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Conductor gallop
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<h2 id="theoretical-analysis">Theoretical analysis</h2> <p>The earliest studies of long wires embedded in a moving fluid motion dates to the late 19th century, when <a href="/facts/Vincenc_Strouhal/ehNxD1y4">Vincenc Strouhal</a> explained "singing" wires in terms of <a href="/facts/Vortex_shedding/hVQpeN7l">vortex shedding</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> Gallop is now known to arise from a different physical phenomenon: <a href="/facts/Lift_(force)/ehQKKaWP">aerodynamic lift</a>. Ice accumulated on the wire destroys the <a href="/facts/Circular_symmetry/gEgSEFxy">circular symmetry</a> of the wire, and the natural up-and-down "singing" motion of a wire changes the <a href="/facts/Angle_of_attack/v5NkMgqu">angle of attack</a> of the iced wire in the wind. For certain shapes, the variation in lift across the different angles is so large that it excites large-scale oscillations.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p><p>Mathematically, an unloaded extended wire in dead air can be approximated as a mass m suspended at height y by a <a href="/facts/Spring_(device)/jknGXzTq">spring</a> with <a href="/facts/Spring_Constant/7hTUcKUm">constant</a> k. If the wind moves with velocity U, then it makes angle α with the wire, where </p><p> tan α = y ˙ U . {\displaystyle \tan {\alpha }={\frac {\dot {y}}{U}}{\text{.}}} </p><p>At large wind velocities, the lift and <a href="/facts/Drag_(physics)/vdl4jF7I">drag</a> induced on the wire are proportional to the square of the wind velocity, but the proportionality constants <i>C</i>L and <i>C</i>D (for a noncircular wire) depend on α: </p><p> F j = 1 2 ρ ( U 2 + y ˙ 2 ) l ⋅ C j ( j ∈ { L , D } ), {\displaystyle F_{j}={\frac {1}{2}}\rho (U^{2}+{\dot {y}}^{2})l\cdot C_{j}\quad \quad \quad {\text{(}}j\in \{{\text{L}},{\text{D}}\}{\text{),}}} </p><p>where ρ is the fluid density and l the length of the wire.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p><p>In principle, the excited oscillation can take three forms: rotation of the wire, horizontal sway, or vertical plunge. Most gallops combine rotation with at least one of the other two forms. For algebraic simplicity, this article will analyze a conductor only experiencing plunge (and not rotation); a similar treatment can address other dynamics. From geometrical considerations, the vertical component of the force must be </p><p> 1 2 ρ l ( U 2 + y ˙ 2 ) ( C L cos α + C D sin α ) ≈ 1 2 ρ l U 2 ( C L | α = 0 − y ˙ U ( C D + ∂ C L ∂ α ) | α = 0 ) , {\displaystyle {\frac {1}{2}}\rho l(U^{2}+{\dot {y}}^{2})(C_{L}\cos {\alpha }+C_{D}\sin {\alpha })\approx {\frac {1}{2}}\rho lU^{2}\left(C_{L}|_{\alpha =0}-{\frac {\dot {y}}{U}}\left.\left(C_{D}+{\frac {\partial C_{L}}{\partial \alpha }}\right)\right|_{\alpha =0}\right){\text{,}}} </p><p>keeping only terms first-order in the regime <i>ẏ</i> ≪ <i>U</i>.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> Gallop occurs whenever the <a href="/facts/Driven_harmonic_motion/7IQNToGt">driving</a> coefficient 1/2<i>ρlU</i> · (<i>C</i>D + ∂<i>C</i>L/∂<i>α</i>)|<i>α</i> = 0 exceeds the natural <a href="/facts/Damping/UW3RsjEd">damping</a> of the wire; in particular, a <a href="/facts/Necessity_and_sufficiency/G45aDCY9">necessary-but-not-sufficient</a> condition is that ( C D + ∂ C L ∂ α ) | α = 0 < 0 . {\displaystyle \left.\left(C_{D}+{\frac {\partial C_{L}}{\partial \alpha }}\right)\right|_{\alpha =0}<0{\text{.}}} This is known as the Den Hartog gallop condition, after the engineer who first discovered it.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a><a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> </p><p>At low wind velocities U, the above analysis begins to fail, because the gallop oscillation couples to the <a href="/facts/Vortex_shedding/hVQpeN7l">vortex shedding</a>.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> </p> <h2 id="flutter">Flutter</h2> <p>A similar aeolian phenomenon is <a href="/facts/Aeroelastic_flutter/xEL9ae5M">flutter</a>, caused by <a href="/facts/Vortex/K6oeqcuX">vortices</a> on the <a href="/facts/Windward_and_leeward/7LebHsNX">leeward</a> side of the wire, and which is distinguished from gallop by its high-frequency (10 Hz), low-amplitude motion.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a><a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> To control flutter, transmission lines may be fitted with <a href="/facts/Tuned_mass_damper/hFFzwrEl">tuned mass dampers</a> (known as <a href="/facts/Stockbridge_damper/JlTPST82">Stockbridge dampers</a>) clamped to the wires close to the towers.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> The use of bundle conductor spacers can also be of benefit. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Aeolian_vibration/xEL9ae5M">Aeolian vibration</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Moore, G. F. (1997), BICC Electric Cables Handbook, Blackwell Publishing, p. 724, ISBN 0-632-04075-0 <a href="0-632-04075-0" target="_blank">0-632-04075-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Guile, A.; Paterson, W. (1978). Electrical Power Systems. Vol. I. Pergamon. p. 138. ISBN 0-08-021729-X. <a href="0-08-021729-X" target="_blank">0-08-021729-X</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Pansini, Anthony J. (2004). Power Transmission and Distribution. Fairmont Press. pp. 204–205. ISBN 0-88173-503-5. <a href="0-88173-503-5" target="_blank">0-88173-503-5</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Ryan, Hugh (2001). High Voltage Engineering and Testing. IET. p. 192. ISBN 0-85296-775-6. <a href="0-85296-775-6" target="_blank">0-85296-775-6</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Pansini, Anthony J. (2004). Power Transmission and Distribution. Fairmont Press. pp. 204–205. ISBN 0-88173-503-5. <a href="0-88173-503-5" target="_blank">0-88173-503-5</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Ryan, Hugh (2001). High Voltage Engineering and Testing. IET. p. 192. ISBN 0-85296-775-6. <a href="0-85296-775-6" target="_blank">0-85296-775-6</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>McCombe, John; Haigh, F. R. (1966). Overhead Line Practice (3rd ed.). Macdonald. pp. 216–219. <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Pansini, Anthony J. (2004). Power Transmission and Distribution. Fairmont Press. pp. 204–205. ISBN 0-88173-503-5. <a href="0-88173-503-5" target="_blank">0-88173-503-5</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Moore, G. F. (1997), BICC Electric Cables Handbook, Blackwell Publishing, p. 724, ISBN 0-632-04075-0 <a href="0-632-04075-0" target="_blank">0-632-04075-0</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Guile, A.; Paterson, W. (1978). Electrical Power Systems. Vol. I. Pergamon. p. 138. ISBN 0-08-021729-X. <a href="0-08-021729-X" target="_blank">0-08-021729-X</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>For an example of a power outage caused by galloping resulting from ice buildup, see: "Delen van Diksmuide en Kortemark zonder stroom" [Parts of Diksmuide and Kortemark without power]. De Krant van West-Vlaanderen (in Flemish). 14 February 2013. Archived from the original on 15 April 2021. <a href="https://web.archive.org/web/20210415151223/https://kw.be/nieuws/delen-van-diksmuide-en-kortemark-uren-zonder-stroom/" target="_blank">https://web.archive.org/web/20210415151223/https://kw.be/nieuws/delen-van-diksmuide-en-kortemark-uren-zonder-stroom/</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Strouhal, V. (1878). "Ueber eine besondere Art der Tonerregung" [On an unusual sort of sound excitation]. Annalen der Physik und Chemie. 3rd series (in German). 5 (10): 216–251. Bibcode:1878AnP...241..216S. doi:10.1002/andp.18782411005. <a href="http://babel.hathitrust.org/cgi/pt?id=uc1.b4433702;view=1up;seq=230" target="_blank">http://babel.hathitrust.org/cgi/pt?id=uc1.b4433702;view=1up;seq=230</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>White, Frank M. (1999). Fluid Mechanics (4th ed.). McGraw Hill. ISBN 978-0-07-116848-9. <a href="978-0-07-116848-9" target="_blank">978-0-07-116848-9</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Den Hartog, J. P. (1985). Mechanical Vibrations. Dover. pp. 299–305 – via Knovel. <a href="https://app.knovel.com/hotlink/pdf/id:kt00AZTMV2/mechanical-vibrations/galloping-electric-transmission" target="_blank">https://app.knovel.com/hotlink/pdf/id:kt00AZTMV2/mechanical-vibrations/galloping-electric-transmission</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Blevins, Robert D. (1990). Flow-Induced Vibration (author's reprint; 2nd ed.). Malabar, Florida: Krieger (published 2001). pp. 104–152. ISBN 1-57524-183-8. <a href="1-57524-183-8" target="_blank">1-57524-183-8</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Blevins, Robert D. (1990). Flow-Induced Vibration (author's reprint; 2nd ed.). Malabar, Florida: Krieger (published 2001). pp. 104–152. ISBN 1-57524-183-8. <a href="1-57524-183-8" target="_blank">1-57524-183-8</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Den Hartog, J. P. (1985). Mechanical Vibrations. Dover. pp. 299–305 – via Knovel. <a href="https://app.knovel.com/hotlink/pdf/id:kt00AZTMV2/mechanical-vibrations/galloping-electric-transmission" target="_blank">https://app.knovel.com/hotlink/pdf/id:kt00AZTMV2/mechanical-vibrations/galloping-electric-transmission</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Blevins, Robert D. (1990). Flow-Induced Vibration (author's reprint; 2nd ed.). Malabar, Florida: Krieger (published 2001). pp. 104–152. ISBN 1-57524-183-8. <a href="1-57524-183-8" target="_blank">1-57524-183-8</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Blevins, Robert D. (1990). Flow-Induced Vibration (author's reprint; 2nd ed.). Malabar, Florida: Krieger (published 2001). pp. 104–152. ISBN 1-57524-183-8. <a href="1-57524-183-8" target="_blank">1-57524-183-8</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Guile, A.; Paterson, W. (1978). Electrical Power Systems. Vol. I. Pergamon. p. 138. ISBN 0-08-021729-X. <a href="0-08-021729-X" target="_blank">0-08-021729-X</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Pansini, Anthony J. (2004). Power Transmission and Distribution. Fairmont Press. pp. 204–205. ISBN 0-88173-503-5. <a href="0-88173-503-5" target="_blank">0-88173-503-5</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>McCombe, John; Haigh, F. R. (1966). Overhead Line Practice (3rd ed.). Macdonald. pp. 216–219. <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> </ol>