. 5 and 6 represent the bending tip-displacement and the axial tip-deflection of a rotating cantilever pipe without a tip mass according to the velocity of fluid and the rotating angular velocity, respectively. In these figures, the horizontal axis scale is the time (s) and the time ðts Þ to reach the steady-state angular velocity is 3 s. Totally, when the rotating angular velocity is constant, as the fluid velocity is increased, the tip-responses (bending tip-displacement and axial tip-deflection) of the rotating cantilever pipe conveying fluid are decreased. That is, the velocity of fluid in pipe controls the rotating cantilever pipe. The tip-responses of the rotating cantilever pipe conveying fluid are more sensitive to the rotating angular velocity than the velocity of fluid. In Fig. 5(b), the difference of maximum bending tip-displacement of a rotating cantilever pipe in the two cases of u ¼ 1 and 2 is about 14.8%. In Fig. 6(b), when u ¼ 0:5, the difference of maximum bending tip-displacement of a rotating cantilever pipe conveying fluid in the two cases of O ¼ 3 and 5 is about 34.1%.

Fig. 4. Comparison between present numerical results and results in Ref. [22]: (a) 1st mode, (b) 2nd mode.

Figs. 7 direction cantilever first and

and 8 show the natural frequencies of each (axial and lateral direction) of a rotating pipe conveying fluid without tip mass for the second mode of vibration, respectively. The

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883

Table 2 Comparison between present numerical results ðO ¼ 0Þ and results in Refs. [22,23] Mp ¼ 1

Without tip mass

Present results Ref. [22, 1997] Ref. [23, 1999]

1st mode

2nd mode

3rd mode

1st mode

2nd mode

3rd mode

3.516 3.516 3.516

22.034 22.034 22.035

61.697 61.697 61.697

1.557 1.557 1.557

16.250 16.250 16.250

50.896 50.896 50.896

Fig. 5. Tip-responses of a rotating cantilever pipe according to fluid velocity ðO ¼ 3Þ. (a) Axial tip-deflection, (b) bending tip-displacement.

Fig. 6. Tip-responses of a rotating cantilever pipe according to fluid velocity ðu ¼ 0:5Þ. (a) Axial tip-deflection, (b) bending tip-deflection.

horizontal axis is the scale of the angular velocity, and the axis of the ordinates is the scale of the natural frequency of a rotating cantilever pipe. In addition, the unit of natural frequency is 1=t. In these figures, the first and second natural frequencies of a cantilever pipe increase as the rotating angular velocity increases. Fig. 7 shows the natural frequency of a rotating cantilever pipe according to the fluid velocity for the axial direction. It is shown that the natural frequency of the rotating cantilever pipe conveying fluid is in inverse proportion to the velocity of fluid for the

first and second mode. In Fig. 8, the effects of the fluid velocity on the natural frequency of lateral direction of a rotating cantilever pipe are shown. In Fig. 8(a), the first mode of vibration, as the fluid velocity increases, the natural frequency of a rotating cantilever pipe is increased. In Fig. 8(b), the second mode of vibration, as the fluid velocity increases, the natural frequency of a rotating cantilever pipe is decreased. The axial tip-deflection of a rotating cantilever pipe conveying fluid according to the angular velocity and a tip

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Fig. 7. Natural frequency of a rotating cantilever pipe according to fluid velocity (axial direction): (a) 1st mode, (b) 2nd mode.

mass are shown in Fig. 9, where the dimensionless velocity of fluid is constant as u ¼ 1, and the dimensionless tip mass are m ¼ 0:05 and 0.3, respectively. As the tip mass and the rotating angular velocity are increased, the axial tipdeflection of a rotating cantilever pipe conveying fluid is increased. Fig. 10 shows the effects of the rotating angular velocity and tip mass on the bending tip-displacement of a rotating cantilever pipe conveying fluid. When the tip mass and fluid velocity are constant, the bending tip-displacement of a rotating cantilever pipe is proportional to the rotating angular velocity. In addition, when the rotating angular velocity is constant, the bending tip-displacement of a rotating cantilever pipe is proportional to the tip mass. In Figs. 11 and 12, the effects of a tip mass on the natural frequency of each direction of a rotating cantilever pipe conveying fluid are depicted for the first mode and the second mode of vibration, respectively. In Fig. 11(a), when the rotating angular velocity is 2 rad/s, the difference of natural frequency of a rotating cantilever pipe in the two cases of m ¼ 0:1 and 0.3 is about 14.5%. It is shown that

Fig. 8. Natural frequency of a rotating cantilever pipe according to fluid velocity (lateral direction): (a) 1st mode, (b) 2nd mode.

Fig. 9. Axial tip-deflection of a rotating cantilever pipe according to angular velocity and tip mass ðu ¼ 1Þ.

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Fig. 10. Bending tip-displacement of a rotating cantilever pipe according to angular velocity and tip mass: (a) m ¼ 0:05; u ¼ 1; (b) m ¼ 0:3; u ¼ 1.

the natural frequency of a rotating cantilever pipe conveying fluid are more sensitive to the tip mass than the rotating angular velocity. In Fig. 12(a), when the rotating angular velocity is zero, the difference of natural frequency of a rotating cantilever pipe in the two cases of m ¼ 0:1 and 0.3 is about 19.8%. When the rotating angular velocity is 5 rad/s, the difference of natural frequency of a rotating cantilever pipe in the two cases of m ¼ 0:1 and 0.3 is about 17.3%. 4. Conclusions In this paper, the influences of the angular velocity, the velocity of fluid and a tip mass are studied on the dynamic behavior of a rotating cantilever pipe conveying fluid by the numerical method. The cantilever pipe is modeled by the Euler–Bernoulli beam theory. The equation of motion is derived by using Lagrange’s equation. In analysis of the dynamic behavior of a rotating cantilever pipe, the modeling involves a stretch deformation instead of two Cartesian deformation variables used conventionally. Also,

Fig. 11. Natural frequency of a rotating cantilever pipe according to tip mass (axial direction, u ¼ 1): (a) 1st mode, (b) 2nd mode.

in this study the effects of the rotating angular velocity, the velocity of fluid, a tip mass and the coupling of these factors on the natural frequencies and tip-displacement of a rotating cantilever pipe conveying fluid are depicted. The main results of this study are summarized as follows: When the rotating angular velocity is constant, as the fluid velocity is increased, the tip-response of the rotating cantilever pipe conveying fluid is decreased. That is, the fluid velocity in pipe controls the rotating cantilever pipe. The tip-responses of the rotating cantilever pipe conveying fluid are more sensitive to the rotating angular velocity than to the effect of the velocity of fluid. On the first mode of vibration, as the fluid velocity increases, the natural frequency of a rotating cantilever pipe is increased. In addition, on the second mode of vibration, as the fluid velocity increases, the natural frequency of a rotating cantilever pipe is decreased. The natural frequency of a rotating cantilever pipe conveying fluid is more sensitive to

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Z ðC3 Þij ¼ f1i ð1Þf2i ð1Þ;

1

f02i ðxÞf2j ðxÞ dx,

ðC4 Þij ¼ 0

(A7,A8) Z

Z

1

f01i ðxÞf01j ðxÞ dx;

ðK1 Þij ¼ 0

1

f001i ðxÞf1j ðxÞ dx,

ðK2 Þij ¼ 0

(A9,A10) Z ðK3 Þij ¼ f1i ð1Þf2j ð1Þ;

1

f1i ðxÞf02j ðxÞ dx,

ðK4 Þij ¼ 0

(A11,A12) Z

Z

1

f01i ðxÞf2j ðxÞ dx;

ðK5 Þij ¼ 0

1

ðK6 Þij ¼ 0

f002i ðxÞf002j ðxÞ dx, (A13,A14)

ðK7 Þij ¼ f02i ð1Þf2j ð1Þ;

Z

1

ðK8 Þij ¼ 0

f002i ðxÞf2j ðxÞ dx, (A15,A16)

Z

1

ð1  xÞðf02i ðxÞf02j ðxÞÞ dx,

ðK9 Þij ¼ 0

Z

(A17)

1

ðK10 Þij ¼ 0

ð1  x2 Þðf02i ðxÞf02j ðxÞÞ dx,

ðK11 Þij ¼ f02i ð1Þf02j ð1Þ,

(A18) (A19)

where (0 ) stands for q=qx. References Fig. 12. Natural frequency of a rotating cantilever pipe according to tip mass (lateral direction, u ¼ 1): (a) 1st mode, (b) 2nd mode.

the tip mass than to the influence of the rotating angular velocity. Appendix A In Eqs. (22)–(33), the matrices are defined as follows: Z 1 ðM1 Þij ¼ f1i ðxÞf1j ðxÞ dx; ðM2 Þij ¼ f1i ð1Þf1j ð1Þ, 0

(A1,A2) Z

1

f2i ðxÞf2j ðxÞ dx;

ðM3 Þij ¼

ðM4 Þij ¼ f2i ð1Þf2j ð1Þ,

0

(A3,A4) Z

Z

1

ðC1 Þij ¼ 0

f01i ðxÞf1j ðxÞ dx;

1

f1i ðxÞf2j ðxÞ dx,

ðC2 Þij ¼ 0

(A5,A6)

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