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Chellapilla 2 Vibrations of Bellows Using FEA Vetomac 2 2002 .pdf



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Title: ICONE 10
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FINITE ELEMENT ANALYSIS OF TRANSVERSE VIBRATIONS OF SINGLE
BELLOWS RESTRAINED AGAINST ROTATION

RADHAKRISHNA.M
Scientist, Design Engineering Division
Indian Institute of Chemical Technology, Hyderabad
India

KAMESWARA RAO.C
Senior Deputy General Manager
Intellectual Property Management Group
Corporate R&D,
Bharat Heavy Electricals Limited, Hyderabad
India

Keywords
Bellows vibration, single bellows, transverse frequency, elastic restraint & finite element

Abstract
The paper presents the results of investigation of transverse vibrations of single bellows
expansion joint restrained against rotation on either end using the finite element method. The
aim of this work is to model flexible U shaped bellows using beam elements. The effect of
rotatory inertia on the natural frequency is included in the beam element matrices. Results
obtained from the numerical analysis are presented and compared with the exact frequencies
for the first three modes of vibration. Experimental results are used for verification. The effect
of variation of internal pressure and velocity on the natural frequencies are also studied and
presented.

Introduction
Bellows expansion joints are commonly used in nuclear /power and chemical piping systems
so as to absorb the axial and transverse displacements due to thermal loading. In order to
design the bellows expansion joint along with the piping system for specified design response
spectra, it is essential to obtain reliable solutions for the transverse vibrations frequencies of
bellows expansion joints. Thus metal bellows have become an important component in a
number of applications.
The formulae presented in the EJMA code and those derived by investigators, [1,4] cover
classical boundary conditions only. It is observed that the flange-bellow-flange junction - as
clamped end conditions or infinitely stiff compared to the bellows stiffness is not a practical
situation compared to the bellows welded on either end in many of the piping systems.
Hence, this assumption of clamped-clamped will lead to over estimation of natural
frequencies. Rao.C.K and Radhakrishna.M [6] analyzed U-shaped bellows subjected to axial
force and internal pressure, and obtained an exact solution for the transverse vibrations of
single bellows that are elastically restrained against rotation on either end using the Bernoulli-

Euler beam theory. The paper presents a finite element representation of single bellows and
to validate the frequencies obtained by exact analysis for first three modes of vibration.
Characteristics of Bellows
The various geometrical dimensional parameters of U-shaped bellows are given in Fig 1. R1
is the meridional radius of the convolution root, R2 is the meridional radius of the convolution
crown and h is the convolution height. Rm is the mean radius of the bellows, that is, the
distance from the bellows centerline to mid convolution height, and t is the bellow material
thickness. It is assumed that t≪R1, R2 and h ≪ Rm. With ‘N’ as number of convolutions the
total length of the bellows is L = 2(R1+R2) N. Therefore, with these assumptions bellows are
considered as equivalent pipe / bar of radius Rm and wall thickness t.

Fig.1: Geometrical Dimensions of bellow

Theory of Free Vibrations of Bellows
The matrix equation for the free vibration of bellows can be written as –
[M] {q} + [K] {q} =0

(1)

Where
{q}

-Generalized coordinates

[M]

- Mass matrix

[K]

- Elastic stiffness matrix

Formulations of Elastic Stiffness Matrix
The strain energy U of a bellow element of length ‘l’ including the effect of rotatory inertia is
given by the relationl

l

U= ½ EI ∫(d y/dx ) .dx − ½ PπRm ∫(d y/d x) 2.dx
2

2 2

0

Where
y

- deflection of the bellow

2

0

(2)

EI

-bending stiffness of bellow

P

-internal pressure and

Rm

-mean radius of bellows

Now on non-dimensionalizing and substituting the following relations, η = x / l and φ = y / l the
above expression of strain energy becomes as follows –
l

l

U= ½ EI/l ∫(d φ/dη ) .dη − ½ PπRm /l∫(dϕ/dη) 2.dη
2

2

2 2

0

(3)

0

Assuming a cubic polynomial expression for ϕ to be of the form –

ϕ=∑arηr

(4)

Now substituting in to equation (1) and replacing the coefficient of ar (r=0,1,2,3), the strain
energy expression becomes -

U= ½ EI/l {ξ} r [K] {ξ}

(5)

ξ is the degree of freedom (DOF)

∴[K] = [K e] − ∆ 2 [K v]

(6)

Where

∆2 = Pπ Rm2 l2 /EI
[K]

-Elastic stiffness matrix

[Ke]

-Velocity stiffness matrix and are defined as follows –

[Ke]

=

12

-6

-12

-6

4

6

2

sym

12

6
4

[Kv]

=

-36

-3

-36

-3

4

-3

-1

sym

-36

3
4

Where

ξ T = {ϕi, ϕi’, ϕi+1, ϕ’ i+1}

(7)

Formulation of Mass and Rotatory Inertia Matrices
The kinetic energy ‘T’ of a bellow element of length including the effects of rotatory inertia is
given by –

l



l



T= ½ ρ A l ∫ (ϕ) dη + ½ ρ J l ∫(ϕ’) 2 dη
3

2

0

(8)

0

The mass of bellows per unit length is computed by using the formula given in equation (9)-

m= ρb 2π Rm [π (R1+R2)+2(h-R1-R2)]t ÷2((R1+R2)

(9)

Where
ρb
- mass density of bellows material
ρf

- mass density of fluid flowing through bellows

A

- area of cross-section of the bellow

Now substituting for ϕ from equation (2) and replacing the coefficient a r (r=0,1,2,3) by the
nodal coordinates the expression becomes-

.
.
T
T= ½ ρAl {ξ} + [M]{ξ}
3

(10)

Where

[M] = [M1] + R λ4 [M2]
R = J / ml2
λ4 = (m l4 Pn2) / EI
The mass moment of inertia of the bellow per unit length is given by-

J = Jxx = Jyy = π Rm3 [(2 h /q + 0.571) t ρb + h/q (2R2 –t) ρf]

(11)

Now the mass matrices M1 and M2 are defined as follows156
M1 = 1/420

22

54

-13

4

13

-3

156

-22
4

-36
M2 = 1/30

-3

-36

-3

4

-3

-1

sym

36

3
4

The matrix equation for free vibration of single bellows is given by-

{[K] - λ 4 [M]} {ξ} = 0
Where

λ 4 = {ρ A l4 Pn2 } / EI

(12)

The matrix eigenvalue equation is solved for the bellows restrained against rotation using the
following boundary conditions

Boundary Conditions
At the end ’A’ - bellows are connected to a pipe nipple, and are considered to have a
rotational stiffness of R1 and R2 at either end as shown in figure 2.

2

1

Fig. 2: Mathematical model of single expansion
joint fixed at both ends (Lateral mode)

At the end A (x = 0)
X (0) = 0 and

(13)

∂2 X (0)
∂ X (0)
---------- = T1 ------∂x 2
∂x

(14)

At the end B (x = L) the boundary conditions are given by X (L) = 0 and

(15)

∂2 X (L)
∂ X (L)
------------ = T2 ------∂x 2
∂x

(16)

T1 =SR1 L / EI

(17)

T2 = SR2 L / EI

(18)

Results and Discussion
A single bellow with the following geometrical and physical parameters is considered –bellows
length L = 0.0693m, mass moment of inertia per unit length, J = 0.001153kgm, EI = 5.078Nm
and total mass m

tot

= 5.138kg/m respectively. The geometrical properties data specified

above is taken from the thesis of Jakubauskas.V.F (5). The Jacobi method is used for solving
the frequency equation.

The first three modes of lateral natural frequencies are obtained for internal pressure of
P=166.0 MPa and are given in Tables 1 & 2 and the same is graphically represented in
figures 3 & 4 respectively. Firstly, it is seen that as internal pressure of the bellows increases,
the lateral mode natural frequency decreases. For example at P =0.0MPa and mode number

N=1, natural frequency obtained is ω =1690.20 rad/s and at P=166.0 MPa and N=1, and
ω=1069.06 rad/s a drastic drop by about 37%. The same is true for even higher order of
mode numbers of N=2 & 3 respectively.

It is also seen that as the value of rotational restraint T (T=T1=T2) -increases from 0.01 to
10

10 , the frequencies tend to increase for all the mode numbers. As the rotational stiffness
value T→α, the frequency value increases by about 54.1% for N=1, P=0.0 MPa and by
68.56% for N=1 & P=166.0 MPa. However, it is observed that there is no change in frequency
4

and it becomes constant from T=10 onwards.

Table1: Natural Frequencies for T=T1=T2 and N=1 to3 & PR=0.0 MPa
#

T

N=1

N=2

N=3

1

0.01

1690.2

4846.58

8112.22

2

0.1

1717.17

4866.0

8126.83

3

1.0

1927.65

5028.8

8252.68

4

10

2821.56

5938.55

9104.02

5

10

2

3544.04

6944.05

10343.6

6

103

3669.43

7138.60

10608.4

7

10

4

3682.84

7159.61

10637.1

10

5

3684.19

7161.73

10640.0

10

6

3684.33

7161.94

10640.3

10

10

7

3684.34

7161.94

10640.3

11

108

3684.34

7161.94

10640.3

12

9

3684.34

7161.94

10640.3

1010

3684.34

7161.94

10640.3

8
9

10

13

F ig 3: Transverse N atural Frequencies of R otationally R estrained Single B ellow s

SR = 0.01 to 1.E +10, Pressure PR = 0.0 M Pa, R IX = 0.04673 & V = 0 m /s
12000
11000
10000

8000
7000
6000
5000
4000

frequency, rad/s

9000

3000
2000
1000
0
1.00E-02

1.00E +00

1.00E +02

1.00E +04

1.00E +06

R otational stiffness, S R

1.00E +08

1.00E+ 10

N=1
N=2
N=3

Table2: Natural Frequencies for T=T1=T2 and N=1 to 3 & PR=166.0 MPa
#

T

N=1

N=2

N=3

1

0.01

1069.06

4468.55

7837.89

2

0.1

1111.2

4489.69

7852.98

3

1.0

1414.32

4665.5

7983.22

4

10

2486.92

5631.27

8862.16

5

10

2

3256.51

6672.85

10131.0

6

103

3385.25

6871.85

10400.1

7

10

4

3398.95

6893.31

10429.2

10

5

3400.33

6895.47

10432.1

10

6

3400.47

6895.69

10432.4

10

7

3400.48

6895.71

10432.5

10

8

3400.48

6895.71

10432.5

12

10

9

3400.48

6895.71

10432.5

13

1010

3400.48

6895.71

10432.5

8
9
10
11

Fig 4: Transverse frequencies of Rotationally Restrained Singles
Bellows for SR=0.01 to 1.E+10 and Pressure PR =166.0 MPa &
V=0 m/s, RIX=0.04673

1.E+04
8.E+03
6.E+03
4.E+03
2.E+03

1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

1.E+08

Rotational Stiffness, SR

0.E+00
1.E+10

Frequency, rad/s

1.E+04

N=1
N=2
N=3

Table 5 gives a comparison of the exact frequency solutions obtained using the finite element
method for a single bellows elastically restrained at both ends with rotational restraint vis-a-vis
to the results presented in the thesis [5] and in paper [6] by the present authors for a
rotationally restrained case.

Table 5 Comparison of Frequency Solutions for T=α and Pmax=166MPa
Mode
#

Exact [6] ω,
rad/s

Thesis [5] ω,
rad/s

FEA, ω
rad/s

Error, %

1

3400.159

3400.334

3400.48

0.004

2

6890.181

6890.356

6895.71

0.077

The first two natural frequencies are calculated for the bellows data as given above using the
frequency formula derived in equation (12) using the FE analysis. It is seen that the results
obtained by the FEA method is found to be in good agreement with the exact method. The
percentage error in the frequency obtained from the present analysis and the exact bisection
method is less and about 0.004%. Therefore, since the percent of error is found to is less it is
precise enough to estimate the natural frequency of single bellows expansion joint using this
formula.

The effect of variation of velocity of flow at V= 1.0, 5.0 and 10 m/s on the first three
frequencies is studied. It is seen that for a constant value of rotational stiffness SR = 10

10

the

frequency increases with increase in the velocity of flow and also with increase in mode
numbers.

Table 6 Frequencies at different velocities and constant value SR= 1010
Mode

V=1.0

V=5.0

V=10.0

Frequency, Hz

Frequency, Hz

Frequency, Hz

N=1

0.08

2.0

7.4

N=2

0.16

4.0

15.5

N=3

0.25

6.0

23.7

Number

It is seen that the effect of variation of pressure and velocity of flow on the natural frequency
is marginal because the dimensions of bellows given in thesis (5) are small. The effect of
variation of pressure and flow velocity on the transverse natural frequencies for modes N=1, 2
& 3 is significant. In order to demonstrate this clearly a large dimensioned bellow is
considered that has the following geometric properties -Db= 1.2m, Rm = 0.6395m, BL=
2

0.254m, EI=5449Nm , J=1485.36kgm and mtot= 912.13kg/m respectively.

Firstly, the effect of velocity of flow for V=1.0 m/s and 5.0 m/s is studied on the natural
frequency. It is found that for a constant value of SR→∝ and velocity of flow increasing the
natural frequency decreases. It is also observed that the frequency, ωn increases with
increase in the mode number N=1, 2 & 3. Table 7 shows the frequencies in radians/s
obtained for different velocities of flow of liquid inside the bellow and figures 5 & 6 depict the
same.

Table 7 Frequencies at SR=1010=∝ & varying velocities
Velocity

V=1.0

V=2.0

V=3.0

V=4.0

V=5.0

N=1

46.71

11.53

5.054

2.8

1.76

N=2

67.38

16.75

7.39

4.13

2.62

N=3

94.77

23.61

10.46

5.867

3.742

100
80
60
40
20
0
1.E-02 1.E+0 1.E+0 1.E+0 1.E+1
1
4
7
0

frequency, rad/s

Fig 5: Transverse Frequency of Single Bellows
Expansion Joint for V = 1.0m/s and N=1,2 &3 &
SR=infinity

N=1
N=2
N=3

Rotational Stiffness, SR

Fig 6: Transverse frequency of Single Bellows Expansion joint
for SR=infinity and V=5.0m/s, N=1, 2 &3
4

3
2.5
2
1.5
1

frequency, radian/sec

3.5

0.5

1.00E-02

1.00E+00

1.00E+02

1.00E+04

1.00E+06

Rotational Stiffness, SR

1.00E+08

0
1.00E+10

N=1
N=2
N=3

Similarly, the effect of variation of internal pressure on natural frequency is investigated for
the same geometrical dimensions of bellows. It is found that for SR→∝ and pressure
increasing from 10.0 MPa to 150.0 MPa, the transverse natural frequencies for the first three
modes of vibration decreases. Table 8 shows the frequencies obtained at different internal
pressures of bellows.

Figures 7 to 10, shows the frequencies obtained by varying the pressures by an interval of
50.0 MPa. It is seen that for P=10.0 MPa & N=1, the percentage increase in frequency for
6

SR=0.01 to SR=10 is 50% and then the frequency remains constant as SR approaches
infinity. At same pressure of 10MPa and N=2, the percentage increase in frequency for
6

SR=0.01 to 10 is 30% and for N=3 the percentage increase in frequency is 25%. Similarly for
a maximum pressure of 160MPa and N=1, the percentage increase in frequency for SR=0.01
6

to 10 is 69%. At same pressure of 160MPa and N=2, the percentage increase in frequency
6

for SR=0.01 to 10 is 33% and for N=3 the percentage increase in frequency is 26%.

Table 8 Frequencies at SR=1010=∝ & varying pressures
Pressure

PR=10MPa

PR=50MPa

PR=100MPa

PR=150MPa

N=1

105.35

20.58

9.97

6.43

N=2

151.41

29.94

14.75

9.69

N=3

212.60

42.28

20.98

13.81

Fig 7: Transverse Frequency of Single Bellows Expansion Joint
for SR=infinity and PR=10.0 MPa, N=1,2&3

200
150
100
50

1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

Rotational Stiffness, SR

1.E+08

Frequency, radian/s

250

0
1.E+10
N=1
N=2
N=3

Fig 8: Transverse Frequency of Single Bellows Expansion Joint
for SR=infinity, PR=50M Pa and N=1,2 &3

45
40
35
30
25
20
15
10
5
N=1
0
1.E-02

1.E+00

1.E+02

1.E+04
1.E+06
Rotational Stiffness, SR

1.E+08

N=2

1.E+10

N=3

Fig 9: Transverse Frequency of Single Bellows Expansion Joint
for SR=infinity, PR=100.0 MPa and N=1,2&3

20
15
10
5

1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

Rotational Stiffness, SR

1.E+08

0
1.E+10

Frequency, radian/s

25

N=1
N=2
N=3

Fig 10: Transverse Frequency of Single Bellows Expansion Joint
for SR=infinity and PR=150.0 MPa and N=1,2 &3
16

12
10
8
6
4

Frequency, rad/s

14

2
1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

1.E+08

0
1.E+10

Rotational Stiffness, SR

N=1
N=2
N=3

Conclusions
Theoretical and experimental comparisons confirm that the finite element method developed
used in this study is well adapted to the dynamic response of bellows gives fairly exact
results. From the various results presented in this paper the following can be concluded-

i.

As the internal pressure of the bellow is doubled the transverse frequency of
vibrations for the first three mode numbers is reduced by nearly 50%.

ii.

Similar kind of analysis has been done to study the effect of velocity of flow on
frequency. It is seen that as the flow velocity is doubled the frequency decreases by
about 4 times for all the first three modes of vibration.

References
1. EJMA – “The Standards of the Expansion joint Manufacturers Association Inc”. New York,
ASME, 1984, p221.
2. Li Ting-Xin, Li Tian-Xiang and Guo Bing-Liang, “Research on Axial and Lateral Natural
Frequencies of Bellows with different end conditions”, International Symposium 86 PVP14, ASME, p367-373.
3. Jakubauskas V.F and Weaver. D.S, “Transverse Vibrations of Bellows Expansion Joints
Part-II: Beam Model Development & Experimental Verification”.
4. Jakubauskas V.F, “Practical Predictions of Natural Frequencies of Transverse Vibrations
of Bellows Expansion Joints”, Mechanika-Kaunas, Technologija, 1998, No.3 (14), p47-52.
5. Jakubauskas.V.F, “Transverse Vibrations of bellows Expansion Joints”- PhD. Thesis –
Hamilton, Ontario, Canada: McMaster University, 1995, PP145-150.
6. Rao.C.K and Radhakrishna.M, “Transverse Vibrations of Single Bellows Expansion Joints
restrained against Rotation”, Presented at Tenth International Conference on Nuclear
Engineering, April 14-18, 2002, USA.


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