10. Extracting Natural Frequencies of Layered Beams Using a Continuous Variation Model and Modal Analysis

This study involves the determination of the bending natural frequencies of beams composed by stacked cells of different materials. The focus is on cases with two cells. The analytical model is based on Euler-Bernoulli theory with the variations from one cell to another modeled via logistic functions. This approach leads to a single differential equation with variable coefficients, which is solved numerically using MAPLE®‘s differential equation solvers. A forced motion method is used. Forcing frequencies are changed until large motions and sign changes are observed, leading to the resonant frequencies. Of interest is the validation of the analytically obtained frequencies via experimental results. Here, an experiment is set in which a simple two-cell beam is analyzed, via modal analysis, in order to verify the analytically calculated frequencies. The beam is excited using an impact hammer and the response is recorded using accelerometers. Mode shapes are also obtained via digital image correlation. The beam, including distinct materials, is composed of one cell made of steel and another made of aluminum. The joining method is discussed and results for fixed-free beams are obtained.