A quasi-zero stiffness triboelectric vibration sensor for high-fidelity vibration characterization

Abstract

Triboelectric nanogenerator-based vibration sensors offer a promising solution for large-scale, distributed vibration monitoring in mechanical equipment and civil infrastructure. However, their performance is often compromised by inherent structural resonance, which introduces frequency-dependent sensitivity and distorts broadband vibration waveforms. In this work, we present a quasi-zero stiffness triboelectric vibration sensor (QZS-TEVS) that extends the post-resonance operational region towards ultralow frequencies, facilitating high-fidelity broadband vibration characterization. Specifically, an ultralow resonant frequency is realized via the QZS characteristic of the tailored harmonic curved beam. This strategic resonance shift broadens the flat-response bandwidth, thereby effectively suppressing resonance-induced distortion. The QZS-TEVS exhibits a nearly constant displacement sensitivity of 0.31 V/μm over 10–300 Hz, with a frequency measurement accuracy of 99.98%. Furthermore, the QZS-TEVS achieves precise reconstruction of arbitrary vibration waveforms. This work shifts the design paradigm from resonance-driven energy maximization to high-fidelity characterization and provides a high-precision scheme for self-powered, multidimensional condition monitoring within the Industrial Internet of Things (IIoT).

Introduction

Vibration stands out as an indispensable metric for evaluating structural integrity and mechanical performance, containing rich information regarding dynamic status. Consequently, the deployment of vibration sensors to extract these critical features has become a fundamental prerequisite for effective condition monitoring in engineering structures and mechanical systems. Moreover, the integration of vibration sensing and the Industrial Internet of Things (IIoT) facilitates the continuous, multidimensional diagnostics of critical industrial and civil infrastructure^1,2. However, the dependence of traditional sensors on batteries results in a limited lifespan and maintenance difficulties, making them unsuitable for the massive and long-term deployment required by large-scale sensor networks. To address these challenges, self-powered vibration sensors based on the active sensing principle have emerged as a promising solution.

Among the emerging self-powered technologies, triboelectric nanogenerators (TENGs), which leverage the coupling effect of contact electrification and electrostatic induction, were first presented by Wang’s group^3,4. Since then, TENGs have been widely recognized as promising solutions for mechanical energy harvesting^5–9 and self-powered active sensing^10–14. The self-generated electrical signals of TENGs precisely reflect the vibration characteristics, forming the basis of the active sensing mechanism. Furthermore, due to their high-efficiency electromechanical energy conversion, TENGs demonstrate exceptional sensitivity and electrical output even under micrometer-scale vibration excitations^15–18, making them ideal for self-powered vibration monitoring and diagnostics.

In practical engineering applications, vibration amplitude serves as a critical parameter for evaluating vibration severity and cumulative fatigue damage. Consequently, enhancing the amplitude sensing capability of triboelectric vibration sensors (TEVSs) across a broad bandwidth is crucial for facilitating the transition from qualitative detection to quantitative characterization. To meet this demand, diverse structural designs have been extensively explored for TEVSs. Among them, cantilever beam structures are widely investigated, owing to their high linearity and sensitivity to external excitations^16,19–21. Similarly, spring-assisted systems with linear stiffness are employed for narrowband vibration measurement centered at specific resonant frequencies^22,23. In contrast to conventional linear resonators with limited response to external excitations, bouncing-ball structures utilize the impact signals of free-moving spheres to effectively monitor low-frequency and multidirectional vibrations^24–28. Magnet-assisted structures employ non-contact mechanisms and address challenges induced by mechanical contact, thereby lowering the detection threshold for weak vibrations^29–31. Furthermore, for applications requiring high sensitivity, flexible film structures have been developed to capture subtle vibrations across a wide frequency range with high signal fidelity^15,32–34. Despite these diverse structural configurations, their operational bandwidth typically encompasses the resonant frequency. While the resonance mechanism improves the sensitivity of TEVSs at specific frequencies, it inherently introduces a non-uniform frequency response within the operational bandwidth. Such frequency-dependent sensitivity variations alter the relative amplitudes of different frequency components, thereby causing severe distortion of the measured vibration waveform. This intrinsic limitation makes TEVSs prone to significant measurement errors when subjected to vibration signals spanning a broad frequency spectrum.

To address this, it is essential to decouple the operational bandwidth from structural resonance, thereby guaranteeing uniform, frequency-independent sensitivity. An ideal solution is to utilize the broad post-resonance region (where the excitation frequency significantly exceeds the resonant frequency) as the operational bandwidth of the TEVSs, which exhibits a naturally flat response that extends far into the high-frequency domain. Furthermore, the resonant frequency of the mechanical structure can be reduced by stiffness tuning, thereby broadening the post-resonance operational bandwidth for accurate vibration characterization. However, in conventional linear structures, reducing stiffness to achieve a low resonant frequency inevitably compromises static stability. Incorporating the quasi-zero stiffness (QZS) mechanism provides an effective strategy to resolve this inherent conflict. While near-zero dynamic stiffness results in an ultralow resonant frequency, high static stiffness ensures that the overall structure is robustly maintained within the QZS region. Generally, the QZS characteristic is realized by the integration of positive and negative stiffness mechanisms, typically exemplified by spring-linkage mechanisms^35–37 and magnetic assemblies^38–40. However, their structural complexity poses significant challenges for sensor miniaturization. Specifically, spring-linkage mechanisms inevitably introduce joint friction and backlash, directly degrading the sensitivity to low-amplitude vibrations, while contact-free magnetic assemblies require strict air-gap tolerances to maintain the desired QZS behavior. In contrast to these complex assemblies, monolithic compliant beams^41–43, fabricated via additive manufacturing, offer an effective and compact solution for QZS mechanisms. This jointless design eliminates friction and assembly errors, while allowing extensive design flexibility to tailor QZS characteristics. However, the non-intuitive relationship between initial geometric parameters and corresponding stiffness characteristics poses significant challenges for the precise design of QZS compliant beams. Therefore, effectively exploiting the QZS characteristic to minimize the lower boundary of the post-resonance region remains challenging, despite its critical importance for broadening the operational bandwidth and achieving precise vibration characterization.

In this work, we propose a quasi-zero stiffness triboelectric vibration sensor (QZS-TEVS) designed for the high-fidelity characterization of vibration waveforms across an ultrawide post-resonance bandwidth. Driven by a simulation-based optimization framework, the geometric parameters of the harmonic curved beam are precisely determined to yield the target QZS characteristic. Consequently, the QZS-TEVS achieves an ultralow resonant frequency, extending the post-resonance operational region towards lower frequencies. In addition, the synergistic integration of the QZS mechanism and the dual-TENG architecture is employed for precise, high-efficiency electromechanical transduction of external dynamic excitations. Experimental results validate the efficacy of the QZS-TEVS in characterizing broadband vibration waveforms while effectively suppressing resonance-related distortion.

Results and discussion

Structure and working principle of the QZS-TEVS

The QZS-TEVS leverages a beam support structure composed of four harmonic curved beams featuring the QZS characteristic. Figure 1a illustrates the principal strain distribution within the beam support structure, as resolved through finite element (FE) analysis. As a monolithic QZS compliant mechanism, the harmonic curved beam with an intuitively wave-like morphology (Fig. 1b) is carefully designed to yield the target force-displacement response. This integrated design strategy delivers a highly compact and lightweight solution, inherently eliminating the assembly errors that commonly arise in multi-component QZS mechanisms. The specific methodology for constructing and optimizing the harmonic curved beam is detailed in Note S1.

Fig. 1.

Structure and working principle of the QZS-TEVS. a Beam support structure and corresponding FE analysis. b The initial shape of the harmonic curved beam. c Target force-displacement response with QZS region. d The low-frequency extension of the post-resonance operational region. e Schematic for external vibration measurement. f Overall working principle of the QZS-TEVS. g A photograph of the assembled QZS-TEVS

Pre-compressed by the proof mass, each curved beam settles into the QZS region of the force-displacement curve, where the dynamic stiffness effectively vanishes (Fig. 1c). In contrast to traditional mechanical components with linear stiffness, the QZS characteristic leads to a significant decrease in the resonant frequency of the system (Fig. 1d). Detailed derivations and theoretical analysis are provided in Note S2. The shift of the resonance peak to as low as 7 Hz substantially minimizes the lower boundary of the post-resonance region, extending the flat-response operational bandwidth into the low-frequency domain. Moreover, as the post-resonance region theoretically extends to infinity, the upper limit of the operational frequency is unbounded. Collectively, these features endow the QZS-TEVS with broadband characterization capabilities and overcome the bandwidth limitations inherent in conventional resonant TEVSs.

The physical basis for characterizing broadband vibrations is the stable inertial reference established within the QZS-TEVS structure. As illustrated in Fig. 1e, within the post-resonance operational region where the excitation frequency $\omega$ significantly exceeds the system’s resonant frequency ${\omega }_n$, the proof mass approximates a quasi-stationary state and effectively decouples from the base motion. Consequently, when the vibration source applies external excitation to the base, this dynamic input is effectively converted into relative motion within the structure. The post-resonance broadband sensing principle and relevant mathematical derivations are detailed in Note S3.

The overall working principle of the QZS-TEVS is illustrated in Fig. 1f. Initially, the harmonic curved beam is optimized to manifest the target QZS behavior, facilitating the precise conversion of mechanical vibrations. Subsequently, the dual-TENG architecture is integrated to further transduce internal relative motion into high-fidelity electrical outputs, thereby realizing self-powered active sensing of vibration signals. A photograph of the assembled QZS-TEVS, which is encased in a transparent acrylic shell to ensure operational stability, is shown in Fig. 1g.

Design of the dual-TENG compensation mechanism

As illustrated in Fig. 2a, the beam support structure of the QZS-TEVS comprises a suspended assembly and a base. The dual-TENG architecture with an integrated compensation mechanism is symmetrically positioned above and below the suspended assembly. Under external vibration excitations, electrical signals are generated through the periodic deformation of the curved beams. For each individual unit of the dual-TENG architecture, fluorinated ethylene propylene (FEP) is chosen as the negative triboelectric material, while the copper tape functions as both the positive triboelectric material and the electrode. An additional copper tape electrode is attached underneath the FEP film. Figure 2b reveals the micro-textured surface morphology of the FEP film treated with 2000-grit sandpaper. Such surface modifications significantly facilitate ion trapping during the subsequent polarization process, thereby enhancing the charge storage capacity and the temporal stability of the surface charge.

Fig. 2.

Design of the dual-TENG compensation mechanism. a Overall structure of QZS-TEVS with the dual-TENG architecture. b Image of the FEP surface treated with 2000-grit sandpaper. c Working process of the dual-TENG architecture in one complete vibration cycle. d FE simulations of the electric potential distribution of an individual TENG at various separation distances of 0.1, 0.2, 0.3, and 0.4 mm

Theoretically, the proof mass remains inertially stationary at its vertical central position within the post-resonance operational bandwidth, where the excitation frequency significantly exceeds the ultralow resonant frequency. In practice, static gravitational offset and asymmetric restoring forces cause the equilibrium position of the proof mass to deviate vertically from the geometric center, leading to asymmetric upper and lower levitation gaps. For illustration, we consider a representative case in which the equilibrium position of the proof mass shifts below the geometric center. This downward shift alters the originally identical gaps in the dual-TENG architecture. Consequently, the lower TENG exhibits a higher displacement sensitivity than the upper TENG due to the enhanced electrostatic induction.

In the dual-TENG compensation mechanism, the lower TENG primarily measures the relative motion of the narrower gap with high sensitivity. Meanwhile, the upper auxiliary TENG captures the corresponding motion of the wider gap as a compensation signal. The final electrical output of the QZS-TEVS is derived by combining these two anti-phase electrical signals via the dual-channel differential approach. The detailed mathematical derivations and underlying principles of the dual-channel differential algorithm are presented in Note S4. This process constructively superimposes the dynamic amplitudes to enhance the overall displacement sensitivity, while eliminating common-mode noise and environmental disturbances. Furthermore, owing to the complementary push-pull layout, the sensitivity variations of the two channels are equal in magnitude but opposite in sign, and thus mutually cancel during the differential combination. Consequently, this collaborative mechanism maintains a constant combined displacement sensitivity, even when the equilibrium position of the proof mass deviates from the geometric center.

The detailed working process of the dual-TENG architecture is illustrated in Fig. 2c. One complete vibration cycle is divided into two parts: the positive half-cycle (from the center upward and back) and the negative half-cycle (from the center downward and back). When the bottom electrode of TENG2 moves upward in the positive half-cycle of the vibration excitation, electrostatic induction drives electrons to transfer from the bottom to the top electrode in TENG2. Conversely, as the bottom electrode of TENG2 moves downward, the direction of electron transfer between the two electrodes in TENG2 is reversed. The negative half-cycle follows an analogous principle, with the symmetrical movement of the electrodes inducing a corresponding electron flow. Under periodic vibration excitations, an alternating current (AC) is generated in the external circuit. Concurrently, the electrodes of TENG1 also generate corresponding compensatory AC signals that are 180° out of phase with those of TENG2. The respective sensitivities of the two TENGs in the dual-TENG architecture may differ, depending on the actual equilibrium position of the proof mass under dynamic excitations. The FE simulations in Fig. 2d show that the electric potential difference of an individual TENG unit scales with its separation distance (specifically at 0.1 mm, 0.2 mm, 0.3 mm, and 0.4 mm), underpinning the sensing mechanism of the QZS-TEVS. Physical experiments further confirm that the electrical outputs of the TENG unit can accurately reflect mechanical displacement variations, thereby supporting the high-fidelity vibration characterization capability of the QZS-TEVS (refer to Note S7).

Optimization and experimental validation of QZS characteristics

Since the beam support structure of the QZS-TEVS operates within the prescribed QZS region for effective measurements, it is of great significance to optimize the initial shape of the curved beam, thereby achieving the target force-displacement response. The compliant beam model is simplified as shown in Fig. 3a, with detailed geometric parameters provided in the Methods section. The left end of the compliant beam is fixed at the coordinate origin $O$, and the right end is constrained by a symmetric boundary condition, allowing only vertical displacement while restricting rotation and horizontal movement. Under the external force $F$, the right end undergoes vertical displacement, and the beam experiences large-deflection deformation around the coordinate origin $O$.

Fig. 3.

Optimization and experimental validation of QZS characteristics. a Simplified model of the compliant beam and constraints. b The QZS characteristic defined by five discrete target points. c Force and stiffness characteristics of the individual QZS compliant beam. d Deflection profile of the harmonic curved beam. e Experimental and simulated deformations of the beam support structure. f Schematic of the equivalent mechanical model and vibration measurement setup. g Experimental displacement waveforms in the time domain

On this basis, we propose a harmonic curved beam as the QZS compliant mechanism, which is inherently composed of a convex curve with positive stiffness and a concave curve with negative stiffness. The centerline profile of the harmonic curved beam is constructed using the fundamental harmonic components, expressed as a linear combination of sine and cosine functions:

$$y_0\left(x\right)={\sum }_{n=1}^N\left(c_{2n-1}\sin \left(2n\pi x\right)+c_{2n}\cos \left(2n\pi x\right)\right),\mathrm{x}\in \left[0,1\right]$$ 1

This harmonic formulation allows for versatile tailoring of the beam’s initial geometry through its coefficients $c_1$, $c_2$, …, $c_{2N}$. However, an increase in the number of coefficients leads to greater complexity in parameter design. Consequently, we focus exclusively on $c_1$ and $c_2$, yielding the following expression:

$$y_0\left(x\right)=c_1\sin \left(2\pi x\right)+c_2\cos \left(2\pi x\right),\mathrm{x}\in \left[0,1\right]$$ 2

The basic curve shape is obtained by scaling down the shape baseline by a factor of $\lambda$ in the y-direction. Subsequently, rotated by an angle $\theta$ and scaled along the axis connecting the two endpoints, the final centerline profile of the harmonic curved beam is determined. The detailed construction process of the curved beam is shown in Figure S1.

The desired QZS characteristic of the compliant beam manifests as a nearly constant reaction force over a specific displacement range, which is defined by the distribution of the five discrete green target points in Fig. 3b. The force-displacement response of the compliant beam is characterized via FE simulations. Subsequently, by integrating this FE model with the genetic algorithm (GA), the proposed framework iteratively determines the optimal structural parameters. The relative squared residual at the $i$-th target point between the simulated reaction force $F_i$ and the target force $F_{\mathrm{target}}$ is formulated as:

$$e_i={\left(\frac{F_i-F_{\mathrm{target}}}{F_{\mathrm{target}}}\right)}^2,i=1,2,\dots ,5$$ 3

Accordingly, the optimization goal is defined as:

$${\min }_{c_1,c_2,\lambda }Res={\sum }_{i=1}^5{e}_i$$ 4

Details of the optimization procedure are provided in Figure S2. Ultimately, a set of optimal parameters for the harmonic curved beam was obtained, where $c_1$ = 0.1414, $c_2$ = -0.0622, and $\lambda$ = 0.5339.

Based on these optimized parameters, the physical prototype was fabricated to experimentally validate the target QZS characteristic, thereby simultaneously verifying the accuracy of the numerical model. The static mechanical behavior of the support structure incorporating harmonic curved beams was characterized using a universal tensile-compression testing machine. Figure 3c presents a comparison of reaction force-displacement curves obtained from static compression experiments and numerical simulations, with the corresponding stiffness-displacement profiles, calculated by differentiating the force-displacement curves, superimposed for reference. The results demonstrate that the experimental data agree well with the numerical simulations, and the system exhibits the desired constant-force and zero-stiffness characteristics within the QZS region, which in turn validates the efficacy of the simulation-based optimization framework. Although a minor hysteretic discrepancy is observed between the experimental and numerical curves—primarily due to material viscoelasticity—it has a negligible impact on the overall QZS characteristics and mechanical repeatability. Quantitatively, the theoretical QZS region (where the stiffness remains below 0.1 × 10³N/m) spans from 2.8 mm to 6.2 mm. Figure 3d presents the deflection profiles of the harmonic curved beam across Δd = 3–6 mm, where the shaded area outlines a prescribed operational range strictly within the theoretical QZS boundaries. To provide a direct visual representation, the experimental and simulated deformations of the beam support structure under specific loading displacements of 3 mm and 6 mm are illustrated in Fig. 3e.

By leveraging the tailored QZS characteristic of the harmonic curved beam, the proof mass (681 g) was specifically configured to constrain the structural equilibrium within the prescribed QZS region, thereby achieving an ultralow resonant frequency and broadening the flat-response operational bandwidth. Crucially, with the curved beam pre-compressed to the static equilibrium position of 4.5 mm, the initial gap of 0.2 mm within the dual-TENG architecture physically restricts the displacement of the proof mass to the range of 4.3–4.7 mm. Within this sufficiently narrow operational range, higher-order nonlinear stiffness effects are mathematically negligible while the dynamic stiffness remains consistently low, thereby preserving the ultralow resonant frequency of the system.

As illustrated in Fig. 3f, the beam support structure is equivalently modeled as a single-degree-of-freedom (SDOF) system. During the random vibration tests, the displacement responses of the base and the proof mass were measured using two commercial laser displacement sensors, respectively. The relative motion signal was subsequently determined as the difference between the mass response and the base excitation. Given that the relative motion exhibits an inherent anti-phase relationship with the base excitation (details in Note S3), the resulting relative motion waveform was phase-inverted to facilitate direct visual comparison. Notably, the displacement of the proof mass remains close to zero, maintaining a quasi-stationary state (Fig. 3g), while the relative motion (blue line) exhibits a strong morphological congruence with the base excitation (black line). The base excitation was a synthesized multi-frequency vibration signal, generated by superimposing multiple discrete frequency components within the 20–40 Hz frequency range. The signal fidelity is quantitatively assessed by a Pearson correlation coefficient of 0.9874, while the measurement accuracy is confirmed by a root mean square error (RMSE) of 0.0510 mm. This underscores the efficacy of the beam support structure in converting vibration excitations into internal relative motion within the prescribed QZS region, laying a solid foundation for the QZS-TEVS to process complex vibration signals.

Output characteristics and sensing performance of the QZS-TEVS

Figure 4a illustrates the vibration test setup for characterizing the electrical output performance of the QZS-TEVS. In this setup, the shaker generated harmonic excitation signals with controllable frequency and amplitude, while the QZS-TEVS was mounted on its push rod. Additionally, a laser displacement sensor was employed to measure the vibration displacement generated by the shaker in a non-contact manner, with its analog output collected by the vibration DAQ system. The peak-to-peak voltage ($V_{PP}$) of the QZS-TEVS as a function of displacement amplitude at 30 Hz is illustrated in Fig. 4b, for various initial gaps (D = 0.1, 0.2, 0.3, and 0.4 mm) between the two triboelectric layers. The maximum output magnitude is attained at an initial gap of 0.2 mm, which is attributed to the combined effects of interfacial potential decay and electrostatic attraction. This reveals a trade-off regarding the initial gap: while a smaller initial distance enhances the electrical output at a fixed amplitude, an excessively narrow distance induces strong electrostatic adhesion that impedes free vibration and reduces the electrical output. Furthermore, the output performance of the QZS-TEVS, configured with various dielectric materials featuring micro-textured surfaces as the negative triboelectric layer, was systematically evaluated at a vibration amplitude of 40 μm and a frequency of 30 Hz. As shown in Fig. 4c, the polarized FEP film exhibits the optimal output performance. Accordingly, the finalized configuration of the dual-TENG architecture incorporates an initial gap of 0.2 mm and the polarized FEP film as the negative triboelectric layer for all subsequent experiments.

Fig. 4.

Output performance characterization of the QZS-TEVS. a Schematic of the experimental test platform. b Peak-to-peak voltage of QZS-TEVS under various initial gaps. c Output voltage signals for different negative triboelectric layers. d The synergistic operation of the dual-TENG architecture. e Comparison of the individual voltage signals of TENG1 and TENG2 at 30 Hz and 40 μm. f The combined signal resulting from superposition. g Peak-to-peak voltage and power densities at 30 Hz and 60 μm under different load resistances. h Output voltage signals and (i) corresponding FFT spectra at the initial stage and after 12 h

The synergistic operation of the dual-TENG architecture under vibration excitation is schematically illustrated in Fig. 4d. The blue arrow indicates the amplitude of external vibration excitation, while the red and black arrows denote the corresponding output voltage amplitudes of TENG1 and TENG2, respectively. In the actual experimental setup, the equilibrium position of the proof mass shifted below the geometric center, consistent with the representative scenario analyzed in the preceding subsection. This asymmetric configuration leads to a dominant electrical output and heightened sensitivity in the lower TENG due to the enhanced electrostatic induction. Meanwhile, the upper TENG with a wider gap exhibits a weaker electrical response, providing a complementary output to achieve dynamic compensation. Based on the dual-channel differential algorithm, calculations reveal that the two individual voltage outputs of TENG1 and TENG2 account for 32% and 68% of the total combined magnitude, respectively (Fig. 4e, f). These results experimentally demonstrate that the dual-channel differential algorithm yields a high-fidelity combined voltage output by suppressing common-mode noise and compensating for sensitivity variation. This ensures the accurate and comprehensive characterization of vibration excitation by the QZS-TEVS.

The $V_{PP}$ and power densities under various load resistances are illustrated in Fig. 4g, measured at a vibration amplitude of 60 μm and a frequency of 30 Hz. As the load resistance increases from 0.1 MΩ to 100 MΩ, the $V_{PP}$ initially increases and subsequently reaches a plateau, indicating a transition from an approximately short-circuit state to an open-circuit state. Meanwhile, the output power exhibits a bell-shaped profile, achieving a maximum of 21.93 mW·m⁻² at a matching impedance of 20 MΩ.

In addition, the QZS-TEVS features a non-contact triboelectric layer configuration, which prevents contact wear during operation and minimizes performance degradation. The output voltage signals and their corresponding Fast Fourier Transform (FFT) spectra, recorded both initially and after 12 h of continuous operation (10 Hz, 40 μm), are illustrated in Fig. 4h, i. The QZS-TEVS exhibits no significant attenuation in output amplitude (with a decay of only 4.49%), while its spectral characteristics remain highly consistent with those of the initial stage. Furthermore, the integrated acrylic shell of the QZS-TEVS serves as a physical barrier, effectively mitigating rapid environmental fluctuations and maintaining a nearly consistent electrical output (Note S9 and Figure S11). These results demonstrate the excellent signal stability and mechanical durability of the QZS-TEVS, underscoring its potential for in-situ vibration monitoring in complex engineering environments. To investigate the performance limits and long-term degradation mechanisms under cyclic fatigue, an extended 24-hour continuous test was conducted, as detailed in Note S11 and Figure S13. The results indicate that while the current prototype is suitable for monitoring applications of up to 12 h, it requires further improvements in structural fatigue resistance and charge stability for longer-term operations.

To validate the performance of the QZS-TEVS in high-fidelity vibration characterization, a series of dynamic excitation experiments was conducted. The voltage signals generated by the QZS-TEVS under different excitation frequencies are illustrated in Fig. 5a, b. The excitation frequency was swept from 6 to 300 Hz while a constant displacement amplitude of 40 μm was maintained. The output reaches its maximum at approximately 7 Hz, which coincides with the resonant frequency of the tailored harmonic curved beam. Beyond the resonant frequency of 7 Hz, the output amplitude tends to stabilize and becomes nearly independent of the vibration frequency. This characteristic facilitates accurate vibration displacement characterization irrespective of frequency variations, thereby ensuring exceptional sensing performance over an ultrawide post-resonance bandwidth. The corresponding short-circuit current response is shown in Figure S10, revealing that the output current increases linearly as the frequency rises from 10 to 300 Hz with the amplitude maintained at 40 μm. To assess the spectral fidelity of the QZS-TEVS, FFT analysis was performed on both the output voltage signal and the reference vibration excitation, as illustrated in Fig. 5c. This confirms that the frequency components of the output voltages align precisely with the excitation spectra, maintaining a high spectral consistency of 99.98%.

Fig. 5.

Dynamic sensing performance of the QZS-TEVS under vibration excitations. Output voltage signals of QZS-TEVS at a fixed amplitude of 40 μm across different frequency ranges: (a) 6–50 Hz and (b) 75–300 Hz. c Relationship between the output voltage frequency and the vibration frequency. d Peak-to-peak voltage as a function of vibration amplitude at 30 Hz. e Peak-to-peak voltage as a function of vibration amplitude at 100 Hz. f Displacement sensitivity as a function of excitation frequency. g Comparison of output voltages under two sets of vibration excitations. h Time-domain displacement signals and (i) corresponding FFT spectra of the reconstructed and reference signals

The linear relationship between the $V_{PP}$ and displacement amplitude at excitation frequencies of 30 Hz and 100 Hz is illustrated in Figs. 5d and e, respectively. The calculated displacement sensitivities are 0.299 V/μm and 0.303 V/μm, while the $R^2$ values (> 0.995) demonstrate the superior linearity of the QZS-TEVS. The validated linear displacement range of the QZS-TEVS is defined as 0–50 μm across the 10–300 Hz frequency range. This upper limit of 50 μm is conservatively established to prevent excessive dynamic accelerations and inertial forces at higher excitation frequencies. Furthermore, the displacement sensitivity was characterized over a broader frequency range of 1–300 Hz (Fig. 5f). At each discrete frequency, output voltage signals were recorded against varying vibration amplitudes to derive the corresponding displacement sensitivity. The frequency step size was reduced in the vicinity of the resonance to obtain more precise and refined sensitivity profiles. The results indicate that the displacement sensitivity increases with frequency below 7 Hz, whereas above 7 Hz, it gradually plateaus and remains consistent across a wide frequency band. Therefore, 10 Hz was selected as the lower bound of the effective operational range because it marks the onset of the stable post-resonance operational region. The upper limit was set to 300 Hz because this defined frequency range encompasses the dominant components in typical structural displacement spectra. The QZS-TEVS achieves a stable average sensitivity of 0.31 V/μm within the 10–300 Hz frequency range, confirming its reliable displacement sensing performance in the post-resonance region. This verified stable displacement sensitivity over 10–300 Hz, together with the inherent flat response of the theoretically unbounded post-resonance region, demonstrates the accurate vibration characterization capability of the QZS-TEVS across an ultrawide bandwidth. Additionally, the equivalent acceleration sensitivity was further derived and experimentally validated against a commercial accelerometer (details provided in Note S12).

Regarding environmental robustness, the displacement sensitivity of the QZS-TEVS under different temperature and humidity conditions was further assessed in Note S10. The QZS-TEVS showed negligible sensitivity drift within the temperature range of 20–45 °C and humidity level below 50% RH, confirming its stable operation under typical ambient conditions. To extend its applicability to harsher environments, however, the integration of protective encapsulation and environmental compensation algorithms remains necessary.

Beyond environmental factors, the physical reliability of the QZS-TEVS is determined by its operational boundaries and potential failure modes. The physical displacement limit of 0.2 mm within the dual-TENG architecture is significantly smaller than the effective QZS range ( ± 1.7 mm), thereby guaranteeing ultralow stiffness during normal operation. Nevertheless, extreme conditions may trigger two primary failure scenarios, as well as nonlinear output characteristics. First, dynamic excitations exceeding the 0.2 mm gap cause physical collision between the TENG layers, resulting in voltage saturation and waveform clipping. Second, static drift of the equilibrium position outside the QZS region causes severe stiffness hardening, resulting in the loss of low-frequency sensing capability. This also introduces significant amplitude attenuation and nonlinear frequency distortions. Consequently, comprehensively defining these operational boundaries provides a guideline for reliable operation, which is crucial for the practical application of the QZS-TEVS.

Moreover, the output voltages under two sets of vibration excitations with different frequencies and amplitudes are presented in Fig. 5g, confirming that the voltage amplitude is linearly proportional to the displacement amplitude while remaining invariant to the excitation frequency. In contrast to resonant-type triboelectric sensors, the consistent and frequency-independent sensitivity of the QZS-TEVS enables direct determination of vibration magnitude from $V_{PP}$ without complex frequency calibration. Subsequently, a synthesized multi-frequency signal (8, 20, 40, 60, and 80 Hz) was employed to emulate complex real-world vibrations. The temporal displacement profiles reconstructed by the QZS-TEVS exhibit close agreement with the reference vibration signals measured by a laser displacement sensor (Fig. 5h). The analysis yields a correlation coefficient of 0.9864 and an RMSE of 7.57 μm, confirming the high-precision sensing performance of the QZS-TEVS. To quantify the accuracy enhancement enabled by the dual-TENG compensation mechanism, a comparative evaluation was performed utilizing only the uncompensated electrical output of TENG2, yielding a degraded correlation coefficient of 0.9787 and a larger RMSE of 9.26 μm. The detailed temporal waveform reconstruction under this uncompensated scheme is provided in Note S13. Regarding the frequency-domain characteristics, the FFT spectra of the QZS-TEVS reconstructed signal and the reference vibration signal are illustrated in Fig. 5i. Notably, while a discernible amplitude deviation occurs near 8 Hz, the QZS-TEVS maintains remarkable amplitude consistency at other dominant frequencies, corroborating the results presented in Fig. 5f. The uniform frequency response across the ultrawide post-resonance spectrum (beyond 10 Hz) demonstrates the efficacy of the QZS-TEVS for high-fidelity vibration characterization across diverse dynamic environments. To evaluate the practical competitiveness of the self-powered QZS-TEVS, comparative experiments were performed against a commercial accelerometer under dynamic broadband excitations. Detailed analyses of the waveform reconstruction accuracy and spectral fidelity are provided in Note S14.

Furthermore, the proposed QZS-TEVS was benchmarked against representative TENG-based vibration sensors reported in recent literature, as summarized in Note S15. It is evident that the QZS-TEVS exhibits competitive dynamic displacement sensing capabilities, particularly in achieving a uniform-response operating bandwidth and high-fidelity waveform reconstruction.

Application demonstration of the QZS-TEVS

To demonstrate the practical application of the proposed QZS-TEVS, an IIoT-based vibration monitoring platform was developed (Fig. 6a). Diverse vibration signals originating from industrial machinery, transportation systems, and civil infrastructure are accurately captured and transduced into electrical signals by the QZS-TEVS. The dual-channel TENG signals from the QZS-TEVS are subsequently acquired by a custom-designed printed circuit board (PCB) (Fig. 6b), which integrates an analog front-end (AFE) for signal conditioning and a high-performance microcontroller unit (MCU) for real-time digitization and processing. The processed data are wirelessly transmitted via a 4G transceiver module to a cloud server, where they can be remotely accessed by user terminals. The remote monitoring dashboard for mobile and desktop platforms is presented in Fig. 6c, which provides an intuitive interface for real-time visualization and vibration analysis in both the time and frequency domains.

Fig. 6.

Application demonstration of the QZS-TEVS. a IIoT-based vibration monitoring platform with signal acquisition, wireless transmission, and multi-terminal visualization. b Photograph of the custom-designed PCB. c Monitoring dashboard for real-time vibration analysis. d Experimental setup for vibration monitoring of an air compressor. e Acquired time-domain signals during different operational phases. f FFT spectrum of the normal operation phase

The experimental setup for vibration monitoring and analysis of a reciprocating air compressor is shown in Fig. 6d. The QZS-TEVS and the custom-designed PCB were securely mounted on the top of the compressor to capture the respective vibration signals generated during the start-up, normal operation, and shutdown phases (see Supplementary Video). The time-domain vibration waveforms were captured and transmitted to the user terminal in real time, as presented in Fig. 6e. These temporal profiles exhibit distinct vibration signatures: the start-up phase (0.2–0.3 s) is characterized by an abrupt rise from the near-zero baseline and short transient peaks; the normal operation phase (0.3–2.1 s) displays continuous oscillations with a relatively stable amplitude envelope; and the shutdown phase (2.1–2.3 s) shows isolated high-amplitude spikes followed by rapid attenuation to a standstill. While these time-domain signals distinguish the operational phases through their amplitude envelopes and transient features, frequency-domain analysis yields deeper spectral insights. The FFT spectrum of the normal operation phase (Fig. 6f) exhibits a marked contrast to the spectral profiles manifested during the transient start-up and shutdown phases (Figure S18). The pronounced distinctions in both temporal profiles and spectral features underscore the robust sensing capability of the QZS-TEVS to comprehensively characterize and accurately discriminate among diverse operational conditions of engineering systems. This integrated platform serves as a foundational framework for large-scale, distributed diagnostics, enabling real-time remote assessment of critical industrial and civil infrastructure.

Conclusion

In this work, a quasi-zero stiffness triboelectric vibration sensor (QZS-TEVS) with the dual-TENG compensation mechanism is developed, leveraging the ultrawide post-resonance bandwidth for high-fidelity vibration characterization. We strategically redefine the lower limit of the post-resonance operational region through the tailored QZS compliant mechanism, with the flat-response bandwidth extended towards lower frequencies. Coupled with the high-efficiency electromechanical transduction of the dual-TENG architecture, the QZS-TEVS demonstrates a consistent displacement sensitivity of 0.31 V/μm and 99.98% spectral agreement across 10–300 Hz. Furthermore, experimental results substantiate its superior fidelity in reconstructing arbitrary vibration waveforms, thereby validating its robust performance across diverse dynamic environments. It is worth emphasizing that the inherent customizability and architectural scalability of QZS compliant mechanisms facilitate tailored adjustments to diverse vibration monitoring scenarios. This work, integrating self-powered nanogenerator technologies with nonlinear structural dynamics, addresses the bottlenecks of power dependency and sensing accuracy within large-scale, distributed Industrial Internet of Things (IIoT) networks.

Materials and methods

Fabrication of the QZS-TEVS

The QZS-TEVS consists of a base, a suspended assembly, two TENGs operating in non-contact mode, and an outer acrylic shell. The suspended assembly comprises a proof mass, a junction plate, and four pre-compressed QZS curved beams. The structural components, particularly the harmonic curved beams, were fabricated using stereolithography (SLA) 3D printing with photosensitive resin (Young’s modulus: 805 MPa, Poisson’s ratio: 0.4). To achieve the desired QZS state, a 681-g tungsten steel block was selected as the proof mass. The top and side panels of the outer shell were laser-cut from acrylic sheets, and all structural components were rigidly fastened using screws. The curved beam featured a length ($L_x$) of 21 mm and a height ($L_y$) of 9 mm, with an out-of-plane thickness ($d$) of 10 mm and an in-plane thickness ($t$) of 0.8 mm; additionally, integrated mounting blocks are employed at both ends for secure screw fixation. Regarding the TENG fabrication, commercially available PET, PTFE, and FEP films (50 μm in thickness and 16 cm² in area) were mechanically abraded with 2000-grit sandpaper to create micro-textured surfaces with increased roughness. Subsequently, the textured films were electrically polarized to achieve high-density ion injection, effectively maximizing the surface charge density (details in Note S5). Copper tape (100 μm in thickness and 16 cm² in area) was selected as both the triboelectric material and the electrode, with lead wires connected for signal acquisition. The overall dimensions of the assembled QZS-TEVS are 96 mm × 96 mm × 40 mm.

Measurement and characterization

The dynamic vibration testing system comprised a shaker (DH40500, DONGHUA Test Co., Ltd.), a signal generator (33500B, Keysight), and a power amplifier (DH5874, DONGHUA Test Co., Ltd.). The analog voltage outputs from laser displacement sensors (HL-G103-A-C5, Panasonic) were captured and digitized using a data acquisition platform (Moku:Lab, Liquid Instruments) to obtain the vibration displacement signals. The vibration acceleration signals were measured via a commercial accelerometer (ZSA1500L, ZiSHU Tech), with its outputs conditioned by a signal conditioner (SRD-1104, Smacq). The QZS-TEVS was rigidly fixed to the shaker via a customized adapter plate. The output voltage of the QZS-TEVS was measured by an oscilloscope (DSOX3024T, Keysight) with a 100 MΩ high-voltage probe, while the short-circuit current was measured by a low-noise current preamplifier (SR570, Stanford Research Systems). Surface morphologies of the treated films were characterized by a metallurgical microscope (M330-HK830, Aosvi). The detailed experimental setup is displayed in Note S6. The frequency spectrum was obtained by performing FFT analysis on the time-domain data after filtering out power-line interference and its harmonics. Unless otherwise specified, all experimental measurements were conducted at 25 °C and 40% relative humidity.

Numerical simulations

To investigate the electromechanical coupling and mechanical characteristics of the QZS-TEVS, FE analysis was performed using COMSOL Multiphysics 6.0. Specifically, the Electrostatics interface was employed to solve Poisson’s equation across the computational domain, determining the spatial electric potential distribution. In the FE model, the FEP dielectric layer (thickness: 50 μm, relative permittivity: 2.1) and the Cu electrodes (thickness: 100 μm) were both configured with an effective area of 16 cm². The triboelectric charge density on the FEP surface was set to 3 × 10⁻⁶C/m². The Beam interface was utilized to derive the nonlinear force-displacement responses of the QZS compliant beam, due to its superior computational efficiency and robustness in handling geometric nonlinearities. Concurrently, the Solid Mechanics interface was applied to calculate the strain distribution within the support structure of the QZS-TEVS, confirming that harmonic curved beams operate within the material’s elastic limit under defined loading conditions. To reduce computational cost, all components other than the QZS compliant beams were modeled as rigid bodies and excluded from the computational domain. The beams were assigned a Young’s modulus of 805 MPa and a Poisson’s ratio of 0.4, consistent with the photosensitive resin used for fabrication.

Remote vibration monitoring platform

For experimental validation, a reciprocating air compressor was employed as the vibration source. The QZS-TEVS was securely mounted on the top casing of the compressor to capture the induced vibrations, while the fabricated PCB simultaneously acquired the corresponding TENG signals at a sampling frequency of 2 kHz. The custom-designed PCB primarily integrated two core components: an AFE circuit and a high-performance MCU (STM32F407VET6, STMicroelectronics). Specifically, the AFE circuit comprised a charge amplifier, a second-order low-pass filter, and an inverting amplifier to ensure signal integrity and measurement accuracy (see Note S16 for details). Following signal conditioning, the MCU performed analog-to-digital conversion and real-time processing tasks. The PCB was powered by a 12 V DC supply and equipped with a 4G transceiver module (Air780EHT, HEZHOU Communication Co., Ltd.) to facilitate stable and wireless data transmission to a cloud server. Subsequently, the transmitted data were remotely accessed and visualized in real time via mobile and desktop terminals connected to the server.

Author contributions

Z. Li and J. Xu conceived the idea and conducted the experiments. X. Chen and Z. Li contributed to the sensor design, system integration and data analysis. L. Zhao supervised the project. Z. Li wrote the paper with contributions from all authors. All the authors discussed the results and commented on the manuscript.

Conflict of interest

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41378-026-01453-4.