'ifietnam Joumai of Mechanics, VAST, Vol. 37, No. 3 (2015), pp. 2 1 7 - 230 IX)L10.15625/0866-7136/37/3/6474
EXTENSION OF DUAL EQUIVALENT LINEARIZATION TECHNIQUE TO FLUTTER ANALYSIS OF TWO
DIIVIENSIONAL NONLINEAR AIRFOILS
Nguyen Minh Triet
VNU University of Engineering and Technology, Hanoi, Vietnam 'E-maU: [email protected]
Received June 26,2015
Abstract Ttiis paper extends the dual equivalent linearization technique (ELT) to obtain flutter speeds of a two-dimensional airfoil with nonlinear stiffness and damping in pitch degree of freedom. Although the use of dual ELT has been investigated in some previous papers, this paper presents an extension of dual ELT usmg the global-local approach, in which the local equivalent linearization coefficients are averaged in the global sense. The numerical calculation shows that the extended dual ELT gives more accurate flutter speeds in comparison with the ones of classical ELT
Keywords: Airfoil flutter, dual equivalent linearization technique, global-local approach, limit cycle oscillation, nonlinearity.
1. INTRODUCTION
The most dramatic physical phenomenon in the field of aeroelasticify is flutter, a dynamic instabilify which often leads to catasfrophic structural faUure. Classical theo- ries of linear flutter have been presented for a long time [1,2]. However, the real system exhibits the nonlinear behavior due to the confrol mechanisms or the connecting parts between wing, pylon, engine, external stores or the large deflection. NonUnear airfoil flutter is a typical seff-excited vibration with rich nonUnear dynamical behaviors, such as limit cycle osciUation (LCO), bifurcation and chaos [3]. Nonlinear aeroelasticify has been a subject of high interest the Uterature is now extensive [4,5], in which many issues are StiU imder active investigation. In the context of nonlinear aeroelastidfy, a LCIO is one of the simplest dynamic bifurcations but is a good general description for many nonlinear aeroelastic behaviors. The LCO may occur once the dynamic stabiUfy (flutter) bound- ary has been exceeded. The LCO of the afrfoils with stiffness nonlinearities has been investigated by several methods such as harmoruc balance method [6,7], center mani- fold theory [8], point fransformation method [9], perturbation-incremental method [10],
© 2015 \^etnam Academy of Science and Technology
218 Nguyen Minh Tnet
precise integration method [11], numerical methods [12,13] and equivalent linearization technique (ELT) [14,15].
The ELT was widely appUed to various nonlinear vibration problems because of its simpUdfy and effectiveness. The approximate solution obtained by the ELT gives some dear physical explanation of the norUinear phenomenon. The most important procedure of ELT is to derive an equivalent linear system based on some certain equivalent condi- tions. In the field of djmamical stochastic nonlinear system, the ELT has a long history of development and a numerous versions of ELT have been proposed [16]. Recentiy, in a series of papers, Anh et al. [17-21] proposed and proved the effectiveness of a so-caUed dual ELT in nonlinear stochastic systems. The dual approach has been also successfuUy appUed to the problem of tuned mass damper design [21-23]. fri [20,23], the global- local approach appUed to the dual criterion has been presented. In this paper, using the global-local approach, we extend the dual ELT by averaging the equivalent linearization coeffidents and apply successfuUy the extended ELT to the flutter LCO problem.
2. EQUATIONS OF MOTION
Fig. 1 shows a classical spring-supported afrfoU section. The model represents an airfoU with a single bending and a single torsional mode. This two-dimensional model is a typical section of a three-dimensional wing, where the spring constants are adjusted to match the actual uncoupled free vibration frequencies of ttie wing, whUe the mass and geomefric properties are taken as tiiose of a typical section. The two-dimensional model can give the approxfrnated critical flutter speed of Uie actual wing with the location of the typical section is generaUy taken in ttie neighborhood of 0.7 span from the root [2].
Fig. 1. AirfoU section supported by vertical and torsional springs
As shown fri Fig. 1, tiie afrfoU semi-diord is denoted as b. Tlie elastic axis of ttie model 15 located at a distance ab irom tiie mid-diord. The mass center (M.C) is located at a distance x b from ttie elastic axis. The plunge deflection (vertical displacement at flie elastic axis) h is positive downward. The pitch angle (angle of twist) « is positive
Extension of dual equivalent linearization technique to flutter analysis of two dimensional nonlinear airfoils
nose-up about the elastic axis. The equation of motion of the aeroelastic model has the form [24-28]
[ mj ntwXab] \h
[mwx^b U
0 cJ Uj + io" fcj [«J + [ 5 ( « , « ) J " I M J '
(1)where m„, is the mass of the wing, mj is the total mass, !« is mass moment of friertia, kh and fca are the linear plunge and pitch stiffness, Cf, and Ca are the Unear damping coef- ficients, g (a, a) is a general nonlinear function of the pitch angle a and its derivative a.
The aerodynamic moment M and Uft L are assumed having the quasi-steady form L = pU^bsci„ [ct-\- — +{
M = pU^hCn,,
(" + 0 +
(2)
where U is the free stream velocity, p is the air density, s is the wing section span, Cj„ and Cma are the aerodynamic lift and moment coefficients per angle of attack. The equation of motion (1) can be transfonned to the following state-space form
0 0 0 0 -ki -k2{U) -ki -kt{U) where the coefficients are defined as
.kh!.
1 0 -Cl (U) - C 3 ( U )
.k,-- 0 1 -C2{U) - C 4 ( U )
-mwXabki, r h 1
a h
BL
\ " 1
0 Pi L V2 J
? ( « , * ) - (3)
'"T-^ff ~ {l^W^a.b) ,fcl = pU^bS , ,n , \ , : !—^ \m}N^a(rCma. + IflC/a) ,^4 =
-pmh
•,p\-- -bmy^Xoi
P2-
'' T '
{'"TCma + WIW^aCjtt),
pUb^S [niwXa^Cma + laCla) {^^ ~ j) + bntw^ttCa pUb^S {niTCmtt + mwXaCli:,) + bmwXctCh pUb^S {mTCma + ^WX^^la) {'^ - I) + ^T^a.
3. EQUIVALENT LINEARIZATION TECHNIQUES 3.1. Classical ELT
Consider the foUowing flutter LCO response
0. = Asm(p, a = Bcosq>, (4) where A and B respectively are the ampUtudes of the pitch angle and its derivative, (p=iot
where CO is the LCO frequency. The nonlinear function g is linearized as
g{a,a) = keOi + Cedc (5)
220 Nguyen Minh Triel
where the equivalent stiffness k and equivalent damping c, are found by a certain op- timal criterion. There are many criteria for this purpose but the most extensively used criterion is the mean square error criterion which requires the foUowmg error be mim- mum
' (g («, 4) - ke" - Cexfdip ^ m i n . (6) The minimum conditions
d I (g(ic,i.)- k,x - Ceifdf a y (g(«,a) - ^f" - Ceinfiif
= 0 , - 2 ^^ = 0 ,
dk. Sc,
yield
h h
(7)I eficf i o?-df using (4) in (7) gives
In 2n kc =—— I g{Asm(p,BcoS(p)smfdip, c, =—^ g{Asinf,B€OSf)cosfd(p (8)
TZA J Tza J
0 0 for a certain type of nonlinear function, the equivalent stiffness kg and damping Ce in (8) are the functions of the pitch amplitude A and pitch velodfy ampUtude B.
3.2. Dual ELT and its extension
In [17-20], Anh et al. proposed a so-caUed dual criterion to obtain a new ELT.
This type of ELT has showed its effectiveness in analyzing a numerous class of nonlinear stochastic systems. In [18], the authors stated that the classical ELT is based on replacing the original nonlinear system by an equivalent linear one. The dual approach does not orUy consider this "forward" replacement but also the "backward" replacement, in which the obtained equivalent linear system is replaced by another nonlinear one that belongs to the same dass as the original nonlinear system. In [19], the authors also proposed the weighted dual criterion, which is used in the foUowing dual criterion of this paper
(1 - p) / {g (a, a) - keO. - CeCi)^d<p + p {\g {oc, a) — ketx - CeCi)^d(p -> min (9) where the first term describes the conventional replacement and second term is its dual replacement, p is a given weighting coeffident. If p = 0 we obtain flie classical ELT. If
Extension of dual ftjuhatenl lm,anzalkm tedmique lo flutter mvdwsis of tuvdnnaismmdrmnlmxiTairfinls 221 p = 1/2 w e obtain the dual ELT presented in [18]. Three minimum conditions of (9)
/ 2JI 2,T \
S I (1 - P) / (X ( t , 4) - *.» - Ceicfdq} + v j (\g («, a) - teO - Ceicfdip j dk,
(
2n Ur \(1 - p) y (g (a,4) - M - Cei.fdip + pj(\g (a,4) - k,ix - Ceiifdcp \ dc,
/ 2.T 2 T
s\(l-p) j {g{'',oc)-k,x- Ceicfdip + p j (\g(ec,ic)- k,ic - dc)'
-- 0, (11)
yield
= 0, (12)
k, = ^ ^ , ce=l^^^, , (13)
1 - p/J 2n 1 - PP 2?
•_o
where p as stated above is a given value between 0 and 1/2 l 0 < p < - J , ^ i s a notation to simplify the formula and is given by
(14) 2n \ /27I \ / 2 r r \ / 2n
Ug'd.p] Uic^dfj U^dJ U^^dip
Using (4) in (13) and (14) gives
2!Z
ke = .^—j^^~-r J g(Asmf,Bcosip)smfdip, (15)
„Bll-pu) / g ( ^ s i n y , B c o s y ) c o s y d y .
I g{Asm(p,Bcosip)sinqidqt | + I g{Asm(p,Bcos(p)cos (16) n / ( ^ ( A s u i ^ , B c o s ^ ) ) ^ d ^
However, the equivalent stiffness and damping (15) obtained from the weighted dual criterion ELT (9) depend on the weighting coefficient p. In this sense, the equivalent stiffness kg and damping Cg in (15) can be considered as local equivalent linearization co- efiidents. It aUows the flexibility in varying the local parameter p. However, ttie main disadvantage of the local solution is that there is stiU no clear way to find the parameter p. Using the global-local approach presented in [20,23], it is suggested that instead of finding a special value of p, one may consider its varying in the global domain of inte- gration. Thus, the equivalent coefficients can be suggested as the mean values of aU local equivalent coefficients as foUows
k,g = 2 f kedp = 1 ( 0 0 1/2 1/2 Ceg = l I Cgdp = 2 I
1 - nA{l
1 - 7 l B ( l
P
- P r i
P
-m)
ls(A
sm cp, B cos q)) sm q1 sm q>, B cos (p) cos a
^-e-^K-D)
JgiA
5ui\j to
2n
/ g {A sin (p,B cos cp) sin q>d<p dp,
dp.
(17)
in which keg and Ceg, respectively, denote the equivalent stiffness and damping obtained by global-local approach. The integrals reduce to
J g{Asmip,i
(18)
In brief, Uie extended dual ELT gives the equivalent stiffness and damping as (18) where the notation p is taken from (16).
4. FLUTTER SPEED PREDICTION AND STABILITY ANALYSIS 4.1. Flutter speed piedictton
We will use ttie equivalent linear coefficients (8) and (18) for further flutter analysis.
The linearized equation of (3) has form
Extension of dual equivalent linearization technique to flutter analysis of two dimensional nonlinear airfoils
0 0 -k.
-k.
0 0 -k2 - Pike -ki - pzk.
1 0 - C l - C 3
0 1 - C 2 - PlC - C 4 - P2C
(19) The response of (19) is convergent when the flow velodfy is smaUer than a critical value (flutter speed). When the flow velodfy is h i ^ e r than flutter speed, the LCO wiU occur.
We can assume the occurrence of following phenomenon. When the flow velocity is smaUer than flutter speed, ttie linear system (19) has two pairs of conjugate poles with negative real parts, which means the convergence of the response. When the flow velocity reaches the flutter speed, a pafr of conjugate poles becomes purely imaginary, which means the occurrence of LCO. Denote flie poles of (19) at flutter speed as ±ico, —^i ± iiO\, where i is the imaginary unit, a? is the LCO frequency as denoted above, ^\, coi are positive real number. TTie characteristic polynomial of (19) wiU has form
P (A) = det 0 0 -kl -k3
0 0 - i z - Pi*.
-ki - p2k.
1 0 - C l - C 3
0 1 - C 2 - PlC -Cl - P2C
-Mt -. (A - ico) (A -I- ia) (A + fl - iwi) (A + fi + iwi),
(20)
in which the variable in the brackets are omitted for simplicity, I4 is the 4 x 4 identity mafrix. Equating the coefficients of the polynomials in (20) gives four algebraic equations to solve four variables co, f i , o^i and U. The solutions can be obtained in the following form
_ Cl + C4 + P2C,
<A = P2*e - ("^2 + piC,)Ci + ki-0J^ + ki+ Cl (Ci + P2C,) - Cl 2 _ -Cik2 + kl (C4 + P2C,) - ki (C2 + piCe)+ kjCj + p2k,Ci - c^pik,
Ci + Ci + p2Ce
kiki - kik2 + kip2k, - k^pik, - oi^pz^c + c'' (cz -I- Pic,) ca
— w^ki +aj^ — co^ki — w^ci (C4 + p2Ce) = 0 By substituting a; from (23) to (24) and by noting that
(21) (22) (23) (24)
Eqs. (23) and (24) change to B^ _ -Cik2 + ki(Ci + p2C,)- A^~
kl (c2 + pic,) + kjCi + piKci ~ cspike C1 + C4 + p2Ce
{kl +ki + CiCi - C2C3 + p2k, + P2C,Ci - pic,cz) (Ci + C4 + P2C,) X (kiCi - csfcz - JTSCZ -t- *:4Ci -I- P2*:iCe + P2keCi - piksc, - pikeCi) -{kiCi - C3k2 - )t3C2 -I- J:4Ci + piklCe - PlklC, + pzkeCj - Czpik,f = (kiki - k3k2 -I- kip2k, - JrspiA:.) (ci -1C4 -I- p2Cef
(25)
(26)
224 Nguyen Mmh Triel
By noting that the equivalent stiffness ke and equivalent damping Ce are ttie func- tions of A and B (as seen ki (8) or (18)) and fcz, h, a (i = 1 , . . . ,4) are the polynomial fimctions of flow velodty U, Eqs. (25) and (26) give tiie relation between Uie flutter speed and the pitch ampUtude A or B. Moreover, Eq. (26) is a polynomial equation of U. After some manipulation, Eq. (26) can reduce to a qufritic equation of U, whose lowest positive root is the flutter speed.
The relation between U and A can be obtafried by the foUowfrig procedure.
- Given a certain value to A.
- By solving the quintic equation (26), we can express U as the function of B. It is noted that the quintic equation is solved very fast and efficiently by the "roots" function fri MATLAB.
- Substitute U as a function of B to (25), we obtafri a norUinear scalar equation with respect to B. It is noted ttiat the nonlinear scalar equation is solved very fast and effidentiy by the "fzero" function in MATLAB.
- After solving tiie nonlinear scalar equation to obtain B, U is determined because it is a function of B as stated above.
- At last, for a certain value of A, we can obtain the corresponding value of U. Then the relation between U and A can be drawn.
The relation between U and B can be obtained in the simUar marmer by changing the roles of A and B.
4.2. Stability condition
For a flutter speed, there can be one or more corresponding pitch amplitude. How- ever, orUy the stable amplitude can occur in practice. The foUowing condition [29] may be used to determine the stability of LCO in the presence of ampUtude perturbations. For a limit cycle with ampUtude A, the linearized system has two eigenvalues with zero real parts (as shown in (20)). SmaU perturbations in the limit cycle amplitude make changes in these two eigenvalues. Denote the change of the real part as Atr. If a positive ampUtude perturbation (AA > 0) results in the negative real parts (Ao" < 0) of the aforementioned eigenvalues, the LCO is stable because the energy is dissipated untfl the ampUtude de- cays to its unperturbed value. SimUarly, a negative ampUtude perturbation (AA < 0) results in the positive real parts (Atr > 0) of the eigenvalues also forms a stable LCO because the ampUtude grows until the unperturbed LCO is again attafried. In brief, the condition
Atr.AA < 0, (27) defines a stable LCO.
In calculation, the stabiUty condition is checked by the foUowing procedure. With a certain value of ttie pitch ampUtude A, ttie flutter speed is obtafried by the procedure presented in section 3.1. Then the flow velodty is fbced at the flutter speed whfle the pitch ampUtude A is given a smaU perturbation AA. The ampUtude of pitch velocity B is obtained by scalar equation ... Then the new roots of the characteristic poljmomial P(A) in (20) are obtained. Two purely imaginary roots now become two conjugate roots with nonzero real part, denoted as A(7. ff two perturbations ACT and AA satisfy the condition (27), the corresponding state of flutter is stable.
Extensum of dual equivalent linearization tedmitfue to flutter analysis of tioo dimensional nonlinear airfoils 225
5. NUMERICAL SIMULATION
In this section, two numerical examples presented in [24] are performed to verify the proposed ELTs. In [24], the numerical examples consider the leading-edge a n d / o r fraUing-edge confrol surfaces. However, because the control problem is beyond ttie scope of this paper, orfly the open loop systems are analyzed. The nonlinear function g {ec, a.) in both examples is expressed in polynomial form u p to 5 * order as foUows
g = k^20c^ + k^oc^ -\- k^ia* -I- k^a^. (28) Substituting the nonlinear form (28) into the formula (16) of p, and the equivalent stiff-
nesses (8) or (17) give the foUowing formulas
3A^fc^3 SA^k^sV 4 8 7
3A2fc„3 , 5A*k^\
where y is the coefficient depending on the ELT used as foUows
* For the dassical ELT: 7 = 1
* For the extended dual ELT: 7 = - -h ^^^~^^ In f 1 - ^"l p p^ \ 2 / 5.1. Example 1
The numerical values of parameters of Example 1 are shown in Tab. 1.
Table 1. Numerical values of parameters of Example 1 A
-0.6847 Ch (kg/s)
27.43 k,2 (N.m)
9.967
Km)
0.135 C|«
6.38 k.3 (N.m)
667.685 ntT (kg)
12.387 Cma -1.16 kai (N.m)
26.569
mw (kg) 2.049 s(m) 0.6 )c.5 (N.m) -5087.931
I. (kg.m2) 0.0558 p ( k g / m ' )
1.225
j : «
0.3314 c, (kg.m'/s)
0.036
kn ( N / m ) 2884.4 t. (N.m)
6.833
The numerical solutions are obtained by the ode45 function in MATLAB. The fol- lowing process is carried out to plot the numerical relation between the flutter speed and the pitch ampUtude:
- Ffrst, an initial condition of pitch angle a is set. Other initial conditions {h ( 0 ) , ft ( 0 ) , ft (0)) are set to zeros.
- GraduaUy increase the flow velodty U. For each flow velodty, the nonlinear differential equations (19) are solved from Os to 120s. ff the response converges, the flow
226 Nguyen Minh Tnet
velocity is smaUer than the flutter speed. When the response starts to make a LCO, the flow velodfy is the flutter speed and the ampUtude of LCO is recorded.
- Increase the initial pitch angle fl;(0) and repeat the step 2. The initial pitch angle is increased untU the pitch ampUtude has no significant changes.
The relation between the initial pitch angle and the flutter pitch amplitude (calcu- lated numericaUy) is shown in Fig. 2. Fig. 3 shows the flutter boundary obtained by nu- merical calculations and two ELTs. Tab. 2 shows the comparisons of some flutter speeds.
0145 T 0 14 1 0 135 I " 0.13
•1. 0.12 2 0 115 • 1 O i l 0.10S • 0.
• • •
1 0.03 0 05 0 07 0.09 O i l Iniliil pitch 1 1 ^ (1(0)
•
• • • •
0.13 0 13
Fig. 2. Relation between initial condition and flutter pitch ampUtude in Example 1
: • - /
Chsscal ELT
^ ^ • - . y / ^ ^ E l i d e d j-^r dualELT
NsmiL (maiker)
Flutter pilch iiqilitude
Fig. 3. Relation between flutter speed and flutter pitch ampUtude in Example 1 5.2. Example 2
The numerical values of parameters of Example 2 are shown in Tab. 3.
The relation between the initial pitch angle and the flutter pitch amplitude (calcu- lated numerically) is shown in Fig. 4. Fig. 5 shows the flutter boundary obtained by nu- merical calculations and two ELTs. Tab. 4 shows the comparisons of some flutter speeds
Extensionc^dualequizialent linearization technique to flutter analysis of two dtmensional nonlinear airfbds Table 1. Comparisons of flutter speeds in Example 1
Numerical calculation (previous paper [24]) Numerical calculation
(this paper) Classical ELT Extended dual ELT
Response at large ampUtude (large nonlinearify) Flutter pitch
ampUtude (rad)
0.1485
Flutter speed (m/s)
11.6 11.69 12.2744 (5% larger than numerical value) 11.5305 (1.36% smaUer than numerical value)
Minimum flutter speed (m/s) 7.9 7.94 7.9484 7.9472 Table 3. Numerical values of parameters of Example 2
A -0.6719 Ci (kg/s)
27.43
*:«2 (N.m) 53.47
Mm) 0.1905 C|«
6.757
*«3 (N.m) 1003
" I T (kg) 15.57
Cm&
-1.162 k„i (N.m)
0
mw (kg) 5.23 s(m) 0.5945 kes (N.m)
0
I . (kg.m2) 0.1419 p ( k g / m ' )
1.225
^tt
0.5721 c„ (kg.m^/s)
0.0184
** ( N / m ) 2844 K (N.m)
12.77
Fi^. 4. Relation between initial condition and flutter pitch amplitude in Example 2
Nguyen Minh Tnet
UnsBbk curves
'--C^
Classical ELT
S.,
Flutte r pitcb anqiUnxk
Fig. 5. Relation between flutter speed and flutter pitch amplitude in Example 2 Table 4. Comparisons of flutter speeds in Example 2
Numerical calculation (previous paper [24]) Numerical calculation
(this paper) Classical ELT Extended dual ELT
Response at large ampUtude (large nonlinearity) Flutter pitch
ampUtude (rad)
0.1746
Flutter speed (m/s)
11.4 11.3 11.4481 (1.31% larger than numerical value) 11.2852 (0.13% smaUer than numerical value)
Minimum flutter speed (m/s) 10.6 10.525 10.5249 10.5248
5.3. Discussions on results
- As seen in Tab. 2 and Tab. 4, the numerical resiflts in this paper are in good agree- ment with those in previous paper [24]. This guarantees the reliabiUfy of the numerical procedure in this paper.
- fri Figs. 2 and 4, there is a jump phenomenon in the relation between the irutial pitch angle and flutter pitch amplitude. When the initial angle is smaU, the flutter pitch amplitude suddenly jumps to a large value. Increase the initial angle more, the flutter pitch ampUtude decreases and tends to a constant value. The jump phenomenon clearly shows the norUinear effect.
- In Figs. 3 and 5, aU the methods show the existence of a minimum flutter speed.
The comparisons in the last column of Tabs. 2 and 4 show that both ELTs can predict the
Exlension of dual etpaoaienl linearization tedmupie to flutter anaiysisf^troo dimensional nonlinaa airfoils 229 minimum flutter speed quite weU. It is noted that the ELTs can find the minimum flutter speed quite simply by solving the quintic equation (24).
- The proposed extended dual ELT based on weighted dual criterion and global- local approach shows its accuracy in the region of large nonlinearify (large amplitude).
As shown in Tabs. 3 and 5, in Examples 1 and 2, the errors of the extended dual ELT are about l / 3 a n d l / 1 0 o f fhe error of the dassical ELT, respectively.
6. CONCLUSIONS
The equivalent linearization technique (ELT) in this paper is improved by a so- caUed weighted dual criterion and global-local approach. The dual criterion consists of "forward" and "backward" replacements with a local weighting coeffident. The ob- tained local equivalent coeffidents are then averaged to be the global one. The proposed ELT then is appUed to predict the flutter limit cyde osciUation of an afrfoU section with nonlinear pitch stiffness and nonlinear pitch damping. Two numerical examples of the nonlinear polynomial pitch stiffness have been carried out. The results show that the pro- posed ELT can reduce the error to about 1/3 (in Example 1) and 1/10 (in Example 2) of the error given by the classical ELT. It is expected that the extended dual ELT can be used as an alternative effective tool for flutter analysis of more compUcate nonUnear systems.
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