Piezoelectric Shunt Stiffness in Rhombic Piezoelectric Stack Transducer with Hybrid Negative-Impedance Shunts: Theoretical Modeling and Stability Analysis

Negative-capacitance shunted piezoelectric polymer was investigated in depth due to its considerable damping effect. This paper discusses the novel controlled stiffness performance from a rhombic piezoelectric stack transducer with three hybrid negative-impedance shunts, namely, negative capacitance in series with resistance, negative capacitance in parallel with resistance, and negative inductance/negative capacitance (NINC) in series with resistance. An analytical framework for establishing the model of the coupled system is presented. Piezoelectric shunt stiffness (PSS) and piezoelectric shunt damping (PSD) are proposed to analyze the stiffness and damping performances of the hybrid shunts. Theoretical analysis proves that the PSS can produce both positive and negative stiffness by changing the negative capacitance and adjustable resistance. The Routh–Hurwitz criterion and the root locus method are utilized to judge the stability of the three hybrid shunts. The results point out that the negative capacitance should be selected carefully to sustain the stability and to achieve the negative stiffness effect of the transducer. Furthermore, negative capacitance in parallel with resistance has a considerably better stiffness bandwidth and damping performance than the other two shunts. This study demonstrates a novel electrically controlled stiffness method for vibration control engineering.


Introduction
Piezoelectric transducers are widely used for vibration control [1][2][3], energy harvesting [4][5][6][7][8], health monitoring [9,10], and sensing. Piezoelectric shunt damping involves the connection of an electrical impedance to terminals of a piezoelectric transducer (PZT), and it was widely studied since Forward carried out a preliminary experimental demonstration of the feasibility of using external electronic circuits to control mechanical vibrations in optical systems [11]. Hagood and von Flotow [12] studied a resistive shunt that is able to dissipate vibrational energy in the form of heat. A piezoelectric patch that consists of a single resonant circuit with an inductor can generate electrical resonance to reduce vibration [13][14][15]. The passive multimode resonant shunts, such as the Hollkamp shunt [16], the current-blocking shunt [17], the current-flowing shunt [18], and the series-parallel shunt [19], were investigated to control multimodal vibrations of host structures. These resonant shunts were applied to control the vibration of a compact disc read-only memory (CD-ROM) [20], a hard disk drive (HDD) disk-spindle system [21][22][23], a smart panel [24], a uniform bimorph beam (simulating chatter) [25], etc. The closed-form solution [26], H∞ [27], matrix inequalities [28], and equal modal damping [29] were employed to optimize the shunt parameters. The pure passive resistive shunt offers negative. The rhombic frame can magnify the output displacement of the piezoelectric transducer like an operation amplifier; the corresponding amplification coefficient can be found in Reference [54]. The terminals of the piezoelectric stack transducer connect to a hybrid NIC shunt, which consists of an adjustable impedance Za and an NIC. The NIC is constructed by an operational amplifier [1,35,55]. The input impedance of the circuit created by the passive impedances Z1, Z2, and Zs is 1 2 s Z ZZ Z =− [43]. If 12 ZZ = , the equivalent impedance of the operational amplifier will be s Z − .
When Zs is replaced with a capacitor, an inductor, or a resistor, we can obtain equivalent negative capacitance, negative inductance, or negative resistance. In the present study, we discuss the effect of negative capacitance on the stiffness performance of PSD. It should be noted that the rhombic frame is used to install the stack. In general, the stiffness of the rhombic frame is very big; thus, it should be carefully designed for vibration isolation application.
where L = T + W − U is the Lagrangian of the system, and Qi is the nonconservative force associated with the generalized coordinates. T is the kinetic energy of the piezoelectric transducer, (2) U is the restored deformation energy of the rhombic frame, where k is the stiffness of the rhombic frame. W is the energy of the piezoelectric stack transducer, 22 33 11 22 where 2 (1 ) pp C C k =− is the capacitance of the transducer under constant extension, and the electromechanical coupling factor of the transducer 2 p k is the efficiency of the conversion of mechanical energy into electricity, which ranges around 0.5 for PZT [1]. The stack includes n discs. The stiffness with short-circuited electrodes is Ka, and d33 is the piezoelectric constant. V is the voltage between the electrodes. Figure 2b is the equivalent model of hybrid shunts. The nonconservative where L = T + W − U is the Lagrangian of the system, and Q i is the nonconservative force associated with the generalized coordinates. T is the kinetic energy of the piezoelectric transducer, U is the restored deformation energy of the rhombic frame, where k is the stiffness of the rhombic frame. W is the energy of the piezoelectric stack transducer, where C p = C(1 − k 2 p ) is the capacitance of the transducer under constant extension, and the electromechanical coupling factor of the transducer k 2 p is the efficiency of the conversion of mechanical energy into electricity, which ranges around 0.5 for PZT [1]. The stack includes n discs. The stiffness with short-circuited electrodes is K a , and d 33 is the piezoelectric constant. V is the voltage between the  Figure 2b is the equivalent model of hybrid shunts. The nonconservative virtual work is related to the damping of the rhombic frame. The dissipated work through the shunt circuit and the external excitation force is as follows: Substituting Equations (2)-(5) into Equation (1), the governing equations of the coupled electromechanical system are as follows: where θ = nd 33 K a is the electromechanical coupling coefficient of the stack, and I s is the current flowing in the circuit. The structural damping coefficient c is represented by c = 2ζω n /m, and the structural damping ratio ζ can be selected between 0.5% and 1%. The natural frequency ω n is represented by (k + K a )/m. virtual work is related to the damping of the rhombic frame. The dissipated work through the shunt circuit and the external excitation force is as follows: Substituting Equations (2)-(5) into Equation (1), the governing equations of the coupled electromechanical system are as follows: where θ = nd33Ka is the electromechanical coupling coefficient of the stack, and Is is the current flowing in the circuit. The structural damping coefficient c is represented by c = 2ζωn/m, and the structural damping ratio ζ can be selected between 0.5% and 1%. The natural frequency ωn is represented by

Model for the Negative-Impedance Shunt
In the time domain, Equations (6) and (7) cannot reflect the stiffness effect of the hybrid shunt. In the Fourier domain, these two equations are as follows: If the shunt is represented by an equivalent impedance Z shown in Figure 2b, we have Is = V/Z. Then, Equation (9) becomes Substituting Equation (10) into Equation (8), the transfer function is as follows:  In the time domain, Equations (6) and (7) cannot reflect the stiffness effect of the hybrid shunt. In the Fourier domain, these two equations are as follows: If the shunt is represented by an equivalent impedance Z shown in Figure 2b, we have I s = V/Z. Then, Equation (9) becomes Substituting Equation (10) into Equation (8), the transfer function is as follows: It can be found that the shunt brings the damping into the piezoelectric stack transducer, which can possibly reduce structural vibration. If Equation (12) is written as then Equation (11) is reorganized as These three equations imply that the introducing of a shunt circuit brings both the damping and the mass effects into the transducer, where m s and c s are defined as the piezoelectric shunt mass (PSM) and piezoelectric shunt damping, respectively. The variation of the mass and stiffness influences the natural frequency of the transducer. Therefore, Equation (14) can also be reorganized as where k s is defined as the piezoelectric shunt stiffness. Equation (18) suggests that the PSS is associated with the excitation frequency, the adjustable impedance, and the capacitance of the shunt circuit. Figure 3 presents the positive and negative capacitance at the complex plane. The horizontal axis represents the resistance, and the vertical axis represents the capacitance. The controlled PSS will be different upon changing the shunt impedance in different quadrants. The negative capacitance −1/jcω can be rewritten as j/cω; it is somewhat like the positive inductance, but the frequency relationship is reciprocal. In previous studies [42,51], the negative capacitance was laid at the first quadrant. In this study, we move the impedance location of the shunt to the second quadrant, and discuss the stiffness and damping effects by changing the value of the negative capacitance.
It can be found that the shunt brings the damping into the piezoelectric stack transducer, which can possibly reduce structural vibration. If Equation (12) is written as then Equation (11) is reorganized as These three equations imply that the introducing of a shunt circuit brings both the damping and the mass effects into the transducer, where ms and cs are defined as the piezoelectric shunt mass (PSM) and piezoelectric shunt damping, respectively. The variation of the mass and stiffness influences the natural frequency of the transducer. Therefore, Equation (14) can also be reorganized as where ks is defined as the piezoelectric shunt stiffness. Equation (18) suggests that the PSS is associated with the excitation frequency, the adjustable impedance, and the capacitance of the shunt circuit. Figure 3 presents the positive and negative capacitance at the complex plane. The horizontal axis represents the resistance, and the vertical axis represents the capacitance. The controlled PSS will be different upon changing the shunt impedance in different quadrants. The negative capacitance −1/jcω can be rewritten as j/cω; it is somewhat like the positive inductance, but the frequency relationship is reciprocal. In previous studies [42,51], the negative capacitance was laid at the first quadrant. In this study, we move the impedance location of the shunt to the second quadrant, and discuss the stiffness and damping effects by changing the value of the negative capacitance.   A resistor R connected in series with a capacitor C s can increase the leakage of the negative resistance, as shown in Figure 4a. R is necessary and should be large enough due to the bias currents flowing from the non-ideal operational amplifier. The parallel resistor and capacitor act like a high-pass filter allowing bias current to flow to ground, thus preventing the capacitor from acquiring a direct current (DC) charge [35]. The equivalent impedance of the NIC circuit is where Assuming that Γ R = 1 and taking the adjustable resistor R s into consideration, the total impedance of this series shunt is If this hybrid shunt connects to the piezoelectric stack transducer, then Equation (15) becomes This equation shows the relationship between the PSS and PSD. The coefficients c s and k s are as follows:

Negative Capacitance in Series with Resistance
A resistor R connected in series with a capacitor Cs can increase the leakage of the negative resistance, as shown in Figure 4a. R is necessary and should be large enough due to the bias currents flowing from the non-ideal operational amplifier. The parallel resistor and capacitor act like a highpass filter allowing bias current to flow to ground, thus preventing the capacitor from acquiring a direct current (DC) charge [35]. The equivalent impedance of the NIC circuit is where ΓR = R1/R2. Assuming that ΓR = 1 and taking the adjustable resistor Rs into consideration, the total impedance of this series shunt is If this hybrid shunt connects to the piezoelectric stack transducer, then Equation (15) becomes This equation shows the relationship between the PSS and PSD. The coefficients cs and ks are as follows:    Figure 4b is the schematic of the negative capacitance in parallel with Rs, where the total impedance of the shunt is

Negative Capacitance in Parallel with Resistance
Substituting into Equation (21) Figure 4b is the schematic of the negative capacitance in parallel with R s , where the total impedance of the shunt is

Negative Capacitance in Parallel with Resistance
Substituting into Equation (21) and simplifying it, we can obtain c s and k s .

Negative Inductance and Negative Capacitance in Series with Resistance
If the equivalent impedance Z s in Figure 2a is replaced by an inductor L s and a capacitor C s in series, and Z 1 and Z 2 are resistors, the schematic is as presented in Figure 5. Assuming that R 1 is equal to R 2 , then the impedance of the NIC is Thus, Substituting Equation (28) into Equation (12), one can obtain

Analysis of the PSS for the Three Hybrid Shunts
We already obtained the PSS ks for the three hybrid shunts. The influence of PSD on PSS is quite important, which determines the design of the controlled stiffness transducer. If ks is divided by cs, c κ is defined as the stiffness and damping ratio of the PSD.
For the negative capacitance in series with the resistance shunt case, according to Equation Error! Reference source not found.,

Analysis of the PSS for the Three Hybrid Shunts
We already obtained the PSS k s for the three hybrid shunts. The influence of PSD on PSS is quite important, which determines the design of the controlled stiffness transducer. If k s is divided by c s , κ c is defined as the stiffness and damping ratio of the PSD. For the negative capacitance in series with the resistance shunt case, according to Equation (31), For the negative capacitance in parallel with the positive resistance shunt case, For the negative inductance and negative capacitance in series with resistance case, Table 1 lists the parameters of the piezoelectric stack transducer and the hybrid shunts that are obtained from the experiment. According to the theoretical model of the PSS and PSD obtained in Section 3, the stiffness performance of the hybrid shunts is discussed below. Table 1. Parameters of the piezoelectric stack and the hybrid shunts.

Parameters (Unit) Value
Piezoelectric charge coefficient, 10 Mass, m (kg) 0.1 Natural frequency of transducer, f n (Hz) 154.9 Figure 6 shows the variation of the natural frequency f n , k s , c s , and κ c with respect to the adjustable resistance R s for the negative capacitance in series with R s shunt. When C s is −0.6 µF, the changes of f n and k s are very small, which means that it is hard to generate the electrically controlled stiffness effect. The corresponding damping effect is also small. When the negative capacitance is −1 µF, f n and k s increase. When the negative capacitance increases to −1.4 µF, which means that the absolute value of negative capacitance approximates to the inherent capacitance of the piezoelectric stack C p , then f n and PSS begin to change in a very large range. The stiffness is a positive value that increases the natural frequency of the transducer. In this case, the damping effect changes with the change of R s , and it can easily find an optimal value. When the negative capacitance is further increased to −2 µF, which means C s is bigger than C p , f n also decreases apparently. In this case, the PSS produces the negative stiffness effect that decreases with the increase of R s . The corresponding damping effect is within an acceptable range. Figure 7 is the variation of the natural frequency f n , k s , c s , and κ c with respect to the adjustable resistance R s for the negative capacitance in parallel with resistance shunt. The negative capacitance for −0.6 µF, −1 µF, −1.4 µF, and −2 µF cases is discussed. When the negative capacitance is increased from −0.6 µF to −1.4 µF, k s is positive, increasing the natural frequency of the transducer, and the natural frequency also increases with the increase of the negative capacitance. The damping performance is excellent when the negative capacitance is −1.4 µF. When the negative capacitance is further increased to −2 µF, f n decreases dramatically. In this case, the PSS produces the negative stiffness effect that decreases with the increase of R s . and PSS begin to change in a very large range. The stiffness is a positive value that increases the natural frequency of the transducer. In this case, the damping effect changes with the change of Rs, and it can easily find an optimal value. When the negative capacitance is further increased to −2 μF, which means Cs is bigger than Cp, fn also decreases apparently. In this case, the PSS produces the negative stiffness effect that decreases with the increase of Rs. The corresponding damping effect is within an acceptable range.  Figure 7 is the variation of the natural frequency fn, ks, cs, and κc with respect to the adjustable resistance Rs for the negative capacitance in parallel with resistance shunt. The negative capacitance for −0.6 μF, −1 μF, −1.4 μF, and −2 μF cases is discussed. When the negative capacitance is increased from −0.6 μF to −1.4 μF, ks is positive, increasing the natural frequency of the transducer, and the natural frequency also increases with the increase of the negative capacitance. The damping performance is excellent when the negative capacitance is −1.4 μF. When the negative capacitance is further increased to −2 μF, fn decreases dramatically. In this case, the PSS produces the negative stiffness effect that decreases with the increase of Rs.  Figure 8 is the variation of the natural frequency, PSS, PSD, and κc with respect to the resistance Rs for the negative inductance/negative capacitance in series with resistance shunt. When the negative capacitance is −1 μF, the changes of fn and ks are small, and the cs is also small, making it hard to control the vibration of the system. When the negative capacitance increases to −1.4 μF, fn and ks begin to change in a very large range. ks is positive, increasing the natural frequency of the transducer. The PSD cs increases apparently and the optimal cs appears when Rs is 165.2 Ω. When the negative capacitance is further increased to −2 μF, fn also decreases like the other two kinds of shunts; ks is also a negative stiffness.  Figure 8 is the variation of the natural frequency, PSS, PSD, and κ c with respect to the resistance R s for the negative inductance/negative capacitance in series with resistance shunt. When the negative capacitance is −1 µF, the changes of f n and k s are small, and the c s is also small, making it hard to control the vibration of the system. When the negative capacitance increases to −1.4 µF, f n and k s begin to change in a very large range. k s is positive, increasing the natural frequency of the transducer. The PSD c s increases apparently and the optimal c s appears when R s is 165.2 Ω. When the negative capacitance is further increased to −2 µF, f n also decreases like the other two kinds of shunts; k s is also a negative stiffness. Figures 6-8 also imply that κ c is very big when c s is small. When c s increases, κ c tends to a small value. This demonstrates that κ c can be used for evaluating the damping effect of the PSD. If we combine f n and k s curves shown in Figures 6-8 together, it can be found that the negative capacitance in parallel with resistance case has a relative stable controlled natural frequency and better stiffness performance compared to the other two hybrid shunts. With this hybrid shunt, κ c curves are straight lines. When the absolute value of negative capacitance is bigger than C p , the controlled stiffness may be negative, which results in the decrease of the natural frequency of the transducer. Conversely, when the absolute value of negative capacitance is smaller than C p , the controlled stiffness is positive, which increases the natural frequency of the transducer. k s increases with the increase of the negative capacitance of the shunt. The natural frequency is determined mostly by the negative capacitance, and the PSS c s is determined by R s . However, when the absolute value of the negative capacitance approximates to C p , R s dramatically influences k s . Consequently, the negative capacitance and the adjustable resistance should be carefully selected to sustain considerable stiffness and damping performance. Figure 7. Controlled stiffness analysis for the negative capacitance in parallel with Rs shunt. Figure 8 is the variation of the natural frequency, PSS, PSD, and κc with respect to the resistance Rs for the negative inductance/negative capacitance in series with resistance shunt. When the negative capacitance is −1 μF, the changes of fn and ks are small, and the cs is also small, making it hard to control the vibration of the system. When the negative capacitance increases to −1.4 μF, fn and ks begin to change in a very large range. ks is positive, increasing the natural frequency of the transducer. The PSD cs increases apparently and the optimal cs appears when Rs is 165.2 Ω. When the negative capacitance is further increased to −2 μF, fn also decreases like the other two kinds of shunts; ks is also a negative stiffness.

Negative capacitance in series with resistance
Note that when s = jω, ω 2 n = (k + K a )/m, then the characteristic function of the piezoelectric stack transducer with the hybrid shunts can be obtained according to Equation (11), For the negative capacitance in series with resistance shunt, when R → ∞, then The inherent capacitance of the piezoelectric stack C p and the negative capacitance C s are all in the microfarad scale; thus, C s C p can be neglected to some extent, and the characteristic function of the closed-loop system is as follows: C p − C s s 2 + 2ςω n C p − C s − θ 2 R s C s /m s + θ 2 /m + ω 2 n C p − C s = 0. The Routh array is The necessary and sufficient condition for the stability of this system is that the first column of the Routh array in Equation (38) is positive.
When C p > C s , the following relationship is required to keep the stability of the control system: Then, one can get The abovementioned equation suggests that Thus, C s should be selected as When C p < C s , the capacitance of the circuit is negative; with the same process, the following condition should be met: Then, we have

Negative capacitance in parallel with positive resistance
For the negative capacitance in parallel with resistance case, when R → ∞, the characteristic function is as follows: Thus, the system should meet the following conditions: Then, we get

Negative inductance and negative capacitance in series with resistance
When Γ R = 1, the characteristic function can be written as When C p < C s , according to the Routh-Hurwitz criterion, we have the following criterion: Therefore, C s should meet the following condition: When C p > C s , with the same process, it can be found that

Root Locus Analysis
We already discussed the stability of the controlled stiffness system according to the Routh-Hurwitz criterion, where some assumptions and simplifications were made to obtain the final limitation expressions of C s . However, this cannot present the whole picture of the influence of shunt parameters. This section discusses the stability of the system with the root locus method.

Negative capacitance C s
This subsection analyzes the root locus of the piezoelectric stack transducer with respect to the negative capacitance C s for the three hybrid shunts. Firstly, the characteristic equation was written in form of the root locus form, allowing an easy simulation with MATLAB.

•
Negative capacitance in series with resistance shunt: 1 − C s mR s C p s 3 + m + R s C p c s 2 + c + kR s C p + θ 2 R s s + k mC p s 2 + cC p s + θ 2 + kC p = 0.
• Negative capacitance in parallel with resistance shunt: 1 + C p − C s mR s s 3 + cR s s 2 + kR s s ms 2 + (c + θ 2 R s )s + k = 0.
• Negative inductance/negative resistance in series with resistance shunt: 1 + C s L s C p ms 4 +(L s C p c−R s C p m)s 3 +(θ 2 L s +L s C p k−m−R s C p c)s 2 −(c+θ 2 R s +R s C p k)s−k mC p s 2 +cC p s+kC p +θ 2 = 0. (56) The root locus of the system with respect to C s was analyzed graphically to evaluate the stability of the system. Figures 9 and 10 present the root locus of the piezoelectric stack transducer with respect to C s for the negative capacitance in series with R s and in parallel with R s cases, respectively. It can be found that the system is stable when C s is within [0. 1,5] µF. The damping improves with the increase of R s . An optimal C s can be found on the root locus curve. The results also imply that the negative capacitance in parallel with R s case has a relatively better damping performance than the negative capacitance in series with R s case. The hybrid negative-capacitance shunts can enhance stability when C s is selected carefully.
The root locus of the system with respect to Cs was analyzed graphically to evaluate the stability of the system. Figures 9 and 10 present the root locus of the piezoelectric stack transducer with respect to Cs for the negative capacitance in series with Rs and in parallel with Rs cases, respectively. It can be found that the system is stable when Cs is within [0. 1,5] μF. The damping improves with the increase of Rs. An optimal Cs can be found on the root locus curve. The results also imply that the negative capacitance in parallel with Rs case has a relatively better damping performance than the negative capacitance in series with Rs case. The hybrid negative-capacitance shunts can enhance stability when Cs is selected carefully.   Figure 11 presents the root locus of the piezoelectric stack transducer with respect to Cs for the negative inductance/negative capacitance in series with resistance when Ls = 10 mH and Rs = 50 Ω. From Equation (56), it can be found that s → ∞ and Cs → ∞ leads the system to be unstable. When s → 0, we have Cs → ∞. Then, the root lies in the real axis. If Cs is used carefully, the system can also be kept stable. In this case, relatively considerable damping can be achieved.  Figure 11 presents the root locus of the piezoelectric stack transducer with respect to C s for the negative inductance/negative capacitance in series with resistance when L s = 10 mH and R s = 50 Ω. From Equation (56), it can be found that s → ∞ and C s → ∞ leads the system to be unstable. When s → 0, we have C s → ∞. Then, the root lies in the real axis. If C s is used carefully, the system can also be kept stable. In this case, relatively considerable damping can be achieved. Figure 10. Root locus of the piezoelectric stack transducer with respect to Cs for the negative capacitance in parallel with Rs. Figure 11 presents the root locus of the piezoelectric stack transducer with respect to Cs for the negative inductance/negative capacitance in series with resistance when Ls = 10 mH and Rs = 50 Ω. From Equation (56), it can be found that s → ∞ and Cs → ∞ leads the system to be unstable. When s → 0, we have Cs → ∞. Then, the root lies in the real axis. If Cs is used carefully, the system can also be kept stable. In this case, relatively considerable damping can be achieved. Figure 11. Root locus of the piezoelectric stack transducer with respect to C s for the negative inductance/negative resistance in series with resistance shunt when L s = 10 mH and R s = 50 Ω.

Adjustable Resistance R s
This subsection analyzes the root locus of the piezoelectric stack transducer with respect to the adjustable resistance R s for the three hybrid shunts. The characteristic equation is also written in the root locus form.

•
Negative capacitance in series with resistance R s : 1 − R s mC s C p s 3 + cC s C p s 2 + θ 2 C s + kC s C p s • Negative capacitance in parallel with resistance R s : • Negative inductance/negative resistance in series with resistance R s : 1 − R s (CpCsms 3 +C p C s cs 2 +(θ 2 +C p k)C s s) L s C p C s ms 4 +L s C p C s cs 3 +(θ 2 L s C s +L s C s C p k+mC p −C s m)s 2 +(C p −C s )cs+(Cp−Cs)k+θ 2 = 0.
Equations (57) and (58) demonstrate that s → ∞ results in R s → ∞. The root lies in the real axis. Figures 12 and 13 show the root locus of the system with respect to R s for negative capacitance in series with R s and in parallel with R s , respectively. The results prove the correctness of the theoretical model. In this case, some roots are positive, which makes the system unstable. In other ranges, the system can maintain stability with the change of C s (C s = 1 µF, 1.4 µF, and 2 µF). When C s = 1.4 µF, we get a considerable damping performance, and the corresponding optimal R s can also be found in Figures 12  and 13. Moreover, the parallel R s case has relatively good damping performance compared to the series R s case. Figure 14 is the root locus of the system with respect to the negative inductance/negative capacitance when C s = 1.4 µF and 2 µF. The result shows that the system is conditionally stable with the change of R s . One should carefully choose R s , C s , and L s .

Piezoelectric Shunt Stiffness
As shown in the theoretical analysis of the PSS and PSD effects in the negative-impedance shunted piezoelectric stack transducer, all three hybrid shunts can achieve the controlled stiffness performance. The frequency response of the system was determined in order to further discuss the influence of PSS and PSD on the vibration control performance. Figure 15 represents the frequency response of the piezoelectric stack transducer for the negative capacitance in series with Rs case. It can be found that the PSS is positive, which increases the natural frequency of the transducer when Cs = 1.4 μF. When Rs = 10 kΩ, the amplitude approximates to the uncontrolled condition. With the decrease of Rs, the amplitude decreases while the natural frequency increases. When Cs = 2 μF, this hybrid shunt can produce the negative stiffness effect, and the natural frequency of the system also decreases. In this case, the amplitude decreases with the increase of Rs. The corresponding optimal Rs can be found from Figure 12. The damping performance of PSD is shown in Table 2; it can be seen that PSD can achieve wonderful damping performance compared with the traditional pure resistive shunt method.

Piezoelectric Shunt Stiffness
As shown in the theoretical analysis of the PSS and PSD effects in the negative-impedance shunted piezoelectric stack transducer, all three hybrid shunts can achieve the controlled stiffness performance. The frequency response of the system was determined in order to further discuss the influence of PSS and PSD on the vibration control performance. Figure 15 represents the frequency response of the piezoelectric stack transducer for the negative capacitance in series with R s case. It can be found that the PSS is positive, which increases the natural frequency of the transducer when C s = 1.4 µF. When R s = 10 kΩ, the amplitude approximates to the uncontrolled condition. With the decrease of R s , the amplitude decreases while the natural frequency increases. When C s = 2 µF, this hybrid shunt can produce the negative stiffness effect, and the natural frequency of the system also decreases. In this case, the amplitude decreases with the increase of R s . The corresponding optimal R s can be found from Figure 12. The damping performance of PSD is shown in Table 2; it can be seen that PSD can achieve wonderful damping performance compared with the traditional pure resistive shunt method. increases. When Cs = 2 μF, this hybrid shunt can produce the negative stiffness effect, and the natural frequency of the system also decreases. In this case, the amplitude decreases with the increase of Rs. The corresponding optimal Rs can be found from Figure 12. The damping performance of PSD is shown in Table 2; it can be seen that PSD can achieve wonderful damping performance compared with the traditional pure resistive shunt method.  As suggested in Figure 7, the PSS is sensitive to R s when C s = 1.4 µF. C s = 1 µF is a better choice. Figure 16 shows the corresponding frequency response of the piezoelectric stack transducer in parallel with R s with the change of C s and R s . When C s = 1 µF, the natural frequency increases, which means PSS is positive for C p > C s , and the amplitude decreases with the decrease of R s . When C s = 2 µF, the natural frequency decreases, which indicates that PSS is negative for C p < C s , and the amplitude decreases with the increase of R s . The amplitude of the transducer can be controlled by the change of R s .  As suggested in Figure 7, the PSS is sensitive to Rs when Cs = 1.4 μF. Cs = 1 μF is a better choice. Figure 16 shows the corresponding frequency response of the piezoelectric stack transducer in parallel with Rs with the change of Cs and Rs. When Cs = 1 μF, the natural frequency increases, which means PSS is positive for Cp > Cs, and the amplitude decreases with the decrease of Rs. When Cs = 2 μF, the natural frequency decreases, which indicates that PSS is negative for Cp < Cs, and the amplitude decreases with the increase of Rs. The amplitude of the transducer can be controlled by the change of Rs.  Figure 17 represents the frequency response of the piezoelectric stack transducer with negative inductance/negative capacitance in series with Rs when Ls = 10 mH. When Cs = 1 μF, the natural frequency increases, which means the PSS is positive. The amplitude decreases with the decrease of Rs. While Cs = 2 μF, the natural frequency decreases, which means the PSS is negative, and the amplitude decreases with the increase of Rs. |x/F| Figure 16. Frequency response of the piezoelectric stack transducer with negative capacitance in parallel with R s . Figure 17 represents the frequency response of the piezoelectric stack transducer with negative inductance/negative capacitance in series with R s when L s = 10 mH. When C s = 1 µF, the natural frequency increases, which means the PSS is positive. The amplitude decreases with the decrease of R s . While C s = 2 µF, the natural frequency decreases, which means the PSS is negative, and the amplitude decreases with the increase of R s .  This implies that Rs can be carefully selected to increase the damping of the system without changing the stiffness of the system, which is important in some special applications. Figures 15-17 also indicate that the bandwidth performance of the negative capacitance in parallel with Rs shunt is better than the other two cases, which can provide considerable controlled stiffness performance. If this transducer is used as an isolator, negative stiffness is a better choice. If one just wants to avoid the resonance of the system, both positive and negative stiffness are acceptable. The previous study by Heuss et al. [52] utilized different combinations of resistant, resonant, and negative capacitance to achieve the tuning of a vibration absorber. The tuning frequency band can be 120 Hz. We can also achieve this performance if negative capacitance and adjusting resistance are carefully designed. Figure 18 shows the time history response of the transducer under sweep sine excitation when Cs is 1.4 μF. When Rs is 1 kΩ, only the response near the resonance is controlled. When Rs increases to 10 kΩ, the natural frequency increases. The response decreases dramatically near the resonance. Furthermore, the low-frequency vibration is also suppressed, and the bandwidth can reach up to 150 Hz. In view of vibration isolation, low-natural-frequency isolators can achieve bandwidth isolation performance when the excitation frequency is bigger than 2 n ω , such as nonlinear vibration isolators [56][57][58], quasi-zero isolators [59,60], etc. These nonlinear vibration isolators can achieve broadband vibration isolation performance with negative dynamic stiffness of nonlinear isolators; however, the vibration suppression in the resonance region is dependent on damping. The proposed PSS can semi-actively decrease the stiffness of linear isolators to improve the vibration performance; therefore, it has application potential in isolation engineering. Furthermore, the PSS can also increase the stiffness of isolators to enhance the vibration suppression performance in the resonance region. This implies that R s can be carefully selected to increase the damping of the system without changing the stiffness of the system, which is important in some special applications. Figures 15-17 also indicate that the bandwidth performance of the negative capacitance in parallel with R s shunt is better than the other two cases, which can provide considerable controlled stiffness performance. If this transducer is used as an isolator, negative stiffness is a better choice. If one just wants to avoid the resonance of the system, both positive and negative stiffness are acceptable. The previous study by Heuss et al. [52] utilized different combinations of resistant, resonant, and negative capacitance to achieve the tuning of a vibration absorber. The tuning frequency band can be 120 Hz. We can also achieve this performance if negative capacitance and adjusting resistance are carefully designed. Figure 18 shows the time history response of the transducer under sweep sine excitation when Cs is 1.4 µF. When R s is 1 kΩ, only the response near the resonance is controlled. When R s increases to 10 kΩ, the natural frequency increases. The response decreases dramatically near the resonance. Furthermore, the low-frequency vibration is also suppressed, and the bandwidth can reach up to 150 Hz. In view of vibration isolation, low-natural-frequency isolators can achieve bandwidth isolation performance when the excitation frequency is bigger than √ 2ω n , such as nonlinear vibration isolators [56][57][58], quasi-zero isolators [59,60], etc. These nonlinear vibration isolators can achieve broadband vibration isolation performance with negative dynamic stiffness of nonlinear isolators; however, the vibration suppression in the resonance region is dependent on damping. The proposed PSS can semi-actively decrease the stiffness of linear isolators to improve the vibration performance; therefore, it has application potential in isolation engineering. Furthermore, the PSS can also increase the stiffness of isolators to enhance the vibration suppression performance in the resonance region.

Conclusions
In this study, we proposed the novel controlled stiffness performance of a rhombic piezoelectric stack transducer with hybrid negative-impedance shunts. The governing equation of the transducer was established according to Lagrange's equation. Piezoelectric shunt stiffness and piezoelectric shunt damping were defined to analyze the stiffness and damping effects of transducer with three kinds of hybrid shunts. The Routh-Hurwitz criterion was employed to get the theoretical selection of negative capacitance. The root locus method was utilized to graphically judge the stability of the proposed three kinds of hybrid shunts. The results demonstrate that the piezoelectric stack transducer can produce both the stiffness and damping effects with hybrid shunts. With the change of negative capacitance, both negative and positive stiffness can also be obtained. Moreover, the negative stiffness effect requires a careful choice of the negative capacitance to sustain the stability of the system. Furthermore, negative capacitance in parallel with resistance demonstrated a considerably better stiffness bandwidth and damping performance than the other two shunts. The proposed PSS can be used to decrease the stiffness to decrease the natural frequency and, thus, to increase the vibration isolation band of linear or nonlinear isolators. Additionally, the PSS can be also used to adjust the stiffness to avoid resonance when the host structure is subjected to harmonic excitations. Future research may focus on experimental investigations of the PSS.

Conflicts of Interest:
The authors declare no conflicts of interest.