Ing. G. Sachs NONLINEAR DYNAMICS AND CON

¨r Flugmechanik und Flugregelung Lehrstuhl fu ¨ t Mu ¨nchen Technische Universita Prof. Dr.-Ing. G. Sachs

NONLINEAR DYNAMICS AND CONTROL OF INTEGRALLY ACTUATED HELICOPTER BLADES

Verfasst von:

cand.-Ing. MATTHIAS ALTHOFF

Betreut von: Dr. Mayuresh Patil (Virginia Polytechnic Institute and State University) Dipl.-Ing. Johannes Traugott (Technische Universität M¨ unchen) Blacksburg, VA U.S.A. - Dezember 2005

Erkl¨ arung

Hiermit erkl¨ are ich, dass ich die vorliegende Arbeit selbständig, ohne fremde Hilfe und nur unter Verwendung der angegebenen Quellen angefertigt habe.

Ort,

Datum,

Unterschrift

Contents 1 Introduction 2 Mathematical Model 2.1 Intrinsic Equations . . 2.2 Constitutive Equations 2.2.1 Passive Beam . 2.2.2 Active Beam .

3

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

5 5 6 6 6

3 Galerkin discretized Beam Model 3.1 Theory of the Galerkin Approach . . . . . . . . 3.2 Energy consistent Weighting . . . . . . . . . . . 3.2.1 Physical Interpretation . . . . . . . . . . 3.2.2 Weighting Functions . . . . . . . . . . . 3.3 Approximated Blade Model . . . . . . . . . . . 3.3.1 Structure of the Comparison Functions 3.3.2 Application to the Beam Model . . . . . 3.4 Comparison Functions . . . . . . . . . . . . . . 3.5 Special Case: Constant Cross Section . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

9 9 10 11 12 13 13 14 17 17

4 Modal Space 4.1 Linear Modal Space . . . . . . . . . . . 4.1.1 Steady State Calculation . . . . 4.1.2 Linearization . . . . . . . . . . . 4.1.3 Natural Modes and Frequencies . 4.2 Nonlinear Modal Space . . . . . . . . . . 4.2.1 Basic Idea . . . . . . . . . . . . . 4.2.2 Nonlinear Conservative Example 4.2.3 Future Work . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

21 21 22 23 23 24 24 25 26

5 Order Reduction 5.1 Order Reduction Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Normal Modes Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Pertubation Modes Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 29 30

6 Simplified Beam Model 6.1 Analytical Comparison Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Comparison Functions fulfilling Boundary Conditions . . . . . . . . . . . . . . . . . . 6.3 Arbitrary Comparison Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 34 35 36

7 Helicopter Blade 7.1 Simple Example . . . . . . . . . 7.2 Helicopter Blade Specification . . 7.2.1 General properties . . . . 7.2.2 Cross-Sectional Flexibility 7.2.3 Actuation . . . . . . . . . 7.2.4 Sensing . . . . . . . . . .

39 39 40 40 40 41 43

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . . . . . .

. . . .

. . . . . . . .

. . . .

. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . and Inertia . . . . . . . . . . . . . . 1

. . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

2

CONTENTS

7.3

Simulation Results . . . . . . . . . . . 7.3.1 Steady State . . . . . . . . . . 7.3.2 Natural Frequencies and Modes 7.3.3 Displacement . . . . . . . . . . 7.3.4 Actuation Response . . . . . . 7.3.5 Reduced Model Performance . 7.3.6 State Space Equations . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

43 43 43 44 46 47 48

8 Preliminary Control Considerations 8.1 System Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 52

9 Stability of Nonlinear Systems 9.1 Lyapunov Stability for Autonomous Systems . . . . . . . . . . . . . . . . . . . . . . 9.2 Proposed Lyapunov function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 55

10 Linear Optimal Control 10.1 LQR Controller . . . . 10.2 Kalman Observer . . . 10.3 LQG Controller . . . . 10.4 Simulation Results . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

57 57 58 59 60

11 Adaptive Damping 11.1 Basic Idea . . . . . . . . . . . . 11.2 Instable Jeffcott Rotor . . . . . 11.3 Helicopter Blade . . . . . . . . 11.3.1 Segment Approximation 11.3.2 Galerkin Approximation

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

63 63 64 65 65 67

12 Further Control Concepts 12.1 Gain Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Input-Output Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Hybrid Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 69 70

13 Conclusion

73

. . . .

. . . .

. . . .

. . . .

A Tensor Fundamentals A.1 Notation and Summation Convention . A.2 Definition of a Tensor . . . . . . . . . A.3 Tensor Algebra . . . . . . . . . . . . . A.4 Tensor Calculus . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

B Alternatively Discretized Beam Model C Software C.1 Documentation . . . . . . . . . . . . . C.1.1 Precalculation . . . . . . . . . C.1.2 Building Model and Controller C.1.3 Simulation . . . . . . . . . . . C.2 Implementation Notes . . . . . . . . . C.2.1 Matlab GUI Fundamentals . . C.2.2 Data Structure of GUI . . . . .

75 75 76 76 77 79

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

81 81 81 82 86 86 86 87

Chapter 1

Introduction The performance of helicopters depends significantly on its blades. To face their importance in the design process, helicopter blade development has to undergo multidisciplinary optimization. Competing design objectives are light weight, aerodynamic efficiency, strength and stiffness. Over the last decades, there have been improvements achieved using passive methodologies such as better materials and structural design. Another approach which is becoming more and more attractive due to improvements of sensors, actuators, control designs and simulation tools is to apply active methodologies.1, 2 Active helicopter blades provide forces and moments induced by actuators to compensate the disturbances occurring from the aero-elastic coupling of the blade. The advantage of actuated helicopter blades is the improvement in its dynamical behavior without compromising the competing aforementioned design constraints for the passive helicopter blade. Secondary effects of the vibration suppression are reduced noise, improved riding quality and longer fatigue life which leads to better cost efficiency. For both design approaches, the passive and the active one, a simulation environment representing the main effects is indispensable. Helicopter blade simulation is afflicted with two challenges. One is that a helicopter blade is a system with significant environment interaction due to aero-elastic coupling. The second challenge is the large deformation of helicopter blades originating from their length and light weight design. Although the blades are stiffened due to centrifugal forces while rotating, a nonlinear mathematical model is necessary to describe the large deformations properly. The nonlinear dynamic solution and control design for an adaptive helicopter blade was presented by Traugott et al.3 The present work is a continuation of that work dealing with • Structural dynamics modelling based on a nonlinear Galerkin approach which is more efficient as compared to FEM • Nonlinear model order reduction techniques to derive a low-order, high fidelity nonlinear model for control design • Nonlinear control design techniques The nonlinear mathematical model used for the simulation of the blade dynamics is presented in chapter 2. As the beam model has no analytical solution, the approximated solution is calculated using a Galerkin approach to discretize the beam model in chapter 3. After introducing the theory of this approach, it is applied in a way that the approximated solution is fulfilling the energy conservation law for passive beams. A simplified solution for beams with constant cross-section is presented, too. For a first insight of the beam model dynamics, the discretized model is transferred into the modal space in chapter 4, receiving free vibration modes and its natural frequencies. Besides the commonly used linear space which uses a linearization of the beam model, the idea of the nonlinear modal space is introduced. In order to receive a low order model for simulation and control design, the discretized beam model is reduced in chapter 5. This reduction will be done using linear free vibration modes and perturbation modes. After the model order reduction, the beam modelling is finished and the solution of a one dimensional beam as a special case is validated 3

4

CHAPTER 1. INTRODUCTION

against the analytical solution for different comparison functions in chapter 6. The full helicopter blade model is then tested in chapter 7. Therefor, the model is compared with a finite element model, and additionally, the full order blade model is validated against a reduced one. The helicopter blade model is characterized to formulate the control objective in chapter 8. As a basis for the upcoming controller design, the stability of nonlinear autonomous systems is presented in chapter 9. In the following chapter 10, a linear optimal controller is designed and simulated. This controller has an optimal performance around the equilibrium, however there is no known stability proof. The next controller design in chapter 11 is designed for global stability. It is based on the idea that the actuators are controlled in a way that they dissipate more energy than comes into the system by boundary conditions. The work is finished with the presentation of further control concepts in chapter 12 and the conclusion in chapter 13.

Chapter 2

Mathematical Model In order to calculate the dynamics of the helicopter blade at low computational costs, a nonlinear beam model developed by Hodges4, 5 is used. This model takes advantage of the one dimensional characteristics of a helicopter blade and is a better choice compared to 3-D finite-element analysis.6

2.1

Intrinsic Equations

The beam model derived by Hodges covers a beam undergoing large deformation and small strain. The beam formulation is intrinsic, i.e. neither displacement nor rotation variables appear in the beam equations. The intrinsic formulation is very compact and is furthermore applicable for general beams (anisotropic, non-uniform, twisted and curved). The measure numbers of the variables in b3

b2

Undeformed State

b1

x1

r

r

r*

u B3 B2

R

Deformed State

s1 B1

R*

R Unwarped Cross Section

Figure 2.1: Schematic of beam undergoing finite deformation and cross-sectional warping the beam model are calculated in the deformed frame (B-frame). This B-frame is orthogonal and defined by the cross section of the deformed beam, see Figure 2.1. The B2 -axis and the B3 -axis lie in the cross-section with the B1 -axis defined by B1 = B2 × B3 . The intrinsic equations for the nonlinear dynamics of the beam are ˜+κ ˜ F 0 + (k ˜ )F + f = P˙ + ΩP ˜+κ ˜ + V˜ P M 0 + (k ˜ )M + (˜ e1 + γ˜ )F + m = H˙ + ΩH ˜+κ V 0 + (k ˜ )V + (˜ e1 + γ˜ )Ω = γ˙

(2.1)

˜+κ Ω0 + (k ˜ )Ω = κ˙ where ( )0 denotes the derivative with respect to the beam reference line and (˙) denotes the absolute time derivative. F and M are the measure numbers of the internal force and moment vector 5

6

CHAPTER 2. MATHEMATICAL MODEL

(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 are calculated in the B–frame, i.e., deformed cross-sectional frame. k = bk1 k2 k3 c is the initial twist/curvature of the beam and e1 = b1 0 0cT . 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, i.e., e ab = a × b.

2.2

Constitutive Equations

The intrinsic beam equations provide four vector equations for eight vector unknowns (F , M , P , H, γ, κ, V , Ω). In order to complete a solvable set of equations, four more vector equations are needed that are called constitutive equations. First, the constitutive equations for a passive beam are presented, which are a special case of the active beam. To get an idea of the derivation of the constitutive equations for the active beam of a helicopter blade, a simple case is considered.

2.2.1

Passive Beam

One of the two constitutive equations relates the generalized forces (F , M ) and the generalized strains (γ, κ) via the beam cross-section stiffness matrix. The beam cross-section inertia matrix relates the generalized momenta (P , H) and the generalized velocities (V , Ω). Both relations are derived from cross-sectional analysis. The constitutive equations are R γ = T κ S

S T

F M

,

G P = T H K

K I

V Ω

(2.2)

where R, S and T are the cross-sectional flexibilities. The cross-sectional flexibilities can be obtained using analytical thin-walled theory7 or computational FEM analysis8 for general configuration. The inertia matrix has the following components:   µ 0 0 G = µI =  0 µ 0  , 0 0 µ



0 ˜  K = −µξ = −µξ˜3 µξ˜2

µξ˜3 0 0

 −µξ˜2 0 , 0

i2 + i3



I= 0 0

0

0



i2 i23  i23 i3

(2.3) where µ, ξ, i2 , i3 , i23 are the mass per unit length, mass center offset and the three cross-sectional mass momenta of inertia per unit length. Throughout this work, only beams with constant cross-section are regarded. In the case of varying cross-sections, the cross-sectional flexibility and inertia are functions of the beam reference line. Although this complication can still be handled by the following concepts, restricting to a constant cross-section is a reasonable simplification.

2.2.2

Active Beam

The constitutive equations of the active beam are an extension of (2.2) to be able to integrate the active forces and moments provided by the actuators. The actuators provided to the integrally actuated helicopter blade regarded in this work are piezo elements. Piezo elements have a frequency response characteristic that allows to operate them at high frequencies. Furthermore, piezo elements can be highly integrated in structures. This technology has been tested in several applications9 and is seen as the most promising solution for vibration control of helicopter blades. An advanced realization of the piezo actuation technology is the ATR blade, a prototype built by NASA Langley Research Center.10 For this blade, piezo fibers have been used that can be integrated into the composite structure of the helicopter blade. The exact specification of the helicopter blade used in this work can be found in section 7.2. In this section, the principle of this technology and the structure of the constitutive equations for an active beam is presented.

2.2. CONSTITUTIVE EQUATIONS

7

Principle Functionality of Piezo Actuated Beams The basic principle of a piezo actuated beam is shown for a case where the piezo element is attached to the beam surface, as shown in figure 2.2. Due to the special crystalline structure of piezoelectric

Fp

Fp

piezo beam

F

F-Fp

M

M-ξ×Fp

Figure 2.2: Piezoelectric element attached to a beam

elements, strain does not only change under an applied force, but also under an applied electric load. As both effects are linear over a wide range, the strain can be calculated by superposition. γ p = Rp F p + DE

F p = Rp −1 γ p − Rp −1 DE

←→

(2.4)

γ p is the piezo strain, F p is the applied force to the piezo element and E is the applied electric field. Rp is the cross-sectional flexibility of the piezo and D the piezoelectric constant. The curvature κP of the piezo element used in this example does not change under an applied electric field E. Additionally, the stiffness of the piezo element under curvature is neglected. Therefore, (2.4) is fully describing the behavior of the piezo element. The generalized strain relations between the piezo element and the beam in figure 2.2 are γ = γb,

κ = κb ,

γ p = γ + ξ˜p κ

(2.5)

where γ b and κb denote the generalized strains of the beam, γ and κ describe the generalized strains of the whole configuration and ξ is the vector lying in the cross-section of the whole configuration connecting the beam to the piezo reference line. With (2.5), (2.4) becomes F p = Rp −1 (γ + ξ˜p κ) − Rp −1 DE

(2.6)

The strain and curvature of the whole configuration is determined using the constitutive equations of the passive beam (2.2) together with the generalized strain relations (2.5). γ (2.5) γ b (2.2) Rb = = T κ κb Sb

Sb Tb

F − Fp M − ξ˜p F p

(2.7)

Next, (2.7) is written in terms of F and M and F p is replaced by (2.6)

F M

Rb = bT S

Sb Tb

−1 p −1 γ R + ˜p p −1 κ ξ R

Rp −1 ξ˜p 0

I γ − ˜p Rp −1 DE κ ξ

(2.8)

Using the linear relation between the electric field E and the voltage applied to the piezo elements uA (E = CuA ) simplifies (2.8) to

F M

R ST

=

S T

−1 γ A A − u κ B

(2.9)

where

R ST

−1

S T

Rb = bT S

Sb Tb

−1

Rp −1 Rp −1 ξ˜p + ˜p p −1 ξ R 0

I A = ˜p Rp −1 DC B ξ

and

(2.10)

8

The constitutive equations R γ = T κ S G P = T H K

CHAPTER 2. MATHEMATICAL MODEL

for the active beam are finally A S F U γ F + ←→ = T M T M V κA K V I Ω

V W

A γ F − κ MA

(2.11)

where U, V and W are the cross-sectional stiffness matrices, F A , M A are the generalized active forces and γ A , κA are the generalized active strains induced by the active elements. A R S FA γ E A = u = T F S T MA κA A (2.12) U V γA F A A = u = T B V W κA MA General Active Beam In a more general case, the cross-sectional flexibilities and the induced generalized forces can be obtained by conducting active cross-sectional analysis using the theory of Patil and Johnson11 (for thin-walled beams) or Cesnik and Palacios12 (for general configuration). However, the structure of the solution for the more general case is the same than in (2.11) as the generalized active strains γ A , κA and the applied voltage uA are still linearly related for the geometrically more complex case. Piezo Elements as Sensors As piezoelectric elements can convert electrical energy to mechanical energy and vice versa, they can also be used as sensors. Similarly to the active elements, the voltage at the sensor elements uS is related linearly to the generalized strains γ and κ. γ S u = O P (2.13) κ

Chapter 3

Galerkin discretized Beam Model The intrinsic (2.1) and constitutive equations (2.11) form a solvable set of equations defining the mathematical beam model. As this model describes the dynamics of a distributed parameter system, it consists of partial differential equations in space and time. Analytical solutions of partial differential equations exist only in very special cases and so, the basic approach of solving these equations is to discretize them in space leading to ordinary differential equations in time. The most commonly used technique for the discretization of partial differential equations is the finite element method .13 This approach was realized for the active helicopter blade investigated in this work by Traugott et al.3, 14 In this work, a Galerkin approach 15 is used. The basic difference between both approaches is that finite elements are discretizing the model locally whereas the Galerkin method is discretizing globally using a basis of globally predefined functions. In this chapter, the theory of the Galerkin approach is presented first. Then, the structure of the global predefined functions is introduced, followed by the application of the Galerkin approach to the beam model. The chapter ends with numerical considerations and a solution for the special case of constant cross-section.

3.1

Theory of the Galerkin Approach

As mentioned above, the Galerkin approach discretizes the solution globally using a basis of predefined functions which are called comparison functions. Let’s assume that the solution of a partial differential equation is dependant on space and time. Then, the approximate solution w(x, t) is build using the Galerkin approach by summing the product of the predefined space dependant comparison functions Φi (x) with time dependant factors qi (t). w(x, t) =

n X

Φi (x)qi (t)

(3.1)

i=1

As the degree of the derivations in the differential equation is not affecting the Galerkin approach, a degree equal to one is assumed for the partial differential equation of w(x, t). f (w(x, t), w(x, ˙ t), w0 (x, t), x, t) = 0;

(3.2)

Substituting w(x, t) by its approximation (3.1) results in f (Φi (x), Φ0i (x), qi (t), q˙i (t), x, t) ≈ 0;

(3.3)

Despite the fact, that (3.3) is not equal to zero, it is demanded that the k integrals of (3.3), each one weighted with a different weighting function Ψk (x) have to be zero. Z L ! {Ψk (x)f (Φi (x), Φ0i (x), qi (t), q˙i (t), x, t)} dx = 0; (3.4) 0

The unknown time functions qi (t) are determined by fulfilling the zero integrals in (3.4) whereas the number of weighting functions Ψk (x) has to equal the number of comparison functions Φi (x) to 9

10

CHAPTER 3. GALERKIN DISCRETIZED BEAM MODEL

determine a unique solution. The k weighted integrals are zero, if the partial differential equation f in (3.4) is orthogonal to each one of the k weighting functions Ψk (x). Orthogonality of two functions g(x) and f (x) over the interval a ≤ x ≤ b is defined as Z

b

g(x)f (x)dx = 0

(3.5)

a

Besides the fact that the solution is unique, a general convergence proof for Galerkin approaches can not be given. However, in the literature, proofs for special cases can be found. Although a convergence proof for the Galerkin approach applied to the beam model is not know, another proof can be performed. Therefor, the assumption in [16, p. 545] is made that the number of comparison functions approaches infinity. In this case, the only way the function f in (3.3) is orthogonal to all Ψk (x) is, if f (Φi (x), qi (t), q˙i (t), t) = 0. This is why an increasing number of comparison functions usually leads to a better approximation.

3.2

Energy consistent Weighting

The quality of the Galerkin discretized model depends on the choice of proper comparison functions Φi (x) and weighting functions Ψk (x) that can be chosen arbitrary. In this chapter, the weighting functions for the beam discreization are selected in a way that the approximated solution fulfills the law of energy conservation if no active elements are used.If active elements are used, no proper weighting functions Ψk (x) to fulfill the law of energy conservation could be found. The energy consistent Galerkin approach uses intrinsic equations (2.1) and boundary conditions that are chosen, without loss of generality, to constrain the generalized velocities at the root (x = 0) and the generalized forces at the tip (x = L) of the beam. The beam variables are not approximated as presented in (3.1), yet, to insure clarity. Consider the following weighted sum of all the differential equations and boundary conditions Z L 0=

0

i h e − F 0 − (e k+κ e)F − f V T P˙ + ΩP i h e + Ve P − M 0 − (e +ΩT H˙ + ΩH k+κ e)M − (ee1 + γ e)F − m i T h + F + FA γ˙ − V 0 − (e k+κ e)V − (ee1 + γ e)Ω i T h + M + MA κ˙ − Ω0 − (e k+κ e)Ω dx

(3.6)

T T − F (0) + F A (0) V (0) − V 0 − M (0) + M A (0) Ω(0) − Ω0 +V (L)T F (L) − F L + Ω(L)T M (L) − M L where, V 0 , Ω0 are the exact linear and angular velocities at the root and F L , M L are the exact force and moment at the tip given by the boundary conditions. Note, that the intrinsic and boundary equations are weighted in a mixed sense. The resulting Galerkin approach is extended to several equations compared to (3.4). In order to increase the number of terms that cancel out in (3.6), V T F 0 and M T Ω0 are integrated by parts. Z Ln Z Ln L L o o 0 0 −V T F 0 − M T Ω0 dx = V T F + M T Ω dx − V T F − M T Ω 0

0

0

0

(3.7)

3.2. ENERGY CONSISTENT WEIGHTING

11

After inserting (3.7) in (3.6), equal terms are labelled by same numbers. Z L 0= 0

e +V T 0 F −V T (e V T P˙ +V T ΩP k+κ e)F −V T f | {z } | {z } | {z } 1

2

3

e +ΩT Ve P −ΩT M 0 −ΩT (e + Ω H˙ +ΩT ΩH k+κ e)M −ΩT (ee1 + γ e)F −ΩT m | {z } | {z } | {z } | {z }| {z } T

=0

1

4

5

6

h i T + (F + F ) γ˙ −F V −F (e k +κ e)V −F T (ee1 + γ k +κ e)V − (ee1 + γ e)Ω e)Ω +F A −V 0 − (e | {z } | {z }| {z } 2 3 6 h i T 0 T k +κ e)Ω +M A −Ω0 − (e k +κ e)Ω + (M + M A ) κ˙ +M T Ω −M T (e | {z } | {z } 4 5 dx A T

T

0

T

−F (0) V (0) +F (0) V 0 − F A (0) | {z } T

T

T

7

T T T V (0) − V 0 −M (0) Ω(0) +M (0) Ω0 − M A (0) Ω(0) − Ω0 | {z } 8

L L +V (L) F (L) −V (L) F L +Ω(L)T M (L) −Ω(L)T M L −V T F −M T Ω {z } | {z } | 0 | {z } | {z 0} T

T

9

10

7,9

8,10

(3.8) After cancelling the terms in (3.8), the equation becomes Z L T T 0= V T P˙ + ΩT H˙ − V T f − ΩT m + (F + F A ) γ˙ + (M + M A ) κ˙ 0

+F

AT

h

i h i AT 0 e e −V − (k + κ e)V − (ee1 + γ e)Ω + M −Ω − (k + κ e)Ω dx 0

+F (0)T V 0 +M (0)T Ω0 − V (L)T F L − Ω(L)T M L − F A (0)

T

3.2.1

T V (0) − V 0 − M A (0) Ω(0) − Ω0 (3.9)

Physical Interpretation

Next, the kinetic and potential beam energy T and U , the power done by the external forces P ext , boundary conditions P bou and actuators P act are introduced, so that (3.9) can be interpreted physically. First, the change of kinetic and potential beam energy T˙ and U˙ is calculated Z T 1 L V G K V T = dx Ω KT I Ω 2 0 Z L T Z Ln o (2.11) G K V˙ V → T˙ = dx = V T P˙ + ΩT H˙ dx T Ω K I Ω 0 0 Z L T 1 U V γ γ dx U= κ VT W κ 2 0 Z L T Z L (2.11) U V γ˙ γ → U˙ = dx = (F + F A )T γ˙ + (M + M A )T κ˙ dx T κ V W κ 0 0 (3.10) The power done by the external generalized forces P ext is Z Lh i P ext = V T f + ΩT m dx

(3.11)

0

P bou , the boundary power of the boundary conditions chosen in this section is P bou = V (L)T F L + Ω(L)T M L − F (0)T V 0 − M (0)T Ω0

(3.12)

12


Finally, the power of the actuators P act is considered. Z Ln o T T act F A γ˙ + M A κ˙ dx P = 0

Z ≈

Ln

FA

0

T

h

i h io T V 0 + (e k+κ e)V + (ee1 + γ e)Ω +M A Ω0 + (e k+κ e)Ω dx | {z } | {z } ≈γ, ˙ see (2.1)

+ FA

T

(3.13)

≈κ, ˙ see (2.1)

T V (0) − V 0 +M A Ω(0) − Ω0 {z } | {z } | ≈0

≈0

The approximation of P act is done using the expressions from the intrinsic equations (2.1) for γ˙ and T T κ. ˙ Furthermore, the terms F A V (0) − V 0 and M A Ω(0) − Ω0 are included in P act as they are only nonzero for active beams. The approximation of P act is exactly fulfilled, if the number of comparison functions in the Galerkin approach is approaching infinity so that the intrinsic equations (2.1) for γ˙ and κ˙ as well as the boundary conditions V (0) = V 0 and Ω(0) = Ω0 are exactly fulfilled. In this case, the intrinsic equations (2.1) and with them the law of energy conservation are fulfilled anyway. Keeping in mind, that P act in (3.13) is approximate, (3.9) is now simplified using (3.10), (3.11), (3.12) and (3.13). T˙ + U˙ ≈ P ext + P bou + P act (3.14) where the approximation sign has to be used because of the inexactness of P act . Equation (3.14) shows the energy balance which is illustrated in figure 3.1 for a helicopter blade. In the case of the passive beam, however, the actuation power P act is zero so that the law of energy conservation in (3.14) is exactly fulfilled. T˙ + U˙ = P ext + P bou (3.15) Note, that the energy level of the approximated solution can still be different from the exact one, although (3.15) is fulfilled, see figure 3.1. E

Pact

No Actuation

Pbou

E

color value t

t E

T+U Wext Wbou Wact

Actuation

Pext

E

t

T+U

Exact Solution

(a) Power balance of the helicopter blade

t Approximated Solution

(b) Energy balance with and without actuation

Figure 3.1: Natural modes compared to analytical solution

3.2.2

Weighting Functions

The result (3.14) of the summation of weighted intrinsic equations and boundary conditions in (3.6) motivates the weighting functions for the Galerkin approach. The space and time dependant weighting functions V , Ω, F + F A and M + M A cannot be used for the Galerkin approach because they have to be space dependant only, see (3.4). However, in the following it is shown that (3.14) still holds if the comparison functions of V , Ω, F + F A and M + M A are used as weighting functions. The first term in (3.6) is exemplarily weighted with the comparison functions V¯k (x) of V (x, t). Z L h i e − F 0 − (e V¯kT (x) P˙ + ΩP k+κ e)F − f dx = 0 (3.16) 0

3.3. APPROXIMATED BLADE MODEL

13

If (3.16) is fulfilled, then Z L X nc 0

h i T ¯ ˙ Vk (x)vk (t) P + . . . dx = 0

(3.17)

k=1

|

{z

≈V

}

is fulfilled, too, where vk (t) are the time function of V¯k (x). The same holds for all other terms in (3.6) and therewith is shown that the comparison functions of the variables can be used instead of the variables themselves.

3.3

Approximated Blade Model

In the previous chapter, a way to set up an energy conservative Galerkin approach for passive beams by choosing proper weighting functions was pointed out. This energy conservative approach is now applied in this chapter.

3.3.1

Structure of the Comparison Functions

As the beam model consists of partial differential equations, the approximated variables need a spacial and time dependant part as presented in (3.1). The approximation of V , Ω, γ and κ is  V¯1l 0 ¯  Vl (x) = 0 V¯2l 0 0  ¯ 1l Ω 0 Pnc ¯ ¯ ¯  0 Ω ≈ l=1 Ω (x)ω (t), Ω (x) = 2l l l l 0 0  γ¯1l 0 Pnc ≈ l=1 γ¯l (x)gl (t), γ¯l (x) =  0 γ¯2l 0 0  κ ¯ 1l 0 Pnc ¯ 2l ≈ l=1 κ ¯ l (x)kl (t), κ ¯ l (x) =  0 κ 0 0

Pnc V (x, t) ≈ l=1 V¯l (x)vl (t),

Ω(x, t)

κ(x, t)

γ(x, t)

   0 v1l  0  , vl (t) = v2l   V¯3l v  3l   0 ω  1l  0  , ωl (t) = ω2l   ¯ 3l Ω ω  3l  0 g1l  0 , gl (t) = g2l   γ¯3l g  3l   0 k1l  0  , kl (t) = k2l   κ ¯ 3l k3l

(3.18)

The numbers 1−3 are the direction indices. As the three dimensional characteristics of the variables is taken for granted, these indices are always left out throughout this work. (¯) indicates the comparison functions of the variables and the small letter variables denote the time functions. As it is very difficult to handle four time functions during the Galerkin approach and for the final structure of the solution, a new time function q(t), covering all other time functions is defined. ql = vl

ωl

gl

kl

T

(3.19)

The approximation for the variables is now written as V (x, t) = ΦVl (x)ql (t), Ω(x, t) = ΦΩ l (x)ql (t), γ γ(x, t) = Φl (x)ql (t), κ(x, t) = Φκl (x)ql (t),

ΦVl (x) = V¯l ΦΩ = 0 l (x) Φγl (x) = 0 Φκl (x) = 0

0 0 0 ¯l 0 0 Ω 0 γ¯l 0 0 0 κ ¯l

(3.20)

Note, that the Einstein notation is used to describe the summation over the same index, see appendix A. This notation will be used from now on, as it is a very compact notation able to describe tensors of any degree. The form of the comparison functions in (3.20) is used to discretize the beam model resulting in a set of ordinary nonlinear differential equations of first degree in the next subsection.

14


3.3.2

Application to the Beam Model

Before the Galerkin approach is applied to the beam model, the constitutive equations (2.11) have to be used to substitute variables in the intrinsic equations (2.1). This will reduce the number of unknown variables to four so that the four intrinsic equations can be solved. In this work, two different substitution possibilities are regarded as both have advantages. In the first substitution scenario, P , H are substituted by V , Ω and F , M are substituted by γ, κ. This particular substitution results in a system with only one input for the actuators (the applied voltage uA ). The second substitution scenario substitutes P , H by V , Ω and γ, κ by F , M . This substitution is used for passive beams because the more problematic system inputs of the actuators (uA and u˙ A ) are zero anyway and flexibility matrices instead of stiffness matrices are appearing in the equations. This makes it possible to simulate stiff systems (zero flexibility values). If stiffness matrices would have to be set up, one needed infinite values. In the following, the first substitution scenario is discretized by the Galerkin approach as shown in chapter 3.2. As this procedure is similar for the second substitution scenario, it is shown in appendix B. The equations of the first substitution scenario (P , H ↔ V , Ω and F , M ↔ γ, κ) are

0=

GV˙ + KΩ˙ − Uγ 0 − Vκ0 − k˜(Uγ + Vκ) ˜ GV + KΩ) − κ ˜ (Uγ + Vκ) + Ω( ˜ )uA + κ + (A0 + kA ˜ AuA − f

0=

KT V˙ + IΩ˙ − VT γ 0 − Wκ0 − k˜(VT γ + Wκ) − e˜1 (Uγ + Vκ) ˜ KT V + IΩ) + V˜ (GV + KΩ) − κ + Ω( ˜ (VT γ + Wκ) − γ˜ (Uγ + Vκ) ˜ + e˜1 A)uA + (˜ + (B0 + kB κB + γ˜ A)uA − m

0=

˜V − e˜1 Ω − κ γ˙ − V 0 − k ˜ V − γ˜ Ω

0=

˜Ω − κ κ˙ − Ω0 − k ˜Ω

(3.21)

Again, like in (3.6), without loss of generality, a beam restricting the generalized velocities at the root (x = 0) and the generalized forces at the tip (x = L) of the beam is used. The weighted boundary conditions at the tip are h i (2.11),(2.12),(3.20) V T L V (L)T F (L) − F L −→ Φk (L) UΦγi (L) + VΦκi (L) qi − Au (L)uA u −F h i (2.11),(2.12),(3.20) Ω T L Ω(L)T M (L) − M L −→ Φk (L) VT Φγi (L) + WΦκi (L) qi − Bu (L)uA − M u

(3.22)

At the root, the boundary conditions are i T (2.11),(3.20) T h F (0) + F A (0) V (0) − V 0 −→ UΦγk (0) + VΦκk (0) ΦVi (0)qi − V 0 i T (2.11),(3.20) T γ T h 0 M (0) + M A (0) Ω(0) − Ω0 −→ V Φk (0) + WΦκk (0) ΦΩ ( 0 )q − Ω i i

(3.23)

The boundary conditions (3.22), (3.23) and the Galerkin discretization of (3.21) as motivated in

3.3. APPROXIMATED BLADE MODEL

15

chapter 3.2 yield to Z L 0=

0

i 0 ˜(UΦγ + VΦκ ) qi −UΦγi − VΦκi 0 − k i i h i T ˜ κ (UΦγ + VΦκ ) + Φ ˜ Ω (GΦV + KΦΩ ) qi qj ΦVk −Φ i j i j j j h h i i 0 VT VT ˜ u u A + ΦV T Φ ˜ κi Au qi uA Φk Au + kA u k u − Φk f

ΦVk + +

T

h

i

V GΦVi + KΦΩ i q˙i + Φk

h

T

i 0 ˜(VT Φγ + WΦκ ) − e˜1 (UΦγ + VΦκ ) qi −VT Φγi − WΦκi 0 − k i i i i h i γ γ γ ΩT ˜ Ω T V Ω V V Ω κ T κ ˜ i (GΦj + KΦj ) − Φ ˜ i (V Φ + WΦj ) − Φ ˜ (UΦ + VΦκj ) qi qj Φk Φi (K Φj + IΦj ) + Φ j i j h h i i T ΩT ˜ u + e˜1 Au uA + ΦΩ T Φ ˜ κi Bu + Φ ˜ γ Au q i u A B0u + kB ΦΩ u k u − Φk m k i

ΦΩ k + +

T

h

i

Ω KT ΦVi + IΦΩ i q˙i + Φk

T

h

i h i 0 T ˜V − e˜1 ΦΩ qi Φγi q˙i + (UΦγk + VΦκk ) −ΦVi − k i h i T ˜ κi ΦVj − Φ ˜ γ ΦΩ (UΦγk + VΦκk ) −Φ i j qi qj

(UΦγk + VΦκk )

T

+

h

i h i T 0 ˜ ΦΩ q i Φκi q˙i + (VT Φγk + WΦκk ) −ΦΩ − k i i h i T γ ˜ κi ΦΩ (VT Φk + WΦκk ) −Φ j qi qj T

(VT Φγk + WΦκk ) + dx

h

i T ΦVi (0) qi + UΦγk (0) + VΦκk (0) V 0 T h Ω i T − VT Φγk (0) + WΦκk (0) Φi (0) qi + VT Φγk (0) + WΦκk (0) Ω0 h i T T VT L + ΦVk (L) UΦγi (L) + VΦκi (L) qi − ΦVk (L)Au (L)uA u − Φk (L)F h i T T γ κ ΩT A ΩT L + ΦΩ k (L) V Φi (L) + WΦi (L) qi − Φk (L)Bu (L)uu − Φk (L)M

−

T h

UΦγk (0) + VΦκk (0)

(3.24) Although the Galerkin approach can be formulated in one equation (3.24), the first two equations and the set of the last two equations are solved separately. This is because only ΦVk or ΦΩ k or (UΦγk + VΦκk ) can be nonzero for a given k, see (3.20). The third and fourth equation of (3.21) are weighted in a mixed way as both weighting functions use Φγk and Φκk . However, this does not worsen the solution significantly, as later tests show. Equation (3.24) can be written in Einstein notation as the desired ordinary differential equation. Note, that we have 12 equations for each k A Aki q˙i + Bki qi + Ckij qi qj + Dk + Eku uA u + Fkiu qi uu + fk + mk = 0

(3.25)

where no BC BC Bki = Bki + Bki

Dk = DkBC Eku =

no BC Eku

(3.26) +

BC Eku

()BC denotes, that the values of these tensors are originating from boundary conditions, where ()no BC denotes that the values are not coming from boundary conditions. Comparing (3.25) with

16


(3.24) yields to the following tensor values. Z L h h i i T ΩT T V Ω Aki = ΦVk GΦVi + KΦΩ K + Φ Φ + I Φ i k i i 0

+(UΦγk + VΦκk )

T

no BC Bki =

Z L 0

ΦVk

T

h

+ΦΩ k

T

h

ΦVk

T

h

h

h i i T Φγi + (VT Φγk + WΦκk ) Φκi dx

i 0 ˜(UΦγ + VΦκ ) −UΦγi − VΦκi 0 − k i i

i 0 ˜(VT Φγ + WΦκ ) − e˜1 (UΦγ + VΦκ ) −VT Φγi − WΦκi 0 − k i i i i h i 0 T ˜V − e˜1 ΦΩ +(UΦγk + VΦκk ) −ΦVi − k i h i T 0 ˜ Ω dx +(VT Φγk + WΦκk ) −ΦΩ i − kΦi Z L Ckij =

0

i V Ω ˜ κi (UΦγ + VΦκj ) + Φ ˜Ω −Φ i (GΦj + KΦj ) j

i γ T V Ω V Ω κ κ ˜Ω ˜V ˜κ T γ ˜γ Φ i (K Φj + IΦj ) + Φi (GΦj + KΦj ) − Φi (V Φj + WΦj ) − Φi (UΦj + VΦj ) h i T ˜ κi ΦVj − Φ ˜ γ ΦΩ +(UΦγk + VΦκk ) −Φ i j h i T γ κ T κ Ω ˜ +(V Φk + WΦk ) −Φi Φj dx

+ΦΩ k

T

h

(3.27) The actuation is determined by Z L h i h i no BC VT 0 ΩT 0 ˜ ˜ Eku = Φk Au + kAu + Φk Bu + kBu + e˜1 Au dx 0

(3.28) no BC Fkiu =

Z L 0

h i h i T T γ κ ˜ κi Au + ΦΩ ˜ ˜ ΦVk Φ Φ B + Φ A dx k i u i u

fk and mk are Z L fk =

0

T − ΦVk f dx (3.29)

Z L mk =

0

ΩT − Φk m dx

The tensors occurring from boundary conditions are T T DkBC = UΦγk (0) + VΦκk (0) V 0 + VT Φγk (0) + WΦκk (0) Ω0 L −ΦVk (L)F L − ΦΩ k (L)M T

BC Bki =

T

h i h i T T T γ κ ΦVk (L) UΦγi (L) + VΦκi (L) + ΦΩ ( L ) V Φ ( L ) + W Φ ( L ) k i i i T h V T h Ω i γ κ T γ κ − UΦk (0) + VΦk (0) Φi (0) − V Φk (0) + WΦk (0) Φi (0) T

(3.30)

T

BC Eku = −ΦVk (L)Au (L) − ΦΩ k (L)Bu (L)

In order to obtain tensors with good numerical condition, proper comparison function have to be chosen. The choice of the comparison functions is described in the following section.

3.4. COMPARISON FUNCTIONS

3.4

17

Comparison Functions

The comparison functions can be chosen arbitrary if they do not restrict the solution, e.g. a comparison function that allows only a zero value at the tip of a cantilevered beam is not appropriate. To be able to fulfill the boundary conditions in the mixed Galerkin approach, the comparison functions have to have a constant part and/or the comparison functions have to be expanded by the steady state. This expansion is shown combined with an order reduction in chapter 5.1. As the steady state is not known first, the comparison functions must have a constant part for a first solution. Besides fulfilling the boundary conditions, it is an advantage to chose orthogonal comparison functions in order to receive a well-conditioned solution. Another plus is the use of functions that are easy to integrate. The comparison functions used in this work are ΦVl (¯ x) = ΨV Pl (¯ x), Ω Ω Φl (¯ x) = Ψ Pl (¯ x), Φγl (¯ x) = Ψγ Pl (¯ x), Φκl (¯ x) = Ψκ Pl (¯ x),

ΨV ΨΩ Ψγ Ψκ

= I 0 0 0 = 0 I 0 0 = 0 0 I 0 = 0 0 0 I

(3.31)

I is the identity matrix and x ¯ is the normalized position of the reference line x. x ¯=

x

L

(3.32)

Pl are shifted Legendre functions [17, pp. 332–357]. By using the normalized reference line position x ¯, one ensures the orthogonality of the shifted Legendre functions Pl (¯ x) which are orthogonal over the interval x ¯ = (0, 1). Z 1 1 Pl (¯ x)Pk (¯ x)d¯ x= δlk (3.33) 2l +1 0 and δlk is the Kronecker delta. Another advantage of Legendre functions is that they are easy to integrate. Legendre functions can be defined in many ways. Here, they are defined recursively as this is also the way the Legendre functions are implemented in the simulation. The first two Legendre functions are defined as P1 = 1,

P2 = 2¯ x−1

(3.34)

The recursive law for the further functions is Pl =

(2(l − 2) + 1)(2¯ x − 1)Pl−1 − (l − 2)Pl−2 l−1

(3.35)

The first five Legendre functions can be seen in figure 3.2. If similar comparison functions would have been chosen, the tensors in (3.25) are badly conditioned as the integral values of the tensor elements become very similar, too. Orthogonal functions are due to its orthogonal character very different from each other and this is why they provide better values for the tensor elements in (3.25). In the next section, the comparison functions introduced in this section, together with the assumption of a constant cross-section are simplifying the tensors of the active beam (3.27) - (3.30).

3.5

Special Case: Constant Cross Section

The helicopter blade used in this work has a constant cross-section. For this special case, the tensors of the discretized beam model (3.27)-(3.30) can be simplified as the constant cross-section parameters can be separated from the integrals. By using the comparison functions introduced in (3.31), the γ κ matrices ΨVl , ΨΩ l , Ψl and Ψl can be seperated from the integrals, too. The only remaining terms

18


1

0.5

P1 P2 P3

0

P4 P5

−0.5

−1 0

0.5 x

1

Figure 3.2: First five Legendre functions

in the integrals are the following

A

R

Sku = d

A

R

Sku

Z L Z 1 Pk (x)Au (x) dx = L Pk (¯ x)Au (¯ x) d¯ x 0

0

Z L Z 1 0 0 = Pk (x)Au (x) dx = Pk (¯ x)Au (¯ x) d¯ x 0

0

Z L Z 1 B Sku = Pk (x)Bu (x) dx = L Pk (¯ x)Bu (¯ x) d¯ x R

0

0

Z L Z 1 B d Sku = Pk (x)B0u (x) dx = Pk (¯ x)B0u (¯ x) d¯ x R

0

0

Z L Z 1 Sk = Pk (x) dx = L Pk (¯ x) d¯ x R

0

R

Dki d

R

Dki

0

(3.36)

Z L Z 1 = Pk (x)Pi (x) dx = L Pk (¯ x)Pi (¯ x) d¯ x 0

0

Z L Z 1 = Pk (x)Pi0 (x) dx = Pk (¯ x)Pi0 (¯ x) d¯ x 0

0

Z L Z 1 A Dkiu = Pk (x)Pi (x)Au (x) dx = L Pk (¯ x)Pi (¯ x)Au (¯ x) d¯ x R

R

B

Dkiu R

Tkij

0

0

0

0

0

0

Z L Z 1 = Pk (x)Pi (x)Bu (x) dx = L Pk (¯ x)Pi (¯ x)Bu (¯ x) d¯ x Z L Z 1 = Pk (x)Pi (x)Pj (x) dx = L Pk (¯ x)Pi (¯ x)Pj (¯ x) d¯ x

R

R

R

where Sk denotes the integral with a single Legendre function Pk (¯ x). Dk and Tk denote the integrals for double and triple Legendre functions. d () denotes that the integral contains a differentiated Legendre function. If actuation force and moment constants A and B are included in the integral, it R R R1 B A and () . Because of the differentiation, the integral 0 . . . d¯ x does not have to is denoted by () x). Additionally, the following symbols are introduced be premultiplied by L as Pi0 (x) = L1 Pi0 (¯ Sk0 = Pk (0),

SkL = Pk (L),

0 = P (0)P (0), Dki k i

L = P (L)P (L) Dki k i

(3.37)

3.5. SPECIAL CASE: CONSTANT CROSS SECTION

19

With (3.36) and (3.37), the tensors in (3.27)-(3.30) can be simplified the following way. h i R h i R T T Aki = ΨV GΨV + KΨΩ Dki + ΨΩ KT ΨV + IΨΩ Dki h i R h i R T T +(UΨγ + VΨκ ) Ψγ Dki + (VT Ψγ + WΨκ ) Ψκ Dki no BC Bki =

Ckij =

i R h i R T ˜(UΨγ + VΨκ ) D −UΨγ 0 − VΨκ 0 d Dki − ΨV k ki h i R h i R T γ0 κ0 d κ ΩT ΩT ˜ T γ −V Ψ − WΨ k(V Ψ + WΨ ) − e˜1 (UΨγ + VΨκ ) Dki +Ψ Dki − Ψ i R i R h h 0 T T ˜ +(UΨγ + VΨκ ) −ΨV d Dki − (UΨγ + VΨκ ) k V − e˜1 ΨΩ Dki i R i R h h 0 T T ˜ΨΩ D +(VT Ψγ + WΨκ ) −ΨΩ d Dki − (VT Ψγ + WΨκ ) k ki ΨV

T

h

ΨV

T

h

i R ˜ κ (UΨγ + VΨκ ) + Ψ ˜ Ω (GΨV + KΨΩ ) T −Ψ kij

i R ˜ Ω (KT ΨV + IΨΩ ) + Ψ ˜ V (GΨV + KΨΩ ) − Ψ ˜ κ (VT Ψγ + WΨκ ) − Ψ ˜ γ (UΨγ + VΨκ ) T Ψ kij h i R T ˜ κ ΨV − Ψ ˜ γ ΨΩ T +(UΨγ + VΨκ ) −Ψ kij R h i T ˜ κ ΨΩ T +(VT Ψγ + WΨκ ) −Ψ kij

+ΨΩ

T

h

(3.38) The actuation for constant cross-section is determined by R R h i R h i R h i R T T A no BC ˜ S A + ΨΩ T d S B + ΨΩ T k ˜ S B + ΨΩ T e˜1 S A Eku = ΨV d Sku + ΨV k ku ku ku ku (3.39) no BC Fkiu =

VT

Ψ

h

˜ κD A + Ψ Ψ kiu R

ΩT

h

˜κ D B + Ψ Ψ kiu

h

T

R

ΩT

R

R

i

ΩT

˜γ D A Ψ kiu i

R

fk and mk are fk = − ΨV f Sk

(3.40) mk = − Ψ

mSk

The tensors occurring from boundary conditions are DkBC =

UΨγ + VΨκ

T 0 0 T V S + VT Ψγ + WΨκ Ω0 S 0

−ΨV F L S L − ΨΩ M L S L T

BC Bki =

ΨV

T

T

h

UΨγ + VΨκ DL + ΨΩ i

T

h

VT Ψγ + WΨκ DL i

T h V i 0 T h Ω i 0 − UΨγ + VΨκ Ψ D − VT Ψγ + WΨκ Ψ D BC Ekm = −ΨV Au (L)S L − ΨΩ Bu (L)S L T

T

(3.41)

20


Chapter 4

Modal Space Linear structural-dynamic problems are mostly analyzed in the modal space described by free vibration modes (normal modes) and its natural frequencies. The free vibration modes are determined by the eigenvectors and the natural frequencies by the eigenvalues of the system [16, chapter 4.6] as shown in chapter 4.1.3. There are two important properties holding for the linear modal solution of conservative systems. • One free vibration mode is described by two state space variables • There exists a state in which the whole structure is vibrating with the same frequency The first property is best regarded for a one degree of freedom system which has one free vibration mode and one natural frequency m¨ x + cx = 0

(4.1)

It can be described by two state space variables x and x. ˙ A n degree of freedom system with n equations of the type of (4.1) consequently has 2n states. The second property holds, as it is imperative for free vibration modes, that all its points are vibrating with the same frequency. As the undisturbed vibration is a superposition of vibrating natural modes, defined by the initial condition, it can be chosen in a way that the structure is vibrating in a single natural mode with its natural frequency. For nonlinear systems, the two presented properties are used to find free vibration modes and natural frequencies. This means, that also for the nonlinear system, it is assumed that two state space variables are necessary for one free vibration mode and additionally, it is assumed that there exists a state in which all state space variables oscillate with the same frequency. From this follows, that the modal vibration of the whole system can be described by two state space variables at a single point. The state space vector at this point is denoted by x∗ . The functional dependence for the complete structure is xn = fn (x∗n )

(4.2)

The concept of nonlinear free vibration modes is presented in chapter 4.2.1. However, it could not be achieved to adopt this concept to the beam model in the short time of this work.

4.1

Linear Modal Space

Before the idea of nonlinear natural modes is presented, the well known linear modal space is established for the discretized beam model (3.25). First, this nonlinear model has to be linearized around the steady state. After the steady state calculation and the linearization are performed, the eigenvalues and eigenvectors are calculated. In the end, the free vibration modes are calculated using the comparison functions in combination with the eigenvectors, whereas the eigenvalues will determine the natural frequencies. 21

22

4.1.1

CHAPTER 4. MODAL SPACE

Steady State Calculation

The steady state of the discretized blade model (3.25) is reached if the time derivation of its state space vector is zero. ! q˙i = 0 (4.3) With (4.3), the steady state equation of (3.25) is 0

0

0 A 0 0 fk0 (qi0 ) = Bki qi0 + Ckij qi0 qj0 + Dk + Eku uA u + Fkiu qi uu + fk + mk = 0

(4.4)

Note, that the beam model function as well as the external force are both denoted by f . Because of the nonlinearities in (4.4), there does not exist an analytical solution. The steady state has to be calculated finding the root of (4.4) iteratively. One of the most capable algorithms for this problem is the Newton-Raphson algorithm. This method converges to the root by approximating its location, using the first term of the Taylor approximation of the steady state equation (linearization). The Taylor series of (4.4) is dfk0 (qi0 ) 0 d2 fk0 (qi0 ) 0 2 ∆qi + ∆qi + . . . 2 dqi0 dqi0

fk0 (qi0 + ∆qi0 ) = fk0 (qi0 ) +

(4.5)

The initial and following iterations for the root approximation are demonstrated in the following. 1st Iteration The Newton-Raphson method assumes, that terms of higher degree than than one in (4.5) can be neglected for a first approximation of the root. The iteration index is denoted by a left raised number i (). df 0 (1 q 0 ) fk0 (1 qi0 + ∆1 qi0 ) ≈ fk0 (1 qi0 ) + k 0 i ∆1 qi0 = 0 (4.6) dqi The Jacobian of (4.6) is 1

0 Jki =

dfk0 (1 qi0 ) 0 = Bki + (Ckij + Ckji )1 qi0 + Fkiu uA u dqi0

∆1 qi0 is calculated from (4.6) as 0 ∆1 qi0 = −1 Jki

The approximated root after the first iteration 2 0 qi

2 0 qi

−1 0 1 0 fk ( qi )

(4.7)

(4.8)

is

= 1 qi0 + ∆1 qi0

(4.9)

nth Iteration The further iterations are analog to the first one. ∆n qi0 calculates as 0 ∆n qi0 = −n Jki

Then,

n+1 0 qi

−1 0 n 0 fk ( qi )

(4.10)

is n+1 0 qi

= n qi0 + ∆n qi0 (4.11) One can choose the iteration to end if H2 fk0 (m qi0 ) < (H2 norm). It can be shown, that the Newton-Raphson algorithm converges quadratically. Unfortunately, this procedure can be unstable near a horizontal asymptote or a local extremum. Unique Solution The steady state solution is unique for statically determinate beams only, e.g. cantilevered beams. For statically overdeterminate and indeterminate beams, like free-free beams or fixed-fixed beams, there exist multiple solutions. Take the steady state solution of a free-free beam, where the generalized forces are zero. It could have any constant linear velocity as a steady state solution so that the solution is not unique anymore. In the case of a fixed-fixed beam, the generalized velocities of the steady state are zero. Here, one can add any generalized force distribution that is still resulting in a steady state solution. The steady state solution is used to linearize the discretized beam equation in the next subsection.

4.1. LINEAR MODAL SPACE

4.1.2

23

Linearization

Similarly to the steady state calculation, the linearization of the discretized beam model (3.25) A A fk (q˙i , qi , uA m , fk , mk ) = Aki q˙i + Bki qi + Ckij qi qj + Dk + Eku uu + Fkiu qi uu + fk + mk = 0

(4.12)

takes advantages of the Taylor series. In order to linearize (4.12), terms of order higher than one are neglected. A fk (q˙i + ∆q˙i , qi + ∆qi , uA u + ∆uu , fk + ∆fk , mk + ∆mk ) = ∂fk ∂fk ∂fk ∂fk ∂fk ∆uA ∆q˙i + ∆qi + ∆fk + ∆mk fk + u + ∂ q˙i ∂qi ∂uA ∂f ∂m k k u ∂ 2 fk ∂ 2 fk ∆qi2 + . . . + 2 ∆q˙i2 + ∂ q˙i ∂qi2 {z } |

(4.13)

neglected

0

A 0 The linearization of (4.12) is done at the steady state solution (q˙i = 0, qi = qi0 , uA u = uu , fk = fk , 0 mk = mk ), denoted by ()∗ . The terms in (4.13) are in the case of the discretized beam model (4.12)

fk ∂fk ∂uA u

= 0, ∗

∂fk ∂ q˙i

= Eku + Fkiu qi0 , ∗

∂fk ∂fk

∗

∂fk ∂qi

∗

∗

∂fk ∂mk

∗

= Aki , = I,

= Bki + (Ckij + Ckji )qi0 + Fkiu uA u

0

=I (4.14)

The linearized discetized beam model (4.12) is finally fk (∆q˙i , ∆qi , ∆uA u , ∆fk , ∆mk ) = h i 0 Aki ∆q˙i + Bki + (Ckij + Ckji )qi0 + Fkim uA ∆qi + Ekm + Fkim qi0 ∆uA m u + ∆fk + ∆mk

(4.15)

This model is used to calculate its eigenvectors and eigenvalues in the following section.

4.1.3

Natural Modes and Frequencies

In this subsection, the eigenvalues and eigenvectors of (4.15) are used to obtain the natural modes and frequencies of the beam. As natural modes and frequencies are determined for non-disturbed, λt passive beams, ∆uA of (4.15), u , ∆fk and ∆mk are set to zero. Inserting the solution ∆qi (t) = ni e known from the theory of linear ordinary differential equations into (4.15) yields to h i 0 Aki λ + Bki + (Ckij + Ckji )qi0 + Fkiu uA ni eλt = 0 u

(4.16)

n h io 0 det Aki λ + Bki + (Ckij + Ckji )qi0 + Fkiu uA =0 u

(4.17)

(4.16) has a solution if

There will be 12 · lmax = 12 · imax eigenvalues λ for which (4.17) is satisfied. The eigenvectors ni are determined inserting the λ into (4.16). As a linear model is used for the eigenvalue and eigenvector calculation, the solution of the linear initial condition response is a superposition of all partial solutions. The vector qi (t) yields after superposition to qi (t) = cl nil eλl t

(4.18)

and cl is a factor that has to be determined by the boundary conditions. The values of the eigenvectors ni and eigenvalues λ are conjugate complex or real. In the case of a real eigenvalue and eigenvector, the partial solution is not a vibration. Vibrations are provided by pairs of conjugate

24


complex eigenvalues and eigenvectors which is shown in the following. ˆ

R

R

I I I λl t I I I nil eλl t + n ˆ il eλl t = eλl t [(nR [(nR il + inil )(cos λl t + i sin λl t)] + e il − inil )(cos λl t − i sin λl t)] R

I I I R I I I = eλl t [nR il cos λ t + inil cos λ t + inil sin λl t − nil sin λl t I I I R I I I + nR il cos λ t − inil cos λl t − inil sin λl t − nil sin λl t] R

I I I = eλl t [2nR il cos λ t − 2nil sin λl t]

(4.19) ˆ denotes a conjugate ()R denotes the real part and ()I the imaginary part of complex values. () complex value. The conjugate complex eigenvectors and values can be combine to the undamped vibrating solutions of (4.15). R ˆ qivib (t) = nil eλl t + n ˆ il eλl t

(4.19),no damping

=

c∗l nil R cos(λIl t)

(4.20)

The solution for the linearized beam model can be written in terms of free vibration modes and natural frequencies (no damping) as V (x, t) = c∗l Vlf vm (x)cos(λIl t), Ω(x, t) = c∗l Ωfl vm (x)cos(λIl t), γ(x, t) = c∗l γlf vm (x)cos(λIl t), κ(x, t) = c∗l κfl vm (x)cos(λIl t),

Vlf vm Ωfl vm γlf vm κfl vm

= ΦVi (x)nil R R = ΦΩ i (x)nil γ R = Φi (x)nil = Φκi (x)nil R

(4.21)

f vm where Vklf vm , Ωfklvm , γkl and κfklvm are free vibration modes (fvb) and λIl are the natural frequencies. The free vibration modes result from comparing (4.21) with (4.20) and (3.20). They are used for model order reduction later in this work. Furthermore, the natural frequencies and modes are a useful tool to analyze the dynamics of the linearized beam model. In the next section the idea of nonlinear natural frequencies and modes is presented.

4.2

Nonlinear Modal Space

In the introduction of chapter 4, the idea of nonlinear normal modes has been pointed out which is that for a nonlinear structural system there exists at least one motion for which all coordinates have the same frequency and damping rate. The shape of this motion is a nonlinear free vibration mode and the frequency its natural frequency. The theory of nonlinear normal modes was presented by Shaw, S. W. and Pierre, C.18 Further papers of the authors deal with the application of this theory to continuous19 and discrete structural systems.20

4.2.1

Basic Idea

Shaw presents his constructive methods for structural problems of the following form x˙ i = yi y˙ i = fi (xj , yj )

(4.22)

where i = 1 . . . n. xi and yi represent displacement and velocity coordinates. As there exist motions in which all coordinates have the same frequency and decay rate, the motion of the whole structure can be described by a single pair of displacement and velocity variables. This pair can be chosen arbitrary, e.g. u = x1 and v = y1 . The motion of the whole structure is then described by this pair xi = Xi (u, v),

yi = Yi (u, v)

(4.23)

The time derivations of (4.23) are x˙ i =

∂Xi ∂Xi u˙ + v, ˙ ∂u ∂v

y˙ i =

∂Yi ∂Yi u˙ + v˙ ∂u ∂v

(4.24)

4.2. NONLINEAR MODAL SPACE

25

Next, (4.24) is inserted into (4.22), note that u˙ = v and v˙ = f1 . ∂Xi (u, v) ∂Xi (u, v) v+ f1 (u, X2 (u, v), . . . , Xn (u, v), v, Y2 (u, v), . . . , Yn (u, v)) = Yi (u, v) ∂u ∂v ∂Yi ∂Yi v+ f1 (u, X2 (u, v), . . . , Xn (u, v), v, Y2 (u, v), . . . , Yn (u, v)) = fi (u, X2 (u, v), . . .) ∂u ∂v (4.25) Finally, the functions Xi (u, v) and Yi (u, v) have to be found. To receive these functions, Shaw uses a series expansion Xi (u, v) = a0i + a1i u + a2i v + a3i u2 + a4i uv + a5i v 2 + . . . Yi (u, v) = b0i + b1i u + b2i v + b3i u2 + b4i uv + b5i v 2 + . . .

(4.26)

(4.25) together with (4.26) form a solvable set of equations for the coefficients aji and bji . These coefficients can be used to generate the normal mode dynamics by substituting them into the above equations. In the following subsection, a short example of a discrete nonlinear conservative system is given.

4.2.2

Nonlinear Conservative Example

For the following system with two cars, two linear springs and one nonlinear spring, the concept of nonlinear normal modes will be shown. The system is visualized in figure 4.1. Without derivation, x1

x2

Figure 4.1: Nonlinear conservative two mass system

the equations of motion are given by x˙ 1 = y1 , x˙ 2 = y2 ,

y˙ 1 = f1 = −x1 (1 + k) − gx31 + kx2 y˙ 2 = f2 = kx1 − x2 (1 + k)

(4.27)

where k is the linear spring stiffness of all springs and g is an additional parameter for the nonlinear spring. The masses of the system are set to one. The variables x1 , x2 , y1 and y2 are chosen like in (4.26) up to cubic terms. x1 = u,

x2 = a1 u + a2 v + a3 u2 + a4 uv + a5 v 2 + a6 u3 + a7 u2 v + a8 uv 2 + a9 v 3

y1 = v,

y2 = b1 u + b2 v + b3 u2 + b4 uv + b5 v 2 + b6 u3 + b7 u2 v + b8 uv 2 + b9 v 3

(4.28)

The derivations of (4.28) are x˙ 1 = y1 , y˙ 1 = f1 ,

x˙ 2 = a1 u˙ + a2 v˙ + 2a3 uu˙ + a4 uv˙ + a4 uv ˙ + 2a5 v v˙ + . . . y˙ 2 = b1 u˙ + b2 v˙ + 2b3 uu˙ + b4 uv˙ + b4 uv ˙ + 2b5 v v˙ + . . .

(4.29)

Inserting (4.28) and (4.29) into (4.27) results in equations consisting of u, v, ai and bi . The coefficients ai and bi for u, v, u2 , . . . are then gathered together and form a set of 18 equations for the coefficients ai and bi . After the solution of these equations, the first normal mode is x1 = u1 , y1 = v1 ,

g[(k − 3)u21 − 3v12 ] u1 2k(k − 4) 3g[(k − 1)u21 − v12 ] y2 = v1 + v1 2k(k − 4) x2 = u1 +

(4.30)

26


and the second one yields x1 = u1 , y1 = v1 ,

g[(3 + 7k)u22 + 3v22 ] u2 2k(4 + 9k) 3g[(1 + 3k)u22 + v22 ] y2 = −v2 + v2 2k(4 + 9k) x2 = −u2 +

After the substitution of the normal modes, the oscillation equations for u1 and u2 are 3 3−k u21 + u˙ 21 = 0 u ¨1 + u1 + gu1 1 + 2(k − 4) 2(k − 4) 3 + 7k 3k u ¨2 + (1 + 2k)u2 + gu2 1 − u22 + u˙ 22 = 0 2(4 + 9k) 2(4 + 9k)

4.2.3

(4.31)

(4.32)

Future Work

As the theory of nonlinear normal modes has been developed for continuous and discrete systems, one could think of applying this technique directly to the set of intrinsic (2.1) and constitutive equations (2.11)(continuous) or the beam model in (3.25)(discrete). However, the beam model is of much higher complexity than the example shown in chapter 4.2.2, whereas this simple example already needs a lot of effort. Furthermore, the nonlinear normal mode theory was shown for structural systems that are described by displacement and velocity variables. The adaption to the beam model that is used here, where we have generalized velocity and generalized strain variables is too complicated and so too time consuming for this work. In the following, only the linear normal modes are used for system analysis and model order reduction, which is presented in the next chapter.

Chapter 5

Order Reduction The discretization of the beam model consisting of the intrinsic (2.1) and constitutive equations (2.11) achieved a simplification as the partial differential equations could be replaced by ordinary differential equations. Along with this simplification comes a model (3.25) of higher order, dependant on the number of comparison functions used for the discretization. As both, the accuracy and the computational costs are increasing with the number of comparison functions, one has to find a compromise for the number of used comparison functions. Another approach to improve the accuracy of the discretized model is to find comparison functions that capture the dynamics in a better way than previously used ones. The model order reduction in this chapter follows this approach by taking improved comparison functions to reduce their number without a loss of accuracy. The resulting reduced model has many advantages that are based on lower computational costs, e.g. faster calculation or the use of analysis tools that are mainly used for low order models. After the presentation of the order reduction framework used in this work, two techniques for the generation of proper comparison functions are described. One of them is based on linear free vibrations modes and the other one on the perturbation of the model.

5.1

Order Reduction Framework

A commonly used technique for the order reduction [21, pp. 1–2] of linear systems, e.g. ¨ + C X(t) ˙ M X(t) + KX(t) = F (t)

(5.1)

is performed applying a coordinate transformation X(t) = T Z(t)

(5.2)

whereas X ∈ Rn , Z ∈ Rm , T ∈ Rnxm and n > m. Additionally to the coordinate transformation (5.2), equation (5.1) is premultiplied with the transpose of the transformation matrix T to receive quadratic matrices. The resulting reduced structural model of (5.1) is ¨ + T T CT Z(t) ˙ T T M T Z(t) + T T KT Z(t) = T T F (t)

(5.3)

The model order reduction framework presented in the following is based on this methodology. The coordinate transformation is performed by the change of comparison functions and the premultiplication by the change of weighting functions. The order reduction process presented in this chapter is equivalent to the generation of a new discretized beam model with improved comparison and weighting functions. In this work, the improved comparison and weighting functions are restricted to be a linear combination Tlr of previously used ones. Furthermore, the steady state is included in the improved 27

28

CHAPTER 5. ORDER REDUCTION

comparison functions so that the steady state solution ql0 stays the same for the reduced model. Its inclusion has also the advantage that the transformed steady state q0r is always zero. V (x, t) = V 0 (x) + ΦVr (x)qr (t) = ΦVl (x)(Tlr qr (t) + ql0 ) Ω 0 Ω(x, t) = Ω0 (x) + ΦΩ r (x)qr (t) = Φl (x)(Tlr qr (t) + ql )

γ(x, t) = γ 0 (x) + Φγr (x)qr (t) = Φγl (x)(Tlr qr (t) + ql0 )

(5.4)

κ(x, t) = κ0 (x) + Φκr (x)qr (t) = Φκl (x)(Tlr qr (t) + ql0 ) γ κ where ΦVr (x), ΦΩ r (x), Φr (x) and Φr (x) are the new comparison and weighting functions.

ΦVl (x) = ΦVr (x)Tlr Ω ΦΩ l (x) = Φr (x)Tlr γ Φl (x) = Φγr (x)Tlr Φκl (x) = Φκr (x)Tlr

(5.5)

The comparison functions of the reduced discretized model are exchanged by substituting ql (t) in (3.25) with (5.6) resulting from the comparison of (5.4) with (3.20). ql (t) = Tlr qr (t) + ql0

(5.6)

The change of the weighting functions can be performed by premultiplying the discretized beam model (3.25) with Tkt . This is possible because the transformation matrix Tlr of the changed weighting functions can be separated from the integrals in all tensors of (3.25) as it is shown exemplary for the Aki tensor. Z L h i h i h i h i T T γ κ T T γ κ T Aki = ΦVk . . . + ΦΩ . . . + ( U Φ + V Φ ) . . . + ( V Φ + W Φ ) . . . dx k k k k k 0

A∗ki

(5.5) ↓ Z L h i h i h i h i T T γ κ T T γ κ T = Tkt ΦVk . . . + Tkt ΦΩ . . . + T ( U Φ + V Φ ) . . . + T ( V Φ + W Φ ) . . . dx kt kt k k k k k 0

=Tkt Aki (5.7) 0

0 0 A Additionally, uA u , fk and mk are split up into the steady state solution uu , fk , mk and its dynamic part. A A0 uA fk = fk + fk0 , mk = mk + m0k (5.8) u = uu + uu ,

After the separation of some variables in (5.8), the change of comparison (5.6) and weighting functions (5.7), the resulting reduced model becomes 0 = Aki Tkt Tir q˙ r (t) + Bki Tkt Tir qr (t) + qi0 + Ckij Tkt Tir qr (t) + qi0 Tjs qs (t) + qj0 A0 A0 (5.9) +Dk Tkt + Ekm Tkt uA + Fkim Tkt uA Tir qr (t) + qi0 m + um m + um 0 0 +Tkt fk + fk + Tkt mk + mk The terms in (5.9) are sorted and recombined h i 0 0 = Aki Tkt Tir q˙ r (t) + Bki + (Ckij + Ckji )qi0 + Fkim uA Tkt Tir qr (t) + Ckij Tkt Tir Tjs qr (t)qs (t) m A + Ekm + Fkim qi0 Tkt uA m + Fkim Tkt Tir um qr (t) + Tkt fk + Tkt mk 0

0

A 0 0 0 + Bki Tkt qi0 + Ckij Tkt qi0 qj0 + Dk Tkt + Ekm Tkt uA m + Fkim Tkt um qi + Tkt fk + Tkt mk {z } | =0,

see (4.4)

(5.10) The reduced beam model is finally A Atr q˙ r + Btr qr + Ctrs qr qs + Etm uA m + Ftrm um qr + Tkt fk + Tkt mk = 0

(5.11)

5.2. NORMAL MODES EXPANSION

29

whereas Atr = Aki Tkt Tir i h 0 Tkt Tir Btr = Bki + (Ckij + Ckji )qi0 + Fkim uA m (5.12)

Ctrs = Ckij Tkt Tir Tjs Etm = Ekm + Fkim qi0 Tkt Ftrm = Fkim Tkt Tir

This reduction technique leads to a reduced model as if a new discretized model with different weighting and comparison functions would have been built. In the next two sections, the transformation matrix T is specified using two different concepts. One generates the transformation matrix T from linear normal modes and the other one from perturbation modes.

5.2

Normal Modes Expansion

One choice of comparison functions that capture the linear dynamics as well as the primary nonlinear couplings are the free vibration modes of the system linearized at the steady state. Free vibration modes provide a very good linearized solution as it includes the exact free vibration modes that have been used for this order reduction. The Transformation Matrix T results from the comparison between the calculation of free vibration modes in (4.21) and the comparison functions in (5.4). ΦVr (x) = ΦVl (x)Tlr Ω ΦΩ r (x) = Φl (x)Tlr Φγr (x) = Φγl (x)Tlr Φκr (x) = Φκl (x)Tlr

←→ ΦVl (x)nlr R R ←→ ΦΩ l (x)nlr ←→ Φγl (x)nlr R ←→ Φκl (x)nlr R

= Vrf vm (x) = Ωfr vm (x) = γrf vm (x) = κfr vm (x)

(5.13)

Additionally, one has to keep in mind, that a pair of conjugate complex eigenvalues and eigenvectors determines one free vibration mode as it was shown in (4.19). Instead of using the real parts of nlr only, the imaginary part of nlr also has to be used to obtain one free vibration mode.22 The choice of the real and imaginary parts of the conjugate complex eigenvectors has the advantage that the elements in T have real values only. The transformation matrix T is finally h i nIl(2·r) Tlr = nR (5.14) l(2·r) ()R denotes the real part and ()I the imaginary part of the eigenvectors vlr and the conjugate complex eigenvectors are arranged as pairs. By this choice of the transformation matrix T, each qr has only two unknowns instead of twelve for ql , see (5.4). A good choice of eigenvector pairs are usually the ones that correspond to the lowest frequencies. As the solution of the linearized model is reused in this approach, the linearized solution calculated with the transformation matrix T in (5.14) consists of the free vibration modes and its natural frequencies used for the order reduction. For example, the eigenvectors nli and eigenvalues λi for a reduced model of forth dimension is         1 + 0i 1 + 0i 0 + 0i 0 + 0i                     0 + 1i 0 − 1i 0 + 0i 0 + 0i n1 = n2 = n3 = n4 = 0 + 0i 0 + 0i 1 + 0i 1 + 0i             (5.15)         0 + 0i 0 + 0i 0 + 1i 0 − 1i λ1 = iω1

λ2 = −iω1

λ3 = iω2

λ4 = −iω2

The exact solution of the most important natural frequencies of the linearized solution is the biggest advantage of this kind of order reduction. Later, it is shown in the case of the helicopter blades, that free vibration modes are also capable of describing the nonlinear behavior of the blades satisfyingly. In the next section, perturbation modes are introduced to complement the free vibration modes in the transformation matrix T.

30

5.3


Pertubation Modes Expansion

Perturbation modes 23, 24 are usually calculated to predict the nonlinear static response of structures. Here, perturbation modes are added as column vectors to the existing normal modes in the transformation matrix T in (5.14) to improve the nonlinear behavior of the reduced model. A reduced model only consisting of perturbation modes will result in a worse performance in terms of the linear behavior as the normal modes are already the perfect solution for the linearized model as it was shown in the previous section. The perturbation modes have their origin in the static perturbation technique.25, 26 In this technique, the static solution of structures is calculated by a varying load parameter ρ. This load parameter changes the strength of an applied external load Q, not its direction. For the nonlinear static problem of the discretized beam model (3.25), the load parameter would occur in the following manner Bki qi0 + Ckij qi0 qj0 + Dk + ρQk = 0 (5.16) Note, that Q is actually a energy due to the integration over the reference line for the Galerkin approximation. Q is caused by active elements and external generalized forces that have to be defined for the perturbation modes. 0

0

A Qk = Eku uA u + Fkiu qi uu + fk + mk

(5.17)

The Taylor series at a chosen external load, e.g ρ = 1 for the static beam model (5.16) is qi0 (ρ) = qi0

+ ρ=1

1 ∂ 3 qi0 1 ∂ 2 qi0 ∂qi0 2 ρ + ρ3 + · · · ρ+ ∂ρ ρ=1 2! ∂ρ2 ρ=1 3! ∂ρ3 ρ=1

(5.18)

Instead of using the derivations in the Taylor series for the calculation of the static response under a varying load parameter, Noor23 and Bauchau24 used them to reduce the order of their nonlinear static structural problems. Here, these derivations are used for the order reduction of the dynamic beam model (3.25). The transformation matrix using perturbation modes is h pR Tpert = l(2·r) lr

pIl(2·r)

i

(5.19)

Together with the normal modes from the previous section, the transformation matrix T becomes with the added perturbation modes T = Tf vm

Tpert

(5.20)

where Tflrvm is the transformation matrix using free vibration modes and Tpert is the transformation lr matrix using perturbation modes. Similarly to the normal modes, one dynamic perturbation mode I consists of two static modes represented by the vectors pR lr and plr . By this choice of the transformation matrix T, each qr has only two unknowns instead of twelve for ql , see (5.4). The vectors pR li are determined by the static perturbation modes (as presented by Noor23 and Bauchau24 ). 0 pR = q l1 i

, ρ=1

pR l3 =

∂qi0 , ∂ρ ρ=1

pR l5 =

1 ∂ 2 qi0 , 2! ∂ρ2 ρ=1

pR l7 =

1 ∂ 3 qi0 , 3! ∂ρ3 ρ=1

···

(5.21) The second vector for each static mode vector can be calculated by substituting it in the following linear eigenvalue problem to calculate the out of phase component. Aki pIl2 + Bki pR l1 = 0 Aki pIl3 + Bki pR l3 = 0

(5.22)

One has to be careful while adding perturbation modes as these modes are not orthogonal to other modes and thus it may lead to ill-conditioned matrices. The derivations necessary for the static perturbation modes can be calculated similarly to [23, Appendix A]. Here, the first three derivations

5.3. PERTUBATION MODES EXPANSION

31

are calculated. They are received by differentiating the equilibrium equation (5.16) with respect to the load parameter ρ. The fist derivative resulting from this procedure is ∂qi0 Bki + (Ckij + Ckji ) qj0 + Qk = 0 ∂ρ −1 ∂q 0 Qk = i − Bki + (Ckij + Ckji ) qj0 ∂ρ

(5.23)

The next derivative is calculated using the previous solution (5.23) ∂ 2 qi0 ∂qj0 ∂qi0 =0 + Bki + (Ckij + Ckji ) qj0 ∂ρ ∂ρ ∂ρ2 −1 ∂qj0 ∂qi0 ∂ 2 qi0 (Ckij + Ckji ) − Bki + (Ckij + Ckji ) qj0 = ∂ρ ∂ρ ∂ρ2

(Ckij + Ckji )

(5.24)

Finally, the third derivative is calculated based on (5.24) 3 0 ∂ 2 qj0 ∂qi0 0 ∂ qi =0 + B + (C + C ) q ki kij kji j ∂ρ2 ∂ρ ∂ρ3 −1 ∂ 2 qj0 ∂qi0 ∂ 3 qi0 − Bki + (Ckij + Ckji ) qj0 3 (Ckij + Ckji ) = 2 ∂ρ ∂ρ ∂ρ3

3 (Ckij + Ckji )

(5.25)

Now, as the perturbation modes are defined, one has to decide how many perturbation modes to use and which load parameter ρ to choose. These decisions are automated in [23, pp. 456–458]. The above formulation is limited to static problems. Bauchau [24, p.32] extended the formulation to accommodate dynamical situation. This is done by choosing the load Q to be associated with a free vibration mode. For this work, the free vibration modes of the internal force and moment would have to be used as external loads Q. The resulting perturbation modes are called dynamic perturbation modes.

32


Chapter 6

Simplified Beam Model In order to get a first idea of how the Galerkin discretized beam model (3.25) works, it is simplified to a cantilevered Euler-Bernoulli beam [16, chapter 7.2] and compared to its analytical solution. The first simplification is that the Euler-Bernoulli beam is passive so that the model for passive beams (B.3) is the better choice compared to (3.25). Aki q˙i + Bki qi + Ckij qi qj + Dk + fk + mk = 0

(6.1)

Actuation and disturbances caused by external forces and moments are not modelled (fk = 0, mk = 0) which reduces (6.1) to Aki q˙i + Bki qi + Ckij qi qj + Dk = 0

(6.2)

Next, (6.2) is linearized as the Euler-Bernoulli beam is restricted to small displacements (qi0 = 0, cantilevered beam). qi0 =0 Aki q˙i + Bki + (Ckij + Ckji )qj0 qi = 0 −→ Aki q˙i + Bki qi = 0

(6.3)

The comparison between the Galerkin approximated solution and the analytical solution is performed for the straight (k = 0), one dimensional beam so that displacements of the reference line are restricted to the x-z plane. As the reference axis matches the elastic axes, there is only occurring bending (κ2 ) in the beam cross-section. Not only the elastic axis matches the reference axis, but also the tension axis. Furthermore, the Euler-Bernoulli beam does not have shear and rotary inertia and the mass center matches the reference line. From there follows that bending in the x-z plane (κ2 ) is achieved by the internal moment M2 and the internal force F3 only. The cross-sectional parameters resulting from these simplifications are     0 0 0 0 0 0 1 0 , G = 0 0 0  , K = 0, I = 0 (6.4) R = 0, S = 0, T = 0 EI 0 0 m 0 0 0 where E is the modulus of elasticity, I the moment of inertia per unit length and m is the mass per unit length. The simplified beam is visualized in figure 6.1. Inserting the beam cross-section parameters (6.4) and k = 0 into (B.5) and (B.7) results in Z L h i h 1 i ¯ ¯ ¯ ¯ M2i dx Aki = V3k mV3i + M2k EI 0 Z L h i h i h i h i (6.5) no BC 0 0 0 0 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Bki = V3k −F3i + Ω2k −M2i + F3i + F3k −V3i − Ω2i + M2k −Ω2i dx 0 h i h i h i h i BC ¯ 2k (0) Ω ¯ 2i (0) + V¯3k (L) F¯3i (L) + Ω ¯ 2k (L) M ¯ 2i (L) Bki = −F¯3k (0) V¯3i (0) − M ¯ 2i , F¯3i and M ¯ 2i are taken from (3.18). The new generalized time function q(t) is where V¯3i , Ω ql = v3l

ω2l 33

f3l

m2l

T

(6.6)

34

CHAPTER 6. SIMPLIFIED BEAM MODEL

(6.5) and (6.6) are the basis for three Galerkin solutions using three different comparison functions in the next sections. The first solution is calculated with the natural modes of the analytical solution, the second one is calculated with comparison functions that automatically fulfill the boundary conditions. Finally, the last solution is calculated with comparison functions that do not automatically fulfill the boundary conditions.

z y x F3,γ3 Reference, elastic, tension and mass center axis

M2,κ2

Figure 6.1: Simplified Beam

6.1

Analytical Comparison Functions

In this section, the analytical solution of a cantilevered Euler Bernoulli beam is presented. The free vibration modes of the analytical solution are then used as comparison functions for the Galerkin approach of the Euler-Bernoulli beam. The free vibration modes of the cantilevered Euler-Bernoulli beam [16, eq 7.163] are Wi (x) = Ai [(sin βi L − sinh βi L)(sin βi x − sinh βi x) + (cos βi L + cosh βi L)(cos βi x − cos βi x)] (6.7) and β is determined by cos βL cosh βL = −1 (6.8) The natural frequencies of the cantilevered Euler-Bernoulli [16, eq 7.157] can be obtained from r EI 2 (6.9) ωi = (iΠ) mL4 where E is the modulus of elasticity, I the moment of inertia per unit length, m the mass per unit length and L is the beam length. By regarding an infinitesimal element of the Euler-Bernoulli beam, the following correspondences are holding V3i (x) = Wi (x),

Ω2i (x) = −Wi0 (x),

M2i (x) = Wi00 (x),

F3i (x) = Wi000 (x) (6.10)

()0 denotes the derivation with respect to x. Next, the analytical solution in (6.10) is used as the comparison functions of the Galerkin approach which are inserted into (6.5). As the analytical BC solution is satisfying the boundary conditions, the tensor Bki is zero. The resulting equations after the insertion of the simplified tensors A, B in (6.3) are Z L h i h i Wk mWi v˙ i + Wk −Wi0000 fi dx = 0 0

Z L h i Wk0 Wi000 (−mi + fi ) dx = 0 0

Z L h i 000 0 Wk Wi (−vi + ωi ) dx = 0 0

Z L i h i h 1 Wk00 Wi00 m ˙ i + Wk00 Wi00 ωi dx = 0 EI 0

(6.11)

6.2. COMPARISON FUNCTIONS FULFILLING BOUNDARY CONDITIONS

35

From (6.11) follows that vi (t) = ωi (t) and that fi (t) = mi (t). Using the relation Wi0000 = β 4 Wi simplifies (6.11) to mv˙ i − β 4 fi = 0 1 ˙ fi + vi = 0 EI

(6.12)

The two first order differential equations in (6.12) can be simplified to one second order differential equation. β 4 EI fï + fi = 0 (6.13) m } | {z ω2

The resulting frequencies from (6.13) are exactly the natural frequencies of the analytical solution, see (6.9). Next, comparison functions are chosen which are different from the analytical free vibration modes but fulfilling the boundary conditions.

6.2

Comparison Functions fulfilling Boundary Conditions

In this section the natural frequencies of the simplified beam model ((6.3), (6.5) and (6.6)) are calculated using comparison functions that fulfill the boundary conditions. The Legendre functions introduced in chapter 3.4 are not used as they do not fulfill the boundary conditions of the cantilevered beam. A set of non orthogonal polynomial comparison functions that fulfill the boundary conditions (V 0 = 0, Ω0 = 0, F L = 0 and M L = 0) is i+1 , V¯3i (x) = Lx ¯ 2i (x) = 1 − x i+1 , M L

¯ 2i (x) = − i+1 x i = −V¯ 0 (x) Ω 3i L L i+1 x i 0 ¯ ¯ F3i (x) = − L L = M2i (x)

(6.14)

After inserting (6.14) into (6.5) one receives a similar solution than in (6.11). Again, vi (t) = ωi (t) ¯ 2i (x) = −V¯ 0 (x) and F¯3i (x) = M ¯ 0 (x). The remaining two equations are and fi (t) = mi (t) as Ω 3i 2i Z L h Z L h i i 00 ¯ 3i V¯3k mV¯3i dxv˙ i + V¯3k −M dxfi = 0 0 0 (6.15) Z L Z L h i h i ¯ 2i dxf˙i + ¯ 2k V¯ 00 dxvi = 0 ¯ 2k 1 M M M 2i EI 0 0 (6.15) can be written in a shorter form as 1 Pki fi = 0 mL2 EI Oki f˙i − 2 Pki vi = 0

Oki v˙ i −

(6.16)

L

with Oki =

1 , k+i+3

Pki =

Γ(k + 2)Γ(i + 2) Γ(k + i + 2)

(6.17)

R∞ where Γ(a) = 0 e−x xa−1 dx. Again, two first order differential equations can be reduced to one second order differential equation because O is invertible. Oki fï −

EI Pki Oki −1 Pki fi = 0 mL4

Inserting the solution fi (t) = ci eωt leads to the equation EI −1 ωt ω 2 Oki − P O P f =0 ki ki ki i ci e mL4

(6.18)

(6.19)

(6.19) has a solution if 2 det ω ¯ Oki − Pki Oki −1 Pki fi = 0

(6.20)

36


Natural Frequency Error 5

Frequency Error in %

0 −5 −10 −15

1st Frequency

−20

2nd Frequency 3rd Frequency

−25

4th Frequency

−30

5th Frequency

−35

6th Frequency

−40 0

5 10 15 Number of Comparison Functions

20

Figure 6.2: Error of Natural Frequencies

q L4 ω. The error of the resulting natural frequencies ω ¯ from (6.20) compared to the where ω ¯ = mEI analytical natural frequencies from (6.9) for a increasing number of comparison functions is plotted in figure 6.2. One can see that the frequency error of a natural frequency becomes negligible small after two comparison functions. In the next section, the convergence of the free vibration modes using comparison functions that do not automatically fulfill the boundary conditions is presented.

6.3

Arbitrary Comparison Functions

The last type of comparison functions tested for the simplified beam are comparison functions that do not fulfill the boundary conditions. To be able to compare the results from the previous section, the comparison functions are chosen similarly to (6.14). i−1 V¯3i (x) = Lx , x i−1 ¯ M2i (x) = 1 − L ,

¯ 2i (x) = x i−1 Ω L i−1 F¯3i (x) = 1 − Lx

(6.21)

Due to the more general comparison functions, vi (t) 6= ωi (t) and fi (t) 6= mi (t), as the first element of the time functions v(t), ω(t), f (t) and m(t) determines the boundary condition which are not exactly fulfilled anymore. The equations that need to be solved follow from (6.5) and (6.21) Z L h Z L h i i h i 0 V¯3k mV¯3i dx v˙ i + V¯3k −F¯3i dx + V¯3k (L) F¯3i (L) fi = 0 0

0

Z L Z L h i h i h i ¯ 2k F¯3i dx fi + ¯ 2k −M ¯0 ¯ 2k (L) M ¯ 2i (L) mi = 0 Ω Ω dx + Ω 2i 0

0

Z L Z L h i h i h i 0 ¯ ¯ ¯ ¯ ¯ ¯ F3k −Ω2i dx ωi + F3k −V3i dx − F3k (0) V3i (0) vi = 0 0

0

Z L Z L h h i h i i ¯ 2k 1 M ¯ 2i dx m ¯ 2k −Ω ¯0 ¯ 2k (0) Ω ¯ 2i (0) ωi = 0 M ˙i+ M dx − M 2i EI 0 0

(6.22)

6.3. ARBITRARY COMPARISON FUNCTIONS

37

(6.22) can be written in a more compact notation as many terms have the same values due to the similar comparison functions (6.21). 1 Pki fi (t) = 0 mL LQki fi (t) + Pki mi (t) = 0 LQki ωi (t) + Pki vi (t) = 0 EI Oki m ˙ i (t) − Pki ωi (t) = 0 Oki v˙ i (t) +

(6.23)

L

where Oki

1 , = i+k−1

Pki =

( 1, Γ(i)Γ(k) Γ(i+k−1) ,

i=1 i 6= 1

Qki =

,

Γ(i)Γ(k) Γ(i + k)

(6.24)

After writing (6.23) in matrix notation, the solution of the time function q ∗ (t) is obtained by the eigenvalue problem  0 0 Oki 0 0 0  0 Oki 0 0 0 0 | {z =A∗

 ˙   1 0 fi (t) mL Pki    0  mi (t) +  LQki 0  vi (t)   0 0 ωi (t) 0 } | {z } | =q˙i∗

0 Pki 0 0

0 0 0 Pki {z

=B∗

 fi (t)  mi (t)  =0 EI − L Pki   vi (t)  ωi (t) LQki } | {z } 0 0



(6.25)

=qi∗

Note, that (6.25) is differently arranged than (6.3) (qi (t) → qi∗ (t), A → A∗ and B → B ∗ ). This rearrangement has been done for this simple case to obtain the clear structure in A∗ and B ∗ . However, for more complicated cases, this rearrangement does not pay off, e.g. by using the full model (3.25). In this section, we are interested in the free vibration modes which are calculated according chapter 4.1.3. In figure 6.3 , the resulting free vibration modes (solid line) are plotted against the analytical solution (dashed line) using only six comparison functions. One can see that the free vibration mode corresponding to the first natural frequency has already converged very well in terms of its shape and its boundary conditions. The natural mode corresponding to the forth natural frequency has still a significant error. By using twelve instead of six comparison functions, the first four natural modes match the exact solution nearly perfectly. In the next chapter, the full model (3.25) will be used to simulate the nonlinear dynamics of the active helicopter blade. Therefor, Legendre functions will be used as comparison functions, which do not fulfill the boundary conditions. But as it was shown for the Euler-Bernoulli beam in this section, the boundary conditions converge satisfyingly.

38


First Natural Modes of Ω

First Natural Modes of V 1 0.8 0.6

0.5 st

nd

2

0

Natural Mode

rd

3 Natural Mode

Ωfvm

Vfvm

1 Natural Mode

0.4

1st Natural Mode

0.2

2nd Natural Mode

0

3rd Natural Mode

−0.2

4th Natural Mode

th

4 Natural Mode −0.5

−0.4 −0.6

−1 0

0.5 x

−0.8 0

1

(a) First four natural modes of V

0.5 x

1

(b) First four natural modes of Ω

First Natural Modes of F

First Natural Modes of M 1

0.8 0.6

0.5 st

1st Natural Mode

1 Natural Mode nd

0.2

2

Natural Mode

rd

0

3 Natural Mode

Mfvm

Ffvm

0.4

2nd Natural Mode

0

3rd Natural Mode

th

4th Natural Mode

4 Natural Mode

−0.2

−0.5

−0.4 −0.6 −0.8 0

0.5 x

1

(c) First four natural modes of F

−1 0

0.5 x

1

(d) First four natural modes of M

Figure 6.3: Natural modes compared to analytical solution

Chapter 7

Helicopter Blade In this chapter, the full discretized beam model (3.25) will be used to simulate the nonlinear dynamics of an actuated helicopter blade. First, simulation results of a simple prismatic beam are presented. The results can be compared to the exact and an FEM solution for a rotating case to get an idea of the accuracy of the more advanced helicopter blade simulation. Next, the used helicopter blade is presented, including structural parameters, actuation and sensing. After that, the steady state solution, the natural frequencies and free vibration modes are shown. Then, the full model is reduced using natural and perturbation modes. The performance of the reduced model is then identified by comparing the natural frequencies of the reduced with the ones of the full model under varying rotational speed and external force. Finally, the transformation of the full and the reduced model to a state space model is presented.

7.1

Simple Example

Before specifying the active helicopter blade, a simple prismatic beam is presented to validate the formulation. This case is using the full passive beam model (B.3) with zero shear and extensional flexibility presented in Table 7.1 and Table 7.2 lists the calculated frequencies and compares the results with exact results27 as well as FEM solution.3 The frequency predictions of a non-rotating as well as rotating beam using the present approach with 10 assumed modes per variable (120 states) are exact to three significant digits. On the other hand, FEM solution with 10 nodes (120 states), is not very accurate leading to errors greater than 10% for the third bending mode. In the case of the active helicopter blade presented next, only simulation results from the FEM solution3 can be utilized to verify simulation results. This motivating example shows that it might come to significantly different results when the Galerkin and the FEM simulations are compared.

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

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 7.1: Prismatic beam data (for model validation purpose) 39

40

CHAPTER 7. HELICOPTER BLADE

Mode (rad/s)

Exact

Present FEM 10 modes 10 nodes Cantilevered Blade: ω = 0 & v = 0 1st bending 2.243 2.243 2.252 2nd bending 14.06 14.06 14.74 3rd bending 39.36 39.36 44.94 1st torsion 31.05 31.05 31.12 2nd torsion 93.14 93.14 95.32 Rotating Cantilevered Blade: ω = 3.189 rad/s & v = 0 1st bending 4.114 4.114 4.110 2nd bending 16.23 16.23 16.88 3rd bending 41.59 41.59 47.12 Rotating Cantilevered Blade with Offset: ω = 3.189 rad/s & v = 51.03 m/s 1st bending 5.703 5.703 5.696 2nd bending 18.72 18.72 19.35 3rd bending 44.50 44.50 49.97 Table 7.2: Beam structural frequencies (for model validation purpose)

7.2

Helicopter Blade Specification

The Active Twist Rotor (ATR) blade1, 2 is a collaborative research effort between the U.S. Army Research Laboratory, at NASA Langley Research Center, and the University of Michigan/MIT. The aeroelastic analysis performed in that project used the thin airfoil theory for the aerodynamic modelling and the Hodges beam theory for the structural modelling. In this work, the aerodynamic coupling is not modelled and the vibration reduction potential is investigated using the Hodges beam theory (chapter 2) only. The ATR blade is the active helicopter blade used in the work of Traugott [14, chap 6]. The parameter values used there are taken from the ATR helicopter blade research. However, these values are slightly different from the ones presented in the paper of Wilbur [2, Table 2]. The helicopter blade model used in this thesis will use the parameter values from the work of Traugott to be able to compare the simulation results. These values are presented in the following subsections.

7.2.1

General properties

For structural purposes, the important general properties of the helicopter blade are the span L = 1.397 m and the operational angular speed Ω03 = 72 rad s−1 . The rotor overspeed is Ωmax = 79 rad 3 s−1 . For more detailed general properties of the ATR blade, see [2, Table 1]

7.2.2

Cross-Sectional Flexibility and Inertia

Due to the thin-walled design of helicopter blades, the cross-sectional flexibilities can be obtained by applying the theory of Patil and Johnson.11 The cross-sectional flexibility matrices are  6.4375 × 10−7 0 R= 0  0 0 S= 1.8621 × 10−4  2.9086 × 10−2  0 T= 0

 0  0 4.4389 × 10−5  . 5.5420 × 10−6  0 0

0 4.9262 × 10−6 0 .

0 0 0

0 2.5038 × 10−2 0

 0  0 −4 9.2640 × 10

(7.1)

7.2. HELICOPTER BLADE SPECIFICATION

41

and the cross-sectional inertia matrices are   6.9310 × 10−1 0 0  0 6.9310 × 10−1 0 G= 0 0 6.9310 × 10−1   . 0 . 4.7999 × 10−4 0 .  0 . . 0 . 0 K= 0 . 0 −4.7999 × 10−4 .   3.7665 × 10−4 0 0  0 6.4630 × 10−6 0 I= 0 0 3.7018 × 10−4

(7.2)

In the next section, the values for the actuation matrices A and B are presented.

7.2.3

Actuation

The actuation of the ATR helicopter blade is provided by active fiber components (AFC). These fiber elements are completely integrated into the composite structure and basically work the same as the piezo elements introduced in chapter 2.2.2, see also figure 2.2. The active fiber components are integrated in the ATR blade in six segments, see figure 7.1. In each segment, there are four different active fiber component layers that can be controlled independently. The generalized strains induced by the active elements are related to the four voltages of one element the following way γA

seg

= Eseg uA

seg

κA

seg

= Fseg uA

seg

(7.3)

with

Eseg Fseg

 8.9562 × 10−9 0 = 0  3.8506 × 10−6 0 = 0

0 2.7843 × 10−8 0 0 1.9155 × 10−6 0

 +1 0  +1 0 +1 2.8536 × 10−8  −1 0  +1 0 +1 8.5769 × 10−8

 +1 +1 +1 −1 −1 +1 −1 +1 −1  +1 −1 +1 +1 −1 −1 +1 +1 +1

(7.4)

seg

where ()seg denotes the restriction to a blade segment. The maximum voltage uA that can be applied is 1500V . From this maximum voltage follows the maximum induced generalized strains as uA

seg

uA

seg

uA

seg

uA

seg

= 1500 +1 +1 +1 = 1500 +1 −1 −1 = 1500 +1 −1 +1 = 1500 +1 +1 −1

T +1 T +1 T −1 T −1

→ γ1A

max

= 5.3737 × 10−5 ,

→ γ2A

max

= 1.6706 × 10−4

→ γ3A

max

= 1.7122 × 10−4 ,

→ κA 2

max

= 1.1500 × 10−2

κA 3

max

= 5.1461 × 10−4

κA 1

max

= 2.2800 × 10−2 (7.5)

Besides the maximum possible induced generalized strains, one can see that γ1A and κA 3 as well as γ3A and κA 1 can not be applied independently. For the actuation tensors of the active beam model (3.28), the matrices A and B have to be set up. The active generalized forces of one segment are determined by (2.12) Eseg A seg seg V FA = U u Fseg (7.6) Eseg A seg seg MA = VT W u seg

F

The induced generalized forces for the whole beam can be described by using a matrix that allocates the 24 voltages to the beam segments. The first four voltages uA are controlling the first four active

42


1 2 4 3

Figure 7.1: Actuators on ATR Blade

fiber layers in the first blade segment. The next four voltages of uA are controlling the second blade segment and so on. In order to describe these affiliations, the boxcar function B(x) is introduced. ( 1, B(x) = 0,

0 ≤ x ≤ L6 otherwise

(7.7)

The generalized strains induced by the actuation can now be described with (7.6) and (7.7) F A = AuA

(7.8)

M A = Bu A where

A = Aseg IB(x) IB(x − 61 L) IB(x − 62 L) . . . IB(x − 56 L) B = Bseg IB(x) IB(x − 16 L) IB(x − 62 L) . . . IB(x − 65 L)

(7.9)

and where I is the identity matrix and Aseg , Bseg are

A

seg

B

seg

= U

= V

T

Eseg

V

Fseg Eseg W Fseg

(7.10)

A, dS A, S B, dS B, D A Because R of the property that A and B are constant within a segment, S and D B in (3.36) can be simplified. Therefore, the following abbreviations are made. R

Z (ξ+ 61 )L Z sk (ξ) = Pk (x) dx = L R

ξL

0

R

R

Pk (¯ x) d¯ x

ξ

R

ξL

R

ξ+ 16

A /B s (ξ) = P (ξ + 1 ) − P (ξ) k k k 6 1 Z Z R (ξ+ 6 )L dki (ξ) = Pk (x)Pi (x) dx = L 0

R

ξ

(7.11) ξ+ 16

Pk (¯ x)Pi (¯ x) d¯ x

7.3. SIMULATION RESULTS

43

With (7.11), the integrals in (3.36) can be written as A

R

Sku d

R

A

Sku B

R

Sku d

R

B

Sku R

A

Dkiu R

B

Dkiu

R 1 = L 0 Pk (¯ x)Au (¯ x) d¯ x R 1 = 0 Pk (¯ x)A0u (¯ x) d¯ x R 1 = L 0 Pk (¯ x)Bu (¯ x) d¯ x R 1 = 0 Pk (¯ x)B0u (¯ x) d¯ x R 1 = L 0 Pk (¯ x)Pi (¯ x)Au (¯ x) d¯ x R 1 = L 0 Pk (¯ x)Pi (¯ x)Bu (¯ x) d¯ x

h R = Aseg Isk (0) . . . h R 0 0 = Aseg I A /B sk (0) h R = Bseg Isk (0) . . . h R 0 0 = Bseg I A /B sk (0) h R = Aseg Idk (0) . . . h R = Bseg Idk (0) . . .

i R Isk ( 56 ) i R 0 0 I A /B sk ( 56 ) i R Isk ( 56 ) i R 0 0 . . . I A /B sk ( 65 ) i R Idk ( 65 ) i R Idk ( 65 )

...

(7.12)

In the next section, the sensing of the active helicopter blade is presented.

7.2.4

Sensing

Piezo elements can be used as actuators and sensors, as described in chapter 2.2.2. For the ATR blade there is no sensor data known. Technically it is not a problem to measure the generalized strains γ and κ at any place of the beam reference line. For the helicopter blade used in this work, five sensors are distributed equidistantly over the blade, where the first sensor is located at the root and the last one at the tip.  γ    Φi (x = 0)  γ(x = 0)             κ(x = 0)   Φκi (x = 0)            Φγi (x = 1 L)   γ(x = 1 L)     4 4    Φκ (x = 1 L) κ(x = 1 L) (3.20) S (2.13) i u = O P = O P qi = Myi qi (7.13) 4 4     .. ..         . .             γ     γ(x = L ) Φ (x = L )     i      κ    κ(x = L) Φi (x = L) where M is the measurement matrix and the sensor matrices O and P are chosen to be identity (O = I, P = I) as the exact sensor data is not known. All the helicopter blade properties are now set up and the blade model simulation results are presented in the next section.

7.3

Simulation Results

Before developing a controller for the active helicopter blade specified in chapter 7.2, its behavior is discussed on simulation results. These simulation results are going to be compared to results gained from the FEM simulation of Traugott.14 Simulation results presented here include steady state solution, natural frequencies and modes, reduced model performance, steady actuation response and helicopter blade displacement.

7.3.1

Steady State

The steady state solution of the linear and angular velocity V and Ω are presented in figure 7.2. It shows the operating rotational speed Ω03 = 72 rad s−1 of the helicopter blade and the resulting linear increase of the velocity V20 up to L × Ω03 = 100.6 m s−1 . Due to the rotation of the helicopter blade, a quadratically decreasing centrifugal force F10 in blade direction occurs, see figure 7.3. The relatively small internal moment M30 arises from the coupling between extension caused by F10 and bending.

7.3.2

Natural Frequencies and Modes

The natural frequencies of the helicopter blades are stated for the rotating and the non-rotating case and compared to the FEM solution of Traugott in Table 7.3. The frequencies are increasing because of rotation which leads to stiffening as the centrifugal force is pulling the blade straight. The bending

44


Steady State Solution of Angular Velocity

Steady State Solution of Linear Velocity 100

70 V01

80

V02

Ω02

50

V03

V0

Ω0

60

Ω01

60

Ω03

40 30

40

20 20

10 0 0

0.2

0.4

0.6

0.8

1

0 0

1.2

0.2

0.4

0.6

0.8

1

1.2

x

x

(a) Steady State of V

(b) Steady State of Ω

Figure 7.2: Steady State Solution of V and Ω

Steady State Solution of Internal Force

Steady State Solution of Internal Moment

3500

0

3000

F01

−0.5

2500

F02

−1

F03

−1.5

F0

M0

2000

−2

1500 1000 500 0 0

0.2

0.4

0.6

0.8

1

1.2

−2.5

M01

−3

M02

−3.5

M03

−4 0

0.2

0.4

x

0.6

0.8

1

1.2

x

(a) Steady State of F

(b) Steady State of M

Figure 7.3: Steady State Solution of F and M

flap frequencies are changing much more than the ones in chord direction as the non-rotating blade is much stiffer in chord than in flap direction. [14, p. 68] Furthermore, the first natural modes change more than the higher ones, as their natural displacement modes have a greater displacement in the tip region of the blade [14, pp. 68 – 69], so that the centrifugal force has a greater effect on the stiffening of the blade. The free vibration modes of the presented natural frequencies are shown in figure 7.4.

7.3.3

Displacement

Until now, the deformation of the helicopter blade has not been regarded as it does not appear in the Hodges equations (2.1) due to its intrinsic character. However, the deformation of the helicopter blade can be of interest, e.g. to extend the helicopter blade model by introducing aero-elastic coupling, where the blade deformation has to be taken into account. The deformation is calculated using the stain-displacement relation. [5, eq. (7),(8)] ˜u) − e1 γ = C(e1 + u0 + k ˜C T − k ˜ κ ˜ = −C 0 C T + C k

(7.14)


Mode (rad/s)

1st 2nd 3rd 4th 5th 6th 1st 2nd 3rd 1st 2nd

bending bending bending bending bending bending bending bending bending torsion torsion

flap flap flap flap flap flap chord chord chord

45

Present FEM Difference 20 modes 10 nodes in % Helicopter Blade: ω=0 13.65 13.68 -0.219 84.47 86.71 - 2.583 232.0 248.8 -6.752 442.5 507.3 -12.77 707.5 885.5 -20.10 1015 1422 -28.62 70.60 70.76 -0.226 425.2 436.1 -2.499 1124 1199 -6.255 339.9 340.4 -0.147 1023 1034 -1.064

Present FEM Difference 20 modes 10 nodes in % Rotating Helicopter Blade: ω = 72 rad/s 75.99 75.97 0.026 199.7 201.8 - 1.041 376.6 394.2 -4.465 610.1 676.3 -9.784 891.4 1032 -13.62 1213 1615 -24.89 76.26 76.35 -0.118 455.7 466.2 -2.252 1159 1233 -6.002 346.4 346.9 -0.144 1021 1073 -4.846

Table 7.3: Helicopter blade structural frequencies (for model validation purpose)

First Natural Modes of Ω

First Natural Modes of V 1st Natural Mode of V1

0.8

2nd Natural Mode of V1

0.6

3 Natural Mode of V1 Ωfvm

2nd Natural Mode of V2

0

rd

3 Natural Mode of V2

−0.2

nd

2

−0.6

1st Natural Mode of Ω2

0.2

2nd Natural Mode of Ω2

0

3rd Natural Mode of Ω2

−0.2

1st Natural Mode of V3

−0.4


0.4

1st Natural Mode of V2

0.2


0.6

rd

0.4 Vfvm

1st Natural Mode of Ω1 0.8

1st Natural Mode of Ω3

−0.4

Natural Mode of V3


−0.6

rd

−0.8 0

3 Natural Mode of V3 0.5

1

−0.8 0

1.5

0.5

x

First Natural Modes of M 1

1st Natural Mode of F1

0.8

1st Natural Mode of M1

2nd Natural Mode of F1

0.6

2nd Natural Mode of M1

rd

nd

2 0

Natural Mode of F2

Mfvm


0.2

3rd Natural Mode of M1

0.5

3 Natural Mode of F1

0.4

1st Natural Mode of M2 2nd Natural Mode of M2

0

3rd Natural Mode of F2

−0.2


−0.4

nd

2

−0.6 1

3rd Natural Mode of M2 1st Natural Mode of M3

−0.5

2nd Natural Mode of M3

Natural Mode of F3

3rd Natural Mode ofF3 0.5


(b) First four natural modes of Ω

First Natural Modes of F

Ffvm

1.5

x

(a) First four natural modes of V

−0.8 0

1

1.5

x

−1 0

3rd Natural Mode of M3 0.5

1

1.5

x

(c) First four natural modes of F

(d) First four natural modes of M

Figure 7.4: Natural modes of helicopter blade

whereas the initial curvature k and the reference line displacement u are measured in the b-frame, see figure 2.1. The other variables are calculated in the B-frame. By using the property of the orthogonal direction cosine matrix matrix C that transposing is equivalent to inverting (CC T = C T C = I),

46


(7.14) can be written as ˜u + e1 − C T (e1 + γ) = 0 u0 + k ˜)C − C k ˜=0 C 0 + (˜ κ+k

(7.15) (7.16)

This equation set could be solved using a Galerkin approach. In this work, however, the deformation is calculated taking advantage of the fact that (7.16) has the analytical solution C = eAx for k = 0 and κ = const. Inserting that solution into (7.16) leads to (A + κ ˜ ) eAx = 0 −→ A = −˜ κ

(7.17)

If the boundary conditions of the helicopter blade are chosen as V 0 = 0, Ω0 = 0, F L = 0 and T T M L = 0 20 0 then κ(x)0 = 0 0.5 0 resulting in an exact solution for C(x). The deformation of the reference line u is then calculated numerically by approximating u0 with u0 = ∆u ∆x x and ∆x = 100 in (7.15). n+1 u = n C T (n γ + e1 ) − e1 ∆x + n u (7.18) The deformation of this steady state is shown in figure 7.5. If κ is not constant along the reference x . line, then C is calculated numerically for different x values with ∆x = 100 n+1 Helicopter Blade Deformation C

= n C + e−˜κ∆x − I;

(7.19)

The deformation of the first three free vibration modes of the helicopter blade as examples for non-constant curvatures are shown in figure 7.6.

0

−0.2

1.2 1 0.8

−0.4

Z Axis

Z Axis

0

0.6

0.1

0.2

0 Y Axis

−0.2 −0.4

0.4 X Axis

0.1

0.2

Y Axis

0.4 0.6 X Axis

(a)

0.8

1

1.2

(b)

Figure 7.5: Deformation of the helicopter blade under constant curvature Beam Deformation

Beam Deformation

Beam Deformation

1.2 0.8

−0.3 0.05 −0.05 Y Axis

0.2

0.6 0.4 X Axis

1

1.2 1

0.02 −0.02

0.8

0.1

0 −0.1 −0.2 Y Axis

(a) 1st bending flap

0.2

0.6 0.4 X Axis

(b) 1st bending cord

Z Axis

1

−0.2

Z Axis

Z Axis

0 −0.1

0.8

0.05 0 −0.05

0.1 0 Y Axis

0.6 0.4 0.2

X Axis

(c) 1st torsion

Figure 7.6: Deformation of normal modes of the helicopter blade

7.3.4

Actuation Response

In this subsection, the effects of the discrete value of the active generalized strains γ A and κA on the simulation are discussed. Principally, it is taken care of the discrete character as for the actuation tensors E and F, the exact values of the discrete functions A(x) and B(x) are used in (3.39). In order to be able to capture the discrete values satisfyingly, a very high number of continuous comparison


47

functions is necessary. Alternatively, one could add comparison functions B(x), see (7.7) that perfectly describe the discrete active generalized strains γ A and κA . However, even if the discrete character can not be captured perfectly, the dynamic solution is still accurate, as the first comparison functions are the most important ones for the dynamic solution because they have the greatest similarity to the first free vibration modes. Problems arise, if the steady state solution of the beam under applied voltage has to be calculated as the solution is discrete in spite of the continuous dynamic solution. This problem is shown for the non-rotating helicopter blade in figure 7.7 where the solid line shows the approximated solution for 20 comparison functions while the dashed line shows the perfect solution. One can see that the response to the actuation is much better for the curvature κ02 than for the strain γ30 as there is only a discontinuity in the first derivation of κ02 . Steady State Solution of Strain

−5

x 10

Steady State Solution of Curvature

5

κ02

0.2 0 0.15 γ03

κ02

−5 γ03

−10

0.05

−15 −20 0

0.1

0.2

0.4

0.6

0.8

1

1.2

x

(a) γ30 under Actuation

0 0

0.2

0.4

0.6

0.8

1

1.2

x

(b) κ02 under Actuation

Figure 7.7: Actuation Response for κ03 = 1.5 × 10−4 in the fist half of the blade

7.3.5

Reduced Model Performance

The performance of a reduced helicopter blade model is tested against the full model in this subsection. This is done by comparing the natural frequencies of the full and reduced model under a varying parameter. One test case is performed by varying the rotational speed from Ω3 = 0 rad s−1 to the rotational overspeed of Ωmax = 80 rad s−1 . In the other test case, different values of 3 the external load f30 under rotation (Ω3 = 72 rad s−1 ) are applied. This external force is applied constantly over the whole blade reference line from values f30 = 0 N m−1 up to f30 = 500 N m−1 . The external load f30 necessary to lift the model helicopter (estimated weight of 28.5 kg )is f30 = 50 N m−1 . First, the full model (discretized by 20 comparison functions) is tested against a model that is reduced by 8 normal modes. This reduces the number of states from 240 to 16. The frequencies for varying rotational speed and external force are presented in figure 7.8. The external force is varied for the rotating blade (72 rad s−1 ). The solid lines are the frequencies of the full model and the dashed ones of the reduced model. One can see that a reduced model that has been reduced using 8 normal modes already provides frequencies that track the ones of the full model satisfyingly. The same tests are now done for a model that is reduced by 6 normal modes and 2 perturbation modes so that the number of state variables is reduced from 240 to 16 again. The first perturbation mode is the deformation under the constant external force f30 =500 N m−1 and the rotation speed Ω03 =72 rad s−1 . The second perturbation is calculated according (5.23) .The results are shown in figure 7.9 and again, the solid line shows the frequencies of the full model and the dashed one shows the frequencies of the reduced model. The perturbation modes improve the accuracy of the important first modes, especially if the rotating speed is varied. The natural frequencies for the rotating blade under varying external force are not affected so much by the change of the order reduction. Besides the improvement in the frequency performance, the use of perturbation modes also leads to a significant damping error. The intrinsic equations (2.1) do not model structural damping so that

48


Natural Frequencies of Full and Reduced System (8 Normal Modes)

Natural Frequencies of Full and Reduced System (8 Normal Modes)

1st Frequency nd

2

300

Frequency

3rd Frequency th

4 Frequency

200

5th Frequency th

6 Frequency

100

0

Frequency in rad/s

Frequency in rad/s

400 400

1st Frequency 2nd Frequency

300

3rd Frequency 4th Frequency 5th Frequency

200

6th Frequency 100

20

40 Ω3

60

0

80

100

200

300

400

500

f03

(a) Frequencies under varying rotational speed

(b) Frequencies under varying external force

Figure 7.8: Frequencies for the full and reduced blade model Natural Frequencies of Full and Reduced System (6 Normal and 2 Perturbation Modes)

Natural Frequencies of Full and Reduced System (6 Normal and 2 Pertutbation Modes)

1st Frequency nd

2

300

Frequency

3rd Frequency th

4 Frequency

200

5th Frequency th

6 Frequency

100

0

Frequency in rad/s

Frequency in rad/s

400 400

1st Frequency 2nd Frequency

300

3rd Frequency 4th Frequency 5th Frequency

200

6th Frequency 100

20

40 Ω3

60

0

80

100

200

300

400

500

f03

(a) Frequencies under varying rotational speed

(b) Frequencies under varying external force

Figure 7.9: Frequencies for the full and reduced blade model

the resulting eigenvalues of the linearized system should lie on the imaginary axis. The eigenvalues of the linearized model reduced by 6 normal modes and 2 perturbation modes are shown in figure 7.10. The 4 eigenvalues with significant damping occur from the perturbation modes. Because of the importance of the correct damping, perturbation modes are not used in the following anymore. 1000

500

0

−500

−1000 −10

0

10

20

30

40

Figure 7.10: Pole-Zero map

7.3.6

State Space Equations

For simulation purposes, the discretized helicopter blade model (3.25) has to be formulated in state space. Having a state space formulation is also a big advantage for the controller design as many


49

design techniques assume that the plant is formulated in state space. Writing the helicopter blade model in state space is simple as the A/A matrix in (3.25) or (5.11) has full rank so that it can be inverted. The state space formulation of the full model is A q˙i = Bki qi + Ckij qi qj + Dk + Eku uA u + Fkiu qi uu + Gki fi + Hki mi yy = Myi qi

where

Bki = −A−1 kh Bhi Fkiu = −A−1 kh Fhiu

Ckij = −A−1 kh Chij Gki = −A−1 ki

Dk = −A−1 kh Dh Hki = −A−1 ki

Eku = −A−1 kh Ehu Myi = Myi

(7.20)

(7.21)

For the reduced model, the difference in the state space equation is the missing D vector and the different measurement equation which follows directly from (5.4). A q˙ i = Bki qi + Ckij qi qj + Eku uA u + Fkiu qi uu + Gki fi + Hki mi

yy = Myi qi + yy0

(7.22)

where Bki = −A−1 kh Bhi Gki = −A−1 kh Thi

Ckij = −A−1 kh Chij Hki = −A−1 kh Thi

Eku = −A−1 kh Ehu Myi = My1 T

Fkiu = −A−1 kh Fhiu yy0 = Myi q 0

(7.23)

For other beam models, the A matrix does not have full rank. This is the case for the prismatic beam in chapter 7.1 using the discretized model for passive beams (B.3). Some of the eigenvalues become infinity due to the infinite shear and extensional rigidity. This system could be simulated by diagonalizing the A matrix using the eigenvectors. The resulting equations are algebraic (zero row vector in A) or differential equations (non-zero row vector in A). After the separation of these two types of equations one gains a set of differential-algebraic equations (DAEs) that can be solved by various simulation programs.

50


Chapter 8

Preliminary Control Considerations Before dealing with control designs, the helicopter blade dynamics is characterized. Based on this characterization, inappropriate control designs are discussed. Finally, the control objective of the helicopter blade is specified in this chapter.

8.1

System Characterization

The helicopter blade model set up in chapter 7.2 does not have structural damping as this is not considered in Hodges intrinsic equations (2.1), where the helicopter blade model origins from. Because of the lack of damping, the eigenvalues of the linearized model are lying on the imaginary axis, see figure 8.1 The damping error with an order of magnitude of 10−8 in figure 8.1 comes from 1500 1000 500 0 −500 −1000 −1500 −2

−1.5

−1

−0.5

0

0.5

1 −8

x 10

Figure 8.1: First 20 poles the Galerkin discretization and numerical errors. From this follows that the helicopter blade is marginally stable. Another way to state the marginal stability is to argument that the helicopter blade does not dissipate or gain energy. Systems that have an energy map that is not convex with a minimum at the origin can be controlled via energy shaping 28, 29 which changes the energy map in a way that it becomes convex. Take an inverted pendulum as described in chapter 9.1. After applying energy shaping, its energy function is modified in a way that it matches the one of the not inverted pendulum which is stable. However, this methodology is not necessary for the helicopter blade as the problem is not the shape of the energy function (9.10) which is already convex, but the lack of energy dissipation. Another property of the helicopter model is its high fidelity. The intrinsic equations (2.1) are describing the nonlinear dynamics very accurate6 and so a system identification,30 e.g. by using 51

52

CHAPTER 8. PRELIMINARY CONTROL CONSIDERATIONS

neuronal nets is not necessary. Furthermore, the system parameters of the helicopter blade (e.g. mass density, flexibility, . . . ) are constant over time. From this characteristics follows that adaptive control 31 is not necessary.

8.2

Control Objective

The helicopter blade controller is designed for a rotational speed of Ω3 = 72 rad s−1 . The task of the controller is to minimize the error of the state variables compared to the steady state solution. Special attention is paid to reduce variable errors that highly contribute to the physical energy of the helicopter blade. Furthermore, it is tried to minimize actuation energy and the saturation of the actuators has to be considered. Although additional physical damping is usually desired in active structural systems, it has to be kept in mind that it may cause instability for rotating systems as shown in chapter 11.2.

Chapter 9

Stability of Nonlinear Systems Lyapunov stability theorems [32, chap 4] give sufficient conditions for the stability of equilibrium points covering autonomous and nonautonomous nonlinear systems. The beauty in Lyapunov stability is the big domain of systems the stability theorems can be applied to. However, systems can be stable even if no Lyapunov stability proof can be found and Lyapunov stability theorems do not provide constructive methods (provide a scheme). To prove Lyapunov stability, one has to find so called Lyapunov functions by trial and error that satisfy the necessary conditions. In the following, the stability theorem for autonomous systems is presented exclusively as no other systems are dealt with in this thesis. Besides this, a Lyapunov function for the helicopter blade and other beam problems is suggested.

9.1

Lyapunov Stability for Autonomous Systems

Autonomous systems are systems, where the time t is not occurring explicitly in the system equations. Time is only arising implicitly in the derivative of state space variables. Consider the autonomous system x˙ = f (x) (9.1) where f : D → Rn is a locally Lipschitz map from a domain D ⊂ Rn into Rn . The function f is satisfying the Lipschitz condition, if

f (x) − f (y) ≤ L x − y (9.2) where x, y ∈ D and L is the Lipschitz constant. The Lipschitz condition basically states that the solution of f (x) is continuously dependent on its input x. Suppose x ¯ ∈ D to be an equilibrium point of (9.1). Next, the stability of this equilibrium is characterized. For convenience, the equilibrium is stated at the origin (¯ x = 0) without loss of generality. Definition: The equilibrium point x = 0 of (9.1) is • stable, if for each > 0, there is δ = δ() > 0 such that

x(0) < δ ⇒ x(t) < ,

∀t ≥ 0

• asymptotically stable if it is stable and δ can be chosen such that

x(0) < δ ⇒ lim x(t) = 0 t→∞

• unstable if it is not stable The three types of stability properties can be illustrated by a pendulum. The pendulum equation x˙ 1 = x2 x˙ 2 = −a sin x1 − bx2 53

(9.3)

54

CHAPTER 9. STABILITY OF NONLINEAR SYSTEMS

has two equilibrium points at x1 = 0, x2 = 0 (pendulum in lower position) and x1 = Π, x2 = 0 (inverted pendulum). Neglecting friction by setting b = 0, the equilibrium where the pendulum is in lower position is stable, as it swings at constant amplitude. In the case of no friction (b = 0), the pendulum energy is constant, whereas in the case of friction (b > 0), the equilibrium of the lower pendulum position is even asymptotically stable due to energy dissipation. The upper equilibrium is unstable for both cases, as no initial position close to this equilibrium can be found to keep the pendulum arbitrary close to the inverted position. The pendulum is gaining energy for every position close to the upper equilibrium. In all three cases, the pendulum energy has been used to determine the stability. Taking advantage of energy concepts is the basic idea of the Lyapunov stability theorems. Note, that the regarded energy does not have to be a physically interpretable one, as the following theorem shows. The theorem and the following proof are taken out of [32, pp. 114–116]. Theorem: Let x = 0 be an equilibrium point for (9.1) and D ⊂ Rn be a domain containing x = 0. Let V : D → R be a continuously differentiable function such that V (0) = 0 and

V (x) > 0 in

V˙ (x) ≤ 0 in

D − {0}

(9.4)

D

(9.5)

D − {0}

(9.6)

Then, x = 0 is stable. Moreover, if V˙ (x) < 0 in then x = 0 is asymptotically stable.

Ωβ

Bδ

D

Br

Figure 9.1: Visualization of sets used for the stability proof

Proof : For a given > 0, choose r ∈ (0, ] such that Br = x ∈ Rn x ≤ r ⊂ D Let α = min||x||=r V (x). Then, α > 0 by (9.4). Take β ∈ (0, α) and let Ωβ = x ∈ Br V (x) ≤ β Then, Ωβ is in the interior of Br (see figure 9.1). The set Ωβ has the property that any trajectory starting in Ωβ at t = 0 stays in Ωβ for all t ≥ 0. This follows from (9.5) since V˙ (x(t)) ≤ 0 ⇒ V (x(t)) ≤ V (x(0)) ≤ β,

∀t ≥ 0

Because Ωβ is a compact set, we conclude that (9.1) has a unique solution defined for all t ≥ 0 whenever x(0) ∈ Ωβ (see [32, Theorem 3.3]). As V (x) is continuous and V (0) = 0, there is δ > 0 such that

x ≤ δ ⇒ V (x) < β

9.2. PROPOSED LYAPUNOV FUNCTION

55

Then, B δ ⊂ Ωβ ⊂ B r and x(0) ∈ Bδ ⇒ x(0) ∈ Ωβ ⇒ x(t) ∈ Ωβ ⇒ x(t) ∈ Br Therefore,

x(0) < δ ⇒ x(t) < r ≤ ,

∀t ≥ 0

which shows that the equilibrium point is stable. For asymptotic stability, see [32, p. 116]. The proof does not only prove stability, but also shows that a solution is bounded if a surface V (x) = c can be found for which V˙ (x) ≤ 0 holds.

9.2

Proposed Lyapunov function

An important challenge of the Lyapunov stability proof is to find a proper Lyapunov function V which is defined in (9.4). In order to introduce a Lyapunov function for the helicopter blade step by step, a cantilevered beam model with zero boundary conditions is considered instead of a rotating helicopter blade. This model has a single equilibrium at q = 0. By choosing the physical energy of the cantilevered beam consisting of kinetic and potential energy T and U as a Lyapunov function V , (9.4) is fulfilled. V =T +U (9.7) Inspired by (3.14), the time derivative of the Lyapunov function can be calculated twofold V˙ = T˙ + U˙

(9.8)

= P bou + P act

whereas P ext is set to zero, as for stability proofs of equilibrium points, the behavior of the system is observed without disturbances. Additionally, it is assumed that P act is exactly known. In the case of the helicopter blade, the reduction of the beam energy is misleading, as the helicopter blade has to have the kinetic energy of the rotating blades to stay in the air. The difference to the cantilevered beam is that the helicopter blade has a nonzero equilibrium q 6= 0. This problem, however can be adapted to the one of the cantilevered beam by shifting the system variables in a way that its new equilibrium is zero. q = q − q0 (9.9) where q is the shifted state space vector and q 0 is the equilibrium of the non-shifted system. The Lyapunov function of the shifted system is then chosen to be V = T∗ + U∗

(9.10)

whereas T ∗ and V ∗ are no physical energies anymore. They are calculated by substituting q by q instead of q + q 0 in (9.7). The same substitution is done for (9.8) V˙ = T˙∗ + U˙∗ ∗

= P bou + P act

∗

(9.11)

The Lyapunov function V and its time derivative for the helicopter blades are basically the same than for the cantilevered beam, except that they use the shifted state vector q. Equation (9.8) holds for any q and so for q, too. However, it should be considered that the Lyapunov function of the helicopter does not allow a physical interpretation anymore.

56

CHAPTER 9. STABILITY OF NONLINEAR SYSTEMS

Chapter 10

Linear Optimal Control Before designing a nonlinear controller, the performance of a LQG controller is tested. LQG design can only be used for linear systems so that the helicopter blade is linearized for the design process. Later, for testing, the controller is implemented in the environment of the nonlinear helicopter blade. The dynamic response of linear controlled plants is determined by pole placement. It is more restricted for lower numbers of available sensor signals. Free pole placement is given for state feedback controlled systems for which many precalculated solutions, e.g. ITAE (Integral Time Multiplied Absolute Error) are available. Placing the poles by this criterium leads to the minimization of the following function Z ∞

J IT AE = t · e(t) dt → min (10.1) 0

where e(t) is the control error of the state variables. The function J IT AE is a cost function which has to be minimized with techniques provided by optimization techniques. Various controllers for linear systems that are calculated by optimizing a cost function have been developed whereas the most promising is the least quadratic regulator (LQR) which is presented in the next section. As the helicopter blade does not provide enough sensor signals for state feedback, a Kalman observer estimating the state vector is designed. The combination of a least squares regulator and a Kalman observer is the previously mentioned LQG regulator.

10.1

LQR Controller

The cost function determining the least square regulator of the linearized helicopter blade model 0 (uA = 0, q0 = 0 and q = q − q 0 ) q˙ = Bq + EuA y = Mq is Z J=

∞

h

(10.2)

i T qT Qq + uA RuA dt → min

(10.3)

0

where Q ≥ 0 and R > 0 are weighting matrices. Q is weighting the error of the controlled system as q = 0 is desired and R is weighting the control effort. Dependant on the control objective, one can chose a high control effort to gain a good control quality or reduce the energy necessary for control leading to a bigger control error. Without loss of generality R is chosen to be R = I for the helicopter blade and Q is chosen to be

Q=

1 α 2

Z 0

L

 V T Φ  ΦΩ   γ Φ  Φκ



G KT 

K I

0 0

0 0

57

0 0

U VT

 V  0 Φ  Ω 0   Φ γ  dx V Φ  W Φκ

(10.4)

58

CHAPTER 10. LINEAR OPTIMAL CONTROL

where the comparison functions are written in matrix notation. After inserting (10.4) in qT Qq of (10.3) one receives  T    G K 0 0 V Z L V T       1 Ω K I 0 0      Ω  dx qT Qq ≈ α (10.5) 2 0 γ   0 0 U V γ  T κ 0 0 V W κ Using (3.10), (10.5) can be written as qT Qq ≈ α (T ∗ + V ∗ )

(10.6)

T ∗ and V ∗ denote the pseudo kinetic and potential energy for a beam with non-zero steady state solution, see (9.10). If the steady state solution is zero, T ∗ and U ∗ are the physical kinetic and potential energy T and U . (10.3) can be written with (10.6) neglecting the approximation error as Z ∞h i T (10.7) α (T ∗ + U ∗ ) + uA uA dt → min J= 0

The compromise between the minimization of T ∗ , U ∗ and the minimization of uA can be adjusted by the single parameter α. The resulting state feedback controller of the minimization of (10.7) is uA u = −Kui qi

(10.8)

K = ET P+

(10.9)

where P+ is the positive definite solution of the Ricatti equation −P B − BT P + P EET P = Q

(10.10)

The derivation33 of the Ricatti equation can be found in various books about linear optimization or linear optimal control.

10.2

Kalman Observer

A Kalman observer33 is designed to estimate the state space vector q. The quality of the estimation is limited due to process and measurement noise v and w. These noises enter the linearized model (10.2) in the following way q˙ = Bq + EuA + Nv y = Mq + w

(10.11)

N= G H f v= m

(10.12)

where

The matrices G and H, as well as f and m are taken from the state space formulation in (7.20) or (7.22). Similar to the LQR design, the Kalman filter is the solution of the optimization of a cost function J. h i 2 J = E e → min (10.13) where E[] is the expectation and e = q − q ˆ. q ˆ is the estimated state space vector. The optimization of the estimation error in (10.13) is calculated under restrictions of the process and measurement noise. v(t) and w(t) are modelled as continuous stochastic values with the properties gaussian, independent, stationary and zero-mean(E[v(t)] = 0, E[w(t)] = 0). This noise is also known as white noise. The covariance intensity matrices V and W are defined as E[v(t)v(τ )T ] = V δ(t − τ ), E[w(t)w(τ )T ] = W δ(t − τ ),

V ≥0 W >0

(10.14)

10.3. LQG CONTROLLER

59

Due to the independence of v(t) and w(t), the matrices V and W are diagonal. As there are no test results available, the values of the process and the measurement noise have to be estimated. The process noise is estimated by assuming that the helicopter blade is mainly perturbed by f2 (x, t) and f3 (x, t). f1 (x, t) and m(x, t) are neglected and set to zero. Here, the indices of f denote the direction of the external force. The disturbances dependent on space and time are generated using Legendre functions, see (3.29) and (3.31). Higher order Legendre functions are expected to have a lower impact on the perturbation than lower order comparison functions. The covariance matrix V f of the first three elements of f (t) is estimated to be   0 0 0 V f = 0 2 0 (10.15) 0 0 5 The complete covariance matrix V is then composed by V f considering that the process noise is mainly composed by the first Legendre functions.   f V ··· 0 0 f · · · 1  0 0 ··· · · · 0 2 V    .. ..  . ..  . .   1 nc f  (10.16) V = 0 V 0 · · · 0 ··· 0  2   0 · · · · · · 0 · · · 0    . ..  ..  .. . . 0 ··· ··· 0 The covariance matrix W is estimated using the stationary system response y in (10.11) of the white noise input v(t) determined by V . The covariance matrix of y is denoted by W f ull . The covariance matrix W is then calculated by picking the diagonal elements of W f ull resulting in W diag and multiplying W diag by a factor β which tunes the sensor noise intensity. W = βW diag

(10.17)

With the covariance matrices V and W , the solution of (10.13) is a Ricatti equation that is similar to (10.10). The observer q ˆ˙ = Bˆ q + EuA + L(y − Mˆ q) (10.18) with the observer gain L L = Π+ C T W −1

(10.19)

where Π+ is the positive definite solution of the Ricatti equation −BΠ − ΠBT + ΠMT W −1 MΠ = NV NT

(10.20)

The derivation33 of the Ricatti equation can be found in various books about linear optimization or linear optimal control.

10.3

LQG Controller

The LQG controller is a combination of a LQR controller and a Kalman observer. Both, the LQR controller and the Kalman observer have been designed for the linearized helicopter blade in the previous sections. In this section, the stability of the LQG controlled nonlinear beam model (figure 10.1) is investigated. The equations of the LQG controlled reduced nonlinear beam model are (7.22)

A q˙ k = Bki qi + Ckij qi qj + Eku uA u + Fkiu qi uu + Gki fi + Hki mi

q ˆ˙ k

(10.18)

=

(10.8)

Bki q î + Eku uA î ) u + Lky (yy − Myi q

uA = −Kui q î u

(10.21)

60


LQG Controller

Nonlinear Plant

Kalman Observer

LQR Controller

Figure 10.1: LQG controlled nonlinear beam model

After introducing the estimation error ei = qi − q î

(10.22)

the undisturbed response of q and e can be written using (10.21) as q˙ k = (Bki − Eku Kui ) qi + (Ckij − Fkiu Kuj ) qi qj + Eku Kui ei + Fkiu Kuj qi ej e˙ k = (Bki − Lky Myi ) ei + (Ckij − Fkiu Kuj ) qi qj + Fkiu Kuj qi ej

(10.23)

After writing (10.23) in matrix notation q˙ k (Bki − Eku Kui ) Eku Kui qi I (Ckij − Fkiu Kuj ) qi qj + Fkiu Kuj qi ej = + e˙ k 0 (Bki − Lky Myi ) ei I (10.24) the Lyapunov function T 1 qk qk (10.25) V = e ek 2 k is defined. Its derivative is T q q˙ k ˙ V = k ek e˙ k T (10.23) qk q = A i + (qk + ek ) (Ckij − Fkiu Kuj ) qi qj + Fkiu Kuj qi ej ek ei where A=

(Bki − Eku Kui ) 0

Eku Kui (Bki − Lky Myi )

(10.26)

(10.27)

A is negative definite because its eigenvalues are all negative which consist of the eigenvalues of the LQR controlled linearized beam and the Kalman observed linearized beam. Both, the eigenvalues of the LQR controlled and the Kalman observed linearized beam are negative.33 det (λI − A) =

det (λI − (Bki − Eku Kui )) {z } |

eigenvalues of the LQR controlled linearized system

·

det (λI − (Bki − Lky Myi )) = 0 | {z }

eigenvalues of the Kalman observer

(10.28) As A is negative definite and the nonlinear terms in (10.26) are of third order, V˙ in (10.26) is negative in a certain area around the equilibrium. The area around the equilibrium in which V˙ < 0 is also called region of attraction.32 However, in this work, a region of attraction could not be specified, only its existence could be shown by (10.26) and (10.28)

10.4

Simulation Results

In this section, the simulation results of an LQG controlled nonlinear helicopter blade are presented. The blade model used for the simulation has been reduced from 20 comparison functions to 8 natural


61

modes. The process noise is chosen as suggested in (10.15), (10.16) and for the sensor noise in (10.17) a β value of 10−5 has been chosen. The α value in (10.4) for the LQR controller has been set to 108 . Furthermore, the actuation voltages have been limited to 1500V. The simulation time is 2 seconds and the controller is switched on after 1 second to observe the changed dynamics, see figure 10.2 and figure 10.3. Especially impressive is the amount of pseudo energy reduction (9.10) that could be achieved. To get an idea of the process noise, the sensor noise and the voltages, exemplary values are plotted in figure 10.4. 180

Blade energy T*+U*

160 140 120 100 80 60 40 20 0 0

0.5

1 Time

1.5

Figure 10.2: Pseudo Energy T ∗ + U ∗

2

62


2

0.04

γ1

0.03

γ2

κ2

1

γ3

κ values at root

γ values at root

0.02

κ1

1.5

0.01 0 −0.01

κ3

0.5 0 −0.5

−0.02

−1

−0.03

−1.5

−0.04 0

0.5

1 Time

1.5

−2 0

2

0.5

(a) Strain at the root

1.5

2

(b) Curvature at the root

40

80 V1

20

V2

60

V3

40

Ω values at tip

30

10 0 −10

Ω1 Ω2 Ω3

20 0 −20

−20

−40

−30 −40 0

0.5

1 Time

1.5

−60 0

2

0.5

(c) Linear velocity at the tip

1 Time

1.5

2

(d) Angular velocity at the tip

Figure 10.3: Values of γ(0), κ(0), V (L) and Ω(L)

−5

800

1000

600

0 −500

x 10

0.5

400 200 0 −200

−1000 −1500 0

1

Sensor Noise

Process Noise

1500

500 Voltages

V values at tip

1 Time

0

−0.5

−400 0.5

1 Time

(a) Voltage uA 1

1.5

2

−600 0

0.5

1 Time

1.5

(b) Exemplary process noise

2

−1 0

0.5

1 Time

1.5

(c) Exemplary sensor noise

Figure 10.4: Exemplary values of voltages, process and sensor noise

2

Chapter 11

Adaptive Damping Although the linear optimal controller is the perfect solution for small disturbances, stability can not be ensured. Therefore, a nonlinear controller is presented in this chapter that is working as a virtual damper with adaptive damping coefficient.

11.1

Basic Idea

The damping coefficient is determined in a way that the virtual damper is dissipating more energy ∗ ∗ ∗ ∗ P act than energy P bou is brought into the system by boundary conditions. P act and P bou are the pseudo actuation and boundary power using the shifted state space vector q, see (9.11). ∗

P act ≤ −P bou

∗

(11.1)

In the case of a discrete model, the actuation power generated by the virtual dampers is T

∗

T

P act = F A V + M A Ω

(11.2)

where V and Ω are the vectors of linear and angular velocities. The actuation force and moment vectors F A and M A of the virtual dampers are ∗

F A = −cF (P bou , V )V

(11.3)

∗

M A = −cM (P bou , Ω)Ω ∗

∗

with the adaptive damping coefficients cF (P bou , V ) and cM (P bou , Ω). These coefficients are chosen in a way to fulfill (11.1). ∗ ∗ ∗ cF (P bou , V ) = α + βH(P bou + δ) P bou (V T V )−1 (11.4) ∗ ∗ ∗ cM (P bou , Ω) = α + βH(P bou + δ) P bou (ΩT Ω)−1 where H(x) is the Heaviside function ( 1, H(x) = 0,

x≥0 x 0, β > 1 and δ > 0. Higher values for α, β and δ make the controller more robust and increase the energy consumption of the actuator on the other hand, too. A greater α leads to more energy dissipation due to the movement of the structure. By increasing β, one can compensate modelling errors of the boundary power that concern its value. Another important 63

64

CHAPTER 11. ADAPTIVE DAMPING

property of the boundary power, the sign change can be made more robust by increasing δ. Finally, the singularities V = 0 and Ω = 0 in (11.6) have to be discussed. High values of F A and M A near these singularities are demanded if P bou 6= 0 to hold (11.1). Due to limitations of the actuation, (11.1) can not be fulfilled in an area around the singularities. However, the controlled system will be closely bounded to the equilibrium, but there is no asymptotic stability guaranteed within that bound.

11.2

Instable Jeffcott Rotor

The basic idea of the adaptive virtual damper compensating the energy brought into the system by its boundary conditions has been presented in the previous section. Before this idea is applied to the helicopter blade, it is shown that an adaptive virtual damper is not necessary if the stability margin is known. This is shown exemplary with a Jeffcott rotor that is regarded as the simplest model to study the flexural behavior of rotors consisting of a point mass that is attached to a massless shaft.34 A sketch of the rotor is given in figure 11.1. Without derivation, the equations of the damped Jeffcott rotor are taken from [34, eq 2.69] A Fx m 0 x ¨ cn + cr 0 x˙ k Ωcr x mΩ2 cos Ωt + + = + (11.7) FyA 0 m y¨ 0 cn + cr y˙ −Ωcr k y mΩ2 sin Ωt where m is the point mass, k the stiffness parameter, the eccentricity and Ω the angular velocity. point mass

FAy FAx ε y massless shaft

x

Ω, M

Figure 11.1: Sketch of the Jeffcott rotor Fx and Fy are forces that can be applied to the point mass m. cn and cr are the non-rotating and rotating damping coefficients. The ratio between these damping coefficients is determining the angular velocity Ω for which the system is still stable (see [34, eq 2.83]). r k cn Ω< 1+ (11.8) m cr The Jeffcott rotor is controlled by the actuation forces Fx and Fy so that the actuation power is determined by A T Fx x˙ act P = (11.9) FyA y˙ The actuation forces FxA and FyA are chosen as A Fx x˙ = −α FyA y˙

(11.10)

with the constant damping coefficient α. Equation (11.10) is inserted into (11.7) m 0 x ¨ c + α + cr 0 x˙ k Ωcr x mΩ2 cos Ωt + n + = (11.11) 0 m y¨ 0 cn + α + cr y˙ −Ωcr k y mΩ2 sin Ωt

11.3. HELICOPTER BLADE

65

which results in the new non-rotational damping c∗n = cn + α and α has to be chosen in a way that (11.8) holds to receive a stable system showing that an adaptive damping coefficient is not necessary. In the next section, the energy based controller is tested on the helicopter blade, where the stability margins are not known.

11.3

Helicopter Blade

Unlike the Jeffcott Rotor, there is no constant damping coefficient known to asymptotically stabilize the helicopter blade so that an adaptive one is designed. The virtual damper of a discrete system presented in (11.2) has to be adapted for the continuous helicopter blade. First, the different ∗ actuation power P act is taken from (3.13) ∗

P act =

Ln

Z

o T T F A γ˙ + M A κ˙ dx

(11.12)

0 ∗

and the boundary power P bou of the helicopter blade is determined by P bou = M3 (0) · Ω03 ∗

(11.13)

The actuation force F A and moment M A are then calculated similarly to (11.3) ∗

F A = −cF (P bou , γ) ˙ γ˙

(11.14)

∗

M A = −cM (P bou , κ) ˙ κ˙ where γ˙ and κ˙ are continuous values and the damping coefficients are calculated as ∗ ∗ ∗ cF (P bou , γ) ˙ = α + βH(P bou + δ) P bou (γ˙ T γ) ˙ −1 ∗ ∗ ∗ cM (P bou , κ) ˙ = α + βH(P bou + δ) P bou (κ˙ T κ) ˙ −1

(11.15)

Problems arise for the realization of (11.14) because the demanded continuous values F A and M A cannot be provided by the piezo fibers in the blade, see chapter 7.2.3. This problem is addressed in two different ways presented in the following. One methodology calculates the constant actuation forces and moments for each section separately. The other one uses a Galerkin approach to determine the actuation voltages.

11.3.1

Segment Approximation

The segment approximation calculates the controller output for each blade segment separately. Instead of using the continuous values of γ and κ, the integrals of each segment are used to determine the segment voltages. This is shown for the first segment. 1 6

R

Z

R

Z

γ(x) ˙ → γ˙ = κ(x) ˙ → κ˙ =

0 0

1 6

L

γ(x) ˙ dx,

1 x ∈ [0, L] 6

κ(x) ˙ dx,

1 x ∈ [0, L] 6

L

(11.16)

The actuation force and moment for the first section become then ∗

R

F A = −cF (P bou , γ˙ )γ˙ ∗

R

R

M A = −cM (P bou , κ˙ )κ˙

R

(11.17)

with the damping coefficients R RT R ∗ ∗ ∗ cF (P bou , γ˙ ) = α + βH(P bou + δ) P bou (γ˙ γ˙ )−1 R RT R ∗ ∗ ∗ cM (P bou , κ˙ ) = α + βH(P bou + δ) P bou (κ˙ κ˙ )−1

(11.18)

66


Although γ˙ and κ˙ have been substituted as shown in (11.16), the stability condition (11.1) is still ∗ holding. To prove this, the actuation power P act is calculated without loss of generality for the first beam segment. act ∗ (11.12)

P

=

Z =

0

1 6

L

1 6

Z

Ln

o (11.17),(11.18) T T F A γ˙ + M A κ˙ dx =

0

( −αγ˙

RT

γ˙ − ακ˙

RT

κ˙ − βH(P

) R T R −1 R T R T R −1 R T ∗ + δ) P bou γ˙ γ˙ γ˙ γ˙ + κ˙ κ˙ κ˙ κ˙ dx

bou ∗

R R R T R −1 R T R RT R RT R −1 R T R T ∗ ∗ γ˙ γ˙ + κ˙ κ˙ κ˙ κ˙ = − αγ˙ γ˙ − ακ˙ κ˙ − βH(P bou + δ) P bou γ˙ γ˙ RT R RT R ∗ ∗ = − α γ˙ γ˙ + κ˙ κ˙ − 2βH(P bou + δ) P bou (11.19) If the controller parameters are chosen as α > 0, β > 1 and δ > 0, the following inequalities can be stated. ∗ ∗ ∗ ∗ P act < −βH(P bou + δ) P bou < P bou (11.20) which proofs that the segment approximation leads to an asymptotically stable P6 controller if only the first segment is used. If all segments are switched on, it is sufficient if seg=1 βseg > 1, where each β of each segment is summarized. The controller output uA of the first blade segment is then obtained using (2.12). T !−1 T

A

A B

u =

R

In the following subsections, γ˙ and κ˙ feedback control including an observer.

R

A B

A B

FA MA

(11.21)

are calculated in the case of state feedback and output

State Feedback If a highly reduced helicopter blade model is used to design the adaptive damping controller, it can be realized as a state feedback controller measuring all states qi . In that case, γ˙ and κ˙ are obtained using the intrinsic equations (2.1) instead of differentiating γ and κ as the differentiation of measurement values is problematic due to sensor noise. R

γ˙ =

Z

1 6

L

γ(x) ˙ dx

0

Z

1 6

L h

Z

1 6

L

(11.16)

=

1 6

Z 0

L

˜+κ V 0 + (k ˜ )V + (˜ e1 + γ˜ )Ω dx

i h i 0 ˜ΦV + e˜1 ΦΩ qi + Φ ˜ κ ΦV + Φ ˜ γ ΦΩ ΦVi + k i i i j i j qi qj dx 0 h i R h i R h i R (11.23) γ Ω κ V ˜ΨV + e˜1 ΨΩ s (0)qi + Ψ ˜ ˜ = ΨVi d si (0)qi + k Ψ + Ψ Ψ i i i j i i j dij (0)qi qj (3.20)

=

(11.22) R

κ˙ =

κ(x) ˙ dx

0

Z

1 6

L h

(11.16)

=

Z 0

1 6

L

˜+κ Ω0 + (k ˜ )Ω dx

i h i 0 ˜ Ω ˜κ Ω ΦΩ i + kΦi qi + Φi )Φj qi qj dx 0 h i R h i R h i R (11.23) d ˜ΨΩ s (0)qi + Ψ ˜ κ )ΨΩ d (0)qi qj = ΨΩ si (0)qi + k i i i j i ij (3.20)

=

11.3. HELICOPTER BLADE

67

and where Z Z (ξ+ 16 )L Pk (x) dx = L sk (ξ) = R

ξL

Pk (¯ x) d¯ x

ξ

Z (ξ+ 16 )L Z d 0 sk (ξ) = Pk (x) dx =

ξ+ 16

R

ξL

R

ξ+ 61

(7.11)

(7.11)

dki (ξ) =

d¯ x

Pk0 (¯ x)

(11.23)

ξ

Z Z (ξ+ 16 )L Pk (x)Pi (x) dx = L ξL

ξ+ 61

Pk (¯ x)Pi (¯ x) d¯ x

ξ

Output feedback with observer If the state space variables are obtained by an observer, γ˙ and κ˙ can be directly taken from the estimated values as they are not influenced by sensor noise. The use of an observer does not guarantee stability anymore. However, the region of attraction of the adaptive damping controller combined with an Kalman observer should be greater than the LQG design as the observer is the more robust element of the LQG design due to ’faster’ eigenvalues. R

Z

R

Z

γ˙ = κ˙ =

11.3.2

1 6

γ(x) ˙ dx

0 0

L

1 6

L

κ(x) ˙ dx

1 6

(11.16)

Z

(11.16)

Z

=

=

0 0

1 6

L L

1 6

(3.20) γˆ˙ (x) dx =

Z

(3.20) κ ˆ˙ (x) dx =

Z

0 0

1 6

L L

R

Φγi qˆ˙i dx

(11.23)

=

Ψγi sk (0)qˆ˙i

Φκi qˆ˙i

(11.23)

R Ψκi sk (0)qˆ˙i

(11.24) dx

=

Galerkin Approximation

The controller output uA can also be obtained by a Galerkin approach   T !−1 T A  Z L A A A F 0= Φvu uA dx u − Φvu B B B MA  0 

(11.25)

where u = v = 1 . . . 32 to receive a unique solution of uA . The disadvantage of this technique is that it cannot be proven that the resulting uA ensures (11.1). This is the reason why only the ’Segment Approximation’ instead of the ’Galerkin Approximation’ has been implemented in this work.

68


Chapter 12

Further Control Concepts In this section, three more interesting control concepts are presented that could not be investigated deeper because of a lack of time.

12.1

Gain Scheduling

The linear optimal controller found in chapter 10 has been designed for a small area around the steady state solution. It was also stated in chapter 10.3 that a region of attraction exists but that it could not be specified in this work. However, it is possible to enlarge this region of attraction by designing various controllers for different operation points and interpolating them. This technique is also called Gain Scheduling. [32, chap 12.5] In the case of the helicopter blade, one could design a simple gain scheduling controller for various rotational speed Ω3 , e.g. 0, 40 and 80 rad s−1 . Ω3 is also called scheduling variable. The state feedback controller gains are denoted by K 0 , K 40 and K 8 0 and the interpolation could be realized linearly. ( 1 − Ω403 K 0 + Ω403 K 40 , Ω3 ∈ [0, 40] 40 Ω3 −40 80 K= (12.1) −40 1 − Ω340 K + 40 K , Ω3 ∈ [40, 80] To improve the quality of the controller, one could choose smaller intervals of the scheduling variable and also more than one scheduling variable which would lead to a dramatic increase of necessary precalculated controllers. A possibility to overcome this is shown in [32, chap 12.5] where a general methodology is presented for an integral controller that is parameterized by the scheduling variable. This continuously gain scheduled controller is designed in a way that the eigenvalues of the linearized controlled system are constant for varying scheduling variable values.

12.2

Input-Output Linearization

The Input-Output Linearization [32, chap 13.2] is basically cancelling out nonlinearities by an adapted input value u so that the changed input-output behavior is the one of a linear system. Input-Output Linearization can be applied to systems of the form [32, eq 13.1] q˙k = fk (qi ) + gk (qi )uv yy = h(qi )

(12.2)

The discretized beam model (7.20) and (7.22) are of that form and in the case of the reduced state space model, the functions in (12.2) can be written as f (qi ) = Bki qi + Ckij qi qj g(qi ) = Eku + Fkiu qi Gki h(qi ) = Myi qi +

yy0

69

Hki

(12.3)

70

CHAPTER 12. FURTHER CONTROL CONCEPTS

and

 A uu  uv = fi   mi

(12.4)

The resulting input-output linearized system is γ

yy = vy

(12.5)

where v is the new input variable of the system and γ denotes the time derivation degree. The first step to receive the system presented in (12.5) is to derive the measurement equation with respect to time until the system input u occurs. The number of necessary time derivations is also known as the relative degree of the system. The beam model has a relative degree of one, as the time derivation of yy occurs after the first derivation. y˙ y = Myk q˙ i = Myk (Bki qi + Ckij qi qj ) + Myk = Oy (qi ) + Pyv (qi )uv

Eku + Fkiu qi

Gki

Hki

uv

(12.6)

Now, the input u is set as −1

uv = Pxv (qi ) (Pxv (qi )Pyv (qi ))

(−Oy (qi ) + vy )

(12.7)

and after inserting (12.7) into (12.5), one obtains the linear system y˙ y = vy

(12.8)

(12.8) can now be controlled with the concepts of linear control. However, there are some problems that arise in the case of the helicopter blade. One of them is that the actual state vector qi is not known so that it has to be estimated using an observer. By using estimated values, the nonlinearities are not cancelled out properly and stability can not be ensured if the controlled input output lin−1 earized model (12.8) is stable. Another problem is that the calculation of Pxv (qi ) (Pxv (qi )Pyv (qi )) needs a lot of computation. Finally, the concept of minimizing the pseudo kinetic and potential energy for the design of an LQR controller (chapter 10.1) can not be applied for the input-output linearized system.

12.3

Hybrid Control

Hybrid Controllers 35 consist of continuous and discrete time elements. In this work, the discrete time element would be a supervisor that switches between different continuous time controllers. This architecture allows better performance compared to a single feedback controller as the supervisor switches to the controllers that has the best performance in the actual domain. A possible hybrid controller for this work could switch between the previously presented linear optimal (chapter 10) and the energy based controller (chapter 11), see figure 12.1. The linear optimal controller is used if the state of the helicopter blade is close to the steady state solution. For big deformations, when the nonlinearities come into play, the blade is controlled by the energy based controller. The resulting controller would unify the good performance near the equilibrium of the linear optimal controller and the stability of the adaptive damping controller for big deformations.

12.3. HYBRID CONTROL

71

LQR Controller Nonlinear Plant Energy Based Controller

Supervisor

Kalman Observer

Figure 12.1: Hybrid controller for nonlinear beam model

72

CHAPTER 12. FURTHER CONTROL CONCEPTS

Chapter 13

Conclusion The simulation and control of the nonlinear beam model developed by Hodges4, 5 has been the focus of this work which is based on similar research done by Traugott et al.3 Here, a Galerkin approach has been chosen to discretize the beam model in contrast to the FEM discretization used by Traugott. Especially for beams without discontinuities, the Galerkin approach shows a faster convergence and better accuracy compared to the FEM solution, see table 7.2. Beams with discontinuities would have to be simulated by solving the continuous segments separately, which has not been investigated in this work. The developed simulation of the Galerkin discretized beam model is limited to beams with constant cross-section and statically determinate boundary conditions. It has an easy to use interface and short response times, e.g. the simulation is capable of building the nonlinear discretized beam model and providing the first natural modes and frequencies of the linearized model within seconds. Another focus in this work, that is also covered by the built simulation, is a nonlinear order reduction. The proposed order reduction by taking linear free vibration modes as comparison functions is easy to implement and provides the exact linear normal frequencies and modes used for the order reduction. The great advantage of the presented order reduction technique is that it can be applied to the unreduced model with a single transformation matrix T. The order reduction made it possible to execute a time marching simulation of the nonlinear model that is not far away from real time speed for highly reduced models. Besides the simulation of the uncontrolled blade, linear optimal controllers have been tested on the nonlinear reduced blade model showing great performance. However, a stability proof using Lyapunov stability could not be found for that controller type. To overcome this problem, adaptive damping was considered as a possible solution to the stability problem. However, due to time limitations, a working simulation of the adaptive damping controller could not be realized. Future work is seen in the calculation of nonlinear normal modes and frequencies that were introduced in this work. They would allow a better order reduction by replacing the linear normal modes used so far. The simulation could be extended to discontinuous beams with statically overdeterminate or indeterminate boundary conditions. Furthermore, aero-elastic coupling could be added to the beam model. The most promising control methodology is seen in hybrid control which could combine the great performance of the linear optimal controller and the stability of the adaptive damping controller as it is outlined in this work.

73

74

CHAPTER 13. CONCLUSION

Appendix A

Tensor Fundamentals As the thesis deals with equations that have elements of higher dimensional space than two, the fundamentals of tensors36 are introduced. Tensors are a powerful tool that describe objects in higher dimensional spaces than matrices. Tensors are generalizations of scalars (zero dimensional tensor), vectors (one dimensional tensor), and matrices (two dimensional tensor).

A.1

Notation and Summation Convention

The elements of a tensor are specified with indices like the elements of a matrix. As the tensors that are used in this thesis are defined in an Cartesian coordinate system, it is sufficient to write the indices subscripted. Tensors can also be used in arbitrary coordinate systems, but then it is necessary to use contravariant (index superscripted) and covariant (index subscripted) tensors. The displacement tensor (vector) of a point in a three dimensional cartesian coordinate system is written in tensor notation as xi where i = 1, 2, 3 instead of x = {x y z}T in matrix notation. The reason for the index notation is that it is easier to add an additional dimension. For this dimension expansion, only an additional index has to be added, e.g. the displacement tensor xi becomes a matrix xij after adding a new index. The compactness of the tensor notation is shown on the example of the equations for the flow of an inviscid fluid [36, p. 4]. ∂u ∂u ∂u ∂u 1 ∂p +u +v +w =− ∂t ∂x ∂y ∂z ρ ∂x ∂v ∂v ∂v 1 ∂p ∂v +u +v +w =− ∂t ∂x ∂y ∂z ρ ∂y ∂w ∂w ∂w ∂w 1 ∂p +u +v +w =− ∂t ∂x ∂y ∂z ρ ∂z

(A.1)

In tensor notation, these equations can be written in one equation. ∂ui ∂ui ∂ui ∂ui 1 ∂p + u1 + u2 + u3 =− ∂t ∂x1 ∂x2 ∂x3 ρ ∂xi

(A.2)

Using the summation sign, the above equation can be written more compact. 3

∂ui X ∂ui 1 ∂p + uj =− ∂t ∂xj ρ ∂xi j=1

(A.3)

With the convention from Einstein, that repeated suffixes in a multiplied term have to be summed without writing the summation symbol the equation is finally written as ∂ui ∂ui 1 ∂p + uj =− ∂t ∂xj ρ ∂xi The Einstein convention is also known as the Summation convention. 75

(A.4)

76

A.2

APPENDIX A. TENSOR FUNDAMENTALS

Definition of a Tensor

A tensor quantity is usually defined in terms of geometrical transformations between coordinate axes. The transformation of a vector A from the original coordinate system to another is A¯m = lmi Ai

(A.5)

lmi = cosθmi

(A.6)

with where θij denotes the angle between the i axis of the original frame and the m axis of the new frame as shown in figure A.1. A tensor A is then defined as a multi-directional quantity which transforms

x2

x2

x1

θ11

x1

Figure A.1: Coordinate transformation

from one set of coordinate axes to another according to the following rule. [36, p. 5] A¯m = lmi Ai ¯ Amn = lmi lnj Aij ¯ Amnp = lmi lnj lpk Aijk

tensor of order one tensor of order two tensor of order three

(A.7)

and similarly for higher orders.

A.3

Tensor Algebra

In this chapter the Tensor Algebra according [36, p. 6] is introduced. The first operation presented is the addition of two tensors of same order. Aij + Bij = Cij

(A.8)

The correctness of that operation is proven by showing that the sum of Aij and Bij is a tensor as it transforms as one. ¯ij C¯ij = Cij lmi lnj = (Aij + Bij )lmi lnj = Aij lmi lnj + Bij lmi lnj = A¯ij + B

(A.9)

The multiplication of two tensors of order p and q yields to a third tensor of order p + q. Aij Bklm = Cijklm

(A.10)

Again, it is shown that the product is a tensor because of its transformation behavior. ¯qrs C¯ijklm = Cijklm lni lpj lqk lrl lsm = Aij lni lpj Bklm lqk lrl lsm = A¯np B

(A.11)

This product is known as outer multiplication. A tensor of order p with two identical suffixes becomes a new tensor of order p − 2, e.g. the tensor Aijk transforms to Aiik = A11k + A22k with i = 1, 2 using the Einstein summation. This process is also called contraction. The combination of outer multiplication and contraction is called inner multiplication. An example for an inner multiplication is Aijk Bij = Ck .

A.4. TENSOR CALCULUS

A.4

77

Tensor Calculus

The tensor calculus found in [36, p. 8] deals with the differentiation and integration of tensors. Partial differentiation of a tensor of order p with respect to the coordinates xj yields to a tensor of order p + 1. For example ∂ ∂ A¯mnp ∂Aijk lmi lnj lpk lql = (Aijk lmi lnj lpk lql ) = ∂xl ∂xl ∂x ¯q

(A.12)

Thus, the differentiation of A transforms under the tensor rule and the order is increased by one as four direction cosines are necessary for the transformation. Integration of a tensor with respect to the coordinate direction yields to a tensor of one order higher unless the integration is combined with a contraction. For example Z Z Z Aij lmi lnj dxk lpk = A¯mn d¯ Aij dxk lmi lnj lpk = xp (A.13)

78

APPENDIX A. TENSOR FUNDAMENTALS

Appendix B

Alternatively Discretized Beam Model As discussed in chapter 3.3.2, there are two favored substitutions to include the constitutive equations (2.11) into the intrinsic equations (2.1) in order to apply the beam discretization via Galerkin approach. Here, the missing substitution scenario P , H ↔ V , Ω and γ, κ ↔ F , M is applied to (2.1) using the constitutive equations of the passive beam (2.2).

0=

] TF + T ˜ GV + KΩ) − (S g GV˙ + KΩ˙ − F 0 − k˜F + Ω( M )F − f

0=

KT V˙ + IΩ˙ − M 0 − k˜M − κ˜ A M − e˜1 F ] TF + T ˜ KT V + IΩ) + V˜ (GV + KΩ) − (S g + Ω( M )M − (Rf F + Sg M )F − m

0=

TF + T g RF˙ + SM˙ − V 0 − k˜V − κ˜ A V − e˜1 Ω − (S] M )V − (Rf F + Sg M )Ω

0=

TF + T g ST F˙ + TM˙ − Ω0 − k˜Ω − (S] M )Ω

(B.1)

The same Galerkin approach pointed out in (3.6) and performed in (3.24) is applied to (B.1). Note that F A = 0, M A = 0 and like in (3.24), without loss of generality, the beam has restricted generalized forces at the tip and generalized velocities at the root. Z L 0=

0

i 0 VT ˜ F −ΦF i − kΦi qi − Φk f h i T V Ω F ^ ] M T F ˜Ω +ΦVk Φ i (GΦj + KΦj ) − (S Φi + TΦi )Φj qi qj ΦVk

T

h

i

+ΦΩ k

T

h

Ω KT ΦVi + IΦΩ i q˙i + Φk

+ΦΩ k

T

h

i T V Ω V Ω M F ^ ] ] ] F M M T F ˜Ω ˜V Φ i (K Φj + IΦj ) + Φi (GΦj + KΦj ) − (S Φi + TΦi )Φj − (RΦi + SΦi )Φj qi qj

V GΦVi + KΦΩ i q˙i + Φk

i

T

h

T

h

i 0 ˜ΦM − e˜1 ΦF qi − ΦΩ T m −ΦM − k i i i k

i 0 ˜ΦV − e˜1 ΦΩ qi −ΦVi − k i i h i T V Ω ] ] ] F M T ΦF + T −(S^ ΦM +ΦF k i i )Φj − (RΦi + SΦi )Φj qi qj +ΦF k

T

h

i

RΦFi + SΦM q˙i + ΦF i k

T

h

79

80

APPENDIX B. ALTERNATIVELY DISCRETIZED BEAM MODEL +ΦM k

T

h

i

ST ΦFi + TΦM q˙i + ΦM i k

T

h

i h i 0 Ω MT ] T ΦF + T ˜ Ω ΦM −ΦΩ −(S^ i − kΦi qi + Φk i )Φj qi qj i

dx (B.2) h i h i T V FT 0 − ΦM T (0) ΦΩ (0) q + ΦM T (0)Ω0 −ΦF ( 0 ) Φ ( 0 ) q + Φ ( 0 )V i i k i k k i k h i h i T VT F VT L ΩT M L +Φk (L) Φi (L) qi − Φk (L)F + Φk (L) Φi (L) qi − ΦΩ k (L)M F M In contrast to (3.24), all equations in (B.2) are solved independently, as only ΦVk , ΦΩ k , Φk or Φk can be nonzero for a given k, see (3.20). Finally, (B.2) can be written in Einstein notation.

Aki q˙i + Bki qi + Ckij qi qj + Dk + fk + mk = 0

(B.3)

where no BC BC Bki = Bki + Bki

(B.4)

Dk = DkBC The tensors in (B.3) are obtained by comparing (B.3) with (B.2). Z L h i h i T ΩT KT ΦVi + IΦΩ Aki = ΦVk GΦVi + KΦΩ i + Φk i 0 h i h i T F M MT T F M +ΦF R Φ + S Φ + Φ S Φ + T Φ dx k i i k i i

no BC Bki

Z L =

0

Z L Ckij =

0

i h i 0 ˜ F + ΦΩ T −ΦM 0 − k ˜ΦM − e˜1 ΦF −ΦF i − kΦi k i i i h i h i FT V0 V Ω MT Ω0 Ω ˜ ˜ +Φk −Φi − kΦi − e˜1 Φi + Φk −Φi − kΦi dx ΦVk

T

h

ΦVk

T

h

T

h

F ] T ΦF + T ˜ Ω (GΦV + KΦΩ ) − (S^ Φ ΦM i j j i i )Φj

i

M F ] ] ] F M T ΦF + T ˜ Ω (KT ΦV + IΦΩ ) + Φ ˜ V (GΦV + KΦΩ ) − (S^ Φ ΦM i j j i j j i i )Φj − (RΦi + SΦi )Φj h i T ^ ] ] ] M )ΦV − (R F +S M )ΦΩ T ΦF + T +ΦF −( S Φ Φ Φ k j j i i i i h i T Ω ] T ΦF + T +ΦM −(S^ ΦM dx k i i )Φj

+ΦΩ k

(B.5) fk and mk are Z L fk =

0

T − ΦVk f dx (B.6)

Z L mk =

0

ΩT − Φk m dx

The tensors occurring from boundary conditions are h i h i T BC V MT Bki = −ΦF (0) ΦΩ k (0) Φi (0) − Φk i (0) h i h i T ΩT M +ΦVk (L) ΦF i (L) + Φk (L) Φi (L) DkBC =

(B.7)

0 M L ΦF (0)Ω0 − ΦVk (L)F L − ΦΩ k (L)M k (0)V + Φk T

T

T

T

For beams with constant cross-section, (B.5) - (B.7) can be simplified analog to chapter 3.5.

i

Appendix C

Software This software chapter is organized in two parts, the documentation and a section with implementation notes. In the documentation, the functionality and the basic structure of the developed software for the simulation and control design of active beams is presented. The implementation notes section introduces the implementation of graphical user interfaces in Matlab and shows the data structure of the software.

C.1

Documentation

The Simulations of the Helicopter blade and other beams for this work have been run in Matlab and Simulink. This chapter gives an overview of the functionality and the design of the developed software. Its architecture (figure C.1) can be broken down into three parts that are run in the following order. First, Legendre functions and integrals have to be precalculated and stored in a file (chapter C.1.1). In a second step, the beam model and controllers are build and analyzed (chapter C.1.2). Finally, the beam model and controllers are imported into the simulation environment Simulink for time marching simulations (chapter C.1.3). Precalculation

Building Model & Controller

Simulation

Legendre integrals and values

•Building and analyzing Galerkin discretized beam models

Time marching

•Designing controllers

Figure C.1: Software Architecture

C.1.1

Precalculation

Before any beam model can be build, the integrals in the beam model tensors, shown in (3.36) have to be calculated in Legendre.m. This function also provides the values of the Legendre functions at 100 equidistant reference points x ¯ = [0, 1] which are necessary to generate free vibration modes later. The Legendre function values and the integrals are stored in a file. The precalculation of the Legendre values and integrals has three advantages. One is that the software becomes much faster, because this data does not have to be calculated when the simulation is started. The second one follows from the fact that the calculation of the Legendre values and integrals is the only software module using the toolbox ’Symbolic Math’. Users who do not have this toolbox still can run the software using the precalculated file. The last advantage of the outsourcing of this program part using the ’Symbolic Math’ toolbox is, that an execution file of the main software can be build using the ’Matlab Compiler’ (which does not support the ’Symbolic Math’ toolbox). Execution files can be run without Matlab. 81

82

C.1.2

APPENDIX C. SOFTWARE

Building Model and Controller

The creation of beam models, controllers and its analysis is provided user friendly by the graphical user interface (GUI) BeamDyn.m. This GUI is organized in three fields ’Generate Model’, ’Analysis’ and ’Generate Controller’, see figure C.2. The menu bar of the GUI provides the menus ’save’, ’load’ and ’set’. Parameters, boundary conditions, external loads and simulation data can be saved and loaded while Legendre values and integrals can only be loaded. The menu ’Set’ is used to specify the beam model. In the following subsections, the functionality of the GUI is described in detail.

Figure C.2: GUI for the generation of the beam model and its controller

Specify Beam Model The beam model is specified using the menu ’Set’. This menu offers the items ’Cross-sectional Parameters’, ’Boundary Conditions’ and ’External Loads’. Choosing one of them opens an associated child GUI. These GUIs are shown in figure C.3. Parameter values of each child GUI are changed in edit boxes and read out using Read EditBox.m. This function takes advantage of the fact that the values of each edit box object are stored in its user data. If a user sets a non-numeric value in one of the edit boxes, the original value is restored. Parameter value changes are not performed if the GUI window is closed or the button ’Cancel’ pressed. The GUI ’Cross Sectional Parameters’ sets the values of the beam length L, the inertia and stiffness matrices G, K, I, U, V, W and the actuation matrices Eseg and Fseg . Instead of providing edit boxes, the actuation matrices are loaded from a file. This is done as the size of the actuation matrix depends on the actuation concept (e.g. 6 voltages per segment instead of 4) so that the number of necessary edit boxes is not known. Boundary conditions are set in the GUI ’Boundary Conditions’, where its type, value and position are set. The boundary conditions have to be chosen in a way that the beam is statically determinate. Furthermore, one can weight the boundary conditions, whereas a high weight increases the accuracy of the boundary conditions and lowers the accuracy of the beam dynamics simultaneously or the other way round for a low weight. 0

External loads f 0 , m0 that are constantly distributed over the beam length and voltages uA of the steady state solution are set in the GUI ’External Loads’. The purpose of this GUI is mainly

C.1. DOCUMENTATION

83

to evaluate the steady state solution. The voltages for the actuation are specified by the resulting steady state strains.

(a) Cross-sectional parameters

(b) Boundary Conditions

(c) External Loads

Figure C.3: Child GUIs setting up the beam model

Create Beam Model The beam model is created pressing the button ’Create System’ which is executing Create System.m performing the following steps. First, the tensors of the discretized beam model are generated. Next, the steady state solution is calculated and the beam model is linearized. Finally, the eigenvalues and -vectors of the linearized model are calculated (see figure C.4). The tensors of the discretized beam Create Beam Model: Create_System.m

Beam Model Tensors: Tensors.m Boundary Condition Tensors: BC.m

Steady State Solution: Steady_ State.m

Linearization: Linearize_ System.m

Eigenvalues & -vectors

Figure C.4: Steps performed building the model model (3.38) - (3.41) are obtained by the function Tensors.m. The tensors specified by boundary BC BC conditions Bki , DkBC and Ekm are obtained from BC.m. The calculation of the steady state solution 0 q (see chapter 4.1.1) is done executing Steady State.m and for the creation of the linearized beam model (4.15), Linearize System.m is called. Finally, the eigenvalue and -vector of the linearized system are calculated in Create System.m. The implementation of the beam model is very close to the theory presented in this work. As the tensors in (3.38) - (3.41) have to be stored during the simulation, huge memory space is required if the number of comparison functions exceeds 10, e.g the C tensor would have 2403 = 13.8 · 106 elements if 20 comparison functions would be used. As many elements of the beam model tensors are zero, sparse matrices are used which only memorize the non-zero elements reducing the amount of required memory space. The tensors C and F are stored as cell arrays of spares matrices. If an updated beam model has to be calculated where only the boundary conditions or the external loads have been changed, the calculation time can be reduced by activating the radiobutton

84


BC BC ’Change BC only’ so that only the tensors Bki , DkBC and Ekm are updated in Tensors.m.

Create Reduced Model The reduced beam model is generated pressing the button ’Create Reduced System’ executing the functions Transformation Matrix.m and Create ReducedSystem.m, see figure C.5. First, Transformation Matrix.m is called providing the transformation matrix T (see (5.19)) which is necessary for the order reduction. It consists of free vibration and perturbation modes that are calculated by the internal functions Freevibration Modes and Perturbation Modes. The internal function Perturbation q0andQ provides the steady state solution q 0 and the load Q of the perturbed system as it is shown in (5.16) and (5.19). The values q 0 and Q are obtained executing the function Create System.m under the boundary conditions and external loads of the specified perturbation (Ω3 = 72 rad s−1 , f30 = 500 N m−1 ). The transformation matrix T is then used by Reduce System.m to reduce the system as presented in (5.12). Afterwards, the same steps than in Create System.m (figure C.4) are executed: Steady state determination, linearization and eigenvalue and -vector calculation. Build Transformation Matrix T: Transformation_Matrix

Free Vibration Modes: Freevibration_Modes

Perturbation Modes: Perturbation_Modes Steady State and Load of Perturbation: Perturbation_q0andQ

Sort Eigenvectors: Sorted_Eigenvectors

Create System under Boundary Conditions of the Perturbed System: Create_System

Create reduced Beam Model: Create_ReducedSystem

Reduce Beam Model Tensors: Reduce_System

Steady State Solution: Steady_State

Linearization: Linearize_ System

Eigenvalues & -vectors

Figure C.5: Steps performed building the reduced model

Analysis The full and reduced beam model can be analyzed by displaying the steady state solution, natural frequencies, their convergence (for different numbers of comparison functions) and free vibration modes. Additionally, natural frequencies of the full and reduced model are compared under a varying beam parameter. The steady state solution is shown when ’Steady State’ is pressed which activates Show SteadyState.m. As the steady state solution is the same for the full and reduced model, see (5.4), the variables V 0 , Ω0 , γ 0 , κ0 , F 0 and M 0 of the steady state solution are calculated according (3.20) with q = q 0 , where q 0 is the steady state solution of the full model and the comparison functions Φl are provided by the function Variable Matrices.m. The steady state deformation of the beam is displayed by Show Deformation.m. The button ’Frequencies’ displays the natural frequencies of the full or reduced system by call-

C.1. DOCUMENTATION

85

ing Show Frequencies.m. The sorted frequencies are provided by Sorted Eigenvectors.m which sorts eigenvectors in a way that the ones with the lowest corresponding natural frequencies come first. The sorted frequencies are returned as well and displayed using the function gui sheet.m which was shared for download in the world wide web. For the full model, the frequencies affecting the variables V , Ω, γ and κ are displayed in an extra table. The natural frequencies of the system using the actual number of comparison functions down to one is displayed by Frequency Convergence.m. This function is activated by the button ’Convergence’. The frequencies are calculated by gradually cancelling the elements of the beam tensors (3.38) - (3.41) that are generated using the comparison function of temporary highest order. The shortened beam model is then linearized by Linearize System.m and the eigenvalues are calculated and sorted. Finally, the natural frequencies are extracted from the sorted eigenvalues and displayed. The button ’Free Vibration Modes’ calls Show FreeVibrationModes.m which shows the free vibration/natural modes of V , Ω, γ, κ, F and M . The free vibration modes of the full beam model are calculated according (3.20) where q = v and v is an eigenvector of the full model. Equation (5.4) shows the calculation of the free vibration modes of the reduced beam model, where q = v and v is an eigenvector of the reduced model. Furthermore, q 0 is set to zero for the calculation of the free vibration modes. The comparison functions Φl in (3.20) and (5.4) are provided by Variable Matrices.m. Before the natural modes are plotted, they are normalized by Normalize so that the highest absolute value is one. The normalization is performed taking the real part of each natural mode that has been divided by the complex value of highest absolute value. By pressing ’Reduced vs Full System’, the full model is tested against the reduced model plotting the first natural frequencies (up to six) under a varying parameter. This functionality is provided by redVSfull.m supporting two test scenarios. In one scenario, the rotation speed Ω3 and in the other one, the external force f3 (constant along the reference line) is varied. The number of steps and the increment of each step of the varying parameter are changed in the provided edit boxes. The remaining parameters of the beam model are not changed so that Ω3 or f3 can be changed under different conditions. The first step of the model comparison is the calculation of the transformation matrix T, see figure C.6. Then, the frequencies of the full model and reduced model are calculated for as many times as it is specified in the edit box labelled ’steps’ by Sorted Frequencies. After each frequency calculation, the varying parameter is changed and finally, the changing frequencies are plotted using Plot Frequencies. Check Performance of Reduced Model: redVSfull Build Transformation Matrix T: Transformation_Matrix

Natural frequencies of Full and Reduced System: Sorted_Frequencies Create Full System: Create_System

Ω3= Ω3+increment or f3=f3+increment

Sorted Full System Eigenvalues: Sorted_Eigenvectors Create Reduced System: Create_ReducedSystem

Plot frequencies: Plot_Frequencies

Sorted Reduced System Eigenvalues: Sorted_Eigenvectors

Figure C.6: Steps performed checking the reduced model performance

86


Controller Before any controller is designed, the Lyapunov function (9.10) has to be build by pressing ’Lyapunov function’. This button activates LQR Weighting that provides the Q matrix in (10.4) to calculate the pseudo energy (9.10). With the Q matrix, the linear optimal controller (LQG) as presented in chapter 10 is build when ’LQG Design’ is pressed. The implementation of the LQG design and the necessary LTI model generation from the linearized model is shown in figure C.7. The LTI system of the full and reCreate LTI Model: Create_LTI

Measurement Tensor: Measurement_Matrix

State Space Tensors (except Measurement Tensor): StateSpace_Transformation

LQG Design: LQG_Design

Create LQR controller: LQR_Design

LQR controlled LTI model: LQR_System

Kalman Observer: Kalman_ Design

Create LQG controller: lqgreg

LQG controlled LTI model: LQG_System

Figure C.7: Steps performed creating the LQG controller

duced model is build by Create LTI. This is done according (7.20) (full model) or (7.22) (reduced model). The measurement tensor M is created by Measurement Matrix and the other beam tensors by StateSpace Transformation. As presented in chapter 10.3, the LQG controller exists of a LQR controller and a Kalman observer. The LQR controller is build according chapter 10.1 with LQR Design and the Kalman observer is build as shown in chapter 10.2 with Kalman Design. Furthermore, the LTI system of the LQR controlled and the LQG controlled linear beam model is build by LQR System and LQG System.

C.1.3

Simulation

The controlled and uncontrolled nonlinear beam is simulated in Simulink. Simulink environments of the nonlinear beam model can be loaded after pressing ’Simulate’. As the beam model is not a LTI system, it had to be implemented as the S-function BladeModel.m. Exemplarily, the environment for the LQG controlled blade is shown in figure C.8.

C.2 C.2.1

Implementation Notes Matlab GUI Fundamentals

The Graphical User Interface (GUI), described in chapter C.1.2 has been developed using ’GUIDE’ (Graphical User Interface Development Environment) which is a development environment of Matlab. After starting GUIDE, one can create the layout of the GUI comfortably by drag&drop of elements, like edit boxes, radio buttons and push buttons. Each element has a number of object properties. One of them is a tag which is used to identify the elements one to one. Another important property is the name of the callback function that is executed if the element is activated. These callback functions are generated in an m-file that is automatically generated when the GUI layout is saved. The layout itself is saved in a ’.fig’-file.

C.2. IMPLEMENTATION NOTES

87

Figure C.8: Simulink environment of LQG controlled helicopter blade

Besides the callback functions, the m-file has an opening function, an output function and a create functions that initialize the elements. The opening function is called when the GUI is started and its purpose is to initialize it. The output function is called, when the GUI is executed after the opening function is completed. This is often unreasonable, e.g. for the child GUIs that are implemented to set the beam model. There, the execution is delayed using uiwait until uiresume is called. The output function is usually used to prepare the data that is passed by the GUI m-file. Another characteristics of GUI m-files is that the whole data generated is usually saved in a structure called ’handles’. This structure is automatically written in the argument list of each callback function. The content of the handles structure used in this work is presented in the next section.

C.2.2

Data Structure of GUI

The handles structure introduced in the previous section is presented in the following, whereas only the process data of the GUI saved in ’handles.Solver’ is shown. The remaining data stored in ’handles’ are identifier for the GUI elements. handles.Solver . BC: boundary conditions . L: true if BC at x=L . Menu: element number of pop up menu . Value: BC value . zero: true if BC at x=0 . BCfactor: weights boundary conditions for Galerkin discretization . cb: true if checkbox for external loads is selected . g1: checkbox for γ1 . g2: checkbox for γ2 . g3: checkbox for γ3 . k2: checkbox for κ2 . d int: D

R

. dd int: d D . ds int: d

R

R

. fullSystem: full blade model

88


. A: A . B: B . C: C . D: D . E: E . F: F . G: G . H: H . B noBC: B noBC . E noBC: E noBC . q0: q 0 . inputs: steady state system inputs . f0: constant f 0 along reference line . g: γ 0 for one active segment . k2: κ02 for one active segment . m0: constant m0 along reference line . u0: uA

0

. linRedSystem: linear reduced blade model . see linFullSystem . linFullSystem: linear full blade model . A: A . B: B . E: E . G: G . H: H . eig vector: v . eig value: λ . LTI . Final: LQR controlled LTI blade . K: state feedback gain . KEST: Kalman estimator . LQR: LQR controlled LTI blade . model: LTI blade model . Process Noise: covariance matrix V . Q: Q matrix for the pseudo energy calculation . Q seg: Qseg matrix for the pseudo energy calculation of one beam segment . RLQG: LQG regulator . Sensor Noise: covariance matrix W . ss tensors: tensors of beam model in state space . maxNrCompFctns: maximum possible number of comparison functions, restricted by the number of precalculated Legendre functions

C.2. IMPLEMENTATION NOTES

89

. NRofFVmodes: number of free vibration modes . NRofPTmodes: number of pertubation modes . NrModes: number of comparison functions . param: beam parameters . G: G . K: K . i2: i2 . i23: i23 . i3: i3 . I: I . L: L . mu: µ . SASg: Eseg . SASk: Fseg . U: U . V: V . W: W . x2: x2 . x3: x3 . pnum: values of Legendre functions at 100 different points between [0,1] . redSystem: reduced blade model . see fullSystem . reduced: actual blade model is a reduced one . redVSfull: parameter for the test of the full against the reduced beam model . type: varying parameter . inc: increment or step size . steps: number of steps R

. sds int: d s R

. ss int: s

0 0 . ss lim: A /B s

R

. step: step increases if more buttons can be enabled, e.g. step=3 means that a reduced system has been calculated . T: transformation matrix for model reduction . t int: T

R

. UpdateOption: update complete beam model or boundary conditions only

90

Table of Symbols

Table of Symbols Symbol

A B C D E F G I i2 ,i3 ,i23 K k L O P R S T U V W

Size 3/18 × na 3/18 × na 1×1 3×1 3/18 × na 3/18 × na 3×3 3×3 1×1 3×3 3×1 1×1 ns × 3/18 ns × 3/18 3×3 3×3 3×3 3×3 3×3 3×3

SI Unit m−1 · s · A s·A m−1 kg −1 · m−1 · s3 · A kg −1 · m−2 · s3 · A kg −1 · m−3 · s3 · A kg · m−1 kg · m kg · m kg m−1 m kg · m2 · s−3 · A−1 kg · m3 · s−3 · A−1 kg −1 · m−1 · s2 kg −1 · m−2 · s2 kg −1 · m−3 · s2 kg · m · s−2 kg · m2 · s−2 kg · m3 · s−2

Description Actuation force constant Actuation moment constant Electric field / voltage relation for piezo element Piezoelectric constant Actuation strain constant Actuation curvature constant Cross sectional inertia Cross sectional inertia Mass moments Cross sectional inertia Initial twist/curvature Beam length Sensing force constant Sensing moment constant Cross sectional flexibility Cross sectional flexibility Cross sectional flexibility Cross sectional stiffness Cross sectional stiffness Cross sectional stiffness

Table C.1: Parameters

Symbol γ κ λ µ ρ ξ Ω ω

Size 3×1 3×1 1×1 1×1 1×1 3×1 3×1 1×1

SI Unit m−1 kg · m−1 m s−1 s−1

Description Strain Curvature Eigenvalue Mass per unit length Load parameter Mass center offset Angular velocity Natural angular velocity

Table C.2: Greek Variables

Table of Symbols

Symbol A B C C c D E E F F f H J K L M m n P P P act P bou P ext Q q T T t U u u V V V W W x x ¯

91

Size 12 · nc × 12 · nc 12 · nc × 12 · nc 12 · nc × 12 · nc × 12 · nc 3×3 1×1 12 · nc × 1 1×1 12 · nc × 12 · na 3×1 12 · nc × 12 · nc × 12 · na 3×1 3×1 1×1 na × nc/nr nc/nr × ns 3×1 3×1 nc/nr × 1 3×1 1×1 1×1 1×1 1×1 12 · nc × 1 12 · nc × 1 1×1 12 · nc × 12 · nr 1×1 1×1 na/ns × 1 3×1 3×1 nf + nm × nf + nm 1×1 1×1 ns × ns 1×1 1×1

SI Unit kg · m2 · s−2 kg · m2 · s−3 kg · m2 · s−4 kg · m2 · s−2 kg · m · s−3 · A−1 A·s kg · m · s−2 A kg · s−2 kg · m · s−1 kg · m2 · s−2 kg · m · s−2 kg · s−1 kg · m2 · s−3 kg · m2 · s−3 kg · m2 · s−3 kg · m2 · s−2 s kg · m2 · s−2 s kg · m2 · s−2 kg · m2 · s−3 · A−1 m m · s−1 m m

Description Dynamic tensor Linear tensor Nonlinear tensor Cosine matrix Constant factor Constant tensor Electric field Linear actuation tensor Internal force Nonlinear actuation tensor External force Angular momentum Cost function State feedback controller gain Observer gain Internal moment External moment Eigenvector Linear momentum Legendre polynomial Power of active elements Power of boundary Power of external loads Perturbation energy Time function Kinetic energy Transformation matrix Time Potential energy Voltage Displacement Linear velocity Covariance matrix Lyapunov function Bernoulli beam displacement Covariance matrix Beam reference line Normalized beam reference line

Table C.3: Latin Variables

Typeface A, B, C, . . . A, B, C, . . . A, B, C, . . . A, B, C, . . .

Domain Active beam Passive beam Reduced beam Beam in state space form

Table C.4: Typefaces

92

Table of Symbols

Symbol 0 ˙ 0 A S b p seg f vm pert norm max 0 L R I ∗ ˜ ˆ BC no BC Φ ¯

Alternative

Description Derivative with respect to beam reference line x Absolute time derivative Steady state Actuator value Sensor value Beam element value Piezo element value Variable restricted to a beam segment Free vibration mode Perturbation mode Normalized value Maximum value Exact boundary condition value at value x = 0 Exact boundary condition value at value x = L Real value of complex variable Imaginary value of complex variable Variable changed Tilde operator Conjugate complex value Dependant on boundary conditions Not dependant on boundary conditions Iteration number Frame Comparison function of Comparison function of

∂ ∂x d dt

real() imag() ×

Table C.5: Sub- and Superscripts

Index k, i, j t, r, s l u y

From 1 1 1 1 1

To nc nr nc na ns

Table C.6: Indices

Constant e1 I ΨV ΨΩ Ψγ Ψκ Ψf Ψm nc nr na ns

Value T 1 0 0

I 0 0 0 0 I 0 0 0 0 I 0 0 0 0 I I I

Description Identity matrix Number of comparison functions Number of comparison functions for reduced system Number of actuation voltages Number of sensor voltages Table C.7: Constants

Bibliography [1] Shin, S. J. and Cesnik, C. E. S., “Helicopter Performance and Vibration Enhancement by TwistActuated Blades,” AIAA, April 2003. [2] Wilbur, M. L., Mirick, P. H., Yeager, W. T., Langston, C. W., Cesnik, C. E. S., and Shin, S., “Vibratory loads reduction testing of the NASA/Army/MIT active twist rotor,” Journal of the American Helicopter Society, Vol. 47, No. 2, Apr. 2002, pp. 123 – 133. [3] Traugott, J. P., Patil, M. J., and Holzapfel, F., “Nonlinear Dynamics and Control of Integrally Actuated Helicopter Blades,” Proceedings of the 13th Adaptive Structures Conference, Austin, Texas, April 2005, AIAA-2005-2271. [4] 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. [5] Hodges, D. H., “Geometrically Exact, Intrinsic Theory for Dynamics of Curved and Twisted Anisotropic Beams,” AIAA Journal , Vol. 41, No. 6, 2003, pp. 1131–1137. [6] Hopkins, A. S. and Ormiston, R. A., “An Examination of Selected Problems in Rotor Blade Structural Mechanics and Dynamics,” Proceedings of the 59th American Helicopter Society Annual Forum, Phoenix, Arizona, May 2003. [7] Johnson, E. R., Vasiliev, V. V., and Vasiliev, D. V., “Anisotropic Thin-Walled Beams with Closed Cross-Sectional Contours,” AIAA Journal , Vol. 39, No. 12, 2001, pp. 2389–2393. [8] Cesnik, C. E. S. and Hodges, D. H., “VABS: A New Concept for Composite Rotor Blade CrossSectional Modeling,” Journal of the American Helicopter Society, Vol. 42, No. 1, January 1997, pp. 27 – 38. [9] Chopra, I., “Review of State of Art of Smart Structures and Integrated Systems,” AIAA Journal , Vol. 40, No. 11, November 2002, pp. 2145–2187. [10] Cesnik, C. E. S., Shin, S. J., and Wilbur, M. L., “Dynamic Response of Active Twist Rotor Blades,” Smart Materials and Structures, Vol. 10, No. 1, Feb 2001, pp. 62–76. [11] Patil, M. J. and Johnson, E. R., “Cross-sectional Analysis of Anisotropic, Thin-Walled, ClosedSection Beams with Embedded Strain Actuation,” Proceedings of the 13th Adaptive Structures Conference, Austin, Texas, April 2005, AIAA-2005-2037. [12] Cesnik, C. E. S. and Palacios, R., “Modeling Piezocomposite Actuators Embedded in Slender Structures,” Proceedings of the 44th Structures, Structural Dynamics, and Materials Conference, Norfolk, Virginia, April 2003, AIAA Paper 2003-1803. [13] Zienkiewicz, O. C., Taylor, R. L., and Zhu, J. Z., The Finite Element Method: Its Basis and Fundamentals, Butterworth-Heinemann, March 2005. [14] Traugott, J., Intrinsic Dynamic Analysis and Control Design of Integrally Actuated Helicopter Blades, Master’s thesis, Technische Universität M¨ unchen, November 2004. [15] Fletcher, C. A. J., Computational Galerkin Methods, Springer, 1984. 93

94

BIBLIOGRAPHY

[16] Meirovitch, L., Principles and Techniques of Vibrations, Prentice Hall, Inc., Upper Saddle River, New Jersey 07458, 1997. [17] Abramowitz, M. and Stegun, A., Handbook of Mathematical Functions, Dover, 1964. [18] Shaw, S. W. and Pierre, C., “Non-Linear Normal Modes and Invariant Manifolds,” Journal of Sound and Vibration, Vol. 150, No. 1, 1991, pp. 170–173. [19] Shaw, S. W. and Pierre, C., “Normal Modes of Vibration for Non-Linear Continuous Systems,” Journal of Sound and Vibration, Vol. 169, No. 3, 1994, pp. 319–347. [20] Shaw, S. W. and Pierre, C., “Normal Modes for Non-Linear Vibratory Systems,” Journal of Sound and Vibration, Vol. 164, No. 1, 1993, pp. 85–124. [21] Qu, Z.-Q., Model Order Reduction Techniques, Springer, 2004. [22] Patil, M. J., “Decoupled Second-Order Equations and Modal Analysis of a General NonConservative System,” Proceedings of the AIAA Dynamics Specialists Conference, Atlanta, Georgia, April 2000, AIAA-2000-1654. [23] Noor, A. K. and Peters, J. M., “Reduced Basis Technique for Nonlinear Analysis of Structures,” AIAA Journal , Vol. 19, No. 4, Apr. 1980. [24] Bauchau, O. A. and Guernsey, D., “On the Choice of Appropriate Bases for Nonlinear Dynamic Modal Analysis,” Journal of the American Helicopter Society, Vol. 38, No. 4, Oct. 1993, pp. 28 – 36. [25] Thompson, J. M. T. and Walker, A. C., “The Nonlinear Pertubation Analysis of Discrete Structural Systems,” International Journal of Solids and Structures, Vol. 4, 1968, pp. 757–768. [26] Walker, A. C., “A Nonlinear Finite Element Analysis of Shallow Circular Arches,” International Journal of Solids and Structures, Vol. 5, February 1969, pp. 97–107. [27] Wright, A. D., Smith, C. E., Thresher, R. W., and Wang, J. L. C., “Vibration Modes of Centrifugally Stiffened Beams,” Journal of Applied Mechanics, Vol. 49, Mar. 1982, pp. 197 – 202. [28] Ortega, R., Spong, M. W., Gomez-Estern, F., and Blankenstein, G., “Stabilization of a Class of Underactuated Mecahnical Systems Via Interconnection and Damping Assignment,” IEEE Transactions on AUtomatic Control , Vol. 47, No. 8, August 2002, pp. 1218–1233. [29] Woolsey, C. A., Reddy, C. K., Bloch, A. M., Chang, D. E., Leonard, N. E., and Marsden, J. E., “Controlled Lagrangian Systems with Gyroscopic Forcing and Dissipation,” European Journal of Control , Vol. 10, No. 5, 2004, pp. 478–496. [30] Ljung, L., System Identification: Theory for the User , Prentice Hall, December 1998. [31] Astrom, K. J. and Wiottenmark, B., Adaptive Control , Prentice Hall, December 1994. [32] Khalil, H. K., Nonlinear Systems, Prentice Hall, 2002. [33] Burl, J. B., Linear Optimal Control , Prentice Hall, November 1998. [34] Genta, G., Dynamics of Rotating Systems, Springer, 2005. [35] McClamroch, N. H. and Kolmanobsky, I., “Performance benefits of hybrid control design for linear and nonlinear systems,” Proceedings of the IEEE , Vol. 88, No. 7, July 2000, pp. 1083– 1096. [36] Tyldesley, J. R., An introduction to tensor analysis for engineers and applied scientists, Longman Group Limited, 1975.