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:
${\mathrm{\chi}}_{f}=\u2225\frac{{\varphi}_{pzt}^{T}\cdot {\varphi}_{tun}}{{\varphi}_{tun}^{T}\cdot {\varphi}_{tun}}\u2225$(20)

Ratio of the infinity modal norm:
${\mathrm{\chi}}_{\mathrm{\infty}}=\frac{{\u2225{\varphi}_{pzt}\u2225}_{\mathrm{\infty}}}{{\u2225{\varphi}_{tun}\u2225}_{\mathrm{\infty}}}$(21)

The mistuned variables in this analysis are random and assumed to have the following form:
${\mathbf{L}}_{\mathrm{i}}={L}_{tun}(1+{\mathrm{\delta}}_{\mathrm{i}})$(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: 10^{4} 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:
${\mathbf{A}}_{s}={A}_{0}\cdot sin\left(\frac{2\mathrm{\pi}d(s-1)}{N}\right)$(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).

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.