# Sensorless calibration of piezoelectric shunts using capacitance measurements

Jens D. Richardt, Boris Lossouarn, Jan Høgsberg, Jean-François Deü

► To cite this version:

Jens D. Richardt, Boris Lossouarn, Jan Høgsberg, Jean-François Deü. Sensorless calibration of piezoelectric shunts using capacitance measurements. Smart Materials and Structures, 2025, 34 (3), pp.035043. 10.1088/1361-665X/adbac8. hal-05459950

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# Sensorless calibration of piezoelectric shunts using capacitance measurements

To cite this article: Jens D Richardt et al 2025 *Smart Mater. Struct.* **34** 035043

View the [article online](#) for updates and enhancements.

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# Sensorless calibration of piezoelectric shunts using capacitance measurements

Jens D Richardt<fnref n="1" />,*, Boris Lossouarn<fnref n="2" />,³, Jan Høgsberg<fnref n="1" />, and Jean-François Deü<fnref n="2" />,³

E-mail: jdari@dtu.dk

Received 10 December 2024, revised 7 February 2025

Accepted for publication 25 February 2025

Published 10 March 2025

<figure><img src="image_2.png" /></figure>

CrossMark

## Abstract

In this work, a new experimental shunt tuning procedure is proposed for piezoelectric shunts targeting a single vibration mode of a flexible structure. The procedure is based on measurements of the dynamic capacitance of the piezoelectric transducer, eliminating the need for additional sensors or actuators. It is demonstrated that all required electromechanical parameters for shunt calibration can be determined from two extremum points on the dynamic capacitance curve. Corrections for the influence of residual modes are included to enable the consistent application of existing tuning expressions for various shunt topologies. The hardware of a digital shunt is used for the measurement of the dynamic capacitance allowing for low-cost implementation. The proposed method is validated numerically and experimentally through the calibration of a series RL shunt using both analog and digital components.

Keywords: piezoelectric shunt damping, shunt calibration, synthetic impedance, dynamic capacitance, vibration control

## 1. Introduction

Piezoelectric shunts were first proposed for damping of vibrations by Forward in 1979 [1] and have been extensively studied in the literature since the introduction of the (series) resonant RL shunt by Hagood and von Flotow in 1991 [2]. Since then, various circuit designs have been developed to target single or multiple vibration modes. For damping of a single vibration mode, circuit designs have been proposed in terms

of resistive shunts [2], capacitive shunts [3], resonant (RL) shunts [2, 4], switching shunts [3, 5, 6], hybrid shunts [7] or shunts using negative capacitance [8]. A similar variety exists for the calibration of the circuits, with different criteria applied, for instance, to obtain maximum modal damping or for minimum response amplitude. For example, for the series RL shunt alone, different explicit calibration expressions are given in [2, 9–16] for a piezoelectric transducer attached to a single-degree-of-freedom (sdof) structure.

The explicit calibration formulas for piezoelectric shunts given in the literature typically depend on the blocked capacitance of the piezoelectric transducer, the natural frequency of the structure with the piezoelectric shunt in the short-circuit (SC) or open-circuit (OC) condition, as well as the so-called modal electromechanical coupling factor (MEMCF) [2, 11, 12]. The MEMCF is often determined experimentally as an effective electromechanical coupling factor (EEMCF),

<figure><img src="image_3.png" /></figure>

<footnote n="1">¹ Department of Civil and Mechanical Engineering, Technical University of Denmark, Kongens Lyngby, Denmark</footnote>

<footnote n="2">² Laboratoire de Mécanique des Structures et des Systèmes Couplés, Conservatoire national des arts et métiers, Paris, France</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>
evaluated from the natural frequencies of the structure with the piezoelectric transducer in the SC and OC conditions, respectively. For an ideal piezoelectric transducer attached to an sdof structure, the MEMCF corresponds to the EEMCF [2, 17]. As demonstrated numerically by Toftekær et al in 2018 [18], the MEMCF and the EEMCF are not identical for flexible structures with multiple vibration modes. Therefore, care must be taken in the definition of the MEMCF when used for shunt calibration.

The blocked capacitance, used in the explicit calibration expressions previously introduced for sdof structures, refers to the capacitance measured at zero strain, corresponding to all vibration modes being blocked in the high-frequency limit. As described by De Marneffe and Preumont in 2008 [19] and Preumont in 2011 [20], the capacitance of a piezoelectric transducer depends on its boundary conditions. Thus, [19, 20] distinguish between the blocked capacitance, the inherited free capacitance where the piezoelectric transducer is free to deform, and the static capacitance measured at low frequencies when the piezoelectric transducer is bonded to the structure. Porfiri et al [21] describe the difficulty of obtaining the blocked capacitance experimentally, as it requires blocking of all deflection modes and proposes a blocked modal capacitance obtained from the static capacitance and EEMCFs. The frequency dependency of the measured capacitance of a piezoelectric transducer bonded to a flexible structure is shown by, for instance, De Marneffe and Preumont in [19] and Berardengo et al in [22]. In practice, the capacitance is often measured experimentally between resonances. However, the frequency at which the capacitance is evaluated is not always well defined, whereby the measured capacitance is only an approximation to the actual blocked capacitance.

As demonstrated for a two-dimensional structure by Gardonio and Casagrande in 2017 [23], the influence of the non-targeted (residual) modes must be accounted for in the calibration of the shunt. Corrections for the influence of residual modes have, for instance, been presented for piezoelectric shunts by Høgsberg and Krenk in [24] using flexibility and inertia correction terms. Berardengo et al [22], include the influence of higher vibration modes by a modal capacitance, measured experimentally between the targeted vibration modes and the next mode, leading to an improved calibration accuracy. As shown by Berardengo et al in [25], the modal capacitance can be obtained by fitting the frequency response of a dynamic capacitance. In 2020, Toftekær et al [26, 27], introduced a modal capacitance with residual mode correction that can be obtained experimentally by measurement of modal charge and voltage amplitudes in the SC and OC conditions, respectively. Other techniques for indirect measurement of the modal capacitance and the MEMCF have been considered by Porfiri et al [21], Delpero et al [17], and Berardengo et al [28] using a variable inductance or negative capacitance connected to the piezoelectric transducer.

In 2000, Fleming et al [29, 30] showed that a synthetic admittance can replace the electrical components of a piezoelectric shunt by use of a digital control unit (DCU) that measures the voltage across the electrodes of the piezoelectric

transducer and prescribes the current. This synthetic admittance, when connected to a piezoelectric transducer, is referred to as a digital shunt and theoretically allows the implementation of any electrical circuit. Several modifications of the digital shunt unit presented by Fleming et al in 2000 [29, 30] have since been proposed, for instance, in [31–35]. In the PhD thesis by Dietrich from 2023 [36] it is shown that the hardware of a digital shunt can be used for measurement of the electrical impedance of the piezoelectric transducer and to calibrate a digital shunt by applying modal analysis tools.

In this paper, the electrical impedance of a piezoelectric transducer attached to a mechanical structure is represented as a frequency-dependent capacitance and a parallel resistance. The paper aims to show that all required parameters for the calibration of a piezoelectric shunt can be determined directly from only two points on the curve of the dynamic capacitance for structures with non-zero damping. The parameters include corrections for residual modes to allow for a consistent calibration of a shunt without other measuring and actuation devices while avoiding detailed numerical data analysis. Although any sufficiently accurate impedance analyzer can be used for the measurement of dynamic capacitance, it is shown that impedance measurements using the hardware of a digital shunt can be performed with sufficient accuracy for the implementation of both a digital shunt and shunts with analogue components. Thus, the proposed method avoids the need for a dedicated impedance analyzer, thereby reducing the cost of a digital shunt implementation.

The structure of the paper is as follows: section 2 describes the electromechanical system and the dynamic capacitance in terms of a modal expansion with the shunt in the SC condition. In section 3 it is shown that the parameters required for calibration of a piezoelectric shunt can be obtained directly from measurements of the dynamic capacitance, including a correction for the influence of residual modes. A numerical example of a cantilever beam in section 4 illustrates the effectiveness of the proposed calibration method. Experimental validation is performed in section 5, along with the introduction of required corrections for measurements of the dynamic capacitance using the hardware of a digital shunt. Final conclusions and perspectives are then presented in section 6.

## 2. Model of the electromechanical system

This section introduces the governing equations for the electromechanical system, which consists of a piezoelectric transducer connected to a flexible structure. These equations are subsequently linked to a dynamic capacitance and resistance, which can be measured experimentally using the digital shunt hardware.

### 2.1. Governing equations

The electromechanical system consisting of a piezoelectric transducer attached to a flexible structure is shown in figure 1,
<figure><img src="image_4.png" /><figcaption>Figure 1. Flexible structure with a piezoelectric transducer.</figcaption></figure>

where *I* is the current flowing through the piezoelectric transducer, and *V* is the voltage across the electrodes. The presented model corresponds to the one used in [28] with a different normalization of the mode shapes. The displacement vector **q** for the discretized structure is represented by a modal expansion with *N* vibration modes for the piezoelectric shunt in the SC condition,

$$ \mathbf{q} = \sum_{n=1}^{N} \mathbf{u}_n r_n \qquad (1) $$

where $r_n$ is the modal coordinate, and $\mathbf{u}_n$ is the mode shape vector of mode *n*. Note that no assumptions are made about the normalization of the mode shape vectors in the following sections.

The modal equation of motion of a damped structure with a piezoelectric shunt can be expressed as

$$ (-\omega^2 + 2i \zeta_n \omega_n \omega + \omega_n^2) r_n - \frac{\chi_n \omega_n^2}{\sqrt{k_n}} V = \frac{\omega_n^2}{k_n} f_n \quad (2) $$

where $\omega$ is the angular frequency, $\omega_n$ is the natural angular frequency with the electrodes of the piezoelectric shunt in the SC condition, $\zeta_n$ is the damping ratio, $k_n$ is the modal stiffness, $f_n$ is the modal load, and $\chi_n$ is a modal coefficient. The modal load is defined as $f_n = \mathbf{u}_n^T \mathbf{f}$, where $\mathbf{f}$ is the external load vector. The modal coefficient is defined as

$$ \chi_n^2 = \frac{(\mathbf{w}^T \mathbf{u}_n)^2}{k_n} \theta^2 \qquad (3) $$

where $\theta$ is the piezoelectric coefficient, and $\mathbf{w}$ is a connectivity vector that describes at which degrees-of-freedom (dofs) the piezoelectric transducer acts on the structure. The expression in (2) is derived from [11] by allowing an arbitrary normalization of the mode shape vector $\mathbf{u}_n$.

Following [28], the governing electrical equation can be written for a piezoelectric transducer in terms of a capacitance $C^\infty$ in parallel with a resistance $R^\infty$. The charge $Q$ at one of the electrodes is then defined as

$$ Q = \left( C^\infty + \frac{1}{i\omega R^\infty} \right) V + \sum_{n=1}^{N} \sqrt{k_n} \chi_n r_n \quad (4) $$

<figure><img src="image_5.png" /><figcaption>Figure 2. Equivalent electrical circuits with a constant capacitance and resistance (a) and with frequency-dependent parameters (b).</figcaption></figure>

whereby $C^\infty$ is the blocked capacitance with constrained transducer boundaries (zero strain), corresponding to the capacitance when the frequency tends to infinity. Similarly, the parallel resistance $R^\infty$ is also associated with constrained transducer boundaries, and typically so large that $1/R^\infty$ is commonly neglected [28]. However, in (4), $1/R^\infty$ is included to allow a coherent connection to the frequency-dependent capacitance that can be measured experimentally, see section 2.2. The equivalent electric circuit to (4) is shown in figure 2(a), where the current *I* is related to the charge *Q* as $I = i\omega Q$. The function $f(\omega)$ in figure 2(a) corresponds to the final summation term in (4) times $-i\omega$, which includes the frequency-dependent influence from the structure.

Assuming that no external loads act on the structure ($f_n = 0$), isolation of the modal coordinate $r_n$ in (2) gives

$$ r_n = \frac{\chi_n \omega_n^2}{\sqrt{k_n} \omega^2 + 2i \zeta_n \omega_n \omega + \omega_n^2} V. \qquad (5) $$

Inserting the modal coordinate in (5) into (4) then gives the charge $Q$ to voltage $V$ ratio

$$ \frac{Q}{V} = \left( C^\infty + \frac{1}{i\omega R^\infty} \right) + \sum_{n=1}^{N} \frac{\chi_n^2 \omega_n^2}{-\omega^2 + 2i \zeta_n \omega_n \omega + \omega_n^2} \qquad (6) $$

The last term in (6) vanishes when $\omega \to \infty$, in which case the blocked capacitance $C^\infty$ can be found as the real part of the charge to voltage ratio $Q/V$. This ratio can be measured experimentally using an impedance meter. However, as described in the introduction, measuring the capacitance with constrained boundaries (corresponding to infinite frequency, $\omega = \infty$) is not feasible in practice. A frequency-dependent capacitance that can be measured experimentally is therefore introduced in the following section.

## 2.2. Dynamic capacitance

The expression for $Q/V$ in (6) can be represented by a frequency-dependent capacitor $C(\omega)$ in parallel with a frequency-dependent resistance $R(\omega)$. This equivalent electric
<figure><img src="image_6.png" /><figcaption>Figure 3. Dynamic capacitance (a) and conductance (b) of a piezoelectric transducer attached to a cantilever beam with a rectangular cross-section, for $\zeta_2 = 0$ (black solid curve), $\zeta_2 = 0.005$ (blue dot-dashed curve) and $\zeta_2 = 0.01$ (red dashed curve).</figcaption></figure>

circuit, shown in figure 2(b), corresponds to the relation

$$ \frac{Q}{V} = C(\omega) + \frac{1}{i\omega R(\omega)} \quad (7) $$

which now only depends on the frequency-dependent capacitance $C(\omega)$ and resistance $R(\omega)$. The former is referred to as the dynamic capacitance. These components can be obtained directly from experimental measurements of the piezoelectric transducer’s electrical impedance at a given frequency $\omega$. The two components are directly isolated from (7) as

$$ C(\omega) = \mathrm{Re}\left(\frac{Q}{V}\right) = \frac{1}{\omega}\mathrm{Im}\left(\frac{I}{V}\right) \quad (8) $$

$$ \frac{1}{R(\omega)} = -\omega \mathrm{Im}\left(\frac{Q}{V}\right) = \mathrm{Re}\left(\frac{I}{V}\right). \quad (9) $$

Inserting (6) into (8) and (9) respectively gives the dynamic capacitance $C(\omega)$ and conductance $1/R(\omega)$ as

$$ C(\omega) = C^{\infty} + \sum_{n=1}^{N} \frac{\chi_n^2 \omega_n^2 (\omega_n^2 - \omega^2)}{(\omega_n^2 - \omega^2)^2 + (2\zeta_n \omega_n \omega)^2} \quad (10) $$

$$ \frac{1}{R(\omega)} = \frac{1}{R^{\infty}} + \sum_{n=1}^{N} \frac{2\chi_n^2 \zeta_n \omega_n^3 \omega^2}{(\omega_n^2 - \omega^2)^2 + (2\zeta_n \omega_n \omega)^2}. \quad (11) $$

The definition of the dynamic capacitance in (10) corresponds to the one used in [28] when the mode shapes are normalized to unit modal mass, and a similar expression is given

for undamped structures in [20]. It is seen that the dynamic capacitance (10) recovers the blocked capacitance $C^{\infty}$ when $\omega \to \infty$, while the second term, representing the interaction with the structure, will influence the measured capacitance for finite values of $\omega$.

An example of the dynamic capacitance and conductance is shown in figure 3 for a piezoelectric transducer on the cantilever beam introduced in section 4. In the undamped state with $\zeta_n = 0$ for $n = 1...N$ (black solid curve), the dynamic capacitance has a pole at the natural frequencies with the shunt in SC condition and a zero at the natural frequencies with the shunt in OC condition [20]. For non-zero damping ($\zeta_n \neq 0$), the dynamic capacitance has a local maximum and local minimum located around the SC natural frequency. These two local extrema of the dynamic capacitance, associated with non-zero damping ($\zeta_n \neq 0$), are used in the following section to estimate the governing modal parameters. The dynamic capacitance $1/R(\omega)$ shows that the transition between the undamped and damped case is discontinuous, as the peak around resonance increases for decreasing damping whereas it is constant in the undamped state. A similar discontinuity can be found for the dynamic capacitance as it is discontinuous around $\omega_s$ in the undamped state, while it is continuous in the damped state. The expressions presented in section 3.3 are therefore only applicable for structures with non-zero damping ($\zeta_n \neq 0$), which is always the case for real structures. Only the dynamic capacitance is considered in the following sections. However, it is noted that the dynamic resistance $R(\omega)$ in (11) recovers $R^{\infty}$ in both frequency limits $\omega = 0$ and $\omega \to \infty$.
### 3. Parameter identification

This section presents a method for parameter estimation from measurements of the dynamic capacitance for structures with non-zero modal damping. First, a single-mode approximation of the dynamic capacitance $C(\omega)$ is constructed, and the EEMCF is included in the expression by consideration of the undamped limit state. Using this single-mode approximation, it is then shown that two points on the measured dynamic capacitance curve can be used for parameter estimation.

#### 3.1. Approximation of dynamic capacitance

The dynamic capacitance $C(\omega)$ is applied for the calibration of a shunt to a targeted vibration mode $n = s$. It is assumed that the term from the primary mode dominates the dynamic capacitance in (10) for frequencies near $\omega_s$ and that the effect from residual modes is frequency independent. Separation of the targeted vibration mode $n = s$ from the residual modes $n \neq s$ in (10), and evaluation of the terms related to residual modes at $\omega = \omega_s$, then leads to the approximation

$$C(\omega) \approx C_s^{\infty} + \frac{\chi_s^2 \omega_s^2 (\omega_s^2 - \omega^2)}{(\omega_s^2 - \omega^2)^2 + (2\zeta_s \omega_s \omega)^2}. \quad (12)$$

The second term in (12) vanishes for $\omega = \omega_s$, and when $\omega \to \infty$. The constant $C_s^{\infty}$, therefore, corresponds to $C(\omega_s)$ and is defined as

$$C_s^{\infty} = C(\omega_s) = C^{\infty} + \sum_{n \neq s}^{N} \frac{\chi_n^2 \omega_n^2 (\omega_n^2 - \omega_s^2)}{(\omega_n^2 - \omega_s^2)^2 + (2\zeta_n \omega_n \omega_s)^2}. \quad (13)$$

The constant $C_s^{\infty}$ is referred to as the blocked modal capacitance because it resembles the blocked capacitance (at infinite frequency) for a piezoelectric transducer attached to an sdof structure with the SC natural frequency $\omega_s$.

#### 3.2. EEMCF

It is convenient to introduce the EEMCF in the expression (12) for the dynamic capacitance, as it is a fundamental parameter in experimental calibration of piezoelectric shunts. The EEMCF is introduced in the undamped case, whereby $\zeta_s = 0$ in (12) gives

$$C(\omega) = C_s^{\infty} + \frac{\chi_s^2 \omega_s^2}{\omega_s^2 - \omega^2}. \quad (14)$$

Following [20], it can be shown that the zeros of the dynamic capacitance in an undamped system correspond to the natural frequencies with the piezoelectric transducer in OC conditions. In this limit, the frequency corresponds to the OC natural frequency ($\omega = \hat{\omega}_s$), whereby $C(\omega) = 0$ gives

$$C_s^{\infty} + \frac{\chi_s^2 \omega_s^2}{\omega_s^2 - \hat{\omega}_s^2} = 0. \quad (15)$$

Thereby, the squared EEMCF is directly obtained as

$$\kappa_s^2 = \frac{\hat{\omega}_s^2 - \omega_s^2}{\omega_s^2} = \frac{\chi_s^2}{C_s^{\infty}}. \quad (16)$$

Substitution into (12) then gives the expression for the capacitance

$$C(\omega) \approx C_s^{\infty} \left( 1 + \kappa_s^2 \frac{\omega_s^2 (\omega_s^2 - \omega^2)}{(\omega_s^2 - \omega^2)^2 + (2\zeta_s \omega_s \omega)^2} \right) \quad (17)$$

near the targeted mode $s$.

Following the definition in [11, 17], the squared MEMCF can be defined as

$$K_s^2 = \frac{\chi_s^2}{C^{\infty}}. \quad (18)$$

In accordance to [2, 17], it is seen from (13), (16) and (18), that the EEMCF $\kappa_s$ corresponds to the MEMCF $K_s$ when the shunt is attached to an sdof structure. The analogy between the EEMCF and MEMCF for an sdof structure allows for direct use of existing calibration expressions developed for sdof structures by replacing the MEMCF $K_s$ with the EEMCF $\kappa_s$ and the blocked capacitance $C_s^{\infty}$ with the blocked modal capacitance $C_s^{\infty}$ in the sdof calibration expressions. Thus, the calibration of a piezoelectric shunt attached to a flexible structure can be performed when the parameters $\omega_s$, $\kappa_s$, and $C_s^{\infty}$ are known. For instance, for a series RL shunt, any of the calibrations presented in [2, 9–16] can be implemented when using $\kappa_s$ and $C_s^{\infty}$ instead of $K_s$ and $C_s^{\infty}$.

The EEMCF $\kappa_s$ and the blocked modal capacitance $C_s^{\infty}$ implicitly contain a correction for the influence from residual modes. For an undamped structure, it can be shown that the correction for residual modes is equivalent to the correction applied numerically in [37], while a similar residual mode correction has been presented for general mechanical vibration absorbers in [38].

#### 3.3. System identification from target points

As illustrated in figure 4, the dynamic capacitance has a local maximum and a local minimum around each natural frequency for structures with non-zero damping. The calibration parameters of the piezoelectric shunt are now determined from these two local extrema.

The derivative $C'(\omega) = dC(\omega)/d\omega$ of the capacitance in (17) is obtained as

$$C'(\omega) = -C_s^{\infty} \left( \frac{2\kappa_s^2 \omega_s^2 \omega}{(\omega_s^2 - \omega^2)^2 + (2\zeta_s \omega_s \omega)^2} + \dots \right. \\ \left. \frac{\kappa_s^2 \omega_s^2 (\omega_s^2 - \omega^2) [-4(\omega_s^2 - \omega^2)\omega + 8\zeta_s^2 \omega_s^2 \omega]}{[(\omega_s^2 - \omega^2)^2 + (2\zeta_s \omega_s \omega)^2]^2} \right). \quad (19)$$

The requirement of non-trivial solutions to the extremum equation $C'(\omega) = 0$ determines the frequencies of the local
<figure><img src="image_7.png" /><figcaption>Figure 4. Local maximum and minimum of the dynamic capacitance with non-zero damping.</figcaption></figure>

maximum ($\omega_s^A$) and minimum ($\omega_s^B$) of mode s in figure 4. These solutions are derived as

$$ (\omega_s^A)^2 = (1 - 2\zeta_s)\omega_s^2, \quad (\omega_s^B)^2 = (1 + 2\zeta_s)\omega_s^2. \quad (20) $$

Isolating $\zeta_s$ in the expression for $(\omega_s^A)^2$ and inserting it into the expression for $(\omega_s^B)^2$ leads to the SC natural frequency

$$ \omega_s = \sqrt{\frac{(\omega_s^A)^2 + (\omega_s^B)^2}{2}} \quad (21) $$

while the modal damping ratio $\zeta_s$ is determined from (20) as

$$ \zeta_s = \frac{1}{2} \left( 1 - \left( \frac{\omega_s^A}{\omega_s} \right)^2 \right) = \frac{1}{2} \frac{(\omega_s^B)^2 - (\omega_s^A)^2}{(\omega_s^B)^2 + (\omega_s^A)^2}. \quad (22) $$

The capacitance at the two local extrema is then found from (17) as

$$ C(\omega_s^A) = C_s^A = C_s^\infty \left( 1 - \kappa_s^2 \frac{1}{4\zeta_s(\zeta_s - 1)} \right) \quad (23) $$

$$ C(\omega_s^B) = C_s^B = C_s^\infty \left( 1 - \kappa_s^2 \frac{1}{4\zeta_s(\zeta_s + 1)} \right) \quad (24) $$

which can be inverted to give the closed-form expressions

$$ \kappa_s^2 = 4\zeta_s \frac{(1 - \zeta_s^2)(C_s^A - C_s^B)}{C_s^A + C_s^B - (C_s^A - C_s^B)\zeta_s} \quad (25) $$

$$ C_s^\infty = \frac{1}{2}((1 - \zeta_s)C_s^A + (1 + \zeta_s)C_s^B) \quad (26) $$

for the squared EEMCF $\kappa_s^2$ and the blocked modal capacitance $C_s^\infty$, respectively.

From the above derivations, the required parameters $\omega_s$, $\kappa_s^2$, and $C_s^\infty$ for shunt calibration can be determined by the following steps:

(i) Measurement of the dynamic capacitance across the electrodes of the piezoelectric shunt.

(ii) Evaluation of the maximum point ($\omega_s^A$, $C_s^A$) and minimum point ($\omega_s^B$, $C_s^B$) in the dynamic capacitance curve around the targeted vibration mode, see figure 4.

(iii) Estimation of $\omega_s$, $\zeta_s$, $\kappa_s^2$, and $C_s^\infty$ from (21), (22), (25) and (26), respectively.

It is noted that the use of the two local extrema for parameter estimation resembles the use of the minimum and maximum impedance in EN50324-2 [39] for estimation of the electromechanical coupling factor in data sheets for piezoelectric transducers. In theory, the modal capacitance $C_s^\infty$ can also be measured experimentally as the capacitance at the SC natural frequency $C(\omega_s)$. However, this is not applicable in practice due to the steep slope of the dynamic capacitance $C(\omega)$ for $\omega \approx \omega_s$, shown in figure 4. As discussed in section 2.2, the dynamic capacitance has a discontinuity in the limit $\zeta_n = 0$. The presented method is, therefore, only valid for structures with non-zero damping $\zeta_n \neq 0$ for $n = 1...N$, which is always the case for real structures.

## 4. Numerical example

The accuracy of the proposed calibration method is illustrated by a numerical example considering a cantilever beam with a single piezoelectric transducer. Two cross-sections with increasing complexity are considered: the rectangular and the right trapezoid cross-sections shown in figure 5. The beam is made of steel, and the piezoelectric material is PIC255. The dimensions of the beam (b) and piezoelectric transducer (p) are $l_b = 100$ mm, $w_b = 30$ mm, $t_b = 0.5$ mm, $l_p = 8$ mm, $w_p = 28$ mm, and $t_p = 0.5$ mm. The beam is clamped at one end, and the piezoelectric transducer is placed 8 mm from the clamped edge. Structural damping is included by Rayleigh damping, such that the damping ratio is $\zeta_2 = 0.003$ for the targeted vibration mode. The influence of the parallel resistance is not included, corresponding to $R^\infty \to \infty$. The beam is modeled in 3D by the finite element method with COMSOL using two elements across the thickness of the piezoelectric transducer and in total 58 722 and 56 225 tetrahedral elements for the rectangular and the right trapezoid cross-sections, respectively. As shown for the rectangular cross-section in figure 3, the first two modes of the beam are bending modes, while the third mode is a torsional mode located near the second bending mode. The following calibration of the piezoelectric shunt targets the second bending mode ($s = 2$).

### 4.1. Parameter estimation

The frequency curve for the dynamic capacitance is determined using (8) by applying a current at one terminal while measuring the voltage across the electrodes. The dynamic capacitance is shown for the rectangular cross-section in figure 6(a) for frequencies near the targeted vibration mode ($s = 2$), where the maximum and minimum points are identified as $(\omega_2^A, C_2^A) = (3386 \text{ rad s}^{-1}, 6.868 \text{ nF})$ and $(\omega_2^B, C_2^B) = (3406 \text{ rad s}^{-1}, 3.715 \text{ nF})$ using a frequency resolution of
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Cantilever beam with the two considered cross-sections: rectangular and right trapezoid. The torsional center is marked by a blue cross in the cross-sections.</figcaption></figure>

<table><caption>**Table 1.** Modal parameters for the cantilever beam with the rectangular and right trapezoid cross-sections, determined using the dynamic capacitance method ('dynamic C') and the reference method ('reference').</caption><thead><tr><th></th><th>ω<sub>2</sub> (rad s<sup>-1</sup>)</th><th>ζ<sub>2</sub> (-)</th><th>√κ<sub>2</sub><sup>2</sup> (-)</th><th>C<sub>2</sub><sup>∞</sup> (nF)</th><th>L (H)</th><th>R (Ω)</th></tr></thead><tbody><tr><td>Rectangular, dynamic C</td><td>3395.9</td><td>0.0030</td><td>0.0599</td><td>5.287</td><td>16.29</td><td>4064</td></tr><tr><td>Rectangular, reference</td><td>3395.8</td><td>0.0030</td><td>0.0599</td><td>5.280</td><td>16.31</td><td>4072</td></tr><tr><td>Trapezoid, dynamic C</td><td>3642.5</td><td>0.0030</td><td>0.0488</td><td>5.323</td><td>14.09</td><td>3073</td></tr><tr><td>Trapezoid, reference</td><td>3642.4</td><td>0.0030</td><td>0.0490</td><td>5.266</td><td>14.25</td><td>3116</td></tr></tbody></table>

0.314 rad s⁻¹. The natural undamped frequency $ω_2$, damping ratio $ζ_2$, squared EEMCF $κ_2^2$, and modal capacitance $C_2^∞$ are then respectively determined by (21), (22), (25), and (26). The parameters are given in table 1 and referred to as dynamic capacitance method because all modal parameters can be obtained directly from measurements of the dynamic capacitance.

For comparison with the estimated values from the dynamic capacitance, the governing parameters in table 1 are also determined by common modal analysis and separate measurements of the blocked modal capacitance. These values are denoted as 'reference' in table 1 because they represent common engineering practice. Initially, the modal parameters $ω_2$, $ζ_2$ and $κ_2$ are obtained from the eigenvalue problems with the shunt in the SC and OC condition and by (16). Similar to the experimental procedures in [22, 40], the blocked modal capacitance $C_s^∞$ is measured between the modal resonance $s$ and $s+1$. For consistency, the reference capacitance $C_2^∞$ in table 1 is measured at the average frequency between the targeted vibration mode $s=2$ and the next visible peak in the measured dynamic capacitance. For the cantilever with rectangular cross-section in figure 6(a), the torsional mode does not affect the dynamic capacitance, whereby the capacitance is measured at the midpoint between the targeted vibration mode ($ω_2 = 3396$ rad s⁻¹) and the third bending mode ($ω_4 = 9353$ rad s⁻¹). Because the blocked modal capacitance for the rectangular cross-section is evaluated far away from the targeted resonance, there is a good agreement between the values obtained by the reference and dynamic capacitance methods in the upper half of table 1. The maximum relative difference for the cantilever beam with the rectangular cross-section is 0.2% for the squared EEMCF.

The parameter estimation methods ('dynamic capacitance' and 'reference') used to obtain the modal values in the upper

half of table 1 are now repeated for the cantilever with the right trapezoid cross-section. The obtained parameters for this non-symmetric cross-section are given in the lower half of table 1. The tapering of the bottom edge moves the torsional center (see blue cross in figure 5) and creates bending-torsion coupling, whereby the resonance for this mode appears in the dynamic capacitance curve in figure 6(b). By the reference method, the blocked modal capacitance is therefore measured at the midpoint between the targeted vibration mode ($ω_2 = 3642$ rad s⁻¹) and the first torsional mode ($ω_3 = 4311$ rad s⁻¹), which results in a relative difference of -1.1% for the capacitance $C_2^∞$ between the reference and the dynamic capacitance method. Furthermore, the inset plot in figure 6(b) around the target frequency $ω_s = ω_2$ (vertical dashed line) shows that the blocked modal capacitance from the dynamic capacitance method (blue dashed line) is very close to $C(ω_s)$ (black solid curve), while the reference method (red dot-dashed line) slightly underestimates the theoretical value $C(ω_s)$. This illustrates that measurements of the dynamic capacitance at the midpoint between two adjacent resonances may not always give entirely accurate estimates of the blocked modal capacitance $C_s^∞ = C(ω_s)$. In theory, the blocked modal capacitance $C_s^∞$ could be measured directly as $C(ω_s)$. However, this is difficult in practice as the steep slope of the dynamic capacitance curve requires a very fine frequency resolution.

## 4.2. Series RL shunt

To illustrate the accuracy of the proposed parameter estimation method, a series RL shunt is calibrated for the cantilever based on the parameters obtained in section 4.1. The ability of the tuned shunt to optimally mitigate the frequency response will thus validate the parameter estimation. The impedance
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 6.** Measured dynamic capacitance (black solid curve) near the targeted vibration mode (vertical dashed line) for the rectangular (a) and right trapezoid cross-section (b). The blocked modal capacitance $C_s^\infty$ is shown for the dynamic capacitance method (blue dashed line) and the reference method (red dot-dashed line).</figcaption></figure>

<table><caption>**Table 2.** Calibration for minimum acceleration, based on derivations in [10].</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>LC<sub>s</sub><sup>∞</sup> ω<sub>s</sub><sup>2</sup></td><td><sup>1</sup>&frasl;<sub>(1+κ<sub>s</sub><sup>2</sup>)<sup>2</sup></sub></td></tr><tr><td>RC<sub>s</sub><sup>∞</sup> ω<sub>s</sub></td><td><sup>3</sup>&frasl;<sub>2</sub> <sup>κ<sub>s</sub><sup>2</sup></sup>&frasl;<sub>(1+κ<sub>s</sub><sup>2</sup>)<sup>3</sup></sub></td></tr></tbody></table>

$Z = R + i\omega L$ of the shunt circuit relates the voltage $V$ and current $I$ as

$$V = -ZI = -(R + i\omega L)I. \qquad (27)$$

The calibration of $R$ and $L$ is based on the fixed-point method for equal acceleration amplitude (accelerance), given in [10] and summarized in table 2. The inductance and resistance are given in table 1 from the modal parameters obtained in

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 7.** Accelerance of the cantilever with a series RL shunt for the rectangular (a) and right trapezoid cross-section (b). The accelerance is shown for the piezoelectric transducer in the SC condition (black dashed curve), as well as the RL shunt calibrated from dynamic capacitance measurements (blue solid curve) and from reference measurements (red solid curve).</figcaption></figure>

section 4.1 using the dynamic capacitance and reference methods for both the rectangular and right trapezoid cross-sections. The accelerance is determined for a point load acting at the torsional center at the tip of the beam and shown for the rectangular cross-section in figure 7(a) and the right trapezoid cross-section in figure 7(b). The accelerance for the calibrated RL shunt based on the dynamic capacitance method (blue solid curve) and the reference method (red solid curve) show a significant vibration reduction compared to the structure's response with the shunt in the SC condition (black dashed curve).

The relative difference in the magnitude of the two maxima for the response with the calibrated RL shunt is approximately 1% for the dynamic capacitance calibration, whereas for the reference calibration it is approximately 4% and 24% for the rectangular and trapezoid cross-sections, respectively. Thus, the dynamic capacitance method gives a more accurate calibration for both cross-sections, while the reference method
is influenced by the nearby torsional mode for the trapezoid cross-section. The small deviation between the optimal tuning and the tuning by the dynamic capacitance method is most likely due to the presence of structural damping [15] and the location of the external load [16] not being accounted for in the calibration. This 1% deviation from the optimal calibration is, however, considered to be sufficiently small when compared to the overall uncertainties associated with the physical implementation of shunts in practice.

## 5. Experimental validation

The methods proposed in section 3 are demonstrated experimentally by calibrating a series RL shunt attached to a woven carbon-epoxy composite blade. As shown in figure 8(a), the blade is clamped at one end, with an array of 40 piezoelectric transducers distributed on one side. The piezoelectric transducers are of the type DuraAct P-876.A15 from PI Ceramic. The series RL shunt is calibrated to target the second bending mode, corresponding to mode three ($s = 3$), at $\omega_s = 1047$ rad s$^{-1}$, using only the transducers in rows 4–6 (from the bottom), which are connected in parallel. In figure 8(a), the three rows of shunted patches are highlighted by the blue box. No additional piezoelectric transducers, sensors, or actuators are used for the shunt calibration.

### 5.1. Setup and phase lag of the digital shunt

The dynamic capacitance is measured with the digital shunt hardware used as an impedance meter. As outlined in the introduction, several modifications have been proposed for the digital shunt hardware first introduced by Fleming et al in 2000 and 2002 [29, 30]. Matten et al [31] modified the circuit in 2014 to secure a 0–3.3 V operating range for the DCU and to protect the electronics by adding high-voltage saturation. This circuit has since been applied for metamaterials by Yi et al in 2020 [41]. In 2020 and 2021, Raze et al [34, 42] applied a similar circuit without saturation protection, using an Arduino Due as the DCU. The circuits described above are all based on the voltage compensation method with the limitation that the electrodes of the piezoelectric transducer cannot be grounded. The performance of the circuit based on the voltage compensation method has been analyzed in detail by Necasek et al in 2016 and 2017 [32, 33]. In 2017, Necasek et al [33] introduced a circuit based on a modification of the Howland current pump, allowing one of the terminals of the piezoelectric transducer to be grounded. A modification of the circuit in [33] is introduced by Sugino et al in 2020 [35] with an additional op-amp to reduce feedback current flowing to the piezoelectric transducer. The circuit introduced in [35] has since been applied for metamaterials by Alshaqqaq et al in 2022 [43], and for nonlinear piezoelectric shunts by Alfahmi et al in 2022 [44]. Other modifications of the Howland current pump have been introduced in 2022 by Raze et al [45] including resistors to attenuate high piezoelectric voltages and in 2023 by Dietrich [36] to

form an R shunt in case of power supply failure. Another circuit design has been composed using instrumental amplifiers by Dal Bo et al in 2019 [46].

In the present approach, the design proposed by Sugino et al in 2020 [35] is utilized due to its simplicity and for the possibility of grounding one of the piezoelectric transducer terminals. For the experimental setup in figure 8, the grounded terminal enables that some transducers can be used for shunting, while others (sharing the same ground) can be applied for actuation or measurements. The schematic of the circuit is shown in figure 8(b). The voltage V across the electrodes corresponds directly to the input voltage of the DCU $V_{in} = V$, while the current I at one of the electrodes is determined from the output voltage $V_{out}$ of the DCU as

$$I = -\frac{R_s}{V_{out}} \qquad (28)$$

where the resistor has a value of $R_s = 10$ kΩ which is chosen to utilize the full input and output range of ±10V for the DCU. As in [35], a parallel resistor $R_p = 1$ MΩ is included to limit the effect of DC bias current. The parallel resistance is chosen sufficiently small that the DC bias current is limited, while being large enough that additional damping is not introduced when the hardware is used as a digital shunt. $R_p$ is considered as a part of the parallel resistance $R_\infty$ of the piezoelectric transducer, and therefore, ideally, does not influence the measurement of the dynamic capacitance $C(\omega)$ following its definition in (10).

In sections 3 and 4, it is assumed that the two signals, I and V, are continuous. However, the input and output of the DCU are discrete signals with a fixed sampling frequency $f_{sample}$. The digital-to-analog converter contains a zero-order hold, which maintains a constant voltage during each sample period [47]. The effect of the zero-order hold is illustrated in figure 9, which shows the measured current through the resistor $R_s$ when the digital shunt emulates a sine wave with a frequency of 200 Hz, at a prescribed amplitude of 0.4 mA, and a sampling frequency of $f_{sample} = 10$ kHz. The zero-order hold introduces phase lag to the impedance measured using the digital shunt hardware. In appendix A, it is shown that an ideal delay of $\tau = 1/(2f_{sample})$ provides a suitable correction for the influence of the phase lag on the measured capacitance for frequencies up to approximately 10% of the sampling frequency. Thus, the corrected dynamic capacitance is determined as

$$C(\omega) = \frac{1}{\omega} \mathrm{Im} \left( \tilde{Z}(\omega)^{-1} e^{-i\omega\tau} \right) \qquad (29)$$

where $\tilde{Z}(\omega)$ is the transfer function of the impedance measured with the DCU.

Similarly, the zero-order hold also affects the system when the digital shunt hardware is used to synthesize the admittance of an analogue circuit. The effect of the zero-order hold on digital shunts has been considered in detail in [48]. In appendix B, an approximate correction is presented for the phase-lag introduced by the zero-order hold and the Forward
<figure><img src="image_13.png" /><figcaption>Figure 8. Experimental setup involving a composite blade covered with an array of piezoelectric patches (a) and electrical circuit of the digital shunt (b).</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 9. Output of the digital shunt with a sampling frequency of $f_{sample} = 10$ kHz, measured at a sampling frequency of 1 MHz as the mean of 5 measurements in each time step.</figcaption></figure>

Euler integration used in the DCU [47]. For a series RL shunt with the impedance $Z = R + i\omega L$, the correction for phase lag leads to a correction of the inductance $L$ and resistance $R$, such that

$$ \tilde{L} = L\cos(2\omega_s\tau) - \frac{R}{\omega_s}\sin(2\omega_s\tau) \quad (30) $$

$$ \tilde{R} = R\cos(2\omega_s\tau) + \omega_s L\sin(2\omega_s\tau) \quad (31) $$

where $\tilde{L}$ and $\tilde{R}$ are the inductance and resistance implemented in the DCU.

## 5.2. Measurement of dynamic capacitance

The dynamic capacitance is measured ten times using the digital shunt hardware by applying a uniform noise current

with an RMS value of 0.3 mA to the piezoelectric transducer and measuring the voltage across the terminals. Each measurement is conducted for 10 min with a sampling frequency of $f_{sample} = 10$ kHz. The data is divided into blocks of 131 072 samples with 50% overlap, a Hanning window is applied to each block, and the frequency response function of the electrical impedance is determined using linear averaging of the blocks. The dynamic capacitance, with a correction for phase lag, is then obtained using (29) with a frequency resolution (frequency range divided by number of samples) of 0.0763 Hz = 0.48 rad s⁻¹. The frequency curve for the tenth measurement is shown in figure 10(a) for frequencies up to 6000 rad s⁻¹ and in figure 10(b) for frequencies near the targeted vibration mode $\omega_s = \omega_3 = 1047$ rad s⁻¹.

The maximum and minimum points for the targeted vibration mode $s = 3$ are determined for each measurement, and the natural undamped frequency $\omega_3$, damping ratio $\zeta_3$, EEMCF $\kappa_3$, and modal capacitance $C_3^\infty$ are then directly found by (21), (22), (25), and (26), respectively. The resulting parameters are provided in table 3, along with their mean and standard deviation. The largest ratio of the standard deviation to the mean value is observed for $\zeta_3$ with a standard deviation corresponding to 8.4% of the mean value 0.0026. A smaller standard deviation is found for $\kappa_3$, corresponding to 3.8% of the mean value 0.075, while the ratio of the standard deviation to the mean value is 0.8% and 0.03% for $C_3^\infty$ and $\omega_3$ with mean values of 482.4 nF and 1047.4 rad s⁻¹, respectively. The maximum and minimum points on the capacitance curve are determined directly as the numerically largest and smallest data points. Therefore, this simplest possible approach directly transfers any noise related to the data points to the presently determined parameters $\omega_3$, $\zeta_3$, $\kappa_3$ and $C_3^\infty$. In further studies, the standard deviation may be reduced by applying polynomial curve fitting near the maximum and minimum points.

The natural frequency, damping ratio, and EEMCF have also been determined by attaching an accelerometer to the
<figure><img src="image_15.png" /><figcaption>**Figure 10.** Measured capacitance measured (black dots) and sdof approximation of the capacitance (blue dashed curve) for frequencies up to 6000 rad s⁻¹ (a) and near the targeted vibration mode (b).</figcaption></figure>

<table><caption>**Table 3.** Measured parameters with the digital shunt hardware, and reference parameters by alternative measurements.</caption><thead><tr><th>Nr.</th><th>ω<sub>3</sub> (rad s<sup>-1</sup>)</th><th>ζ<sub>3</sub> (−)</th><th>√κ<sub>3</sub><sup>2</sup> (−)</th><th>C<sub>3</sub> (nF)</th><th>L (H)</th><th>R (Ω)</th></tr></thead><tbody><tr><td>1</td><td>1047.9</td><td>0.0027</td><td>0.077</td><td>489.0</td><td>1.84</td><td>182.3</td></tr><tr><td>2</td><td>1047.7</td><td>0.0030</td><td>0.080</td><td>479.8</td><td>1.87</td><td>192.6</td></tr><tr><td>3</td><td>1047.7</td><td>0.0025</td><td>0.074</td><td>483.6</td><td>1.86</td><td>177.3</td></tr><tr><td>4</td><td>1047.2</td><td>0.0025</td><td>0.074</td><td>482.9</td><td>1.87</td><td>177.1</td></tr><tr><td>5</td><td>1047.2</td><td>0.0030</td><td>0.080</td><td>482.1</td><td>1.87</td><td>191.6</td></tr><tr><td>6</td><td>1046.9</td><td>0.0027</td><td>0.077</td><td>478.8</td><td>1.88</td><td>186.3</td></tr><tr><td>7</td><td>1047.2</td><td>0.0025</td><td>0.073</td><td>488.4</td><td>1.85</td><td>174.6</td></tr><tr><td>8</td><td>1047.4</td><td>0.0023</td><td>0.071</td><td>482.0</td><td>1.87</td><td>170.4</td></tr><tr><td>9</td><td>1047.2</td><td>0.0025</td><td>0.075</td><td>474.6</td><td>1.90</td><td>183.1</td></tr><tr><td>10</td><td>1047.2</td><td>0.0025</td><td>0.074</td><td>483.2</td><td>1.87</td><td>178.8</td></tr><tr><td>Mean</td><td>1047.4</td><td>0.0026</td><td>0.075</td><td>482.4</td><td>1.87</td><td>181.4</td></tr><tr><td>Std</td><td>0.30</td><td>2.2 · 10<sup>-4</sup></td><td>2.9 · 10<sup>-3</sup></td><td>4.2</td><td>0.017</td><td>7.2</td></tr><tr><td>Reference</td><td>1046.6</td><td>0.0027</td><td>0.073</td><td>479.7</td><td>1.88</td><td>177.6</td></tr></tbody></table>

structure and performing classical modal analysis tests with the shunt in the SC and OC limit states. The EEMCF is calculated from (16), and the capacitance is measured at the center frequency 1815 rad s⁻¹ between the 2nd and 3rd bending modes (modes 3 and 5) using a Keysight E4980AL Precision LCR Meter. The reference measurements are presented in table 3, showing a maximum relative deviation of 3% from the mean values determined from the capacitance measurements. It is highlighted that the parameters obtained using the reference method also contain uncertainty and that using the center frequency for the capacitance may not necessarily give an accurate correction for the influence from residual modes.

### 5.3. Response of series RL shunt

A series RL shunt is calibrated for the 2nd bending mode (s = 3) of the composite blade based on the parameters obtained from the capacitance measurements with the digital

shunt hardware presented in table 3. Similar to the numerical example in section 4, the present experimental calibration is based on the fixed-point method for equal acceleration amplitude [10], summarized in table 2. The inductance L and resistance R for each measurement are shown in table 3, with mean values L = 1.87H and R = 181.4Ω.

The obtained calibration can be tested directly with the digital shunt. The sampling frequency of the shunt is chosen as f<sub>sample</sub> = 50kHz which is five times the measuring frequency in section 5.2 while still manageable by the DCU. The parameters (L and R) are corrected to account for phase lag according to (30) and (31). The experimental response with the calibrated RL shunt (black curve) is shown in figure 11, with a sweep of the inductance in (a) and of the resistance in (b). The response displays the acceleration at the top of the blade when the two bottom rows of piezoelectric transducers (red box) in figure 8(a) are used to excite the structure by white noise. As seen from the calibration expressions in table 2, variations of
<figure><img src="image_16.png" /><figcaption>**Figure 11.** Acceleration measured experimentally at the tip of the blade normalized by voltage (accelerance) for: *L* = (1.82 : 0.01 : 1.92) H. and *R* = 185Ω in (a), and *L* = 1.87H. and *R* = (135 : 10 : 235)Ω in (b). The black curve show the calibrated value (*L* = 1.87H, *R* = 181.4Ω).</figcaption></figure>

*L* are inversely proportional to variations in *C*<sub>s</sub><sup>∞</sup> and ω<sub>s</sub><sup>2</sup> for κ<sub>s</sub><sup>2</sup> ≪ 1, while *R* is inversely proportional to *C*<sub>s</sub><sup>∞</sup> and ω<sub>s</sub>, and proportional to κ<sub>s</sub>. The sweep of inductance and resistance in figure 11 is therefore directly related to changes in the properties of the electromechanical system.

The calibrated RL shunt (L = 1.87H and R = 181.4Ω) leads to a reduction of the maximum acceleration amplitude of 88.5% compared to the undamped case with the electrodes of the piezoelectric transducer in the SC condition. To evaluate the accuracy of the calibration, the optimal calibration for minimum amplitude has been determined by a parameter sweep. The parameter sweep gives L = 1.88H and R = 155Ω, which corresponds to a relative deviation of 0.53% for the inductance and -14.6% for the resistance with respect to the parameters obtained from the calibration. The optimal calibration (blue dot-dashed curve) is shown in figure 12 alongside the calibrated RL shunt (black solid curve) and the response with the shunt in the SC condition (black dashed curve). The optimal calibration (L = 1.88H and R = 155Ω) results in a reduction of the acceleration amplitude by 7.5% compared to the initial calibration (L = 1.87H and R = 181.4Ω). The two calibrations are almost identical, with a notable difference for the resistance (R). This difference in R is most likely caused by inherited losses in the piezoelectric transducer.

Finally, figure 12 also shows the response of an analog RL shunt with L = 1.87H and R = 180.4Ω. There is a good agreement between the responses obtained with the analog components (red dotted curve) and the digital shunt hardware (black solid curve). This illustrates that the calibration procedure using the digital shunt hardware also enables a direct implementation of analog shunts. Additionally, the calibration was repeated without the correction for phase delay introduced in appendix A. This resulted in mean values of L = 1.82H and R = 176.7Ω, which can be seen to result in a significant detuning in figure 11(a). This highlights the importance of correcting for phase delay when using the digital shunt hardware for capacitance measurements.

<figure><img src="image_17.png" /><figcaption>**Figure 12.** Frequency response for the shunt in the SC condition (black dashed curve), calibrated shunt with analog components (red dotted curve) and digital components (black solid curve), and optimal calibration of the digital shunt (blue dot-dashed curve).</figcaption></figure>

**6. Conclusions and perspectives**

Accurate calibration of piezoelectric shunts using analog or digital components necessitates an accurate estimation of the electromechanical properties of the system. This paper presents the estimation of the SC natural frequency, modal damping ratio, EEMCF, and effective modal capacitance based on measurement of the dynamic capacitance.

The parameters can be derived directly from the positions of two extremum points on the frequency curves of the dynamic capacitance for a structure with non-zero damping. The dynamic capacitance curve can be determined experimentally through a single measurement using the hardware of a digital shunt, allowing for parameter estimation without additional sensors or actuators. The estimated parameters
allow for direct implementation of analog or digital shunts targeting a single vibration mode while accounting for the influence from residual vibration modes.

The accuracy of the proposed method has been demonstrated numerically for a cantilever beam and validated experimentally through calibration of a series RL shunt attached to a composite blade. By comparing the frequency response of the acceleration at the tip of the composite blade, it is found that the calibration method gives a sufficiently accurate tuning of a series RL shunt when the effect of phase lag is included.

In future work, the proposed calibration method will be used to investigate the influence of the inherited losses of the piezoelectric transducer on the accuracy of the shunt calibration. It is expected that a correction for the inherited losses will result in better agreement between the response of the structure and the ideal response in the calibration.

## Data availability statement

All data that support the findings of this study are included within the article (and any supplementary files).

## Acknowledgment

The composite blade has been manufactured by Safran, using a resin transfer molding (RTM) process. Thanks to Safran for allowing the use of the composite blade for the experimental analysis conducted at Cnam.

## Appendix A

The Digital Control Unit (DCU) requires a single input and output of ±10V with a sufficiently high sampling rate. In the present work the DCU consists of a Dspace MicroLabBox, while the electrical circuit of the digital shunt hardware includes two OPA445 operational amplifiers and one AMP03 differential amplifier. The MicroLabBox and the AMP03 differential amplifier have a fast settling time, given in the data sheets as 1 µs. The datasheet does not directly provide the settling time of the OPA445 operational amplifiers. However, it is expected to be of the same order of magnitude since the rise time is 0.1 µs for ±200mV. The fast settling time of the digital shunt hardware is illustrated in figure 9 for a sampling frequency of $f_{sample} = 10$ kHz. It is therefore assumed that the continuous current $I$ corresponds to a step function for the sample time $1/f_{sample} \gg 1$ µs.

From [47], the transfer function of a zero-order hold can be found for an ideal step function as

$$E(\omega) = \frac{1 - e^{-i2\omega\tau}}{i2\omega\tau} \qquad (32)$$

where $\tau = 1/(2f_{sample})$.

The transfer function of the physical current $I(\omega)$ can then be approximated from the transfer function of the prescribed current $\tilde{I}(\omega)$ in the DCU, resulting in the expression

$$I(\omega) = \tilde{I}(\omega) E(\omega). \qquad (33)$$

The following expression for dynamic capacitance and conductance can then be found from (8) and (9) as

$$C(\omega) = \frac{1}{\omega} \mathrm{Im} \left( \tilde{Z}(\omega)^{-1} E(\omega) \right) \qquad (34)$$

$$\frac{1}{R(\omega)} = \mathrm{Re} \left( \tilde{Z}(\omega)^{-1} E(\omega) \right) \qquad (35)$$

where the impedance measured with the DCU is $\tilde{Z}(\omega) = \tilde{I}(\omega)/V(\omega)$.

The dynamic capacitance and conductance are shown in figure 13 for frequencies $f = \omega/(2\pi)$ up to the Nyquist frequency $f_{sample}/2$ for the tenth measurement in section 5.2. The measurements are compared with a measurement (red dashed curve) obtained with a dedicated LCR meter with a stepped-sine signal (Keysight E4980AL Precision LCR Meter). While the correction with the zero-order hold in (32) gives a significant improvement of the accuracy of the capacitance measurement (magenta curve) compared to the direct measurement with the digital shunt hardware (black curve), it is seen that a slightly better approximation (blue curve) is found by introducing an ideal delay

$$E(\omega) = e^{-i\omega\tau} \qquad (36)$$

such that the measured capacitance becomes

$$C(\omega) = \frac{1}{\omega} \mathrm{Im} \left( \tilde{Z}(\omega)^{-1} e^{-i\omega\tau} \right). \qquad (37)$$

This expression is applied because of its simplicity, which enables the formulation of simple correction expressions in appendix B.
<figure><img src="image_18.png" /><figcaption>Figure 13. Dynamic capacitance for the shunt patches on the composite blade. Measurements done with the digital shunt unit with no corrections for phase lag (black curve), with correction by ideal zero-order hold (magenta curve), and assuming an ideal delay (blue curve), as well as with an LCR-meter (red dashed curve).</figcaption></figure>

## Appendix B

The effect of the zero-order hold in the DCU on the accuracy and stability of a digital shunt has been described in detail in [48], including corrections for resonant RL shunts. This appendix presents a simplified correction for the delay in the digital shunt.

As described in appendix A, the effect of the ZOH on the current output of the DCU can be approximated as an ideal delay $e^{-i\omega\tau}$ in (36). Furthermore, Forward Euler integration is used for the discretization of the transfer function of the shunt, introducing additional delay in the current output. Assuming that the delay imposed by the zero-order hold can be described as an ideal delay $e^{-i\omega\tau}$, the corrected transfer function of the current can be expressed as

$$I(\omega) = \tilde{I}(\omega) e^{-2i\omega\tau} \quad (38)$$

based on the transfer function of the prescribed current $\tilde{I}(\omega)$. The required shunt impedance of the DCU can then be found as

$$\tilde{Z}(\omega) = Z(\omega) e^{-2i\omega\tau} \quad (39)$$

based on the ideal impedance of the shunt $Z(\omega)$. While more detailed corrections can be done for the influence of the DCU, the present correction is applied for its simplicity.

For a series RL shunt with the impedance $Z(\omega) = R + i\omega L$, this leads to the following corrections

$$\tilde{L} = L\cos(2\omega_s\tau) - \frac{R}{\omega_s}\sin(2\omega_s\tau) \quad (40)$$

$$\tilde{R} = R\cos(2\omega_s\tau) + \omega_s L\sin(2\omega_s\tau) \quad (41)$$

where $\tilde{L}$ and $\tilde{R}$ are the inductance and resistance implemented in the DCU to recover the desired values of $L$ and $R$. The accuracy of the presented correction is shown in figure 14 for a digital shunt emulating a series RL shunt with $L = 1.87$ H and $R = 180.4$ $\Omega$ for varying sampling frequencies.
<figure><img src="image_19.png" /><figcaption>**Figure 14.** Comparison of an analog shunt to digital shunts with different sampling frequencies without correction for delay (a) and with correction for delay (b).</figcaption></figure>

ORCID iDs

Jens D Richardt https://orcid.org/0009-0000-1849-8888

Boris Lossouarn https://orcid.org/0000-0001-7382-3137

Jan Høgsberg https://orcid.org/0000-0002-1098-3245

Jean-François Deü https://orcid.org/0000-0002-0107-8024

References

[1] Forward R L 1979 Electronic damping of vibrations in optical structures *Appl. Opt.* **18** 690–7

[2] Hagood N W and von Flotow A 1991 Damping of structural vibrations with piezoelectric materials and passive electrical networks *J. Sound Vib.* **146** 243–68

[3] Davis C L and Lesieutre G A 2000 Actively tuned solid-state vibration absorber using capacitive shunting of piezoelectric stiffness *J. Sound Vib.* **232** 601–17

[4] Wu S-Y 1996 Piezoelectric shunts with a parallel R-L circuit for structural damping and vibration control *Proc. SPIE* **2720** 259–69

[5] Clark W W 2000 Vibration control with state-switched piezoelectric materials *J. Intell. Mater. Syst. Struct.* **11** 263–71

[6] Richard C, Guyomar D, Audigier D and Bassaler H 2000 Enhanced semi passive damping using continuous switching of a piezoelectric device on an inductor *Proc. SPIE* **3989** 288–99

[7] Tang J and Wang K W 2001 Active-passive hybrid piezoelectric networks for vibration control: comparisons and improvement *Smart Mater. Struct.* **794** 794–806

[8] Behrens S, Fleming A J and Moheimani S O R 2003 A broadband controller for shunt piezoelectric damping of structural vibration *Smart Mater. Struct.* **12** 18–28

[9] Caruso G 2001 A critical analysis of electric shunt circuits employed in piezoelectric passive vibration damping *Smart Mater. Struct.* **10** 1059–68

[10] Yamada K, Matsuhisa H, Utsuno H and Sawada K 2010 Optimum tuning of series and parallel LR circuits for passive vibration suppression using piezoelectric elements *J. Sound Vib.* **329** 5036–57

[11] Thomas O, Ducarne J and Deü J-F 2012 Performance of piezoelectric shunts for vibration reduction *Smart Mater. Struct.* **21** 015008

[12] Høgsberg J and Krenk S 2012 Balanced calibration of resonant shunt circuits for piezoelectric vibration control *J. Intell. Mater. Syst. Struct.* **23** 1937–48

[13] Soltani P, Kerschen G, Tondreau G and Deraemaeker A 2014 Piezoelectric vibration damping using resonant shunt circuits: an exact solution *Smart Mater. Struct.* **23** 125014

[14] Ikegame T, Takagi K and Inoue T 2019 Exact solutions to $H_\infty$ and $H_2$ optimizations of passive resonant shunt circuit for electromagnetic or piezoelectric shunt damper *Trans. ASME. J. Vib. Acoust.* **141** 1–15

[15] Soltani P, Kerschen G, Tondreau G and Deraemaeker A 2017 Tuning of a piezoelectric vibration absorber attached to a damped structure *J. Intell. Mater. Syst. Struct.* **28** 1115–29

[16] Toftekær J F and Høgsberg J 2021 On the inclusion of structural loading and damping in piezoelectric shunt tuning *J. Sound Vib.* **498** 115960

[17] Delpero T, Bergamini A E and Ermanni P 2013 Identification of electromechanical parameters in piezoelectric shunt damping and loss factor prediction *J. Intell. Mater. Syst. Struct.* **24** 287–98

[18] Toftekær J F, Benjeddou A, Høgsberg J and Krenk S 2018 Optimal piezoelectric resistive-inductive shunt damping of plates with residual mode correction *J. Intell. Mater. Syst. Struct.* **29** 3346–70

[19] De Marneffe B and Preumont A 2008 Vibration damping with negative capacitance shunts: theory and experiment *Smart Mater. Struct.* **17** 035015

[20] Preumont A 2011 *Vibration control of active structures* 3rd edn (Springer)

[21] Porfiri M, Maurini C and Pouget J 2007 Identification of electromechanical modal parameters of linear piezoelectric structures *Smart Mater. Struct.* **16** 323–31

[22] Berardengo M, Thomas O, Giraud-Audine C and Manzoni S 2016 Improved resistive shunt by means of negative capacitance: new circuit, performances and multi-mode control *Smart Mater. Struct.* **25** 075033

[23] Gardonio P and Casagrande D 2017 Shunted piezoelectric patch vibration absorber on two-dimensional thin structures: tuning considerations *J. Sound Vib.* **395** 26–47
[24] Høgsberg J and Krenk S 2017 Calibration of piezoelectric RL shunts with explicit residual mode correction *J. Sound Vib.* **386** 65–81

[25] Berardengo M, Manzoni S, Thomas O and Vanali M 2018 Piezoelectric resonant shunt enhancement by negative capacitances: optimisation, performance and resonance cancellation *J. Intell. Mater. Syst. Struct.* **29** 2581–606

[26] Toftekær J F and Høgsberg J 2020 Multi-mode piezoelectric shunt damping with residual mode correction by evaluation of modal charge and voltage *J. Intell. Mater. Syst. Struct.* **31** 570–86

[27] Toftekær J F and Høgsberg J 2020 Experimental validation of piezoelectric shunt tuning with residual mode correction: damping of plate-like structures *J. Intell. Mater. Syst. Struct.* **31** 1220–39

[28] Berardengo M, Manzoni S, Høgsberg J and Vanali M 2021 Vibration control with piezoelectric elements: the indirect measurement of the modal capacitance and coupling factor *Mech. Syst. Signal Process.* **151** 107350

[29] Fleming A J, Behrens S and Moheimani S O R 2002 Optimization and implementation of multimode piezoelectric shunt damping systems *IEEE/ASME Trans. Mechatronics* **7** 87–94

[30] Fleming A J, Behrens S and Moheimani S O R 2000 Synthetic impedance for implementation of piezoelectric shunt-damping circuits *Electron. Lett.* **36** 1525–6

[31] Matten G, Collet M, Cogan S and Sadoulet-Reboul E 2014 Synthetic impedance for adaptive piezoelectric metacomposite *Proc. Technol.* **15** 84–89

[32] Nečásek J, Václavík J and Marton P 2016 Digital synthetic impedance for application in vibration damping *Rev. Sci. Instrum.* **87** 024704

[33] Nečásek J, Vaclavik J and Marton P 2017 Comparison of analog front-ends for digital synthetic impedance device *Proc. 2017 IEEE Int. Workshop of Electronics, Control, Measurement, Signals and Their Application to Mechatronics, ECMSM 2017*

[34] Raze G, Jadoul A, Guichaux S, Broun V and Kerschen G 2020 A digital nonlinear piezoelectric tuned vibration absorber *Smart Mater. Struct.* **29** 015007

[35] Sugino C, Ruzzene M and Erturk A 2020 Digitally programmable resonant elastic metamaterials *Phys. Rev. Appl.* **13** 1

[36] Dietrich J 2023 Vibration mitigation of bladed structures using piezoelectric digital vibration absorbers *PhD Thesis*
University of Liege

[37] Richardt J D, Lossouarn B, Høgsberg J and Deü J-F 2024 Calibration of multiple shunted piezoelectric transducers with correction for residual modes and shunt interactions *JVC/J. Vib. Control* **31** 24–36

[38] Høgsberg J, Lossouarn B and Deü J-F 2024 Tuning of vibration absorbers by an effective modal coupling factor *Int. J. Mech. Sci.* **268** 109009

[39] CENELEC 2002 EN 50324-2:2002: *Piezoelectric Properties of Ceramic Materials and Components—Part 2: Methods of Measurement—Low Power* (CENELEC European Committee for Electromechanical Standardization)

[40] Watine Y, Lossouarn B, Gabillet C, Astolfi J-A and Deü J-F 2024 Mitigation of hydrofoil torsional flow induced vibrations by resonant piezoelectric shunt *Ocean Eng.* **313** 119598

[41] Yi K, Matten G, Ouisse M, Sadoulet-Reboul E, Collet M and Chevallier G 2020 Programmable metamaterials with digital synthetic impedance circuits for vibration control *Smart Mater. Struct.* **29** 035005

[42] Raze G 2021 Piezoelectric digital vibration absorbers for multimodal vibration mitigation of complex mechanical structures *PhD thesis* Liege

[43] Alshaqqaq M, Sugino C and Erturk A 2022 Programmable rainbow trapping and band-gap enhancement via spatial group-velocity tailoring in elastic metamaterials *Phys. Rev. Appl.* **17** 1

[44] Alfahmi O, Sugino C and Erturk A 2022 Duffing-type digitally programmable nonlinear synthetic inductance for piezoelectric structures *Smart Mater. Struct.* **31** 095044

[45] Raze G, Dietrich J, Lossouarn B and Kerschen G 2022 Shunts vs. networks: tuning and comparison of centralized and decentralized piezoelectric vibration absorbers *Smart Mater. Struct.* **31** 115006

[46] Dal Bo L, Gardonio P, Casagrande D and Saggini S 2019 Smart panel with sweeping and switching piezoelectric patch vibration absorbers: experimental results *Mech. Syst. Signal Process.* **120** 308–25

[47] Franklin G F, Powel J D and Workman M L 1998 *Digital Control of Dynamic Systems* 3rd edn
(Ellis-Kagle Press)

[48] Raze G, Dietrich J and Kerschen G 2022 Onset and stabilization of delay-induced instabilities in piezoelectric digital vibration absorbers *J. Intell. Mater. Syst. Struct.* **33** 2033–48