Wednesday, March 25, 2026
Dr. Alok Sinha Professor of Mechanical Engineering The Pennsylvania State University
One dimensional continuous structures include longitudinal vibration of bars, torsional vibration of shafts, and transverse vibration of beams. For uniform one-dimensional structures, there are analytical solutions. However, many engineering structures are not uniform and they can also be spatially discontinuous. In general, natural frequencies and mode shapes for these nonuniform structures are computed by approximate techniques such as Rayleigh-Ritz, Galerkin, finite element and transfer matrix methods.
In this talk, a new method will be presented to compute natural frequencies and mode shapes of an one dimensional continuous structure with arbitrary nonuniformities, discontinuities and constraints. The method is based on the computation of spatial state transition matrix, which is independent of boundary conditions. Then, the equations for computing natural frequencies and mode shapes are developed in terms of the state transition matrix for any boundary conditions. The method can be described as “almost closed-form solution” in this digital age.
The method will be illustrated via many examples: longitudinal vibration of rectangular bars, torsional vibration of circular shafts, Euler-Bernoulli beam, axially moving beams, flutter of one dimensional panel, Timoshenko beams , and fundamental partial differential equation for mistuned bladed disk vibration.