Chemical Engineering May 2012 - 67

FFT Plot for Velocity
70
60
50
40
30
20
10
Initial
Final
pressed as:
(8)
180 184 188 192
Frequency, Hz
196 200 204 208
FIGURE 11. After modiication, the maximum
vibration velocity was reduced drastically
Mathematical background
The basic pipe configuration is shown
in Figure 1. Considering the Bernoulli-Euler
formulation and structural
damping, the dynamic equation
of motion in the frequency domain [6]
is as follows:
(1)
(2)
(3)
Equation (1) pertains to steady-state
vibrations with the frequency dependence
on ω. Here, the variables Û, M0,
ML are complex, arising out of the
Fourier Transform. BCs (Equation (3))
imply that the excitation at the ends
is by moments, which is the source
of vibration of the pipe in this span.
The damping component has been expressed
in terms of the loss factor η [7],
which is a function of ω. The solution
of Equation (1) (which is also termed a
wave solution [6,7]) can be written as:
The complex coefficients can be
obtained by the following matrix
system:
GX = V
where G is the matrix
The SRSS method has been used
(9)
for the computation of the resultants
for stress and the reaction forces. This
method is simple, reasonably accurate
and also widely used. Alternatively, for
a more rigorous analysis, other combinations
for cumulative fatigue evaluation,
such as the rain-flow counting
method or the more recent Dirlik's
method [8] may be used.
The number of measurement points
(10)
(11)
(12)
Here, X is the solution vector and V
is the vector comprising the displacement
measurements (from the FFT)
at points x1 and x2 in the span. It can
be observed that the determinant of G
is nonzero. Hence, G is invertible and
X can be solved uniquely as:
X = G-1V
(13)
After the coefficients are obtained,
other response quantities like velocity
and stress can be computed. For stress,
we have the expression as follows:
(14)
(5)
The complex coefficients A,B,C and D
are independent of x, but dependent
on ω. The first two terms of Equation
(5) represent travelling waves from the
left and right ends respectively. The
last two terms represent evanescent
waves that rapidly decay away from the
boundaries. The complex wave number
k may be expressed as follows [7]:
(6)
(7)
Here, kre is the wave number for the
undamped case and kim may be exThe
stress function is a complex
quantity and has a continuous dependence
on frequency, which varies
theoretically from −∞ to ∞. However,
for practical purposes, the response is
dominated by some finite number N
modes or frequencies. We can define
the total stress as the square root sum
of squares (SRSS) combination of the
individual components.
(15)
Here, σi= σ(x,ωi). Because the quantity
is complex, the modulus has been
used for the combination. In the same
vein, the end reactions may be obtained
as follows:
(16)
may be reduced to one. This is because
of the exponential terms in the matrix
G. One of the coefficients, C or D, becomes
negligible and we are left with
three coefficients.
■
Edited by Scott Jenkins
References
1. Saha, S. Estimation of Point Vibration Loads
for Industrial Piping. Journal of Pressure
Vessel Technology, Vol. 131, 2009, ASME,
New York.
2. Moussa, W.A., Abdel Hamid. A.N. On the
Evaluation of Dynamic Stresses in Pipelines
Using Limited Vibration Measurements and
FEA in the Frequency Domain. Journal of
Pressure Vessel Technology, Vol. 121, 1999,
ASME, New York.
3. Dobson, B.J. and Rider, E., A Review of the
Indirect Calculation of Excitation Forces
from Measured Structural Response Data.
Jour. Mech. Eng. Sci. 204, 1990.
4. Wachel, J.C. Piping Vibration and Stress,
Proc. Machinery Vibration Monitoring &
Analysis, Vibration Institute, USA, 1981.
5. ASME -OM. Code for Operation and Maintenance
of Nuclear Power Plants, ASME, New
York, 2004.
6. MacDaniel, J. and others, A Wave Approach
to Estimating Frequency-Dependent Damping
Under Transient Loading, Journal of
Sound and Vibration, Vol. 231, 2000.
7. Goyder, H. Method and Applications of Structural
Modeling from Measured Structural
Frequency Response Data. Journal of Sound
& Vibration. Vol.68(2),1980.
8. Dirlik, T., Ph.D. Thesis. Application of Computers
to Fatigue Analysis, Warwick University,
1985.
Author
S. Saha is is currently the
head of the piping engineering
dept. at Reliance Refinery
(Jamnagar, India; Email:
subratap.saha@ril.com). His
area of specialization is finite
element analysis (FEA), as
well as stress and dynamic
analysis of mechanical and
structural systems. He has
wide consultancy experience
in piping design for the refinery,
petrochemical and power (both nuclear
and conventional) industries. Dr. Saha holds a
B.Tech. (Hons.) degree in mechanical engineering
from the Indian Institute of Technology
(Kharagpur, India) and a Ph.D. from the Indian
Institute of Technology (Kanpur, India). He has
several publications in international journals
and conferences.
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Peak velocity, mm/s
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