Variable-order finite elements for nonlinear, intrinsic ... - dept.aoe.vt.edu

Variable-order finite elements for nonlinear, intrinsic, mixed beam equations Mayuresh J. Patil∗ Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061-0203 and Dewey H. Hodges† Georgia Institute of Technology Atlanta, Georgia 30338-0150

Abstract

analytical thin-walled theory (Refs. 3, 4) or computational FEM analysis (Refs. 5, 6) for general configurations. Rotor blade dynamics analysis can be broken up into four parts:

The paper presents variable-order finite elements for nonlinear, mixed, fully intrinsic equations for both non-rotating and rotating blades. The finite element technique allows for hp-adaptivity. Results show that these finite elements lead to very accurate solutions for the static equilibrium state as well as for modes and frequencies for infinitesimal motions about that state. The results based on the Galerkin approximation (which is a special case of the present approach) are more accurate as compared to finite elements with the same number of variables. Furthermore, the accuracy of the finite elements increases with the order of the finite element. Cubic elements seem to be the optimal combination of accuracy and complexity.

• Rotor blade partial differential equations; • Techniques for solving those equations (including discretization); • Nonlinear model-order reduction schemes; • Application to aeroelasticity, active blades and control design. The focus of the present work is on the second item, i.e. discretization and solution of the geometricallyexact, intrinsic beam equations.

Introduction Rotor blade equations The analysis of rotor blades is one of the most interesting and complex parts of rotorcraft design and development. The goal of this paper is to present a variable-order finite element solution to the intrinsic equations for the nonlinear dynamics of rotor blades (Refs. 1, 2). The research assumes that a suitable cross-sectional analysis is available for beams of arbitrary geometry and material distribution. These cross-sectional properties can be calculated using an

There are a number of geometrically-exact formulations for the nonlinear dynamics of beams that can be used for rotor blade analysis (Refs. 7–9). The present work is based on the “fully intrinsic” formulation developed by Hodges (Ref. 1), which can be written in a simple matrix form with only second-degree nonlinearities. To say that these equations are intrinsic is to say that they are independent of displacement and rotation measures. They were used along with generalized strain- and velocity-displacement measures to solve for blade dynamics, yielding excellent agreement with experimental results as presented in Ref. 10. Recently, a set of generalized strain-velocity compatibility relations were derived by Hodges (Ref. 2) which, along with the equations of motion, make up a complete set of equations that can be solved without using displacement/rotation measures for certain

∗Assistant Professor, Department of Aerospace and Ocean Engineering. E-mail: [email protected] †Professor, School of Aerospace Engineering. Fellow, AHS. E-mail: [email protected]

Presented at the 62nd Annual Forum and Technology Display of the American Helicopter Society International, Phoenix, Arizona, May 9 – 11, 2006. c

2006 by Mayuresh J. Patil and Dewey H. Hodges.

1

loading and boundary conditions. Although they incorporate all the nonlinearities and anisotropic couplings, these equations are very simple.

blades.

Nonlinear, Intrinsic Beam Equations

Discrete equations of motion

The nonlinear, intrinsic, mixed equations for the dynamics of a general (non-uniform, twisted, curved, anisotropic) beam undergoing small strains and large deformation are given below,

The equations described above are partial differential equations in space and time for 12 variables (force, moment, velocity and angular velocity vectors). The solution of these equations requires discretization in space to convert the equations to ordinary differential equations in time. For example, one may use finite elements or a series of assumed functions. It is also possible to use a combination of the two, leading to variable-order (or hp) finite elements. A simple nonlinear finite element representation of the blade equations was presented in Ref. 2. The FEM equations were used successfully to conduct nonlinear dynamic analysis and control design of integrally actuated helicopter blades (Ref. 11). A new nonlinear, energy-consistent, Galerkin approach has been developed recently and is presented in Ref. 12. The Galerkin approach leads to a highly accurate solution with low computational cost. The application of the nonlinear Galerkin approach is possible because of the simplicity of the intrinsic equations. Since the highest degree nonlinearity is quadratic, the Galerkin integrals can be evaluated exactly without resorting to numerical quadrature. This contributes to the method’s accuracy and efficiency. The Galerkin approach leads to the exact nonlinear solution (within 5 decimal places) with as few as 10 assumed functions per variable. Also it is clear (Ref. 12) from the slow convergence of low-order FEM approach, that one would require very large number of finite element nodes to generate results with the same order of accuracy as those generated by the Galerkin approach using 10 functions per variable. The nonlinear Galerkin approach leads to an approximate solution that is more accurate for a given level of computation (or more efficient for a given level of accuracy). This is especially important if one needs to conduct numerous optimizations, simulations and control scenarios. The limitation of the Galerkin approach is the handling of discontinuities. Thus, when one has a complex helicopter blade with multiple changes in the properties of the blade along the span then it will be more accurate to break the beam into multiple finite elements, to each of which one can apply the Galerkin approach. This would lead to variable-order finite elements. It is the focus of this research to develop a variable-order finite element scheme to optimally model future helicopter

e F 0 + (e k+κ e)F + f = P˙ + ΩP (1) 0 e + Ve P M + (e k+κ e)M + (ee1 + γ e)F + m = H˙ + ΩH (2) 0 V + (e k+κ e)V + (ee1 + γ e)Ω = γ˙ (3) Ω0 + (e k+κ e)Ω = κ˙

(4)

where ( )0 denotes the partial derivative with respect to the axial coordinate of the undeformed beam, and (˙) denotes the partial derivative with respect to time. F (x, t) and M (x, t) are the measure numbers of the internal force and moment vector (cross-section stress resultants), P (x, t) and H(x, t) are the measure numbers of the linear and angular momentum vector (generalized momenta), γ(x, t) and κ(x, t) are the beam strains and curvatures (generalized strains), V (x, t) and Ω(x, t) are the linear and angular velocity measures (generalized speeds), and f(x, t) and m(x, t) are the external force and moment measures. Measure numbers of all variables except for k are calculated in the B frame, i.e. the deformed beam cross-sectional frame. k(x) = bk1 (x) k2 (x) k3 (x)c is the initial twist/curvature of the beam. The measure numbers of k are in the undeformed beam cross-sectional frame. Finally, e1 = b1 0 0cT . The first two equations in the above set are the equations of motion (Ref. 1) while the last two are the intrinsic kinematical equations (Ref. 2) derived from the generalized strain- and velocity-displacement equations. The cross-sectional stress resultants are related to the generalized strains via the cross-sectional beam stiffnesses/flexibilities. These cross-sectional properties can be calculated using an analytical thin-walled theory (Refs. 3, 4) or computational FEM analysis (Refs. 5, 6) for general configuration. Such an analysis gives the following linear constitutive law,      R S γ F = T (5) κ S T M where R(x), S(x), and T(x), are the cross-sectional flexibilities of the beam. This linear constitutive law is valid only for small strain, but the global deformations still may be large. 2

L1 x0

L2

x1

L3

Ln

Ln−1

x2

x3

n xn−1 x

xn−2

Figure 1: Beam discretization as weighting functions. Let the solution in the ith element be given by, V i , Ωi , F i and M i . We require that the solution satisfying (approximately) the equations of motion, the kinematic equations and the boundary conditions given above. In addition we require that the the continuity equations be satisfied (approximately) between adjacent elements.

The generalized momenta are related to the generalized speeds via the cross-sectional beam inertia, #     "   G K V P µ∆ −µξe V = = (6) H Ω Ω KT I µξe I where µ(x), ξ(x), I(x) are the mass per unit length, mass center offset (vector in the cross-section from the beam reference axis to the cross-sectional mass center), and mass moment of inertia per unit length respectively. Usually, the constitutive laws are used to replace some variables in terms of others. Here it was decided to express the generalized strains in terms of the cross-section stress resultants, allowing easy specification of zero flexibility, and the generalized momenta in terms of generalized speeds, allowing easy specification of zero inertia. Thus, the primary variables of interest are F , M , V and Ω. Finally the boundary conditions need to be specified. For the given beam of length L, there will be two boundary conditions at each end: V (0, t) = V0 Ω(0, t) = Ω

0

V (L, t) = V

L

Ω(L, t) = ΩL

or F (0, t) = F0 0

or M (0, t) = M

L

V i (Li , t) = V i+1 (0, t)

or M (L, t) = ML

(10)

(11)

Ω(0, t) = Ω0

(12)

L

(13)

V (0, t) = V F (L, t) = F

L

M (L, t) = M

i

i

i+1

i

i

M (L , t) = M

(0, t) (0, t)

i+1

(0, t)

(15) (16) (17) (18)

Note that the above continuity conditions will be modified for any node at which there is a concentrated mass, a rigid body, a nodal force or moment, or a kink in the axis. Now consider the following weighting of the equations of motion, the kinematical equations, continuity conditions, and the boundary conditions: *Z i  n L h i X T fi P i − F i 0 − (kei + κei )F i − f i V i P˙ i + Ω i=1

0

+ Ωi

T

h

fi H i + V fi P i − M i 0 H˙ i + Ω − (kei + κei )M i − (ee1 + γei )F i − mi

For ease of presentation, we consider a helicopter blade clamped at its root. It should be noted that the formulation as well as the conclusions that would be presented are general enough to be applicable to all possible boundary conditions. Thus, the boundary conditions are 0

i+1

F (L , t) = F

(8) (9)

i

Ω (L , t) = Ω

(7)

or F (L, t) = F

i

+ Fi

T

+ Mi

+

T

i

0 γ˙ i − V i − (kei + κei )V i − (ee1 + γei )Ωi + h i i i0 i i e e i i κ˙ − Ω − (k + κ )Ω dx

h

i

n−1 X

  F i+1 (0, t) V i (Li , t) − V i+1 (0, t)

i=1

  + M i+1 (0, t) Ωi (Li , t) − Ωi+1 (0, t)   + V i (Li , t) F i (Li , t) − F i+1 (0, t)    + Ωi (Li , t) M i (Li , t) − M i+1 (0, t)     − F 1 (0, t)T V 1 (0, t) − V0 − M 1 (0, t)T Ω1 (0, t) − Ω0   + V n (Ln , t)T F n (Ln , t) − FL   + Ωn (Ln , t)T M n (Ln , t) − ML = 0 (19)

(14)

Energy Consistent Weighting Let us assume that the beam is discretized into n elements as shown in Fig. 1. To create a finite element model we need to choose trial functions as well 3

Note that the constitutive equations are not included, as these equations are satisfied exactly. Since the continuity conditions and the boundary conditions are satisfied weakly we will not get exact satisfaction of the boundary conditions and the variables will not be continuous at the nodes. Integrating by parts and then simplifying the expression one obtains n Z X i=1

Li

h

iT

V

iT

˙i

P + Ω H˙

i

i

as V i (xi , t) = Ωi (xi , t) = F i (xi , t) =

dx

0

+

=

n Z X

Li

i=1 0 n Z Li X i=1 n

h

F

iT

i

γ˙ + M

iT

i

M i (xi , t) =

i

κ˙ dx

m X j=0 m X j=0 m X j=0 m X


Pj x ¯i v j,i (t)

(24)


Pj x ¯i ω j,i (t)

(25)


Pj x ¯i f j,i (t)

(26)


Pj x ¯i mj,i (t)

(27)

j=0 i

h

V

iT i

iT

i

where x ¯i = Lxi , and v j,i , ω j,i , f j,i and f j,i are column matrices of the unknowns of the formulation, corresponding to the ith element and j th order. With i = 1, 2, . . . , n and j = 0, 1, . . . , m, we have a total of 12(m + 1)nvariables. The FEM equations for the ith element can be derived based on the energy-conserving integral equation, Eq. (20), as

i

f + Ω m dx

0

 + V (Ln , t)T FL + Ωn (Ln , t)T ML  − F 1 (0, t)T V0 − M 1 (0, t)T Ω0 = 0

(20)

The first term above is the rate of change of kinetic energy, and the second is the rate of change of potential energy. The third is the rate of work done (power) due to applied forces in the interior of the beam, and the fourth is the power due to applied forces at the boundaries. The equation states that the rate of change of energy of the beam is equal to the rate of work done on the beam. Thus, the above weighting of all the equations leads to an energy balance, on the basis of which we derive the FEM equations.

Z

Li

P

k



Gi P j v˙ j,i + Ki P j ω˙ j,i




0


^ l ω l,i Gi P j v j,i + Ki P j ω j,i − P j 0 f j,i +P   (28) ^ T − kei + Si P l f l,i + Ti^ P l ml,i P j f j,i    −f i dxi + P k (1) P j (1)f j,i − P j (0)f j,i+1 = 0

Z

Variable-order FEM

Li

Pk



T

Ki P j v˙ j,i + Ii P j ω˙ j,i



0

  ^ l ω l,i Ki T P j v j,i + Ii P j ω j,i +P
^ l v l,i Gi P j v j,i + Ki P j ω j,i − P j 0 mj,i +P (29)   ^ T l l,i j j,i e ^ i i i l l,i − k +S P f +T P m P m    j j,i i ^ ^ i l l,i i l l,i − ee1 + R P f + S P m P f − m dxi   +P k (1) P j (1)mj,i − P j (0)mj,i+1 = 0

The independent trial functions used are the shifted Legendre polynomials (Ref. 13), denoted by P j (¯ x), which constitute a complete set of orthogonal polynomials that are orthogonal over the shifted interval 0≤x ¯ ≤ 1, so that Z

1

P j (¯ x)P k (¯ x)d¯ x=

0

δjk 2i + 1

(21)

These polynomials can be obtained from the following recursive relations: P 0 (¯ x) = 1

P 1 (¯ x) = 2¯ x−1 i

P i+1 (¯ x) =

(2i + 1)(2¯ x − 1)P (¯ x) − iP i+1

i−1

Z



 0 P Ri P j f˙j,i + Si P j m ˙ j,i − P j v j,i 0   ^ T l l,i e ^ i i i l l,i − k +S P f +T P m P j v j,i    j j,i ^ ^ i l l,i i l l,i − ee1 + R P f + S P m P ω dxi   +P k (0) P j (1)v j,i−1 − P j (0)v j,i = 0

(22) (¯ x)

Li

(23)

Expanding all 12 variables in terms of these polynomials, one finds that the unknowns can be written 4

k

(30)

Table 1: Blade data Span 16 m Chord 1 m Mass per unit length 0.75 kg/m Mom. Inertia (50% chord) 0.1 kg m Spanwise elastic axis 50% chord Center of gravity 50% chord Bending rigidity 2 × 104 N m2 Torsional rigidity 1 × 104 N m2 Bending rigidity (chordwise) 4 × 106 N m2 Shear/Extensional rigidity ∞ Akj Li (Ri f˙j,i + Si m ˙ j,i ) − B kj v j,i

Li

  0 T P k Si P j f˙j,i + Ti P j m ˙ j,i − P j ω j,i 0    ^ T − kei + Si P l f l,i + Ti^ P l ml,i P j ω j,i dxi   +P k (0) P j (1)ω j,i−1 − P j (0)ω j,i = 0

Z

^ T ^ i ml,i )v j,i −Akj Li kei v j,i − C kjl Li (Si f l,i + T

(31)

−P0k P0j v j,i + P0k P1j v j,i−1 = 0

In the above equations Ri , Si , Ti are the crosssectional stiffness coefficients for the ith element, Gi , Ki , Ii are the cross-sectional inertia coefficients for the ith element, ki is the initial curvature for the ith element, and f i , mi define the loading for the ith element. In the equations, summation is assumed over indices j and l. Thus we have a set of equation for each i (element) and k (order), giving us a total of 12(m + 1)n equations for as many unknowns. We need to calculate the Galerkin integrals so as to obtain the equations in a form suitable for solution. For demonstration, we assume that the crosssectional properties, the initial twist and curvature, and the distributed loading are all constant within each element. With the above assumptions, the FEM equations for the ith element can be derived and we obtain the discretized equations of motion as kj

i

i j,i

A L (G v˙

i

(34)

^ ^ i f l,i + S i ml,i )ω j,i −Akj Li ee1 ω j,i − C kjl Li (R

T Akj Li (Si f˙j,i + Ti m ˙ j,i ) − B kj ω j,i

^ T ^ i ml,i )ω j,i −Akj Li kei ω j,i − C kjl Li (Si f l,i + T

(35)

−P0k P0j ω j,i + P0k P1j ω j,i−1 = 0 In the above equations summation is implied over indices j and l. And Akj , B kj , C kjl and Dk are nondimensional integrals, given by

B kj = C kjl =

Z

Z

1

Z

Akj =

P k (¯ x)P j (¯ x)d¯ x

(36)


0 x P k (¯ x) P j (¯ x) d¯

(37)

P k (¯ x)P j (¯ x)P l (¯ x)d¯ x

(38)

0 1

0 1

0

Dk =

j,i

+ K ω˙ )

Z

1

P k (¯ x)d¯ x

(39)

0

g l,i (Gi v j,i + Ki ω j,i ) − B kj f j,i +C kjl Li ω Now, representing all the system unknowns of the ith element as

(32) ^ T ^ i ml,i )f j,i −Akj Li kei f j,i − C kjl Li (Si f l,i + T −Dk Li f i + P1k P1j f j,i − P1k P0j f j,i+1 = 0

 0,i  v (t)       ω 0,i (t)         f 0,i (t)         0,i   m (t)         ..   . i q (t) = ..       .    m,i  v (t)         ω m,i (t)         m,i     f (t)   m,i   m (t)

T

Akj Li (Ki v˙ j,i + Ii ω˙ j,i ) T

g l,i (Ki v j,i + Ii ω j,i ) +C kjl Li ω l,i (Gi v j,i + Ki ω j,i ) − B kj mj,i +C kjl Li vf

(33)

^ T ^ i ml,i )mj,i −Akj Li kei mj,i − C kjl Li (Si f l,i + T ^ ^ i f l,i + S i ml,i )f j,i −Akj Li ee1 f j,i − C kjl Li (R −Dk Li mi + P1k P1j mj,i − P1k P0j mj,i+1 = 0 5

(40)

Table 2: Blade structural frequencies Mode (rad/s) Exact n=9 m=1 n=3 m=3 n=1 m=9 Cantilevered Blade: ω = 0 & v = 0 1st bending 2.243 2.243 2.243 2.243 2nd bending 14.06 14.03 14.06 14.06 3rd bending 39.36 39.22 39.38 39.36 1st torsion 31.05 31.05 31.05 31.05 2nd torsion 93.14 93.17 93.14 93.14 Rotating Cantilevered Blade: ω = 3.189 rad/s & v = 0 1st bending 4.114 4.114 4.114 4.114 2nd bending 16.23 16.21 16.23 16.23 3rd bending 41.59 41.44 41.62 41.59 Rotating Cantilevered Blade with Offset: ω = 3.189 rad/s & v = 51.03 m/s 1st bending 5.703 5.703 5.703 5.703 2nd bending 18.72 18.69 18.72 18.72 3rd bending 44.50 44.33 44.53 44.50

and,  1  q (t)        q 2 (t)       .  . . q(t) =    .       ..     n  q (t)

method, commonly referred to as h-version, consisting of linear shape functions. Finally, the case of m = 3 and n = 3 is a good balance between these two extreme approaches. Here cubic polynomials are used to represent the variables in each of three elements. The results are quite good with negligible errors. Recall that a finite element formulation in which the order of the polynomials with elements as well as the number of elements both vary is an hpversion finite element. Figure 2 presents the convergence of results for the first bending mode of a non-rotating blade. Figure 2(a) shows the convergence of frequencies with increase in the order of the system. The red line corresponds to the h-version, and the blue line corresponds to the p-method. Finally, the green dots correspond to hp-method. As expected, for this simple case, the Galerkin approximation (the p-version) is the best of the three. It should be noted that the Galerkin approximation cannot be easily applied to variable geometry/material systems unless the integrals are recalculated taking into account the variable geometry/material. Thus, the Galerkin approach becomes computationally intensive for a general configuration. Figure 2(b) shows the convergence of the various methods. The Galerkin approximation is seen to reach the exact result with error of the order of machine precision using eighth-order polynomials. The first-order finite-elements have a good convergence. The convergence is approximately third-order, i.e for every doubling of the number of finite-elements, the error decreases by a factor of 8. Figure 2(c) shows the convergence of the error in first frequency for finite elements of various orders. The linear finite element shows third-order conver-

(41)

the governing equations consist of N = 12n(m + 1) equations and unknowns. The equations can then be written as Aji q˙ i + Bji qi + Cjik qi qk + Dj = 0

(42)

where summation is assumed over the i and k indices.

Results The equations were solved using the variable-order FEM for a simple prismatic beam case presented in Table 1. Table 2 lists the calculated frequencies and compares the results with exact results (Ref. 14). The frequency predictions of a non-rotating as well as rotating beam using the present approach with mn = 9 is shown. For m = 9 and n = 1, i.e. a single, highorder element, the frequencies are obtained to three significant digits for both the bending and torsion modes. This approach is equivalent to the Galerkin approach discussed in Patil and Althoff (Ref. 12) and to formulations commonly referred to p-version. On the other hand, for m = 1 and n = 9, the maximum number of the crudest possible elements for mn = 9, the solution is not as accurate, leading to errors greater than 0.3% for the third bending mode. This last approach corresponds to lowest-order FEM 6

0

2.3

10

error in frequency

frequency

2.25 2.2 2.15 2.1

n=1 m=1 n=m

2.05 2 0

5

10

n × (m+1)

15

−5

10

−10

10

−15

10

n=1 m=1 n=m

2

20

(a) Convergence of first bending frequency

5

n × (m+1)

10

20

(b) Error in first bending frequency

0

modeshape (vertical velocity)

error in frequency

10

−5

10

−10

10

2

m=1 m=2 m=3 m=4 5

n × (m+1)

10

20

1 0.8 0.6 0.4 0.2

n = 1; m = 4 n = 2; m = 2 n = 4; m = 1

0 0

0.25

0.5

0.75

1

spanwise location

(c) Error in first bending frequency

(d) Convergence of first bending mode shape

Figure 2: Results for the first bending mode of a non-rotating cantilevered blade gence, quadratic finite elements show a fifth-order convergence and cubic finite elements show a seventhorder convergence. Finally, the quartic finite elements show a whopping ninth-order convergence, i.e. for every doubling of the number of finite elements there is a reduction in the error by a factor of 512. Finally, Fig. 2(d) shows the convergence of mode shape for three implementations. Again, the mode shape obtained by the Galerkin approach is the closest to the exact mode shape. The mode shape predicted by the lowest-order, linear finite elements shows deviation from the exact mode shape and is discontinuous as expected (because the continuity conditions are weakly satisfied). Also, the mode shapes do not satisfy the boundary condition exactly due to weak satisfaction of the boundary conditions as well.

Figure 3 shows the corresponding plots for the first torsion mode. For the torsional modeling the Galerkin approximation again leads to the most accurate results while the accuracy increases with the increase in the order of the finite element. The order of the relative error is third-order, fifth-order, seventhorder and ninth-order for the linear, quadratic, cubic and quartic elements respectively. Figure 4 shows the results for the first bending mode of a rotating blade with root velocity. For the rotating blade, the static steady-state solution is non-trivial. Thus, the accuracy of the frequencies obtained from linearizing about the nonlinear steady state is dependent on the accuracy of the steady-state solution and the accuracy of the linearized perturbation. We obtain the exact steady-state solution for 7

0

10

33

frequency

32.5

error in frequency

n=1 m=1 n=m

32 31.5

−5

10

−10

10

31 −15

10 30.5 0

5

10

n × (m+1)

15

n=1 m=1 n=m

2

20

(a) Convergence of first torsion frequency

5

n × (m+1)

10

20

(b) Error in first torsion frequency

0

modeshape (pitch ang vel)

error in frequency

10

−5

10

−10

10

2

m=1 m=2 m=3 m=4

1 0.8 0.6 0.4 0.2

n = 1; m = 4 n = 2; m = 2 n = 4; m = 1

0

5

n × (m+1)

10

20

0

0.25

0.5

0.75

1.0

spanwise location

(c) Error in first torsion frequency

(d) Convergence of first torsion mode shape

Figure 3: Results for the first torsion mode of a non-rotating cantilevered blade

Conclusions

finite-elements of second and higher order. The errors are in general higher for the rotating blade as compared to the non-rotating blade. Furthermore, the rate of convergence for the rotating blade is slightly slower than that of the non-rotating blade.

A variable-order finite element technique is presented in the paper. This technique is based on the geometrically-exact, intrinsic formulation developed by Hodges. The results presented show that one can obtain approximately third-order, fifth-order, seventh-order and ninth-order convergence for the linear, quadratic, cubic and quartic finite elements. It is recommended that one use quadratic- or higher-order finite elements for better approximation of the mode shape. The cubic finite elements are especially good balance of accuracy, computational requirement and applicability to general configurations.

Not yet addressed in the present study is the impact of bandedness or sparsity of the matrices in the variable-order finite element formulation. For low-order elements, the coefficient matrices are very sparse, and though we have not yet done so, one can take advantage of this. Although the number of unknowns in the high-order elements is less for a given level of accuracy, the bandwidth increases for such elements. An assessment of this aspect of the method’s accuracy/efficiency will be addressed in a later paper. 8

0

6

10

error in frequency

frequency

5.8 5.6 5.4 5.2

n=1 m=1 n=m

5

0

5

10

n × (m+1)

15

−5

10

−10

10

n=1 m=1 n=m

20

2

(a) Convergence of first bending frequency

5

n × (m+1)

10

20

(b) Error in first bending frequency

0

modeshape (vertical velocity)

10

error in frequency

−2

10

−4

10

−6

10

m=1 m=2 m=3 m=4

−8

10

−10

10

2

5

n × (m+1)

10

20

1 0.8 0.6 0.4 0.2

n = 1; m = 4 n = 2; m = 2 n = 4; m = 1

0 0

0.25

0.5

0.75

1.0

spanwise location

(c) Error in first bending frequency

(d) Convergence of first bending mode shape

Figure 4: Results for the first bending mode of a rotating cantilevered blade with offset

References 1

Beams with Embedded Strain Actuation,” Proceedings of the 13th Adaptive Structures Conference, AIAA, Reston, Virginia, April 18 – 21, 2005, AIAA Paper 2005-2037.

2

5 Cesnik, C. E. S. and Hodges, D. H., “VABS: A New Concept for Composite Rotor Blade Cross-Sectional Modeling,” Journal of the American Helicopter Society, Vol. 42, No. 1, January 1997, pp. 27 – 38.

Hodges, D. H., “A Mixed Variational Formulation Based on Exact Intrinsic Equations for Dynamics of Moving Beams,” International Journal of Solids and Structures, Vol. 26, No. 11, 1990, pp. 1253 – 1273. Hodges, D. H., “Geometrically-Exact, Intrinsic Theory for Dynamics of Curved and Twisted Anisotropic Beams,” AIAA Journal , Vol. 41, No. 6, June 2003, pp. 1131–1137.

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Yu, W., Hodges, D. H., Volovoi, V. V., and Cesnik, C. E. S., “On Timoshenko-Like Modeling of Initially Curved and Twisted Composite Beams,” International Journal of Solids and Structures, Vol. 39, No. 19, 2002, pp. 5101 – 5121.

3

Volovoi, V. V. and Hodges, D. H., “Theory of Anisotropic Thin-Walled Beams,” Journal of Applied Mechanics, Vol. 67, No. 3, Sept. 2000, pp. 453 – 459. 4

7

Patil, M. J. and Johnson, E. R., “Cross-sectional Analysis of Anisotropic, Thin-Walled, Closed-Section

Borri, M. and Mantegazza, P., “Some Contributions on Structural and Dynamic Modeling of Heli9

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Patil, M. J. and Althoff, M., “Energy-consistent, Galerkin approach for the nonlinear dynamics of beams using mixed, intrinsic equations,” Proceedings of the AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, AIAA, Reston, Virginia, May 1 – 4, 2006. 13

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