Mechanical metastructures have been reported11. J. D. Hobeck and D. J. Inman, in A Tribute Conference Honoring Daniel Inman ( SPIE, 2017), Vol. 10172, p. 95. to offer potential for passive vibration suppression by utilizing the out-of-phase motions between the periodically attached local resonators and the primary structure to form a bandgap around the resonating frequency of these local resonators. The concept builds on the traditional low degree-of-freedom vibration absorber technique, drawing inspiration from the extensively studied phononic metamaterials,2–42. H. H. Huang, C. T. Sun, and G. L. Huang, Int. J. Eng. Sci. 47, 610 (2009). https://doi.org/10.1016/j.ijengsci.2008.12.0073. J. Zhou, K. Wang, D. Xu, and H. Ouyang, Phys. Lett. A 381, 3141 (2017). https://doi.org/10.1016/j.physleta.2017.08.0204. X. Zhang and Z. Liu, Appl. Phys. Lett. 85, 341 (2004). https://doi.org/10.1063/1.1772854 to create a distributed locally resonating system for vibration suppression with potential applications in the aerospace, civil, and automobile industries. Extensive research works have explored linear resonators to broaden bandgaps by tailoring the combination of these absorbers' natural frequencies, such as the metastructure chains or beams with graded local resonators,5,65. G. Hu, A. C. M. Austin, V. Sorokin, and L. Tang, Mech. Syst. Signal Process. 146, 106982 (2021). https://doi.org/10.1016/j.ymssp.2020.1069826. M. Alshaqaq and A. Erturk, Smart Mater. Struct. 30, 015029 (2021). https://doi.org/10.1088/1361-665X/abc7fa zigzag oscillators with low natural frequency,11. J. D. Hobeck and D. J. Inman, in A Tribute Conference Honoring Daniel Inman ( SPIE, 2017), Vol. 10172, p. 95. internally coupled local resonators,77. G. Hu, L. Tang, and R. Das, J. Appl. Phys. 123, 055107 (2018). https://doi.org/10.1063/1.5011999 chiral lattice,88. R. Zhu, X. N. Liu, G. K. Hu, C. T. Sun, and G. L. Huang, J. Sound Vib. 333, 2759 (2014). https://doi.org/10.1016/j.jsv.2014.01.009 and X-shaped oscillators.99. Z. Wu, W. Liu, F. Li, and C. Zhang, Mech. Syst. Signal Process. 134, 106357 (2019). https://doi.org/10.1016/j.ymssp.2019.106357Considering the variety of nonlinear dynamic features, such as multistability, intra/interwell chaotic behavior, multi-periodic response, etc., introducing nonlinearity into the local resonators has the potential to widen the resonating frequency range, thus, to broaden the vibration attenuation bandwidth. However, although single degree-of-freedom nonlinear absorbers have been extensively studied for vibration control, very little effort has been devoted to nonlinear locally resonating metastructures for vibration suppression until recently.10–1710. C. Cai, J. Zhou, K. Wang, D. Xu, and G. Wen, Sound Vib. 540, 117297 (2022). https://doi.org/10.1016/j.jsv.2022.11729711. A. Casalotti, S. El-Borgi, and W. Lacarbonara, Int. J. Non-Linear Mech. 98, 32 (2018). https://doi.org/10.1016/j.ijnonlinmec.2017.10.00212. D. P. Vasconcellos and M. Silveira, J. Vib. Acoust. 142, 061009 (2020). https://doi.org/10.1115/1.404719713. Y. Xia, M. Ruzzene, and A. Erturk, Appl. Phys. Lett. 114, 093501 (2019). https://doi.org/10.1063/1.506632914. Y. Xia, M. Ruzzene, and A. Erturk, Nonlinear Dyn. 102, 1285 (2020). https://doi.org/10.1007/s11071-020-06008-415. J. A. Mosquera-Sánchez and C. De Marqui, Nonlinear Dyn. 105, 2995 (2021). https://doi.org/10.1007/s11071-021-06785-616. A. Banerjee, E. P. Calius, and R. Das, Int. J. Non-Linear Mech. 101, 8 (2018). https://doi.org/10.1016/j.ijnonlinmec.2018.01.01317. B. Bao, D. Guyomar, and M. Lallart, Mech. Syst. Signal Process. 82, 230 (2017). https://doi.org/10.1016/j.ymssp.2016.05.021 Examples involve combining local resonators with compliant quasi-zero-stiffness,1010. C. Cai, J. Zhou, K. Wang, D. Xu, and G. Wen, Sound Vib. 540, 117297 (2022). https://doi.org/10.1016/j.jsv.2022.117297 monostable cubic nonlinearity,11,1211. A. Casalotti, S. El-Borgi, and W. Lacarbonara, Int. J. Non-Linear Mech. 98, 32 (2018). https://doi.org/10.1016/j.ijnonlinmec.2017.10.00212. D. P. Vasconcellos and M. Silveira, J. Vib. Acoust. 142, 061009 (2020). https://doi.org/10.1115/1.4047197 bistable nonlinearity,13,1413. Y. Xia, M. Ruzzene, and A. Erturk, Appl. Phys. Lett. 114, 093501 (2019). https://doi.org/10.1063/1.506632914. Y. Xia, M. Ruzzene, and A. Erturk, Nonlinear Dyn. 102, 1285 (2020). https://doi.org/10.1007/s11071-020-06008-4 essentially cubic nonlinearity,1515. J. A. Mosquera-Sánchez and C. De Marqui, Nonlinear Dyn. 105, 2995 (2021). https://doi.org/10.1007/s11071-021-06785-6 impact-induced piecewise nonlinearity,1616. A. Banerjee, E. P. Calius, and R. Das, Int. J. Non-Linear Mech. 101, 8 (2018). https://doi.org/10.1016/j.ijnonlinmec.2018.01.013 or nonlinear circuit1717. B. Bao, D. Guyomar, and M. Lallart, Mech. Syst. Signal Process. 82, 230 (2017). https://doi.org/10.1016/j.ymssp.2016.05.021 to attenuate the transmission or shift the bandgap.Separately, harnessing vibration energy has received ever-growing attention in the past decade, with the eventual goal of sustainably powering standalone low-power microelectronics, such as wireless sensors for structural health monitoring purpose. For instance, bistability1818. A. Ibrahim, S. Towfighian, and M. I. Younis, J. Vib. Acoust. 139, 051008 (2017). https://doi.org/10.1115/1.4036503 and parametric resonance1919. W. Yang and S. Towfighian, J. Sound Vib. 446, 129 (2019). https://doi.org/10.1016/j.jsv.2019.01.038 have been exploited as valid techniques for energy harvesting enhancement in terms of power bandwidth, power level, and excitation threshold. In particular, harnessing the localized vibration energy within the distributed local resonators has the potential for developing dual-functional metastructures for simultaneous vibration suppression and energy harvesting. Sugino and Erturk2020. C. Sugino and A. Erturk, J. Phys. D: Appl. Phys. 51, 215103 (2018). https://doi.org/10.1088/1361-6463/aab97e showed that the vibration suppression capability of a metastructure beam is not significantly diminished while harnessing useful energy. Hu et al. studied the influence of electromechanical coupling on the linear bandgap2121. G. Hu, L. Tang, A. Banerjee, and R. Das, J. Vib. Acoust. 139, 011012 (2017). https://doi.org/10.1115/1.4034770 of a metastructure. Bukhari and Barry2222. M. Bukhari and O. Barry, J. Sound Vib. 473, 115215 (2020). https://doi.org/10.1016/j.jsv.2020.115215 studied a nonlinear metamaterial chain with linear local resonators. Hobeck and Inman11. J. D. Hobeck and D. J. Inman, in A Tribute Conference Honoring Daniel Inman ( SPIE, 2017), Vol. 10172, p. 95. investigated low-frequency vibration suppression and power generation using a zigzag metastructure beam. Chen et al.2323. Z. Chen, Y. Xia, J. He, Y. Xiong, and G. Wang, Mech. Syst. Signal Process. 143, 106824 (2020). https://doi.org/10.1016/j.ymssp.2020.106824 employed a metamaterial plate with synchronized charge extraction circuit for energy harvesting. These studies have highlighted the feasibility and potential of nonlinear metastructures in achieving the dual function of vibration suppression and power generation.However, one challenge with the recently reported nonlinear metastructures is that their vibration attention bandwidth is strongly amplitude-dependent. Xia et al.13,1413. Y. Xia, M. Ruzzene, and A. Erturk, Appl. Phys. Lett. 114, 093501 (2019). https://doi.org/10.1063/1.506632914. Y. Xia, M. Ruzzene, and A. Erturk, Nonlinear Dyn. 102, 1285 (2020). https://doi.org/10.1007/s11071-020-06008-4 showed that, while the transmission peaks are attenuated using the bistable resonators, the bandgap entirely vanishes at high accelerations. Therefore, there is a strong need to develop amplitude-robust nonlinear metastructures where the transmission reduction and bandgap widening are less susceptible to excitation intensity. Moreover, very little is known on how to simultaneously improve vibration suppression and energy harvesting with strong nonlinearities and to provide a quantitative evaluation of the dual-objective performance for the two competing and almost contradictory goals. This study presents a combination of bistable and monostable-hardening mechanisms in the local resonators to achieve well-balanced performance, which enables wide bandgap at high accelerations, attenuation of transmission peaks, and generation of power over a broad bandwidth at low frequencies (The concept of the system is schematically shown in Fig. 1, consisting of a primary structure under base excitation, represented by a cantilevered beam, and multiple cantilevered oscillators attached symmetrically onto the primary beam to serve as vibration absorbers as well as energy harvesters. A magnet is attached to the free end of each oscillator. Around each oscillator, another magnet with repelling dipole is fastened to a rigid frame that is fixed on the primary beam. By adjusting the distances between the magnets, the oscillators can be tailored to exhibit various nonlinearities. When the distance between magnets is reduced below a certain threshold, the oscillator transitions from being monostable hardening to bistable. Piezoelectric transducers are bonded to each oscillator near the fixed end to convert the strain energy into electricity. The energy harvesting circuit is connected across the two electrodes of the transducer. In this study, four nonlinear configurations are investigated and compared, as shown in Fig. 1, including, case A—all bistable oscillators; case B—all monostable-hardening oscillators; case C—alternate bistable and monostable-hardening oscillators; case D—half-half arrangement, with bistable oscillators for the first half-beam near the fixed end and monostable oscillators for the second half-beam near the free end. The underlying principle for cases C and D is to avoid the scenario where all resonators suffer large-amplitude interwell chaotic oscillation when the base excitation is high, which has been found to cause the bandgap to vanish.1313. Y. Xia, M. Ruzzene, and A. Erturk, Appl. Phys. Lett. 114, 093501 (2019). https://doi.org/10.1063/1.5066329 Two configurations are investigated for each case: in the uniform metastructure, the local resonators have the same resonant frequency; while in the graded metastructure, the natural frequencies follow a spatially increasing pattern, which will be presented in the theoretical model.To evaluate the performance of the nonlinear dual-functional metastructure, an electromechanically coupled model is developed based on a lumped cell-chain. As an example, Fig. 2 shows the model for case C. The governing equation of the ith cell is given by M0üi+K02ui−ui+1−ui−1+C02u̇i−u̇i+1−u̇i−1+K1iui−wi+K3iui−wi3+C1u̇i−ẇi−Θvi=0i=1,2,…,n−1M0üi+K0ui−ui−1+C0u̇i−u̇i−1+K1iui−wi+K3iui−wi3+C1u̇i−ẇi−Θvi=0 i=nM1ẅi+K1iwi−ui+K3iwi−ui3+C1ẇi−u̇i+Θvi=0 i=1,2,…,nCpν̇i+νiRL−Θẇi−u̇i=0 i=1,2,…,n,(1)where M0, C0, and K0 are the mass, damping, and stiffness of the discrete cell for the primary beam, respectively; M1, C1, K1i, and K3i are the mass, damping, linear, and cubic stiffness of the ith local resonator, respectively; ui and wi represent the absolute displacements of the primary mass and resonator mass, respectively; Θ is the electromechanical coupling term; vi is the voltage; Cp is the piezoelectric capacitance; and RL is the load resistance. The overdot denotes the derivative with respect to time t. The base excitation displacement u0 is expressed as u0=f cos Ωbt. The oscillator is monostable hardening when K1>0 and K3>0, and bistable when K1K3>0. Equation (1) is further nondimensionalized by letting Ω0=K0M0,μ=M1M0, ζ0=C02M0Ω0, ζ1=C12M1Ω1, h=K0K31, ωb=ΩbΩ0, ωi=Ω1Ω0,pi=uih, qi=wih,Vi=CpΘhvi,rL=RLCpΩ0, z=fh, ke=Θ2CpK0, k1i=K1iK0, k3i=K3ih2K0, and τ=Ω0t. The nondimensional governing equation is transformed to p̈i+2ζ02ṗi−ṗi+1−ṗi−1+2pi−pi+1−pi−1+2μωiζ1ṗi−q̇i+k1ipi−qi+k3ipi−qi3−keVi=0 i=1,2,…,n−1p̈i+2ζ0ṗi−ṗi−1+pi−pi−1+2μωiζ1ṗi−qi̇+k1ipi−qi+k3ipi−qi3−keVi=0 (i=n)μq̈i+2μωiζ1q̇i−ṗi+k1iqi−pi+k3iqi−pi3+keVi=0 (i=1,2,…,n)V̇i+VirL−q̇i−ṗi=0 i=1,2,…,n,(2)where the overdot now denotes the derivative with respect to the rescaled nondimensional time τ. The base acceleration is p̈0=A cos ωbτ with A=fωb2/h. Since we consider two locally resonating mechanisms, we have k1i > 0 for the monostable-hardening oscillators and k1iωi=k1i/μ (k1i > 0) and ωi=−2k1i/μ (k1iTable I.

TABLE I. System parameters.

PropertiesValueζ0, ζ10.015, 0.01μ0.2ωiuniform[0.750, 0.750, 0.750, 0.750, 0.750, 0.750]graded[0.625, 0.669, 0.716, 0.766, 0.819, 0.877]h0.1001 (uniform), 0.1131 (graded)ke0.0056rL0.2161k3i1(i = 1, 2…,6)We first consider the uniform configuration. The predicted transmittance and voltage outputs are shown in Fig. 3, with ωi (i = 1, 2,…,6) being tuned identically at 0.750. The target frequency is set to be the second resonance of the primary chain at 0.706. The transmittance is taken as pṅ/p0̇, which is the ratio of the velocity amplitude of the primary mass in the nth cell to that of the input base excitation. The voltage is calculated by V¯=1n∑i=1nVi, representing the average voltage from each resonator. As shown in Figs. 3(a)–3(d), with the increase in the base acceleration amplitude A0, the four nonlinear configurations show different variation features. The bandgap in case A entirely vanishes at high accelerations (e.g., A = 9.44 × 10−3); the primary structure experiences considerable vibration at the initial bandgap, similar to the supratransmission phenomenon in acoustic wave propagation. Case B is controlled by the monostable cubic hardening effect, where the left transmission peak is higher than the linear case. On the contrary, cases C and D attenuate the transmission peaks and maintain a bandgap at high accelerations. In terms of the generated voltage [Figs. 3(e)–3(h)], broad bandwidths are formed for all four cases, especially at high accelerations.To further increase the bandgap at high accelerations, graded metastructure chains are subsequently investigated with ωi=ω1(1+Δ)i−1. This pattern ensures that as ωi increases, the difference between the neighboring ωi (i.e., ωi+1−ωi) also increases. Such a design aims for a continuously merged wide bandgap, since the higher the absorber's resonance, the wider the bandgap (regardless of the target frequency). Remarkably, widened bandgaps are achieved at low accelerations as expected, as shown in Fig. 4. The bandgap results from the out-of-phase oscillation between the primary structure and the local resonator. The variation in the linearized natural frequency of the local resonators leads to a combination of bandgaps contributed by each oscillator, achieving a widened bandgap of the graded metastructure. Even at high accelerations, cases C and D produce a bandgap comparable to that of a uniform linear metastructure [Figs. 4(c) and 4(d)]. The voltage bandwidths in Figs. 4(e)–4(h) are further widened compared to the uniform condition. It is apparent that cases C and D attenuate the resonance amplitudes around the bandgap, maintaining a substantial bandgap even at high accelerations, while simultaneously provide wide voltage bandwidths to better facilitate self-powered onboard microelectronics.To quantitatively evaluate the efficiency of the two simultaneous capabilities of energy harvesting and vibration suppression, a weighted index Perfection Rate (PR)24,2524. M. Rezaei, R. Talebitooti, and W.-H. Liao, Int. J. Mech. Sci. 207, 106618 (2021). https://doi.org/10.1016/j.ijmecsci.2021.10661825. P. Firoozy, M. I. Friswell, and Q. Gao, Int. J. Mech. Sci. 163, 105098 (2019). https://doi.org/10.1016/j.ijmecsci.2019.105098 is defined by assigning weighting factors to each functionality, given by where G is the bandgap size, W is the power bandwidth, defined to be the frequency range where the total power [Pavg=∑i=1nViRMS2/rL] is above a predetermined power threshold, and α and β are the weighting factors for the vibration suppression and energy harvesting capability, respectively. The subscript lu donates linear uniform metastructure. Glu and Wlu are adopted at rL = 2.161. The power threshold here is taken as the half-power point of Pavg from the linear uniform metastructure around ωb = 0.9.Figures 5(a)–5(e) depict the variation of PR with the root mean square (RMS) base acceleration ARMS and load resistance rL for the uniform nonlinear metastructures, with [α β] = [0.5 0.5], [0.2 0.8], [0.8 0.2], [1 0], and [0 1], respectively; Fig. 5(f) shows the peak transmittance (PT) value over the whole considered frequency range. Inspecting Fig. 5(d), where vibration suppression is given a full priority, we can see that the largest bandgap can be obtained at high accelerations by using an optimal rL. For example, with case C at ARMS = 0.010, PRmax is 0.7778 at an optimal rL = 0.684; however, at rL = 0 (i.e., short-circuit condition, corresponding to a pure mechanical metastructure without the energy harvesting functionality), PR is 0.2778. This indicates that integrating electromechanical coupling in the nonlinear oscillators for energy harvesting helps widen the bandgap by 180%.The bandgap in case A entirely vanishes at high accelerations [Fig. 5(d)], while case B fails to attenuate the peak transmittance [Fig. 5(f)], endangering the primary structure with large-amplitude oscillations; on the contrary, cases C and D are efficient in maintaining bandgap as well as attenuating peak transmittance. Take case C at ARMS = 0.010 as an example: it keeps the bandgap at 77.78% of the linear size [PRmax = 0.7778, Fig. 5(d)], compared to case A where the bandgap reduces to zero; at this optimal rL, it has a peak transmittance of 6.37, compared to case B where the minimum peak transmittance is 17.52, corresponding to a 63.6% reduction. A further comparison between cases C and D is given in Figs. 5(a)–5(c). When energy harvesting and vibration suppression are of equal importance (Fig. 5(a), [α β] = [0.5 0.5]), or when energy harvesting is prior to vibration suppression (Fig. 5(b), [α β] = [0.2 0.8]), case D is superior than case C with higher PR yielded in a larger range of ARMS and rL. However, with vibration suppression priority (Fig. 5(c), [α β] = [0.8 0.2]), case C outperforms case D at high ARMS.The results of the graded configuration are shown in Fig. 6. Again, cases C and D outperform A and B in both bandgap formation and transmittance mitigation [Figs. 6(d) and 6(f)], while also generating wideband power [Fig. 6(e)]. Moreover, the values of PRmax for vibration suppression and power bandwidth are substantially higher than those in their uniform counterparts, increasing from 1.03 to 1.67 and 2.43 to 3, respectively, as can be seen by comparing Figs. 5(d) and 5(e) and Figs. 6(d) and 6(e).Experiments are then conducted for the uniform metastructure. The experimental setup is shown in Fig. 7. Six pairs of aluminum cantilevered oscillators are evenly positioned along an 850 × 50 × 3 mm3 primary aluminum beam. A pair of repulsive magnets are attached to the free end of the oscillator and the acrylic glass frame around each oscillator. When the distance d between the magnet pairs is decreased below a certain threshold dcr, the resonator can be transformed from the monostable-hardening status with one equilibrium position to the post-buckled bistable status with two symmetrical equilibrium positions. Replacing the magnet on the frame by a non-magnetic equal mass will change the system to a linear one. The first three natural frequencies of the primary beam are measured to be 2.9, 14.9, and 40.6 Hz. In this study, 14.9 Hz is set to be the target frequency for demonstrating the vibration suppression function. The linearized natural frequencies of all pre-buckled and post-buckled local resonators are tuned at Ω = 13.0 Hz. In the linear metastructure, with the target natural frequency at 13.0 Hz, the size of the linear local resonators is set to be L × b×t = 97.5 × 20 × 0.5 mm3, and the mass at the free end including the magnet and fastening frame is Mtip = 6.4 g. The natural frequency of the linear oscillators is experimentally measured to decrease with increasing L, which is also theoretically expected since ωLR=Keff/Meff, where Keff = EI/L3 and Meff = 33(L × b×t)/140+Mtip. In the nonlinear metastructure, after the introduction of a repulsive magnet at the acrylic glass frame surrounding each oscillator, the linearized natural frequency changes because the repulsive magnetic force also induces a correspondingly negative linear stiffness in addition to the cubic nonlinear stiffness. Therefore, L is decreased to increase the overall linear stiffness, compared to the linear metastructure case. By manually adjusting L, the linearized natural frequency of the monostable-hardening oscillators in pre-buckled condition and the intrawell natural frequency of the bistable oscillators in post-buckled condition are maintained at the target frequency.

On each oscillator, a 28 × 7 × 0.3 mm3 piezoelectric sheet (MFC M2807-P2 from Smart Materials Corp.) is attached. To focus on the vibration-to-electricity conversion capability, a simple AC energy harvesting circuit consisting of a pure load resistance of RL = 510 kΩ is employed across the piezoelectric electrodes for all oscillators. A displacement sensor (Wenglor CP24MHT80) is used to measure the displacement at the free end of the primary beam. The prototype is excited by a vibration shaker (TIRA S51120) where the harmonic base acceleration is measured by an accelerometer and controlled by an amplifier (TIRA BAA500). The output voltage across RL is measured using the NI9229 data acquisition module (National Instruments Corp.). In the test, the transmittance is defined as un (RMS)/u0(RMS)