THEORETICAL STUDY ON PIPE OF TAPERED THICKNESS WITH AN INTERNAL FLOW TO ESTIMATE NATURAL FREQUENCY

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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 7, Issue 2, March-April 2016, pp. 105–120, Article ID: IJMET_07_02_014
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THEORETICAL STUDY ON PIPE OF
TAPERED THICKNESS WITH AN
INTERNAL FLOW TO ESTIMATE
NATURAL FREQUENCY
Nawal H. Al Raheimy
College of Engineering / Mechanical Department,
Babylon University / Babil-Iraq
ABSTRACT
This research study the effect of tapered thickness on the free transverse
vibration of clamped – free pipe which have uniform circular cross section
conveying water by using Raighly –Ritz method in the two case, the first
involves the pipe have a constant wall thickness (t1) at clamped end equal to
(1mm & 2mm) while the thickness (t2) at free end changes according to the
ratio (t2/t1=0.2, 0.4, 0.6, 0.8, 1). In the second case the thickness at free end
(t2) is constant (1mm & 2mm) whereas the thickness at clamped end (t1)
changes at ratio (t1/t2=0.2, 0.4, 0.6, 0.8, 1). The pipe has a constant inner
radius (Ri) of (1 cm or 2 cm) and different values of length (1m & 2m). This
study shows in the 1st
case the critical velocity (Vc) of the fluid can be
decreased at the increase in length of the pipe at the same values of Ri & t1 but
the value of critical velocity is increased with increasing t1, Ri and the
thickness ratio (t2/t1) at the same length of the pipe. In addition at absence the
flow of water the natural frequency of system is decreased with the increase in
the ratio of thickness (t2/t1) and length of pipe, whereas Ri and t1 are
decreased. In the 2nd
case the dynamic behaviors of the system at the same
that in the 1st
case except that the natural frequency increase with increasing
the thickness ratio t1/t2. At any formation of the pipe for uniform section the
natural frequency decreased when the velocity of water increased from zero to
critical velocity. Results are compared with those available in literature and
are found to be in excellent agreement.
Key words: Cantilever Pipe, Internal Flow, Tapered Thickness
Cite this Article: Nawal H. Al Raheimy, Theoretical Study on Pipe of
Tapered Thickness with an Internal Flow to Estimate Natural Frequency
International Journal of Mechanical Engineering and Technology, 7(2), 2016,
pp. 105–120.
http://www.iaeme.com/currentissue.asp?JType=IJMET&VType=7&IType=2

Nawal H. Al Raheimy
1. INTRODUCTION
The vibrations resulting from fluid flow causing noisy problems occur in a wide range
in industrial field from civil engineering, chemical processing, aerospace and marine
structures. Nabeel and et.al. [1]
, fluid flow and the pipe line will be an interactive
system dynamics where coupled by the force of fluid exerted on the pipe this force
causes the deformation pipe thus change the direction the flow also change fluid
force. Chol [2]
investigated the natural frequencies of piping system under effect fluid
velocity and coriolis force. It is obtained that at certain critical velocities causing
buckling–type instability for different boundary conditions. Alaa [3]
studied the effect
of the fluid flow through a pipe with restriction affect the dynamic behavior on the
vibration of system. Wang [4]
investigated the static and dynamic behavior pipes
conveying fluid mathematically by using finite difference method. Shintaro [5]
investigated experimentally the vibration of hanging tube conveying fluid with
varying the length of the tube. Marijonas [6]
investigated flow induced vibration in
rotation pipe conveying fluid in hypothesis that the fluid is incompressible and in
viscid by using non linear equations of motion which is derived by finite elements
method. Kuiper[7]
gave analytical proof of stability of pipe transmitted fluid in
clamped pinned by at low speed by using a plug flow model after consideration a
tensioned Euler – Bernoulli beam in arrangement. Muhsin [8]
studied the effect of
boundary conditions of pipes on the natural frequency of the system conveying fluid
at different diameter, length, pipe materials and velocity of fluid by using beam
theory. Ivan[9]
investigated the flow induced vibration at uniform and tapered
thickness in different boundary conditions (clamped – clamped & pined – pined ) by
using finite elements method. A. Marzani and et.al. [10]
used Winkler – type elastic
foundation to study it is effect on the stability pipe fluid conveying fluid at transverse
motion to determine the flutter velocity. Ali [11]
studied the dynamic manners of a pipe
transmission fluid at laminar flow taking into consideration general boundary
conditions as complaint material with linear and rotational springs. Shankarachar
and et.al. [12]
investigated the dynamic behavior of pipe conveying fluid the frequency
equations is derived for classical boundary conditions where the frequency of system
decreased with increasing the velocity of flow.
In this paper, can be obtained the frequency by using approximate form which
represented by Rayleigh – Ritz method of cantilever pipe with an internal flow which
have tapered thickness in the two cases which have different ratio between thickness
at clamped and free end, estimated the natural frequency of vibrations at different
values of inner radius, the thickness at clamped and free end, different values of
velocity flow of water and different values of length.
2. THEORETICAL ANALYSIS
Figures (1) show the uniform cross section of clamped – free pipe at tapered thickness
of length L, inner radius Ri, the thickness at clamped end t1, and at free end t2 can be
derived

Theoretical Study on Pipe of Tapered Thickness with An Internal Flow To Estimate Natural
Frequency
Fig (1-a) Cantilever pipe of tapered Fig (1-b) Cantilever pipe of tapered
Thickness t2/t1 ≤ 1 Thickness t 1/t 2≤1
From Fig.(1-a) :- (tx – t2 ) / (L-x) = (t1 – t2) /L (1-a)
From fig (1-b):-(tx – t1 ) / x = (t2 – t1) /L (1-b)
After simplify above relations yields:
tx = t1(1-x/L )+ t2 (x/L) (2)
In tapered thickness of pipe at length of part of pipe (x), A(x) = π ( 22
ixo RR  ) = 2 π Ri tx,
where Rxo = (Ri+tx), therefore mp(x) = ρp * A(x), and
I(x) = π/4( 44
ixo RR  ), therefore  43223
464
4
)( xixixix tRtRtRtxI 

.
Now the procedure of Rayeigh-Ritz to is applied derive the natural frequency for
transverse motion of tapered cross section of cantilever pipe. Let us use the simple
two term approximation Benoray [13]
.
)()( 2211 xycxycYr  (3)
3
2
2
1 












L
x
c
L
x
cYr (4)
By using above equations the values of mij and kij can be estimated Benoraya[13]
:

L
jiji dxyyxmm
0
)( (5)
''
0
''
)( j
L
iji yyxIEk  (6)
After integration equation (5) according to pipe where the pipe is empty from fluid
can be yielded:-
m11p = 2π p Ri L [t1/30 + t2/6], m12p = 2π p Ri L [t1 /42 + t2/7]
m22p = 2π p Ri L [t1/56 + t2/8], m12p = m21p. (5-a)

Nawal H. Al Raheimy
fm (x) = f *Af (x) therefore fm (x) = π f Ri
2
Now after using equation (5) and integration yields:
{m11f =mf* L/5 , m12f =mf *L/6 =m21f, m22 =mf* L/7 } (5-b)
Now the employment superposition between equations (5-a) and equations (5-b)
will be obtained:
fp mmm 111111  , fp mmm 121212  , fp mmm 212121  , fp mmm 222222  (7)
After integration of equation (4) the following relations which represent the
stiffness of pipe as follows:-
   
 
  

















4
22
3
1
2
2
2
12
3
1
4
1
3
2
2
212
2
1
3
1
3
221
2
1
2
21
3
311
5
1
22
tttttttt
ttttttR
ttttRttR
L
E
k i
ii
p

(8)














































4
22
3
1
2
2
2
1
1
102
3
1
4
1
3
2
2
212
2
1
3
1
3
221
2
1
2
21
3
312
6
1
15
2
15
1
30
1
5
1
20
3
10
1
20
1
4
4
1
6
1
12
1
6
3
1
6
1
4
3
tttttttt
ttttttR
ttttRttR
L
E
k i
ii
p

(9)
pp kk 1221  (9-a)














































4
22
3
1
2
2
2
12
3
1
4
1
3
2
2
212
2
1
3
1
3
221
2
1
2
21
3
322
7
1
42
4
105
6
35
1
105
1
3
2
5
2
5
1
15
1
5
6
5
3
5
1
3
1
9
tttttttt
ttttttR
ttttRttR
L
E
k i
ii
p

(10)
Now can be write the other relations of mass and stiffness in the matrix form as
follow :






12
2
12
11
2
11
mk
mk
np
np








22
2
22
12
2
12
mk
mk
np
np








2
1
c
c
=






0
0
(11)
or in general matrix notation as :
       02
 cMK n (12)
The evaluation of this determinant provides an estimation of the two fundamental
natural frequencies ω1
2
and ω2
2
for the pipe carrying fluid which is not moved. In
order to complete the natural frequency of pipe when the fluid moves at any velocity,

Frequency
firstly the critical velocity of flow should be determined for uniform cantilever pipe
from the flowing equation Ivan[9]
,
Vc =
L
875.1
ff AIE / (13)
Thus the natural frequency (ω) of a pipe at any velocity of fluid can be found from
the following equation:
2
1 








c
f
n V
V


Blivens [14]
(14)
Thus Vf is represented the velocity of the flow.
3. RESULTS AND DISCCUSION
The construction of pipes effect on the practical application subsequently effect on
quality performance. Table (1) shows comparison of the natural frequency of the first
mode for transverse free vibrations of pipe comparison of the natural frequency of the
first mode in the different values of velocity of flow from Vf =0 to Vf = Vc where
those are compared between the Rayleigh-Ritz method in the present work and the
finite element method (FEM) in Ivan[9]
at clamped-free boundary for uniform pipe,
D=0.01m, t=0.0001m, L=2m. The results based on the main properties of material
E=207 Gpa, ρ=8000 kg/m3
. Figures (2 &3) show that the first mode of vibration of
tapered thickness in absence flow (Vf =0) as a function of the ratio (t2/t1) obtain for the
Rayleigh – Ritz method for variation values of inner radius (Ri), the length of pipe (L)
and thickness at clamped end ( t1). It is clearly seen that the natural frequency
increased with the increased in thickness (t1) and the inner radius (Ri), This manners
illustrated the strain energy of structure increased with increase in the thickness and
the radius therefore that is caused increased the stiffness of system. In the same
figures the natural frequency decreased with increase in the ration of thickness (t2/t1)
and the length (L) that is causes increasing in the mass for the pipe and the water
which caused an increase in the kinetic energy of the structure. In the figures ( 4 to 9)
the natural frequency as a function with the velocity of flow (Vf) of water for of pipe
at different values of (t2/t1), Ri, ti &L where the natural frequency decreased with
increased in the velocity of flow. Figures (10 & 11) show that the first mode of
vibration of tapered thickness in absence flow (Vf =0) as a function of the ratio (t1/t2)
for variation values of inner radius (Ri), the length of pipe (L) and thickness at free
end ( t2). Can be seen that the natural frequency increased with the increase in
thickness (t2), the inner radius (Ri) and ratio of thickness (t1/t2).This behavior
illustrated the strain energy of structure increased with increase in the thickness and
the radius. In the same figures the natural frequency decreased with increased in the
length of pipe as like in the above case. Figures (12 to 19) show the natural frequency
also decrease with increased the velocity of flow of all different structures of pipe
because of velocity of flow impose pressure on the wall and caused deformation of
the pipe therefore caused decrease in the elasticity of pipe and the natural frequency
of the system.

Nawal H. Al Raheimy
Table 1 Natural frequency (rad/sec) of transverse vibrations of pipe in different value of
velocity of flow
Velocity Vf (m/s) R.R.M. F.E.M. Difference δ %
0 7.99 7.794 2.45 %
2 7.81 7.5968 2.73 %
4 7.26 6.9807 3.847 %
6 6.23 5.8549 6.021 %
8 4.39 3.8825 11.56 %
9 2.47 1.9897 19.445 %
Vc=9.5872 0 0 0
δ= [(R-Ritz method– FEM method)/ R-Ritz method] *100%
Figure 2 Natural frequency for 1st mode as a function of thickness ratio (t2/t1) in different
values of radius & length, absence flow and t1=1mm.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Thickness ratio (t2/t1)
0
40
80
120
160
200
240
280
Naturalfrequencywn(rad/sec)
1st. mode, t1=1mm, Vf=0
L=1m, Ri=0.01m
L=1m, Ri=0.02m
L=2m, Ri=0.01m
L=2m, Ri=0.02m

Frequency
Figure 3 Natural frequency for 1st
mode as a function of thickness ratio (t2/t1) in different
mode as a function of velocity of flow Vf in different
values of thickness ratio (t2/t1) at one meter length, radius =0.01m and t1=1mm.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
25
50
75
100
125
150
175
200
225
250
275
300
325
350
L=1m, Ri=0.01m
L=1m, Ri=0.02m
L=2m, Ri=0.01m
L=2m, Ri=0.02m
0 20 40 60 80 100 120 140
Velocity of flow Vf (m/sec)
0
20
40
60
80
100
120
140
160
180
L=1m, Ri=0.01m, t1=1mm
t2/t1=0.2
t2/t1=0.4
t2/t1=0.6
t2/t1=0.8
t2/t1=1

Nawal H. Al Raheimy
values of thickness ratio (t2/t1) at two meter length, radius =0.01m and t1=1mm.
0 20 40 60 80 100 120 140
Velocity of flwo Vf (m/sec)
0
20
40
60
80
100
120
140
160
180
200
L=1m, Ri=0.02m, t1=1mm
t2/t1=0.2
t2/t1=0.4
t2/t1=0.6
t2/t2=0.8
t2/t1=1
0 5 10 15 20 25 30 35 40 45 50
0
5
10
15
20
25
30
35
L=2m, Ri= 0.01m, t1=1mm
t2/t1=0.2
t2/t1=0.4
t2/t1=0.6
t2/t0.8
t2/t1=1

Frequency
0 10 20 30 40 50 60 70
0
10
20
30
40
50
60
L=2m, Ri=0.02m, t1=1mm
t2/t1=0.2
t2/t1=0.4
t2/t1=0.6
t2/t1=0.8
t2/t1=1
0 10 20 30 40 50 60 70 80
0
5
10
15
20
25
30
35
40
45
L=2 m, Ri=0.01m, t1=2mm
t2/t1=0.2
t2/t1=0.4
t2/t1=0.6
t2/t1=0.8
t2/t1=1

Nawal H. Al Raheimy
0 10 20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
60
70
L=2m, Ri=0.02m, t1=2mm
t2/t1=0.2
t2/t1=0.4
t2/t1=0.6
t2/t1=0.8
t2/t1=1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
20
40
60
80
100
120
140
160
180
200
220
240
L=1m, Ri=0.01m
L=1m, Ri=0.02m
L=2m, Ri=0.01m
L=2m,Ri=0.02m

Frequency
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
25
50
75
100
125
150
175
200
225
250
275
300
L=1m, Ri=0.01m
L=1m, Ri=0.02m
L=2m, Ri=0.01m
L=2m, Ri=0.02m
0 10 20 30 40 50 60 70 80 90 100
0
20
40
60
80
100
120
L=1m, Ri=0.01m, t2=1mm
t1/t2=0.2
t1/t2=0.4
t1/t2=0.6
t1/t2=0.8
t1/t2=1

Nawal H. Al Raheimy
0 20 40 60 80 100 120 140
Velocity of floww Vf (rad/sec)
0
20
40
60
80
100
120
140
160
180
L=1m, Ri=0.01m, t2=2mm
t1/t2=0.2
t1/t2=0.4
t1/t2=0.6
t1/t2=0.8
t1/t2=1
0 20 40 60 80 100 120 140
0
20
40
60
80
100
120
140
160
180
200
L=1m, Ri=0.02m, t2=1mm
t1/t2=0.2
t1/t2=0.4
t1/t2=0.6
t1/t2=0.8
t1/t2=1

Frequency
0 20 40 60 80 100 120 140 160 180 200
0
25
50
75
100
125
150
175
200
225
250
L=1m, Ri=0.02m, t2=2mm
t1/t2=0.2
t1/t2=0.4
t1/t2=0.6
t1/t2=0.8
t1/t2=1
0 5 10 15 20 25 30 35 40 45 50
Velocity of floww Vf(m/sec)
0
5
10
15
20
25
30
35
L=2m, Ri=0.01m, t2=1mm
t1/t2=0.2
t1/t2=0.4
t1/t2=0.6
t1/t2=0.8
t1/t2=1

Nawal H. Al Raheimy
0 10 20 30 40 50 60 70
Velocity of floww Vf (m/sec)
0
5
10
15
20
25
30
35
40
45
Naturalfequencywn(rad/sec)
L=2m, Ri=0.01m, t2=2mm
t1/t2=0.2
t1/t2=0.4
t1/t2=0.6
t1/t2=0.8
t1/t2=1
0 10 20 30 40 50 60 70
0
5
10
15
20
25
30
35
40
45
50
Naturalfrequencuwn(rad/sec)
L=2m, Ri=0.02m, t2=1mm
t1/t2=0.2
t1/t2=0.4
t1/t2=0.6
t1/t2=0.8
t1/t2=1

Frequency
values of thickness ratio (t1/t2) at tw0 meter length, radius =0.02m and t2=2mm.
4. CONCLUSION
The flowing conclusions can be deduced from the results of the present study, the
natural frequency of pipes conveying flow of fluid at thickness ratio t2/t1≤1 decrease
with increased the ratio of thickness against that pipes which thickness ratio t1/t2≤1
where the natural frequency increased with increasing the ratio of thickness. In the
other hand the increasing of inner radius of the system will rise the natural frequency
but the increasing the length of the pipe caused reduced the natural frequency also
increasing the velocity of flow caused decreasing the frequency of the system.
LIST OF SYMBOLS
A1 Cross section area at of pipe clamped end (m2
).
A2 Cross section area of pipe at free end (m2
)
A(x) Cross section area of pipe at part of length (x) (m2
)
Af Cross section area of fluid (m2
)
c1 & c2 Constants
E Modulus of elasticity (N/m2
)
L Length of the pipe (m)
I Second moment of area (m4
)
I(x) Second moment of area at part of length(x) (m4
)
mf Mass of fluid per unit length (kg/m)
mp (x) Mass of pipe per part of length x (kg/m)
t1 Thickness of pipe at clamped end (mm)
t2 Thickness of pipe at free end (mm)
tx Thickness of pipe at any part of length of pipe
Ri Inner radius of pipe (m).
Ro1 Outer radius of pipe at clamped end (m)
0 10 20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
60
70
L=2m, Ri=0.02m, t2=2mm
t1/t2=0.2
t1/t2=0.4
t1/t2=0.6
t1/t2=0.8
t1/t2=1

Nawal H. Al Raheimy
Ro2 Outer radius of pipe at free end (m)
Rxo Outer radius of pipe at part of length x
Vf Velocity of fluid (m/sec)
Vc Critical velocity of fluid flows in the pipe (m/sec).
x Length of part of pipe (m).
Yr Displacement (amplitude of pipe (m)
ρp Mass density of pipe material (kg/m3
)
ρf Mass density of fluid in the pipe (water) (kg/m3
)
ω Natural frequency of pipe at velocity of flow Vf (rad/sec)
ωn Fundamental natural frequency of pipe in absence of flow (rad/sec)
REFRENCESES
[1] Nabeel K. Abid Al-Sahib a, Adnan N. Jameel b, Osamah F. Abdulateef a*,
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