Abstract
The Exterior-Matrix Method (EMM) is applied to the in-plane dynamical equations of a shallow elastic catenary. The transfer matrices of the composite system are expanded so that asymptotic analysis can be done for the general case. The main idea is to find the 4-by-4 transfer matrices for each span of the cable system, and the transfer matrices for the joint, which could be an insulated pole. Then, by using the exterior matrices, the 6-by-6 matrices which correspond to the transfer matrices can be found. At first it seems surprising to use 6-by-6 matrices instead of the 4-by-4 transfer matrices, but with the exterior matrices, the determinant of the product need not be taken. This allows one to immediately take approximations of the matrices without worry that the highest-order terms will cancel. As a bonus to using the exterior matrices, an angle for the insulator will be allowed, which generalizes the work done by Simpson (Proc IEE 113(5):870–878, 1966). The practical use of this method is illustrated by finding the normal frequencies of a symmetrical 3-span system.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Irvine M (1981) Cable structures. Dover Publications, Inc, New York
Paulsen W (2007) The exterior matrix method for sequentially coupled fourth order equations. J Sound Vibr 308: 1–32
Triantafyllou MS, Grinfogal L (1986) Natural frequencies and modes of inclined cables. J Struct Eng 112(1): 139–148
Grinfogel L (1984) Dynamics of elastic taut inclined cables. Thesis presented to the Massachusetts Institute of Technology, Cambridge, MA, June 1984, in partial fulfillment of the requirements for the degree of Master of Science
Paulsen W, Slayton G (2006) Eigenfrequency analysis of cable structures with inclined cables. Appl Math Mech 27(1): 37–49
Simpson A (1966) Determination of the inplane natural frequencies of multispan transmission lines by a transfer matrix method. Proc IEE 113(5): 870–878
Paulsen W (1995) Eigenfrequencies of curved Euler–Bernoulli beam structures with dissipative joints. Q Appl Math 53(2): 259–271
Paulsen W (1997) Eigenfrequencies of the non-collinearly coupled Euler–Bernoulli beam system with dissipative joints. Q Appl Math 55(3): 437–457
Rienstra SW (2005) Nonlinear free vibrations of coupled spans of overhead transmission lines. J Eng Math 53: 337–348
Yuan J, Dickinson SM (1992) On the use of artificial springs in the study of the free vibrations of systems comprised of straight and curved beams. J Sound Vibr 153(2): 203–216
Luongo A, Romeo F (2005) Real wave vectors for dynamic analysis of periodic structures. J Sound Vibr 279: 309–325
Yong Y, Lin YK (1989) Propagation of decaying waves in periodic and piecewise periodic structures of finite length. J Sound Vibr 129(2): 99–118
von Flotow AH (1986) Disturbance propagation in structural networks. J Sound Vibr 106(3): 433–450
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Franklin, M., Paulsen, W. Calculating in-plane frequencies of multispan cables using the Exterior-Matrix Method. J Eng Math 67, 289–306 (2010). https://doi.org/10.1007/s10665-010-9360-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10665-010-9360-5