Closed-Form Formula of the Transverse Dynamic Stiffness of a Shallowly Inclined Taut Cable

Table 2

Comparison of values of dynamic stiffness at different locations.

Method^1	0.05	0.1	0.25	0.5	0.75	0.9	0.95
One 𝐾11^2	1.4807×10^6	7.2566×10^5	3.3728×10^5	2.4966×10^5	3.3100×10^5	7.1567×10^5	1.4695×10^6
Two (𝐾11)	1.4690×10^6	7.1449×10^5	3.2801×10^5	2.4356×10^5	3.2801×10^5	7.1449×10^5	1.4690×10^6
Three (𝐾11)	1.5323×10^6	7.4813×10^5	3.4502×10^5	2.5732×10^5	3.4728×10^5	7.5560×10^5	1.5478×10^6
Relative error^3 (%)	3.3659	3.0044	2.2433	2.9752	4.6871	5.2844	5.0586
One 𝐾22^2	114.47	111.96	110.67	110.40	110.69	111.99	114.50
Two (𝐾22)	114.47	111.95	110.66	110.38	110.66	111.95	114.47
Three (𝐾22)	127.29	124.55	123.18	122.93	123.28	124.70	127.39
Relative error (%)	10.07	10.11	10.15	10.19	10.21	10.20	10.11

Note: ¹method one is the analytic method considering sagging effects, which corresponds to (42). Method two is the analytic method without considering sagging effects, which corresponds to (48). Method three is the finite element method. ²𝐾₁₁ and 𝐾₂₂ are the first and second terms on the main diagonal of matrix, respectively. ³The relative error indicates the difference between the results obtained via method one and method three.