**4. Synchronized switch damping on negative capacitance and inductance (SSDNCI)**

This section describes a novel combination of a negative capacitance together with the well-known SSDI-technique schematized in Figure 3, which has recently also been studied by other research groups [11]. It is intended to combine the adaptive ability of the SSDI with the enhanced performance and coupling of a negative capacitor. Again the calculations will be performed using nondimensional parameters,

$$\delta = \frac{\mathsf{C\_{e}}}{\mathsf{C\_{P}}}, \quad \omega\_{\mathrm{el}} = \frac{1}{\sqrt{(1+\delta)L\mathsf{C\_{P}}}}, \quad \tilde{\zeta} = \sqrt{1+\delta}\frac{\mathsf{R}}{2}\sqrt{\frac{\mathsf{C\_{P}}}{L}} = \sqrt{1+\delta}\tilde{\zeta}\_{0\prime} \quad \text{\(\tau = \omega\_{\mathrm{el}}t.\)}\tag{12}$$

The electrical damping ratio of the *LR*-branch of the circuit is again termen *ζ*0, while the overall electrical damping ratio with negative capacitance is *ζ*. The capacitance ratio *δ* can be set by choosing appropriate values for the negative capacitance. For a positive external capacitance, the parameter *δ* is positive, for a negative capacitance it is negative. The electrical resonance frequency as well as the electrical damping ratio both depend on the capacitance ratio *ζ*. Setting *δ* = 0 results in the standard SSDI technique without negative capacitance, with the corresponding electrical damping ratio *ζ*0. Obviously, the negative capacitance influences the electrical resonance frequency as well as the damping ratio. Especially the latter one is important, as the damping ratio should be as small as possible. Using a negative capacitance reduces the damping ratio.

The switching network is a nonlinear system, but it can be considered as linear during the periods with open and closed switch. When the switch is closed, the shunt impedance *Z*cl reads

$$Z\_{\rm cl} = \frac{1}{\frac{1}{Ls^2 + Rs} + \frac{1}{1/\mathcal{C}\_{\rm e}}} = \frac{Ls^2 + Rs}{\mathcal{C}\_{\rm e}Ls^2 + \mathcal{C}\_{\rm e}Rs + 1}.\tag{13}$$

Substitution of *Z*cl into (3) and representation in terms of the non-dimensional parameters yields

6 Will-be-set-by-IN-TECH

*Z*

*S*

**4. Synchronized switch damping on negative capacitance and inductance**

This section describes a novel combination of a negative capacitance together with the well-known SSDI-technique schematized in Figure 3, which has recently also been studied by other research groups [11]. It is intended to combine the adaptive ability of the SSDI with the enhanced performance and coupling of a negative capacitor. Again the calculations will

, *<sup>ζ</sup>* <sup>=</sup> <sup>√</sup>

The electrical damping ratio of the *LR*-branch of the circuit is again termen *ζ*0, while the overall electrical damping ratio with negative capacitance is *ζ*. The capacitance ratio *δ* can be set by choosing appropriate values for the negative capacitance. For a positive external capacitance, the parameter *δ* is positive, for a negative capacitance it is negative. The electrical resonance frequency as well as the electrical damping ratio both depend on the capacitance ratio *ζ*. Setting *δ* = 0 results in the standard SSDI technique without negative capacitance, with the corresponding electrical damping ratio *ζ*0. Obviously, the negative capacitance influences the electrical resonance frequency as well as the damping ratio. Especially the latter one is important, as the damping ratio should be as small as possible. Using a negative

The switching network is a nonlinear system, but it can be considered as linear during the periods with open and closed switch. When the switch is closed, the shunt impedance *Z*cl

> <sup>=</sup> *Ls*<sup>2</sup> <sup>+</sup> *Rs C*e*Ls*<sup>2</sup> + *C*e*Rs* + 1

1/*C*<sup>e</sup>

1 + *δ R* 2

*<sup>C</sup>*<sup>p</sup>

*<sup>L</sup>* <sup>=</sup> <sup>√</sup>

1 + *δζ*0, *τ* = *ω*el*t*. (12)

. (13)

*L*

*R*

*<sup>u</sup>*<sup>p</sup> *<sup>C</sup>*<sup>e</sup>

**Figure 3.** SSDNCI network.

be performed using nondimensional parameters,

, *<sup>ω</sup>*el <sup>=</sup> <sup>1</sup> 

capacitance reduces the damping ratio.

(1 + *δ*)*LC*<sup>p</sup>

*<sup>Z</sup>*cl <sup>=</sup> <sup>1</sup> 1 *Ls*<sup>2</sup>+*Rs* <sup>+</sup> <sup>1</sup>

**(SSDNCI)**

*<sup>δ</sup>* <sup>=</sup> *<sup>C</sup>*<sup>e</sup> *C*p

reads

$$
\begin{bmatrix} m & 0 \\ a\delta L & (1+\delta)L \end{bmatrix} \begin{bmatrix} \ddot{q} \\ \ddot{Q} \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ a\delta R & (1+\delta)R \end{bmatrix} \begin{bmatrix} \dot{q} \\ \dot{Q} \end{bmatrix} + \begin{bmatrix} c + \frac{a^2}{\mathsf{C}\_p} \ \frac{a}{\mathsf{C}\_p} \\ \frac{a}{\mathsf{C}\_p} \ \frac{1}{\mathsf{C}\_p} \end{bmatrix} \begin{bmatrix} q \\ Q \end{bmatrix} = \begin{bmatrix} F(t) \\ 0 \end{bmatrix}.\tag{14}$$

The standard SSDI network without negative capacitance is the special case with *δ* = 0.

In this study, a harmonic mechanical vibration *q* is assumed, and the influence of the shunted piezoceramics upon the vibration is neglected. This is fulfilled in good appromixation for systems where the piezoelectric coupling is not excessively large. Therefore, the second equation from (14) can be rewritten, summarizing all terms with *q* as excitation on the right side,

$$(1+\delta)L\ddot{Q} + (1+\delta)R\dot{Q} + \frac{1}{\mathcal{C}\_{\mathbb{P}}}Q = -(a\delta L\ddot{\eta} + a\delta R\dot{\eta} + \frac{a}{\mathcal{C}\_{\mathbb{P}}}\eta).\tag{15}$$

When the switch is open, the piezoceramics is connected to the negative capacitance, *<sup>Z</sup>*iso <sup>=</sup> <sup>1</sup> *C*e . Inserting this impedance into (3) yields

$$
\begin{bmatrix} m \ 0 \\ 0 \ 0 \end{bmatrix} \begin{bmatrix} \ddot{\eta} \\ \ddot{Q} \end{bmatrix} + \begin{bmatrix} c + \frac{\kappa^2}{\mathsf{C}\_p} & \frac{\kappa}{\mathsf{C}\_p} \\ \frac{\kappa}{\mathsf{C}\_p} & \frac{1}{\mathsf{C}\_p} + \frac{1}{\mathsf{C}\_v} \end{bmatrix} \begin{bmatrix} \eta \\ Q \end{bmatrix} = \begin{bmatrix} F(t) \\ 0 \end{bmatrix}. \tag{16}
$$

The charge *Q* is directly coupled with mechanical displacement *q*. During the switch open period, the charge *Q* changes according to

$$Q(t) = -\frac{\kappa}{\mathsf{C}\_{\mathsf{P}}} \frac{1}{\frac{1}{\mathsf{C}\_{\mathsf{p}}} + \frac{1}{\mathsf{C}\_{\mathsf{s}}}} q(t) + \mathsf{C} = -\kappa \frac{\delta}{1 + \delta} q(t) + \mathsf{C}\_{\mathsf{s}} \tag{17}$$

where *C* is the offset of charge signal, which still has to be determined. For a harmonically excited system, it can be calculated by assuming that the voltage signal is periodic with the same period time. It is therefore sufficient to consider one half period time of excitation only. Every half period consists of a period of open switch and of closed switch. Here the magnitude change of switch open circuit and switch closed circuit are defined respectively as Δ*Q*open and Δ*Q*close. The steady state is then characterized by

$$
\Delta Q\_{\text{open}} + \Delta Q\_{\text{close}} = 0. \tag{18}
$$

This equation basically means that the voltage is the same after each period time. In the case of harmonic excitation with an amplitude of *q*ˆ and excitation frequency of Ω, the mechanical displacement, velocity and acceleration can be expressed as

$$q(t) = \not\!\!\!/ \cos(\Omega t), \quad \not\!\!\/ = -\not\!\!\/\Omega \sin \Omega t, \quad \not\!\!\/= -\not\!\!\/\Omega^2 \sin \Omega t. \tag{19}$$

The change of charge Δ*Q*open is proportional to the change in displacement during switch open period. With the absolute charge value after inversion (which is the initial condition of the closed switch period) termed *Q*0, and the absolute value before inversion *Q*∗, we can write

$$
\Delta Q\_{\text{open}} = Q\_0 - Q^\* = -2a \frac{\delta}{1+\delta} \hat{\eta}\_{\prime}
$$

$$
Q\_0 = \frac{1}{2} \Delta Q\_{\text{open}} + \mathcal{C} = -a \frac{\delta}{1+\delta} \hat{\eta} + \mathcal{C}.\tag{20}
$$

Compared to the mechanical periodic time, the electric periodic time is normally very short. Additionally, the switching occurs at the times when the deformation *q* is maximized, which means that the velocity is zero. It is demonstrated in [14], that it is therefore feasible to neglect the change of mechanical signals during the time the switch is closed. Thus we can approximate the right side of (15) with the following terms: *q*(*t*) = *q*ˆcos(Ω*t*) ≈ *q*ˆ, *q*˙ = <sup>−</sup>*q*ˆΩsin(Ω*t*) <sup>≈</sup> 0, *<sup>q</sup>*¨ <sup>=</sup> <sup>−</sup>*q*ˆΩ2cosΩ*<sup>t</sup>* ≈ −*q*ˆΩ2. As a result, the right side of the differential equation becomes a constant,

$$(1+\delta)L\ddot{Q} + (1+\delta)R\dot{Q} + \frac{1}{\mathcal{C}\_{\text{P}}}Q = \mathfrak{a}\mathfrak{f}(\delta L\Omega^2 - \frac{1}{\mathcal{C}\_{\text{P}}}).\tag{21}$$

The solution of (21) is the superposition of the general solution and the particular solution. The particular solution can be obtained with the Duhamel integral. After some mathematical calculations, the value of charge at *τ*∗, which is the moment of opening the switch, is obtained as

$$Q(\mathbf{r}^\*) = -\mathbf{e}^{-\pi\tilde{\zeta}}Q\_0 - a\hat{q}(1 + \mathbf{e}^{-\pi\tilde{\zeta}}), \quad Q^\* = |Q(\mathbf{r}^\*)|. \tag{22}$$

The difference between *Q*∗ and *Q*<sup>0</sup> is the magnitude change of charge for closed switch Δ*Q*close. Combining all results, the stationary value of charge *Q*<sup>0</sup> and the constant component *C* are obtained as

$$Q\_0 = a\sharp(\frac{1+\mathbf{e}^{-\pi\tilde{\omega}}}{1-\mathbf{e}^{-\pi\tilde{\omega}}} - \frac{2\delta}{1+\delta}\frac{1}{1-\mathbf{e}^{-\pi\tilde{\omega}}}), \quad \mathbb{C} = \frac{1}{1+\delta}a\sharp\frac{1+\mathbf{e}^{-\pi\tilde{\omega}}}{1-\mathbf{e}^{-\pi\tilde{\omega}}}.\tag{23}$$

The results for *C* and *Q*<sup>0</sup> are the absolute values, their signs periodically change so that they are always in antiphase with the velocity *q*˙. Further on, this result can be approximated for low damping *ζ* � 1,

$$\mathcal{C} \approx \frac{1}{1+\delta} \mathfrak{a} \mathfrak{d} \frac{2}{\pi \zeta} = (1+\delta)^{-\frac{3}{2}} \mathfrak{a} \mathfrak{d} \frac{2}{\pi \zeta\_0}.\tag{24}$$

Equation (24) demonstrates that the stationary charge is increased for *δ* < 0, which means that only a negative capacitance increases the charge buildup. Especially when *δ* approaches −1, the constant *C* is theoretically infinity. The negative capacitance is an active analog circuit, so in practice the stationary charge cannot be infinitely high due to the limited maximal output of the operational amplifier. Additionally, the overall capacitance has to be positive in order to keep the electrical network stable. Therefore, the theoretical available range of the negative capacitance is the same as for the *LRC* shunt circuit

$$-\mathcal{C}\_{\rm P} < \mathcal{C}\_{\rm e} < 0 \quad \text{or} \quad -1 < \delta < 0. \tag{25}$$

The time signals of the SSDI and the SSDNCI with different capacitance ratios are given in Figure 4. For a clear illustration of the switching times *t*cl and *t*op, the inversion of charge does not occur instantaneously, as it is assumed in the calculations. Obviously, a larger negative capacitance increases the charge amplitudes as compared to the SSDI technique (*δ* = 0). Finally, the dissipated energy *E*diss per vibration period, which is a measure of the damping

8 Will-be-set-by-IN-TECH

Δ*Q*open + *C* = −*α*

Compared to the mechanical periodic time, the electric periodic time is normally very short. Additionally, the switching occurs at the times when the deformation *q* is maximized, which means that the velocity is zero. It is demonstrated in [14], that it is therefore feasible to neglect the change of mechanical signals during the time the switch is closed. Thus we can approximate the right side of (15) with the following terms: *q*(*t*) = *q*ˆcos(Ω*t*) ≈ *q*ˆ, *q*˙ = <sup>−</sup>*q*ˆΩsin(Ω*t*) <sup>≈</sup> 0, *<sup>q</sup>*¨ <sup>=</sup> <sup>−</sup>*q*ˆΩ2cosΩ*<sup>t</sup>* ≈ −*q*ˆΩ2. As a result, the right side of the differential

> 1 *C*p

The solution of (21) is the superposition of the general solution and the particular solution. The particular solution can be obtained with the Duhamel integral. After some mathematical calculations, the value of charge at *τ*∗, which is the moment of opening the switch, is obtained

The difference between *Q*∗ and *Q*<sup>0</sup> is the magnitude change of charge for closed switch Δ*Q*close. Combining all results, the stationary value of charge *Q*<sup>0</sup> and the constant component

1

The results for *C* and *Q*<sup>0</sup> are the absolute values, their signs periodically change so that they are always in antiphase with the velocity *q*˙. Further on, this result can be approximated for

Equation (24) demonstrates that the stationary charge is increased for *δ* < 0, which means that only a negative capacitance increases the charge buildup. Especially when *δ* approaches −1, the constant *C* is theoretically infinity. The negative capacitance is an active analog circuit, so in practice the stationary charge cannot be infinitely high due to the limited maximal output of the operational amplifier. Additionally, the overall capacitance has to be positive in order to keep the electrical network stable. Therefore, the theoretical available range of the negative

The time signals of the SSDI and the SSDNCI with different capacitance ratios are given in Figure 4. For a clear illustration of the switching times *t*cl and *t*op, the inversion of charge does not occur instantaneously, as it is assumed in the calculations. Obviously, a larger negative capacitance increases the charge amplitudes as compared to the SSDI technique (*δ* = 0).

*πζ* = (<sup>1</sup> <sup>+</sup> *<sup>δ</sup>*)<sup>−</sup> <sup>3</sup>

<sup>1</sup> <sup>−</sup> <sup>e</sup>−*πζ* ), *<sup>C</sup>* <sup>=</sup> <sup>1</sup>

1 + *δ*

*<sup>α</sup>q*<sup>ˆ</sup> <sup>2</sup>

*δ* 1 + *δ q*ˆ,

> *δ* 1 + *δ*

*<sup>Q</sup>* <sup>=</sup> *<sup>α</sup>q*ˆ(*δL*Ω<sup>2</sup> <sup>−</sup> <sup>1</sup>

*<sup>Q</sup>*(*τ*∗) = <sup>−</sup>e−*πζQ*<sup>0</sup> <sup>−</sup> *<sup>α</sup>q*ˆ(<sup>1</sup> <sup>+</sup> <sup>e</sup>−*πζ* ), *<sup>Q</sup>*<sup>∗</sup> <sup>=</sup> <sup>|</sup>*Q*(*τ*∗)|. (22)

1 + *δ αq*ˆ

<sup>2</sup> *<sup>α</sup>q*<sup>ˆ</sup> <sup>2</sup> *πζ*0

− *C*<sup>p</sup> < *C*<sup>e</sup> < 0 or − 1 < *δ* < 0. (25)

*C*p

1 + e−*πζ*

*q*ˆ + *C*. (20)

). (21)

<sup>1</sup> <sup>−</sup> <sup>e</sup>−*πζ* . (23)

. (24)

Δ*Q*open = *Q*<sup>0</sup> − *Q*<sup>∗</sup> = −2*α*

*<sup>Q</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> 2

(1 + *δ*)*LQ*¨ + (1 + *δ*)*RQ*˙ +

equation becomes a constant,

as

*C* are obtained as

low damping *ζ* � 1,

*Q*<sup>0</sup> = *αq*ˆ(

1 + e−*πζ* <sup>1</sup> <sup>−</sup> <sup>e</sup>−*πζ* <sup>−</sup> <sup>2</sup>*<sup>δ</sup>*

capacitance is the same as for the *LRC* shunt circuit

*<sup>C</sup>* <sup>≈</sup> <sup>1</sup> 1 + *δ*

**Figure 4.** Time signals of the electrical charge for different capacitance values *δ*.

performance, can be obtained by integrating the product of piezoelectric force and mechanical velocity over a mechanical period time *T*mech,

$$E\_{\rm diss} = -\alpha \int\_{t^\*}^{t^\* + T\_{\rm mach}} u\_\mathbf{P}(t) \dot{\eta}(t) \,\mathrm{d}t. \tag{26}$$

When the charge inversion occurs nearly instantaneously, it is sufficient to consider the time with open switch only. With above results the piezovoltage can be obtained as

$$
\mu\_{\rm P}(t) = \frac{\mathfrak{a}}{\mathfrak{C}\_{\rm P}} q(t) + \frac{\mathcal{Q}(t)}{\mathcal{C}\_{\rm P}} = \frac{\mathfrak{a}}{\mathcal{C}\_{\rm P}} \frac{1}{1 + \delta} q(t) + \frac{\mathcal{C}}{\mathcal{C}\_{\rm P}}.\tag{27}
$$

Inserting (27) into (26), the expression of dissipated energy is rewritten as

$$E\_{\rm diss} = -\frac{a}{\mathcal{C}\_{\rm p}} \int\_{t^\*}^{t^\* + T\_{\rm mach}} \left( a \frac{1}{1 + \delta} q(t) \dot{q}(t) + \mathcal{C} \dot{q} \right) \, \mathrm{d}t = -\frac{a}{\mathcal{C}\_{\rm p}} \int\_{t^\*}^{t^\* + T\_{\rm mach}} \mathcal{C} \dot{q} \, \mathrm{d}t. \tag{28}$$

As it is shown in (28), the amount of dissipated energy only depends on the charge offset *C*. Therefore the aim in the design of the nonlinear shunt network is to maximize the offset of the charge. Another way to illustrate the damping performance is the hystereisis cycle, in which the piezoelectric voltage or force is drawn versus the deformation. Periodic vibrations are characterized by closed loops, and the energy dissipation is proportional to the enclosed area. Fig. 5 depicts the hysteresis loops for the standard SSDI (*δ* = 0) and the SSDNCI with two different capacitance ratios. The voltage amplitude immediately before inversion is maximal, <sup>±</sup>*u*ˆp, and after inversion, <sup>∓</sup>*u*ˆpe−*πζ* . For the case of an instantaneous voltage inversion, the hysteresis cycles are parallelograms. The slope of these lines is proportional to the force factor

*α*. However, for the extension of the area, only the voltage amplitude, i.e. the charge offset, is relevant. Clearly, a negative capacitance has a positive effect in both states, therefore resulting in a higher charge offset. Inserting (24) into (28) we can get the expression of the dissipated

**Figure 5.** Hysteresis cycles for different capacitance values *δ*.

energy per period,

$$E\_{\rm diss} = 4 \frac{a^2}{C\_{\rm P}} \eta^2 \frac{1 + \mathbf{e}^{-\pi \zeta}}{1 - \mathbf{e}^{-\pi \zeta}} \frac{1}{1 + \delta}. \tag{29}$$

The increase in dissipated energy has the same trend as for the charge offset. Comparing with SSDI shunt, the dissipated energy is scaled by (1 + *δ*)<sup>−</sup> <sup>3</sup> <sup>2</sup> . For a linear *LRC* shunt, the dissipated energy is scaled by 1/(1 + *δ*), see also in [15].

#### **5. Optimized switching law for bimodal excitation**

The assuption of a harmonic excitation is not valid for all situations. In many cases, the signal also contains additional frequencies. In order to discuss the influence of more general excitations, in the following a bimodal excitation is considered, which contains two frequencies Ω<sup>1</sup> and Ω<sup>2</sup> with Ω<sup>2</sup> > Ω1,

$$q(t) = \not{q}\_1 \cos\left(\Omega\_1 t\right) + \not{q}\_2 \cos\left(\Omega\_2 t + \varphi\right). \tag{30}$$

Both signals have in general different amplitudes and a phase shift between them.

It is obvious that the standard switching law, which means switching at the maxima of the first mode, does not yield optimal results anymore. One can show that - using the standard switching law - the dissipated energy per vibration period is exactly the same as

**Figure 6.** Time signals for standard and enhanced SSDI with bimodal excitation.

for a monoharmonic excitation with frequency Ω<sup>1</sup> only,

10 Will-be-set-by-IN-TECH

*α*. However, for the extension of the area, only the voltage amplitude, i.e. the charge offset, is relevant. Clearly, a negative capacitance has a positive effect in both states, therefore resulting in a higher charge offset. Inserting (24) into (28) we can get the expression of the dissipated

> *δ* = −0.7 *δ* = −0.5 *δ* = 0

−*q*ˆ 0 *q*ˆ Displacement *q*

> <sup>2</sup> 1 + e−*πζ* <sup>1</sup> − <sup>e</sup>−*πζ*

The increase in dissipated energy has the same trend as for the charge offset. Comparing

The assuption of a harmonic excitation is not valid for all situations. In many cases, the signal also contains additional frequencies. In order to discuss the influence of more general excitations, in the following a bimodal excitation is considered, which contains two

It is obvious that the standard switching law, which means switching at the maxima of the first mode, does not yield optimal results anymore. One can show that - using the standard switching law - the dissipated energy per vibration period is exactly the same as

1 1 + *δ*

*q*(*t*) = *q*ˆ1 cos (Ω1*t*) + *q*ˆ2 cos (Ω2*t* + *ϕ*). (30)

. (29)

<sup>2</sup> . For a linear *LRC* shunt, the

0

−*u*ˆp

frequencies Ω<sup>1</sup> and Ω<sup>2</sup> with Ω<sup>2</sup> > Ω1,

**Figure 5.** Hysteresis cycles for different capacitance values *δ*.

with SSDI shunt, the dissipated energy is scaled by (1 + *δ*)<sup>−</sup> <sup>3</sup>

**5. Optimized switching law for bimodal excitation**

dissipated energy is scaled by 1/(1 + *δ*), see also in [15].

*<sup>E</sup>*diss <sup>=</sup> <sup>4</sup> *<sup>α</sup>*<sup>2</sup>

*C*p *q*ˆ

Both signals have in general different amplitudes and a phase shift between them.

<sup>−</sup>*u*ˆpe−*πζ*

energy per period,

*u*ˆp

*u*ˆpe−*πζ*

Voltage

$$E\_{\rm diss} = 4 \frac{a^2}{C\_{\rm p}} \rho\_1^2 \frac{1 + \mathbf{e}^{-\pi \zeta}}{1 - \mathbf{e}^{-\pi \zeta}} \,\mathrm{}\tag{31}$$

which is the result for the SSDNCI circuit with *δ* = 0. Therefore more sophisticated switching laws have been developed, which target to extract energy from the higher frequency oscillations and use it to increase the damping of the main mode [17].

The new switching law described in the following is defined according to these positions:


For such a switching law it is assured that the voltage induced by the second mode is added to the value caused by the first mode. Figure 6 shows a comparison of the standard and the enhanced switching law for a biharmonic excitation. The higher frequency is recognizable

**Figure 7.** Amplification of dissipated energy with enhanced SSDI technique versus amplitude ratio *r*<sup>A</sup> and frequency ratio *r*f.

in the high frequency oscillations during the open switch phases. One can realize that in the standard switching law the switching always occurs exactly during the first mode extrema. At these moments, the voltage at the piezoceramics might be increased or decreased by the influence of the higher frequency, so that in mean this effect cancels out. With the enhanced switching law, the switch is always triggered when the second mode is maximum and augments therefore the voltage buildup. However, the switching is no longer occuring in phase with the first mode velocity, which reduces slightly the energy dissipation. For more details the reader is referred to [12].

Obviously the increase in energy dissipation grows with the second mode amplitude. But also the frequency ratio *r*<sup>f</sup> = Ω2/Ω<sup>1</sup> between the first and second mode has an influence. The higher the second frequency, the smaller is the period time *T*<sup>2</sup> and therefore the timeframe. This means that the second mode maximum is in average closer to the first mode maximum, which is ideal for the energy dissipation. Figure 7 shows the amplification of energy dissipation versus the frequency ratio *r*<sup>f</sup> and the amplitude ratio *r*<sup>A</sup> = *q*ˆ2/*q*ˆ1. It can be concluded that for a given frequency ratio *r*<sup>f</sup> (this ratio is approximately 2*π* = 6.26 for the clamped beam), the energy dissipation grows linearly with the second mode amplitude. Additionally, the energy dissipation grows with a higher second frequency. Theoretically, for very low second mode amplitude, this enhanced switching law actually might give less damping than the standard law (the borderline is marked by a red line). This is due to the non-optimal phase shift of the switching signal, which is not in exact antiphase with the first mode velocity anymore. But these regions are practically not very relevant.
