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1. Sule, S, Nwofor T.C / International Journal of Engineering Research and Applications (IJERA)
ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2306-2310
Improving Natural Frequency Of Vibration In Overhead
Transmission Lines With Concentrated Masses
Sule, S and Nwofor T.C
Department of Civil and Environmental Engineering, University of Port Harcourt,
Rivers State, Nigeria. P.M.B 5323
ABSTRACT
In this paper, the natural frequencies of concentrated masses are assumed to undergo vertical
vibration of a transmission line subjected to three oscillation about their mean positions. The degree of
unequal and equal concentrated masses at equal freedom of the transmission line is equal to the
interval are compared. The natural frequencies of number of concentrated masses.
vibration in both cases are predicted based on the In this paper, the dynamic analysis of a
assumption that the total kinetic and potential single span transmission line subjected to unequal
energy of the vibrating masses is constant in the and equal concentrated masses is carried out using
course of the system’s oscillation. The natural energy approach. The results of the dynamic analysis
frequencies of vibration of a transmission line of a transmission line subjected to unequal
subjected to unequal masses (control) were concentrated masses are serving as the control points.
compared with those of equal concentrated The formulated energy model is computationally
masses. The natural vibration frequency was simple and can be handled manually most specially
found to increase by 29% in the first vibration when fewer number of concentrated masses are
mode, 27% in the second vibration mode but involved.
deceased by 4.96% in the third vibration mode
when the transmission line was subjected to equal 2.0 MODEL DEVELOPMENT
concentrated masses. Consider a transmission line of span L
suspended between two transmission towers and
Keywords: Natural frequencies, transmission line, carrying concentrated masses m1 , m 2 , . . . , m n as
degree of freedom, concentrated masses, vibrating
shown in Figure 1. The tension T in the transmission
masses.
line is assumed to be constant in the course of the
system‟s oscillation.
1.0 INTRODUCTION
An overhead transmission line is the
medium through which electricity moves from the m1 m2 m3 m4 mn
point of generation to the points of utilization. The
distribution system moves electricity from the
transmission line to where it is used by customers at
home and business areas. Transmission lines are
made from cables of aluminium alloy which are
suspended by towers in a row.
Vibration of transmission lines in due to
wind excitation causes oscillation of large amplitude
in overhead transmission lines [1-8]. This large
amplitude vibration is a very dangerous phenomenon
that causes instability of the overhead transmission L
lines [9-13]. For example, the large amplitude
displacement of transmission lines resulting from Figure 1: A transmission line and subjected to n
wind excitation normally occurs when one of the concentrated masses
natural frequencies of vibration is excited leading to
resonance. This short circuits the overhead
m1 , m 2 , . . . , m n .
transmission lines as a result of entanglement of Let X i , i  1, 2, 3, . . . , N represents the chosen
lines. Dynamic analysis of a transmission line coordinate that describes the configuration of the
subjected to wind induced forces is of paramount
importance to engineers as the end result is transmission line in Figure 1. X i is assumed to be
devastating to human lives. zero at equilibrium position. During self-excited
The lumping of concentrated masses on the vibration, the various parts of the above transmission
transmission line as vibration dampers results in line undergo instantaneous velocities given by:
discretizing the transmission line in to segments. The
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2. Sule, S, Nwofor T.C / International Journal of Engineering Research and Applications (IJERA)
ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2306-2310
Vi  X 1 , X 2 ,...,X N 
   T 
2X T 0
(1) (11)
For small amplitude vibration of the or
transmission line, the potential and kinetic energy of 
AX  BX  0
vibration can be approximated by a quadratic surface (12)
given by: Let
X (t )  y j cos t
n n
K .E    a ij X i X j
 
i j (13)
(2) be the solution of equation (12).
n n where:
P.E   bij X i X j yj = Amplitude of displacement of a particular
i 1 j 1 concentrated mass.
(3)  = Natural vibration frequency of a transmission
In matrix form, equation (2) and (3) can be written line.
as: X (t ) = - y j sin t

 
K .E  X T AX (14)
(4) X (t ) = -  2 y j cost

P.E  X T AX (15)
(5) Equations (14) and (15) represent the velocity and
where acceleration of a particular concentrated mass on the
a ij , bij represent the elements in the ith row and jth transmission line.
Substituting equations (14) and (15) into equation
column of the matrix respectively.
(12) and factorizing gives:
  Ay j  By j cost  0
A, B = n X n symmetric matrices corresponding to
2
kinetic and potential energies of the oscillating
masses. (16)
Applying the conservation of mass, the total energy Therefore,
of the oscillating masses is constant.
cost  0
Therefore,
(17)
K .E  P.E  constant or
(6)
The rate of change of total energy given by equation   2 Ay j  By j  0
(6) w.r.t. time is zero. (18)
From equation (18), we have:
 K .E  P.E  0
d
dt B   Ay 2
j 0
(7) (19)
Substituting for K.E. and P.E. in equation (7) using For non-trivial solution, the determinant of equation
equations (4) and (5) transforms equation (7) to: (19) must be zero.
d T 
dt

X AX  X T BX  0  Therefore,
B 2 A 0
(8) (20)
Differentiating equation (8) w.r.t. t gives: Equation (20) is the frequency equation of a
dt

d T  T

X AX  X BX  X T AX  X T AX  X T BX  X T BX  0
      transmission line under self-excited oscillation.
For a transmission line carrying n concentrated
(9) masses (Figure 2),
Without loss of generality, it is assumed that:
   
1 1

2 1
 2
 1

B  Ty12  T y1  y 2  T y 2  y 3  ...  T y N 1  y N  
2
X  X T , X T  X , and X T  X and simplifying 2 2 2 2
transforms equation (9) to : (21)
and
dt
    
X AX  X BX  2 X T AX  2 X T BX  2 X T AX  BX   0
d T  T   1

1 1
2
1
A  m1 y12  m2 y 2  m3 y 3  ...  mn y n
2 2
(10) 2 2 2 2
Therefore,
(22)
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3. Sule, S, Nwofor T.C / International Journal of Engineering Research and Applications (IJERA)
ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2306-2310
m1 m2 m3 mn 2m1   Ty 1  T  y1  y 2 
y
T T T T T (23)
T
y1 y2 y3 yn m 2  T  y1  y 2   T  y 2  y 3 
y
(24)
Figure 2: A Transmission line with constant tension 3m 3  T  y 2  y 3   Ty 3
y
carrying n concentrated masses. (25)
Simplifying and arranging equations (23) – (25)
3.0 RESULTS OF DYNAMIC ANALYSES gives:
A transmission line subjected to three 2m1 1 2 Ty1  Ty 2  0
y
unequal and equal concentrated masses for numerical
(26)
study.
m2 2 2 Ty 2  Ty1  Ty 3  0
y
2m m 3m (27)
3m3 2 Ty 3  Ty 2  0
y
(28)
In matrix form, equations (26) – (28) can be written
as:
2m 0 0   1 
y 2  1 0  y1  0
0 m 0      T  1 2  1   y   0
   y2     2  
0 0 3m  3 
  y  0  1 2   y 3  0 
    
(29)
From equations (21) and (22) and using equation (29)
the kinetic and potential energy of symmetric
matrices A and B are given by:
l l l l  2m 0 0 
ml 
A 0 m 0 
4l 2  
0
 0 3m

Figure 3: An overhead transmission carrying
(30)
unequal concentrated masses for numerical study
and
(control).
2 1 0
T
 1
m m m
B   1 2 
2
0
 1 2 
(31)
Substituting for A and B in equation (20) gives:
2 1 0  2 m 0 0 
   ml
2
T 0
1 2  1 m 0  0
2 2  
0
 1 2 0
 0 3m 

(32)
Let
ml  2

T
l l l (33)
l
Equation (32) now transforms to:
4l
2 1 0  2 0 0
   
Figure 4: An overhead transmission carrying equal
concentrated masses for numerical study.
  1 2  1    0 1 0 0
0 1 2  0 3
From Figure 3, the equations of motion of    0 
the vibrating unequal concentrated masses are; (34)
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4. Sule, S, Nwofor T.C / International Journal of Engineering Research and Applications (IJERA)
ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2306-2310
2  2 1 0 T
For  2  2,  2  1.4142 Rad / sec.
 1 2 1 0 ml
0 1 2  3 T
For 3  3.414 ,  3  1.8477 Rad / sec.
(35) ml
From equation (35), 4.0 DISCUSSION OF RESULTS
3  4.67 2  5.33  1.33  0 Table 1: Comparison of results of dynamic analysis
(36) of a transmission line subjected to unequal and equal
Using Newton Raphson approximation, the roots of concentrated masses.
equation (36) are: NATURAL FREQUENCY
(RAD/SEC)
1  0.352 ,  2  1.24 , 3  3.78
1 2 3
From equation (33),
Unequal 1 1 1
ml  2 T  2 T  2 T  2
 concentrat 0.5932 1.1135 1.9442 
T ed  ml   ml   ml 
masses(co
T
For 1  0.352 , 1  0.5932 Rad / sec. ntrol)
ml Equal 1 1 1
T  2 T  2 T  2
T
concentrat 0.7655 1.4142 1.8477 
For  2  1.24 ,  2  1.1135 Rad / sec. ed masses  ml   ml   ml 
ml
T
For 3  3.78,  3  1.9442 Rad / sec. The dynamic analyses of a transmission
ml subjected to unequal and equal concentrated masses
From Figure 4, m1  m2  m3  m has been presented. From Table 1, it can be seen that
The frequency equation is: for a transmission line subjected to unequal
concentrated masses (control), the natural frequency
2 1 0 1 0 0 of vibration is improved by 29% in the first vibration
T  ml  2  
1  1  0  0 mode, 27% in the second vibration mode but
2 0 1
decreased by 4.96% in the third vibration mode when
2 2
0 1 2
0
 0 1 unequal concentrated masses are replaced by equal
(37) concentrated masses.
Again, using equation (33),
5.0 CONCLUSION
2 1 0 1 0 0 The dynamic analyses of a transmission line
   
1 2  1    0 1 0  0 subjected to unequal and equal concentrated masses
using energy approach has been presented. From the
0 1 2 0 1
   0  results, it can be concluded that to improve the
(38) natural frequency of vibration, equal concentrated
masses at equal spacing should be lumped on the
2 1 0 transmission line at equal interval as the obtained
 1 2 1  0 results showed a significant improvement compared
with those of the control, most especially in the first
0 1 2 and second vibration modes. The damping
(39) characteristic of the transmission line is thus
Expansion and evaluation of the above determinant improved.
gives: The formulated model can be used in the
dynamic analysis of a multistory building having
3  62  10   4  0 irregular floor masses and column stiffnesses.
(40)
From equation (40), REFERENCES
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T
For 1  0.586 , 1  0.7655 Rad / sec .
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Chopra, A.K. Dynamic of Structures:
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Hall; 2001.
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5. Sule, S, Nwofor T.C / International Journal of Engineering Research and Applications (IJERA)
ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2306-2310
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