Endurance of dampers for electric conductors

International Journal of Fatigue 23 (2001) 21–28 www.elsevier.com/locate/ijfatigue

Endurance of dampers for electric conductors A. Lara-Lo´pez *, J. Colı´n-Venegas University of Guanajuato, Prolongacion Tampico S/N, Salamanca, Gto. Mexico C.P. 36730, Mexico Received 12 December 1998; received in revised form 6 January 2000; accepted 10 July 2000

Abstract Dampers are used to suppress vibration in transmission lines, reducing the probability of failure due to fatigue. Dampers dissipate energy due to friction between wires of a strand cable subjected to reversible bending. In the present article a method to calculate the life of dampers subjected to a know vibration is presented. Such a method is directly related to a standard fatigue test for dampers. The proposed method is oriented for design or analysis of dampers regarding endurance requirements. The dynamic response of dampers is predicted by an analytical model and related to the stress analysis of the strand cable. Analytical results are compared to experimental data.  2001 Elsevier Science Ltd. All rights reserved. Keywords: Fatigue; Damper; Conductors

1. Introduction The problem of vibrations in transmission lines induced by wind and the attenuation of such vibrations by dampers have been investigated during a long period [1,2]. Fig. 1 shows a damper for eolian vibrations, which consists of two weights suspended from the ends of a short steel cable (messenger cable) having a clamp in the middle. Damping is caused by friction between wires of the messenger cable. A fundamental analysis for the dynamic response of dampers and transmission lines including dampers was reported by Claren and Diana [3]. Most recently, an analysis for the capacity of dampers for energy dissipation is presented by Richardson [4]. However, little research have been reported on the analysis of stresses and endurance of dampers. Such dampers are subjected to reversible bending with high

risk to fail due to fatigue of the messenger cable. The mechanism of failure in such a strand cable is very complex. However, several authors have produced simplified models for analysis of cables. Costello [5] reported an analysis of cables assuming that such cables pass around a sheave of a given diameter under a combination of bending and axial load. In such a model, forces of contact between wires are not considered. Therefore friction is not considered. It was also reported that the wire subjected to the largest stress is the one in the center. Dampers are subjected to an endurance test for acceptance. In such a test, the damper is subjected to severe conditions that may occur in real practice. In this article, dynamic conditions for the acceptance test are assumed to determine the expected life of dampers. Relationships presented in this article would aid damper design and may be applied to the design of accelerated tests of dampers.

2. Dynamic analysis

Fig. 1. Aeolian vibration damper type stockbridge. * Corresponding author. Tel.:+52-91-473-31534; fax: +51-19-47330433. E-mail address: [email protected] (A. Lara-Lo´pez).

The dynamic performance of dampers may be experimentally analyzed by excitation similar to the vibrations on the transmission line. One critical condition occurs when excitation frequency is one of the natural frequencies of the damper. Fig. 2 illustrates the experiment to determine the performance of dampers and it shows also forces on the messenger cable caused by excitation

0142-1123/01/$ - see front matter  2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 0 ) 0 0 0 7 2 - 4

A. Lara-Lo´pez, J. Colı´n-Venegas / International Journal of Fatigue 23 (2001) 21–28

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L is the length of the messenger cable, and G is the distance from o⬘ to the center of mass of the weight. The general solution of Eq. (3) is {D}⫽[U]{h}

(5)

where [U] is the modal matrix and {h} is the uncoupled solution which is given by: hr⫽Hrei(wt−br)

(6)

where Hr is given by: Hr⫽

(−U11+U21G)mw2

冑

w2r (1−R2r)2+m2

r⫽1,2

(7)

w is the excitation frequency, wr the rth natural frequency and Fig. 2. (a) Endurance test with a frequency of the vibratory table equal to the first natural frequency f1 and amplitude of 25.4/f1 (mm). (b) Forces on the menssenger cable and on the weights.

m w Rr⫽ ; br⫽tan−1 r⫽1,2 wr 1−R2r

(8)

If ar ⫽

4K−mw2r U1r ⫽ r⫽1,2 2KL−mGw2r U2r

of the vibration table. Assuming that displacements of the table and damper weight are small, the equation of motion for the damper mass may be established as follows [3]:

the eigenvalues are given by:

¨ }⫹[C]{D ˙ }⫹[K]{D}⫽{F(t)} [M]{D

Ur1⫽

(1)

Where [M], [C] and [K] are inertial, damping and stiffness matrixes respectively and {D} and {F(t)} are the displacement and the exciting force vectors, respectively. Due to the fact that the mechanism of dissipation of energy is nearly equal to a hysteresis damping phenomenon, it is possible to apply the equivalent viscous damping theory as Meirovitch [6]. Therefore each component of the damping matrix is written in the general form: Kij Cij ⫽m w

冪m+J a −2a mG r⫽1,2 1

o

u⫽UaAmaxsin(wt⫺bu)

(11)

f⫽UaAmaxROTmaxsin(wt⫺bf)

(12)

where: Amax⫽

冑U

11

(3)

(13)

H1cosb1+U12H2cosb2)2+(U11H1sinb1+U12H2sinb2)2

bu⫽tan−1

¨ }⫹(1⫹im)[K]{D}⫽{F(t)} [M]{D

(10)

r

and then the displacement of the center of mass and the rotation of the mass are:

(2)

where m is the dimensionless coefficient of damping for the messenger cable. Since {D} is harmonic, Eq. (1) may be written as follows:

2 r

(9)

U11H1sinb1+U12H2sinb2 U11H1cosb1+U12H2cosb2

ROTmax⫽

冑(U

21

(14) (15)

H1cosb1+U22H2cosb2)2+(U21H1sinb1+U22H2sinb2)2 −(U21H1sinb1+U22H2sinb2) U21H1cosb1+U22H2cosb2

Solving the eigenvalues problem, the expression for natural frequencies is the following:

bf⫽tan−1

w21,2⫽

Attention was concentrated on the determination of the stiffness K and the damping coefficient m for the messenger cable [7]. The stiffness is given by the summation of individual wire stiffnesses:

2K

L 2m (Jo⬘+ −LGm)± 3

(4)

冪

L2m L2m (jo⬘+ −LGm)2− (J −mG2) 3 3 o⬘ mJo⬘−m2G2

Where Jo⬘ is the moment of inertia of the damper weight mass relative to the point of contact of it with the messenger cable, m is the mass of one weight of the damper,

3nEIa 3EIc K⫽ 3 ⫽ 3 L L

(16)

(17)

where n is the number of wires, Ia is the second moment of inertia of the cross section area of an individual wire

A. Lara-Lo´pez, J. Colı´n-Venegas / International Journal of Fatigue 23 (2001) 21–28

23

about its neutral axis, and Ic=nIa. The dimensionless coefficient of damping is obtained by solving the following simultaneous equations for R1 and m which are approximately valid for the first mode of vibration. R1 is the ratio of the vibration frequency to the first natural frequency.

4 Bj ⬘2⫽⫺2KaLAmaxsinbu⫹ KaL2ROTmaxsinbf 3

m tan−1m⫹tan−1 ⫽b 1−R1 u

(18)

1+m2 X ⫽ 2 2 2 (1−R1) +m Ua

Values of Amax, bu, ROTmax and bφ are calculated using Eqs. (13)–(16). The maximum stress is given by the following equation

(19)

冋册

2

Values of X, Ua and bu are obtained experimentally from the oscilloscope for a frequency of excitation close to the first natural frequency.

(26)

and A1⬘ f1⫽arctan I⫽i,j B1⬘

(27)

s⬘⫽UaDmax

(28)

where Dmax⫽

冑

d Al⬘2+Bl⬘2 2Ia

I⫽i,j

(29)

3. Endurance analysis The principal assumption was that all wires act independently of each other and the maximum bending stress will always occur in the center wire [5]. Such a central wire is bent as a beam which boundary displacements are obtained from the dynamic analysis. Such boundary linear and angular displacements are the same for each wire. Hence, for the purpose of stress calculation, it will be necessary to analyze only the central wire considering the values of the displacement u and rotation f at the end point of the messenger cable. Such variables were related to the shear force P and bending moment Ms (Fig. 2). The bending stress on the wire at the point of attachment between the clamp and cable i is calculated by the equation: (PL+Ms)d si⫽ 2Ia

(20)

The bending stress on the wire at the point of attachment between the weight and cable j is calculated by equation: Msd sa⫽ 2Ia

(21)

An equation for the bending stress for each instant is the following [7]:

冑

Uad A1⬘2+B1⬘2 s1⫽ sin(wt⫺fm) I⫽i,j 2Ia

The value of the maximum stress is necessary for the calculation of the damper life under conditions of the fatigue test. The damper must resist 107 cycles. Such value is indicative that the level of stress generated in the wire during the damper test would be less than the endurance limit and the damper theoretically would have infinite life. A first analysis is based on the stress life approach also called the classical theory of fatigue. Fig. 3 shows sA as a possible value for stress which is in the range of infinite life. If the test is run with a bending stress sB, the cable may have a duration less than 106 cycles. This period is shorter than 3 days of test for most dampers. Using the classical theory of fatigue for the number of cycles between 103 and 106, as in [8], the following equation is obtained: 1 LogN⫽ log(UaDmax)⫺C I⫽i,j b

The influence of the amplitude of excitation on the stress and on the duration is given by Eq. (30). Stress correcting factors Ka =0.64 and Kc=0.95 for surface finish and miscellaneous effects respectively need to be used [9]. Assuming a more general excitation amplitude in terms of the excitation frequency 25.4q/2f, where f is the

(22)

Where the values of A1 and B1 for the points i and j are respectively: 2 A1⬘2⫹2KaLAmaxcosbu⫹ KaL2ROTmaxcosbf 3

(23)

2 B1⬘2⫽⫺2KaLAmaxsinbu⫹ KaL2ROTmaxsinbf 3

(24)

4 Aj ⬘2⫹2KaLAmaxcosbu⫹ KaL2ROTmaxcosbf 3

(25)

(30)

Fig. 3.

Typical diagram S–N.

A. Lara-Lo´pez, J. Colı´n-Venegas / International Journal of Fatigue 23 (2001) 21–28

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first natural frequency in Hz and q is only a selected number, the following equation relating duration and amplitude of excitation is obtained.

冋

冉冊 册

q Dmax N⫽ 0.004762 f Su

1 − 0.157

(31)

For the particular case of the test, the vibration amplitude of the excitation table is 25.4/2f (mm). Dampers are accepted if a sample of these endure such vibration conditions for 107 cycles, according to laboratory requirements [10]. To reduce the duration of the current test, it is necessary to cause a stress corresponding to finite life. In such a case the bending stress on the wire should be higher than the endurance limits. Such stress may correspond to a fatigue life in the order of 105 cycles. However, to submit the damper to relatively high stress levels, corresponding to the 105 cycles of life, it is necessary to estimate the influence of the plastic strain. This type of behavior generally has been called low cyclic fatigue or cyclic strain fatigue. The transition from low cycle fatigue behavior to high cycle fatigue behavior generally occurs in the range of about 104 to 105 cycles. The equations of strain-based approach to fatigue are the following [8,9,11]:

冉 冊

⌬e ⌬s ⌬s (s⬘)⫽ ⫹ 2 2E 2K⬘

1 n⬘

(32)

s⬘f (e⬘f)n⬘

Table 1 Characteristics of a damper

d G J K L m N Su m s⬘f e⬘f n⬘ b c

Values Wire diameter, m Distance between o and o⬘, m Moment of inertia about the center mass kg m2 Stiffness of the cable N/m Length of cable, m Mass of one weight, kg Number of wires Ultimate strength, MPa Dimensionless damping Fatigue strength coefficient [8], MPa Fatigue ductility coefficient [8] Cyclic strain hardening exponent [8] Fatigue strength exponent [8] Fatigue ductility exponent [8]

(34)

where N is the number of cycles to failure and the other parameters are defined in Table 1. Such terms are parameters of fatigue that also depend on the material. The fatigue strength coefficient b given in Table 1 is affected by the stress correcting factors for the endurance limits by the equation [11]:

冉 冊

s⬘f −log Se b⫽ log(2Ne)

(35)

Where Se is the endurance limit affected by stress correcting factors and Ne is equal to 106. It is important to mention that the value of the parameter b also affects the value of K⬘ in Eq. (33) and therefore the value of total amplitude of strain ⌬e/2 is also influenced by b. In fact n⬘, b and c are related by [8,10,11]. (36)

(33)

where ⌬e/2 is the total strain amplitude, ⌬s/2 is the stress amplitude (s⬘), E is the modulus of elasticity and s⬘f , ⑀⬘f and n⬘ are parameters of fatigue defined in Table 1 which respective values depend on the material [8,12].

Symbol

⌬e s⬘f (N)⫽ (2N)b⫹e⬘f(2N)c 2 E

b n⬘⫽ c

and, K⬘⫽

The first term of the right side of Eq. (32) is the strain elastic amplitude and the second term is the strain plastic amplitude. The relation between the total amplitude of strain and stress amplitude can be transformed in other relations between total strain amplitude and life given by:

0.00287 0.007 0.0160 4741 0.205 3.907 19 1174 0.678 1174 0.5 0.17 ⫺0.085 ⫺0.5

A simplified version of Eq. (34) is the equation of universal slopes which is [8,9]: Su ⌬e⫽3.5 (N)−0.12⫹efN0.6 E

(37)

where N is the number of cycles to failure, Su is the ultimate strength and ef are parameters whose values are defined in Table 1. The transition fatigue life is derived by equating the terms of plastic amplitude of strain and the elastic amplitude of strain obtaining the following equation [8,11]:

冉 冊

e⬘fE Nt⫽ s⬘f

1 (b−c)

(38)

Therefore, both strain life and stress life approaches are needed to calculate endurance life under any conditions of excitation.

4. Application and experimental analysis A commercial damper was submitted to the fatigue test, following the method previously exposed. The stiff-

A. Lara-Lo´pez, J. Colı´n-Venegas / International Journal of Fatigue 23 (2001) 21–28

ness coefficient of the damper were calculated by Eq. (17). For determination of the dimensionless damping coefficient, it is necessary to establish one experiment as shown in Fig. 9. One velocity sensor is located on the center of mass of the damper weight and one more is located on the vibratory table which is tuned to the first natural frequency of the damper. Analytically, natural frequencies may be calculated from Eq. (4). Experimentally, the natural frequencies may be tuned searching for maximum displacement of the center of mass. Signals from both sensors are amplified and recorded in an oscilloscope as shown in Fig. 9(b). From these graphics, it is possible to determine the phase angle between the excitation and response bu and amplitudes X for the center of mass and Ua for the vibratory table. Such values were

25

Table 2 Comparison of experimental and calculated data Symbol

Parameter

f1

First natural 5.0 frequency, Hz Second natural 21.0 frequency, Hz Frequency of 5.0 excitation, Hz Amplitude of 0.0027 excitation, m Amplitude of 0.0039 response, m Angle of phase 93° Stress in one wire, MPa Number of ⬎107 cycles

f2 F Ua X ε s N

Experimental

Calculated 5.1 21.8 5.1 0.0027 0.004 43° 68 Long life

bu=93° X=0.0039 m Ua=0.0027 m Eqs. (18) and (19) were solved, obtaining the following values for m, the ratio R1 and the response of the system: m=0.678 R1=0.98 X⫽0.0039sin(⍀t⫺93)

(39)

For this particular case, ⍀ is equal to 5.1 Hz. Analytical and experimental responses are compared in Fig. 10, showing a maximum error of 1.6%. Once the dynamic responses of the damper are known, it is possible to calculate its life. For this particular case, the ultimate strength Su for the messenger steel cable is 1174 MPa. Its endurance limit is modified by the previous correction factors as follows [8,9]: Se⫽(0.64)(0.95)(0.5)(1174)⫽356.6MPa

(40)

The maximum stress generated in the cable is calculated by Eq. (28) giving the value of 68 MPa. Then the safety factor is 5.23. In fact the damper resisted the fatigue test (Nⱖ107). Experimental data and calculated values are shown in Table 2. As described above, the fatigue test of the damper was done at the first natural frequency of excitation f1 and at an amplitude equal to 25.4/f1 (mm). However, it is interesting to investigate the stress on the wire and the life of the damper if the frequency of excitation is equal to the second natural frequency, maintaining constant the amplitude of excitation. Fig. 4 shows the curves of the stress on points i and j for the damper. The value of the stress at the second natural frequency is greater than the stress at the first natural frequency. In addition the stress

Fig. 4. Calculated stress on individual wire as function of excitation frequency on points i and j.

on the point J is greater than the stress on the point i for the second natural frequency. Therefore, it is possible to find amplitudes of excitation which lead to the same value of the stresses as that of the first natural frequency, when exciting with a higher frequency. Experimental test showed that the dimensionless coefficient of damping decreases linearly respect to the amplitude of excitation. For this damper such relation is the following [7]: mB⫽119.2Ua⫹0.99

(41)

Fig. 5 shows an experimental curve of m versus q. Based on the previous equations, relationships of stress versus q and number of cycles versus q are plotted in Figs. 6 and 7 respectively. Values of q between 2.4 and 2.92 (5.98–7.26 mm of base displacement) produce a range of stresses between 358 and 968 MPa. The corresponding range of the number of cycles according to the strain life approach using Eq. (34) is between 1,000,000 and 50.4 cycles [curve (a)]. On the other hand, using the classical theory the number of cycles for values of q

A. Lara-Lo´pez, J. Colı´n-Venegas / International Journal of Fatigue 23 (2001) 21–28

26

Fig. 5.

Fig. 6.

Value of m versus q.

Stress on the single wire versus factor q.

between 2.4 and 2.92 is between 1,000,000 and 1742 cycles [curve (c)]. For the interval of low cycles (number cycles Nⱕ105) the corresponding interval of q is 2.5⬍q⬍3 [curve (a)]. Fig. 7 also shows the comparison between curves of life being calculated using the strain life approach, universal slope equation and the classical theory. Based on the strain life approach, a life equal to 105 cycles corresponds to the conditions of test Ua equal to 6.31 mm (q=2.54). The duration of the test would be approximately 5.45 h instead of 23 days as in the current test. If the life is equal to 5×105 cycles, the condition of test is Ua approximately equal to 6.06 mm (q=2.44) and the duration of the test would be approximately 23 h. A small difference in the values of amplitudes may be noted. Fig. 8 shows the total amplitude of strain versus number of cycles to failure, calculated using both Eq. (32) and Eq. (34) showing a total coincidence between them. Also, the corresponding terms of amplitudes of strain elastic and strain plastic are plotted. On the other hand, the value of transition fatigue life Nt calculated by Eq. (38) is equal to 37,485 cycles. The values presented in Table 3 show that the stress and duration are very sensitive to changes in excitation amplitude Ua. To run the fatigue test under conditions such as those presented in Table 3 or those shown in Figs. 6 and 7, a machine with an amplitude precision of ±0.01 mm would be required (Figs. 9 and 10).

Fig. 7. Number of cycles of life N versus factor q (a) strain life approach, (b) universal slope equation, (c) classical theory.

Fig. 8. Total amplitude of strain, amplitude of elastic strain, and amplitude of elastic strain versus cycles to failure calculated by Eqs. (32) and (34) are coincidents.

5. Conclusions The determination of the stiffness of the cable as an elastic beam and the dimensionless coefficient of damping from experimental response resulted in a relatively simple procedure. The method used for calculation of the life according

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Table 3 Stress on individual wire and cycles to failurea Ua[10⫺3] m

q 2.40 2.42 2.46 2.50 2.54 2.60 2.70 2.82 2.90 2.92 a

5.98 6.01 6.11 6.21 6.31 6.46 6.70 7.0 7.21 7.26

m 0.278 0.272 0.260 0.249 0.237 0.219 0.180 0.154 0.130 0.124

s⬘ MPa 358 368 391 416 444 491 589 756 918 968

N1 [106] 1 0.726 0.399 0.216 0.114 0.042 0.007 0.00058 0.000085 0.000050

N2 [106] 1 0.814 0.456 0.305 0.201 0.105 0.032 0.0063 0.0017 0.0012

N1 cycles calculated by the classical theory: Eq. (30); N2 cycles calculated by the strain life approach: Eq. (34).

Fig. 10. Experimental and analytical response for the center of mass of damper and vibratory table.

Fig. 9. (a) Damper on vibratory table with a velocity sensor located on a vertical line across the center of mass of the weight. (b) Response signal. One curve corresponds to the velocity sensor on the center of mass of the weight and the other to the velocity sensor on the vibratory table.

to the specifications of the test of dampers produced results similar to those obtained during the fatigue test. The method used for calculation of the stress and life for the messenger cable for a given amplitude of excitation may be used to reduce the duration of the test or to aid design of dampers. Stress on wires of messenger cable and its life are very sensitive to the changes of amplitude of excitation Ua, being in the range of finite life for high cycles (105⬍N⬍106) the following 5.96 mm⬍Ua⬍6.36 mm. For the low cycles approach the amplitude of the vibration table is in the interval 6.36⬍Ua⬍7.45 mm (N⬍105). Therefore it is necessary that the dynamic properties of dampers which are to be tested must be as equal as possible between them.

Acknowledgements The authors gratefully acknowledge the support of The Laboratory for Test of Equipment and Materials (LAPEM) of the Federal Commission of Electricity at Irapuato, Me´xico that made it possible to carry out the experimental investigation. Thanks to the facilities and support of the Faculty of Mechanical and Electrical Engineering of the University of Guanajuato at Salamanca the present research project was completed.

References [1] Stockbridge GH. Electr World 1925:1305. [2] Gilbert/Common Wealth. Transmission Line Reference Book. Wind-Induced Conductor Motion, Electric Power Research Institute, Inc. Palo Alto California, USA. [3] Claren R, Giorgio D. IEEE Trans Power App Sys 1969;88(12):1741–71. [4] Richardson AS. Performance requirements for vibration dampers. Electric Power Systems Research 1996;36:21–8.

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[5] Castello GA. J Energy Resour Technol 1983;105:337–40. [6] Meirovitch L. Analytical methods in vibrations. New York: McMillan, 1969. [7] Lara LA, Colı´n VJ. Fatigue analysis of aeolian-vibration dampers. Internal report (in Spanish), University of Guanajuato, Me´xico, 1995. [8] Banantine AJ, Comer JJ, Handrock JL. Fundamentals of metal fatigue analysis. Englewood Cliffs, NJ: Prentice Hall, 1990.

[9] Dieter GE. Mechanical metallurgy. New York: McGraw-Hill, 1988. [10] Federal Commission for Electricity (CFE). Stockbridge dampers for transmission lines. Standard CFE 511BO-36, 1994. [11] Dowling NE. Mechanical behavior of materials. Englewood Cliffs, NJ: Prentice Hall, 1999. [12] Jastrzehski ZD. The nature and properties of engineering materials. 3rd ed. New York: John Wiley and Sons, 1987.