Explicit Green's Function Approach to Forced Vertical Vibrations of Circular Disk on Semi‐Infinite Elastic Space

Abstract

Arbitrary modes of vertical vibrations of a circular disk on an elastic half space is investigated mathematically. The problem is formulated with an integral equation, which describes explicitly the relation between the vertical displacement of the disk and the normal stress of contact. This equation has long remained difficult to handle, because its integral kernel has been represented as an integral of a singular function over an infinite interval. This integral kernel, the Green's function of the problem, can be reduced to definite integrals of some higher analytic functions over finite intervals. These higher functions, also defined as a definite integrals of trigonometric functions over a finite interval, are closely similar to the Bessel function and are easily evaluated. The integral equation accompanied by the new Green's function affords a basis of analysis; one can readily take the dynamic interaction into account by solving this equation simultaneously with the equation of motion of the disk or some other structure whose bottom is circular. Complicated structures can be treated if an appropriate discretization procedure is introduced. If the method presented here for the first time is applied to the problem of a rigid disk for demonstration, it gives identical solutions to the conventional theories.