**2. Modeling of mechanical structures with a shunted piezoceramics**

In order to obtain general results which can be transferred to various mechanical structures, the structure is reduced to a one degree of freedom oscillator. The piezoceramics is shunted to an arbitrary electrical impedance *Z*, as shown in Figure 1. The equations of motion for this general case read

$$
\begin{bmatrix} m \ 0 \\ 0 \ 0 \end{bmatrix} \begin{bmatrix} \ddot{q} \\ \ddot{Q} \end{bmatrix} + \begin{bmatrix} d \ 0 \\ 0 \ 0 \end{bmatrix} \begin{bmatrix} \dot{q} \\ \dot{Q} \end{bmatrix} + \begin{bmatrix} c + \frac{a^2}{\mathsf{C}\_\mathbb{P}} & \frac{a}{\mathsf{C}\_\mathbb{P}} \\ \frac{a}{\mathsf{C}\_\mathbb{P}} & \frac{1}{\mathsf{C}\_\mathbb{P}} \end{bmatrix} \begin{bmatrix} q \\ Q \end{bmatrix} = \begin{bmatrix} F(t) \\ -u\_\mathbb{P} \end{bmatrix} \tag{1}
$$

where the parameters *m*, *d* and *c* denote respectively the modal mass, damping and stiffness. Stiffness *c* is herein the sum of the mechanical stiffness *c*mech and the stiffness of the piezoelement *c*p. *F*(*t*) represents the external force, *u*<sup>p</sup> is the voltage at the electrodes of piezoceramics, *C*p the capacitance of the piezoceramics and *α* the force factor which can be deduced from geometry and characteristics of the piezoceramics and the mechanical structure. The variables *q* and *Q* are respective the modal displacement and electrical charge.

**Figure 1.** Single Degree-of-Freedom oscillator with piezoceramics and shunt circuit.

When the piezoceramics is shunted to an electrical circuit, the voltage *u*p depends on the charge *Q* as well as the impedance *Z* of the shunt,

$$
\mu\_{\rm P} = ZQ.\tag{2}
$$

In this context, *Z* describes the relationship between voltage and charge rather than between voltage and current. Inserting (2) into (1), the generalized equation of a shunt damping system reads as

$$
\begin{bmatrix} m \ 0 \\ 0 \ 0 \end{bmatrix} \begin{bmatrix} \ddot{q} \\ \ddot{Q} \end{bmatrix} + \begin{bmatrix} d \ 0 \\ 0 \ 0 \end{bmatrix} \begin{bmatrix} \dot{q} \\ \dot{Q} \end{bmatrix} + \begin{bmatrix} c + \frac{a^2}{C\_\mathbb{P}} & \frac{a}{C\_\mathbb{P}} \\ \frac{a}{C\_\mathbb{P}} & \frac{1}{C\_\mathbb{P}} + Z \end{bmatrix} \begin{bmatrix} q \\ Q \end{bmatrix} = \begin{bmatrix} F(t) \\ 0 \end{bmatrix}. \tag{3}
$$

This equation is the basis for all further calculations.

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In order to enhance the limited damping performance of passive shuntings, active elements have been proposed. The mostly studied element is a negative capacitance, which can be realized by a negative impedance converter circuit [5]. Initially considered by Forward [3], a negative capacitance proves to be able to increase the effective piezoelectric coupling factor. Especially the combination of passive networks with active elements is a promising approach. This class of networks is called 'active-passive hybrid piezoelectric network' (APPN) by Tang [21]. Most prominent APPN networks are a negative capacitance with a resistor and a negative

The drawback of these linear resonant networks is that they all must be tuned to a certain frequency, which has to be known in advance and which should not change during operation. For many applications they are therefore not suitable. In these cases adaptive, non-linear networks are a better choice. The most common one is the 'synchronized switch damping on inductor' (SSDI) technique, which consists of an *LR*-branch and a switch that can connect and disconnect the network to the electrodes of the piezoceramics [10]. For the case of monoharmonic excitation the switch is closed at the moments of maximum deformation of the piezoceramics. In this moment, the electrical charge is inverted via the inductance. The inductance value is very small in order to realize a fast inversion. When fully inverted, the switch is closed so that the charge cannot flow anymore. During the following half period of excitation the charge stays nearly constant, so that the piezoceramics generates a force acting against the deformation velocity. The resulting force signal is nearly rectangular shaped. Like for the passive *LR* shunting the damping strongly depends on the electrical damping ratio, which can be set by the resistance value. A small damping results in a good inversion of the charge, which amplifies the stationary charge amplitudes and the dissipated energy. The adaptive capability of the SSDI technique comes from the triggering of the switching times. Therefore, typically one additional sensor is used which monitors the vibration of the mechanical structure. Due to this triggering, the force signal from the piezoceramics is always in phase with the structure vibration and the performance is only minimally dependend on

**2. Modeling of mechanical structures with a shunted piezoceramics**

 *q*˙ *Q*˙ + *c* + *<sup>α</sup>*<sup>2</sup> *C*<sup>p</sup> *α C*<sup>p</sup>

The variables *q* and *Q* are respective the modal displacement and electrical charge.

In order to obtain general results which can be transferred to various mechanical structures, the structure is reduced to a one degree of freedom oscillator. The piezoceramics is shunted to an arbitrary electrical impedance *Z*, as shown in Figure 1. The equations of motion for this

where the parameters *m*, *d* and *c* denote respectively the modal mass, damping and stiffness. Stiffness *c* is herein the sum of the mechanical stiffness *c*mech and the stiffness of the piezoelement *c*p. *F*(*t*) represents the external force, *u*<sup>p</sup> is the voltage at the electrodes of piezoceramics, *C*p the capacitance of the piezoceramics and *α* the force factor which can be deduced from geometry and characteristics of the piezoceramics and the mechanical structure.

*α C*<sup>p</sup>

1 *C*<sup>p</sup>  *q Q* = *F*(*t*) −*u*<sup>p</sup>

, (1)

capacitance with an inductor and resistor.

the excitation frequency.

general case read

 *q*¨ *Q*¨ + *d* 0 0 0

### **3. Optimization of resonant** *LR***-shunting for damped mechanical systems**

Let us first consider a resonant *LR*-shunt with impedance *Z* = *Ls*<sup>2</sup> + *Rs*. Substituting this term into (3) leads us to

$$
\begin{bmatrix} m \ 0 \\ 0 \ L \end{bmatrix} \begin{bmatrix} \ddot{q} \\ \ddot{Q} \end{bmatrix} + \begin{bmatrix} d \ 0 \\ 0 \ R \end{bmatrix} \begin{bmatrix} \dot{q} \\ \dot{Q} \end{bmatrix} + \begin{bmatrix} c + \frac{a^2}{\mathcal{C}\_\mathbb{P}} \ \frac{a}{\mathcal{C}\_\mathbb{P}} \\ \frac{a}{\mathcal{C}\_\mathbb{P}} \ \frac{1}{\mathcal{C}\_\mathbb{P}} \end{bmatrix} \begin{bmatrix} q \\ Q \end{bmatrix} = \begin{bmatrix} F(t) \\ 0 \end{bmatrix}. \tag{4}
$$

For maximum damping performance the circuit parameters *L* and *R* have to be tuned in such a way that the system has double eigenvalues. For the undamped case (*d* = 0) the optimal parameters are well known.

With the help of normalized, non-dimensional parameters, the equations can be written in a more compact form. In detail, the generalized piezoelectric coupling coefficient *K*, the eigenfrequency of the system with isolated electrodes *ω*iso, the electrical eigenfrequency *ω*el, the electrical damping ratio *ζ*<sup>0</sup> of the *LR*-branch and the frequency ratio *η*el are introduced,

$$K^2 = \frac{a^2}{c\mathcal{C}\_\mathcal{P} + a^2}, \quad \omega\_{\rm el} = \sqrt{\frac{1}{\mathcal{C}\_\mathcal{P}L}}, \quad \omega\_{\rm iso}^2 = \frac{c + a^2/\mathcal{C}\_\mathcal{P}}{m},$$

$$\mathcal{L}\_0 = \frac{R}{2}\sqrt{\frac{\mathcal{C}\_\mathcal{P}}{L}}, \quad \eta\_{\rm el} = \frac{\omega\_{\rm el}}{\omega\_{\rm iso}} = \sqrt{\frac{m}{a^2L}}\mathcal{K}.\tag{5}$$

The generalized piezoelectric coupling coefficient *K* is a measure of the effectiveness of the piezoceramics. It depends on the piezoceramics characteristics as well as on the structure vibration form. The shunt parameters *L*, *R* are substituted by non-dimensional parameters *η*el, *ζ*0. The matrix **A** of the corresponding state-space system then reads as

$$\mathbf{A} = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & -\mathbf{K}^2 & 0 & 0 \\ -\eta\_{\rm el}^2 & -\eta\_{\rm el}^2 & 0 & -2\eta\_{\rm el}\zeta\_0 \end{bmatrix} \tag{6}$$

and the characteristic equations become

$$\det\left(\mathbf{A} - \lambda\mathbf{I}\right) = \lambda^4 + 2\zeta\_0\eta\_{\rm el}\lambda^3 + \left(1 + \eta\_{\rm el}^2\right)\lambda^2 + 2\zeta\_0\eta\_{\rm el}\lambda + \left(1 - K^2\right)\eta\_{\rm el}^2 = 0. \tag{7}$$

Double eigenvalues are obtained when the shunt parameters are tuned according to

$$
\eta\_{\rm el,opt} = \frac{1}{\sqrt{1 - K^2}} \approx 1, \quad \zeta\_{0,\rm opt} = K. \tag{8}
$$

These equations basically state that the inductance has to be tuned in such a way that the electrical resonant frequency nearly equals the mechanical resonant frequency. The damping ratio of the circuit must match the generalized piezoelectric coupling coefficient *K*. The resulting damping ratio from such a shunting then becomes

$$D = \frac{K}{2\sqrt{1 - K^2}} \approx \frac{K}{2}.\tag{9}$$

This equation proves the importance of the generalized piezoelectric coupling coefficient *K*, as the damping performance grows with *K*. Figure 2 shows the influence of the network parameters upon the location of the complex eigenvalues.

For the damped mechanical oscillator, the same strategy can be followed to optimize the system. Normalizing the mechanical damping by *D*<sup>m</sup> = *<sup>d</sup>* <sup>2</sup>*mω*iso the characteristic equation of the system becomes

$$
\lambda^4 + \left(2\zeta\_0 \eta\_{\rm el} + 2D\_{\rm m}\right) \lambda^3 + \left(1 + \eta\_{\rm el}^2 + 4\zeta\_0 \eta\_{\rm el} D\_{\rm m}\right) \lambda^2 + \left(2\zeta\_0 \eta\_{\rm el} + 2\eta\_{\rm el}^2 D\_{\rm m}\right) \lambda + \left(1 - K^2\right) \eta\_{\rm el}^2 = 0. \tag{10}
$$

The results for the optimal network parameters *η*el,opt and *ζ*0,opt as well as the resulting damping performance are very lengthy terms. It is useful to express them in a Taylor series,

**Figure 2.** Eigenvalue of mechanical 1 DOF oscillator with *LR*-shunted piezoceramics.

and only consider the first elements of the series. This gives

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the electrical damping ratio *ζ*<sup>0</sup> of the *LR*-branch and the frequency ratio *η*el are introduced,

� 1 *<sup>C</sup>*p*<sup>L</sup>* , *<sup>ω</sup>*<sup>2</sup>

*ω*iso = � *m α*2*L*

The generalized piezoelectric coupling coefficient *K* is a measure of the effectiveness of the piezoceramics. It depends on the piezoceramics characteristics as well as on the structure vibration form. The shunt parameters *L*, *R* are substituted by non-dimensional parameters

> 0 01 0 0 00 1 <sup>−</sup><sup>1</sup> <sup>−</sup>*K*<sup>2</sup> 0 0

> > el 0 −2*η*el*ζ*<sup>0</sup>

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

*<sup>λ</sup>*<sup>2</sup> + <sup>2</sup>*ζ*0*η*el*<sup>λ</sup>* +

� <sup>1</sup> <sup>−</sup> *<sup>K</sup>*<sup>2</sup> � *η*2

<sup>1</sup> <sup>−</sup> *<sup>K</sup>*<sup>2</sup> <sup>≈</sup> 1, *<sup>ζ</sup>*0,opt <sup>=</sup> *<sup>K</sup>*. (8)

<sup>2</sup> . (9)

<sup>2</sup>*mω*iso the characteristic equation

iso <sup>=</sup> *<sup>c</sup>* <sup>+</sup> *<sup>α</sup>*2/*C*<sup>p</sup>

*<sup>m</sup>* ,

*K*. (5)

, (6)

el = 0. (7)

*<sup>K</sup>*<sup>2</sup> <sup>=</sup> *<sup>α</sup>*<sup>2</sup>

*<sup>ζ</sup>*<sup>0</sup> <sup>=</sup> *<sup>R</sup>* 2

and the characteristic equations become

det(**<sup>A</sup>** <sup>−</sup> *<sup>λ</sup>***I**) <sup>=</sup> *<sup>λ</sup>*<sup>4</sup> <sup>+</sup> <sup>2</sup>*ζ*0*η*el*λ*<sup>3</sup> <sup>+</sup>

*cC*<sup>p</sup> <sup>+</sup> *<sup>α</sup>*<sup>2</sup> , *<sup>ω</sup>*el <sup>=</sup>

*<sup>L</sup>* , *<sup>η</sup>*el <sup>=</sup> *<sup>ω</sup>*el

*η*el, *ζ*0. The matrix **A** of the corresponding state-space system then reads as

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

<sup>−</sup>*η*<sup>2</sup> el <sup>−</sup>*η*<sup>2</sup>

> � 1 + *η*<sup>2</sup> el �

Double eigenvalues are obtained when the shunt parameters are tuned according to

*<sup>D</sup>* <sup>=</sup> *<sup>K</sup>* 2

el + 4*ζ*0*η*el*D*<sup>m</sup>

These equations basically state that the inductance has to be tuned in such a way that the electrical resonant frequency nearly equals the mechanical resonant frequency. The damping ratio of the circuit must match the generalized piezoelectric coupling coefficient *K*. The

<sup>√</sup><sup>1</sup> <sup>−</sup> *<sup>K</sup>*<sup>2</sup> <sup>≈</sup> *<sup>K</sup>*

This equation proves the importance of the generalized piezoelectric coupling coefficient *K*, as the damping performance grows with *K*. Figure 2 shows the influence of the network

For the damped mechanical oscillator, the same strategy can be followed to optimize the

� *λ*<sup>2</sup> + �

The results for the optimal network parameters *η*el,opt and *ζ*0,opt as well as the resulting damping performance are very lengthy terms. It is useful to express them in a Taylor series,

<sup>2</sup>*ζ*0*η*el + <sup>2</sup>*η*<sup>2</sup>

el*D*<sup>m</sup> � *λ*+ � <sup>1</sup> <sup>−</sup> *<sup>K</sup>*<sup>2</sup> � *η*2 el = 0. (10)

**A** =

*<sup>η</sup>*el,opt <sup>=</sup> <sup>1</sup> √

resulting damping ratio from such a shunting then becomes

parameters upon the location of the complex eigenvalues.

� 1 + *η*<sup>2</sup>

of the system becomes

*<sup>λ</sup>*<sup>4</sup> + (2*ζ*0*η*el + <sup>2</sup>*D*m) *<sup>λ</sup>*<sup>3</sup> +

system. Normalizing the mechanical damping by *D*<sup>m</sup> = *<sup>d</sup>*

�*C*<sup>p</sup>

$$
\eta\_{\rm el,opt} \approx \frac{1}{\sqrt{1 - K^2}} + \frac{K}{1 - K^2} D\_{\rm m\prime}
$$

$$
\zeta\_{0,\rm opt} \approx K + \sqrt{1 - K^2} D\_{\rm m\prime}
$$

$$
D \approx \frac{1}{2} \frac{K}{\sqrt{1 - K^2}} + \frac{4 - 3K^2}{4 - 4K^2} D\_{\rm m\cdot} \tag{11}
$$

These approximations are valid for small mechanical damping ratios *D*m, which is practically fulfilled in most cases. The equations clearly show the trend when mechanical damping is included. For the undamped case, *D*<sup>m</sup> = 0, the results are per definition equal to the values obtained in (8) and (9). But additional mechanical damping leads to a slight increase of the optimum electrical resonant frequency *η*el and damping ratio *ζ*0,opt. Naturally also the resulting damping performance grows with additional mechanical damping. One can realize that for high mechanical damping *D*m (compared to the coupling coefficient *K*), the overall damping converges the mechanical damping, *D* ≈ *D*m, and the additional damping caused by the shunted piezoceramics is negligible.

**Figure 3.** SSDNCI network.
