Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook

Results obtained by applying the method of stochastic averaging to random vibration problems are discussed. This method is applicable to a variety of problems involving the response of lightly damped systems to broad-band random excitations. Solutions pertaining to both linear and non-linear vibrations are reviewed, and it is shown that the technique enables, in the case of parametric excitation, stability criteria to be established. Some results which have been obtained relating to the first-passage reliability problems are also surveyed. Various applications of the theory to engineering problems are outlined.

Stochastic averaging: An approximate method of solving random vibration problems

Free wave propagation in periodically supported, infinite beams

Wave reflection and transmission in beams

Shunted piezoelectric patches are periodically placed along rods to control the longitudinal wave propagation in these rods. The resulting periodic structure is capable of filtering the propagation of waves over specified frequency bands called stop bands. The location and width of the stop bands can be tuned, using the shunting capabilities of the piezoelectric materials, in response to external excitations and to compensate for any structural uncertainty. A mathematical model is developed to predict the response of a rod with periodic shunted piezoelectric patches and to identify its stop band characteristics. The model accounts for the aperiodicity, introduced by proper tuning of the shunted electrical impedance distribution along the rod. Disorder in the periodicity typically extends the stop bands into adjacent propagation zones and, more importantly, produces the localization of the vibration energy near the excitation source. The conditions for achieving localized vibration are established and the localization factors are evaluated for different levels of disorder on the shunting parameters. The numerical predictions demonstrate the effectiveness and potentials of the proposed treatment that requires no control energy and combines the damping characteristics of shunted piezoelectric films, the attenuation potentials of periodic structures, and the localization capabilities of aperiodic structures. The theoretical investigations presented in this paper provide the guidelines for designing tunable periodic structures with high control flexibility where propagating waves can be attenuated and localized.

https://iopscience.iop.org/article/10.1088/0964-1726/10/5/314/pdf

Attenuation and localization of wave propagation in rods with periodic shunted piezoelectric patches

A new mathematical theory is proposed to analyze the propagation of fatigue crack based on the concepts of fracture mechanics and random processes. The time-dependent crack size is approximated by a Markov process. Analytical expressions are obtained for the probability distribution of crack size at any given time and the probability distribution of the random time at which a given crack size is reached, conditional on the knowledge of the initial crack size. Examples are given to illustrate the application of the theory, and the results are compared with available experimental data.

A stochastic theory of fatigue crack propagation

Dynamics of Beam-Type Periodic Structures

A brief survey of transfer matrix techniques with special reference to the analysis of aircraft panels

Propagation of decaying waves in periodic and piecewise periodic structures of finite length

A structure designed to be spatially periodic in its configuration cannot be exactly periodic due to material, geometric, and manufacturing variabilities. Such variabilities are random, and their presence (often referred to as disorder) can reduce the ability of the structure to transmit waves from one location to another. The phenomenon is known as vibration confinement or localization. A new perturbation scheme is developed in this paper on the basis of probability theory to calculate the average exponential decay rate of wave transmission with respect to the distance of transmission, called the localization factor. Account is also taken of structural damping. The new scheme permits successive improvement of accuracy, making it applicable to either weak, strong, or moderate localization. Moreover, the analysis is based on a generic periodic structure; thus, it is not restricted to a specific set of governing equations. These are achieved by taking into account reflections from nearby disordered cell-units successively and substituting the ensemble average for the sequential average of certain statistical properties of the cell units on the basis of spatial ergodicity. Application of the method is illustrated by an example, and the results are compared with Monte Carlo simulations.

Y. K. Lin

Papers

A stochastic theory of fatigue crack propagation

Dynamics of Beam-Type Periodic Structures

A brief survey of transfer matrix techniques with special reference to the analysis of aircraft panels

Propagation of decaying waves in periodic and piecewise periodic structures of finite length

Localization of wave propagation in disordered periodic structures