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Bragg scattering in periodic media generates bandgaps, frequency bands where waves attenuate rather than propagate. Yet, a finite periodic structure may exhibit resonance frequencies within these bandgaps. This is caused by boundary effects introduced by the truncation of the nominal infinite medium. Previous studies of discrete systems determined existence conditions for bandgap resonances, although the focus has been limited to mainly periodic chains with free–free boundaries. In this paper, we present closed-form existence conditions for bandgap resonances in discrete diatomic chains with general boundary conditions (free–free, free–fixed, fixed–free, or fixed–fixed), odd or even chain parity (contrasting or identical masses at the ends), and the possibility of attaching a unique component (mass and/or spring) at one or both ends. The derived conditions are consistent with those theoretically presented or experimentally observed in prior studies of structures that can be modeled as linear discrete diatomic chains with free–free boundary conditions. An intriguing case is a free–free chain with even parity and an arbitrary additional mass at one end of the chain. Introducing such an arbitrary mass underscores a transition among a set of distinct existence conditions, depending on the type of chain boundaries and parity. The proposed analysis is applicable to linear periodic chains in the form of lumped-parameter models, examined across the frequency spectrum, as well as continuous granular media models, or similar configurations, examined in the low-frequency regime.

No abstract

A parametric study is developed to investigate the effect of geometry, material stiffness and the rotational motion on the coupled flapwise bending and torsional vibration modes of a wind turbine blade. The assumed modes method is used to discretize the derived kinetic and potential energy terms. Lagrange’s equations are used to derive the modal equations from the discretized terms, which are solved for the vibration frequencies. The parametric study utilizes dimensional analysis techniques to study the collective influence of the investigated parameters by combining them into few non-dimensional parameters, thus providing deeper insight to the physics of the dynamic response. Results would be useful in providing rules and guidelines to be used in blade design.