Stable and Unstable Motion induced by Forced Vibrations
|If one tries to stand a thin rod (e.g. a pencil)
vertically on its end on a horizontal table without constraints it proves
impossible. The slightest perturbation tips it over. However it is not too
difficult to keep it from toppling over if one is allowed to move its lower end
about in space. In fact a small amplitude oscillatory motion at the lower end
will keep it vertically stable if the frequency of oscillation is right. The
longer the rod the lower the frequency must be in order for this trick to work.
If the rod is suspended from it upper end which is then set into horizontal
oscillatory motion the rod can be made to execute an altogether different kind
of motion that may become very complex. Indeed at some frequency the rod may
loop over and over the vertical in a propeller motion.
The ability of an external intermittent or oscillatory motion to both amplify or stabilise the motion of a system is apparent in a wide variety of phenomena. Around 1930 it was noticed that overhead electrical power lines sometimes began to execute large swaying motion in light winds (less that 5mph) with sleet.
In some cases cables between pylons were displaced more than 20 feet. A controversy arose as to whether this was due to some strange electrical effect (such as coronal discharge into the air) or mechanical forcing by the air. The mystery was compounded by the observation that sometimes adjacent cables did not share this motion which came to be called "galloping". One suggestion was that the presence of sleet in the environment was not fortuitous and that ice on the cable was forming to change its shape from a stranded cylinder. With a modified profile the aerodynamic forces of lift and drag on the iced cable due to even a light wind were sufficient to amplify its swaying motion and set the cable into a "galloping" mode. This suggested the introduction of damping devices that are now in place on all overhead power lines worldwide and serve to eliminate this kind of motion. However from time to time wind and rain induced vibrations reappear and their effects continue to puzzle. Modern long span cable-stay bridges are particularly prone to such subtle effects.
The effects of moving air on solid bodies can be both subtle and dramatic. If one draws a solid rod through a tray of milk one can observe a swirling in the fluid in the wake of the motion.
If the motion is rapid enough these swirls form "vortices" in which eddies of fluid appear to generate isolated tornadoes of fluid with their own overall translational motion. As the vortices escape from the rod they cause it to react slightly. The impulsive reactions occur as each vortex is shed from the fluid in contact with the rod. If the solid were free to react to these forces as it moves through the fluid it would execute an oscillatory response. This is why a falling leaf still air dances in such a delicate way as it falls to earth and why a flag flaps in the way it does in a stiff breeze.
The effects of such vortex-induced vibrations are very important to take into account in the design of large span suspension bridges and under-sea drilling operations.
The origin of the collapse of the Tacoma-straits suspension bridge in 1940 was probably a wind-vortex induced torsional vibration of the bridge-deck that in turn excited a lateral mode of the entire bridge. The ensuing resonance caused the bridge to collapse catastrophically.
In undersea oil and gas exploration the drill-string is connected to a floating rig and passes down a hollow "marine-riser" that is fixed both to the floating rig and the seabed. The "marine-riser" protects the rotating drill-string from the sea, keeps the drill cuttings debris from escaping into the environment and offers a conduit for the collection of oil and gas. Unlike on-shore drilling systems one now has to reckon with the vibrational effects of both drill-string and the surrounding "marine-riser".
Although the marine-riser is kept in a state of mechanical tension to minimise it lateral motion the effects of vortex shedding along its length can be important especially if there a steady horizontal sea current that varies with depth. The reaction forces on the marine-riser as the vortices are shed cause a perceptible vibration to occur that in time produce "fatigue" in the steel of the riser and require that it be replaced before it breaks. (Any attempt to make the riser of a more flexible material is likely to risk the excitation of enhanced lateral vibrations induced by internal hydrodynamic forces as well as the fluid forces due to sea currents.)
More striking is the phenomenon of "lock-in". The frequency of vortex shedding depends on the overall speed of the sea current and the diameter of the marine-riser. As this frequency approaches the natural frequency of one of the natural modes of the riser the structure and the fluid suddenly behave in unison and lock each other into a grand vibration with a common frequency that persists. This behaviour is hazardous since the energy of the sea currents now exploit an enhanced route into the motion of the riser that now sustains motion with damaging fatigue stresses.
There is a more subtle effect that affects marine risers analogous to the amplification of sleet and rain wind induced vibrations on cables. In those cases we saw the amplification of persistent small forcing excitations into a large-scale motion of the structure. Since the surface of the sea is not smooth as the rig rises and falls it will cause an under-sea riser to stretch in harmony as it is fixed to the seabed. This in turn sets up axial stresses that travel down the riser and reflect off the junction where the riser terminates on the seabed, returning to the surface. Although these axial vibrations may be small in amplitude and irregular in time their repetition triggers an amplification mechanism that offers a new channel in which external energy can be directed into the motion of the structure.
If the nature of the surface fluctuations enter a critical domain this energy slips effortlessly into the riser and deforms its shape. This triggers a lateral motion that, along with the vortex shedding effects, contributes to accelerated fatigue rates in the riser.