Elastic Beams with Identical Buckling and Vibration Mode Shape

In this paper we discuss the problem of determining the shape of simply supported beams with added masses to achieve identical buckling and vibration mode shapes. Motivation for this work stems from the design of a new type of hydraulic switching valve utilizing a parametrically excited beam as actuation element. An analytical expression for the necessary thickness of the beam is developed, ths shape of the buckling mode and the eigenvalues (i.e. buckling force and vibration frequency) are determined by solving a Sturm-Liouville Eigenvalue Problem numerically.