Tension in Fluttering Flags

Abstract

When a flag flutters, tension is dynamically induced by the two-dimensional vibratory motion. The dynamic process, involving centrifugal forces due to the curved path of the trailing edge, is similar to the whipping of an oscillating rope, and accounts for most of the drag-force observed at the flagpole attachment (luff). Conversely, the induced tension, combined with the curvature of the fabric, opposes the pressure forces from the flow field and extracts momentum from it. In order to estimate post-critical flag and panel flutter amplitudes, it is necessary to compute the structural stiffening due to dynamically induced tension. Tension in typical flag flutter motion, consisting of a traveling wave growing in amplitude as is progresses towards the trailing edge (leech), is obtained by approximate analysis, using a Computer Algebra System. The time-averaged tension depends on the square of the velocity amplitude of the oscillating fabric; the distribution of time-averaged tension is shown for a typical flag flutter motion. An estimate of the tension fluctuations is developed: the fluctuations are small (relative to the average tension) at locations several wavelengths from the leech, but are important near the leech. The general P.D.E. of motion is obtained from Hamilton’s principle. The induced-tension term in the governing P.D.E. derives from the in-plane kinetic energy of the flag motion. Dynamically induced tension is shown to be important if the stiffness of the fabric is low: an order-of-magnitude criterion is presented.