The Vibration of a Layered Rotating Planet and Bryan’s Effect

Because, among other seismological observations, it is important to be able to predict the location of the vibrating pattern of an earthquake, in this chapter we take the first tentative steps towards including "Bryan’s effect" in a mathematical model for the seismic vibration pattern of a layered slowly rotating vibrating planet or moon. Historically, G.H. Bryan observed (in his article (Bryan, 1890)) that when a vibrating structure is subjected to an inertial rotation, the vibrating pattern rotates within the structure at a rate proportional to (but in general not equal to) the inertial angular rate. This effect has come to be known as " Bryan’s effect". It is interesting to note that (Rayleigh, 1894) mentions Bryan’s effect in § 233, but thereafter investigations of Bryan’s effect appear to have lain dormant for about 75 years, reappearing in connection with a resonator gyroscope in 1965 in the small Delco Wakefield, MA, USA R&D facility, according to (Rozelle, 2009). Rozelle states that the resonator gyroscope "has been utilized in many applications over its developmental lifetime: aircraft navigation, strategic missile navigation, underground borehole navigation, communication satellite stabilization, precision pointing, and in deep space missions". The effect may be useful in understanding the dynamics of pulsating stars in astrophysics and this was mentioned in the paper (Shatalov et al., 2009) in which the theoretical background on this chapter was discussed in general terms. For the constant of proportionality, Bryan made the following calculation for a body consisting of a ring or cylinder: