Parametric resonance of fractional multiple-degree-of-freedom damped beam systems

This article investigates the problem of viscoelastic beams characterized by a fractional constitutive model and dynamically excited by longitudinal harmonic point loads. Damped systems of this kind undergo parametric resonance for specific values of the amplitude and frequency of the load. The study addresses the definition of the instability regions of the system when damping is related to a fractional-order derivative of transverse displacements. By using a Galerkin variational approach on multiple-degree-of-freedom beam systems, differently constrained, the problem is formulated as a set of fractional differential equations. The harmonic balance method is then extended to the fractional problem to determine the first region of instability of such systems. The results are presented in the form of stability regions in the parameter space. They are discussed for different constraint conditions, fractional order, and degree of damping. Particularly, the order of the fractional operator seems to affect the shape and the size of such regions. It is found that a fractional order in the range 0–0.5 shifts the regions of instability to high excitation frequencies, while it is conversely for a fractional order in the range 0.5–1, thus changing the resonant properties of the system. A comparison with the undamped conservative solution is made, which confirms that the presence of diffuse dissipation forces generally increases the stability margins of the system.