Diagnostics of Piezoelectric Bending Actuators Subjected to Varying Operating Conditions

Abstract

In applications of piezoelectric actuators and sensors, the dependability and particularly the reliability throughout their lifetime are vital to manufacturers and end-users and are enabled through condition-monitoring approaches. Existing approaches often utilize impedance measurements over a range of frequencies or velocity measurements and require additional equipment or sensors, such as a laser Doppler vibrometer. Furthermore, the non-negligible effects of varying operating conditions are often unconsidered. To minimize the need for additional sensors while maintaining the dependability of piezoelectric bending actuators irrespective of varying operating conditions, an online diagnostics approach is proposed. To this end, time- and frequency-domain features are extracted from monitored current signals to reflect hairline crack development in bending actuators. For validation of applicability, the presented analysis method was evaluated on piezoelectric bending actuators subjected to accelerated lifetime tests at varying voltage amplitudes and under external damping conditions. In the presence of a crack and due to a diminished stiffness, the resonance frequency decreases and the root-mean-square amplitude of the current signal simultaneously abruptly drops during the lifetime tests. Furthermore, the piezoelectric crack surfaces clapping is reflected in higher harmonics of the current signal. Thus, time-domain features and harmonics of the current signals are sufficient to diagnose hairline cracks in the actuators.

1. Introduction

Due to the absence of mechanical wear parts and other characteristics such as miniaturization capability, piezoelectric materials have found numerous applications as sensors and actuators, such as in textile knitting machines, automotive fuel injectors, medical dosing systems, wire bonding, welding, energy harvesting and structural and health monitoring (SHM) applications [1,2,3,4]. In these applications, the dependability and particularly the reliability throughout the lifetime of piezoelectric sensors and actuators are vital to manufacturers and end-users. To this end, diagnostics and prognostics, enabled through condition monitoring, can be employed. Diagnostics involves detection, isolation and identification of anomalies, faults or failure [5,6]. Diagnostics, and specifically fault detection, can enable prognostics, which involves predicting the remaining useful life time until failure [5,6].

In SHM applications, Park et al. [2,7] monitored the degradation of piezoelectric sensors by detecting changes in the slope of the susceptance, that is, the imaginary part of measured electrical admittance. Taylor et al. [8] normalized the susceptance to account for the degradation of piezoelectric sensors under time-varying conditions. Mueller and Fritzen [9,10] provide an overview of piezoelectric actuator and sensor diagnostics methods in SHM applications, specifically via electromechanical impedance measurements or features derived thereof. After temperature compensation, Liang et al. [11] employed characteristic changes in wave propagation measurements to diagnose a piezoelectric actuator and sensor network. Jiang et al. [12,13,14] trained machine learning (ML) techniques, such as k-nearest neighbors and support vector machines, with features extracted from measured electrical admittance to identify different piezoelectric sensor faults and structural degradation. Based on the time-reversal process and a self-sensing circuit, Lee and Sohn [15] diagnosed piezoelectric sensor faults, even under varying temperature and structural conditions, via features derived from voltage signals. However, the influence of time-varying operating conditions was not considered in their study. In an ultrasonic application, the characteristics of the measured small-signal electrical admittance also reflect the presence of a crack in the power transducer [3]. In an energy harvesting application, Avvari et al. [16] monitored the degradation of a piezoelectric transducer via impedance, displacement and specifically via voltage measurements. Although initial damage to the transducer could be detected by a significant decrease in the measured output voltage, the influence of varying operating conditions, such as external force, was not investigated. Kimotho et al. [4] extracted features from velocity measurements to track the degradation of piezoelectric bending actuators and mapped features derived from electric current through ML techniques to predict their remaining useful life (RUL). Bender [17] derived initial model parameters from electrical and mechanical admittance curves and stiffness values from velocity measurements to diagnose piezoelectric bending actuators.

In summary, the proposed methods often utilize impedance or admittance measurements over a range of frequencies or velocity measurements to monitor the piezoelectric sensors, actuators and transducers. To minimize the need for additional equipment or sensors, such as a laser Doppler vibrometer, while maintaining the dependability of piezoelectric bending actuators irrespective of time-varying operating conditions and external force, an online diagnostics approach is proposed, while Bender [17] diagnosed cracks in piezoelectric bending actuators via a Butterworth and Van Dyke surrogate model, this paper focuses on a data-driven approach. Additionally, external forces are taken into account in a comprehensive diagnostics of piezoelectric bending actuators. Both approaches enable online diagnostics of cracks, without interruptions of the experiments due to further measurements, e.g., impedance measurements over a range of frequencies. However, the evaluation of a real-time diagnostics is not part of this paper. In this paper, varying voltage amplitude and external force via immersing the actuator in a viscous fluid is considered, while other operating and environmental conditions, such as humidity and temperature, are not considered. During operation, the anomaly is detected by evaluating time-domain features and fundamental frequency extracted from continuously acquired current signals of a resonant-driven piezoelectric bending actuator. Furthermore, features from higher harmonics of the current signals allow crack isolation in at least one piezoelectric layer of the piezoelectric bending actuator. Owing to the clamping situation, mode of operation and resulting mode shape, hairline crack(s) typically occur near the clamped region of the investigated bending actuator.

The paper is structured in four further sections. In Section 2, possible failure modes, their causes and effects are identified. In Section 3, the lumped-parameter model of a piezoelectric actuator is presented. Moreover, the framework for diagnosing the actuator is developed. Based on the identified failure modes and presented methodology, the further analysis and experimental evaluation is laid out in Section 4. Lastly, Section 5 concludes with some limitations of the current study, paving the way for further research.

2. Failure Mode and Effects Analysis

To develop and conduct a suitable diagnostics approach, possible failure modes of the studied actuators must be identified. Therefore, a failure mode and effects analysis (FMEA) was performed based on the literature [18,19,20,21,22,23,24,25,26] and lifetime tests published in [17]. In this FMEA failure modes, causes and effects of the focused actuators are evaluated. The following FMEA in

Table 1. FMEA of the piezoelectric bending actuator.

Failure Mode	Failure Effect	Failure Cause
Piezoelectric ceramics
Crack	Reduced stiffness, increased resistance, consequent crack in the electrode	Mechanical load
Depolarization	Decreased piezoelectric characteristics, flatter resonance peaks, resonance and anti-resonance frequency coincide, loss of functionality	Operation with an electrical voltage opposite to the actuator’s polarization, mechanical load opposite to the actuator’s polarization, elevated temperature
Leakage current	Overload of electronic components in the measurement chain, such as the piezoelectric amplifier, loss of functionality	Elevated unipolar electric field strength, electromigration, contamination, humidity
Electrode
Crack	Increased contact resistance, increased thermal load, reduced capacity, decreased active area	Crack of the piezoelectric ceramics, mechanical load
Burn	Reduced capacity, decrease in the active area	Thermal load
Passive middle layer
Crack	Altered damping and stiffness characteristics	Mechanical load
Copper overhang
Soldering failure	Partial or complete loss of electrical connection and loss of functionality	Improper soldering, mechanical vibrations, thermal load
Adhesive
Delamination	Decrease in bending movement, loss of functionality	Mechanical load

is structured by the layers of the piezoelectric bending actuator: two piezoelectric ceramics, six electrodes, a passive middle layer based on carbon fiber reinforced polymer (CFRP), adhesive and a copper overhang to electrically contact the inner electrodes, as shown in Figure 1. However, no failure mode of the copper overhang was found, and therefore none is mentioned in Table 1.

As part of a typical risk analysis [22], failure modes are evaluated regarding importance and probability of occurrence. Therefore, all of the previously named failure modes of the piezoelectric ceramics, the electrodes, the passive middle layer and the adhesive were evaluated. Since all failure modes influence the actuator’s characteristics and thereby the expected functionality, all but one failure mode were classified as “important”. In addition, the leakage current failure mode of the piezoelectric ceramics was classified as “very important”, as it can affect safety through additional damage to the control electronics caused by an electrical short circuit, for example. Regarding the probability of occurrence, the failure mode crack of piezoelectric ceramics or electrodes was classified as “very probable” based on Ruschmeyer’s work [20], while the electrode’s failure mode burning could be classified as “very probable” based on Denzler’s work [19], it was only classified as “probable” in this work. This assessment was made because of the lower voltage amplitudes of almost a factor of five in the following use case compared to the 200 $\mathrm{V}$ Denzler applied. Depolarization is rather unlikely if the application is operated correctly. Cracks in the passive middle layer and delamination of the adhesive is not evident in the literature. Lastly, soldering failure and leakage current is not expected in the controlled laboratory experiments. To conclude, cracks in the piezoelectric ceramics and electrode layers are relevant for the use case presented in Section 4. Therefore, the following parts of the paper focus on these failure modes and the associated effects.

3. Methodology

Generally, an electromechanical system and specifically a piezoelectric bending actuator can be modeled according to a four-pole model [28,29], with an electrical port consisting of an oscillating voltage $u\left(t\right)$ and current $i\left(t\right)$ and a mechanical port consisting of the velocity $v\left(t\right)$ and force $F\left(t\right)$. The four-pole model and corresponding lumped-parameter model of a piezoelectric actuator actuating near its fundamental resonance frequency are depicted in Figure 2. In Figure 2b, $C_e$ and $R_e$ denote the electrical capacitance and resistance, respectively. $m_m$ denotes the effective vibrating mass of the actuator, $d_m$ denotes the internal mechanical damping, $c_m$ denotes the mechanical stiffness and $\alpha$ denotes the electromechanical coupling factor.

The relationship between the inputs and the outputs of a piezoelectric actuator is given in Equation (1), where $\omega$ denotes the actuating angular frequency and circumflex (hat) denote complex amplitude of the harmonic signals [28,30]. In the following analysis, only the absolute value of the complex amplitudes is considered.

$$\left[\begin{array}{c}\widehat{\underline{i}}\\ \widehat{\underline{v}}\end{array}\right]=\left[\begin{array}{cc}\frac{{\alpha }^2}{j·\omega ·m_m+d_m+\frac{c_m}{j·\omega }}+\frac{1}{R_e+\frac{1}{j·\omega ·C_e}}& \frac{\alpha }{j·\omega ·m_m+d_m+\frac{c_m}{j·\omega }}\\ \frac{\alpha }{j·\omega ·m_m+d_m+\frac{c_m}{j·\omega }}& \frac{1}{j·\omega ·m_m+d_m+\frac{c_m}{j·\omega }}\end{array}\right]·\left[\begin{array}{c}\widehat{\underline{u}}\\ \widehat{\underline{F}}\end{array}\right].$$

(1)

The input vector in Equation (1) consists of the amplitudes of the voltage $\widehat{\underline{u}}$ and force $\widehat{\underline{F}}$, and the output vector consists of the amplitudes of the current $\widehat{\underline{i}}$ and the velocity $\widehat{\underline{v}}$. Hence, since the aim of this paper is to diagnose a piezoelectric actuator without additional sensors and equipment, the driving voltage and current, typically acquired for controlled applications, were continuously monitored. For an unloaded piezoelectric actuator, that is, for $\widehat{\underline{F}}=0$, the current is directly proportional to the voltage. This implies that for a healthy piezoelectric actuator, when the driving voltage amplitude increases, the measured current amplitude increases, and the resonance frequency decreases. Alternatively, when the driving voltage amplitude decreases, the measured current amplitude decreases, and the resonance frequency increases. Thus, any deviation in the current, without a proportional deviation in the voltage, suggests that the internal parameters of the actuator must have changed, for example, due to a crack. Furthermore, as a result of a crack in a piezoelectric layer, the stiffness reduces, and consequently the resonance frequency decreases. In summary, and as depicted in Figure 3, anomalous behavior in the piezoelectric actuator can be detected by comparing the measured current to the driving voltage or resonance frequency. Specifically, anomalous behavior occurs when a feature of the measured current signal, such as its amplitude, simultaneously decreases with the resonance frequency.

Furthermore, in the case of a crack, higher harmonics causes the signal to become distorted, due to non-linearity resulting from so-called “clapping” of the crack surfaces [31,32,33]. Thus, to isolate a crack in the piezoelectric actuator, a higher harmonic spectrum of the measured current signal was investigated, as outlined in the following section. The 3rd, 5th and 7th harmonics were analyzed in particular.

4. Experimental Setup and Results

Experiments were conducted with cantilever piezoelectric bending actuators, as seen in Figure 4a, to evaluate the proposed methodology. The piezoelectric bending actuators with protective coating are from Johnson Matthey of type Type1 (427.0085.11F) and have a dimension of $50 \mathrm{m}\mathrm{m}\times 7.2 \mathrm{m}\mathrm{m}\times 0.81 \mathrm{m}\mathrm{m}$. When clamped, the effective length is approximately 38 $\mathrm{m}$$\mathrm{m}$, as seen in Figure 4a.

In the experiments, a crack typically occurs in the piezoelectric ceramic layer, between the clamped region and the protective coating. The crack occurrence region is approximately 1 $\mathrm{m}$$\mathrm{m}$, as depicted in Figure 4b. Accelerated tests were conducted in this study, where the piezoelectric bending actuators were operated near resonance to achieve large velocity amplitudes at low excitation voltage. The first bending mode occurs at the first natural frequency, which implies a maximum deflection at the tip of the bending actuator and high potential for crack development near the clamped end.

Since the internal properties of the actuator, such as the stiffness and correspondingly the resonance frequency, can change over time as a consequence of degradation or loading situation, the operating frequency should be adjusted accordingly during operation. Thus, to guarantee a near-resonance operation at its first natural frequency, a measurement setup with frequency control is considered, as depicted in Figure 5. A phase-locked loop (PLL) control was adopted in the experiments, such that the frequency was tracked via the phase difference between the sinusoidal current and voltage signals [34]. Prior to the experiments, the initial resonance frequency was measured using an impedance analyzer and adjusted during operation via bidirectional communication between the impedance analyzer and a control PC, as seen in Figure 5. The current was acquired via a Tektronix A6312 current clamp and a corresponding AM 503B current probe amplifier.

Furthermore, the input voltage signal was acquired with TT-SI 9002 differential probe from TESTEC Elektronik GmbH, Frankfurt, Germany. The signals were acquired with a sampling frequency of 30 $\mathrm{k}$$\mathrm{Hz}$, a measurement duration of $0.5$ $\mathrm{s}$ and a measurement interval of 10 $\mathrm{s}$ with a PC oscilloscope. In the following subsections, one evaluation result is presented each for an experiment with varying voltage amplitude without external force and for an experiment with external force, since the mechanical and electrical properties of piezoelectric actuators can be influenced by external excitations [35].

4.1. Varying Voltage Amplitude without External Force

To evaluate the proposed methodology for situations where the input voltage amplitude might vary over time, the setup for the unloaded cantilever piezoelectric actuator, as previously described, is considered. The root-mean-square amplitude (RMS) for the measured input voltage $u\left(t\right)$ and output current $i\left(t\right)$ is shown in Figure 6a. In general, over the number of cycles, the current RMS varies directly according to the input voltage RMS. However, the current RMS slightly drops at about 650,000 cycles, even where there is no concurrent significant decrease in the input voltage RMS. The spectrum analysis results for the measured input voltage $u\left(t\right)$ and output current $i\left(t\right)$ are shown in Figure 6b and Figure 6c, respectively. During the experiment, the actuating frequency is not constant, but lies in the range [190, 200] $\mathrm{Hz}$. Thus, in the figures, the frequencies on the y-axis are normalized to the actuating frequency, that is, the first resonance frequency, to reveal the harmonics. As can be seen in Figure 6b, the first harmonic, which corresponds to the actuating resonance frequency, is dominant in the input voltage signal. However, as can be seen in Figure 6c, higher harmonics, and dominantly odd harmonics, that is, harmonics of order 3, 5, and 7, are also clearly visible in the current signal, due to non-linearity of the crack surfaces. Specifically, according to Solodov et al. [33] clapping of crack surfaces typically results in odd and even harmonics, while predominantly odd harmonics are attributed to friction non-linearity of the crack surfaces.

As presented in the introductory section, the measured admittance or impedance can also be employed to detect cracks in piezoelectric actuators. Thus, to highlight the contributions of the present study, the electrical admittance measured before and after the experiment and the impedance derived from the continuously monitored current and voltage signals are presented in Figure 7a and Figure 7b, respectively. As seen in the top sub-figure of Figure 7a, the minimum absolute value of the electrical admittance significantly decreases after the experiment as a result of a crack. Furthermore, the resonance frequency, that is, the frequency at $0^{\circ }$, also decreases. As a consequence of a crack, the minimum phase also decreases, as seen in the bottom sub-figure of Figure 7a. However, as investigated and shown in [36], such softening effect is also evident due to varying mechanical and electrical excitation amplitudes. For example, the measured resonance frequency in Figure 7a is higher than the range of frequencies in the experiments because, as opposed to the electrical admittance measurement, the voltage amplitudes are significantly higher in the experiments. Admittance or impedance measurements over a range of frequencies are typically acquired with small signals, that is, low driving voltage, and consequently often require an interruption of operation. Furthermore, measurements before and after a possible crack have to be acquired at similar excitation amplitudes for direct comparison. As seen in Figure 7b, without thorough investigation with consideration of the underlying operating condition, cracks cannot be detected only utilizing the impedance derived from the continuously monitored current and voltage signals.

The top sub-figure of Figure 8a shows the current RMS alongside the actuating resonance frequency, since the actuating resonance frequency can be influenced by an external force or a change in the internal parameters of the actuator, for example, due to a possible crack. Specifically, from the resulting FMEA in Table 1, a reduced stiffness is an effect of a crack in a piezoelectric ceramic layer, which in turn leads to a decrease in the resonance frequency. A further effect of a crack in a piezoelectric ceramic layer is a consequent crack in the electrode, which leads to a decreased active area and hence to a decreased current. As can be deduced from Figure 8a, the frequency also simultaneously decreases, as the current RMS abruptly drops at about 650,000 cycles. To detect such an anomaly during operation, the signum function of the difference of consecutive feature values, that is, the current RMS and actuating frequency, can be evaluated, as depicted in the bottom sub-figure of Figure 8a. As can be inferred from the bottom sub-figure of Figure 8a, over the number of cycles, the evaluated signum function is either zero or opposite for both signals, except for at about 650,000 cycles, where the evaluated signum function values are the same. The fluctuation in frequency from about 420,000 to about 450,000 cycles is attributed to the control system and does not falsify the anomaly detection procedure.

To isolate a crack in the cantilever piezoelectric actuator, the amplitude values of the third, fifth and seventh harmonics are normalized to the amplitude value of the fundamental component, as in Equation (2).

$${\tilde{I}}_n=\frac{{\widehat{I}}_n}{{\widehat{I}}_1},$$

(2)

where ${\tilde{I}}_n$ is the normalized amplitude value of the nth harmonic current signal, ${\widehat{I}}_n$ is the amplitude value of the nth signal and ${\widehat{I}}_1$ is the amplitude value of the fundamental component.

At about 650,000 cycles, the normalized amplitude value of the third harmonic (${\tilde{I}}_3$) slightly decreases, and the sum of the normalized amplitude values of the fifth and seventh harmonics (${\tilde{I}}_5+{\tilde{I}}_7$) simultaneously significantly increases, as can be inferred from Figure 8b. This in turn corresponds to the abrupt decrease in the current RMS. Thus, from the results and the preceding FMEA, the anomaly can be classified as a crack in at least a piezoelectric layer.

Resulting images from post-experimental microscopic crack analysis are presented in Figure 9, where the red arrows show the crack location. As can be seen in the top sub-figure of Figure 9a and in the bottom enlarged sub-figure of Figure 9a, there is a transgranular hairline crack in the piezoelectric layer close to the protective coating. The crack resulted from the alternating voltage applied to the cantilever piezoelectric bending actuator and the consequent high stress in the vicinity of the clamped region.

Although not shown here, the crack goes through the entire width of the piezoelectric bending actuator. Figure 9b shows the piezoelectric bending actuator from the side view. However, due to the microscopic nature of the crack, the resulting crack is not visible without further magnification. On a closer inspection, in the bottom sub-figure of Figure 9b, the resulting crack can be seen in a piezoelectric layer and ends towards the CFRP layer.

4.2. Varying Voltage Amplitude with External Force

An external force has a significant influence on the characteristics of the piezoelectric actuator, such as increased damping and consequently reduced resonance frequency, depending on the force. Thus, the aim of this experimental study was to investigate the applicability of the proposed methodology in such a case. In some studies [37,38,39,40], it has been shown that damping increases and the resonance frequency of a piezoelectric actuator decreases when the actuator is immersed within viscous fluid. Thus, in this study, an external force was exerted on the cantilever piezoelectric bending actuator via immersing the actuator in a viscous fluid.

During the experiment, the piezoelectric bending actuator was unloaded and actuated with a sinusoidal voltage at the first resonance frequency to accelerate degradation, while in operation and after a crack occurred, the actuator was immersed in a glass beaker containing a mixture of several Reely silicone oils with a resulting viscosity between 4000 $\mathrm{m}$$\mathrm{Pa}$ $\mathrm{s}$ and 6000 $\mathrm{m}$$\mathrm{Pa}$ $\mathrm{s}$ at room temperature. The initial depth of the immersed actuator was approximately 11 $\mathrm{m}$$\mathrm{m}$ out of an effective length of approximately 38 $\mathrm{m}$$\mathrm{m}$, as shown in Figure 10a and labeled as position A in Figure 10b. Furthermore, to investigate the influence of further increased damping, the actuator depth in the viscous liquid was increased in two steps, as labeled in positions B and C in Figure 10b. The maximum depth of actuator within the viscous fluid is approximately 15 $\mathrm{m}$$\mathrm{m}$ from an effective length of approximately 38 $\mathrm{m}$$\mathrm{m}$.

The measured signals were analyzed, as in the previous case study in Section 4.1. The resulting features from the measured signals are depicted in Figure 11. As seen in the first sub-figure of Figure 11a, the RMS of the measured input voltage $u\left(t\right)$ was increased from approximately 39 $\mathrm{V}$ to 41 $\mathrm{V}$ at about 330,000 cycles. The RMS of the measured output current $i\left(t\right)$ concurrently increases, while the frequency decreases, as can be inferred from the second sub-figure of Figure 11a. At about 360,000 cycles, although there is no following significant change in the voltage, the current and frequency decrease significantly. As previously stated, immersing the actuator in the viscous liquid typically leads to increased damping and decreased frequency. Thus, as a consequence, the current also decreases at about 440,000 cycles, as seen in the second sub-figure of Figure 11a. At about 970,000 cycles, the current increases because the actuator is elevated from the viscous liquid.

In employing the proposed diagnostics approach, an anomaly is detected at about 360,000 cycles because the current decreases and the frequency simultaneously decreases, without a significant change in the voltage. A further anomaly is detected at about 440,000 in region A, as the actuator is initially immersed in the viscous liquid. However, at this point, it is impossible to discern whether the anomaly is due to a crack or external damping. Thus, further analysis for fault isolation is required. The analysis results of higher harmonics of the current signal is presented in Figure 11b. The first detected anomaly is isolated as a crack because the sum of the normalized amplitude values of the fifth and seventh harmonics (${\tilde{I}}_5+{\tilde{I}}_7$) increases, while the normalized amplitude value of the third harmonic (${\tilde{I}}_3$) decreases. The second anomaly at about 430,000 cycles is not isolated as a crack because the sum ${\tilde{I}}_5+{\tilde{I}}_7$ abruptly decreases after an initial increase and ${\tilde{I}}_3$ slightly increases. As can be inferred from Figure 11b, the sum ${\tilde{I}}_5+{\tilde{I}}_7$ decreases at about 580,000 cycles and 840,000 cycles as the the actuator is being immersed in positions B and C. As the actuator is elevated from the viscous liquid at about 970,000 cycles, the sum ${\tilde{I}}_5+{\tilde{I}}_7$ abruptly increases, while ${\tilde{I}}_3$ decreases because the clapping of the opposite crack surfaces is not further damped by the viscous liquid.

The microscopic crack analysis of the actuator after the experiment shows a crack in the piezoelectric layer, which occurred in the vicinity of the clamped region, as in the previous experiment. The red arrows show the crack location from the top and side view as shown in Figure 12a and Figure 12b, respectively. Although a crack is only clearly visible in one of the piezoelectric layers, a crack might also be present in the second layer, due to the alternating voltage applied to the cantilever piezoelectric bending actuator, which results in an equal maximum deflection in both directions.

5. Conclusions

Based on a FMEA, the failure modes crack of piezoelectric ceramics and electrodes are identified as relevant for piezoelectric bending actuators. To maintain the dependability of piezoelectric bending actuators irrespective of varying operating conditions or the influence of external force, a data-driven diagnostics approach was proposed. To this end, the measured driving input voltage and output current, typically acquired for controlled applications, are continuously monitored. As opposed to admittance or impedance measurements over a range of frequencies that typically require an interruption of operation and similar excitation amplitudes for fault detection, the proposed methodology relies on the combined analysis of voltage and current signals and higher harmonics thereof to detect and isolate hairline cracks in a piezoelectric bending actuator subjected to time-varying voltage amplitudes. An anomaly is detected when a time-domain feature extracted from the measured current signal simultaneously decreases with the actuating resonance frequency. Furthermore, a crack is isolated in at least a piezoelectric layer when the sum of the normalized amplitude values of the fifth and seventh harmonics (${\tilde{I}}_5+{\tilde{I}}_7$) increases, while the normalized amplitude value of the third harmonic (${\tilde{I}}_3$) decreases. From the experimental studies of a resonant-driven actuator, it can be concluded that features, in the time and frequency domains, derived from the measured output current are sufficient to diagnose a piezoelectric bending actuator even at time-varying voltage amplitudes and under external force conditions.

As an outlook, the influence of other time-varying operating and environmental conditions, such as temperature, can be considered. Further failure modes can be analyzed in more detail, e.g., structural failures of the host structure and possible effects on the higher harmonics. Additionally, the suitability of the proposed approach for online diagnostics can be validated via further experiments.