Chapter 5. Reliability-Oriented Optimal Design of Intentional Mistuning for Bladed Disk with
6.2 System Model and Mode Localization Characterization
Configuration of the piezoelectric circuitry network to achieve both mode delocalization and vibration suppression of a bladed disk is shown in Figure 6.1 (Zhang and Wang 2002; Yu and Wang 2009), where each piezoelectric shunt circuit, which consists of piezoelectric transducer (PZT), inductor, and resistor, is integrated onto individual blades. Once vibration energy is converted to electrical energy through PZT, the energy is stored in inductor. Resistor in piezoelectric shunt circuit then dissipates the energy, acting as damper to the system while performing vibration suppressions of blades. By coupling the circuits through capacitive elements to form network, vibration delocalization is achieved by evenly distributing the stored energy to the blades.
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Figure 6.1 Coupled system of bladed disk with piezoelectric circuitry network
The first step to optimize circuit parameters is to derive the equation of motion for the coupled system.
The transversal displacement of the j-th blade is approximated as (Wang and Tang 2008)
,
j j
w x t x q t (6.1) where is the first local blade mode without the piezoelectric circuit. The discretized equations of motion for the coupled system can be derived using Hamilton’s principle and the assumed mode method, which are
1 1 2
j j j c j j c j j j j
mq cq kq k q q k q q k Q f (6.2a)
k Q2 jk q1 j Vaj (6.2b) where m, c, k, kc, fj, k1, and k2 are mass, damper, stiffness of blade, coupling stiffness, external force, inverse of the capacitance of piezoelectric patch, and electro-mechanical coupling coefficient, which are given by
2 2
0 0
b b
l l
b b p p
m A dx A Hdx, 2
0 lb
gcb dx
"2 "2
0 0
b b
l l
b b p p
k E I dxE I Hdx , kcks2 xs
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33
1 2
p p p
k A w l
, "
2 31
0
1
lb p p p
k F h Hdx
w l
where qj is the generalized mechanical displacement of the j-th blade, and Vaj and Qj are the voltage across and charge flowing to the piezoelectric patch attached to the j-th blade. Here H H x xlH x xr
where H is the Heaviside step function, and xl and xr are location of left and right ends of piezoelectric patch. Based on the circuit configuration in Figure 6.1 and Kirchhoff’s current law, we have
1
a j j
Q Q Q
1
b j j
Q Q Q
which, in virtue of the voltage law, lead to
1 1
aj j j a j j a j j
V LQ RQ k Q Q k Q Q (6.3) where ka is the inverse of the coupling capacitance. Substituting Equation (6.3) into Equation (6.2b) yields the discretized equations of motion of the electro-mechanically coupled system in Figure 6.1 as
1 1 2
j j j c j j c j j j j
mq cq kq k q q k q q k Q f (6.4a)
1 1 1 2 0
j j a j j a j j j j
LQ RQ k Q Q k Q Q k Q k q (6.4b) For the original bladed disk without piezoelectric circuitry, the equation of motion is given as
1 1
j j j c j j c j j j
mq cq kq k q q k q q f (6.5) In the above derivations, it is assumed that the bladed disk is ideally periodic. On the other hand, uncertainty inevitably exist in blades of the system in its realistic application. As it is the common practice in the localization study, it is assumed in this study that uncertainties of the bladed disk are characterized by the
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stiffness variations, and the mass remains unaffected. The stiffness of the j-th blade with uncertainty can be then written as
kj k k (6.6) The external load is a time harmonic excitation that consists of a summation of all spatial harmonics with equal weightings. For each spatial harmonic excitation, there is a fixed phase shift between adjacent substructures, but the amplitude of force on all substructures is the same (Duan et al. 2016).
1 2 2 1 1 2 1
1
1 / 2 1
N N
T j t j j j j
N
C
f f f e e e e e
N
(6.7)
2 1
i
D i N
, C0,1,...,N/ 2 1
where i is the phase shift for a given D, which is engine order. Localization level of the bladed disk is defined as following (Tang and Wang 2008).
/
k L
c
S k k
(6.8)
where k is the random variation of blade stiffness. Generalized electro-mechanical coupling coefficient of the coupled system is defined as the following (Tang and Wang 2008).
2 1
k
kk (6.9) Depending on level of localization, bladed disks are subject to various degrees of vibration localization.
One of its detrimental effects is localization of vibration modes. Let there be the bladed disk, whose condition is listed in Table 6.1, in the presence of 2.5% of random uncertainty in bladed stiffness, and the corresponding localization level is 0.025 / 0.025 1 . The 1st, 3rd, and 20th vibration modes of the localized system are shown in Figure 6.1, where they are compared with vibration modes of the ideal system. For the perfectly periodic system, all the modes are extended ones that essentially exhibit spatial harmonic patterns,
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while vibration occurs only in small number of blades in the localized system. We can quantify localization of each vibration mode based on the definition of modal assurance criterion localization factor (MACLF) (Chandrashaker et al. 2016), i.e.
T 2
T T
1 2
MACLF max
b
j k
j k N
j j k k
v b
v v b b (6.10) where, in this study, vj is the j-th vibration mode of the localized system, and bk is the k-th vibration mode of the perfectly periodic system. Localized modes of rotationally periodic structure often undergo mode swapping that numbering of the modes does not always correspond to the mode numbers of the system with non-localized modes. Thus the maximum MACLF value is chosen in Equation (6.10) to pair the correct modes. Owing to its definition, MACLF is near 1 for the non-localized system, and it is more close to 0 for the more localized system. MACLF for the 1st, 3rd, and 20th modes for the localized system in Figure 6.2 are 0.1603, 0.1700, and 0.2702, respectively. Another detrimental effect of vibration localization is the serious amplification of the forced vibration response under engine-order excitation. The maximum blade tip displacements for the perfectly periodic and the mistuned system are shown in Figure 6.3.
Mode localization and the maximum vibration response can be both alleviated and reduced, respectively, by integrating piezoelectric circuitry network into host structure of the bladed disk. To maximize their effects, vibration suppression and mode delocalization each requires unique tuning of piezoelectric circuit parameters. In this study, methods are individually developed to optimize piezoelectric circuit parameters for each vibration suppression and mode delocalization. The multi-objective optimization is then developed by integrating both methods together.
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(a) (b) (c)
Figure 6.2 Vibration modes of ideally periodic and localized systems. (a) 1st mode; (b) 3rd mode; and (c) 20th mode
(a) (b)
Figure 6.3 Maximum blade tip displacement under 10th engine order excitation. (a) Perfectly periodic system; (b) Localized system
Table 6.1 Parameters used in computer simulation
b 20
N lb 0.2253 m
p 0.08
l m wb 0.1001 m 0.075
wp m xl 0.005 m 0.085
xr m hb0.0302 m 0.008
hp m b 7.8335 10 kg/m 3 3
3 3
1.250 10 kg/m
p Eb 1.9818 10 N/m 11 2
10 2
7.1246 10 N/m
Ep h316.3076 10 N/C 9
6 33 9.4240 10 V
Rc 0.025