If the amplitudes are small enough for Hooke's law to be valid, then the local tension is 



T= T(, + EAe (A4) 



and the local potential energy is equal to the product of the average local tension and the local strain. 



Expanding the local strain in a power series of dy/dx and neglecting terms higher than fourth order 



results in 



1/27^0 



+ 



EA - n 



dx. 



(A5) 



the local potential energy due to stretching. For all practical cases EA >> Tq and the contribution to 

 potential energy from stretching is 



J' 



-'0 



+ 



EA 



dx. 



\/2To 

 The kinetic energy of the cable in terms of the virtual mass density p is given by 



>/^r^- If' 



and the generalized work due to structural damping per unit length is 



dx 



C-is^SyS:. 



'0 =^ dt 

 After Hamilton's principle is applied, the final equation of planar cable motion becomes 



(A6) 



(A7) 



(A8) 



EipL+pA^ 



Tn + 



3EA 



dx' ^' dt ''■ 



It is convenient to transform this equation by 



niTX - 



-, y 



JL ; = 



(A9) 



(AlO) 



L ' ^ Y' 

 where n is the mode number, Y is the antinode amplitude, and w is the frequency of vibration. The 



resulting equation is 



d'y ^ EIn\^ d^y ■ 

 9/2 pAw'L^ dx^ 



Ton'rr' ^ ^EY'n'ir^ 



pAL' 



Ipoj'L' 



dx\ 



d'y ^ 8 dy 



Bx' 



2tt dt 



0, 



(All) 



in which 8 is the log decrement of the vibration (8 = Itt^s for small damping). The fluid forces that 

 act on a resonantly vibrating, taut cable are usually included on the right hand side of the equation of 



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