We discuss the rich dynamics of an inverted pendulum kept from falling
over by an anelastic flexure. The anelastic character of the flexure
is dramatically emphasized by the pendulum. We report measurements sho
wing internal friction in one flexure to be independent of frequency o
ver a band of three to four decades. This anelastic behavior in flexur
es can limit the sensitivity in gravitational wave detectors and in ot
her precision measurements.