The effects of longitudinal steady translational motion on the vibration frequencies and modes of simple Euler beams are outlined. An example shows that the natural frequencies decrease monotonically with increasing translation velocity, while the mode shapes are complex, indicating significant phase disparities from point to point across the span. Divergence is shown to occur when the translation velocity is greater than the fundamental flexural wave propagation velocity. More general equations of motion, governing the translating Timoshenko beam under axial loading, are derived in Appendix 1.

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A. Simpson, Dip.Tech.(Eng.), Ph.D. SynopsisThe dynamical equations of a 'shallow' elastic catenary, oscillating in its own plane, are formulated and solved generally. Using the solutions, a transfer matrix is derived which is generally applicable to any shallow elastic catenary-whether symmetrical with regard to its lowest point or not. The oscillations of composite systems comprising shallow catenaries can thus be treated by standard transfer matrix methods.On the assumption that the insulator strings hang vertically at equilibrium, a special transfer matrix of reduced order is obtained, and this can be used as the basis of the vibration analysis of a wide variety of multispan lines. The practical use of this matrix is illustrated by a computer calculation of the natural frequencies and normal modes of a symmetrical 3-span system. A successful comparison is made between these results and those obtained from a previous 'lumped parameter' calculation.In an Appendix the frequency equations of symmetrical shallow elastic catenaries, under a variety of end conditions, are enumerated. ListA = B, C = D = E = * PATEL, M. R.: 'Electromagnetic fields applied to electrical machines with particular reference to induction machines', M.E. dissertation, Gujarat University, India, 1965, pp. 62 and 166

SummaryThe stability of a smooth circular cylinder, free to translate horizontally and vertically against linear springs in the wake from an identical neighbouring cylinder, is studied using quasi-static aerodynamic derivatives and simple flutter theory. It is found that at spacing values between ten and twenty cylinder diameters (typical of the spacings employed on “bundled” overhead transmission lines) classical flutter of the leeward cylinder can occur in a certain critical range of wind speeds at certain orientations of this cylinder in the wake. However, the occurrence of flutter appears to be conditional on a positive difference of natural frequency between vertical and horizontal motions of the leeward cylinder in still air.Classical static instability (divergence) of the leeward cylinder is also shown to be possible over the entire “incidence” range in the wake, but this occurs in a much higher wind speed range than that associated with flutter.

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