Rigorous analytical analysis of resonant Euler-Bernoulli beams with constant thickness and polynomial width

We report a novel exact closed-form solution of the Euler-Bernoulli beam equation expressible in terms of Meijer G-functions. This solution allows for analytically studying the natural frequencies and mode shapes of a very general class of beams characterized by both a polynomially varying flexural beam bending stiffness EI(x) and beam cross section A(x), but a constant EI(x)=A(x)-ratio. Its application is exemplarily demonstrated on cantilevers characterized by a uniform thickness and a spatially narrowing width of either linear form (i. e., trapezoid cantilevers) or describable by a power function of higher order. The analytically deduced results are validated by computer numerical simulations and compared to test measurements carried out on micromachined thin-film cantilevers.