COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
A New Multi-harmonic Method for Predicting the Forced Response of
Mistuned Bladed disks with Dry Friction Damping
Erming He, Hongjian Wang*
College of Aeronautics, Northwestern Polytechnical University, Xi’an, 710072 China
Email: [email protected], [email protected]
Abstract Bladed disks often have blade-to-blade dampers to reduce their resonant vibration. The inevitable mistuning
of blade properties can cause great changes on the forced response of the bladed disk structures. In this paper, based on
the harmonic balance method, an efficient multi-harmonic (MH) method is presented for studying effects of mistuning
on resonant features of bladed disks with blade-to-blade dry friction damping. The method can predict the forced
response of bladed disks in frequency domain, and is validated by numerical integration method in time domain.
Analysis of forced response features is performed by the MH method for both tuned and mistuned systems with
various coupling, viscous damping, dry friction damping and blade stiffness mistuning etc. The results demonstrate
that: for the weakly coupled bladed disk with blade-to-blade dry friction damping, mistuning can reduce resonant
amplitudes of the system significantly; when the viscous damping and dry friction are all smaller, mistuning can
broaden the resonant frequency region, and produce multiple resonant peaks.
Key words: dry friction damping; mistuned bladed disk; forced response; multi-harmonic method; resonant features.
INTRODUCTION
The mistuning of blades in the turbine engines can cause localized vibrations, and increase the resonant amplitudes of
the bladed disk significantly. Mistuning refers to variations of the blade geometry due to procession and wear etc. these
variations, even small, can increase blade amplitudes significantly, and can lead to high cyclic fatigue (HCF) failure,
Srinivasan [1] discussed this problem in detail. In order to reduce the vibration of blades, designers often employ
shrouded blades or install friction dampers between blades. These two methods have a same effect of adding friction
damping between blades, and the friction in general is nonlinear. Because the friction interface is not parallel to the
direction of vibratory motion, and some other complicated factors, the analysis of forced responses of bladed disks
with blade-to-blade friction damping is not easy, Griffin [2] described this situation. The simplified mass-string model
is extensively used to investigate qualitatively the vibration of bladed disks with dry friction damping [2-4]. In the
analysis methods, though numerical integration can analyze responses of bladed disks with nonlinear damping, its
large amount of computation time make it unsuitable for parametric analysis. The harmonic balance method (HBM)
can give the steady-state solution in the frequency domain, and has been used extensively to predict the forced
response of bladed disks with dry friction [3, 5-9]. Wei and Pierre [3] used the method to analyze the effects of dry
friction on nearly cyclic structures. Sanliturk etc [4] obtained the conclusion that the HBM can effectively predict the
response of turbine blades through the investigation of turbine blades with friction dampers. Li Lin [5] studied the
effects of dry friction angle on statistics of dynamic characteristics of mistuned shrouded blisk. Petrov and Ewins
investigated the effects of scatter of contact interface characteristics on the vibration of bladed disks.
Due to the complicated performance of dry friction on bladed disks, the knowledge of the forced response features of
bladed disks with blade-to-blade dry friction is not sufficient. Many previous methods for analyzing vibration equation
with nonlinear dry friction only use one harmonic which is not accurate. In this paper, based on HBM, an efficient
multi-harmonic method which can use fast Fourier transformation (FFT) is presented to predict the forced response of
bladed disks with blade-to-blade dry friction. By using this method, the resonant features of bladed disks are studied
and summarized for tuned and mistuned bladed disks under various system parameters.
⎯ 1149 ⎯
MOTION EQUATION OF BLADED DISKS
The motion equation of bladed disks with blade-to-blade dry friction damping can be expressed as:
..
.
.
M x + C x + Kx = f (t ) + g(x, x )
(1)
where, x(t) is displacement vector, M,C,K,f,g are mass, viscous damping, stiffness, excitation and blade-to-blade
friction matrices respectively. In the analysis of forced response of bladed disks, the excitation often has the harmonic
form:
f j (t ) = f 0 cos(ωt + ψ j )
(2)
where, ψ j = 2π r ( j − 1) N , j=1,…N. N is the number of blades. r is engine order excitation. The blade-to-blade
friction force g is stick-slip friction. For the jth blade, the friction is determined by the displacements and velocities of
blades j-1, j, j+1. Considering the characteristics of the stick-slip, its dynamic model for numerical calculation can be
written as:
.
.
.
.
.
.
.
g j ( x j −1 , x j , x j +1 ) = h j , j −1 ( x j − x j −1 ) + h j , j +1 ( x j − x j +1 )
(3)
where, h j , j −1 and h j , j +1 are the frictions between j-1 and j, and between j and j+1 respectively.
.
.
.
1 .
⎧
d
(
x
=
−
j − x j −1 ), x j − x j −1 ≤ λ
1
,
j
−
j
.
.
⎪⎪
λ
h j , j −1 ( x j − x j −1 ) = ⎨
.
.
.
.
⎪= − d j −1, j sgn( x j − x j −1 ), x j − x j −1 > λ
⎪⎩
( 4)
The form of h j , j +1 is the similar to h j , j −1 . d j is given by a coefficient of friction times normal force to the sliding
surfaces, and its value is d 0 when all friction dampers are identical. λ is a numerical parameter.
MUTI-HARMONIC METHOD FOR ANALYZING THE RESPONSE OF BLADED DISKS
The motion equation of bladed disks is analyzed by balancing multiple harmonic coefficients of the system. Assume
the solution for Eq. (1) as:
x j (t ) =
X j ,c 0
2
NH
+ ∑ X j ,cn cos nωt + X
n =1
sin nωt
(5)
j , sn
where, NH is the number of harmonics. The dry friction can be expressed by Fourier series:
g j (t ) =
D j ,c 0
2
NH
+ ∑ D j ,cn cos nωt + D j ,sn sin nωt
(6)
n =1
The coefficients are:
D j ,cn ( X ) =
D j , sn ( X ) =
.
.
.
ω 2π / ω
cos(
n
ω
t
)
g
(
x
,
x
,
x
1
j
−
j
j +1 ) dt
j
π ∫0
.
.
.
ω 2π / ω
sin(
n
ω
t
)
g
(
x
j −1 , x j , x j +1 ) dt
j
π ∫0
(7)
(8)
substituting Eq.(5) and Eq.(6) into Eq.(1), balancing the coefficients of all orders of harmonics, after some
rearrangements, we can get:
S ⋅ X − F − D( X ) = 0
(9)
where, the coefficients of harmonics are rearranged by components and orders:
⎯ 1150 ⎯
X T = ( X 1T,c 0 , X 2T,c 0 ,..., X NT ,c 0 , X 1T,c1 , X 2T,c1 ,..., X NT ,c1 , X 1T, s1 , X 2T, s1 ,..., X NT , s1 ,..., X NT , sNH )
(10)
D T = ( D1T,c 0 , D2T,c 0 ,..., D NT ,c 0 , D1T,c1 , D2T,c1 ,..., D NT ,c1 , D1T, s1 , D2T, s1 ,..., D NT , s1 ,..., D NT , sNH )
(11)
The excitation matrix F and coefficient matrix S can be obtained by some simple transformations for Eq.(1) and
Eq.(2). The solution vector X is found by an iterative Newton-Raphson procedure. The residual matrix R(X) is:
R( X ) = S ⋅ X − F − D ( X )
(12)
the iterative equations are:
J ( X ) ⋅ ΔX = − R( X ) , ΔX = − J ( X ) −1 R( X ) , X k +1 = X k + ΔX
(13)
where, J(X) is Jacobian matrix:
J(X ) =
∂R
∂D
=S−
∂X
∂X
(14)
In order to calculate D(X) and ∂D / ∂X by numerical FFT, the functions are discretized with NP points per period. NP
is the power of 2 for acceleration of transformations. The time step is:
Δt =
2π
ωNP
(15)
The displacement and velocity are:
.
.
x k = x (kΔt ) , x k = x (kΔt ) , k=1,…,NP-1
(16)
The coefficients of discretized solution can be obtained by:
D j ,cn ( X ) =
.
.
.
2π
ω NP −1
cos(
nk
)
g
(
x
,
x
,
x
j
−
1
j
j +1 )
∑ NP
j
π k =0
(17)
.
.
.
2π
ω NP −1
D j , sn ( X ) = ∑ sin(
nk ) g j ( x j −1 , x j , x j +1 )
NP
π k =0
(18)
where, g j is given by Eq.(3). Eq.(17) and Eq.(18) can be solved by FFT at a same time.
Matrix ∂D / ∂X has (2NH+1)×(2NH+1) blocks, each block matrix is composed of N×N blade matrices, and the size
of these matrices is determined by DOF of the corresponding blade. The matrices are classified to four types:
∂Dcn / ∂X cm , ∂Dcn / ∂X sm , ∂Dsn / ∂X cm , ∂Dsn / ∂X sm .
∂D j ,cn ( X )
∂X j ,cm
∂D j ,cn ( X )
∂X j −1,cm
∂D j ,cn ( X )
∂X j +1,cm
.
.
.
∂g j ,k ( x j −1,k , x j ,k , x j +1,k )
2π
ω NP −1
nk )
= ∑ cos(
NP
∂X j ,cm
π k =0
.
.
.
∂g j ,k ( x j −1,k , x j ,k , x j +1,k )
2π
ω NP −1
= ∑ cos(
nk )
NP
∂X j −1,cm
π k =0
.
.
(19)
(20)
.
∂g j ,k ( x j −1,k , x j ,k , x j +1,k )
2π
ω NP −1
= ∑ cos(
nk )
NP
∂X j +1,cm
π k =0
where,
⎯ 1151 ⎯
(21)
∂g j ,k
=
∂X j ,cm
∂h j , j −1,k
∂X j ,cm
.
.
∂ ( x j ,k − x j −1,k
.
.
∂h j , j +1,k
∂ ( x j ,k − x j −1,k )
⋅
+ .
.
∂X j ,cm
∂( x − x
)
∂h j , j −1,k
=(
∂X j ,cm
.
∂h j , j −1,k
=
∂h j , j +1,k
+
j ,k
∂h j , j +1,k
+
.
.
∂ ( x j ,k − x j −1,k )
In the similar way,
.
,
∂X j −1,cm
.
(22)
.
)⋅
∂ ( x j ,k − x j +1,k )
∂g j ,k
j +1, k
.
∂ ( x j ,k − x j +1,k )
⋅
∂X j ,cm
)
∂ x j ,k
∂X j ,cm
∂g j ,k
can also be obtained.
∂X j +1,cm
From Eqs. (3-5):
.
2π
∂ x j ,k
= − mω sin(
mk )
∂X j ,cm
NP
(23)
.
.
⎧
/
λ
,
d
x
x
−
−
j
j −1 ≤ λ
⎪⎪ j −1, j
∂h j , j −1,k
=⎨
.
.
.
.
∂ ( x j ,k − x j −1,k ) ⎪
0, x j − x j −1 > λ
⎪⎩
(24)
.
.
⎧
/
λ
,
d
x
−
j − x j +1 ≤ λ
,
1
j
j
+
⎪⎪
∂h j , j +1,k
=⎨
.
.
.
.
∂ ( x j ,k − x j +1,k ) ⎪
0, x j − x j +1 > λ
⎪⎩
(25)
Then:
∂D j ,cn ( X )
∂X j ,cm
=
∂h j , j −1,k
ω NP −1
2π
2π
nk ) ⋅ (−mω sin(
mk ))( .
cos(
∑
.
NP
NP
π k =0
∂( x − x
j −1, k
j ,k
∂h j , j +1,k
+
)
.
)
.
(26)
∂ ( x j ,k − x j +1,k )
After some trigonometry transformations, the final expression is:
∂D j ,cn ( X )
∂X j ,cm
=−
mω 2
2π
k =0
∂X j , sm
∂X j ,cm
and
∂h j , j −1,k
2π
∑ (sin NP (n + m)k − sin NP (n − m)k )(
∂D j ,cn ( X ) ∂D j , sn ( X )
,
2π
NP −1
∂D j , sn ( X )
.
.
∂h j , j +1,k
+
∂ ( x j ,k − x j −1,k )
.
.
(27)
)
∂ ( x j ,k − x j +1,k )
can also be obtained in the similar way.
∂X j , sm
Then we can also calculate the parameters:
∂D j ,cn ( X )
∂X j −1,cm
,
∂D j ,cn ( X )
∂X j −1, sm
,
∂D j , sn ( X )
∂X j −1,cm
,
∂D j , sn ( X )
∂X j −1, sm
,
∂D j ,cn ( X )
∂X j +1,cm
parameters can all be efficiently solved by the FFT method.
,
∂D j ,cn ( X )
∂X j +1,cm
,
∂D j ,cn ( X )
∂X j +1,cm
and
∂D j ,cn ( X )
∂X j +1,cm
. These
EXAMPLES AND DISCUSSIONS
The real bladed disk with blade-to-blade dry friction is simulated by the mass-string model, as Fig.1. the parameters of
the system are defined as: N is the number of blades, m is the mass of blades, k is the blade stiffness, k c is coupling
between blades, c is viscous damping, ω is excitation frequency, ω b is the nominal natural frequency of blades,
⎯ 1152 ⎯
Ω = ω / ω b is frequency parameter, σ is standard deviation of mistuning, ξ = c / 2 km is viscous damping ratio,
R 2 = k c / k is the strength of coupling between blades, hi ,i −1 is the dry friction between the ith and i-1th blades, g i is
the resultant dry friction forces of the ith blade, p = f 0 / d 0 is the force ratio of excitation to dry friction, xi is the
displacement of the ith blade.
xj-1
kc
kc
kc
hj,j+1
hj-1,j
k
c
xj+1
x
m
k
m
c
k
m
c
Figure 1: The model of the bladed disk with blade-to-blade dry friction damping
Two examples are employed to validate the MH method presented in this paper. The forced responses of the bladed
disk with blade-to-blade dry friction damping are simulated by the MH method and the fourth order Runge-Kutta
numerical integration (NI) method. The engine order excitation r =1, and the responses of the first blade are shown in
Fig. 2.
It can be seen from Fig. 2 that, for the weakly coupled system (Fig. 2(a)), one harmonic can give accurate result; for the
strongly coupled system with strong dry friction damping, the accurate result can only be obtained with including
higher order harmonics (Fig. 2(b), NH=3). These examples indicate that the MH method can be used to analyze the
forced response of bladed disks with blade-to-blade dry friction damping. The following examples all use
three-harmonics for solving the maximum forced responses of bladed disks with blade-to-blade dry friction, and the
engine order excitation r=1.
(a) R=0.1, ξ =0.001, Ω =1, p=1.5
(b) R=0.5, ξ =0.004, Ω =1, p=1
Figure 2: Comparison of the response results between the MH method and the NI method
RESPONSES OF THE TUNED BLADED DISKS
In tuned bladed disks, the properties of all blades and friction dampers are identical respectively. The strength of
coupling and viscous damping are two key factors which affect the characteristics of maximum responses of bladed
disks. Besides these two factors, the force ratio is also considered in following examples.
1. Effects of coupling strength Fig. 3 displays the effects of coupling strength on the maximum responses of tuned
bladed disks. It is found that increasing coupling strength shifts the maximum resonant regions into higher frequencies.
Besides that, we can also find that coupling strength has a different way of effect on maximum resonant amplitudes for
different system’s viscous damping. For small viscous damping (Fig. 3(c)), resonant amplitudes of the system are
similar with each other for different coupling strength, the maximum amplitude is around the medium coupling region;
when the viscous damping is larger (Fig. 3(a, b, d)), the resonant amplitude decreases with an increase of coupling
strength.
⎯ 1153 ⎯
(a) ξ =0.004, p=1.5
(b) ξ =0.004, p=3
(c) ξ =0.001, p=1.5
(d) ξ =0.008, p=1.5
Figure 3: Effects of coupling strength on maximum amplitudes of the tuned bladed disk
For various viscous damping ratios and coupling strengths of the tuned system, larger force ratios (value p) produce
larger resonant amplitudes, but have little effects on resonant frequencies.
2. Effects of viscous damping It is observed from Fig. 4 that the resonant amplitudes of the tuned system are much
sharp when the viscous damping of the system is small. Increasing viscous damping decreases resonant amplitudes
significantly. The reductions of resonant amplitudes are similar for both weakly and strongly coupled systems.
(a) R=0.1, p=1.5
(b) R=0.5, p=1.5
Figure 4: Effects of viscous damping on maximum amplitudes of the bladed disk
RESPONSES OF THE MISTUNED BLADED DISKS
The mistuning of the bladed disk is considered as the blade stiffness mistuning, and its distribution complies with the
normal distribution function. The degree of mistuning σ is 4%. Figs. 5-7 display the features of maximum amplitudes
of the mistuned bladed disk with various viscous damping, coupling strengths and the force ratios. With small viscous
damping, weak coupling and large dry friction, the resonant amplitude of the mistuned bladed disk deceases
significantly due to the existence of mistuning. This resonant amplitude reduction suggests that the mistuning likely be
a benefit for suppressing the resonant vibration in this specific case (Fig. 5(a)). On the other hand, mistuning broadens
the resonant region. When the dry friction is small, mistuning can produce multiple resonant peaks (Fig. 5(b)), and
⎯ 1154 ⎯
create larger resonant amplitudes in low frequency regions. These results indicate that the smaller dry friction damping
is not good for suppressing resonant vibrations of the mistuned system; even the maximum amplitude is less than that
of the tuned system. In the strongly coupled system (Figs. 5(c, d)), the resonant frequencies and the amplitudes
converge to those of the tuned system respectively, and the resonant peaks far from the resonant frequency of the tuned
system are damped out.
(a) R=0.1, ξ =0.001, p=1.5
(b) R=0.1, ξ =0.001, p=3
(c) R=0.5, ξ =0.001, p=1.5
(d) R=0.5, ξ =0.001, p=3
Figure 5: Effects of small viscous damping on maximum amplitudes of the mistuned bladed disk ( ξ =0.001)
(a) R=0.1, ξ =0.004, p=1.5
(b) R=0.1, ξ =0.004, p=3
(c) R=0.3, ξ =0.004, p=1.5
(d) R=0.5 ξ =0.004, p=1.5
Figure 6: Effects of the medium viscous damping on maximum amplitudes of the mistuned bladed disk ( ξ =0.004)
⎯ 1155 ⎯
With medium viscous damping (Fig. 6), resonant amplitudes of the mistuned weakly coupled system are less than
those of its corresponding tuned system (Figs. 6(a, b)). In smaller viscous damping case (Fig. 6(b)), the resonant region
of the mistuned system is broader than that of the tuned system, but the resonant amplitudes are not sharper than those
of the tuned system. Increasing the coupling, the resonant amplitudes of the mistuned system exceed the resonant
amplitudes of the tuned system, but the resonant amplitudes of the mistuned system are reduced with further increasing
of the coupling, as shown in Fig. 6(d) compared with Fig. 6(c). These findings indicate that the stronger coupling has
an effect of decreasing resonant amplitudes of the mistuned bladed disk when the viscous damping of the system is in
medium range.
With larger viscous damping (Fig. 7), resonant amplitudes of the mistuned weakly coupled system approach the
resonant amplitudes of its corresponding tuned system (Figs. 7(a, b)), and the resonant region of the mistuned system
is still broader. In the mistuned strongly coupled system, the trend of the resonant responses of the mistuned system is
similar to those in the medium viscous damping case, but with less sharp resonant amplitudes.
(a) R=0.1 ξ =0.008, p=1.5
(b) R=0.1 ξ =0.008, p=3
(c) R=0.3 ξ =0.008, p=1.5
(d) R=0.5 ξ =0.008, p=1.5
Figure 7: Effects of the large viscous damping on maximum amplitudes of the mistuned bladed disk ( ξ =0.008)
CONCLUSIONS
In this paper, based on the HBM, we present an efficient multi-harmonic method to analyze the resonant responses of
mistuned bladed disks with blade-to-blade dry friction damping. The resonant characteristics is simulated and
analyzed with various system parameters for both tuned and mistuned systems. The investigation indicates that:
(1) The MH method presented in this paper can predict efficiently the forced response of bladed disks with
blade-to-blade dry friction. The simulation results of the method are proved better than those of previous methods
which only include one harmonic.
(2) In the tuned bladed disk, stronger coupling can increase the resonant frequencies and decrease resonant amplitudes
when the viscous damping is larger. It has little effects on resonant amplitudes when the viscous damping is small.
(3) With the small viscous damping and weak coupling, mistuning can reduce the resonant amplitude of the bladed
disk; when dry friction is smaller, mistuning can produce multiple resonant peaks and make the resonant region
broader. Increasing coupling, resonant frequencies and amplitudes of mistuned system approach those of the tuned
system respectively.
(4) With larger viscous damping, resonant amplitudes of mistuned weakly coupled system are less than those of the
tuned system, and the resonant region of the mistuned system is still broader. Increasing viscous damping makes
resonant peaks close to those of the tuned system; with strong coupling, resonant amplitudes of the mistuned system
⎯ 1156 ⎯
exceed those of the tuned system. Meanwhile, smaller dry friction leads to larger resonant amplitudes, but doesn’t
change the trend of resonant responses of the mistuned system.
Acknowledgement
This work is supported by the National Natural Science Foundation of China (No.50275121)
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⎯ 1157 ⎯