Dynamic Response of a Beam With Absorber Exposed to a Running Force: Fractional Calculus Approach

This paper analyzes the transverse vibration of Bernoulli-Euler homogeneous isotropic simply-supported beam. The beam is assumed to be fractionally-damped and attached to a single-degree-of-freedom (SDOF) absorber with fractionally-damping behavior at the mid-span of the beam. The beam is also exposed to a running force with constant velocity. The fractional calculus is introduced to model the damping characteristics of both the beam and absorber. The Laplace transform accompanied by the used decomposition method is applied to solve the handled problem with homogenous initial conditions. Subsequently, curves are depicted to measure the dynamic response of the utilized beam under different set of vibration parameters and different values of fractional derivative orders for both of the beam and absorber. The results obtained show that the dynamic response decreases as both the damping-ratio of the absorber and beam increase. The results reveal that there are critical values of fractional derivative orders which are different from unity. At these optimal values, the beam behaves with less dynamic response than that obtained for the full-order derivatives model of unity order. Therefore, the fractional derivative approach provides better damping models for fractionally-damped structures and materials which may allow researchers to choose suitable mathematical models that precisely fit the corresponding experimental models for many engineering applications.