Planar vibrations of rigid structure on kinematic supports after A.M. Kurzanov

The author carries out parametric studies for the equation of planar vibrations of rigid structure resting on kinematical rolling supports with planar bottom (after A.M. Kurzanov). Both support and the surface below are assumed rigid; no sliding assumed. Varied parameter is the width of the bottom. Horizontal structural acceleration is studied. Three variants of the possible behavior are shown: (i) minor rocking with little decrease in response accelerations as compared to the initial excitation; considerable rocking with considerable decrease in the response accelerations; intensive rocking leading to the overturn of the supports. In vertical direction there appear shocks (infinite accelerations) during gap closings of the supports. The importance of the problem for the seismic response analysis of the unanchored items is noted. The author gives recommendations for the experimental program, aimed to obtain data about damping both for rotation and for the gap closing, and also about the impact of the flexibility of the supports and underlying surface.

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

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.