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# International Journal of Turbo & Jet-Engines

Ed. by Sherbaum, Valery / Erenburg, Vladimir

IMPACT FACTOR 2018: 0.863

CiteScore 2018: 0.66

SCImago Journal Rank (SJR) 2018: 0.211
Source Normalized Impact per Paper (SNIP) 2018: 0.625

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2191-0332
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Volume 35, Issue 1

# Theoretical Study of the Vibration Suppression on a Mistuned Bladed Disk Using a Bi-periodic Piezoelectric Network

Lin Li
• Collaborative Innovation Center for Advanced Aero-Engine, School of Energy and Power Engineering, Beihang University, 405Group, No.37, Xueyuan Road, Haidian District, Beijing 100191, PR China
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• Other articles by this author:
/ Pengcheng Deng
• Corresponding author
• Collaborative Innovation Center for Advanced Aero-Engine, School of Energy and Power Engineering, Beihang University, 405Group, No.37, Xueyuan Road, Haidian District, Beijing 100191, PR China
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• Other articles by this author:
/ Jiuzhou Liu
/ Chao Li
Published Online: 2018-02-15 | DOI: https://doi.org/10.1515/tjj-2016-0028

## Abstract

The paper deals with the vibration suppression of a bladed disk with a piezoelectric network. The piezoelectric network has a different period (so called bi-period) from that of the bladed disk and there is no inductor in it. The system is simulated by an electromechanical lumped parameter model with two DOFs per sector. The research focuses on suppressing the amplitude magnification or reducing the vibration localization of the mistuned bladed disk. The dynamic equations of the system are derived. Both mechanical mistuning and electrical mistuning have been taken into account. The Modified Modal Assurance Criterion (MMAC) is used to evaluate the vibration suppression ability of the bi-periodic piezoelectric network. The Monte Carlo simulation is used to calculate the MMAC of the system with the random mistuning. As a reference, the forced responses of the bladed disk with and without the piezoelectric network are given. The results show that the piezoelectric network would effectively suppress amplitude magnification induced by mistuning. The vibration amplitude is even smaller than that of the tuned system. The robustness analysis shows that the bi-periodic piezoelectric network can provide a reliable assurance for avoiding the forced response amplification of the mistuned bladed disk. The amplified response induced by the mechanical mistuning with standard deviation 0.2 can be effectively suppressed through the bi-periodic piezoelectric network.

## 1 Introduction

Based on above researches, this paper focuses on suppressing the amplified vibration of the mistuned bladed disk through the piezoelectric network without inductors. The piezoelectric patches are distributed in circumference and connected by the circuits, which forms a bi-periodic and electromechanical system. Here, the term ‘bi-period’ means that the period of the piezoelectric network may be different from that of the bladed disk. Obviously, for this kind of system, the connection type of the network plays an important role to the dynamic behavior of the coupling system. In fact, the bi-periodic design could be considered as a certain kind of intentional mistuning. The objective of this research lies in the acquisition of the theory about the periodic piezoelectric network, with which the forced vibration of the mistuned bladed disk could not be amplified, even smaller than that of the tuned counterpart. As well as the robustness of the system to mistuning is improved. The paper is organized as follows. The dynamic equations based on a lumped parameter model with a bi-periodic piezoelectric network are given (Section 2). The Modified Modal Assurance Criterion (MMAC) is used to evaluate the ability of the network to suppress vibration (Section 3). Then the effect of the mistuned piezoelectric network is analysed (Section 4). Finally, the robustness of the electromechanical system is examined (Section 5).

## 2.1 The lumped parameter model

The lumped parameter model shown in Figure 1 is used to represent a bladed disk having N sectors. In the model, ${m}_{b}$ and ${m}_{d}$ represent the blade mass and the disk mass of a sector respectively, ${k}_{b}$ and ${k}_{d}$ are the blade stiffness and the disk stiffness of a sector respectively, ${k}_{c}$ represents the mechanical coupling stiffness between two adjacent sectors. The mechanical damping c between the blade and the disk is considered as a viscous damping. Every n sectors a piezoelectric patch with a shunt circuit is set on the connection between the blade and the disk. In addition, the piezoelectric shunt circuit of each sector is connected in parallel or in series to form a network (as shown in Figure 2). The original mechanical system with N periods becomes a bi-periodic and electromechanical system. The bi-period number is p = N/n, p is an integer. No inductors are considered in the piezoelectric network for a reason: inductors that could significantly affect the mechanical performance of the system are too large to be made in practice.

Figure 1:

Lumped parameter model of bladed disk structure.

Figure 2:

The piezoelectric network in parallel and in series.

## 2.2 No electric circuit connection among sectors

In the model shown in Figure 1, there are two kinds of sectors, with or without the piezoelectric shunt circuit composed of the resistor R and the capacitor ${C}_{e}$ (as shown in Figures 3 and 4). Each sector is connected with adjacent sector through a mechanical coupling stiffness ${k}_{c}$ (as shown in Figure 3). The displacements of the blade and the disk in the sector are represented by ${\mathbf{x}}_{b}$ and ${\mathbf{x}}_{d}$ respectively. The dynamic equations of the sector without the piezoelectric shunt circuit are just as the equation of the ordinary bladed disk. For the sectors with the piezoelectric shunt circuit, the dynamic equations have been given by many related studies [32, 38, 39, 40, 41]. So the dynamic equations based on the model in Figure 1 can be written directly as follows: $\left\{\begin{array}{c}{m}_{b}{\stackrel{¨}{\mathbf{x}}}_{b\mathrm{i}}+c\left({\stackrel{˙}{\mathbf{x}}}_{bi}-{\stackrel{˙}{\mathbf{x}}}_{di}\right)+\left({k}_{b}+{k}_{pzt}\right){\mathbf{x}}_{bi}-\left({k}_{b}+{k}_{pzt}\right){\mathbf{x}}_{di}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\mathrm{\eta }{\mathbf{D}}_{\mathrm{i}}={\mathbf{F}}_{bi}\left(t\right)\\ {m}_{d}{\stackrel{¨}{\mathbf{x}}}_{di}+c\left({\stackrel{˙}{\mathbf{x}}}_{di}-{\stackrel{˙}{\mathbf{x}}}_{bi}\right)-\left({k}_{b}+{k}_{pzt}\right){\mathbf{x}}_{bi}-{k}_{c}{\mathbf{x}}_{d\left(i-1\right)}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\left({k}_{d}+{k}_{b}+2{k}_{c}+{k}_{pzt}\right){\mathbf{x}}_{di}-{k}_{c}{\mathbf{x}}_{d\left(i+1\right)}+\mathrm{\eta }{\mathbf{D}}_{\mathrm{i}}={\mathbf{F}}_{di}\left(t\right)\\ R{\stackrel{˙}{\mathbf{D}}}_{\mathrm{i}}+\left({C}_{e}^{-1}+{C}_{pzt}^{-1}\right){\mathbf{D}}_{\mathrm{i}}-\mathrm{\eta }{{\mathbf{x}}_{b}}_{\mathrm{i}}+\mathrm{\eta }{{\mathbf{x}}_{d}}_{\mathrm{i}}={\mathbf{U}}_{\mathrm{i}}\left(t\right)\\ \phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}i=1,n+1,2n+1,\cdots \\ {m}_{b}{\stackrel{¨}{\mathbf{x}}}_{bi}+c\left({\stackrel{˙}{\mathbf{x}}}_{bi}-{\stackrel{˙}{\mathbf{x}}}_{di}\right)+{k}_{b}\left({\mathbf{x}}_{bi}-{\mathbf{x}}_{di}\right)={\mathbf{F}}_{bi}\left(t\right)\\ {m}_{d}{\stackrel{¨}{\mathbf{x}}}_{di}+c\left({\stackrel{˙}{\mathbf{x}}}_{di}-{\stackrel{˙}{\mathbf{x}}}_{bi}\right)-{k}_{b}{\mathbf{x}}_{bi}-{k}_{c}{\mathbf{x}}_{d\left(i-1\right)}+\left({k}_{d}+{k}_{b}+2{k}_{c}\right){\mathbf{x}}_{di}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-{k}_{c}{\mathbf{x}}_{d\left(i+1\right)}={\mathbf{F}}_{di}\left(t\right)\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}i\ne 1,n+1,2n+1,\cdots \end{array}$(1)in which a dot above a letter means differentiation with respect to time t, D is the electric charge, ${k}_{pzt}$ is the short-circuit stiffness of the piezoelectric patch, ${C}_{pzt}$ is the inherent capacitance of the piezoelectric patch, F is the external exciting force, in general, it is harmonic in space corresponding to an engine order excitation, U is the voltage boundary.

Figure 3:

Two kinds of sector modes.

Figure 4:

Shunt circuit in piezoelectric networks.

## 2.3 Piezoelectric shunt circuits of sectors connected with each other

If p (p = N/n) piezoelectric shunt circuits are connected with each other in parallel (as shown in Figure 2(a)), then the 3rd equation in eq. (1) becomes: $\begin{array}{rl}R\sum _{k=1}^{\left(p-1\right)n+1}{\stackrel{˙}{\mathbf{D}}}_{k}& +\left({C}_{e}^{-1}+{C}_{pzt}^{-1}\right)\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{D}}_{\mathrm{k}}-\mathrm{\eta }\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{x}}_{bk}\\ & +\mathrm{\eta }\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{x}}_{dk}=\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{U}}_{k}\left(t\right)\end{array}$(2)

According to KCL (Kirchhoff’s Current Law), the current of each sector has the following relationship: $\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{I}}_{k}\left(t\right)=0$(3)

Meanwhile, according to KVL (Kirchhoff’s Voltage Law), each sector has the same voltage, that is: ${\mathbf{U}}_{\mathbf{k}}\left(t\right)=\mathbf{U}\left(t\right)\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\mathrm{k}=1,n+1,\cdots ,\left(p-1\right)n+1$(4)

Besides, the differential relationship between the current and electric charge is written as: ${\mathbf{I}}_{k}\left(t\right)=\frac{d{\mathbf{D}}_{k}}{dt}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}k=1,n+1,\cdots ,\left(p-1\right)n+1$(5)

Integrating eq. (5) upon time, according to eq. (3), there is the equation: $\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{D}}_{k}\left(t\right)=0\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}$(6)

Substituting eqs (4) and (6) into eq. (2), the following relation is obtained: $-\mathrm{\eta }\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{x}}_{bk}+\mathrm{\eta }\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{x}}_{dk}=\mathbf{U}\left(t\right)\phantom{\rule{1pt}{0ex}}$(7)

Therefore, eq. (1) can be simplified by eliminating ${\mathbf{U}}_{\mathrm{i}}$ based on eq. (7).

Then the dynamic equations of the model with the piezoelectric network in parallel (as shown in Figures 1 and 2(a)) are written as: $\left\{\begin{array}{c}{m}_{b}{\stackrel{¨}{\mathbf{x}}}_{b\mathrm{i}}+c\left({\stackrel{˙}{\mathbf{x}}}_{b\mathrm{i}}-{\stackrel{˙}{\mathbf{x}}}_{d\mathrm{i}}\right)+\left({k}_{b}+{k}_{pzt}\right){\mathbf{x}}_{b\mathrm{i}}-\left({k}_{b}+{k}_{pzt}\right){\mathbf{x}}_{d\mathrm{i}}-\mathrm{\eta }{\mathbf{D}}_{\mathrm{i}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}={\mathbf{F}}_{b\mathrm{i}}\left(t\right)\\ {m}_{d}{\stackrel{¨}{\mathbf{x}}}_{d\mathrm{i}}+c\left({\stackrel{˙}{\mathbf{x}}}_{d\mathrm{i}}-{\stackrel{˙}{\mathbf{x}}}_{b\mathrm{i}}\right)-\left({k}_{b}+{k}_{pzt}\right){\mathbf{x}}_{b\mathrm{i}}-{k}_{c}{\mathbf{x}}_{d\left(\mathrm{i}-1\right)}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\left({k}_{d}+{k}_{b}+2{k}_{c}+{k}_{pzt}\right){\mathbf{x}}_{d\mathrm{i}}-{k}_{c}{\mathbf{x}}_{d\left(\mathrm{i}+1\right)}+\mathrm{\eta }{\mathbf{D}}_{\mathrm{i}}={\mathbf{F}}_{di}\left(t\right)\\ R{\stackrel{˙}{\mathbf{D}}}_{\mathrm{i}}+\left({C}_{e}^{-1}+{C}_{pzt}^{-1}\right){\mathbf{D}}_{\mathrm{i}}-\mathrm{\eta }{{\mathbf{x}}_{b}}_{\mathrm{i}}+\mathrm{\eta }{{\mathbf{x}}_{d}}_{\mathrm{i}}+\mathrm{\eta }\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{x}}_{bk}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\mathrm{\eta }\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{x}}_{dk}=0\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\mathrm{i}=1,n+1,2n+1,\cdots \\ {m}_{b}{\stackrel{¨}{\mathbf{x}}}_{bi}+c\left({\stackrel{˙}{\mathbf{x}}}_{bi}-{\stackrel{˙}{\mathbf{x}}}_{di}\right)+{k}_{b}\left({\mathbf{x}}_{bi}-{\mathbf{x}}_{di}\right)={\mathbf{F}}_{bi}\left(t\right)\\ {m}_{d}{\stackrel{¨}{\mathbf{x}}}_{di}+c\left({\stackrel{˙}{\mathbf{x}}}_{di}-{\stackrel{˙}{\mathbf{x}}}_{bi}\right)-{k}_{b}{\mathbf{x}}_{bi}-{k}_{c}{\mathbf{x}}_{d\left(i-1\right)}+\left({k}_{d}+{k}_{b}+2{k}_{c}\right){\mathbf{x}}_{di}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-{k}_{c}{\mathbf{x}}_{d\left(i+1\right)}={\mathbf{F}}_{di}\left(t\right)\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\mathrm{i}\ne 1,n+1,2n+1,\cdots \end{array}$(8)

Likewise, the dynamic equations of the model with the piezoelectric network in series (as shown in Figures 1 and 2(b)) can be obtained: $\left\{\begin{array}{c}{m}_{b}{\stackrel{¨}{\mathbf{x}}}_{bi}+c\left({\stackrel{˙}{\mathbf{x}}}_{b\mathrm{i}}-{\stackrel{˙}{\mathbf{x}}}_{d\mathrm{i}}\right)+\left({k}_{b}+{k}_{pzt}\right){\mathbf{x}}_{bi}-\left({k}_{b}+{k}_{pzt}\right){\mathbf{x}}_{d\mathrm{i}}-\mathrm{\eta }{\mathbf{D}}_{\mathrm{i}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}={\mathbf{F}}_{b\mathrm{i}}\left(t\right)\\ {m}_{d}{\stackrel{¨}{\mathbf{x}}}_{d\mathrm{i}}+c\left({\stackrel{˙}{\mathbf{x}}}_{d\mathrm{i}}-{\stackrel{˙}{\mathbf{x}}}_{b\mathrm{i}}\right)-\left({k}_{b}+{k}_{pzt}\right){\mathbf{x}}_{b\mathrm{i}}-{k}_{c}{\mathbf{x}}_{d\left(i-1\right)}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\left({k}_{d}+{k}_{b}+2{k}_{c}+{k}_{pzt}\right){\mathbf{x}}_{d\mathrm{i}}-{k}_{c}{\mathbf{x}}_{d\left(i+1\right)}+\mathrm{\eta }{\mathbf{D}}_{\mathrm{i}}={\mathbf{F}}_{d\mathrm{i}}\left(t\right)\\ R\stackrel{˙}{\mathbf{D}}+\left({C}_{e}^{-1}+{C}_{pzt}^{-1}\right)\mathbf{D}-\mathrm{\eta }\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{x}}_{b\mathrm{k}}+\mathrm{\eta }\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{x}}_{d\mathrm{k}}=0\\ \phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\mathrm{i}=1,n+1,2n+1,\cdots \\ {m}_{b}{\stackrel{¨}{\mathbf{x}}}_{b\mathrm{i}}+c\left({\stackrel{˙}{\mathbf{x}}}_{b\mathrm{i}}-{\stackrel{˙}{\mathbf{x}}}_{d\mathrm{i}}\right)+{k}_{b}\left({\mathbf{x}}_{b\mathrm{i}}-{\mathbf{x}}_{d\mathrm{i}}\right)={\mathbf{F}}_{b\mathrm{i}}\left(t\right)\\ {m}_{d}{\stackrel{¨}{\mathbf{x}}}_{d\mathrm{i}}+c\left({\stackrel{˙}{\mathbf{x}}}_{d\mathrm{i}}-{\stackrel{˙}{\mathbf{x}}}_{b\mathrm{i}}\right)-{k}_{b}{\mathbf{x}}_{b\mathrm{i}}-{k}_{c}{\mathbf{x}}_{d\left(i-1\right)}+\left({k}_{d}+{k}_{b}+2{k}_{c}\right){\mathbf{x}}_{d\mathrm{i}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-{k}_{c}{\mathbf{x}}_{d\left(i+1\right)}={\mathbf{F}}_{d\mathrm{i}}\left(t\right)\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}i\ne 1,n+1,2n+1,\cdots \end{array}$(9)

## 2.4 Normalizing process

It is convenient to use the non-dimensional variables for a general study. The non-dimensional equations can be obtained by introducing the following non-dimensional parameters into the corresponding equations: $\left\{\begin{array}{c}\mathrm{\mu }={m}_{d}/{m}_{b}\\ \mathrm{\tau }=\mathrm{\omega }t\\ {\mathrm{\omega }}_{n}^{2}={k}_{b}/\phantom{{k}_{b}{m}_{b}}{m}_{b}\\ \mathrm{\lambda }=\mathrm{\omega }/\phantom{\mathrm{\omega }{\mathrm{\omega }}_{n}}{\mathrm{\omega }}_{n}\\ {\mathrm{\gamma }}_{d}={k}_{d}/{k}_{b}\\ {\mathrm{\gamma }}_{c}={k}_{c}/{k}_{b}\\ {\mathrm{\gamma }}_{pzt}={k}_{pzt}/{k}_{b}\\ {\mathrm{\gamma }}_{pc}={\mathrm{\eta }}^{2}{C}_{pzt}/\phantom{{\mathrm{\eta }}^{2}{C}_{pzt}{k}_{b}}{k}_{b}\\ {\mathrm{\gamma }}_{e}={\mathrm{\eta }}^{2}{C}_{e}/\phantom{{\mathrm{\eta }}^{2}{C}_{e}{k}_{b}}{k}_{b}\end{array}\begin{array}{c}\begin{array}{c},\end{array}\end{array}\left\{\begin{array}{c}\mathrm{\xi }=c/\phantom{c\sqrt{{k}_{b}{m}_{b}}}\sqrt{{k}_{b}{m}_{b}}/\phantom{c/\phantom{c\sqrt{{k}_{b}{m}_{b}}}\sqrt{{k}_{b}{m}_{b}}2}2\\ \mathrm{\epsilon }={\mathrm{\eta }}^{2}/\phantom{{\mathrm{\eta }}^{2}\left({\mathrm{\omega }}_{n}^{2}Rc\right)}\left({\mathrm{\omega }}_{n}^{2}Rc\right)\\ \mathrm{\delta }={\mathbf{F}}_{b1}/\phantom{{\mathbf{F}}_{b1}{k}_{b}}{k}_{b}\\ {\mathbf{y}}_{b\mathrm{i}}={\mathbf{x}}_{b\mathrm{i}}/\phantom{{\mathbf{x}}_{b\mathrm{i}}\mathrm{\delta }}\mathrm{\delta }\\ {\mathbf{y}}_{d\mathrm{i}}={\mathbf{x}}_{d\mathrm{i}}/\phantom{{\mathbf{x}}_{d\mathrm{i}}\mathrm{\delta }}\mathrm{\delta }\\ {\mathbf{q}}_{\mathrm{i}}=\mathrm{\eta }{\mathrm{D}}_{\mathrm{i}}/\phantom{\mathrm{\eta }{\mathbf{D}}_{\mathrm{i}}{\mathbf{F}}_{b1}}{\mathbf{F}}_{b1}\\ {\mathbf{P}}_{b\mathrm{i}}\left(t\right)={\mathbf{F}}_{b\mathrm{i}}\left(t\right)/\phantom{{\mathbf{F}}_{b\mathrm{i}}\left(t\right){\mathbf{F}}_{b1}}{\mathbf{F}}_{b1}\\ {\mathbf{P}}_{d\mathrm{i}}\left(t\right)={\mathbf{F}}_{d\mathrm{i}}\left(t\right)/\phantom{{\mathbf{F}}_{d\mathrm{i}}\left(t\right){\mathbf{F}}_{b1}}{\mathbf{F}}_{b1}\end{array}$(10)

Introducing eq. (10) into eqs (8) and (9), the latter can be written in non-dimensional forms respectively as follows: $\left\{\begin{array}{c}{\mathrm{\lambda }}^{2}{\stackrel{¨}{\mathrm{y}}}_{b\mathrm{i}}+2\mathrm{\lambda }\mathrm{\xi }\left({\stackrel{˙}{\mathbf{y}}}_{b\mathrm{i}}-{\stackrel{˙}{\mathbf{y}}}_{d\mathrm{i}}\right)+\left(1+{\mathrm{\gamma }}_{pzt}\right){\mathbf{y}}_{b\mathrm{i}}-\left(1+{\mathrm{\gamma }}_{pzt}\right){\mathbf{y}}_{d\mathrm{i}}-{\mathbf{q}}_{\mathrm{i}}={\mathbf{P}}_{b\mathrm{i}}\left(\mathrm{\tau }\right)\\ {\mathrm{\lambda }}^{2}\mathrm{\mu }{\stackrel{¨}{\mathbf{y}}}_{d\mathrm{i}}+2\mathrm{\lambda }\mathrm{\xi }\left({\stackrel{˙}{\mathbf{y}}}_{d\mathrm{i}}-{\stackrel{˙}{\mathbf{y}}}_{b\mathrm{i}}\right)-\left(1+{\mathrm{\gamma }}_{pzt}\right){\mathbf{y}}_{b\mathrm{i}}-{\mathrm{\gamma }}_{c}{\mathbf{y}}_{d\left(i-1\right)}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\left(1+{\mathrm{\gamma }}_{d}+{\mathrm{\gamma }}_{pzt}+2{\mathrm{\gamma }}_{c}\right){\mathbf{y}}_{d\mathrm{i}}-{\mathrm{\gamma }}_{c}{\mathbf{y}}_{d\left(i+1\right)}+{\mathbf{q}}_{\mathrm{i}}={\mathbf{P}}_{d\mathrm{i}}\left(\mathrm{\tau }\right)\\ \frac{\mathrm{\lambda }}{2\mathrm{\xi }\mathrm{\epsilon }}{\stackrel{˙}{\mathbf{q}}}_{\mathrm{i}}+\left({\mathrm{\gamma }}_{e}^{-1}+{\mathrm{\gamma }}_{pc}^{-1}\right){\mathbf{q}}_{\mathrm{i}}-{\mathbf{y}}_{b\mathrm{i}}+{\mathbf{y}}_{d\mathrm{i}}+\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{y}}_{b\mathrm{k}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{y}}_{d\mathrm{k}}=0\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}i=1,n+1,2n+1,\cdots \\ {\mathrm{\lambda }}^{2}{\stackrel{¨}{\mathbf{y}}}_{b\mathrm{i}}+2\mathrm{\lambda }\mathrm{\xi }\left({\stackrel{˙}{\mathbf{y}}}_{b\mathrm{i}}-{\stackrel{˙}{\mathbf{y}}}_{d\mathrm{i}}\right)+{\mathbf{y}}_{b\mathrm{i}}-{\mathbf{y}}_{d\mathrm{i}}={\mathbf{P}}_{b\mathrm{i}}\left(\mathrm{\tau }\right)\\ {\mathrm{\lambda }}^{2}\mathrm{\mu }{\stackrel{¨}{\mathbf{y}}}_{d\mathrm{i}}+2\mathrm{\lambda }\mathrm{\xi }\left({\stackrel{˙}{\mathbf{y}}}_{d\mathrm{i}}-{\stackrel{˙}{\mathbf{y}}}_{b\mathrm{i}}\right)-{\mathbf{y}}_{b\mathrm{i}}-{\mathrm{\gamma }}_{c}{\mathbf{y}}_{d\left(i-1\right)}+\left(1+{\mathrm{\gamma }}_{d}+2{\mathrm{\gamma }}_{c}\right){\mathbf{y}}_{d\mathrm{i}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-{\mathrm{\gamma }}_{c}{\mathbf{y}}_{d\left(i+1\right)}={\mathbf{P}}_{d\mathrm{i}}\left(\mathrm{\tau }\right)\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}i\ne 1,n+1,2n+1,\cdots \end{array}$(11)

for the parallel network, and $\left\{\begin{array}{c}{\mathrm{\lambda }}^{2}{\stackrel{¨}{\mathbf{y}}}_{b\mathrm{i}}+2\mathrm{\lambda }\mathrm{\xi }\left({\stackrel{˙}{\mathbf{y}}}_{b\mathrm{i}}-{\stackrel{˙}{\mathbf{y}}}_{d\mathrm{i}}\right)+\left(1+{\mathrm{\gamma }}_{pzt}\right){\mathbf{y}}_{b\mathrm{i}}-\left(1+{\mathrm{\gamma }}_{pzt}\right){\mathbf{y}}_{d\mathrm{i}}-\mathbf{q}={\mathbf{P}}_{b\mathrm{i}}\left(\mathrm{\tau }\right)\\ {\mathrm{\lambda }}^{2}\mathrm{\mu }{\stackrel{¨}{\mathbf{y}}}_{d\mathrm{i}}+2\mathrm{\lambda }\mathrm{\xi }\left({\stackrel{˙}{\mathbf{y}}}_{d\mathrm{i}}-{\stackrel{˙}{\mathbf{y}}}_{b\mathrm{i}}\right)-\left(1+{\mathrm{\gamma }}_{pzt}\right){\mathbf{y}}_{b\mathrm{i}}-{\mathrm{\gamma }}_{c}{\mathbf{y}}_{d\left(i-1\right)}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\left(1+{\mathrm{\gamma }}_{d}+{\mathrm{\gamma }}_{pzt}+2{\mathrm{\gamma }}_{c}\right){\mathbf{y}}_{d\mathrm{i}}-{\mathrm{\gamma }}_{c}{\mathbf{y}}_{d\left(i+1\right)}+\mathbf{q}={\mathbf{P}}_{d\mathrm{i}}\left(\mathrm{\tau }\right)\\ \frac{\mathrm{\lambda }}{2\mathrm{\xi }\mathrm{\epsilon }}\stackrel{˙}{\mathbf{q}}+\left({\mathrm{\gamma }}_{e}^{-1}+{\mathrm{\gamma }}_{pc}^{-1}\right)\mathbf{q}-\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{y}}_{b\mathrm{k}}+\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{y}}_{d\mathrm{k}}=0\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\\ \phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}i=1,n+1,2n+1,\cdots \\ {\mathrm{\lambda }}^{2}{\stackrel{¨}{\mathbf{y}}}_{b\mathrm{i}}+2\mathrm{\lambda }\mathrm{\xi }\left({\stackrel{˙}{\mathbf{y}}}_{b\mathrm{i}}-{\stackrel{˙}{\mathbf{y}}}_{d\mathrm{i}}\right)+{\mathbf{y}}_{b\mathrm{i}}-{\mathbf{y}}_{d\mathrm{i}}={\mathbf{P}}_{b\mathrm{i}}\left(\mathrm{\tau }\right)\\ {\mathrm{\lambda }}^{2}\mathrm{\mu }{\stackrel{¨}{\mathbf{y}}}_{d\mathrm{i}}+2\mathrm{\lambda }\mathrm{\xi }\left({\stackrel{˙}{\mathbf{y}}}_{d\mathrm{i}}-{\stackrel{˙}{\mathbf{y}}}_{b\mathrm{i}}\right)-{\mathbf{y}}_{b\mathrm{i}}-{\mathrm{\gamma }}_{c}{\mathbf{y}}_{d\left(\mathrm{i}-1\right)}+\left(1+{\mathrm{\gamma }}_{d}+2{\mathrm{\gamma }}_{c}\right){\mathbf{y}}_{d\mathrm{i}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-{\mathrm{\gamma }}_{c}{\mathbf{y}}_{d\left(\mathrm{i}+1\right)}={\mathbf{P}}_{d\mathrm{i}}\left(\mathrm{\tau }\right)\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\mathrm{i}\ne 1,n+1,2n+1,\cdots \end{array}$(12)

for the series network.

## 2.5 Random mistuned system

As mentioned above, the mechanical mistuning of a bladed disk is random and unavoidable in general, due to manufacturing tolerance or in-service wear. Without loss of generality, it is assumed that the mechanical coupling stiffness between sectors is mistuned in this study, which means that the parameter ${\mathrm{\gamma }}_{c}$ is not a constant along the circumference. In addition, due to the similarity of the research for the piezoelectric network in parallel and in series, only the formula for the network in parallel would be given next. The dynamic equations of the system in this case become eq. (13), the only change induced by mistuning lies in the second equation compared to eq. (11): $\left\{\begin{array}{c}{\mathrm{\lambda }}^{2}{\stackrel{¨}{\mathbf{y}}}_{b\mathrm{i}}+2\mathrm{\lambda }\mathrm{\xi }\left({\stackrel{˙}{\mathbf{y}}}_{bi}-{\stackrel{˙}{\mathbf{y}}}_{d\mathrm{i}}\right)+\left(1+{\mathrm{\gamma }}_{pzt}\right){\mathbf{y}}_{b\mathrm{i}}-\left(1+{\mathrm{\gamma }}_{pzt}\right){\mathbf{y}}_{d\mathrm{i}}-{\mathbf{q}}_{\mathrm{i}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}={\mathbf{P}}_{b\mathrm{i}}\left(\mathrm{\tau }\right)\\ {\mathrm{\lambda }}^{2}\mathrm{\mu }{\stackrel{¨}{\mathbf{y}}}_{d\mathrm{i}}+2\mathrm{\lambda }\mathrm{\xi }\left({\stackrel{˙}{\mathbf{y}}}_{d\mathrm{i}}-{\stackrel{˙}{\mathbf{y}}}_{b\mathrm{i}}\right)-\left(1+{\mathrm{\gamma }}_{pzt}\right){\mathbf{y}}_{b\mathrm{i}}-{\mathrm{\gamma }}_{c\left(\mathrm{i}-1\right)}{\mathbf{y}}_{d\left(\mathrm{i}-1\right)}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\left(1+{\mathrm{\gamma }}_{d}+{\mathrm{\gamma }}_{pzt}+{\mathrm{\gamma }}_{c\left(\mathrm{i}-1\right)}+{\mathrm{\gamma }}_{c\mathrm{i}}\right){\mathbf{y}}_{d\mathrm{i}}\phantom{\rule{1pt}{0ex}}-{\mathrm{\gamma }}_{c\mathrm{i}}{\mathbf{y}}_{d\left(\mathrm{i}+1\right)}+{\mathbf{q}}_{\mathrm{i}}={\mathbf{P}}_{d\mathrm{i}}\left(\mathrm{\tau }\right)\\ \frac{\mathrm{\lambda }}{2\mathrm{\xi }\mathrm{\epsilon }}{\stackrel{˙}{\mathbf{q}}}_{\mathrm{i}}+\left({\mathrm{\gamma }}_{e}^{-1}+{\mathrm{\gamma }}_{pc}^{-1}\right){\mathbf{q}}_{\mathrm{i}}-{\mathbf{y}}_{b\mathrm{i}}+{\mathbf{y}}_{d\mathrm{i}}+\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{y}}_{b\mathrm{k}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}{\mathbf{y}}_{d\mathrm{k}}=0\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\mathrm{i}=1,n+1,2n+1,\cdots \\ {\mathrm{\lambda }}^{2}{\stackrel{¨}{\mathbf{y}}}_{b\mathrm{i}}+2\mathrm{\lambda }\mathrm{\xi }\left({\stackrel{˙}{\mathbf{y}}}_{b\mathrm{i}}-{\stackrel{˙}{\mathbf{y}}}_{d\mathrm{i}}\right)+{\mathbf{y}}_{b\mathrm{i}}-{\mathbf{y}}_{d\mathrm{i}}={\mathbf{P}}_{b\mathrm{i}}\left(\mathrm{\tau }\right)\\ {\mathrm{\lambda }}^{2}\mathrm{\mu }{\stackrel{¨}{\mathbf{y}}}_{d\mathrm{i}}+2\mathrm{\lambda }\mathrm{\xi }\left({\stackrel{˙}{\mathbf{y}}}_{d\mathrm{i}}-{\stackrel{˙}{\mathbf{y}}}_{b\mathrm{i}}\right)-{\mathbf{y}}_{b\mathrm{i}}-{\mathrm{\gamma }}_{c\left(\mathrm{i}-1\right)}{\mathbf{y}}_{d\left(i-1\right)}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\left(1+{\mathrm{\gamma }}_{d}+{\mathrm{\gamma }}_{c\left(i-1\right)}+{\mathrm{\gamma }}_{c\mathrm{i}}\right){\mathbf{y}}_{d\mathrm{i}}\phantom{\rule{1pt}{0ex}}-{\mathrm{\gamma }}_{c\mathrm{i}}{\mathbf{y}}_{d\left(i+1\right)}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}={\mathbf{P}}_{d\mathrm{i}}\left(\mathrm{\tau }\right)\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\mathrm{i}\ne 1,n+1,2n+1,\cdots \end{array}$(13)

For an electromechanical and cyclic symmetric system, besides the mechanical mistuning, it should be paid attention to the effect of the electrical mistuning. For a periodic piezoelectric network without inductor, the electrical mistuning may come from the piezoelectric material, the capacitor and the resistor. Therefore, in the following text, the small and random variation of the stiffness of the piezoelectric patch (${\mathrm{\gamma }}_{pzt}$), the inherent capacitance of the piezoelectric patch (${\mathrm{\gamma }}_{pc}$), and the resistance ($\mathrm{\epsilon }$) of the network will be taken into account. The variation of ${\mathrm{\gamma }}_{pzt}$ only changes the first two equations in eq. (11); that of ${\mathrm{\gamma }}_{pc}$ and $\mathrm{\epsilon }$ can only induce changes in third equation in eq. (11). In addition, the impact of ${\mathrm{\gamma }}_{pc}$ and ${\mathrm{\gamma }}_{e}$ are coherent to the network because of their series connection. Then it is enough to take one variation of the two capacitances into account. When electrical parameters vary along the circumference (mistuned) respectively, the changed equations are listed as follows:

With mistuned resistance ($\mathrm{\epsilon }$), the third equation of eq. (11) becomes: $\begin{array}{rl}\frac{\mathrm{\lambda }}{2\mathrm{\xi }{\mathrm{\epsilon }}_{\mathrm{i}}}{\stackrel{˙}{\mathbf{q}}}_{\mathrm{i}}& -\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}\left(\frac{\mathrm{\lambda }}{2\mathrm{\xi }{\mathrm{\epsilon }}_{k}}{\stackrel{˙}{\mathbf{q}}}_{\mathrm{k}}\right)+\left({\mathrm{\gamma }}_{e}^{-1}+{\mathrm{\gamma }}_{pc}^{-1}\right){\mathbf{q}}_{\mathrm{i}}-{\mathbf{y}}_{b\mathrm{i}}+{\mathbf{y}}_{d\mathrm{i}}\\ & +\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}\left({\mathbf{y}}_{b\mathrm{k}}-{\mathbf{y}}_{d\mathrm{k}}\right)=0\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}i=1,n+1,\cdots \end{array}$(14)

With mistuned inherent capacitance of the piezoelectric patch (${\mathrm{\gamma }}_{pc}$), the third equation of eq. (11) becomes: $\begin{array}{rl}\frac{\mathrm{\lambda }}{2\mathrm{\xi }\mathrm{\epsilon }}{\stackrel{˙}{\mathbf{q}}}_{\mathrm{i}}& +\left({\mathrm{\gamma }}_{e}^{-1}+{\mathrm{\gamma }}_{pc\mathrm{i}}^{-1}\right){\mathbf{q}}_{\mathrm{i}}-\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}\left({\mathrm{\gamma }}_{pc\mathrm{k}}^{-1}{\mathbf{q}}_{k}\right)-{\mathbf{y}}_{b\mathrm{i}}\\ & +{\mathbf{y}}_{d\mathrm{i}}+\frac{1}{p}\sum _{k=1}^{\left(p-1\right)n+1}\left({\mathbf{y}}_{b\mathrm{k}}-{\mathbf{y}}_{d\mathrm{k}}\right)=0\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\mathrm{i}=1,n+1,\cdots \end{array}$(15)

With mistuned stiffness of the piezoelectric patch (${\mathrm{\gamma }}_{pzt}$), the first two equations of eq. (11) become: $\left\{\begin{array}{c}{\mathrm{\lambda }}^{2}{\stackrel{¨}{\mathbf{y}}}_{b\mathrm{i}}+2\mathrm{\lambda }\mathrm{\xi }\left({\stackrel{˙}{\mathbf{y}}}_{b\mathrm{i}}-{\stackrel{˙}{\mathbf{y}}}_{d\mathrm{i}}\right)+\left(1+{\mathrm{\gamma }}_{pzt\mathrm{i}}\right){\mathbf{y}}_{b\mathrm{i}}-\left(1+{\mathrm{\gamma }}_{pzt\mathrm{i}}\right){\mathbf{y}}_{d\mathrm{i}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-{\mathbf{q}}_{\mathrm{i}}={\mathbf{P}}_{b\mathrm{i}}\left(\mathrm{\tau }\right)\\ {\mathrm{\lambda }}^{2}\mathrm{\mu }{\stackrel{¨}{\mathbf{y}}}_{d\mathrm{i}}+2\mathrm{\lambda }\mathrm{\xi }\left({\stackrel{˙}{\mathbf{y}}}_{d\mathrm{i}}-{\stackrel{˙}{\mathbf{y}}}_{b\mathrm{i}}\right)-\left(1+{\mathrm{\gamma }}_{pzt\mathrm{i}}\right){\mathbf{y}}_{b\mathrm{i}}-{\mathrm{\gamma }}_{c}{\mathbf{y}}_{d\left(i-1\right)}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\left(1+{\mathrm{\gamma }}_{d}+{\mathrm{\gamma }}_{pzt\mathrm{i}}+2{\mathrm{\gamma }}_{c}\right){\mathbf{y}}_{d\mathrm{i}}-{\mathrm{\gamma }}_{c}{\mathbf{y}}_{d\left(i+1\right)}+{\mathbf{q}}_{\mathrm{i}}={\mathbf{P}}_{d\mathrm{i}}\left(\mathrm{\tau }\right)\\ \phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\mathrm{i}=1,n+1,2n+1,\cdots \end{array}$(16)

## 3 The vibration suppression effect of the bi-periodic piezoelectric network

The objective of integrating the piezoelectric network into a bladed disk is to suppress its amplified vibration, which is induced by mistuning. In order to evaluate the effectiveness of the piezoelectric network, the Modified Modal Assurance Criterion (MMAC for short in the following text) [23] is used in this study. It doesn’t need to calculate the dynamic response of the system directly, while it just needs to consider the changed modal information such as the modal stiffness, the modal damping and the modal force.

The MMAC is defined as follows, $MMAC=\frac{{\mathrm{\chi }}_{f}\cdot {\mathrm{\chi }}_{\mathrm{\infty }}}{{\mathrm{\chi }}_{\mathrm{\xi }}\cdot {\mathrm{\chi }}_{\mathrm{\omega }}}$(17)

in which, the four variables ${\mathrm{\chi }}_{\mathrm{\xi }}$, ${\mathrm{\chi }}_{\mathrm{\omega }}$, ${\mathrm{\chi }}_{f}$, ${\mathrm{\chi }}_{\mathrm{\infty }}$ are the ratio of the modal damping, the ratio of the modal stiffness, the ratio of the generalized force, and the ratio of the infinity modal norm between the changed system and the original system.

The signification of the MMAC can be found from the derivation of the formula (17). Actually, it is the ratio between the resonant response of the changed structure and that of the original structure. They are both excited by the same force, parallel to the corresponding mode. As a result, if MMAC<1, the changed structure gets a lower resonant response compared to the original structure.

For these two systems: the tuned bladed disk without a bi-periodic piezoelectric network (with subscript “tun”) and the mistuned bladed disk with a bi-periodic piezoelectric network (with subscript “pzt”), four variables in the formula (17) have expressions as follows:

Ratio of the modal damping: ${\mathrm{\chi }}_{\mathrm{\xi }}=\frac{{\mathrm{\xi }}_{pzt}}{{\mathrm{\xi }}_{tun}}$(18)

Ratio of the modal stiffness: ${\mathrm{\chi }}_{\mathrm{\omega }}=\frac{{\mathrm{\omega }}_{pzt}^{2}}{{\mathrm{\omega }}_{tun}^{2}}$(19)

Ratio of the generalized force: ${\mathrm{\chi }}_{f}=∥\frac{{\varphi }_{pzt}^{T}\cdot {\varphi }_{tun}}{{\varphi }_{tun}^{T}\cdot {\varphi }_{tun}}∥$(20)

Ratio of the infinity modal norm: ${\mathrm{\chi }}_{\mathrm{\infty }}=\frac{{∥{\varphi }_{pzt}∥}_{\mathrm{\infty }}}{{∥{\varphi }_{tun}∥}_{\mathrm{\infty }}}$(21)

The mistuned variables in this analysis are random and assumed to have the following form: ${\mathbf{L}}_{\mathrm{i}}={L}_{tun}\left(1+{\mathrm{\delta }}_{\mathrm{i}}\right)$(22)where ${L}_{tun}$ is the tuned variable, ${\mathbf{L}}_{\mathrm{i}}$ is the mistuned variable, $\mathrm{\delta }$ follows the normal distribution, with zero mean and different standard deviations. In the numerical simulation, $\mathrm{\delta }$ is generated by a pseudo-random generator.

The following numerical analysis is based on the model with the sector number N = 30 (as shown in Figures 1 and 2). The mechanical parameters (non-dimensional) are taken as follows: $\mathrm{\mu }=426$, ${\mathrm{\gamma }}_{d}=1.1$, ${\mathrm{\gamma }}_{c}=492$, and $\mathrm{\xi }=0.01$.

The values come from the literature [6]. Without considering the piezoelectric network, a modal analysis of such a mechanical system is performed at first. The frequency-nodal diameter index diagram of the system is obtained and shown in Figure 5. The 5-nodal-diameter mode is in the frequency veering region, which is a coupling vibration between the blade and the disk. A lot of researches have shown that a bladed disk is much more sensitive to mistuning at the frequency veering region, which means it easy to get an amplified vibration response. For this reason, the 5-nodal-diameter mode is taken as the focus in the following research.

Figure 5:

Diagram of natural frequency versus nodal diameter index.

To examine the vibration suppression ability of a bi-periodic piezoelectric network to the mistuned model (sector number N = 30), all possible bi-periods should be taken into account in the evaluation. The 5-nodal-diameter mode and related modal parameters, necessary for the calculation of the MMAC, can be obtained from the solution of the last two equations of eq. (13), letting the subscript i vary from 1 to N. The electrical parameters are set as follows: ${\mathrm{\gamma }}_{pzt}=0.01$, ${\mathrm{\gamma }}_{pc}=0.005$, ${\mathrm{\gamma }}_{e}=+\mathrm{\infty }$, and $\mathrm{\epsilon }=5$. Here ${\mathrm{\gamma }}_{e}=+\mathrm{\infty }$ simulates the case without the external capacitor in the piezoelectric network. To obtain the MMAC, the Monte Carlo simulation is performed as follows: 104 mistuning cases are generated to calculate a MMAC corresponding to one bi-periodic number p, with the standard deviation $\mathrm{\sigma }=0.05$. The result is plotted in Figure 6, in which the vertical axis is the MMAC within 99 % confidence interval. Figure 6(a) is the MMAC evaluation to the piezoelectric network in parallel, Figure 6 (b) is that in series. The figure shows that the piezoelectric network both in parallel and in series can attenuate vibration of the mistuned system. For both networks, the effect of the vibration suppression is different from one period to another. Especially the effect becomes very small for some periods such as p = 10, 5 and 2. This phenomenon can be explicated from the expression of the nodal-diameter mode of a cyclic symmetric structure written as: ${\mathbf{A}}_{s}={A}_{0}\cdot sin\left(\frac{2\mathrm{\pi }d\left(s-1\right)}{N}\right)$(23)

in which: ${A}_{0}$ is the amplitude of every sector; N is the number of sectors; d is the nodal diameter index; s is the sector index. The position making ${\mathbf{A}}_{s}=0$ is just the nodal diameter. Of course, for some modes the nodal diameter is just between sectors, but for others the nodal diameter is on the sector. Taking p = 10 for an example, considering a tuned system with 30 sectors, in the mode with 5 nodal diameters, the nodal diameters appear on the sectors with number 1, 4, 7, 10,...,28, these are also the sectors on which piezoelectric patches are set. As a result, for this mode, the piezoelectric network with the bi-period number p = 10 doesn’t have an effect on vibration suppression. Obviously, for an arbitrary p, in order to suppress vibration of a d-nodal-diameter mode, the periodic number of the piezoelectric network should satisfy the following inequality $\frac{2\pi \cdot d}{p}\ne \text{h}\cdot \pi ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{h}=0,1,2,\cdots$(24)

Taking d = 5 as an example, it is easy to obtain that, if p = 2, almost no effect will be found. In the mistuning case, the pure nodal diameter vibration doesn’t exist, which means that the nodal diameter doesn’t exist and the piezoelectric patches can always capture vibration at any position. However, because mistuning is small, vibration is of quasi nodal-diameter form. Therefore, if the periodic number p doesn’t satisfy the inequality eq. (24), the effect of the piezoelectric network should be small (but not zero). In conclusion, the bi-periodic piezoelectric network can always attenuate vibration of the mistuned system. But in order to obtain an ideal effect, the layout of the piezoelectric network needs designing carefully. In addition, it is worth to highlight that, the case where p equals to N is a good choice but should not be the best one.

Figure 6:

MMAC of the system with bi-periodic piezoelectric network.

In Figure 7, it is given a comparison of the blade’s forced response between the mechanical mistuned model and that with a bi-periodic piezoelectric network, where the standard deviation of the random mistuning in ${\mathrm{\gamma }}_{c}$ is $\mathrm{\sigma }=0.1$. The blade’s response of the corresponding tuned system is also given in the figure as a reference. The exciting force with the spatial wave number 5 is chosen to excite the 5-nodal-diameter modal vibration, which is more sensitive to mistuning as mentioned before. The resonant response is amplified about 30 % in the figure (the black curve), compared to the corresponding response of the tuned system (the red curve). Therefore, it can be observed that after integrating a piezoelectric network, the response of the blade is significantly attenuated (the pink curve), even smaller than that of the tuned system. The result shows that the integration of a bi-periodic piezoelectric network can provide a reliable assurance for avoiding the forced response amplification of the mistuned bladed disk.

Figure 7:

Frequency-response curves of mistuned bladed disk (N = 30).

## 4 The effect of the random mistuned piezoelectric network

It is no doubt that the integration of a piezoelectric network will introduce some new mistuning in the system. In a bladed disk with a piezoelectric network, there are not only the mechanical mistuning but also the electrical mistuning. In this section, a random mistuning in the parallel piezoelectric network is taken into account. The objective is to make clear whether a random mistuning occurred in the piezoelectric network would lead the resonant vibration amplification. To highlight the purpose, the mechanical system (the bladed disk) is assumed to be tuned. Therefore, when the MMAC is used to evaluate the effect, the bladed disk with the mistuned piezoelectric network is taken as the changed system in the formula (17). In the following study, two bi-periodic numbers (p = 30 and p = 15) are taken into consideration. The MMAC within 99 % confidence interval is calculated for the cases with mistuned resistance $\mathrm{\epsilon }$, mistuned capacitance ${\mathrm{\gamma }}_{pc}$ and mistuned stiffness ${\mathrm{\gamma }}_{pzt}$, respectively. The results are shown in Figures 8–10. From the figures it can be seen that almost all MMAC factors are less than 1, which means that even the piezoelectric network is mistuned, it is capable of suppressing vibration, and of course it doesn’t cause any amplification.

Figure 8:

MMAC of the system with mistuning of $\mathrm{\epsilon }$.

Figure 9:

MMAC of the system with mistuning of ${\mathrm{\gamma }}_{pc}$.

Figure 10:

MMAC of the system with mistuning of ${\mathrm{\gamma }}_{pzt}$.

## 5 The robustness of the system

In actual, the random mistuning is always existed in both mechanical elements and electrical elements for an electromechanical system such as a bladed disk with a piezoelectric network. What aero-engineers are concerned with is the robustness of the system. In this section, the performance of a bladed disk with a bi-periodic piezoelectric network is firstly examined, in which both mechanical elements (the coupling stiffness) and electrical elements (piezoelectric patches, resistors, capacitors) are assigned different random mistuning levels, which would simulate different designing margin. Then the robustness of the system is examined by considering a new (additional) mistuning of mechanical elements, which can simulate a certain configuration of an engine in service.

All four parameters (${\mathrm{\gamma }}_{c}$, ${\mathrm{\gamma }}_{pc}$, $\mathrm{\epsilon }$ and ${\mathrm{\gamma }}_{pzt}$) are mistuned, whose mistuning levels assumed to have the same standard deviation, changing from 0.01 to 0.10. The same random simulation as described in section 4 has been performed to obtain a MMAC. The evaluation is made by taking the tuned mechanical system without the piezoelectric network as a reference in the formula (17). Figure 11 gives some estimations of the MMAC within 99 % confidence interval for two bi-period numbers: p = 30 and p = 15. The result shows that the bi-periodic piezoelectric network can avoid the amplified resonant vibration, as well as it provides a big margin during the designing period.

Figure 11:

MMAC of the system with mistuning of multiple parameters.

In Figure 12, it gives the estimation of the MMAC within 99 % confidence interval for two bi-period numbers: p = 30 and p = 15, where the mistuning level (standard deviation) assigned to $\mathrm{\epsilon }$, ${\mathrm{\gamma }}_{pc}$, ${\mathrm{\gamma }}_{pzt}$ is 0.1, while that assigned to ${\mathrm{\gamma }}_{c}$ is from 0.11 to 0.20. It simulates a growth of mistuning, induced by wear in service for example. The system as the reference in the formula (17) is the same as Figure 11. The result shows that even the standard deviation of the mechanical mistuning attains 20 %, the resonant vibration of the mistuned system can be effectively suppressed because of the bi-periodic piezoelectric network, which proves a good robustness of the system.

Figure 12:

MMAC of the system with mistuning of multiple parameters in which mistuning of ${\mathrm{\gamma }}_{c}$ increases.

## 6 Conclusions

The vibration suppression of a mistuned bladed disk with a piezoelectric network has been investigated. There is no inductor in the piezoelectric network, and its period may be different from that of the bladed disk. An electromechanical lumped parameter model is taken into consideration in this study. The impact of both the mechanical mistuning and the electrical mistuning to the dynamic response of the system has been studied. The MMAC is used to evaluate the effect of the piezoelectric network. The obtained result shows that integrating a piezoelectric network with the mistuned blade, the vibration responses are significantly attenuated, even smaller than that of the tuned system. Especially a bi-periodic piezoelectric network can provide a reliable assurance for avoiding the forced response amplification of the mistuned bladed disk. The amplified response induced by a mechanical mistuning can be effectively suppressed by the bi-periodic piezoelectric network, which also has a good robustness to the mistuned bladed disk.

Further research will be developed in experiments to validate the obtained results.

## Nomenclature

$A$

modal amplitude of a sector

$c$

damping of a sector

$d$

nodal diameter index

${k}_{b}$

${k}_{c}$

coupling stiffness between sectors

${k}_{d}$

stiffness between disk and ground

${k}_{pzt}$

short-circuit stiffness of a piezoelectric material

${m}_{b}$

${m}_{d}$

disk mass of a sector

$n$

number of substructure

$p$

bi-periodic number

$\mathbf{q}$

non-dimensional electrical charge

$t$

time

${\mathbf{x}}_{b}$

${\mathbf{x}}_{d}$

displacement of disk

${\mathbf{y}}_{b}$

${\mathbf{y}}_{d}$

non-dimensional displacement of disk

${C}_{e}$

capacitance in circuit

${C}_{pzt}$

intrinsic capacitance of a piezoelectric material

$\mathbf{D}$

electrical charge

${\mathbf{F}}_{b}$

${\mathbf{F}}_{d}$

exciting force on disk

$\mathbf{I}$

electrical current

${\mathbf{L}}_{\mathrm{i}}$

mistuned variables

${L}_{tun}$

tuned variables

$N$

total number of sectors

${\mathbf{P}}_{b}$

${\mathbf{P}}_{d}$

non-dimensional excitation on disk

$R$

resistance in circuit

$\mathbf{U}$

voltage

## Greek symbols

${\mathrm{\gamma }}_{c}$

non-dimensional coupling stiffness

${\mathrm{\gamma }}_{d}$

non-dimensional disk stiffness

${\mathrm{\gamma }}_{e}$

non-dimensional capacitance in circuit

${\mathrm{\gamma }}_{pc}$

non-dimensional intrinsic capacitance of a piezoelectric material

${\mathrm{\gamma }}_{pzt}$

non-dimensional short-circuit stiffness of a piezoelectric material

$\mathrm{\delta }$

stationary response

$\mathrm{\epsilon }$

non-dimensional resistance in circuit

$\mathrm{\eta }$

electromechanical coupling factor

$\mathrm{\lambda }$

non-dimensional frequency of excitation

$\mathrm{\mu }$

mass ratio of disk to blade

$\mathrm{\xi }$

mechanical damping ratio

${\mathrm{\xi }}_{pzt}$

modal damping of mistuned system with a piezoelectric network

${\mathrm{\xi }}_{tun}$

modal damping of tuned system without a piezoelectric network

$\mathrm{\sigma }$

standard deviation

$\mathrm{\tau }$

non-dimensional time

${\mathrm{\chi }}_{f}$

ratio of relative generalized force

${\mathrm{\chi }}_{\mathrm{\xi }}$

ratio of relative modal damping

${\mathrm{\chi }}_{\mathrm{\omega }}$

ratio of relative modal stiffness

${\mathrm{\chi }}_{\mathrm{\infty }}$

ratio of relative infinity norm

$\mathrm{\omega }$

frequency of excitation

${\mathrm{\omega }}_{n}$

natural frequency

${\mathrm{\omega }}_{pzt}$

frequency of mistuned system with a piezoelectric network

${\mathrm{\omega }}_{tun}$

frequency of tuned system without a piezoelectric network

${\varphi }_{pzt}$

mode shape of mistuned system with a piezoelectric network

${\varphi }_{tun}$

mode shape of tuned system without a piezoelectric network

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Accepted: 2016-05-18

Published Online: 2018-02-15

Published in Print: 2018-03-26

Citation Information: International Journal of Turbo & Jet-Engines, Volume 35, Issue 1, Pages 17–28, ISSN (Online) 2191-0332, ISSN (Print) 0334-0082,

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© 2018 Walter de Gruyter GmbH, Berlin/Boston.

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