The Long Span Problem in the Analysis of Conductor Vibration Damping

Transcription

1 770 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 5, NO., APRIL 000 The Long Span Problem in the Analysis of Conuctor Vibration Damping Charles B. Rawlins, Fellow, IEEE Abstract In analysis of vibration amping of overhea spans, expecte severity is etermine on the basis of the balance of vibration power supplie by the win through vortex sheing, against issipation by self amping in the conuctor an by amping evices. Conventional practice is to approximate the vibration of the span as a series of sine-shape loops, all of equal amplitue, for purposes of estimating win input an self amping. However, for long spans requiring efficient amping, most of the loops are not sine shape, an loop amplitue varies along the span. The effects of these ifferences are explore by representing the vibration as opposite-moving waves that grow as they travel. Nonlinearity in the win excitation an self amping functions is taken into account. Results show that so-calle long span effects significantly reuce require span-en amping in long spans. Inex Terms Aeolian, vibration, spans, power balance, waves, traveling, win, self amping, amping, conuctor. NOMENCLATURE A Single amplitue of wave traveling in +x ir. B Single amplitue of wave traveling in 0x ir. c Wave velocity, see (6) Conuctor iameter E Energy per cycle of vibration E Tot Store energy in span EI Conuctor flexural rigiity f Frequency Hz h Damping ratio; see (5) L Span length m Conuctor mass per unit length P Vibration power p Power per unit length of conuctor Power issipate by perfect span-en amping setup; see (7) r Net istribute resistance acting on conuctor S Inverse staning wave ratio; see (8) S D Require amping, ignoring long span effects, see (3) S I Require amping, accounting for long span effects s S=x T Conuctor tension y Displacement of conuctor ue to vibration ^y Local single peak amplitue y min Noe amplitue single peak; see () y max Free loop amplitue single peak; see (0) y 0 Amplitues of A an B waves at unampe en of span, or at mile of symmetrically ampe span. Manuscript receive January 7, 999; revise September 5, 999. The author is with Alcoa Conuctor Accessories, Spartanburg, SC Publisher Item Ientifier S (00) y 00 Conuctor y=@x ^y 00 Peak local curvature p Z 0 Characteristic impeance, Tm Real propagation constant; see (5), (6) r Reuce ecrement of win input power rs r for staning sine loops (8) rv r for traveling waves (3) s Logarithmic ecrement for sine-shape loops " Hysteritic loss tangent for conuctor self amping (37) Loss factor; see (5) Mass ensity of air! Circular frequency, f Subscripts: C; c Conuctor self amping D; Span-en amping s Sine-shape loops W; w Win power input v Traveling waves I. INTRODUCTION CONVENTIONAL practice in analysis of aeolian vibration of overhea conuctors relies on the balance of vibration energy or power in a span. For single conuctor spans, the elements of this balance are the power supplie by the win, P w ; power lost to self amping in the conuctor, P c ; an the power P D issipate at the en of the span by a amping arrangement. The balance is expresse [], P D = P W 0 P C : () It may also be expresse in terms of energy per cycle: E D = E W 0 E C : () Although it is generally the case that several vibration frequencies are present in actual spans at any particular time, resulting in beats in the vibration pattern, analysis usually assumes single-frequency, steay vibration in evaluating the power balance. This approach is compelle by the fact that ata on P w, P c an P D are obtaine from win tunnel an laboratory tests that are feasible only for single frequency test conitions. The vibration of a span is comprise of traveling waves. These waves receive, store an transmit the vibration power from the win. For the case where they eliver power to span-en amping, they must grow as they travel, harvesting the win s energy as they go, so the wave amplitues must vary with position in the span. Since there are waves moving in both irections, both growing in their respective irections /00$ IEEE

2 RAWLINS: ANALYSIS OF CONDUCTOR VIBRATION DAMPING 77 of travel, their variations with span location ten to balance out, an conventional practice assumes that the amplitue of vibration in the free loops is constant along the span. Uner that assumption, win input an self amping are taken to be uniformly istribute over the span. Win input per unit length, p w, an self amping per unit length, p c, are then constant over the span, an the power balance can be written, P D = L (p w 0 p c ): (3) It is convenient to normalize (), () an (3) by iviing them by some reference power or energy to rener them imensionless. There are two ways to o this in current use. For energybalance. the ivisor is taken as the total energy, potential an kinetic, store in the span, usually ignoring what is store in the amper. If pure sine-loop vibration is assume, this energy is, E Tot = m 4! y max L: (4) Parameters are then efine [], [3]: h = 4 E = E Tot E (5) E Tot calle respectively amping ratio an loss factor. Energy balance may then be expresse, h D = h w 0 h c or D = w 0 c : (6) Fig.. Stroboscopic view of vibration loop. The wave power lost in reflection is the power absorbe by the amper, so P D = P A 0 P B : (0) On the basis of (9), if the amper were to allow no reflection (B =0), then P D =. Thus, is the power that woul be issipate by a perfect amper, a perfect wave absorber, given the frequency an free loop amplitue. The ratio of power actually issipate to that which woul be issipate by a perfect amper, P D =, is calle amping efficiency. In view of (8), (9) an (0), P D = is equal to the inverse staning wave ratio, S. Power balance may then be expresse, Conceptually, the issipation provie by the amper is sprea uniformly along the conuctor as though it were aitional self amping. Power balance () may be normalize by iviing by, where [], = Z 0! y max : (7) or, Letting, P D = P w 0 S D = S w 0 S c : P c () Some explanation of this choice may be worthwhile. Although ampers are inee energy issipation evices, the function of a amping setup on an overhea span is that of a wave absorber. At an unampe support, the waves carrying the energy of vibration are simply reflecte back into the span to continue growing. The function of the amper is to absorb some of the wave energy uring reflection, offsetting any growth it might have ha since its last trip up an own the span. If the amplitue of the wave approaching the amper is A, an that of the reflecte wave is B, the reflection ratio B=A gives a measure of the effectiveness of the amper as a wave absorber. A closely relate measure is, 0 B S = A + B A = A 0 B A + B : (8) S is calle inverse staning wave ratio because it is the inverse of the ratio of free loop amplitue, y max = A + B, to the noal amplitue, y min = A 0 B. These amplitues are illustrate in Fig.. If the powers conveye by the waves are P A an P B, respectively, then it can be shown [] that, P A 0 P B = A 0 B A + B : (9) s w = p w an s c = p c () S D = L (s w 0 s c ): (3) Eqs. (6) an (3) are alternative methos of normalizing the balance-of-energy or -power relationship in a span. The latter is more convenient, particularly for the present iscussion, for two reasons. First, it keeps the istribute elements of (), P w an P c separate from the concentrate element P D. Division of () by the store energy istribute over the span E Tot mixes the two types. If, as we shall iscuss below, y max varies along the span, the efinition (4) is replace by an integral, an the physical meaning of h D an D becomes somewhat obscure. In contrast, is a point parameter, efine by (7) at the place where the amping setup interfaces with the rest of the span. Its meaning is not affecte by the variation of y max along the span. Secon, the parameter P= measures the amping actually available from a amping setup, an also the amping emane by the span, as a ratio to what is possible: a perfect amper. That can be illuminating. Note, for example, that the left han sie of (3) cannot excee unity, because that is all a perfect amper can eliver. The right han sie can achieve any value, given a long enough span. Eviently, there can be spans

3 77 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 5, NO., APRIL 000 that simply cannot be ampe from the ens because they are too long. II. THE LONG SPAN PROBLEM There are actually two long span problems. One is the practical problem of how to amp them. The other is how to analyze them. This paper concerns the secon problem. Available amper technology is far from being able to provie amping setups having amping efficiencies approaching unity over a significant range of vibration frequencies. Even when such setups are carefully tailore through laboratory span testing, few can reliably maintain even S D = 0:5 over the range of frequencies where the most amaging vibration occurs in long spans. It was this practical limitation that le to aoption of so-calle in-span amping for many long river crossings [4]. In-span amping places groups of ampers at intervals along the span, say at the one-thir points, to effectively ivie long spans into several shorter, an more manageable subspans. Other approaches to the practical problem inclue: employing conuctor materials, principally steel [5], with greater resistance to fatigue; an configuring amping arrangements to make the span more tolerant of high amplitue vibration, principally by iviing wave reflection at span ens among a number of points so that high bening stresses o not occur at any of them [6]. Thus, there are resources available for ealing with the practical problem, even though they may entail consierably increase cost in installation an perhaps laboratory testing. The analytical problem is that of etermining when these resources nee to be applie. The ifficulty with this problem is that threshol conitions for triggering their use lie in the range of line esign parameters where the approximations that unerlie (6) an (3) begin to break own. Specifically, the spans are long enough that there is significant variation of y max.in aition, there is substantial eparture of loop shapes from the sinusoi assume in (4), an use in obtaining laboratory ata on p c [7] an certain sets of win tunnel ata on p w [8], [9]. Inverse staning wave ratios consierably greater than zero occur. Both the variation of y max an the eviation in S cause p w an p c to vary over the span. The purpose of this paper is to present a metho for taking these eviations into account. III. LINEAR ANALYSIS OF THE PROBLEM It is helpful to begin by exploring the gross effects of large span length through a linear approximation. Aeolian vibration is self-excite, that is, the vibration itself brings into being the forces that cause it. The Kármán vortices that supply the forces ten to be ranomly out of phase at points more than a few conuctor iameters apart, so the forces within a vibration loop ten to cancel out. The conuctor s motion tens to synchronize the vortex phases so that their forces act in concert, an this tenency is greater, the larger the amplitue [0], []. Such self-excite forces are often represente by a negative mechanical resistance for purposes of analysis. In the case of aeolian vibration, this negative resistance varies with vibration amplitue, making the phenomenon nonlinear. However, for the moment, it is convenient to treat it as constant, with negative resistance r w per unit length of conuctor. Self amping, which may be represente by a positive mechanical resistance, also varies with amplitue an thus is nonlinear. However, again, it is convenient to ignore that variation for the moment an assign a value r c to it. The net mechanical resistance acting on the conuctor is r = r w 0 r c. Now, the equation of motion for a vibrating conuctor is, + y =0: The effect of conuctor stiffness is neglecte, since it is generally small, an is negligible in the context of the linearization approximation. This equation amits solutions, y = y 6 e 6(0j(!=c))x e j!t (5) where, when r= p Tm is small, r T c = m an = r 4 p Tm (6) p Z 0 = Tm is the characteristic impeance of the conuctor. If the origins of x an t are chosen for convenience, the motion of the conuctor can be written, i y = hy + e (0j(!=c))x + y 0 e 0(0j(!=c))x e j!t : (7) This represents a traveling wave moving in the positive x irection an growing exponentially at the rate, an another moving negatively, an growing at the same exponential rate in its irection of travel. In what follows, it is convenient to consier a span with amping at only one en, even though long spans always have amping at both ens. The iscussion can be applie to such spans simply by consiering the unampe en in what follows to correspon to the mile of a fully ampe span, where ientical amping setups are applie at the two ens. The reflecte B wave from one en simply becomes the incient A wave for the other en as it passes that point. Suppose then that the span has amping at only one en. At the other, there is complete wave reflection, so the y + an y 0 waves have equal amplitues (but opposite phase) there, the magnitue of which we call y 0. Then, i y = y 0 he (0j(!=c))x 0 e 0(0j(!=c))x e j!t : (8) The local amplitues of the two waves can be written, A = y 0 e x B = y 0 e 0x : (9) The local free-loop amplitue, as a function of position in the span, is, y max (x) =A + B = y 0 0 e x + e 0x The noe amplitue is, =y 0 cosh x: (0) y min (x) =A 0 B = y 0 0 e x 0 e 0x =y 0 sinh x: ()

4 RAWLINS: ANALYSIS OF CONDUCTOR VIBRATION DAMPING 773 From (0) an (), the inverse staning wave ratio is, S(x) = y min(x) =tanhx: () y max (x) Then the amping efficiency require of the amper to maintain steay vibration is, S(L) = tanh L: (3) Note that for sine-shape loops the power transferre to the conuctor by r, average over a loop, is, p = 4 r! y max : (4) Thus, from (7) an (), the effective value of is (s w 0s c )=. Substituting that in (3) an referring to (3), we have, S(L) = tanh S D : (5) In the linear case, the effects of variations in loop shape an free loop amplitue over the span can be accounte for simply by calculating require amping efficiency ignoring those variations, an then taking the hyperbolic tangent. Now, the hyperbolic tangent of any real number is always less than unity. This means that the amping efficiency require of a amping arrangement to limit the vibration to any particular level is always less than unity. That, in turn, means that there is no span that is too long to amp, even with amping at only one en, contrary to the inication of (3). Of course this result is of only acaemic interest, because amping setups with efficiencies approaching unity are not practically available. It oes suggest, however, that (3) may overstate the require amping for conitions that are within reach of what is available. The iea that, even theoretically, there is no span too long to amp is counter-intuitive. The apparent anomaly is explaine by the fact that, when require amping efficiency approaches unity, the amplitue at the unampe en of the span approaches zero. For example, if the span is so long that the require amping, S(L) = 0:99, so that, from (3), L = tanh 0 0:99 = :647, then cosh L =7:09. Thus from (0), y 0 = y max (L)=7:09. But y 0 is the free loop single amplitue at the unampe en ot the span, y max (0). The amper is vibrating at a high amplitue, issipating substantial power, while the other en of the span an, in fact, much of its length, is vibrating at much smaller amplitue an consequently gathering only small power from the win. The fact that the A an B waves grow at the same exponential rate, since they are both expose to the same r, has an interesting an useful consequence. From (9), A B = y 0 : (6) This means that the prouct of their local amplitues is constant over the span, an equal to their prouct at the unampe en of the span. This result oes not epen upon r being constant over the span. The span can be ivie into increments short enough that r is essentially constant within each one, so A B is preserve over the increment. The step change in r between increments causes a slight reflection of each wave, but the effect is small enough to ignore. IV. WIND INPUT AND SELF DAMPING FOR TRAVELING WAVES However, the problem is not linear. Both r w an r c vary with local amplitue, which varies not only along the span ue the variation in y max, but within iniviual vibration loops. Furthermore, since the inverse staning wave ratio S varies along the span, the loop shape, which is characterize by y min =y max, oes also. In orer to trace the growth of the waves for the nonlinear case, it is necessary to have expressions for r w an r c as functions of local amplitue, ^y. Now, ata on win input an self amping is generally presente for pure sine-shape loops. Reporte, an generally the measure values are the integrals of power over a loop, ivie by loop length [7] [9]. Thus, this ata nees to be e-integrate to fin the power that woul occur at the iniviual amplitues foun within the loop, i. e., p w (^y) an p c (^y). These can then be integrate over the nonsinusoial loop shapes an varying free loop amplitues that actually occur in the span. The following subsections construct analytical expressions for local r w an r c, to make that possible. A. Win Power Input For win input to sine-shape loops, power from the win per unit length of conuctor may be expresse as [], p ws = f 3 4 Fnc(y max =): (7) The form of the equation is base on imensional consierations. More conventionally [3], [8], it is expresse in terms of reuce amping or reuce ecrement, rs, efine by, rs = m ws (8) where ws is the logarithmic ecrement ue to win excitation for sine-shape loops. For small values of ws such as foun in aeolian vibration, the power from the win to sine-shape loops, per unit length, is, p ws f ws E Tot L : Substituting from (4) an (8), we have, p ws = f 3 4 ymax rs : (9) It is useful to express rs as a polynomial, rs = b 0 + b y max ymax + b + : (30) The functions corresponing to (9) an (30), if available for traveling waves, woul take the form, ^y p wv = f 3 4 rv (3) rv = a 0 + a ^y 3 ^y ^y + a + a 3 + : (3) The factor on the right in (3) is ue to the fact that, given the frequency an amplitue, E Tot is twice as large for traveling waves as it is for sine loops.

5 774 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 5, NO., APRIL 000 TABLE I ILLUSTRATIVE COEFFICIENTS FOR (3) If we use (3) an (3) to integrate p wv over a loop we get, Z = p ws = f 3 4 ^y rv ^y x: (33) 0 When comparing (33) an (9), it turns out that the coefficients a i an b i are relate in the following manner. Fig.. Net propagation constant Finch ACSR at 36% RS. Now " an to a lesser extent EI are functions of K. It is convenient to efine, For purposes of illustration, we will use a set of coefficients, erive from [4], where the effect of win turbulence with intensity of 5% has been impose through calculation upon the win tunnel ata. The coefficients b i from the sine loop win tunnel tests were e-integrate by iviing them by the above factors, to yiel the a i coefficients for the traveling wave case liste in Table I. Since, locally, then, from (3), p = r (w^y) (34) r w = f rv : (35) B. Conuctor Self Damping Energy is lost to self amping through hysterisis as the conuctor is flexe back an forth by the vibration [5]. The energy require per unit length to ben the conuctor to peak local curvature ^y 00 is, E b = EI (^y00 ) : (36) The fraction of E b that is lost to hysterisis per cycle is calle the loss tangent, ". Thus, self amping power is, p cv = f " EI (^y00 ) : (37) From (8), the peak local curvature is, ^y 00 =! c ^y: (38) The nonimensional peak local curvature, equivalent to the strain at the conuctor surface, is, K = ^y00 =! ^y c : (39) "EI 4 = c 0 + c K+ c K + : (40) Then from (37) an the first part of (39), p cv =f K 0c 0 + c K+ c K + : (4) Laboratory ata on self amping is obtaine in tests where the vibration is in the form of staning sine-shape loops [7]. Thus, the polynomial corresponing to (4) from the laboratory tests must be e-integrate as was one for win input. The same coefficient multipliers are applicable. From (4) an (34), r c 4 f 3 4 c 4 (c 0 + c K): (4) C. Net Input From (6), (35), the secon part of (39) an (4), = f rv (^y=) 0 4 f 4Z 0 c 4 c0 + c 4 f c ^y : (43) Fig. shows, base on (43), for Drake ACSR at 5% RS using the values from Table I for a i, an the following values for the coefficients c i in (4): c 0 =5MPa c =0:4 TPa: The curve in the ^y= 0 f plane efine by the toes of the -profiles gives the limiting amplitue for traveling wave growth as a function of frequency. V. THE NONLINEAR CASE The -profiles in Fig. illustrate clearly that cannot be constant over a span where there is a significant range of variation in ^y=, such as implie by (0). Thus, the solution (5) of (4) can apply only locally, where is effectively constant. The

6 RAWLINS: ANALYSIS OF CONDUCTOR VIBRATION DAMPING 775 Fig. 3. Calculate span-en amping require to limit vibration to various levels of fymax, ignoring long span effects, an proviing for them through integration. 500 m span of Finch ACSR at 36% RS. span-wie variations in y max an y min given in (0) an () are not accurate, an must be etermine instea through integration of the net power per unit length along the span. The integration starts at the unampe en, where some free loop single amplitue y 0 is assume, an the A an B wave amplitues are both equal to y 0. Then, for each increment x along the span, is calculate from the local amplitue via (43) an the A an B amplitues are incremente by, A =A x B =B x: (44) At each step in the integration, it is necess recalculate ^y=, taking into account not only the changes in A an B but also the changing position in the vibration loop. The necessary relationship is the profile of the loop, illustrate in Fig., an this profile is given by [], r ^y = A + B 0 AB cos 4x : (45) The amount of calculation involve is reuce by taking avantage of (6). Integration over the span yiels the values of A an B at the amper en. These are then use to calculate y max = A + B there an, via (8), the amping efficiency S require to hol that amplitue. Orinarily, the value of S for a specifie y max at the amper en is esire, so it is necessary to integrate repeately, iterating on y 0, to get the esire solution. Because of the complicate nature of (43) an (45), an analytical solution like that for the linear case (8) cannot be given. Instea, the integration must be one numerically. Fig. 3 shows a typical set of results. The soli curves in Fig. 3 show the amping efficiency require at each en of the span, to limit the severity of vibration to various levels of fy max at the (ampe) ens of the span. fy max is a parameter that is proportional to the bening stress at a fixe clamp [6], that type of support generally being the worst fatigue environment a conuctor can face. To put the values in Fig. 3 in context, the enurance limit for ACSR is liste in [6] as 8 mm/s. A safety factor of or 3 is usually sought, in part Fig. 4. Ratio of require amping efficiency, accounting for long span effects, SI, to that ignoring them, SD. to minimize wear an risk of fatigue in the amping setup, so fy max of 50 mm/s might be a practical target level. The ashe curves in Fig. 3 show the require amping efficiencies S D, as calculate by (3), i. e., assuming that y max is constant over the span an that the win input an self amping are those for pure sine-shape loops. It is evient from Fig. 3 that long span effects reuce the level of amping actually require, an the reuction becomes quite significant when (3) calls for S D > 0:5. This result comes about whether the require high amping is ue to seeking conservative, low levels of fy max, or to aopting a certain value of fy max an consiering longer an longer spans. Even at the amping levels S D of 0. to 0.3 of concern in typical overlan spans, long span effects reuce require amping by 0% or more. The lack of an analytical solution to the integration of (44) makes it har to generalize on long span effects. However, the results of numerical integration have been foun to fall in a rather simple pattern that is relatively insensitive to most parameters of the problem. That pattern is shown in Fig. 4. Results of iniviual integrations are shown by iamons an triangles, an represent the ratio of require amping efficiency with long span effects accounte for, S I,toS D where they are not. The soli curve is the correction for long span effects surmise in (5) for the linear case. The results in Fig. 4 cover several conuctors of quite ifferent strength to weight ratio, several tension levels expresse as T=m, an several values of fy max. Inspection of the etaile results shows that the lower fringe of the scatter ban is associate with low frequency sies of curves such as shown in Fig. 3, an the upper fringe with the high frequency sies. Points corresponing to the crests of the curves are a little above the lower fringe. Although the pattern in Fig. 4 is insensitive to most parameters, it is influence by the turbulence level assume in win. Figs., 3 an 4 pertain to 5% turbulence, which is near the low en of the range experience by overhea spans. At higher turbulence levels, the scatter ban of Fig. 4 is higher. The amelioration ue to nonlinear effects is reuce. At 5% turbulence, it is roughly half as great.

7 776 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 5, NO., APRIL 000 For spans, such as river crossings, long enough to require span-en amping near the limits of what is feasible in practice, the reuction may be as much as 5%. In typical overlan spans, the reuction may be as much as 0%. Fig. 4. Averaging r over a loop. The ifference in Fig. 4 between the scatter ban an the tanh S D curve is attributable to the nonlinearity of reflecte in (43). The reason nonlinearity has this effect is illustrate in Fig. 5. This figure shows one of the profiles from Fig. converte to net istribute resistance r via (6), plotte as a function of ^y. Now, the power elivere to the loop is the integral of (34) over the loop. Thus, average power is proportional to the average of r, weighte accoring to the istribution of ^y. Note that ^y ranges between y min an y max. Thus, for sine-shape loops, where y min =0, the average is taken over the range a c in Fig. 5, weighte accoring the sine loop istribution shown: ymax sin x=. For a loop of the same amplitue, y max,but where y min > 0, ^y must be taken from (45). This istribution is shown for S = 0:33 in the figure. It has greater kurtosis than that for the sine loop, thus giving increase weight to the high-amplitue, low r range. More importantly, it averages r over the range b c, where the simple average of r is less than for a c. The result is that the ownwar slope of the profiles takes the most vigorous range of out of play as S becomes larger. xin the linear case, is constant; it has no slope. The scatter ban of Fig. 4 merges into the linear solution: tanh S D. When turbulence is greater than 5%, nonlinearity has less effect because the ownwar slope of the rv function in (43) turns out to be less. VI. CONCLUSIONS Long span effects reuce the levels of span-en amping require to limit the severity of vibration to particular values, compare to requirements estimate from conventional technology ignoring those effects. There are two effects. One arises from the significant variation in vibration wave amplitues that occurs over long spans. The other is cause by nonlinearity in the self-excititation mechanism. The effects are aitive. REFERENCES [] J. S. Tompkins, L. L. Merrill, an B. L. Jones, Quantitative Relationships in Conuctor Vibration Damping, AIEE Transactions on Power Apparatus & Systems, pt., vol. 75, pp , 956. [] G. Diana, M. Falco, A. Curami, an A. Manenti, A Metho to Define the Efficiency of Damping Devices For Single an Bunle Conuctors Of EHV an UHV Lines, IEEE Trans. on Power Delivery, vol. PWRD-, no., pp , Apr [3] Denis. U. Noiseux, Similarity laws of the internal amping of strane cables in transverse vibrations, IEEE Trans. on Power Delivery, vol. 7, no. 3, pp , July 99. [4] C. B. Rawlins, In-Span Damping Cuts Vibration, in Electrical Worl, Aug., 96. [5] L. H. J. Cook an R. A. McLachlan, Jervis inlet overhea crossing, in AIEE Conf., Paper no. CP [6] M. Ervik, Vibration amping on long Fjor crossings Theoretical investigations, IEEE Transactions on Power Apparatus an Systems, vol. PAS-00, no. 4, pp , Apr. 98. [7] IEEE Guie on Conuctor Self-Damping Measurements, IEEE St [8] C. B. Rawlins, Win Tunnel Measurements of the Power Imparte to amoel of a Vibrating Conuctor, IEEE Trans. on Power Apparatus an Systems, vol. PAS-0, no. 4, pp , Apr [9] D. Brika an A. Laneville, A Laboratory Investigation of the Aeolian Power Imparte to a Conuctor Using a Flexible Circular Cyliner, IEEE Trans. on Power Delivery, vol., no., pp. 45 5, Apr [0] M. Preiswerk, Contribution to the Solution of Vibration Problems on Aerial Lines, in CIGRE, Paris, 937, Paper no.. [] G. H. Toebes, Fluielastic Features of Flow Aroun Cyliners, in Proceeings, Int l. Research Seminar on Win Effects on Builings an Structures, vol. II, Ottawa, Canaa, Sept. 5, 967, pp [] F. B. Farquharson an R. B. McHugh Jr., Win Tunnel Investigation of Conuctor Vibration Using Rigi Moels, AIEE Trans. on Power Apparatus & Systems, pt. III, vol. 75, pp , 956. [3] O. M. Griffin, Vibrations an Flow-Inuce Forces Cause by Vortex Sheing, in Proceeings, Symposium on Flow-Inuce Vibrations, vol., 984, pp. 3. [4] C. B. Rawlins, Moel of Power Imparte by Turbulent Win to Vibrating Conuctor, Alcoa Conuctor Proucts Company, Spartanburg, SC, Tech. Note no. 3, Nov [5] D. U. Noiseux, Formulation of Internal Losses of Strane Conuctors by Means of a Complex Flexural Rigiity, in International Symposium on Overhea Conuctor Dynamics. Toronto, June 8 9, 98. [6] Win Inuce Conuctor Motion. Palo Alto, CA: Transmission Line Reference Book, Electric Power Research Institute, 980, ch.. Charles B. Rawlins (M 6 SM 79 F 8) was born in Annapolis, M. on July 4, 98. He grauate from Johns Hopkins University, Baltimore, M. with B. E. egree, an receive his M. S. from Clarkson University in 965. He joine Alcoa Laboratories in 949, an specialize in overhea conuctor ynamics throughout his career. He conucte research in aeolian vibration, galloping an wake-inuce oscillation of overhea conuctors, fatigue an aeroynamic characteristics of conuctors, an other relate areas. He is the author of a number of papers an articles, an co-author of the EPRI Transmission Line Reference Book, Win-Inuce Conuctor Motions. He hols nine patents. Mr. Rawlins is member an past chairman of the Working Group on Conuctor Dynamics of the Subcommittee on Towers, Poles an Conuctors. He is a member of CIGRE an is U.S. Member of Working Group -, Mechanical Behavior of Conuctors an Fittings. He is a member of ASME an Sigma Xi.