Planar Vibrations of Rigid Structure Resting on Kinematical Supports of Yu. D. Cherepinsky

Abstract

The author derives the equation of motion for a structure resting on
 kinematical pendulum supports of Yu.D.Cherepinsky. Both structure and supports are
 assumed to be rigid; no sliding is assumed during rolling. Two components of seismic
 excitation are considered (horizontal one and vertical one). Equation of motion for free
 vibrations looks like that of the free vibrations for massive pendulum support standing
 alone (it was studied earlier). It is fact the equation of motion for pendulum, but center
 of rotation, inertia moment and stiffness are varying with time. This equation may be
 simplified to the linear one by skipping the second order terms. The equation of
 motion for seismic response after linearization is the extension of the Mathieu-Hill’s
 equation, where horizontal component is responsible for the right-hand part (in the
 conventional Mathieu-Hill’s equation it is zero), and vertical component creates
 parametric excitation in the left-hand part. In fact, vertical seismic acceleration
 modifies gravity acceleration g, which controls the effective natural frequency for
 pendulum. Thus, there might appear dynamic instability (though without infinite
 response due to the finite duration of excitation). The author presents numerical
 example.