NONLINEAR DYNAMICS AND CONTROL OF ...

NONLINEAR DYNAMICS AND CONTROL OF INTEGRALLY ACTUATED HELICOPTER BLADES Johannes P. Traugott1 , Mayuresh J. Patil2 , and Florian Holzapfel1 1

2

Institute of Flight Mechanics and Flight Control, Technische Universit¨at M¨ unchen, Boltzmannstr. 15, D-85747 Garching, Germany e-mail: [email protected] / [email protected]

Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0203 USA e-mail: [email protected]

Key words: Active Helicopter Blades, Nonlinear Finite Elements, Intrinsic Formulation, Control Design Abstract: A set of nonlinear, intrinsic equations describing the dynamics of beam structures undergoing large deformations is presented. The intrinsic kinematical equations are derived for the general case of a moving beam. Active force/strain terms are added to the equations to take into account active components The equations are then discretized into finite elements, transformed into state-space form and finally decomposed into modes. Actuation and sensor models are established before implementing a simulation model in Matlab/SIMULINK. The model is validated by comparison with exact, analytical results and then utilized to analyze the dynamic behavior of an active helicopter blade. Beside the analysis of the inherent dynamics of this system in terms of eigenvalues and vectors, the modal controllability of the blade is discussed under the influence of rigid body motion. In a final step, the design of a MIMO controller based on full-state optimal control (LQR approach) and optimal state estimation (Kalman filter) is presented with the aim to add vibrational damping to the weakly damped system. The closed loop properties are validated by both analytical methods and simulation runs. 1

Introduction

Vibration and noise are persistent problems in helicopters. Reduction of vibration and noise is one of the goals in the design of next generation vehicles. Smart materials provide a way to address this issue.[1] Embedded strain actuation can be used to reduce the blade vibration, minimize blade-vortex interaction, decrease noise and improve stability and response characteristics of the helicopter. The design and development of new ‘smart’ blades requires accurate modeling of the active blade, coupled with appropriate control design methodology. Active blade models can be developed based on pure three dimensional finite-elements. This approach is quite computationally intensive for preliminary design or control synthesis. A beam model of the helicopter blade is an efficient alternative and thus an effective choice.[2] The beam model requires cross-sectional stiffness (flexibility) parameters as well as strain measures induced by the embedded actuation devices as inputs. These quantities can be obtained by means of a two dimensional cross-sectional analysis.[3] Therefore the three dimensional structural problem is split up into a one dimensional, nonlinear beam dynamics model based on an off-line two dimensional cross-sectional analysis. This approach leads to a low-order, high-fidelity model for control design.

1

The goal of the paper at hand is to present a practical realization and implementation of such a simulation model. This model is then used to analyze the control potential of a given active blade configuration and to design an optimal MIMO controller providing additional vibrational damping to the structure. 2 2.1

Blade Model The Equations

2.1.1 Beam Equations of Motion b3 Cross Section

Undeformed State

b2 b1

Reference Line

r Reference Point

u

B3 B2 Deformed State

Reference Line

B1 Unwarped Cross Section

Figure 1: Frames and reference lines of the beam model

The blade is modeled as an active beam. The nonlinear equations of motion for the dynamics of a straight beam undergoing rigid-body motion (constant translational and angular velocity) are given by,[4] e F0 + κ eF + f = P˙ + ΩP e + Ve P M0 + κ eM + (ee1 + γ e)F + m = H˙ + ΩH

(1) (2)

where ( )0 denotes the derivative with respect to the undeformed beam reference line and (˙) denotes the absolute time derivative. F and M are the measure numbers of the internal force and moment vector (generalized forces), P and H are the measure numbers of the linear and angular momentum vector (generalized momenta), γ and κ are the beam strains and curvatures (generalized strains), V and Ω are the linear and angular velocity measures (generalized velocities), and f and m are the external force and moment measures. All measure numbers refer to the B–frame of the deformed cross-section as illustrated in Figure 1. This figure shows the undeformed reference line of an initially straight beam and the displaced reference line of the deformed beam. Further, a reference cross-section is depicted in its initial and deformed position. Two cartesian coordinate frames are set up: The b–frame of the undeformed beam and the B–frame of the deformed beam. The unit vector b1 of the b–frame is tangential to the undeformed reference line; b2 and b3 are

2

defining the plane of the reference cross-section. The origin of the B–frame is the origin of the b–frame translated by the displacement vector u. The unit vector B1 is orthogonal to the non-wrapped translated and rotated reference cross-section. Note that B1 is not necessarily tangential to the deformed reference line because the displaced cross-section does not have to be orthogonal to the new reference line (Euler-Bernoulli approximation is not made; shear deformation is allowed). The tilde operator transforms a vector a to a matrix e a so as to effect a cross product when left-multiplied to the vector b :        0 −a3 a2 b1  a1  b1  0 −a1  b2 = a × b; a = a2 , b = b2 . e a b =  a3 (3)       −a2 a1 0 b3 a3 b3 Equations (1) and (2) are valid for initially twisted and curved beams with closed crosssections undergoing large global deformations (the beam elastic curvature κ in the above equation has to be replaced by the total curvature to be valid for an initially curved/twisted beam). The effects of restrained warping due to boundary conditions is not considered as it can be neglected for beams with closed cross-sections in most cases.[4]. In the following, only initially straight beams with closed cross-sections will be of interest. 2.1.2 Beam Kinematical Relations Generalized strain-displacement relations and generalized velocity-displacement relations are kinematic equations which relate the generalized strains to generalized displacements and the generalized velocities to generalized displacements respectively. These equations for a moving beam are,[4] γ = C (e1 + u0 ) − e1 0

T

κ e = −C C V = C [v + ω e (r + u) + u] ˙ T T e = −CC ˙ Ω + Cω eC

(4) (5) (6) (7)

where, u is the displacement vector, i.e. the structural deformations measures indicated in the undeformed b–frame, C is the direction cosine matrix that transforms the measure numbers of a vector from the undeformed b–frame to the deformed B–frame, v and ω are the constant linear and angular velocity measures of a reference point in the undeformed frame, and r are the measures in the deformed frame of the position vector from the reference point to any point on the beam reference axis, see Figure 1. The above set of kinematical equations can be used along with the equations of motion, Eqs. (1) and (2), and the constitutive equations (as presented below) to form a complete set of equations describing the behavior of the beam. However, the goal of the present work was to use intrinsic equations, i.e., equations without displacements (u) or rotation (C) variables. Such a set of intrinsic kinematical equations is derived for a beam without rigid-body motion (v = ω = 0) in Ref. [5]. Based on a procedure similar to that presented in Ref. [5], intrinsic equations are derived for a moving beam. The derivation is presented in the Appendix, which gives the intrinsic equations (generalized strain-velocity) to be, V0+κ eV + (ee1 + γ e)Ω = γ˙ 0 Ω +κ eΩ = κ. ˙

3

(8) (9)

2.1.3 Cross-sectional Constitutive Equations The generalized forces are related to the generalized strains via the cross-sectional beam stiffnesses. In the present paper, actuation is provided to the beam by embedded strain actuators and thus the resulting strains are a function of the external mechanical forces as well as of the voltage (electric field) applied to the actuators. Active cross-sectional analysis is performed using the theory of Ref. [6] (for thin-walled beams) or Ref. [7] (for general configuration). Such an analysis gives the following, linear constitutive law,        γ R S F γa = T + (10) κ S T M κa where, R, S, T , are the cross-sectional flexibilities of the beam and γa , κa are the induced generalized strains and curvatures due to embedded strain actuation. Being linear, the present material law is only valid for small local strains, which can, however, lead to large global deformations as they occur in helicopter blades. The generalized momenta are related to the generalized velocities via the cross-sectional beam inertia, #    " e P µ∆ −µξ V = (11) e H Ω µξ I where, µ, ξ, I 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.

F, M 6 LAE

Equations of Motion

2× 3 PDE

Constitutive Equations I

Kinematical Equations Partial Differential Equations

LAE:

Linear Algebraic Equations

6 LAE

Constitutive Equations II

γ, κ PDE:

P, H

V,Ω

2× 3 PDE

Figure 2: The equations describing the dynamics of the beam. The cancelled quantities are replaced in the actual calculations by the corresponding constitutive relation.

With the constitutive equations, Eq.(10) and (11), the kinematical equations, Eq. (8) and (9), and the equations of motion, Eq. (1) and (2), the dynamics of the beam are described completely. Figure 2 illustrates the basic relations between these equations at a glance. Due to their intrinsic nature, they do not exhibit translational (u) or rotational (C) displacement variables. This circumstance contributes essentially to the compact and easy-to-implement character of the whole approach. However, for analyzing, visualization and validation purpose, the actual displacement often is of substantial interest. Hence, C and u can readily be calculated after the actual solution is found by making use of the

4

strain-displacement relation, Eqs. (4) and (5). When actually solving the resulting system of equations, the linear algebraic equations can directly be incorporated in the partial differential equations by substituting the generalized strains γ and κ by the generalized forces F and M and the generalized momenta P and H by the generalized velocities V and Ω, see Figure 2. Hence, the reduced vector of primary variables X can be written as follows:  T X = F T , M T , V T , ΩT . (12) The generalized strains γ and κ and the generalized momenta P and H are therefore considered as secondary variables. Even though the actual choice of the primary and secondary variables does not change the original system of equations, it has an important influence on the practical implementation of the equations to a final finite element simulation model, as it will be addressed later on. 2.1.4 Finite Element Discretization To solve the above set of equations, the beam is discretized into finite elements. The equations for each element are obtained by discretizing the differential equations such that energy is conserved.[5] For example, consider a variable Y . Let the nodal values of the variable after discretization be represented by Yb n , where the superscript denotes the node number, and the hat denotes that it is a nodal value. For the element n of length ∆l this results in Yb n+1 − Yb n ∆l n+1 b Y + Yb n n Y = . 2 Y0 =

(13) (14)

For a beam discretized to N elements, the primary equations of the nth element, Eqs. (1), (2), (8) and (9), can be written as   n   f      1n       f 2 b n, X b n+1 = fn X = n f       3n     f4 

bn+1 −Fbn fn n F fn F n + f n − P˙n − Ω +κ P ∆l n n cn+1 −M cn M n n f f + κ M + (ee1 + γ )F + mn ∆l b n+1 −Vb n V fn V n + (ee1 + γfn )Ωn − γ˙n +κ ∆l n+1 b n b Ω −Ω fn Ωn − κ˙n +κ ∆l

˙ n fn n fn n −H −Ω H −V P

     

=0

    

n = 1, 2, . . . , N (15)

where, as defined above, the barred quantities correspond to the constant values of the variables in the element interior while the hatted quantities are nodal values. The barred and hatted quantities of the primary variables of the nth element X n are related as  n   Fbn+1 +Fbn      2  F n   M  cn+1 +M cn     n M 2 X = = Vb n+1 +Vb n . (16) n    V      2  n   Ωb n+1 +Ωb n   Ω 2

The barred secondary variables (γ, κ, P, V ) are related to the barred primary variables as stated above in the cross-sectional constitutive laws, Eqs. (10) and (11).

5

2.2

The Simulation Model

The transformation of the present approach from a set of equations written on paper to a ready-to-use simulation tool was accomplished in the scope of this work using Matlab V 6.5.0 R13. To solve the dynamic equations, the routines provided by SIMULINK (ode23s) were used. But before heading to the actual simulation and to control design, the basic equations have to be supplemented by actuator and sensor models and brought in a form suitable for dynamic simulation. 2.2.1 Actuator and Sensor Models Actuation is provided to the beam by embedded strain actuators driven by an applied electric field, i.e. by voltage. In order to establish a closed control loop, the deformation of the structure is measured by strain sensors. Hence, for setting up a simulation model suitable for control design, the generalized actuation strains γa , κa have to be related to the actual control input, i.e. to the applied voltage ua . Vice versa, the resulting generalized strains γ, κ have to be transformed to the output signal, i.e. to the sensor voltage us . This task can be accomplished in a linear range by introducing the actuation matrix AM n and the sensor matrices SM n and SAn for the nth element element:  n γa (17) = AM n una κna where una is the vector of voltage applied to the actuators integrated in the nth element. Accordingly, the voltage measured by the sensors of the nth element uns can be obtained by means of the sensor matrices SM n and SAn : ) ( bn X n n + SAn una . us = SM (18) n+1 b X One notes that SAn represents a feed-through term directly relating the actuation voltage una to the sensor voltage uns . This stems form the choice of the primary variables as defined in Eq. (12). X n does not contain strain or curvature variables but only generalized forces. Analogous to thermal deformation, the deformation caused by the actuators, i.e. by una , does not lead to any resulting internal forces and therefore has no immediate impact on X n . (For the steady case: X 6= f (ua )). 2.2.2 Solution The set of finite element equations presented above is a set of nonlinear differential equations in time. The steady solution X0 for a specified external load w0 = {f0 T , m0T }T and actuation field ua,0 can be calculated by solving the nonlinear steady equations (no time derivatives) using the Newton-Raphson method. Now, the dynamic equations can be linearized about any steady solution to obtain a set of equations in state-space form: (

b˙ X u˙ A

)

" =

−J ˆ˙−1 Jua,0

0

0

X0

| us =

−J ˆ˙−1 JXˆ 0

X0

{z

System Matrix A

  SM SA | {z }

#



}

 b X ua | {z }

"



State Vector

+

−J ˆ˙−1 Ju˙ a,0

−J ˆ˙−1 Jw0

I

0

X0

|

{z

#

X0

Input Matrix B

}



 u˙ a w | {z }

Input Vector



b X . ua

Output Matrix C

(19)

6

Here, JY0 is the Jacobian of the complete set of finite element equations f n as defined in Eq. (15) with respect to Y in the point of linearization. When approximating f n by the b ua , linear section of a Taylor series, the linearized equations are not only a function of X, and w but as well of the rate of change of the actuation voltage u˙ a as {γ, ˙ κ} ˙ T = f (u˙ a ), see Eq.(17). Because of that it becomes necessary to introduce the virtual input u˙ a and, consequently, to enhance the state vector by ua to transform the linearized equations to the usual state-space form. Thus, the final model does not show an explicit feed-through matrix as the feed-through term SA is already integrated in the output matrix C. b and the For reducing the order of Eq. 19 a modal transformation and reduction of X corresponding system sub-matrices is performed. In order to avoid complex numbers, the matrix of eigenvectors Θ used for the transformation is replaced by the real matrix ΘR with the same column space. This leads to a new system (sub-)matrix of the following shape:   ζ1 ω1 0 0 ... 0 0 −ω1 ζ1 0 0 ... 0 0     0 0 ζ2 ω2 . . . 0 0     0  −1 −1 0 −ω ζ . . . 0 0 2 2 −ΘR J ˆ˙ JXˆ0 ΘR =  (20)  X0  ..  .. .. .. . .  . . 0 0  . . .    0 0 0 0 . . . ζ12N/2 ω12N/2  0 0 0 0 . . . −ω12N/2 ζ12N/2 with ζi and ωi being the real and imaginary parts of the 4 · 3 · N conjugate complex eigenvectors λi = ζi ± iωi . Simulation Model Steady State

Disturbance

 f 1  1  m  w= M   f n   m n 

Xˆ 0 Modal Trafo & Reduction

Z0 us

B

u&a

ua ,0

State Space Model

∫ x&dt

Command

A ∫ u& dt

Output

C ˆ X

ua

a

Figure 3: Realization of the state-space model

Figure 3 illustrates the principle assembly of the above to a executable simulation model with the vector of applied external forces and moments w as disturbance input, the rate of change of the actuator voltage u˙ a as virtual command input and the sensor voltage us as only “visible” system output. 2.2.3 Preliminary Validation As a preliminary validation of the structural dynamic model, the frequencies of a cantilevered as well as rotating beam are presented and compared with published results.[8]

7

Span Chord Mass per unit length Mom. Inertia (50% chord) Spanwise elastic axis Center of gravity Bending rigidity Torsional rigidity Bending rigidity (chordwise)

16 m 1m 0.75 kg/m 0.1 kg m 50% chord 50% chord 2 × 104 N m2 1 × 104 N m2 4 × 106 N m2

Table 1: Blade data (for model validation purpose)

Mode (rad/s) Present: 10 elem Exact % difference Cantilevered Blade: ω = 0 & v = 0 1st bending 2.250 2.243 0.3 nd 2 bending 14.61 14.06 3.9 3rd bending 43.78 39.36 11.2 1st torsion 31.11 31.05 0.2 2nd torsion 94.90 93.14 1.9 Rotating Cantilevered Blade: ω = 3.189 rad/s & v = 0 st 1 bending 4.111 4.114 -0.1 2nd bending 16.75 16.23 3.2 rd 3 bending 45.97 41.59 10.5 Rotating Cantilevered Blade with Offset: ω = 3.189 rad/s & v = 51.03 m/s 1st bending 5.697 5.703 -0.1 nd 2 bending 19.22 18.72 2.7 rd 3 bending 48.83 44.50 9.7 Table 2: Blade structural frequencies (for model validation purpose)

8

The structural as well as inertial data for the test blade are given in Table 1. The structural frequencies are presented in Table 2. The results show very accurate estimation of the first few modes using just 10 elements for the non-moving blade (ω = v = 0), the rotating blade (ω = 3.189 rad/s, v = 0 m/s) and the rotating blade undergoing additional translation (ω = 3.189 rad/s, v = 51.03 m/s). 3

Application to an Active Blade

The Active Twist Rotor (ATR) is a prototype rotor blade designed for vibration reduction using embedded anisotropic piezoelectric actuators. It is developed, built, and tested in a joint program between NASA Langley, US Army Research Office and MIT. The main emphasis of the ATR project is to add vibration damping to the blade via twist actuation and aeroelastic torsion–bending coupling. In the scope of the present study, the control potential of the ATR prototype without aerodynamic forces shall be investigated. ACTIVE REGION E-Glass 0°/90° AFC +45° (±1500 V) E-Glass +45°/-45° AFC -45° (±1500 V) E-Glass 0°/90° NOSE E-Glass 0°/90° S-Glass 0° E-Glass +45°/-45° E-Glass 0°/90°

FAIRING E-Glass 0°/90°

Non-Structural MASS 4.85 mm

WEB E-Glass 0°/90° E-Glass 0°/90°

47.75 mm 107.70 mm

Figure 4: ATR profile and ply lay-up

3.1

Configuration

The radius of the 4 fully articulated blades of the ATR is 1.397m and the airfoil section is a NACA 0012 as depicted in Figure 4. The blade is made from E-glass, S-Glass and Active Fiber Composites (AFC), exhibiting significant structural and inertial bending–extension and shear–torsion coupling. For mass balancing, lead blocks are placed at the leading edge and at the web. Actuation is provided by 24 AFCs, 4 AFCs embedded along the inner and outer side of the upper and lower surface of the blade at alternating ±45◦ orientation angles at 6 span-wise locations. The maximum actuation voltage is ±1500 V and the maximum resulting actuation in terms of generalized strains is γ(1)a,max = 5.5e−5 (extension), κ(1)a,max = 2.3e − 2 (Torsion), κ(2)a,max = 1.2e − 2 (flap-wise bending) and κ(3)a,max = 5.1e − 4 (chord-wise bending). The nominal rotational frequency of the rotor is 11.46 Hz (72.0 rad/s) leading to a centrifugal loading of 738.5 g at the tip. Most of the above and further details about the ATR program can be found in Ref. [9]. 3.2

Modeling and Modal Analysis

The ATR blade is modeled is discretized using 12 equidistant finite elements with each actuator extending over 2 elements. The actuation matrix per element AM n is set so as to result in the maximum achievable strains when commanding the corresponding maximum voltage. All simulations presented in the following are run for the blade rotating with

9

Internal Forces

Internal Moments

3500

0

3000 -1

2500 2000

F1 [N] F2 [N] F3 [N]

1500 1000

M [Nm] 1 M2 [Nm] M [Nm]

-2

3

-3

500 0

0

-4

0.5 1 length along B1-axis [m]

0

Translational Speeds

0.5 1 length along B1-axis [m] Rotational Speeds

100

70 60

80

50 60 V1 [m/s] V2 [m/s] V [m/s]

40 20 0

Ω [rad/s] 1 Ω2 [rad/s] Ω [rad/s]

40 30

3

0

3

20 10 0

0.5 1 length along B -axis [m]

0

1

0.5 1 length along B -axis [m] 1

Figure 5: Steady states for ω = 72.0 rad/s

ω 0 rad/s

72.0 rad/s

% Diff.

Class [Hz] [rad/s] Class [Hz] [rad/s]

1 bf1 2.2 13.68 bf1 12.1 75.97 455

2 bc1 11.3 70.76 bc1 12.2 76.35 7.9

3 bf2 13.8 86.71 bf2 32.1 201.8 133

4 bf3 39.6 248.8 tors1 55.2 346.9 1.9

5 tors1 54.2 340.4 bf3 62.7 394.2 58

6 bc2 69.4 436.1 bc2 74.2 466.2 6.9

7 bf4 80.7 507.3 bf4 108 676.3 33

8 bf5 141 885.5 bf5 164 1032 17

9 tors2 165 1034 tors2 171 1073 3.8

10 bc3 191 1199 bc3 196 1233 2.8

11 bf6 226 1422 bf6 257 1615 14

72 1.04e5 6.5e5 1.04e5 6.5e5 0.0

Table 3: The first eigenfrequencies of the ATR, discretized to 12 finite elements, for the static and the rotating case

the nominal frequency of 72.0 rad/s, which leads to the steady internal forces Fi , Mi (extension–bending coupling) and velocities Vi , Ωi as depicted in Figure 5. The high nominal rotational frequency also has an important impact on the dynamic behavior of the structure, i.e. on its modeshapes (bfi - ith bending mode in flap direction, bci - ith bending mode in chord direction, torsi - ith torsional mode, exti - ith extensional mode) and its eigenfrequencies as listed in Table 3 for the rotating and the non-rotating case. The comparison of the eigenfrequencies clearly shows the stiffening effect arising from the rotation for the lower bending modes, especially for those in flap direction, bfi . This phenomenon can be explained by two effects: Unlike the more strain based higher order modes, the lower order modes are mainly displacement based, i.e., relatively small strains leading to large deflections. Therefore these modes are more sensitive to rotational acceleration. Figure 6 illustrates the idea of strain and displacement based modes for two different bending modes of the ATR. Furthermore, the structure is more flexible in flap direction than in chord direction and hence the relative stiffening of the modes in flap direction is significantly higher than the chord-wise stiffening. When calculating the eigenfrequencies of the ATR by means of the system matrix of the state-space system as given in Eq. 19, one expects purely imaginary poles as damping is

10

Bending 1 (flap direction) 0

Bending 5 (flap direction)

-0.1

0.04 0.02 0 -0.02

-0.15

0

-0.05

0.2

-0.2 -0.25

0.4

-0.3

0.6 -0.35

0.8

0 0.2 0.4

1 0.6 0.8

1.2

1 1.2 -0.05 0

-0.05 0

0.05

(a) 1st bending mode in flap direction (ω = 72.0 rad/s)

0.05

(b) 5th bending mode in flap direction (ω = 72.0 rad/s)

Figure 6: Visualization of eigenmodes of the ATR (simplified profile)

not taken into account in the present approach. However, the present formulation is mixed (strains and velocities) and does not lead to symmetric mass/stiffness and anti-symmetric damping matrices as seen in a pure displacement formulation. Thus the discretization affects the real as well as imaginary part of the eigenvalues. Thus positively damped eigenfrequencies as depicted in Figure 7 are encountered. The damping ratios are very low for the non-rotating case (Φd,min = −1.17e − 15), as seen in the left-hand side of Figure 7 and can be attributed to numerical error. But the damping is unacceptably high for the rotating case (Φd,min = −2.37e − 3) and cause severe simulation problems. This effect can not be explained by numerical inaccuracies but is a general problem of the present approach, where damping is not ruled out per definition, as it is in the classical linear theory, but simply not considered at all. Because of that, the inevitable modeling error due to discretization not only affects the eigenfrequencies but the damping ratios as well. Knowing that, the problem can be solved for the linearized equations by setting the real parts ζi of the eigenvalues in the system sub-matrix from Eq.(20) to zero. This correction represents the enforcement of the alignment of the system poles on the imaginary axis and was performed during the present work. But, the foregoing does not help the simulation model based on the complete non-linear equations. The only way to realize such a model is the use of energy-decaying algorithms when solving the finite element equations. 4

Control Design

To reduce structural vibrations of the ATR blade, causing both fatigue and noise, the augmentation of structural damping is the primary goal of the present control design. A very accurate and reliable mathematical model of the MIMO (Multiple Input Multiple Output) system is available in a linearized, modal transformed form. Hence, the application of a LQR approach in combination with a Kalman filter not only promises satisfying results but in a way suggests itself: For the Kalman filter, the order of the observation model readily can be reduced to a reasonable degree and the individual modes can be penalized directly by the diagonal elements of the respective weighting matrix in the LQR cost functionals.

11

ATR Poles (Ω3 = 0Hz)

5

x 10

6

6

4

4

2

2

0

-2

-4

-4

-6

-6

-3

-2

-1

0

1

x 10

0

-2

-8 -4

ATR Poles (Ω3 = 11.5Hz)

5

8

Imaginary Axis

Imaginary Axis

8

-8 -400

2

-200

0

-11

Real Axis

200

400

Real Axis

x 10

Figure 7: Damped poles of the ATR for the non-rotating (left hand side) and the rotating case.

4.1

Controllability Analysis

The maximum strains γa , κa achievable with the given actuators already have been presented in Section 3 3.1. Considering these relatively small strains and keeping in mind the additional stiffening effect due to the high rotational speeds of the rotor, one can expect restricted controllability of the lower order bending modes (especially in flap direction). This is confirmed by a detailed controllability analysis performed on the ATR making use of the Popov-Belevitch-Hautus (in short PBH) test, as described in the literature such as Ref. [10]. Figure 8(a) illustrates this general decrease of controllability with the increase of rotation ω for the first bending modes in flap direction bfi . Further the decrease of controllability with higher eigenfrequencies for both the non-rotating and the rotating case comes to light clearly for bending in both directions. The latter effect is due to the strain based actuation devices: Their influence is much higher on the modes of higher or-3

1.4

2.5

x 10

ω = 0 Hz ω = 11.5 Hz

1.2

-3

-3

x 10

1.2

Modal Controllability

x 10

bf1 (ω = 11.5 Hz)

ω = 0 Hz ω = 11.5 Hz

bf (ω= 0 Hz) 1

1

2

bc (ω = 11.5 Hz) 1

bf2 (ω = 11.5 Hz)

1 0.8

1.5

0.8

0.6

0.6

1 0.4

0.4 0.5 0.2 0

0.2

1

2

3 bfi

4

0

1

2 bci

3

0

1

2

3

4

5

AFC Station

(a) Controllability of the first bending modes in flap (left) and chord (right) direction

(b) Modal controllability per AFC station

Figure 8: Controllability of the ATR

12

6

Mode Class Φd × 1e−2 Mode Class Φd × 1e−2

1 bf1 0.89 7 bf4 5.43

2 bc1 5.76 8 bf5 4.94

3 bf2 4.58 9 tors2 5.98

4 tors1 3.91 10 bc3 5.03

5 bf3 5.09 11 bf6 5.17

6 bc2 5.39 12 ext1 5.20

13 tors3 5.32

Table 4: Damping of the closed loop

der which are more strain based by themselves. When determining the penalty matrices for the LQR design, particular problems arising from the weak controllability of the first eigenmode in flap direction bf1 have to be expected. Figure 8(b) shows the allocation of controllability of the modes bf1 , bc1 and bf2 spread over the 6 AFC packs along the blade (1 AFC pack comprises 4 AFCs at the 4 sides of a specific cross-section). One notes that the very weak controllability of bf1 is even further decreased for the rotating case and virtually limited to only the actuators at the blade root, due to the modified shape of this mode as depicted in Figure 6(a). Fortunately, a more evenly distributed controllability is already given for bf2 and the controllability of the first bending mode in the stiffer chord direction bc1 does not cause problems and neither do other modes of higher frequency. Closed Loop Poles (72 Mode Plant) 1800

1600

1400

Imaginary Axis

1200

1000

800

600

400

200

0 -500

-400

-300 Real Axis

-200

-100

0

Figure 9: Poles of the closed loop

4.2

Control Design

The controller is designed for the blade rotating with the nominal frequency of 72.0 rad/s. The system is reduced to 13 modes, with the frequencies given in Table 3. The weighting matrices for the calculation of the LQR gain matrix where obtained assuming the command input ua , the disturbance input w and the measurement output us to be superimposed by a white noise with realistic standard deviations. The achieved damping of the closed loop for the first 13 modes of the system is illustrated in Figure 9 and listed in detail in Table 4. The purely real poles depicted in Figure 9 are the eigenvalues related to the artificial states ua of the closed loop system. The achieved damping ratios Φd of the physical modes of about 0.05 (expect for the first bending mode bf1 ) represent a significant augmentation compared to the natural damping of comparable structures, which often exhibit damping ratios as small as 0.008.[10] When using a higher order model for the plant in the closed loop, spillover, i.e. the destabilization of higher order modes by the

13

controller, can be analyzed. For the present configuration, the worst spillover damping ratios for poles close to the controller bandwidth do not exceed -0.0021 and therefore do not cause severe problems, either when simulating or for practical applications. 4.3

Simulation Results

In order to validate the control design, the ATR response to a representative white noise in the applied external forces is simulated for both the open loop and the closed loop over a time period of 1s. The measured voltage us as well as the commanded voltage ua are assumed to be noisy. The resulting time history of the internal forces F and moments M and the generalized speeds V and Ω in an individual node in the center of the blade are shown in Figure 10. The significant reduction of the oscillation amplitude demonstrates the efficiency of the final controller. 5

Conclusion

The paper presents a set of intrinsic beam equations for the dynamic behavior of cantilevered active blades subject to rigid body motion and large global deformations. The equations have been used to establish and implement an environment in Malab/SIMULINK for simulation control design. The possibility to reduce the number of primary equations and variables by directly incorporating the linear constitutional equations in the approach has been pointed out. As shown, the actual choice of primary variables implies the introduction of the rate of change of the commanded actuator voltage as virtual system input and avoids an explicit feed-through matrix in the resulting state-space model. The present simulation environment has been validated successfully for a given beam by comparison with exact analytical results before heading to the analysis of the Active Twist Rotor configuration. When calculating the eigenfrequencies of the ATR, the problem of positively damped poles has been encountered. This phenomenon can be explained by the nature of the equations used, which do not rule out damping per definition and thus not only imply frequency but as well damping errors when applied in discretized form. The problem has been solved effectively by shifting the poles onto the imaginary axis by transforming to the modal system matrix form. However, when aiming to directly simulate with a nonlinear model, energy decaying integration algorithms have to be used. The controllability analysis performed on the ATR has clearly shown the strong impact of rotation upon the behavior of the structure and served as the starting point for the design of the LQR controller. Significant vibrational damping can be provided to the structure and has been demonstrated by analytical analysis and simulation runs of the closed loop. The presented work comprises all steps starting from the set up of the continuous system equations, then to finite elements discretization and the implementation in MATLAB, and finally to the analysis of a given configuration and the final design of an optimal MIMO controller. During this whole process, both the accuracy and the compactness of the used equations, based on the absence of any displacement or rotation variables, have come to light clearly. Therefore the present approach is thought to represent a valuable and effective alternative supporting design, analysis and simulation of beam like structures, such as (wind)turbine blades, high aspect-ratio wings of modern gliders or, as in this case, helicopter blades.

14

Internal Forces in Node 6

Internal Moments in Node 6 15

dF [N] 1 dF2 [N] dF [N]

60 40

3

20

dM [Nm] 1 dM2 [Nm] dM [Nm]

10

3

5

0

0

-20

-5

-40 -10 -60 -15

-80 0

0.2

0.4 0.6 time [s]

0.8

1

0

Translational Speeds in Node 6 10

0.4 0.6 time [s]

0.8

1

Rotational Speeds in Node 6

dV1 [m/s] dV2 [m/s] dV3 [m/s]

5

0.2

d: [rad/s] 1 d:2 [rad/s] d:3 [rad/s]

100 50

0

0

-5

-50 -100

-10 0

0.2

0.4 0.6 time [s]

0.8

1

0

0.2

0.4 0.6 time [s]

0.8

1

(a) Noise response of the open loop Internal Moments in Node 6

Internal Forces in Node 6 3

dF1 [N] dF [N] 2 dF3 [N]

10 5

dM1 [Nm] dM2 [Nm] dM3 [Nm]

2 1

0

0

-5

-1

-10

-2 -3 0

0.2

0.4 0.6 time [s]

0.8

1

0

Translational Speeds in Node 6 4

2

0.4 0.6 time [s]

0.8

1

Rotational Speeds in Node 6

dV1 [m/s] dV2 [m/s] dV3 [m/s]

3

0.2

d:1 [rad/s] d:2 [rad/s] d:3 [rad/s]

20 10

1 0

0

-1

-10

-2 -20

-3 0

0.2

0.4 0.6 time [s]

0.8

1

0

0.2

0.4 0.6 time [s]

(b) Noise response of the closed loop Figure 10: Noise response of the open and the closed loop

15

0.8

1

Appendix Consider the generalized strain-displacement equations,   γ = C e1 + u0 + e ku − e1

(21)

κ e = −C 0 C T + C e kC T − e k

(22)

and, the generalized velocity-displacement equation, V = C [v + ω e (r + u) + u] ˙ e = −CC ˙ T + Cω Ω eC T

(23) (24)

Even though the present paper analyzes straight helicopter blades, the derivation of the intrinsic kinematical equations is kept general and includes a constant initial twist/curvaure (k). To derive intrinsic equation, the generalized displacement variables (u and C) have to be removed from the above equations. To do so, the strain-displacement equations are differentiated w.r.t. time and the velocity-displacement equations are differentiated w.r.t. space. Thus for the linear strain and velocity,     γ˙ = C˙ e1 + u0 + e ku + C u˙ 0 + e k u˙ (25) V 0 = C 0 [v + ω e (r + u) + u] ˙ + C [v 0 + ω e (r0 + u0 ) + ω e 0 (r + u) + u˙ 0 ]

(26)

By subtracting the first equation from the second, the term containing u˙ 0 drops out. In addition, using the following, u0 = C T (γ + e1 ) − e1 − e ku T

(27)

u˙ = C V − ω e (r + u) − v C 0 = −e κC + C e k−e kC

(28)

e + Cω C˙ = −ΩC e r0 = e1 − e kr

(30)

0

ω = −e kω 0 v = −e kv the expression becomes,     V 0 − γ˙ = −e κC + C e k−e kC v + ω e (r + u) + C T V − ω e (r + u) − v   n  h io f T e e e e + C −kv + ω e e1 − kr + C (γ + e1 ) − e1 − ku − kω(r + u)  n h i o e + Cω − −ΩC e e1 + C T (γ + e1 ) − e1 − e ku + e ku   − Ce k CT V − ω e (r + u) − v   e (e1 + γ) =− κ e+e k V +Ω

(29) (31) (32) (33)

(34)

In the above simplification the following identity for (f) operator was used, f −e ωe k−e kω + e ke ω=0

16

(35)

Similarly, for the curvature and angular velocity, e˙ = −C˙ 0 C T − C 0 C˙ T + C˙ e κ kC T + C e k C˙ T e 0 = −CC ˙ T − CC ˙ 0T + C 0 ω Ω eC T + C ω e0C T + C ω e C 0T

(36) (37)

Again subtracting the first equation from the second,   T   f 0 e e e e e e e Ω − κ˙ = − −ΩC + C ω e −e κC + C k − kC + −e kωC T κC + C k − kC ω eC T − C e  T   T e e e e e + Cω e −e κC + C k − kC + −e κC + C k − kC −ΩC + C ω e    T e + Cω e + Cω − −ΩC e e kC T − C e k −ΩC e e κ+e e = Ω(e k) − (e κ+e k)Ω z g }| { e + k) = Ω(κ (38) Thus, the intrinsic equations can be written as: Ω0 + (e κ+e k)Ω = κ˙ V 0 + (e κ+e k)V + (ee1 + γ e)Ω = γ˙

(39) (40)

Acknowledgement The authors would like to acknowledge some thoughtful technical discussions with Prof. Dewey Hodges from Georgia Tech. WE would also like to thank Prof. Carlos Cesnik and Dr. Rafael Palacios from University of Michigan for the ATR cross-sectional data. Further, we would like to acknowledge Prof. Dr.-Ing. Gottfried Sachs from the Technische Universit¨at M¨ unchen for his support of the author J. Traugott. References [1] M. L. Wilbur, P. H. Mirick, W. T. Yeager, C. W. Langston, C. E. S. Cesnik, and S. Shin. Vibratory loads reduction testing of the nasa/army/mit active twist rotor. Journal of the American Helicopter Society, 47(2):123 – 133, Apr. 2002. [2] A. S. Hopkins and R. A. Ormiston. An examination of selected problems in rotor blade structural mechanics and dynamics. In Proceedings of the 59th American Helicopter Society Annual Forum, Phoenix, Arizona, May 2003. [3] V. V. Volovoi, D. H. Hodges, C. E. S. Cesnik, and B. Popescu. Assessment of beam modeling methods for rotor blade applications. Mathematical and Computer Modelling, 33(10 – 11):1099 – 1112, May – June 2001. [4] D. H. Hodges. A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams. International Journal of Solids and Structures, 26(11):1253 – 1273, 1990. [5] D. H. Hodges. Geometrically exact, intrinsic theory for dynamics of curved and twisted anisotropic beams. AIAA Journal, 41(6):1131–1137, 2003.

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[6] M. J. Patil and E. R. Johnson. Cross-sectional analysis of anisotropic, thin-walled, closed-section beams with embedded strain actuation. In Submitted for presentation at the 46th Structures, Structural Dynamics, and Materials Conference, 2004. [7] C. E. S. Cesnik and R. Palacios. Modeling piezocomposite actuators embedded in slender structures. In Proceedings of the 44th Structures, Structural Dynamics, and Materials Conference, Norfolk, Virginia, April 2003. AIAA Paper 2003-1803. [8] D. H. Hodges and M. J. Rutkowski. Free-vibration analysis of rotating beams by a variable-order finite-element method. AIAA Journal, 19(11):1459 – 1466, 1981. [9] C.E.S. Cesnik, S.J. Shin, and M.L. Wilbur. Dynamic Response of Active Twist Rotor Blades. Smart Materials and Structures, 10:62–76, 2001. [10] A. Preumont. Vibration Control of Active Structures - An Introduction. G.M.L. Gladwell, Waterloo, Ontario, Candada, 2001.

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