This paper aims at presenting a survey of the fractional derivative acoustic wave equations, which have been developed in recent decades to describe the observed frequency-dependent attenuation and scattering of acoustic wave propagating through complex media. The derivation of these models and their underlying elastoviscous constitutive relationships are reviewed, and the successful applications and numerical simulations are also highlighted. The different fractional derivative acoustic wave equations characterizing viscous dissipation are analyzed and compared with each other, along with the connections and differences between these models. These model equations are mainly classified into two categories: temporal and spatial fractional derivative models. The statistical interpretation for the range of power-law indices is presented with the help of Lévy stable distribution. In addition, the fractional derivative biharmonic wave equations governing scattering attenuation are introduced and can be viewed as a generalization of viscous dissipative attenuation models.
The three-dimensional free vibration and time response of rotating functionally graded (FG) cantilevered beams are studied. Material properties of functionally graded beams are assumed to change gradually through both the width and the thickness in power-law form. The second-kind Lagrange’s equations are used in conjunction with the Ritz method to derive the comprehensive coupling dynamic equations for the axial, chordwise, and flapwise motions. The trial functions of deformations are taken as the products of the Chebyshev polynomials and the corresponding boundary functions. Nonlinear coupling deformations are considered to capture the dynamic stiffening effect due to the rotating motion. The influences of the material gradient index and rotational speed on modal characteristics are investigated by the state space method. The eigenvalue loci veering phenomena with modal conversions are exhibited. The time responses indicate that the deformations of rotating functionally graded beams are greatly affected by the material gradient index. It is shown that for large deformation problems, using Chebyshev polynomials is more efficient in computing precision and robustness than using other polynomials.
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