In this paper, we establish Z-eigenvalue inclusion theorems for general tensors, which reveal some crucial differences between Z-eigenvalues and H-eigenvalues. As an application, we obtain upper bounds for the largest Z-eigenvalue of a weakly symmetric nonnegative tensor, which are sharper than existing upper bounds. 2 ,. .. , x m−1 n ] T. (λ, x) is called an H-eigenpair if both of them are real. Definition 1.2. Let A be an m-order n-dimensional tensor. We say that (λ, x) ∈ C × (C n \{0}) is an E-eigenpair of A if Ax m−1 = λx and x T x = 1. (λ, x) is called a Z-eigenpair if both of them are real.

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Acoustic metamaterials, capable of both sound insulation and energy harvesting, composed of Helmholtz resonators and built-in decorated membranes are proposed. It is theoretically and experimentally proved that the proposed metamaterial can obtain two sound insulation bands where the sound energy is also efficiently harvested at the same frequency simultaneously. The peak of the output power can reach up to 244 nW when the incident sound pressure is 1 Pa, which is about 14 times larger than structures composed of single membrane reported before with the same input. Moreover, considerable energy harvesting with average output power of about 20 nW can be achieved within wider frequency ranges (with bandwidth up to 90 Hz) by choosing proper parameters. Additionally, the hybrid resonance mechanism of the proposed metamaterials is explained well with an equivalent theoretical model. These novel properties of the proposed metamaterial can be applied in many potential applications in noise control and energy harvesting.

Embedded wireless sensing networks (WSNs) provide effective solutions for structural health monitoring (SHM), where how to provide long-term electric power is a bottle-neck problem. Piezoelectric vibration energy harvesting (PVEH) has been widely studied to realize self-powered WSNs due to piezoelectric effect. Structural vibrations are usually variable and exist in the form of elastic waves, so cantilever-like harvesters are not appropriate. In this paper, one kind of two-dimensional (2D) piezoelectric metamaterial plates with local resonators (PMP-LR) is investigated for structural vibration energy harvesting. In order to achieve low-frequency and broadband PVEH in SHM, it is highly necessary to study dynamic characteristics of PMP-LR, particularly bandgaps. Firstly, an analytical model is developed based on the Kirchhoff plate theory, and modal analysis is done by using the Rayleigh–Ritz method. Then, effects of geometric and material parameters on vibration bandgaps are analyzed by finite element-based simulations. In the end, experiments are carried out to validate the simulated results. The results demonstrate that the location of bandgaps can be easily adjusted by the design of local resonators. Therefore, the proposed method will provide an effective tool for optimizing local resonators in PMP-LR.

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