Dynamic Response of Elastic Rods Under Parametric Excitations

The problem of dynamic behavior of parametrically excited mechanical elements, viz., elastic rods and linkages in mechanisms and boring bars, is presented using the WKB solutions of a general second-order partial differential equation. The differential equations are solved by using first a plane wave type solution leading to the “regular WKB” solution and then through the solutions of Airy integral type employing the “generalized WKB” method. The analytical techniques are illustrated by considering the lateral vibrations due to an exponentially decaying axial force on a pin-ended rod. It is shown that the regular and generalized WKB solutions are complementary, i.e., where the regular WKB method fails, the generalized WKB solution yields satisfactory results.