Design of mechanical metamaterials for simultaneous vibration isolation and energy harvesting

Metamaterials are artificial materials engineered to have extraordinary mechanical,^1,2 optical,^3,4 thermal,^5,6 or electromagnetic^7,8 properties not found in nature. They are usually made of conventional materials, e.g., metals and plastics, and arranged into periodic patterns to achieve these exceptional properties. For instance, Fang et al. have demonstrated one-dimensional ultrasonic metamaterials through using an array of subwavelength Helmholtz resonators with carefully designed acoustic inductance and capacitance.² Near the resonant frequency of the Helmholtz resonators, these metamaterials displayed an effective negative dynamic modulus,² which sounds counterintuitive and has not been realized in nature. With the help of these unusual properties, e.g., effective negative dynamic modulus and density, many important applications, including negative refraction,^9,10 invisibility cloaking,^11–13 and superlensing below the diffraction limit,^3,14,15 could be achieved through these metamaterials. More interestingly, these unusual properties of metamaterials originate from their rationally designed microstructures rather than the constituting materials. Thus, the shape, size, orientation, and arrangement of internal microstructures can be engineered to create exceptional properties of metamaterials.^16–21 Similar to metamaterials, phononic crystals also acquire their properties from the spatial arrangements of their building blocks.²² However, the design of phononic crystals relies on the Bragg diffraction from the periodic arrangements of the building blocks.²³ Thus, the characteristic length of microstructures is comparable to the corresponding wavelength.²⁴ In contrast, metamaterials utilize the building block with the dimension much smaller than the wavelength, thus allowing the manipulation of the electromagnetic and acoustic waves at deep subwavelength scale.^10,25

In addition to electromagnetic metamaterials, the concept of mechanical metamaterials has attracted considerable interest in recent years. For example, Wang et al. demonstrated that the acoustic bandgap can be controlled and tuned through the buckling of microstructures.²⁶ Moreover, the buckling of microstructures is also found to be able to induce the auxetic behavior of 3D soft mechanical metamaterials, creating a class of materials named “Bucklicrystals”.²⁷ Through direct lattice transformation, Bückmann et al. explored the mechanical cloaking behavior of 2D lattices.²⁸ They found that a 2D discrete lattice can be used to cloak a void with respect to static uniaxial compression, after the spatial coordinate transformation. All these works shed light on manipulating the microstructure design of mechanical metamaterials for enabling a wide variety of applications, particularly in wave filtering, manipulation, and focusing.^29–33 On the other hand, the piezoelectric material based energy harvesting devices also rely on the design of their microstructures.^34–36 For instance, bi-stable structures and different beam shapes have been used to enhance the vibration energy harvesting efficiency.^36,37 Nevertheless, it has not been realized to combine the wave filtering and vibration energy harvesting in a single device.

In this letter, we report a design of mechanical metamaterials that enable simultaneous vibration isolation and energy harvesting, through finite element analysis (FEA) and 3D printing-assisted experimental validation. Such a mechanical metamaterial design compromises a large array of free-standing cantilevers being physically attached to a primary structural frame and thus allows direct coupling of bulk elastic waves propagating through the structural frame and the intrinsic harmonic vibration of the cantilevers. A strongly coupled state occurs when the frequency of the bulk elastic wave matches with the harmonic of the cantilever. The vibration energy carried by the bulk wave is then transferred into the kinetic energy of the resonating cantilevers and localized. Macroscopically, it results in the stop-band as the bulk wave propagation being suppressed. By integrating the piezoelectric thin film into the cantilever beams, the kinetic energy can be further converted into electric energy for powering other devices. In this way, two distinct functions, vibration isolation and energy harvesting, are achieved simultaneously through the designed mechanical metamaterials. This study reveals the energy harvesting mechanisms of microstructured mechanical metamaterials through a deep understanding of the interplay between bandgaps and the mechanical-electrical coupling in the electroactive polymer materials.

To demonstrate our concept, we use the standard unit cell of a regular square lattice, as shown in Fig. 1(a), as the primary structural frame. Then, additional cantilevers, denoted as the auxiliary structure, are attached to the primary structure [cf. Figs. 1(b) and 1(c)]. The detailed dimensions of these unit cells are given in the supplementary material (Fig. S1). The band structures of the original and modified square lattices are investigated through FEA. In this process, the homogenized mass matrix, $M$⁠, and generalized stiffness matrix, $K$⁠, are generated through standard finite element discretization. Then, the band characteristics of a periodic structure will be evaluated using a finite-element-based unit-cell technique based on Floquet-Bloch theory.^38–40 The equations of motion are given by $K−ω2Mq=0$⁠, where $ω$ is the frequency and $q$ is a vector of the generalized nodal degree of freedom (d.o.f.). After the d.o.f.s are partitioned and proper wave number dependent propagation conditions are imposed, the dynamics problem is recast as $Krξ1,ζ2−ω2Mrξ1,ζ2qr=0$⁠, where $Krξ1,ζ2$ and $Mrξ1,ζ2$ are reduced matrices that depend on the two-dimensional wave number vector $ξ=ξ1,ζ2$⁠. Then, the solution of this eigenvalue problem is done by sweeping the wave vector along the contour of the irreducible Brillouin zone [cf. Fig. 1(a)] and solving for the corresponding $ω$⁠. The result is coined as a band structure or a band diagram and is a set of dispersion relation curves, which represent the propagation modes allowed in the structure and corresponding frequency ranges. The existence of bandgaps is verified by isolating the frequency intervals in which no modes are allowed, where elastic waves are stopped from propagating as a result of wave interference. The detailed mathematical formulation for this problem has been given in our recent study on the band structure of mechanical metamaterials.³⁹ 

According to the above method, the band structures of the original and modified square lattices are obtained through FEA, as demonstrated in Fig. 1. As it has been discussed by Fleck and co-workers,⁴¹ the original square periodic lattice does not have a complete bandgap in the low frequency regime and other periodic lattices, e.g., honeycomb and triangular honeycomb, display bandgaps in the high frequency regime. These bandgaps are induced by the scattering of the longitudinal wave due to the periodic variations of the modulus and density in the space. Thus, the longitudinal wavelength is on the same order of the unit cell size (mm), indicating that the corresponding bandgap frequency will be on the order of kHz. However, the mechanical excitations and related vibration noises are characterized by low frequencies (about 100 Hz), which require the bandgaps in the low frequency regime for filtering.^42,43 To resolve this issue, modifications are made on these original periodic lattices to introduce additional bandgaps, as shown in Figs. 1(b) and 1(c). The cantilevers are attached to the primary structure, forming extra microstructures. Due to the local resonant behaviors of these auxiliary structures, the bandgaps are created in the low frequency regime [cf. Figs. 1(b) and 1(c)]. A similar idea has been applied to design sonic composite materials with spherical metal inclusions embedded into the epoxy matrix.²⁴ The surfaces of inclusions are coated by silicon rubber for local resonant to introduce complete bandgaps in the low frequency regime. Note that the location of the bandgaps can be easily changed by the design of the auxiliary structure, which will be explained in detail.

To further understand the band structures of original and modified square lattices, the mode shapes at critical frequencies are given in Fig. 2. For the original square lattice, the resonant of the primary structure occurs in the high frequency regime, as the fundamental resonance frequency of a pinned-pinned flexural lattice beam is given by $ω1=πEI/ρL4/2$⁠, where $E$ and $ρ$ are Young's modulus and density of the material, respectively. $L$ and $I$ denote the length and cross-sectional inertia of the beam, respectively. Consequently, the lattice beam deformation displays the first pinned-pinned flexural mode at $ω1$ (1613 Hz). Comparing the band structures between the original and modified square lattices, it immediately reveals that the bandgaps in the low frequency regime arise from the additional cantilevers (auxiliary structure). The fundamental bending natural frequency of these cantilever beams is given by $Ω1=3.5156EI/ρL4/2π$⁠,⁴⁴ where $L$ and $I$ are the length and moment of inertia for the cantilever, respectively. Plugging in the corresponding material properties and geometric parameters of auxiliary, we can get $Ω1=720$Hz, which indeed falls into the bandgap range. Moreover, the deformations are mainly localized in the auxiliary at this frequency, featuring the first-mode-like deformed shape [cf. Figs. 2(b) and 2(c)]. In short, the mechanical vibration energy has been localized and transferred into the kinetic energy of the auxiliary in the bandgap frequency range. Besides, the free ends of these cantilevers can be attached with masses to further reduce the lower bandgap frequency from 715 Hz to 146 Hz, with similar deformation modes [see Figs. 1(c) and 2(c)]. The reduced bandgap frequency has profound effects on our design since the low frequency mechanical vibration (<300 Hz) is more abundant in nature.^42,43 Our current work mainly focused on the demonstration of the complete bandgap by the coherent coupling among the bulk wave and the localized resonances. As a result, the bandgap is limited to the natural resonance of the free-standing cantilevers. It is possible to further broaden the complete bandgap by implementing the following strategies: (1) cascading multiple bandgap structures to synthesis a wider bandgap, such as bi-stable structures for cantilevers;³⁷ (2) introducing the randomness into the dimension of cantilevers; and (3) incorporating the active tuning of the cantilever resonance through the piezoelectric shunting technique.⁴⁵ 

For an isotropic medium with Young's modulus $E*$⁠, shear modulus $G*$⁠, bulk modulus $K*,$ and density $ρ*$⁠, the phase velocities along the longitudinal and transverse directions are $Cl=(G*+K*)/ρ*$ and $Ct=G*/ρ*$⁠, respectively. For the square and modified square lattices, their homogenized bulk modulus $K*$ and shear modulus $G*$ are $K*=Eρ¯/4$ and $G*=Eρ¯3/16$ ⁴¹, respectively, where $ρ¯$ is the relative density (⁠$ρ*=ρ¯ρ$⁠) and $ρ¯=43/λ$ for the square lattice. Here, $λ$ is the slenderness ratio defined as $λ=23L/d$⁠. $L$ and $d$ are the length and thickness of the cell walls in the square lattice, respectively. Considering $L=35 mm$⁠, $d=1 mm$⁠, $E=1900 MPa$⁠, and $ρ=1.099 g/cm3$⁠, the phase velocities are $Cl=20798 m/s$ and $Ct=594 m/s$ for the square lattice. For the modified squared lattice with additional masses, the relative density $ρ¯$ will be enlarged due to the auxiliary, while the bulk and shear moduli are not affected. Thus, the phase velocities have been reduced to $Cl=12068 m/s$ and $Ct=345 m/s$⁠. Clearly, the characteristic length of microstructures is much smaller than the corresponding wavelength, demonstrating the unique capability of mechanical metamaterials. In addition, we further explore the material damping property in the bandgap region. As shown in the supplementary material (Fig. S2), the material damping can slightly lower the frequency range of bandgaps, without changing the major characteristics.

From the energy harvesting point of view, the above deformation mode is ideal for a beam-like piezoelectric device. When the attached cantilevers display the first-mode-like bending deformation, the maximum stress should be observed at the root of the beam, which can enhance the performance of the piezoelectric device.^46,47 Then, we can design the modified square lattice structure with polyvinylidene difluoride (PVDF) thin films being attached to the surface of the cantilever structure, as shown in Fig. 3(a). The designed mechanical metamaterial is an assembly of 10 × 10 unit cells of the modified squared lattice. Here, PVDF was selected for its ideal properties including low density (1.78 g/cm³) with high stretchability (10–20%), flexibility (1000 MPa), and excellent piezoelectric properties (mechano-electrical conversion, 14.4 V/N).⁴⁸ Therefore, in the bandgap frequency range, the vibration energy has been localized and transferred into the kinetic energy of the auxiliary structure, which can be further converted into electric energy through the piezoelectric response of the integrated PVDF thin film transducers. With full scale FEA simulation on this mechanical material, we find that the transmission of the wave can be fully absorbed within the bandgap region estimated by unit cell analysis, as given in the supplementary material (Fig. S3).

As a proof-of-concept demonstration, the above mechanical metamaterial is fabricated using a Fortus 400mc fused deposition modeling machine (3D printing) with acrylonitrile butadiene styrene (ABS) plastic, as shown in Fig. 3(b). This specific machine was selected because of its large available build volume of 406 × 355 × 406 mm³, relatively low cost, and a high precision of 127 μm well-suited to fabricate the mechanical metamaterials we designed. After fabrication, measurements of each of the corresponding elements of the mechanical metamaterial were taken and compared with the original design. The smallest feature, the thickness of cantilevers (auxiliary), is about 0.63 mm, which is close to our targeted value and within the +/− 5% tolerance. Other parts of the design that had larger feature dimensions were all within a +/− 5% tolerance after fabrication.

The experimental setup, as given in the supplementary material (Fig. S4), is designed to simulate environmental vibration sources, closely mimicking our conceptual design in FEA [Fig. 3(a)]. To represent the general waveform of the vibration, a control signal is selected and generated using a function generator (Agilent 33120A). The control signal is then amplified using a power amplifier (Bruel and Kjaer No. 2718) before entering into an electromagnetic shaker (LDS V203). The resulting shaker's waveform and force output are monitored with a LabVIEW program using an ICP^® force sensor (PCB Piezotronics 208C01) mounted to the electromagnetic shaker drive output. An aluminum fixture and clamps are used to position and fix the metamaterial under testing. To measure energy absorbed within the metamaterial, a uniaxial accelerometer (PCB Piezotronics No. 333B50) is mounted on the free end. The surfaces of cantilevers are attached with PVDF films (thickness 28 μm) to convert the localized kinetic energy into electric energy, as shown in Fig. 3(b). The thin PVDF film was selected for its low rigidity to have minimal impact on the resonating behavior of the cantilever. Finally, the voltage output of the PVDF film is measured across a resistive load of 1 MΩ and recorded with a LabVIEW program.

Separate vibration sweeps are performed to study the capability of vibration isolation for the designed mechanical metamaterials. These sweeps represent the frequency response of a forced, harmonic vibration, with a sinusoidal input over a frequency sweep in the range of 50–300 Hz. The signal is controlled using a custom LabVIEW program with proportional gain feedback control to maintain a constant input force, so as to decouple the test stand dynamics from the structure dynamics. The difference between the input force and output acceleration across the device is measured and displayed in Figs. 4(a) and 4(b). Within the bandgap frequency range, the vibration energy has been greatly absorbed and trapped within the metamaterial. In the meantime, the kinetic energy has been localized on the resonating cantilevers (auxiliary structure), as we have observed in the FEA (results not shown here). Note that the square lattice is anisotropic materials,⁴¹ and both the compression (longitudinal wave) and shear (transvers wave) loadings are tested (cf. Figs. S4 and S5 in the supplementary material). The vibration isolation has been confirmed to be independent on the wave direction, indicating that a complete bandgap is formed in this frequency range.

The voltage output across the PVDF film is then collected into the LabVIEW program at a sampling rate of 10 kHz over the course of 10 wavelengths at the target frequency and averaged to obtain the final peak to peak result graphed. To generate a clear measurable signal from the PVDF film, an input force of up to 5.34 N was applied to the system. From these tests, the natural frequency and voltage output for the vibration energy harvester are found experimentally, as displayed in Figs. 4(c) and 4(d). The location of the cantilevers is denoted as “X,Y,” where X and Y represent the number of cantilevers in perpendicular and parallel directions to excitation, respectively (cf. Fig. S6 in the supplementary material). Within the bandgap frequency range (146–171 Hz), the peaks of voltage and power output have been observed. The maximum voltage of 0.22 V was measured across a 1 MΩ load and had a corresponding power of 0.05 μW. Here, we should emphasize that the voltage and power outputs rely on tunable device properties including the tip mass, the beam dimensions, and the thickness of the PVDF films. In the present study, we chose the thinnest available 28 μm thick PVDF films, so they would have a negligible effect on the cantilever's natural frequency, enabling us to easily study the pure bandgap performance. To further increase the voltage and power output per beam on this device, a larger tip mass, a thicker piezoelectric film, and thinner/longer cantilever beams within each unit cell could be implemented. Besides, the output voltage can also be enlarged by adding the cantilevers in parallel, which leads to a much higher voltage output of about 1.75 V, as depicted in Fig. S7 in the supplementary material. Extracting the electric energy from the system will effectively increase the energy dispersion, thus further enhancing the vibration isolation characteristic of the mechanical metamaterials.

In recent years, the need for multifunctional electromechanical structures and material systems with the capability to harvest energy from low frequency mechanical vibration (<300 Hz) has grown significantly in response to the proliferation of portable electronic devices, wireless sensors, and micro-electro-mechanical systems and the related demand for sustainable and reliable power sources.^49–51 The extremely low duty cycle of these systems pushes the power source requirement into the μW range.⁵² Therefore, the current study provides a viable alternative to harvest ambient energy for self-powered systems, eliminating the needs for replacing batteries and creating low-maintenance, autonomous systems. Moreover, the energy generation mechanism involved in our concept exploits internal deformation states. Devices based on this idea minimize the risk of damaging the piezoelectric materials as a result of shocks or friction forces, as the active microstructure is embedded in the system and therefore shielded from the external environment.

In summary, we demonstrate a design of mechanical metamaterials for simultaneous vibration energy isolation and energy harvesting through numerical simulations (FEA) and a 3D printing-assisted experimental study. The mechanical metamaterial facilitates the strong interaction between the bulk elastic wave and the localized harmonic resonance, which results in the formation of a complete bandgap. While the bulk wave propagation being strongly suppressed within the stop-band, the vibrational energy has been transferred and trapped into the kinetic energy of auxiliary, which is further converted into electric energy through the integrated piezoelectric components. Thus, the proposed mechanical metamaterials are particularly useful for advanced aerospace and mechanical engineering applications due to their multifunctional capabilities for mechanical wave filtering and energy harvesting, with the dimension smaller than the wavelength of the elastic waves.

See supplementary material for detailed dimensions of designed unit cells, additional simulation results for band structures and the force transmission spectrum of mechanical materials, the experimental setup, and results on energy harvesting performance of mechanical metamaterials.

The support of this research by National Science Foundation (IDR-1130948) is gratefully acknowledged. This research was supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology. Y.L. is grateful for support from the Department of Mechanical Engineering at the University of Connecticut.