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By neglecting the sub-span motions between the conductors, a bundle is modelled as an equivalent single conductor so that the initiation conditions for galloping, periodic and quasi-periodic states and their stability conditions are considered by taking advantage of previous achievements for a single conductor.
In particular, we scrutinize the occurrence of buckled solutions in connection with the length of each sub-span of the bridge.
The existence of buckled solutions is investigated in connection with the length of the sub-spans.
Figure 1 The joint connecting two consecutive sub-spans and the intermediate pier.
We prove that they exist provided that the lengths of the sub-spans are properly chosen.
Now we consider a bridge with a single pier at x = x 1 and two sub-spans.
In this section we generalize the problem to a bridge with N sub-spans and M = N − 1 piers.
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Remark 3.3 In the limit κ → 0, from (2.9) it follows ε κ → 1 / 2, so that buckled solutions exist even if all sub-spans are equal.
The main results of this paper concern the steady states analysis of a bridge with N = 1, 2, 3 sub-spans and are stated in Section 2.
Solutions u ( 2 m ) + and u ( 2 m ) −, m = ( M + 1 ) / 2 = N / 2, are assumed to change the sign alternately on the sub-spans.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com