Fractional-Order Mathematical Models for Vibration Analysis in Smart Structural Systems

Abstract

Adaptive intelligent and sensing structures are novel architecture of structures that comprise new breath in structural engineering and permit better performance, greater durability and a superior security in the transforming working conditions of the structure. Nonlocal, hereditary and memory-dependent behavior such as that observed when viscoelastic damping occurs, adaptive actuation and nonlinear stiffness responses are usually difficult to capture with standard vibration analysis using integer-order models alone. In order to defeat these inadequacies, the paper theorizes a holistic conceit of fractional-order mathematical modeling that will examine vibration of smart structural systems. It can be recognized that the methodology can extend the classical theory that investigated viscoelastic damping and energy dissipation to the level of the fractional-order by the application of the Caputo operator in the determination of governing equations. Beam-like structures are solved analytically, with beam-excitation under harmonic excitation, and computer codes under numerical simulation and employing finite element formulations which are combined with a fractional operator. A systematic comparison to the classical model reveals that the fractional-order model better captures resonance frequencies changes, amplitude decay and dynamical stability, over an excitation space. Two are provided, one piezoelectric embedded smart beam and an actuator system made of shape memory alloy(SMA). The model suggested in this paper suits the experimental modal data in both cases and can also predict the vibration response behavior which would be underestimated by integer-order models. Additionally, parametric study shows how the fractional order can influence the damping characteristics of the systems that can be incorporated in optimization problems of the design and active vibration suppression. The paper concludes that not only is the concept of the fractional-order modeling mathematically well-founded, but also practically significant to the development of applications in structural health-monitoring, aerospace engineering, and robotics/civil infrastructure. This contribution connects theoretical fractional calculus to the practice of engineering, such that fractional-order models become a potentially radical instrument to next-generation smart structural design and vibration control.