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

DOI: 10.37153/2618-9283-2020-3-5-18

Authors:  
Rubric:     Theoretical and experimental studies   
Key words: seismic input, pendulum kinematical supports, equations of motion
Annotation:
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. Used Books:

1.     Tyapin A.G. Svobodnye kolebaniya zhestkoi kinematicheskoi opory Yu. D. Cherepinskogo. Seismostoikoe stroitel’stvo.Bezopasnost’ sooruzhenii. 2020, no.2, pp.18-31 [In Russian] DOI: 10.37153/2618-9283-2020-2-18-31

2.     Vibratsii v tekhnike. Spravochnik. V 6-i t. T 1. Kolebaniya lineinyh system.Pod redaktsiei V.V. Bolotina. M.: Mashinistroenie.1978, 352 p. [In Russian]

3.     Spravochnik po matematike (dlya nauchnyh rabotnikov i inzhenerov). G. Korn, T. Korn. M.: Nauka.1974.832 p.

4.     Bate K., Vilson E. Chislennye metody analiza i metod konechnyh elementov. M.: Nauka.1982. 448 p.

5.     Seismic Analysis of Safety-Related Nuclear Structures and Commentary. ASCE4-16. Reston, Virginia, USA. 2017.