Variable-order finite elements for nonlinear, intrinsic, mixed beam equations

Transcription

1 Variable-order finite elements for nonlinear, intrinsic, mixed beam equations Mayuresh J. Patil 1 Dewey H. Hodges 2 1 Department of Aerospace and Ocean Engineering Virginia Polytechnic Institute and State University, Blacksburg, Virginia 2 School of Aerospace Engineering Georgia Institute of Technology, Atlanta, Georgia 62nd Annual Forum and Technology Display of the American Helicopter Society International Phoenix, Arizona, May 9 11, 26

2 Rotor Blade Analysis Beam Equations continuous PDEs nonlinear Solution of Beam Equations discrete in space ODEs nonlinear Basis Reduction low-order, high-fidelity? nonlinear Application aeroelasticity control MDO

3 Beam Equations Well established geometrically-exact formulations Borri and Mantegazza (1985) Bauchau and Kang (1993) Hodges (199) Simo and Vu-Quoc (1988) Ordering schemes have been shown to result in unnecessary approximations, potentially erroneous results, lengthy and cumbersome equations Hodges and Dowell (1974) Crespo da Silva and Hodges (1986)

4 Beam Solution Introduction Various solutions techniques FEM (most common) nonlinear Galerkin (SDM) hybrid (presented here) Various formulations in FEM displacement-based or mixed different variables various shape functions certain aspects of the equations are exactly satisfied

5 Fully Intrinsic Equations Equations Discretization Solution F + ( k + κ)f + f = Ṗ + ΩP M + ( k + κ)m + (ẽ 1 + γ)f + m = Ḣ + ΩH + Ṽ P V + ( k + κ)v + (ẽ 1 + γ)ω = γ Ω + ( k + κ)ω = κ Geometrically-exact Intrinsic Second-order nonlinear

6 Equations Discretization Solution Algebraic, Linear, Local Constitutive Laws Replace γ & κ using F & M { } [ ] { } γ R S F = κ S T T M { } [ ] { } P G K V = H K T I Ω Replace P & H using V & Ω Primary variables: F, M, V Ω

7 Boundary Conditions Equations Discretization Solution V (, t) = V Ω(, t) = Ω F(L, t) = F L M(L, t) = M L Hingeless rotor blade is easy to model Analysis of hinged rotor blade requires the inclusion of the coning angle

8 FEM Discretization Introduction Equations Discretization Solution L 1 L 2 L 3 L n 1 L n x x 1 x 2 x 3 x n 2 x n 1 x n n is the number of finite elements For i = 1,, n 1: V i (L i, t) = V i+1 (, t) Ω i (L i, t) = Ω i+1 (, t) F i (L i, t) = F i+1 (, t) M i (L i, t) = M i+1 (, t)

9 Weighted Residual Introduction Equations Discretization Solution nx DZ L i nv h i T Ṗ i + Ωi e P i F i (e k i + κi e )F i f ii i=1 + Xn 1 i=1 + Ω i Th Ḣ i + e Ωi H i + f V i P i M i (e k i + e κi )M i (ee 1 + e γi )F i m ii + F i T h γ i V i (e k i + e κi )V i (ee 1 + e γi )Ω ii + M i T h κ i Ω i (e k i + e κi )Ω iio dx ie D F i+1 (, t) + V i (L i, t) h V i (L i, t) V i+1 (, t) h F i (L i, t) F i+1 (, t) i i h i + M i+1 (, t) Ω i (L i, t) Ω i+1 (, t) ie + Ω i (L i, t) F 1 (, t) T h V 1 (, t) V i M 1 (, t) T h Ω 1 (, t) Ω i h M i (L i, t) M i+1 (, t) + V n (L n, t) T h F n (L n, t) F Li + Ω n (L n, t) T h M n (L n, t) M Li =

10 Energy Conservation Equations Discretization Solution n i=1 L i [ V i T Ṗ i + Ω i T Ḣ i] dx + = n L i i=1 L i n i=1 [ F i T γ i + M i T κ i] dx [ V i T f i + Ω i T m i] dx + [ V n (L n, t) T F L + Ω n (L n, t) T M L F 1 (, t) T V M 1 (, t) T Ω ] =

11 Equations Discretization Solution Variable-Order FEM Expansion V i (x i, t) = F i (x i, t) = mx x P i j v j,i (t) Ω i (x i, t) = j= mx x P i j f j,i (t) M i (x i, t) = j= m is the order of finite elements P j are shifted Legendre functions: mx j= mx j= P ( x) = 1 P 1 ( x) = 2 x 1 P i+1 ( x) = (2i + 1)(2 x 1)Pi ( x) ip i 1 ( x) i + 1 Z 1 P j ( x)p k ( x)d x = δ jk 2i + 1 P j x i ω j,i (t) P j x i m j,i (t)

12 Discretized Equations Equations Discretization Solution Z L i P k G i P j v j,i + K i P j ω j,i + P l ω l,i G i P j v j,i + K i P j ω j,i P j f j,i ek h i + S i T P l f l,i + T i P l m l,i P j f j,i f dx i i + P k (1) P j (1)f j,i P j j,i+1i ()f = Z L i P K k i T j j,i P v + I i P j ω j,i + P l ω l,i K i T P j v j,i + I i P j ω j,i S i T P l f l,i + T i P l m l,i P j m j,i h ee1 + R i P l f l,i + S i P l m l,i P j f j,i m dx i i + P k (1) P j (1)m j,i P j j,i+1i ()m = + P l v l,i G i P j v j,i + K i P j ω j,i P j m j,i ek i + Z L i Z L i P k R i P j ḟ j,i + S i P j j,i ṁ P j j,i v ek i + S i T P l f l,i + T i P l m l,i P j v j,i h ee1 + R i P l f l,i + S i P l m l,i P j ω dx j,i i + P k () P j (1)v j,i 1 P j j,ii ()v = P k S i T P j ḟ j,i + T i P j ṁ j,i P j ω j,i ek i + +P k () S i T P l f l,i + T i P l m l,i P j ω dx j,i i h P j (1)ω j,i 1 P j j,ii ()ω =

13 Equations Discretization Solution Constant Properties within an Element A kj L i (G i v j,i + K i ω j,i ) + C kjl L i g ω l,i (G i v j,i + K i ω j,i ) B kj f j,i A kj L i e k i f j,i C kjl L i ( S i T f l,i + T i m l,i )f j,i D k L i f i + P k 1 Pj 1 f j,i P k 1 Pj f j,i+1 = A kj L i (K i T j,i v + I i ω j,i ) + C kjl L i g ω l,i (K i T j,i v + I i ω j,i ) +C kjl L i f v l,i (G i v j,i + K i ω j,i ) B kj m j,i A kj L i e k i m j,i C kjl L i ( S i T f l,i + T i m l,i )m j,i A kj L i ee 1 f j,i C kjl L i ( R i f l,i + S i m l,i )f j,i D k L i m i + P k 1 Pj 1 mj,i P k 1 Pj mj,i+1 = A kj L i (R i ḟ j,i + S i ṁ j,i ) B kj v j,i A kj L i e k i v j,i C kjl L i ( S i T f l,i + T i m l,i )v j,i A kj L i ee 1 ω j,i C kjl L i ( R i f l,i + S i m l,i )ω j,i P k Pj vj,i + P k Pj 1 vj,i 1 = A kj L i (S i T ḟ j,i + T i ṁ j,i ) B kj ω j,i A kj L i e k i ω j,i C kjl L i ( S i T f l,i + T i m l,i )ω j,i P k Pj ωj,i + P k Pj 1 ωj,i 1 = Z 1 Z 1 A kj = P k ( x)p j ( x)d x B kj = P k ( x) P ( x) j d x Z 1 Z 1 C kjl = P k ( x)p j ( x)p l ( x)d x D k = P k ( x)d x

14 Final Equations Introduction Equations Discretization Solution A ji q i + B ji q i + C jik q i q k + D j = Order of the system N = 12n(m + 1) Second-order nonlinear equations Steady-state equation: B ji q i + C jik q i q k + D j = Jacobian: J = B ji + C jik q k + C jki q k

15 Equations at the Steady-State Equations Discretization Solution  ji ˆq i + ˆB jiˆq i + Ĉjikˆq iˆq k + ˆD j =  ji = A ji ˆB ji = B ji + C jik q k + C jki q k Ĉ jik = C jik Linearized equations for stability:  ji ˆq i + ˆB jiˆq i =

16 Blade Properties Introduction First Bending Mode First Torsion Mode First Bending (Rotating) Span 16 m Chord 1 m Mass per unit length.75 kg/m Mom. Inertia (5% chord).1 kg m Spanwise elastic axis 5% chord Center of gravity 5% chord Bending rigidity N m 2 Torsional rigidity N m 2 Bending rigidity (chordwise) N m 2 Shear/Extensional rigidity

17 Blade Frequencies Introduction First Bending Mode First Torsion Mode First Bending (Rotating) Mode (rad/s) Exact n = 9 m = 1 n = 3 m = 3 n = 1 m = 9 Cantilevered: ω = & v = 1 st bending nd bending rd bending st torsion nd torsion Rotating Cantilevered: ω = rad/s & v = 1 st bending nd bending rd bending Rotating Cantilevered with Offset: ω = rad/s & v = 51.3 m/s 1 st bending nd bending rd bending

18 First Bending Mode First Torsion Mode First Bending (Rotating) Convergence of Frequency (1 st bending) frequency n = m = 1 n = m n (m+1)

19 Error in Frequency Introduction First Bending Mode First Torsion Mode First Bending (Rotating) 1 error in frequency n = 1 m = 1 n = m n (m+1)

20 Convergence of Error First Bending Mode First Torsion Mode First Bending (Rotating) 1 error in frequency m = 1 m = 2 m = 3 m = n (m+1)

21 Modeshape Introduction First Bending Mode First Torsion Mode First Bending (Rotating) modeshape (vertical velocity) n = 1; m = 4 n = 2; m = 2 n = 4; m = spanwise location

22 First Bending Mode First Torsion Mode First Bending (Rotating) Convergence of Frequency (1 st torsion) n = 1 m = 1 n = m frequency n (m+1)

23 Error in Frequency Introduction First Bending Mode First Torsion Mode First Bending (Rotating) 1 error in frequency n = 1 m = 1 n = m n (m+1)

24 Convergence of Error First Bending Mode First Torsion Mode First Bending (Rotating) 1 error in frequency m = 1 m = 2 m = 3 m = n (m+1)

25 Modeshape Introduction First Bending Mode First Torsion Mode First Bending (Rotating) modeshape (pitch ang vel) n = 1; m = 4 n = 2; m = 2 n = 4; m = spanwise location

26 First Bending Mode First Torsion Mode First Bending (Rotating) Convergence of Frequency (1 st bending-rotating) frequency n = 1 m = 1 n = m n (m+1)

27 Error in Frequency Introduction First Bending Mode First Torsion Mode First Bending (Rotating) 1 error in frequency n = 1 m = 1 n = m n (m+1)

28 Convergence of Error First Bending Mode First Torsion Mode First Bending (Rotating) 1 error in frequency m = 1 m = 2 m = 3 m = n (m+1)

29 Modeshape Introduction First Bending Mode First Torsion Mode First Bending (Rotating) modeshape (vertical velocity) n = 1; m = 4 n = 2; m = 2 n = 4; m = spanwise location

30 Conclusions Introduction A variable-order finite element is presented for geometrically-exact, intrinsic, mixed beam equations Galerkin approximation shows best convergence but can be computationally expensive for beam with varying properties The elements shows variable order of convergence: linear: 3 rd order quadratic: 5 th order cubic: 7 th order quartic: 9 th order Good approximation of modeshapes for quadratic and higher-order elements Cubic elements seems to be a good balance of accuracy, computational requirement and applicability to general configurations.