A remarkable feature of the vibration of the mistuned bladed disk is the amplitude magnification occurred in certain blades, which is also called vibration localization. However, due to manufacturing tolerance, assembling tolerance or in-service wear etc., mistuning is generally random and unavoidable. Therefore, how to mitigate the amplitude magnification of the bladed disk has been a problem drawing high attention of aero-engine engineers and researchers for many years. Up till now, most of the researches have focused on modeling the mistuned bladed disk [1–4], developing the reduced calculation method [[5–7], 1], describing the vibration localization [8, 9], as well as predicting the worst response amplitude [10, 1, 2] etc. The study on suppressing the localized or magnified vibration is still relatively rare. In recent decades, the intentional mistuning is proposed to reduce the random mistuned response. Griffin  introduced an intentional mistuning into a random mistuned bladed disk, by setting two types of blades (±10 % difference from the original weight) in alternating slots around the disk. The results of the Monte Carlo simulation showed that the vibration amplitude of the worst-case blade was reduced significantly. Rzadkowski  found that the harmonic mistuning may reduce the stress response of the random mistuned bladed disk greatly. Castanier [15, 16, 17, 18] indicated that some intentional mistuning patterns could reduce the maximum resonance amplitude magnification factor effectively and make the bladed disk insensitive to the random mistuning. Mignolet [19, 20] calculated the forced response of bladed disks with harmonic and partial mistuning, the results demonstrated that many features matched not only qualitatively but also quantitatively on bladed disks with random mistuning. Kenyon  also studied the effect of the harmonic mistuning on the sensitivity to small perturbations in the bladed disk. Jones  found that a linear pattern of intentional mistuning was very effective to reduce the amplitude magnification induced by the random mistuning. Wang  attempted to reduce vibration of the mistuned bladed disk through the mechanical bi-periodic design. However, it’s obvious that the intentional mistuning through the mechanical design would increase the difficulty of processing, manufacturing and assembling. Furthermore, it also should be investigated whether it would cause some new mistuning problems.
The development of the smart material science and technology opens a new way to solve the difficult problems in engineering. The smart material has been used to suppress the vibration of the mistuned bladed disk. Livet [24, 25] introduced an optimized piezoelectric shunt circuit into turbomachinery blades, and did some numerical simulations and experimental tests. The results verified that vibration can be reduced significantly when piezoelectric shunt circuits were mounted on a real structure. Min [26, 27] made a numerical and experimental study on rotating composite fan blades with the piezoelectric shunt damping. It showed a great potential to reduce vibration of blades under centrifugal loading. Kauffman  used piezoelectric materials to reduce vibration of turbomachinery bladed disks, which was governed by three parameters. An increase in any of them would control the structural response. Zhou  applied nonlinear piezoelectric shunt circuits to mistuned bladed disks. A sound damping performance was achieved. Mokrani  proposed a kind of parallel piezoelectric shunt damping for rotationally periodic structures. It was turned to be very effective when the targeted mode was close to harmonic circumferentially. Some researchers studied the application of the piezoelectric network in periodic or nearly periodic structures such as bladed disks for many years. In the literatures [31, 32], the piezoelectric shunt circuit was introduced to the bladed disk, four kinds of shunt layouts were taken into account, and a good damping effect was attained. 80 % of vibration could be reduced under the semi-active control condition. Furthermore, the external circuits were connected to form a piezoelectric network [33, 34, 35, 36]. The results showed that it was effective to delocalize vibration. Yu also made a study on electrical elements , based on a lumped parameter model with two DOFs per sector. The results showed that even the electrical elements were largely mistuned, the piezoelectric network could effectively reduce and delocalize vibration too. The remaining problem is that the network needs large inductors, which is difficult to be realized in practice. Li  tried to change the energy propagating path of a cyclic-periodic structure by designing the piezoelectric network, which could finally reduce vibration. In order to highlight this purpose, the mechanical coupling among the sectors was not considered. However, for a cyclic-periodic structure such as the bladed disk, the mechanical coupling must be taken into consideration.
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 Dynamic equations of the system with a bi-periodic piezoelectric network
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, and represent the blade mass and the disk mass of a sector respectively, and are the blade stiffness and the disk stiffness of a sector respectively, 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.
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 (as shown in Figures 3 and 4). Each sector is connected with adjacent sector through a mechanical coupling stiffness (as shown in Figure 3). The displacements of the blade and the disk in the sector are represented by and 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: (1)in which a dot above a letter means differentiation with respect to time t, D is the electric charge, is the short-circuit stiffness of the piezoelectric patch, 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.
2.3 Piezoelectric shunt circuits of sectors connected with each other
According to KCL (Kirchhoff’s Current Law), the current of each sector has the following relationship: (3)
Meanwhile, according to KVL (Kirchhoff’s Voltage Law), each sector has the same voltage, that is: (4)
Besides, the differential relationship between the current and electric charge is written as: (5)
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: (10)
for the parallel network, and (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 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): (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 (), the inherent capacitance of the piezoelectric patch (), and the resistance () of the network will be taken into account. The variation of only changes the first two equations in eq. (11); that of and can only induce changes in third equation in eq. (11). In addition, the impact of and 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 (), the third equation of eq. (11) becomes: (14)
With mistuned inherent capacitance of the piezoelectric patch (), the third equation of eq. (11) becomes: (15)
With mistuned stiffness of the piezoelectric patch (), the first two equations of eq. (11) become: (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)  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, (17)
in which, the four variables , , , 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: (18)
Ratio of the modal stiffness: (19)
Ratio of the generalized force: (20)
Ratio of the infinity modal norm: (21)
The mistuned variables in this analysis are random and assumed to have the following form: (22)where is the tuned variable, is the mistuned variable, follows the normal distribution, with zero mean and different standard deviations. In the numerical simulation, is generated by a pseudo-random generator.
The values come from the literature . 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.
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: , , , and . Here 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 . 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: (23)
in which: 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 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 (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.
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 is . 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.
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 , mistuned capacitance and mistuned stiffness , 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.
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 (, , and ) 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.
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 , , is 0.1, while that assigned to 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.
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.
modal amplitude of a sector
damping of a sector
nodal diameter index
stiffness between blade and disk
coupling stiffness between sectors
stiffness between disk and ground
short-circuit stiffness of a piezoelectric material
blade mass of a sector
disk mass of a sector
number of substructure
non-dimensional electrical charge
displacement of blade
displacement of disk
non-dimensional displacement of blade
non-dimensional displacement of disk
capacitance in circuit
intrinsic capacitance of a piezoelectric material
exciting force on blade
exciting force on disk
total number of sectors
non-dimensional excitation on blade
non-dimensional excitation on disk
resistance in circuit
non-dimensional coupling stiffness
non-dimensional disk stiffness
non-dimensional capacitance in circuit
non-dimensional intrinsic capacitance of a piezoelectric material
non-dimensional short-circuit stiffness of a piezoelectric material
non-dimensional resistance in circuit
electromechanical coupling factor
non-dimensional frequency of excitation
mass ratio of disk to blade
mechanical damping ratio
modal damping of mistuned system with a piezoelectric network
modal damping of tuned system without a piezoelectric network
ratio of relative generalized force
ratio of relative modal damping
ratio of relative modal stiffness
ratio of relative infinity norm
frequency of excitation
frequency of mistuned system with a piezoelectric network
frequency of tuned system without a piezoelectric network
mode shape of mistuned system with a piezoelectric network
mode shape of tuned system without a piezoelectric network
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About the article
Published Online: 2018-02-15
Published in Print: 2018-03-26