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Chellapilla 5 Axial Vib Bellows TWS 2004 .pdf


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Thin-Walled Structures 42 (2004) 415–426
www.elsevier.com/locate/tws

Axial vibrations of U-shaped bellows with
elastically restrained end conditions
M. Radhakrishna a, C. Kameswara Rao b,
a

b

Design Engineering Group, Indian Institute of Chemical Technology, Hyderabad, India
Intellectual Property Management Group, Corporate R & D, Bharat Heavy Electricals Limited,
Vikasnagar, Hyderabad 500 093, India
Received 10 June 2003; received in revised form 19 June 2003; accepted 4 July 2003

Abstract
Previous work by Li et al. in the area of axial vibrations of bellows dealt with fixed end
conditions. However, it is seen on several occasions that bellow ends are welded to a small
pipe spool that has a lumped mass such as a valve or an instrument. Hence, the present
paper aims at finding out the effect of elastically restrained ends on the axial natural frequencies. The analysis considers finite stiffness axial restraints on the bellows, i.e. solving the
set of equations with non-homogeneous boundary conditions. Two bellow specimens are
considered for comparison having the same dimensions as taken by Li in his analysis. The
transcendental frequency equation deduced is accurate as the first, second and third mode
frequencies computed are in close agreement to the ones obtained by Li.
# 2003 Elsevier Ltd. All rights reserved.
Keywords: Bellow vibration; Single bellows; Axial frequency and elastic restraint

1. Introduction
Gerlach [1] was the first to have developed a simplified method of computing the
natural frequencies of bellows. He represented a bellow having ‘N’ convolutions by
a system consisting of 2N 1 identical masses and connected to a 2N identical
springs. By calculating the elemental mass, added mass and elemental stiffness, the
axial natural frequencies of bellows were determined. It is reported that very good
predictions were observed for the lowest vibration modes. EJMA [2] also provides


Corresponding author. Fax: +91-40-3776320.
E-mail address: ckrao_52@yahoo.com (C. Kameswara Rao).

0263-8231/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/S0263-8231(03)00130-7

416

M. Radhakrishna, C. Kameswara Rao / Thin-Walled Structures 42 (2004) 415–426

a method for calculating the axial natural frequency of single and double bellows
by assuming the bellows as a continuous elastic rod. However, the method does
not take into account the mass resulting from simple translation of the fluid
between undeformed convolutions.
Jakubauskas derived an expression for the axial natural frequency of bellows. He
assumes the total fluid mass, mf, per unit length to comprise three components—
rigid convolution motion in axial direction, mf1, convolution distortion component,
mf2, and the third component associated with the motion in return flow in central
area of cross-section of bellows, mf3. The distortion component for a half-convolution was determined using finite element analysis. The percentage error in frequency obtained by the EJMA modified method was compared with the results of
experiments and was found to be far less than the percentage error in frequency
obtained either from standard EJMA or Gerlach methods [3].
The paper contributed by Li et al. [4] presents equations to calculate the axial
and lateral natural frequencies of single bellows with three types of end conditions—one end fixed and other end free, one end fixed and other end attached to
weight and both ends of the bellows are fixed. The theoretical results were then
compared with the experiments performed on bellow specimens having different
geometrical parameters. Though the error corresponding to the experimental value
was found to be reasonably close to theoretical value, the end conditions represented by Li do not represent most of the practical situations.
It is seen on several occasions that bellow ends are welded to a small pipe spool
that has a lumped mass such as a valve or an instrument. Hence, the present paper
aims at finding out the effect of elastically restrained ends on the axial natural frequencies of a physical system as shown in Fig. 1. A weight, W0, is attached as

Fig. 1. Pipe with end condition.

M. Radhakrishna, C. Kameswara Rao / Thin-Walled Structures 42 (2004) 415–426

417

Fig. 2. Dimension of below.

shown and is connected to a spring of stiffness, ks. It is assumed that a straight
pipe is partially fixed at the bottom and connected to a spring of stiffness, kp. Fig. 2
depicts the various geometrical parameters of the U-shaped bellows used in the
analysis.

2. Axial natural frequency of bellows
The differential equation to express the axial vibration for the straight pipe is
given by [4]:
@2u
@2u
¼ a2 2
2
@t
@x

ð1Þ

where ‘u’ is the axial displacement of the pipe (mm), ‘T’ is the time (s), E is the
elastic modulus of bellow material (MPa), g is the gravitational acceleration
2
3
(9806.65 p
mm/s
ffiffiffiffiffiffiffiffiffiffiffi ), m is the weight per unit volume of the bellow material (N/mm )
and a ¼ Eg=m, respectively.

3. Formulation and analysis
When considering the axial vibrations of a single bellow expansion joint, the
boundary conditions for the system are given by
@uðxÞ
¼ kp u
@x
@u W0 @ 2 u
þ
AE
þ ks uðxÞ ¼ 0
@x
g @t2

AE

ð2Þ
ð3Þ

The exact solution of the differential equation is written as follows
uðx; TÞ ¼ ½Csinb þ Dcosbx eixT

ð4Þ

418

M. Radhakrishna, C. Kameswara Rao / Thin-Walled Structures 42 (2004) 415–426

where
b ¼ xi =a

ð5Þ

and xi is the angular frequency in radian/s, C and D are the integration constants.
Now applying the boundary conditions and substituting T ¼ 0, we get
uðxÞ ¼ Csinbx þ Dcosbx

ð6Þ

@uðxÞ
¼ bfCcosbx Dsinbxg
@x

ð7Þ

Substituting for @u=@x and u(x) in Eq. (2), we get
AE bfCcosbx Dsinbxg ¼ kp fCsinbx þ Dcosbxg
Applying x ¼ 0 in Eq. (8), we get


C ¼ kp =b AE D

ð8Þ

ð9Þ

We know
uðx; TÞ ¼ uðxÞ eixT

ð10Þ
2

2

Now substituting for u(x), @u=@x and @ u=@t in Eq. (3)



AEfbðCcosbx DsinbxÞg þ ks W0 x2 =g ðCsinbx ¼ DcosbxÞ ¼ 0
ð11Þ
Applying x ¼ L in Eq. (8), where L is the bellow length, we get
AEfbðCcosbL DsinbLÞg



þ ks W0 x2 =g ðCsinbL þ DcosbLÞ ¼ 0
Rearranging the terms in Eq. (12), we get




C AE bcosbL þ ks W0 x2 =g sinbL




¼ D AE bsinbL ks W0 x2 =g cosbL
Substituting the value of C in (13)


AEb kp þ ks W0 x2 =g
tanbL ¼
ðbAEÞ2 þ kp ðks W0 x2 =gÞ
Dividing (13) through out by AE and multiplying by L, we get


b kp L=AE þ ks L=AE W0 Lx2 =gAE
tanbL ¼
ðbLÞ2 þ kp L=AEðks L=AE W0 Lx2 =gAEÞ

ð12Þ

ð13Þ

ð14Þ

ð15Þ

Let
Tp ¼ kp L=AE

ð16Þ

M. Radhakrishna, C. Kameswara Rao / Thin-Walled Structures 42 (2004) 415–426

419

where Tp is physically the ratio of the axial pipe stiffness to the effective bellow
stiffness
Ts ¼ ks L=AE

ð17Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðEgÞ=m

ð18Þ

1=a ¼ W0 =G

ð19Þ



where a is the ratio of bellow mass to the end mass
bL ¼ xi ðL=aÞ

ð20Þ

Now, substituting for the value of ‘a’ in Eq. (20), we get
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
bL ¼ xi G=ðg kn Þ

ð21Þ

where G ¼ LAm, A is the effective area of cross-section of bellow in mm2 and kn is
the theoretical stiffness of bellows obtained using the equation given in EJMA.
Assuming the exact equation as
uðx; TÞ ¼ fCsinxi x=a þ Dcosxi x=ageixT

ð22Þ

Applying the boundary conditions as mentioned above and repeating the procedure, the final frequency equation is obtained as follows


bi a 1 þ Ts =Tp b2i =Tp
ð23Þ
tanbi ¼


a 1=Tp Ts þ b2i

4. Different end conditions
4.1. Case 1
One end of the bellows is fixed and the other end is free (ks ¼ 0, Ts ¼ 0, W0 ¼ 0
and Tp ¼ 1): substituting the above conditions in Eq. (23), the resulting frequency
equation is written as Eq. (27)
bi tanbi ¼ a



bi a Ts þ Tp b2i
tanbi ¼


a 1 Ts Tp þ Tp b2i

ð24Þ
ð25Þ

We know
fi ¼ xi =2p

ð26Þ

therefore
fi ¼ 49:5ði 0:5Þ

pffiffiffiffiffiffiffiffiffiffiffi
kn =G

ð27Þ

420

M. Radhakrishna, C. Kameswara Rao / Thin-Walled Structures 42 (2004) 415–426

Table 1
Tp

bi, N ¼ 1

bi, N ¼ 2

bi, N ¼ 3

0.01
0.1
1.0
10.0
102
103
104
105
106
107
108
109

1.502
1.5023
1.5022
1.5044
1.5644
1.5701
1.5707
1.5707
1.5707
1.5707
1.5707
1.5707

3.17331
3.4761
4.4934
4.6910
4.71026
4.71217
4.71236
4.71238
4.71238
4.71238
4.71238
4.7123

6.3465
6.8862
7.7725
7.8412
7.8527
7.853
7.8539
7.8539
7.8539
7.8539
7.85398
7.8539

A computer code in FORTRAN is developed to find the solution of the frequency
equation ((23) and (25)) and compute the value of bi. The values of bi for the
modes N ¼ 1; 2 and 3 and increasing order of Tp are given in Table 1.
By considering one end of the bellows to be fixed and the other end free, the
value of bi for Tp ) a is 1.5707 for N ¼ 1, 4.7123 for N ¼ 2 and 7.8539 for N ¼ 3,
respectively. The weight of the bellow W 0 ¼ 0, a ¼ 0 and ks and Ts ¼ 0. It is seen
that as the value of elastic restraint Tp of the pipe increases from 0.01 to 108, the
frequencies tend to increase for all the mode numbers. As the stiffness value
T ! 1, the frequency value increases by about 54% for N ¼ 1, P ¼ 0 MPa and by
69% for N ¼ 1 and P ¼ 166 MPa. However, it is observed that there is no change
in frequency and it becomes constant from T ¼ 104 onwards. The same is reflected
in Figs. 3 and 4, respectively.
4.2. Case II
If both ends of the bellows are fixed (W 0 ¼ 0 and ks ¼ 1), the frequency equation is written as




bi
1=Ts þ 1=Tp b2i a Ts Tp
tanbi ¼



1= Ts Tp 1 þ b2i =Ts a
fi ¼ 49:5i

pffiffiffiffiffiffiffiffiffiffiffi
kn =G

ð28Þ

ð29Þ

where ‘i’ is the order number of frequency, i ¼ 1; 2; 3; etc: The values of bi for the
mode numbers N ¼ 1; 2 and 3 and different values of Tp are given in Table 2.
The value of bi for clamped–clamped condition for Tp ) 1 is 3.1415 for N ¼ 1,
6.2831 for N ¼ 2 and 9.4247 for N ¼ 3, respectively. The weight W 0 ¼ 0, a ¼ 1
and ks and Ts ¼ 1.

M. Radhakrishna, C. Kameswara Rao / Thin-Walled Structures 42 (2004) 415–426

421

Fig. 3. Axial vibrations of single bellow expansion joint (one end fixed and other free) (bi for
N ¼ 1; 2 and 3, W 0 ¼ 0, Ts ¼ 0 and a ¼ 1).

4.3. Case III
This represents a practical situation where one end of the bellows is welded to a
pipe and the other end is attached to a lumped weight, W0 (W 0 6¼ 0 and ks ¼ 0). It
is commonly noticed that in a pipeline having a bellow, also has a valve for controlling the flow or an instrument for measuring the flow. Therefore, the values of
bi for this type of end condition are different from the other two cases and are
obtained by varying Tp and keeping 1/a (0.1, 1, 100 and 1000) a constant value. It
depends on the value of a. The same is computed and given in Tables 3–6, respectively. We know that a ¼ G=W 0 or 1=a ¼ W 0 =G where G ¼ G 1 þ G 2 , where G1 is
the weight of the bellow material and G2 is the weight of liquid.

Fig. 4. Axial vibrations of single bellow expansion joint (both ends fixed) (bi for N ¼ 1; 2 and 3,
W 0 ¼ 0, Ts and a ¼ 1).

422

M. Radhakrishna, C. Kameswara Rao / Thin-Walled Structures 42 (2004) 415–426

Table 2
Tp

bi, N ¼ 1

bi, N ¼ 2

bi, N ¼ 3

0.01
0.1
1.0
10.0
102
103
104
105
106
107
108
109

1.57713
1.63199
2.02875
2.86277
3.11049
3.13845
3.14127
3.14156
3.14158
3.14159
3.14159
3.14159

4.71451
4.733518
4.91318
5.76055
6.22105
6.27690
6.28257
6.28312
6.28317
6.28318
6.28318
6.28318

7.85525
7.86669
7.97866
8.70831
9.33172
9.41536
9.42383
9.42468
9.42476
9.42477
9.42477
9.42477

Table 3
1=a ¼ 0:1 and Ts ¼ 1
Tp

bi, N ¼ 1

bi, N ¼ 2

bi, N ¼ 3

0.01
0.1
1.0
10.0
102
103
104
105
106
107
108

1.435
1.4899
1.8964
2.8417
3.1101
3.13845
3.14127
3.14156
3.14158
3.14159
3.14159

4.3076
4.3239
4.4899
5.5990
6.2185
6.2768
6.2825
6.28312
6.28317
6.28318
6.28318

7.2289
7.2366
7.3172
8.228
9.323
9.4152
9.4238
9.4246
9.4247
9.4247
9.4247

Table 4
1=a ¼ 1:0 and Ts ¼ 1
Tp

bi, N ¼ 1

bi, N ¼ 2

bi, N ¼ 3

0.01
0.1
1.0
10.0
102
103
104
105
106
107
108

0.8645
0.9014
1.2077
2.5293
3.1072
3.1384
3.14127
3.14156
3.14158
3.14159
3.14159

3.42583
3.42775
3.44823
3.82916
6.18340
6.27665
6.28312
6.28317
6.28317
6.28318
6.28318

6.43733
6.43765
6.44095
6.48289
8.98748
9.42382
9.42468
9.42468
9.42476
9.42477
9.42477

M. Radhakrishna, C. Kameswara Rao / Thin-Walled Structures 42 (2004) 415–426

423

Table 5
1=a ¼ 100 and Ts ¼ 1
Tp

bi, N ¼ 1

bi, N ¼ 2

bi, N ¼ 3

0.01
0.1
1.0
10.0
102
103
104
105
106
107
108

0.1003
0.1047
0.1411
0.3311
1.0031
3.0809
3.1412
3.1415
3.14158
3.14159
3.14159

3.14477
3.14477
3.14477
3.14480
3.1455
3.2236
3.14124
3.14156
3.14158
3.14159
3.14159

6.28477
6.28477
6.28477
6.28478
6.28481
6.285316
6.283119
6.283119
6.283178
6.28318
6.28318

Table 6
1=a ¼ 1000 and Ts ¼ 1
Tp

bi, N ¼ 1

bi, N ¼ 2

bi, N ¼ 3

0.01
0.1
1.0
10.0
102
103
104
105
106
108

0.0317
0.0331
0.0447
0.1048
0.3177
1.0003
1.1123
1.3112
1.4234
1.5671

3.1419
3.14190
3.14190
3.14190
3.14191
3.14191
3.14151
3.14155
3.14158
3.14159

6.28334
6.28334
6.28334
6.28334
6.28334
6.28339
6.28308
6.28317
6.28318
6.28318

Fig. 5. Axial vibrations of single bellow expansion joint (one end of the bellows fixed and other end
attached to weight) (bi for N ¼ 1; 2 and 3 and a ¼ 10).

424

M. Radhakrishna, C. Kameswara Rao / Thin-Walled Structures 42 (2004) 415–426

Fig. 6. Axial vibrations of single bellow expansion joint (one end fixed and other end attached to
weight) (bi for N ¼ 1; 2 and 3 and a ¼ 1:0).

Tables 3 and 4 present bi values for 1/a of 0.1 and 1.0, respectively. It is seen
from Figs. 5 and 6 that the values of bi in both the cases tend to be constant from
Tp ¼ 105 onwards. However, for lower values of 1/a of 0.01 and 0.001, which
means the bellow weight is greater than the weight attached (G > W 0 ), the values
of bi are almost constant for N ¼ 1 and 2 and vary for N ¼ 1 only. The same is
shown in Figs. 7 and 8, respectively.

Fig. 7. Axial vibrations of single bellow expansion joint (one end fixed and other attached to a weight)
(bi for N ¼ 1; 2 and 3 and a ¼ 0:01).

M. Radhakrishna, C. Kameswara Rao / Thin-Walled Structures 42 (2004) 415–426

425

Fig. 8. Axial vibrations of single bellow expansion joint (one end fixed and other end attached to a
weight) (bi for N ¼ 1, 2 and 3 and a ¼ 0:001).

5. Theoretical verification
Five numbers of U-shaped bellow specimens are used that have the same geometrical dimensions as considered by Li et al. [4] and are given in Table 7. The two
specimens have austenitic stainless steel as the material of construction. Db represents the inside diameter of cylindrical tangent, w is the convolution depth, q is the
convolution pitch, n is the number of plies and t is the nominal thickness of the
bellow material before forming.
Table 7
Geometrical dimensions of bellows (in mm)
SP

Db

w

q

n

t

ˇ
N

1
6

322.5
192

24.5
26

22.4
22

1
1

0.49
0.5

9
9

SP, specimen number.
Table 8
SP
Air
1
6
Water
1

EXP

Li

Mod

%Error

37.5
40.0

40.4
38.6

40.5
39.8

0.27
0.45

27.5

27.5

25.5

7.0

426

M. Radhakrishna, C. Kameswara Rao / Thin-Walled Structures 42 (2004) 415–426

Table 8 presents the frequencies obtained by the present modified for single bellow expansion joint. The frequencies are in close agreement to the experimental
and theoretical results of Li for air and water, respectively.
6. Conclusions
A theoretical model has been developed for determining the axial frequencies.
The effect of considering the pipe as elastic and partially restrained and the influence of elastic restraints on axial frequencies has not been previously dealt. It is
found that the axial frequencies obtained from the present analysis are in close
agreement to the ones determined by Li both by theoretical formulations and
experiments. The percentage error is 0.27% and 0.45% for specimen numbers 1 and
6 in air and 7% in water.
References
[1] Gerlach CR. Flow induced vibrations of metal bellows. ASME J Eng Ind 1969;91:1196–202.
[2] EJMA. The Standards of the Expansion Joint Manufacturers Association, Inc. 6th ed. 1995.
[3] Jakubauskas VF. Added fluid mass for bellows expansion joints in axial vibrations. ASME J Pressure Vessel Technol 1999;121:216–9.
[4] Li T-X, et al. Natural frequencies of U-shaped bellows. Int J Pressure Vessel Piping 1990;42:61–74.


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