Analytical and numerical solutions for vibration of a functionally graded beam with multiple fractionally damped absorbers

The paper investigates the vibration response of functionally graded (FG) and viscoelastic/fractionally damped beams located on the Pasternak foundation. The vibration suppression of this simply supported FG beam subjected to a point harmonic load is studied by attaching fractionally damped absorbers in order to minimize the beam deflection at its natural frequencies. First, a new method is developed and utilized to obtain the solution of the governing equations analytically in the Laplace domain. Subsequently, by taking the inverse Laplace transform via contour integration, the solution in the time domain is analytically derived and compared with the numerical results obtained by the Stehfest's method and the Talbot's method. For short time duration, as expected, a good agreement is observed when using the former method, and for longer periods, the latter method provided a good agreement. Several FG beams are analyzed and the effects of the viscoelastic properties of the material and the Pasternak foundation stiffness are evaluated. Optimization of the H-2 norm of the FG beam deflection at the full wide frequency band with respect to mass, stiffness and fractional damping parameters of the absorbers is also investigated. When the dimensionless excitation frequency is large enough and the damping coefficient of the absorber is significantly greater than that of the FG beam, the H-2 norm of the beam deflection is influenced by the value of the absorber damping order. This work also specifically focuses on the resonance response of the viscoelastic/fractionally damped beam under harmonic excitation and the effects of the absorber parameters on vibration suppression characteristics of the FG beam.