Power Technology and Engineering

Vol. 38, No. 5, 2004

EVALUATION OF PERFORMANCE EFFICIENCY OF VIBRATION DAMPER ON A CONDUCTOR S. V. Trofimov1 Translated from Élektricheskie Stantsii, No. 5, May 2004, pp. 64 – 70.

The method suggested makes it possible to determine maximum amplitudes of standing vibration waves, slopes of the conductor with respect to horizontal, bending moments, and cutting forces in any cross section of the conductor and of the flexible element of the vibration damper with the help of “SVT — Conductor with Vibration Damper” software for any grade of conductor, earth wire, or vibration damper (GPG). Comparative evaluation of parameters at the places of exit of the conductor from the supporting clamp and from the clamp of the vibration damper is performed. Keywords: vibration, vibration amplitude, vibration damper, efficiency of vibration damper.

Wind-induced vibrations affect to this or that degree any conductor of an overhead transmission line (OL). The vibrations cause fatigue damage of both internal and external lays of conductors. Dangerous fatigue is the most probable at the places of entrance of conductors into supporting clamps, into retaining clamps on the wires of vibration and galloping dampers, and into connectors of different type (SOAS and SAS). In is stressed in [1] that intense vibration of wires in over 108 vibration cycles has never lead to damage of OL conductors outside clamps, which means that a conductor itself is absolutely reliable under the action of standing vibration waves. In order to prevent fatigue damage of conductors at places of their exit from supporting clamps, vibration dampers of type GPG with loads axisymmetric with respect to the longitudinal axis of the flexible element are mounted in accordance with [2]. If the vibration has a low intensity, one damper is mounted per one span. If the vibration intensity is enhanced, the number of GPG vibration dampers per one span is increased (two dampers or two main dampers of one size and two additional dampers of another size), which reliably protects the conductor at the outlet from supporting clamp. The efficiency of a vibration damper can be evaluated in comparison with the degree of decrease in the maximum values of alternating bending moments acting in the most fatigue-dangerous section of the conductor in the span. The comparison should be made with allowance for the constant conductor tension, for the equality of maximum amplitudes in antinodes of vibration half-waves, and for the minimum 1

differences in the values of eigenfrequencies. Thus, in order to evaluate the efficiency of protection of a conductor from vibration at the outlet from a supporting clamp ensured by a GPG vibration damper we should determine the proportion of the bending moments in the cross section of the conductor without vibration damper and in the cross section of the conductor with vibration damper at the places of its entrance into the supporting clamp and evaluate the degree of danger of these bending moments. In the first approximation we can describe the “conductor – conductor with vibration damper – conductor” system as a system consisting of an elastic rod (homogeneous within one segment) rigidly fixed at one end and hinge-supported at the other end, which is stretched with a constant force T (Fig. 1), and two elastic rods rigidly fixed in the body of the clamp of vibration damper and equipped with rigidly fixed loads at their free ends. The bending stiffness of the conductor E1J1, which depends in the general case on the conductor tension T and on the bending deformations of the conductor, can be assumed to be constant in the first approximation for low bending deformations. The clamp of the vibration damper, which is rigidly fixed on the conductor, is equivalent in the computational mechanical model to a segment of the rod that has, in contrast to the main length of the conductor, a mass per unit length, bending stiffness, and other characteristics. The flexible element of the vibration damper (a steel wire cable) is also treated in the mechanical model as a homogeneous elastic rod (Fig. 1). In the first approximation for low bending deformations the bending stiffness of the flexible element of the vibration damper is also assumed to be constant. The loads of the vibration damper have symmetric shape with re-

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Evaluation of Performance Efficiency of Vibration Damper on a Conductor O

T

l1

x

l4 l2

l5

Ji is the moment of inertia of the cross section of the conductor or of the flexible element of the vibration damper relative to its central axis in the ith segment;

l3

ì W 1 ( x , t ) at 0 £ x £ l1 ; ï W ( x , t ) at l < x £ l ; 1 2 ïï 2 W i ( x , t ) = í W 3 ( x , t ) at l 2 < x £ l 3 ; ï W ( x , t ) at - l + l £ 0 £ l ; 4 4 1 1 ï ïî W 5 ( x , t ) at l 2 £ x < l 2 + l 5 ,

Fig. 1. Mechanical model of a conductor with a GPG vibration damper protecting the conductor from fatigue damage at the place of its entrance into the supporting clamp.

spect to the longitudinal axis of the flexible element and are fixed rigidly at the ends of the flexible element. The masses of the loads are M1 and M2. The loads have moments of inertia I1 and I2 relative to the central axes passing through the centers of the cross sections of the elastic cable at the outlets from the bodies of the loads. The difference in the methods of fixation of the ends of the elastic rod (conductor) makes it possible to compare the effect of fixation of the rod ends on the values of bending moments in the end zones of the “conductor – conductor with vibration damper – conductor” system. With allowance for the mentioned description the conductor in a span is broken into three homogeneous computational segments, i.e., the fist segment with length l1 from the place of rigid fixation to the clamp of the vibration damper, the second segment with length l2 – l1 coinciding with the length of the conductor in the body of the damper clamp, and the third segment with length l3 – l2 coinciding with the length of the conductor from the damper clamp to the hinged support. The fourth computational segment with length l4 is the length of the first functional part of the flexible element of the vibration damper with a load with mass M1 and a moment of inertia I1 relative to the central axis passing through point x = l1 – l4. The fifth computational segment l5 is the length of the second functional part of the flexible element of the vibration damper with a load mass M2 and its moment of inertia I2 relative to the central axis passing through point x = l2 + l5 (Fig. 1). The displacements in the vertical plane from the position of static equilibrium of points of longitudinal axes of the conductor and of parts of the flexible element of the vibration damper are described by two homogeneous differential equations with an accuracy sufficient for practical purposes [3], i.e., for the conductor Ei J i

¶ 4W i ¶x 4

-T

¶ 2W i ¶x 2

+ r i Fi

¶ 2W i ¶t 2

= 0, (i = 1, 2, 3);

¶ 4W i ¶x

4

+ r i Fi

¶ 2W i ¶t 2

= 0, i = 4, 5,

(2)

where W1(x, t), W2(x, t), and W3(x, t) are vertical displacements from the position of static equilibrium of the point of the longitudinal axis of the conductor with coordinate x at moment t; W4(x, t) and W5(x, t) are vertical displacements from the position of static equilibrium of the point of the longitudinal axis of the flexible element of the vibration damper with coordinate x at moment t; ñi is the reduced density of the conductor or of the flexible element of the vibration damper on the ith segment; Fi is the cross sectional area of the ith segment of the conductor or of the flexible element of the vibration damper. The equation of free transverse vibrations of a compound elastic rod was solved by the method of separation of variables [3] in which the solution for each of the segments is sought in the form ¥

W i ( x , t ) = å V j ( x )j j ( t ),

(3)

ì C 1 e k1 x + C 2 e - k1 x + C 3 sin k 2 x ï + C 4 cos k 2 x at 0 £ x £ l1 ; ï k 3x ï C 5 e + C 6 e - k 3 x + C 7 sin k 4 x ï + C 8 cos k 4 x at l1 < x £ l 2 ; ï k 5x ï C e + C 10 e - k 5 x + C 11 sin k 6 x V j ( x) = í 9 + C 12 cos k 6 x at l 2 < x £ l 3 ; ï k ï C 13 e 7 x + C 14 e - k 7 x + C 15 sin k 8 x ï + C 16 cos k 8 x at - l 4 + l1 £ 0 £ l1 ; ï k ï C 17 e 9 x + C 18 e - k 9 x + C 19 sin k 10 x ï + C 20 cos k 10 x at l 2 £ x < l 2 + l 5 , î

(4)

i =1

where

(1a)

for parts of the elastic elements of the vibration damper Ei J i

299

(1b)

where Ei is the reduced modulus of elasticity of the material of the conductor or of the flexible element of the vibration damper in the ith segment of the conductor (i = 1, 2, 3, 4, 5),

where Vj(x) is a function of variable x determining the jth eigenvibration (standing vibration wave) of the rod system consisting of a conductor and a damper fixed on it, which corresponds to the jth eigenfrequency ùj (j = 1, 2, 3, ...); k1, k2, ..., k10 are eigenvalues corresponding to the jth eigenfrequency ùj; C1, C2, ..., C20 are constant factors determining the jth eigenvibration of transverse vibrations of the rod system; öj(t) = sin ùjt are functions of the variable t determining the variation of the vibration amplitude in time (j = 1, 2, 3, ...).

300

S. V. Trofimov

The forms of the eigenvibrations (standing vibration waves) are determined for each eigenfrequency of vibration of the rod system ùj with allowance for the boundary and dynamic conditions at the ends of the rods entering the system and for the conditions of matching of the solutions at the boundaries of the segments (see Fig. 1). For example, since each jth transverse eigenvibration of the rod system meets the boundary conditions, we will omit the subscript j in the formulas presented below. The boundary conditions for rigid fixation of the end of a rod at x = 0 are

The dynamic conditions at the ends of the flexible elements of the vibration damper have the form

ì V 1 ( 0 ) = 0; ï dV í 1 = 0. ïî dx x = 0

ì d 2V 5 2 dV 5 = E5J 5 ; ï I 2w dx x = l2 + l5 dx 2 x = l2 + l5 ï í 3 ï - M 2 w 2V 5 ( l 2 + l 5 ) = E 5 J 5 d V 5 ï dx 3 x = l2 + l5 î

(5)

The conditions for the hinge-supported end of a rod at x = l3 are ì V 3 ( l 3 ) = 0; ï d 2V 3 í = 0. ï dx 2 x = l 3 î

(6)

The conditions for matching of the solutions at the boundaries of the segments of the rod system have the form ì V1 ( l1 ) = V 2 ( l1 ); ï V ( l ) = V ( l ); 2 1 ï 4 1 dV ï dV1 = 2 ; dx x = l1 ï dx x = l1 ï dV dV ï 4 = 2 ; í dx x = l1 dx x = l1 ï 2 ï E 1 J 1 d V1 + E4 J 4 ï dx 2 x = l1 ï 3 ï E J d V1 + E4 J 4 1 1 ï dx 3 x = l1 î

(7) d 2V 4 dx 2

= E2J 2 x = l1

d 3V 4 dx 3

= E2J 2 x = l1

d 2V 2 dx 2

; x = l1

d 3V 2 dx 3

x = l1

at x = l1 and ì V 2 ( l 2 ) = V 3 ( l 2 ); ï V ( l ) = V ( l ); 2 2 ï 4 2 dV dV 2 ï = 3 ; dx x = l2 ï dx x = l2 ï dV dV ï 2 = 5 ; í dx x = l2 dx x = l2 ï 2 ï E2 J 2 d V2 = E3J 3 ï dx 2 x = l2 ï 3 ï E J d V2 = E3J 3 2 2 ï dx 3 x = l2 î at x = l2.

(8) 2

d V3 dx

2

+ E5J 5 x = l2

d 3V 3 dx 3

+ E5J 5 x = l2

2

d V5 dx 2

; x = l2

d 3V 5 dx 3

x = l2

ì 2 dV 4 = E4 J 4 ï -I 1w dx x =- l4 + l1 ï í ï M 1w 2V 4 ( -l 4 + l1 ) = E 4 J 4 ï î

d 2V 4 dx 2

; x =- l4 + l1

d 3V 4 dx 3

(9)

x =- l4 + l1

at x = –l4 +l1 and the form

(10)

at x = l2 + l5. The computation of the bending moments acting in the cross sections of the conductor at the place of its exit from the supporting clamp at x = 0 and at the inlet to and at the outlet from the clamp of the vibration damper at x = l1 and x = l2, as well as at the exits of the flexible elements of the damper from the bodies of the loads of the damper and from the casing of the clamp keeping the damper on the conductor was made for AS 120/19 wire and GPG-1,6-11-450 damper. The total length of a span with conductor is 200 m. According to the GOST 839–80 State Standard [4], the AS 120/19 conductor has a diameter dc = 15.2 mm and consists of 26 aluminum wires with diameter dal = 2.4 mm each and a steel core consisting of 7 steel wires with diameter dst = 1.85 mm each. The mass per unit length of the conductor is equal to the mass per unit length of the rod in the first and third segments and amounts to ñ1F1 = ñ3F3 = 0.471 kg/m. The design bending stiffness of the conductor in these segments is determined as the sum of the stiffnesses of the steel and aluminum wires composing the steel core and the aluminum lays and having a relative freedom of displacement relative to each other, i.e., E1J1 = E3J3 = 3.77 N × m2. The design bending stiffness of the second segment of the rod E2J2 is determined as the sum of the stiffnesses of the conductor itself, of the flexible element of the vibration damper, and of the clamp of the vibration damper, i.e., E2J2 = 5000 N × m2. The mass per unit length of the second segment of the rod ñ2F2 is determined as a sum of the masses per unit length of the conductor, of the flexible element of the vibration damper, and of the clamp of the vibration damper, i.e., ñ2F2 = 16.7 kg/m. In accordance with [2] vibration dampers GPG-1,6-11-450 or GVN-3-17, which possess virtually coinciding amplitude and frequency characteristics, are used to protect the AS 120/19 conductor from fatigue damage at the outlet from the supporting clamp. GPG-1,6-11-450 dampers are mounted on AS 120/19 conductors at a distance from the exit of the conductor from the supporting clamp equal to

Evaluation of Performance Efficiency of Vibration Damper on a Conductor

301

W, m 0.011

0.85, 0.9, or 0.95 m [2], i.e., the coordinates of the damper are calculated with a step of 0.05 m. A GPG-1,6-11-450 damper has two axisymmetric loads with M1 = M2 = 1.6 kg. The mass of a GPG-1,6-11-450 damper is 4.5 kg; the total length of the vibration damper is 450 mm. The moments of inertia of the loads I1 and I2 with respect to the central axes lying in the cross sections of the flexible elements of the damper at the outlet from the bodies of the loads are I1 = I2 = 0.002 kg × m2. The functional lengths of the flexible elements of the vibration damper l4 = l5= 0.142 m. The diameter of a flexible element dT = = 11 mm. The width of the clamp of the vibration damper coincides with the width of the clamped flexible element, i.e., l2 = 60 mm. The elastic elements of the vibration damper are made of a steel cable (GOST 3063–80) consisting of 19 steel wires (18 steel wires with a diameter of 2.2 mm and one central wire with a diameter of 2.4 mm). The mass per unit length of a flexible element is 0.6274 kg/m. The bending stiffness of a flexible element of the vibration damper is determined as a sum of the stiffnesses of its steel wires, i.e., E4J4 = E5J5 = 4.46 N × m2. VNIIÉ has developed “SVT — Vibration Damper on Conductor” software that permits determination of maximum amplitudes of standing vibration waves W(x) and slopes ø(x) of a conductor and a flexible element of vibration damper, bending moments Mbend(x) in cross sections of the conductor and of the flexible element of the vibration damper, and the values of cutting forces Qc(x) for any conductor of grade AS and any vibration damper with axisymmetric loads. Figure 2 presents the results of computations of maximum amplitudes of vibrations of the “conductor with vibration damper” system at l1 = 0.8 m, which correspond to the 50th eigenfrequency ù50 and the amplitude Amax = 0.01 m in the intermediate half-wave antinode. This vibration mode of a “conductor with vibration damper system” appear due to forced vibrations of AS 120/19 conductor in a wind flow with velocity v = 1.47 m/sec. As a rule, a wind with such velocity causes stable vibration of the given kind of conductor [2]. The values of maximum amplitudes of standing vibration waves of a conductor Wc(x) obtained with the help of the developed software show that two kinds of vibration halfwaves can appear in the presence of a vibration damper of

1

0.009 0.007 0.005

2 3

0.003 0.001 –0.001

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0 x, m

–0.003 –0.005

Fig. 2. Maximum amplitudes of vibration of points on the longitudinal axes of the conductor and of the flexible element of the vibration damper: 1, conductor Wc(x); 2, left-hand side of the flexible element of the vibration damper Wd(x) on the left of the damper clamp; 3, right-hand side of the flexible element of the vibration damper Wd(x) on the right of the damper clamp.

the conductor, which differ from each other by the length or by the maximum amplitude in the half-wave antinodes (see curve 1 in Fig. 2). The first kind of vibration half-wave is represented by the first vibration half-wave from the place of rigid fixation 0.5l*50 , which includes the conductor segment of about 0.05 m where the conductor experiences the action of an edge effect and the conductor segment on which the clamp of the vibration damper is mounted. The second kind of vibration half-wave 0.5l050 is an intermediate half-wave having a regular sinusoidal form. In the absence of vibration damper on a conductor two kinds of vibration half-waves arise too. The fist kind is the first vibration half-wave 0.5l** 50 that includes the conductor segment of about 0.05 m where it experiences the edge effect due to rigid fixation. The second kind of vibration half-wave 0.5l00 is an intermediate half-wave of a regular sinusoidal 50 form. The lengths of different vibration half-waves for the “conductor with damper” system for several variants of placement of damper in the vibration half-wave and in the absence of damper are presented in Table 1. The length of the half-wave 0.5l*50 nearest to the place of rigid fixation increases considerably (by 7 – 10%) with the appearance of vibration damper on the conductor. In accordance with the recommendations of [2], a damper is mounted at a distance l1 = 0.8 – 1.0 m from the place of exit of the conductor from the supporting clamp; the length of the half-wave 0.5l*50 increases with l1 (see Table 1).

TABLE 1. Lengths of Vibration Half-Waves with Frequency ù50 for AS 120/19 Conductor with GPG-1,6-11-450 Vibration Damper in a 200-m Span Element Conductor without vibration damper Conductor with vibration damper

l 1, m

ù50, rad/sec

f50, Hz

0.5l*50, m

0.5l050, m

— 0.8 0.9 1.0

114.5 114.6 114.6 114.5

18.23 18.26 18.25 18.22

4.0190 4.2961 4.3479 4.4014

3.9995 3.9939 3.9929 3.9918

302

S. V. Trofimov

ø, min 190

the appearance of two additional zones or possible fatigue damage of the conductor at the places of its entrance into the clamp of the vibration damper. The maximum slopes of the axis of the flexible element y d50 ( x ) close to the places of the entrance of the flexible ele-

2

140 90 1

40 –10

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

x, m

–60 –110 3 –160 –210

Fig. 3. Maximum slopes of longitudinal axes of the conductor and of the flexible element of vibration damper in the first half-wave of vibrations of the “conductor with vibration damper” system, which correspond to the 50th eigenfrequency of vibrations ù50 and to the amplitude in the antinode of the intermediate half-wave Amax = 0.01 m: 1, conductor; 2, left-hand side of the flexible element of vibration damper, on the left of the damper clamp; 3, right-hand side of the flexible element of vibration damper, on the right of the damper clamp.

The form of the half-wave closest to the rigidly fixed end on the side of the main span coincides with a regular sinusoid virtually until the place where the damper is mounted, but the differences of the part of the first half-wave 0.5l*50 between the place of rigid fixation and the damper clamp are very substantial, which has to be taken into account in the determination of the bending moment in the cross section of the conductor within the rigid fixation. The maximum amplitudes of vibrations of the flexible eld ement of the vibration damper W 50 ( x ) are substantially

ment into the bodies of the loads of the vibration damper are much larger than the values y c50 corresponding to the same coordinates (see curves 2 and 3 in Fig. 3). However, the values of y d50 and y c50 at the places of the entrance of the conductor and of the flexible element into the clamp of the vibration damper are pairwise equal, i.e., y d50 ( l1 ) = y c50 ( l1 ) and y d50 ( l 2 ) = y c50 ( l 2 ). We evaluated the efficiency of operation of a vibration damper mounted for protecting a conductor from fatigue damage at the outlet from the supporting clamp using the values of the bending moment M *bend in the cross section of the conductor at the place of fixation, the bending moments in the cross sections of the conductor at the entrance into the ) and on the clamp of the vibration damper on the left (M cl bend right (M cr ), the bending moments in the cross sections of bend the flexible elements of the vibration damper at the inlets to the damper clamp on the left (M dl ) and on the right bend ( M dr ), the bending moments in the cross section of the bend flexible elements of the vibration damper at the inlets to the bodies of the loads on the left (M loadl ) and on the right bend ( M loadr ) of the damper clamp at maximum amplitudes at the bend

the places of entrance of the conductor and of the flexible element into the clamp of the vibration damper are pairwise d c d d equal, i.e., W 50 ( l1 ) = W 50 ( l1 ) and W 50 ( l 2 ) = W 50 ( l 2 ).

antinodes of the standing vibration waves W(x) = 0.01 m corresponding to ö50(x), i.e., for the 50th eigenvibration and the respective frequencies ù50 = 114.5 – 114.6 rad/sec or f50 = 18.22 – 18.26 Hz. All the computational data presented below were obtained for the specified mode of vibration of the conductor with vibration damper or without it. The results of the computations of maximum values of , M cr , M dl , M dr , bending moments M *bend , M cl bend bend bend bend

Figure 3 presents computed data for maximum slopes of the longitudinal axis ø50(x) of the “conductor with vibration damper” system with respect to horizontal for the 50th eigenvibration ö50(x) and for maximum amplitude in the antinode of the intermediate vibration half-wave Amax = 0.01 m. Analyzing the computed slopes y c50 ( x ) of the conductor we es-

The maximum bending moment in the cross section of the conductor in the absence of vibration damper occurs at the place of exit of the conductor from the place of rigid fixation (Mbend = 1.53 N × m, see Table 2); the eigenfre0 quency of the vibrations of the conductor f 50 = 28.23 Hz,

tablished that two additional zones of marked variation of the slopes with respect to the horizontal on the left and on the right of the damper clamp with a length of 0.03 – 0.05 m were similar to the zone of abrupt variation of the slope ø50(x) near the place of rigid fixation, whereas the lengths of all the three zones were about equal (see curve 1 in Fig. 3). The changes in the slopes of the longitudinal axis of the conductor in the zones close to rigid fixation and at the entrance of the conductor into the damper clam were commensurable. This means that the installation of a vibration damper causes

and the eigenvibration of the conductor contains 50 vibration half-waves. Installation of a vibration damper reduces the maximum value of the bending moment in the conductor at the outlet from the place of rigid fixation by a factor of 1.38 – 1.4. When the vibration damper is mounted at a distance l d* = 0.8 and l d** = 0.9 m from the outlet from the place of rigid fixation, the vibration eigenfrequencies increase to * ** = 18.26 Hz and f 50 = 18.25 Hz; the eigenvibrations of f 50

c higher than the values of W 50 ( x ) corresponding to the same d c coordinates; however, the values of W 50 ( x ) and W 50 ( x ) at

, and M loadr are presented in Table 2. M loadl bend bend

Evaluation of Performance Efficiency of Vibration Damper on a Conductor

the “conductor with vibration damper” system in the span also contain 50 vibration half-waves. Upon the installation of vibration damper in the “conductor with vibration damper” system the latter acquires two additional zones dangerous with respect to fatigue damage, i.e., the cross sections of the conductor at its entrance into the damper clamp on the left an on the right. Comparing the values of bending moments in the conductor at the inlet and M cr we established that to the damper clamp M cl bend bend the bending moment in the cross section of the conductor on the right of the vibration damper clamp M cr was 5 times bend higher than that in the conductor on the left of the damper clamp. It should be noted that the bending moments at the entrance of the conductor into the vibration damper clamp and M cr are more than 1.5 times lower than the M cl bend bend bending moments in the cross section of the conductor at the outlet from the place of rigid fixation M kbend (see Table 2 and Fig. 4). In our case this means that the GPG-1,6-11-450 vibration damper mounted on AS 120/19 conductor at a distance l1 = 0.8 – 1.0, ensures reliable and efficient protection of the conductor vibrating in a frequency range close to f50 = 18.2 Hz from fatigue damage. Maximum values of bending moments in the cross section of the flexible element of the vibration damper depend on the length of the flexible elements, on the masses of the loads M1 and M2 and the moments of inertia I1 and I1, and on the bending stiffness E4J4 and E5J5. Optimum bending moments can be chosen with the help of the “SVT — Conductor with Vibration Damper” software. The values of the bending moments in the cross sections of the flexible element should be minimized in order to ensure the appropriate fatigue resistance of the flexible element of the vibration damper. It should be noted that in contrast to the conductor the flexible element is not stretched by the tension force T and therefore , M dr , M loadl , can withstand high bending moments M dl bend bend bend and M loadr . bend

303

Mbend, N · m 3 2 2 1

1 0 –1

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0 x, m

3

–2 –3 –4 –5 –6

Fig. 4. Maximum bending moments in cross sections of the conductor and of the flexible element of the vibration damper in the first vibration half-wave of the “conductor with vibration damper” system corresponding to the 50th eigenfrequency of vibration ù50 at the amplitude Amax = 0.01 m of the intermediate half-wave: 1, conductor; 2, flexible element of vibration damper, on the right of the clamp; 3, flexible element of vibration damper, on the right of the clamp.

For a reliable analysis of the efficiency of the use of vibration dampers similar computations should be performed for all eigenfrequencies of the “conductor with vibration damper” system. This will give a full picture of bending moments in dangerous zones of the conductor with vibration damper. CONCLUSIONS 1. We have developed a mathematical model, an algorithm, and “SVT — Conductor with Vibration Damper” software for computing the deflection of a conductor and the bending moment in the conductor and in flexible elements of the vibration damper for any type of conductor with allowance for the effect of the mass per unit length, the bending stiffness of the conductor and of the flexible element of the vibration damper, the masses and moments of inertia of the damper loads, and the conductor tension.

TABLE 2. Bending Moments Acting in Various Cross Sections of AS 120/19 Conductor and of the Flexible Element of GPG-1,6-11-450 Vibration Damper Vibrating in a 200-m Long Span at a Frequency ù50 at Maximum Aptitude of Vibration of the Intermediate Half-Wave on the Right of the Damper Amax = 0.01 m Moment, N × m Element

l, m

Conductor without vibration damper Conductor with vibration damper

— 0.8 0.9 1.0

k M bend

cl M bend

cr M bend

dl M bend

loadl M bend

dr M bend

loadr M bend

ih M bend

1.53 1.06 1.04 1.02

— 0.088 0.053 0.019

— 0.60 0.61 0.61

— 3.79 4.20 4.60

— 1.21 1.33 1.44

— 5.20 5.56 5.90

— 1.47 1.58 1.69

0.023 0.023 0.023 0.023

Note. l is the distance from the rigidly fixed end to the middle of the clamp of the vibration damper, M kbend is the maximum bending moment in is the maximum bending moment in the cross section of the conductor in the cross section of the conductor at the place of rigid fixation, M ih bend the intermediate vibration half-wave on the right of the damper.

304

2. Installation of GPG-1,6-11-450 vibration damper on AS 120/19 conductor vibrating at a frequency f = 18.2 Hz at a distance of 0.8 – 1.0 m from the place of exit of the conductor from the supporting clamp decreases the bending moment in the cross section of the conductor at the outlet from the supporting clamp by a factor of 1.5. 3. Maximum bending moments in cross sections of the flexible element of vibration damper at inlets to the clamp or to the loads of the vibration damper are comparable with the bending moment in the conductor at the place of rigid fixation or even exceed the latter.

S. V. Trofimov

REFERENCES 1. E. P. Nikiforov, “Use of spiral protectors for protecting OL conductors from fatigue damage,” Élektr. Stantsii, No. 6 (2002). 2. Methodological Recommendations on Routine Protection of Conductors and Earth Wires of 35 – 750-kV Overhead Transmission Lines from Vibrations and Subvibrations. RD 34.20.189–90, Izd. ORGRÉS, Moscow (1991). 3. I. M. Babakov, The Theory of Vibrations [in Russian], Gos. Izd. Tekhniko-Teoreticheskoi Literatury, Moscow (1958). 4. State Standard GOST 839–80. Bare Conductors for Overhead Transmission Lines. Specification, Izd. Standartov, Moscow (1982).