March 2005 TAE 955

Aeroelastic Analysis of Propellers Part 2 - “Flutter” Analysis

By

Y. Yadykin, V. Tenetov, I. Weissberg and A. Rosen

2

March 2005

Aeroelastic Analysis of Propellers

Part 2 - “Flutter” Analysis

By

Y. Yadykin, V. Tenetov and A. Rosen Faculty of Aerospace Engineering

Technion – Israel Institute of Technology Haifa 32000, Israel

I. Weissberg Aero Design & Development LTD Park Rehovot, P.O.B 565 Rehovot

TAE NO. 955

3

Abstract This document presents a theoretical model and an associated computer program, that are used to predict flutter of a propeller blade, operating in a subsonic incoming flow. The model is based on a two-dimensional unsteady strip theory in conjunction with a finite element structural model of the blade.

A FEM structural software is used. The generalized aerodynamic forces are based on the two-dimensional subsonic theory of Theodorsen, and are applied in a strip theory manner with appropriate modifications.. Parametric studies are presented, illustrating the effects on flutter of the rotational speed, cruise Mach number and structural damping. The analysis is applied to the SR2-EC propeller. This propeller includes eight straight blades and it was designed for a high Mach number cruise, up to Mach 0.8. The SR2-EC blade is characterized by large twist angles (between the root and the tip sections) and thin airfoil sections (especially at the tip of the blade).

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List of Contents Subject Page Abstract 3 List of Symbols 5 List of Tables 9 List of Figures 9 1. Introduction 11 2. Finite Element Model 13

2.1 The FEM Coordinate System 13 2.2 A Finite Element Modeling of the Blade 13

3. Aeroelastic Model 15 3.1 Equations of the Motion 15 3.2 Linearization of the Equations 16

3.4 Generalized Aerodynamic Forces 23

4. Flutter Analysis 28

5. Computer Code 31 5.1. Calculation of the Mode Shapes 33 5.2 Calculation of the Generalized Mass 33 5.3. Calculation of the Lift-Curve Slope 34

6. Results and Discussions 35 6.1. The SR2-EC Propeller 35 6.2 Results and Discussion 38

7. Conclusions 50 References 51 Appendix A: Explanation of Nonlinearities in the Basic Equation 52

A.1. Introduction to Nonlinearities 52 A.2. Stress Stiffening 52

A.3. Spin Softening 53 Appendix B: Matrices 54 Appendix C: Aerodynamic Forces and Moments Acting on an Oscillating Airfoil 56 Appendix D: Calculating the matrix [ ]H 61 Appendix E: Calculating the matrix [ ]AP 68 Appendix F: Calculating the matrix [ ]0T 71

5

List of Symbols [ ] [ ] [ ]2g1g0g ,,, ,, ΑΑΑ generalized aerodynamic matrices

cmA coefficient of the distribution of the aerodynamic loads

a distance between the mid-chord and the point where h is measured b semi-chord [ ] [ ] [ ]321 CCC ,, aerodynamic matrices defined in Appendix C [ ]dC viscous damping matrix CL lift coefficient [ ]nC matrix of the coefficients of a transformation CLd design lift coefficient c chord, c=2b c/d chord ratio of a blade cross-section D drag force per unit span of the blade

D∆ perturbation of the drag force d propeller diameter

αddCL lift curve-slope )(σF Theodorsen’s function, ( ) )iG(σΘ(σ)F +=σ

)( u,uu,F aerodynamic cross-sectional loads vector CFF centrifugal force vector CFF∆ perturbation of the centrifugal force vector

GRF gyroscopic force vector GRF∆ perturbation of the gyroscopic force vector

)t∆,( 0 ,F uu∆ perturbation of the aerodynamic loads vector

[ ]H reduced modal matrix h plunging displacement vector

0nh amplitude of the plunging motion

[ ]Ι identity matrix

αααα ΙΙΙΙ ,,, hhhh unsteady aerodynamic coefficients, defined in Appendix C J total number of degrees of freedom of the FEM model of the blade

0J advance ratio, dNV60J0 = K total number of degrees of freedom per each node k index of degree of freedom, K1 ≤≤ k [ ]0K stiffness matrix for the steady-state position

[ ]LK linear elastic stiffness matrix [ ]CFK centrifugal (spin) "softening" matrix in physical coordinates

( )[ ]uK nonlinear stiffness matrix in physical coordinates [ ]gK generalized stiffness matrix

0L total number of nodes of the blade

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i index of degrees of freedom of the FEM model, Pi1 ≤≤ L aerodynamic lift force per unit span of the blade nl strip length along the reference line

[ ]0M physical mass matrix, see Appendix B [ ]gM generalized mass matrix M aerodynamic moment per unit span of the blade

cM number of finite elements in the chordwise direction cm index of elements in the chordwise direction, cc Mm1 ≤≤

N speed of propeller rotation, rpm ( )

cmmN1 parameter representing a distribution of the aerodynamic loads

mN number of nodes per a finite element

mn index of considered nodes of the finite element, mm Nn1 ≤≤

sN total number of strips along the blade n index of strips along the blade, sNn1 ≤≤ P number of degrees of freedom of the FE model of the blade [ ]AP matrix of the distribution of the aerodynamic loads over the blade [ ] nMP matrix of the distribution of the aerodynamic loads over the strip

( ) uuu ,,P aerodynamic nodal force vector ( ) 0uP steady-state aerodynamic nodal force vector

nnnn S,R,Q ,P unsteady aerodynamic force coefficients acting on the nth strip Q vector of the aerodynamic loads

tgQ generalized aerodynamic force vector q vector of the generalized coordinates

0q amplitude of the motion described using the generalized coordinates R radius of the propeller

0u steady-state blade deflection at the grid points

r radial coordinate along the reference line S total number of modes used in the analysis s index of modes s=1, 2, 3, …,S [ ])(T 0u general transformation matrix, usually [ ])( 0uT is denoted [ ]0T [ ] nMT transformation matrix for the nth strip of the blade, see Appendix F

[ ]cmFT transformation matrix for the cm th finite element, see Appendix F

[ ]enT transformation for en th node, see Appendix F

t time ct 0 thickness ratio of a blade cross-section.

V aircraft velocity 1w~~ , 2w~~ , 3w~~ linear displacements at the finite element node 4w

~~ , 5w~~ , 6w

~~ rotational displacements at the finite element node

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XYZ global coordinates system

xyz blade fixed coordinates system

Rrx = dimensionless radial coordinate of the blade cross-section

Greek Symbols α effective angle of attack α rotation deflections vector

0nα amplitude of the rotation perturbation

210 βββ ,, Euler's angles of transformation β0.75R blade pitch angle at the three-quarter radius of the blade µ real part of an eigenvalue ν imaginary part of an eigenvalue

L∆ perturbation of the aerodynamic lift force M∆ perturbation of the aerodynamic moment

( ) ,t,∆∆ 0 uuP perturbation of the aerodynamic nodal force vector

( ) tu∆ vector of the perturbation of vibratory deflections

E0 Young’s modulus sζ modal damping

λ frequency of the blade oscillations

0λ reference frequency π constant, 3.14159 ρ air density

oρ blade material density σ reduced frequency, Vb /λ=σ [ ]Φ modal matrix of the FEM φ eigenvector ϕ phase angle [ ]Ψ matrix of the aerodynamic forces acting on the blade [ ]0n ,Ψ , [ ] [ ]2n1n ,, , ΨΨ strip aerodynamic matrices

[ ]2ω eigenvalue matrix. Ω angular speed, Ω=2πN/60 (rad/sec)

sω frequency of the sth mode Subscripts 0 steady state value CF centrifugal RG gyroscopic g generalized (modal) L linear n n th strip of the blade

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mn n mth node of the finite element

cm m c th finite element along the chord s s th mode of the blade perturbations st stiffness Superscripts G global system coordinates ( )Τ transpose

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List of Tables

Table 1. The geometry of a blade of the SR2-EC propeller 37 Table 2. Equivalent material properties of the SR2-EC blade 37

List of Figures Figure 1a. Coordinates System for a blade 14 Figure 1b. Section A-A showing rigid plunging (h) and pitching (α ) motions

for a strip 14 Figure 2. Flow Chart 32 Figure 3 .Global coordinate system for the SR2 blade 36 Figure 4. The geometry of a blade of the SR2-EC propeller 38 Figure 5. Mode shapes calculated with respect to the blade coordinates system:

β0.75R =55.4 deg, Ω=6000 rpm. 40

Figure 6. Variation of the angle of attack along the blade for various rotational speeds: β0.75R =55.4 deg, Mach number M=0. 41 Figure 7. Variation of the lift-curve slope along the blade for various rotational speeds: β0.75R =55.4 deg, Mach number M=0. 42 Figure 8. Variation of damping and frequency vs. the rotational speed: β0.75R =55.4 deg., ξ=0. 43 Figure 9. Variation of damping and frequency vs. the rotational speed: β0.75R =55.4 deg., ξ=0,002. 44 Figure 10. Variation of damping and frequency vs. the rotational speed β0.75R =55.4 deg., ξ=0,02. 45 Figure 11. Variation of damping and frequency vs. the rotational speed: β0.75R =55.4 deg., ξ=0,2. 46 Figure 12. Variation of the angle of attack along the blade for various Mach numbers: β0.75R =55.4 deg, Ω=6000 rpm 47

Figure 13. Variation of the lift-curve slope along the blade for various Mach

numbers: β0.75R =55.4 deg, Ω=6000 rpm. 48 Figure 14. Variation of damping and frequency vs. the free-stream Mach number: β0.75R =55.4 deg., ξ=0.2. 49

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Figure D1. 3D quadratic shell element 63 Figure D.2. Matrix [ ]Φ Components 67 Figure E.1. The scheme of the numbering of the nodes of the finite element 69

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1. Introduction

This document describes an extension of the static aeroelastic analysis of propeller blades that was described in Ref. 1

The major goals of propeller design are to maximize aerodynamic efficiency, minimize noise and assure structural integrity. Often aerodynamic and acoustic requirements result in designs with thin, swept, and twisted blades, having low aspect ratio and high solidity, compared to conventional propellers [2]. These blades operate in subsonic, transonic, and possibly supersonic flows.

The above described properties of advanced blades add complexity to the understanding of aeroelastic phenomena and the development of appropriate aeroelastic models. Since the blades are thin and flexible, the influence of deflections due to centrifugal and aerodynamic loads cannot be ignored. The aeroelastic problem is inherently nonlinear, requiring the application of a geometric nonlinear theory of elasticity. As indicated above these blades have large sweep and twist, that couple blade's bending and torsion deflections. These structures are plate-like structures because of their low aspect ratio. These characteristics lead to using a finite element structural model that accounts for centrifugal softening/stiffening effects and possibly for Coriolis effects [3-6]. The centrifugal softening terms are important because of the large blade sweep and flexibility. Because of these unique features, it is impossible to use directly existing aeroelastic analyses of conventional propellers or helicopter blades.

Classical flutter of propellers occurred, unexpectedly, during wind tunnel tests of a model (designated SR-5) with ten highly swept titanium blades [2]. Reference 2 presents experimental data of the SR-5 model and correlation of the data with theory.

In the analysis of Ref. 3, the aerodynamic model is based on a two-dimensional unsteady theory, with a correction for blade sweep, while the structural model is an idealized swept beam. In Ref. 5, the model of Ref. 3 is improved by using blade normal modes, calculated using a finite-element plate model of the blade. The analytical results are compared with the data of the SR-5 model. The correlation between theory and experiment, in Refs.3-5, is varied between poor to good.

Additional subsonic wind tunnel flutter results, obtained during the tests of another composite blade model, SR3C-X2, are presented in Ref. 7. A two-dimensional steady and unsteady aerodynamic theory for a blade having a subsonic leading edge, are presented in Ref. 8, and the theory is used for predicting the flutter speed of the wind tunnel model, reported in Ref. 7.

The specific objectives of the research are:

(1) To develop a flutter analysis method that uses a two-dimensional aerodynamic model.

(2) To conduct parametric studies in order to understand the effect of steady airloads on the: frequencies, mode shapes and flutter speed. Also to study the effect of blade pitch angle and blade structural damping - on the flutter speeds.

(3) To validate the analytical model by correlating calculated and measured flutter speeds.

(4) To examine the limitations of a two-dimensional unsteady aerodynamic theory for the analysis of propellers.

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In the present approach the unsteady flow is modeled as a small perturbation superimposed on a uniform steady flow. The unsteady non-linear aerodynamic equations are linearized about the steady flow [8], resulting in linear unsteady aerodynamic equations, that include the effects of the steady inertia and aerodynamic loading.

The flutter analysis presented herein is based on the method of Ref. 4. It uses the elastic modes in conjunction with a two-dimensional aerodynamic theory in a stripwise manner. The flutter analysis is carried out in three steps:

(1) A geometric nonlinear structural analysis of the rotating blade is performed using a finite element model. This analysis provides the steady-state deformed configuration of the propeller blade.

(2) The natural frequencies and mode shapes of the blade, in its deformed state, are calculated, based on the results of step (1).

(3) The unsteady aerodynamic loads and the stability characteristics of the propeller blade are calculated.

To achieve the objectives, a computer program is developed. The computer code combines both, structural and aerodynamic models. The code is used to analyze a straight bladed propeller (SR-2), designed to operate at high Mach numbers.

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2. Finite Element Model The structural analysis is based on the use of a Finite Element Model (FEM) to determine the blade structural behavior. The blades experience deflections due to the action of centrifugal and aerodynamic loads. The blade FEM model is built by using three dimensional shell or other elements, and equivalent material properties in the case of composite materials. The computer program, which was developed during this study, builds the FEM of the blade.

2.1 The FEM Coordinates System A global coordinates system, shown in Fig. 1, is used for the description of the blade

geometry and the calculations of the loads. This is a right-hand Cartesian coordinates system, where; The Z-axis is co-linear with the propeller axis of rotation and is positive in the direction of incoming flow. The Y-axis is in the plane of rotation. The X-axis is collinear with the pitch change axis, lies in the plane of rotation and points towards the blade tip. The blade pitch angle is denoted, β0, and it can be varied by rotating the blade about its pitch change axis. The angle β0 is usually specified at the tree quarters radius section. The propeller rotates about the Z axis with an angular speed Ω

2.2 A Finite Element Modeling of the Blade

The blade is represented by its mid surface. Boundary conditions are defined at the blade root. The blade if affected by two types of loads: the inertia and aerodynamic loads. The inertia forces are treated directly by the FEM software. The aerodynamic loads are calculated using a two-dimensional aerodynamic theory in a strip-wise manner. A more detailed description of the aerodynamic model appears in what follows.

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Figure1a. Coordinates System for a Blade

Section A-A

Figure1b. Section A-A showing rigid plunging (h) and pitching ( α )

motions for a strip

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3. Aeroelastic Model Aeroelasticity deals with the interaction between the structural and aerodynamic behavior. The purpose of an aeroelastic analysis is to combine the formulation of the structural dynamics and aerodynamic models, in a consistent manner, in order to study the combined aeroelastic behavior. Two-dimensional aerodynamic strip theory is used in the present study in order to calculate the aerodynamic loads. The unsteady, two-dimensional, aerodynamic loads in subsonic flow are obtained by using the theory of Ref. 8.

3.1. Equations of Motion The right hand Cartesian coordinates system used for deriving the equations of motion of a rotating propeller blade, is shown in Fig.1. The propeller rotates about the Z-axis which is aligned with the freestream direction. The X–axis is aligned along the blade pitch-axis and the Y- axis is perpendicular to the Z-X plane.

The structural model of the blade is presented by a FEM, as described in the previous section.

The aeroelastic equations of motion of the blade can be written as [10]:

[ ] [ ] [ ][ [ ] ( )[ ]] =+−++ uuKKKuu CFLdC0M

( ) u,uu,PFF GRCF +−= (1)

u represents the blade deflections at the grid points of the finite-element model, [ ]0M is the symmetric inertia matrix [9-11], , [ ]dC the viscous damping matrix, [ ]LK the linear elastic stiffness matrix, [ ]CFK the centrifugal (spin) "softening" matrix, and ( )[ ]uK the nonlinear geometrical (stress) stiffness contribution. CFF , GRF and ( ) uuuP ,, are the force vectors related to the centrifugal, gyroscopic and aerodynamic loads, respectively.

Note, that [ ]CFK and CFF are functions of 2Ω [see Eqs. B.2 - B.4], [ ]LK and [ ]CFK are symmetric matrices [9], [ ]LK is positive definite in nature [9].

In what follows, for clarity, u , u , and u will be replaced by u ,u , and u , respectively.

The vector of aerodynamic loads can be expressed as:

( ) [ ] [ ] )()( u,uu,uu,uu, FPTP A= (2)

[ ])(uT is the a transformation matrix that rotates the aerodynamic loads to the directions of the FE model, namely the rotation between the aerodynamic and global coordinates systems. [ ]AP is the matrix that describes the distribution of the resultant cross-sectional aerodynamic loads over the blade. )( u,uu,F is the aerodynamic cross-sectional loads vector, which is a function of the displacement vector u and its time derivatives, u and u .

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At each cross-section, n, there is a lift force, nL , drag force, nD , and an aerodynamic moment, nM , per unit span of the blade. These loads are functions of the displacement vector, u and its time derivatives u and u , )u,uu,(nD ,

)u,uu,(nL , and )u,uu,(nM .

The vector )( u,uu,F is defined as:

sss NNNnnn111 M,L,D,,M,L,D,,M,L,D)(F =Τu,uu, (3)

where sN is the number of strip elements along blade ( sNn1 ≤≤ ).

In general, because of relatively large deflections, there is a need to use the geometric nonlinear theory of elasticity. Thus the strain and displacement relations are nonlinear. The stiffness matrix is a function of nodal displacements and, hence, is nonlinear. The level of the geometric nonlinear theory of elasticity that will be used here, as well as in the FEM software, is the one in which elongations and shears are negligible compared with unity. This explicit consideration of the geometric nonlinear theory of elasticity provides the additional geometric differential stiffness due to centrifugal stiffening terms. The displacement dependent centrifugal "softening" terms are included in the matrix [ ]CFK , which is linear [9].

3.2. Linearization of the Equations Equation (1) is nonlinear and is used to calculate the: steady-state deflections, frequencies, mode shapes and flutter speed. An appropriate solution method includes a direct integration of the equations in the time domain, but it is computationally inefficient. Common practice is to perturb the equations about a steady state configuration.

Consider the case of a propeller where the shaft rotates at a constant angular velocity ( 0=Ω ) and the incoming flow is in the opposite Z direction. As a result of loads acting on it, the blade deforms. This deformation is described by the vector 0u . The applied loads are steady, thus 0u is not a function of time. The purpose now is to derive the equations for small vibrations superimposed on these predeformations. If the small vibrations are described by the vector ( )t∆u , the resultant deformation vector, ( )tu , equals:

( ) ( )tt uuu ∆0 += (4)

It is assumed that the perturbations are small and so nonlinear terms of perturbations are neglected. Thus the nonlinear terms of Eq. (1) can be presented as:

( )[ ] ( )[ ] ( )[ ] ( )[ ][ ] uuuuuuu ∆0000 KKKK ∆++≅ (5)

( )[ ] ( )[ ] ( )[ ]∆u,uuu 00 TTT ∆+= (6)

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( ) ( ) t,∆)( uuuu,uu, ,FFF 00 ∆+= (7)

( ) ( ) ( ) t∆,00 ,PPP uuuu,uu, ∆+= (8)

Substituting Eqs. (6) and (7) into Eq. (2) and linearization, result in:

( ) ( )[ ] ( )[ ][ ] [ ] +∆+= )(∆ A 0uuuuu,uu, FP,TTP 00

( )[ ] [ ] )t,∆(A uuu ,FPT 00 ∆+ (9)

The steady state vector of the aerodynamic loads, is:

( ) ( )[ ] [ ] )(FPT A 000 uuu =P (10)

The perturbation vector of the aerodynamic loads, is:

( ) ( )[ ] [ ] +≅ )(,∆∆,t,∆∆ 0A00 uuuuu FPTP

( )[ ] [ ] ),(FPT t∆,0A0 uuu ∆+ (11)

In the present analysis, the term ( )[ ] [ ] ( ) 00 ∆, uuu FPT A∆ in the equation (11) will be neglected. This term is important in the case of divergence [1].

Based on the above assumptions, Eq. (11) may be written as:

( ) ( )[ ] [ ] )t∆(,t,∆∆ A0 ,FPTP u,uuuu 00 ∆= (12)

where the perturbation of the aerodynamic cross-sectional loads vector, )t∆( ,F u,u0∆ , is defined as follows:

sss NNN222111)t∆( M,L,D,,M,L,D,M,L,D,F ∆∆∆∆∆∆∆∆∆=∆ Τu,u0 (13)

Substituting Eqs. (4)-(9) into Eq. (1), leads to two sets of equations: Equations for the basic state, 0u , and another set of equations for the perturbation , u∆ :

[ ][ [ ] ( )[ ]] ( ) 000 uuu PFKKK CFCFL +=+− (14)

and:

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[ ] [ ] [ ] [ ] ( )[ ] ( )[ ][ ] =∆++−+∆+∆ uuuuu 00 KKKK CFL ∆CM d0

( ) GRCF FFt ∆−∆+∆= ,P u∆,u0 (15)

The elements of the matrices [ ]0M , [ ]LK , [ ]CFK , ( )[ ]0uK , ( )[ ]0uK∆ , and ( )[ ]0uT are constants [9]. The matrices [ ]0M , [ ]CFK , CFF , GRF , and ( )[ ]0uT

are described in Appendices B and F. In what follows, for the sake of simplicity, the matrix ( )[ ]0uT will be denoted [ ]0T . In the present analysis, the terms CFF∆ and GRF∆ will be neglected

The steady state solution, for a given rotational speed and Mach number, is obtained by solving Eq.(14). This solution was discussed in Ref.1.

Once the steady state deflections are known, the natural frequencies and mode shapes are calculated by solving the homogeneous part of Eq. (15), assuming: that there are no damping and external forces. The equation of motion of an undamped system, expressed in a matrix form, using the above assumptions, is:

[ ] [ ] 0u =+ ∆Ku∆M 00 (16)

The stiffness matrix [ ]0K includes the regular elastic stiffness, differential stiffness due to centrifugal stiffening loads and steady-state aerodynamic loads:

[ ] [ ] [ ] ( )[ ] ( )[ ][ ]00 uu K∆KKK CFL ++−=0K (17)

For a linear system, that has J degrees of freedom, there are in general J natural modes and J natural frequencies. Since the system is undamped, the vibrations are harmonic. Thus, the solution of Eq. (16) is of the form:

tiss

se∆ ωφ=u , ( s=1, 2, 3, ..., J ) (18)

sω is the natural circular frequency (radians per unit time) of the sth mode, t is the

time and 1−=i . The vector sφ represents the corresponding mode shape.

That is:

sJs3s2s1s ,,,, ,,,, φφφφ=φ Τ (19)

Substitution of Eq. (18) into Eq. (16) produces a set of homogeneous algebraic equations:

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[ ] [ ]( ) 0MK 02

0 =φω− ss (20)

For a nontrivial solution of Eq. (20), the following condition should exist, that yields the characteristic equation of the free vibrations:

[ ] [ ][ ] 0MK 02

0 =ω− sdet (21)

Expansion of this determinant yields a polynominal in which the term of highest order is ( ) J2

sω . Equation (21) presents an eigenvalue problem. sω are the eigenvalues, while sφ are the eigenvectors. If the eigenvalues of a system are known, the eigenvectors, namely mode shapes, may be calculated from the homogeneous algebraic equations (20). Because there are J eigenvalues, there will also be, in general, J corresponding eigenvectors. The eigenvectors are defined up to a factor that multiplies all the elements of the vector. For the eigenvectors (modes) the following relations exist [12]:

[ ] 0M 1s02s =φφ Τ 12 ss ≠ (22)

and:

[ ] 0K 2s01s=φφ Τ 21 ss ≠ (23)

Equations (22) and (23) represent orthogonality conditions of the natural modes of vibration. From Eq. (22) we see that the eigenvectors are orthogonal with respect to [ ]0M , and Eq. (23) indicates that they are also orthogonal with respect to [ ]0K .

For cases where 21 ss = , the following relationships hold [12]:

[ ] sgs0sMM =φφ Τ (24)

and

[ ] sgs0sKK =φφ Τ (25)

sgM and sgK are constants that depend upon the manner in which the eigenvectors

s

φ normalized.

Assume that in order to analyse the problem one decides to use only S normal modes, ( JS1 ≤≤ ). Then the perturbation can be expressed as a superposition of these normal modes:

[ ] qu Φ=∆ (26)

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where

[ ]

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

φφφ

φφφ

φφφ

=×

SP,sP,1P,

S,s,1,

S1,s1,11,

SPΦ , (27)

Here, P1 ≤≤ , KLP 0 ×= , [ ]Φ is the matrix of eigenvectors, the dimension of which is SP × . S is the number of modes used in the analysis, 0L is the total number of nodes, K is the number of degrees of freedom at each node. The assembly of the elements of φ is shown in Appendix D

q is the vector of generalized coordinates, defined as follows:

Ss21 qqqq ,,,,, ……=Τq (28)

Equation (20) can be written as:

[ ] [ ] [ ] [ ] [ ]200 M ωΦ=ΦΚ (29)

The matrix [ ]2ω in Eq. (29) is a diagonal matrix of order S:

[ ]

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

ω

ωω

ω

=

2S

23

22

21

2

000

000000000

ω

………………

………

(30)

The matrix [ ]2ω is referred to as the eigenvalue matrix.

Multiplying Eq. (29) from the left by [ ]ΤΦ and using the relationships (24) and (25), result in:

[ ] [ ] [ ]2gg MK ω= (31)

[ ]gΜ and [ ]gK are diagonal square matrices of order S, defined as:

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[ ]

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=Μ

Sg

sg

1g

g

M

M

M

(32)

and

[ ]

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

Sg

sg

1g

g

K

K

K

K (33)

Hence,

2ssgsg ωMK = (34)

As indicated above, the vectors

sφ are defined up to a factor that multiplies all the

elements of the vector. It is convenient to choose these vectors such that:

[ ] [ ] [ ] [ ]Ι=ΦΦ Τ0M (35)

where [ ]Ι is an identity matrix of order S. It is clear that if Eq. (35) is satisfied, then:

[ ] [ ] [ ] [ ]20 ωK =ΦΦ Τ (36)

Damping plays a minor role in the response of a system to a periodic forcing

function when the frequency of the excitation is not near a resonance [12]. However, for a periodic excitation with a frequency at or near a natural frequency, damping is of prime importance und must be taken into account. When its effects are not known in advance, damping should be included in a vibratory analysis until its importance is known. The equations of motion of our system, Eq. (15), may be written as:

[ ] [ ] [ ] ( ) t∆∆uKu∆Cu∆M 0d0 ,,P 0 uu∆=++ (37)

Equation (37) can be expressed using principal coordinates, by the same transformation that was used for the undamped homogeneous system (29). Thus, using the principal coordinates, equation (37) becomes:

[ ] [ ] [ ] ggdg QKCM =++ qqq (38)

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where the matrices [ ]gM and [ ]gK are given by Eqs. (32) and (33). The vector gQ is defined as:

[ ] ( ) t∆Qg ,,P 0 uu∆Φ= Τ (39) The symbol [ ]gC in Eq. (38) represents a damping matrix:

[ ] [ ] [ ] [ ]ΦΦ= ΤdgC C (40)

The nature of damping in physical systems is not well understood. The simplest approach [14] consists of assuming that the equations of motion of the damped system are also uncoupled. In other words, the eigenvectors are assumed to be orthogonal not only with respect to [ ]M and [ ]K , but also with respect to [ ]dC , thus:

[ ] 0C 2sd1s=φφ Τ 21 ss ≠ (41)

For cases where 21 ss = , the following relationship holds [12]:

[ ] sgsdsCC =φφ Τ (42)

sgC is a constant that depends upon the normalization of the eigenvector

sφ .

In the matrix form, Eq. (42) can be written as:

[ ] [ ] [ ] [ ]gC=ΦΦ ΤdC (43)

[ ]gC is a diagonal square matrix of order S, defined as:

[ ]

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

Sg

sg

1g

g

C

C

C

C (44)

.

In order to make Eq. (37) analogous to a one-degree system [12], the following notation is introduced:

sssg

sg2

M

Cωξ= , 2

ssg

sg

MK

ω= , and sgsg

sg QMQ ~= (45)

Finally the equations of perturbations become:

sgssssss Q~qωqωξ2q 2 =++ (s=1, 2, 3, ...,S ) (46)

23

Here, ss ωξ is the modal damping constant of the sth normal mode. sξ is the corresponding modal damping ratio. It is possible to obtain experimentally or to assume the damping ratio sξ for the natural modes of vibration [14].

3.4 Generalized Aerodynamic Forces The problem of calculating the aerodynamic forces acting on a two-dimensional airfoil moving in a simple harmonic motion about an equilibrium position (of an incoming uniform flow-fixed wing), was analyzed by Theodorsen [8]. In [8] the lift and moment per unit span, due to blade motion, are linearly related to the displacements and their derivatives with respect to time. The drag force is ignored, that is 0=∆ D . The expressions for the lift perturbation, L∆ , and moment perturbation, M∆ , per unit span, are given in Appendix C and are derived for airfoil sections, using the airfoil system of coordinates. With the sign convention of both, the plunging and leading edge displacement due to pitching, positive upward (Fig. 1), the lift and moment, at the cross-section r of the blade, for a specified frequency, using the notation of reference 8, are:

( ) ( )[ ]⎭⎬⎫

⎩⎨⎧

−−−++−−=∆ αFbV2α

bV1Faαa

bhF

bV2

bhbρπtr,L 2

23 )σ()σ(12)σ( (47)

( )⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ ++=∆ αa

81h

bVFa

21

b2h

babρπtr,M 24 )σ( +

⎭⎬⎫

⎟⎠⎞

⎜⎝⎛ ++⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+−+ α

bVFa

212α

bVFa

412

21a 2

22 )σ()σ( (48)

Here, a is the nondimensional distance between the mid-chord point and the point where h is measured.. Vbλ=σ is the reduced frequency, λ is the circular frequency of oscillations, b is half chord of the blade cross-section and V is the velocity of incoming air flow The quantities h and α represent the plunging and pitching displacements of the blade cross-section and are functions of time. The lift deficiency function, )σ(F , is defined in Appendix C.

Expressions (47) and (48) can be written in a matrix form as:

( )( ) [ ] [ ] [ ]

⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

∆∆

αh

Cαh

Cαh

Ctr,M

tr,L321 (49)

The aerodynamic matrices [ ]1C , [ ]2C , [ ]3C are defined in Appendix C.

In the present aerodynamic analysis the blade is divided, in the spanwise direction, into sN strip elements. Each strip has two "aerodynamic" degrees of freedom: plunging displacement, nh , (motion perpendicular to chord), and a pitching (torsion) displacement, nα .

24

As shown in Appendix C, the plunging and pitching motion of cross-section n, can be expressed as:

ti0n

0n e

h λ

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

α=

⎭⎬⎫

⎩⎨⎧

n

n

αh

(50)

Then the lift and moment acting on the nth oscillating strip section can be expressed as:

[ ] [ ] [ ]( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

αΨ+Ψλ+Ψλ−=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∆

∆

n

n0n1n2n

2

n

n hi ,,,M

L (51)

The matrices 2n ,Ψ , 1n ,Ψ and 0n ,Ψ are complex square matrices of order 2, defined in Appendix C.

The plunging, nh , and pitching, nα , displacements of the nth blade strip in equation (51), are expressed in terms of normal modes and normal coordinates, as:

[ ] qnH=⎭⎬⎫

⎩⎨⎧

n

n

αh

(52)

where the matrix [ ]nH is defined as follows:

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡=

Sn,n,2n,1

Sn,n,2n,1Sx α.......αα

h..........hh2nH (53)

Details about the matrix [ ]nH appear in Appendix D

After substituting Eq. (52) into Eq.(51), the lift, ( )tL n∆ , and the moment, ( )tM n∆ , due to blade motions nh and nα , can be rewritten as:

( )( ) [ ] [ ] [ ]( ) [ ] ( )ti

t

tn0n1n2n

2

n

n qHM

L,,, Ψ+Ψλ+Ψλ−=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∆

∆ (54)

Once the lift, ( )tL n∆ , and the moment, ( )tM n∆ , are known, the perturbations of the aerodynamic forces, Eq.(13), acting on the entire blade, can be expressed as:

25

ss NNnn11 000)t∆( M,L,,,M,L,,,M,L,,,F 0 ∆∆∆∆∆∆=∆ Τuu (55)

By using Eq. (54), Eq. (55) can be written as:

[ ] [ ] [ ]( ) [ ] ( )tHi)t∆( 0122 quu Ψ+Ψλ+Ψλ−=∆ ,,F 0 (56)

The individual matrices 2Ψ , 1Ψ , 0Ψ and [ ]H are:

[ ]

[ ]

[ ]

[ ] ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡Ψ

⎥⎦

⎤⎢⎣

⎡Ψ

⎥⎦

⎤⎢⎣

⎡Ψ

=Ψ

2N

2n

21

N3xN32

s

ss

000

000

000

,

,

,

(.57)

[ ]

[ ]

[ ]

[ ] ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡Ψ

⎥⎦

⎤⎢⎣

⎡Ψ

⎥⎦

⎤⎢⎣

⎡Ψ

=Ψ

1N

1n

11

N3xN31

s

ss

000

000

000

,

,

,

(58)

[ ]

[ ]

[ ]

[ ] ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡Ψ

⎥⎦

⎤⎢⎣

⎡Ψ

⎥⎦

⎤⎢⎣

⎡Ψ

=Ψ

0N

0n

01

N3xN30

s

ss

000

000

000

,

,

,

(59)

The strip aerodynamic matrices 2n ,Ψ , 1n ,Ψ and 0n ,Ψ are presented in Appendix C by Eqs. (C.17), C.19), and C.24).

26

The matrix [ ]H is defined as follows:

[ ]

[ ]

[ ]

[ ] ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡

=×

s

S

N

n

1

SN3

H0

H0

H0

H (60)

The method of calculating the elements of the matrix [ ]H is described in detail in Appendix D.

The perturbation in the aerodynamic forces, ( ) t0 ,P , uu ∆∆ , acting on the oscillating blade, according to equation (12), can be written as:

( ) 1P0 t

×∆∆ ,P , uu =

= [ ] [ ] [ ] [ ] [ ]( ) [ ] 1SS3330122

3PAPP0 Hisss ×××××

Ψ+Ψλ+Ψλ− qsNNNNPT (61)

The matrix [ ]0T , is of order PP × and expressed as:

[ ]

[ ]

[ ]

[ ] ⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=×

ss NNM

nnM

11M

PP0

T

T

T

T

,

,

,

(62)

Expression for the matrix [ ]

nnMT,

, ( sNn1 ≤≤ ), appear in Appendix F.

The matrix [ ]AP can be expressed as follows:

27

[ ]

[ ]

[ ]

[ ] ⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=×

ss

s

N,N

,

1,1

N3

P

P

P

P

M

nnM

M

PA

00

00

00

(63)

The matrix [ ]AP is of order sN3P × . The matrices [ ] nnM ,P are defined in Appendix E.

The generalized aerodynamic force vector ( ) tgQ , (Eq. (39)), can be written as:

( ) [ ] ( ) t,,PtQg uu0 ∆∆Φ= Τ (64)

Substitution of Eq.(61) into Eq.(64) gives the following expression for the vector ( ) tgQ :

( ) [ ] [ ][ ] [ ] [ ] [ ]( ) [ ] ( )tHit 0122

A0Τ

g qΨ+Ψλ+Ψλ−Φ= ΡTQ (65)

Rewriting ( ) tgQ in a compact matrix form, gives:

( ) [ ] [ ] [ ]( ) 1SSxS0gSxS1gSxS2g2

1Sg it ××Α+Αλ+Αλ−= q,,,Q (66)

The matrices [ ]2g ,Α , [ ]1g ,Α and [ ]0g ,Α are given by:

[ ] [ ] [ ] [ ] [ ] [ ] SN3N3N32N3PAPP0PS2g ssssH ××××

Τ

×ΨΦ=Α ΡT, (67)

[ ] [ ] [ ] [ ] [ ] [ ] SN3N3N31N3PAPP0PS1g ssssH ××××

Τ

×ΨΦ=Α ΡT, (68)

[ ] [ ] [ ] [ ] [ ] [ ] SN3N3N30N3PAPP0PS0g ssssH ××××

Τ

×ΨΦ=Α ΡT, (69)

28

4. Flutter Analysis

The equations of motion of the blade (with the viscous damping matrix [ ]dC ) that were obtained in the previous section, can be written as follows (see Eqs.(38) ):

[ ] [ ] [ ] SSSxSgSSXSgSSxSg C gQ=Κ++Μ qqq (70)

The order of Eq. (70) depends on the number of modes which are used in Eq. (26).

This number is determined by performing numerical experiments, as it will be explained later. Since the blades of the propeller are assumed to be identical, the same equation is obtained for each blade.

In accordance with Eq. (66), Eq. (70) can be rewritten in following form:

[ ] [ ] [ ] =Κ++Μ SG

SxSgSG

SxSgSG

SxSg C qqq

= [ ] [ ] [ ] SG

SxS0g

2

750GSxS1g

750GSxS2g

2

Vb

Vbi q⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Α⎟⎟⎠

⎞⎜⎜⎝

⎛σ

+Α⎟⎟⎠

⎞⎜⎜⎝

⎛σ

+Α−λ ,.

,.

, (71)

Here, [ ]GgΜ , [ ]g

gC , [ ]GgΚ , [ ]G

2g ,Α , [ ]G1g ,Α are the generalized mass, viscous

damping, stiffness and aerodynamic matrices, expressed in the generalized system of coordinates, (superscript G).

[ ] [ ] [ ] [ ]ΦΦ=Μ Τ0

Gg M (72)

[ ] [ ] [ ][ ]ΦΦ=Κ Τ

0G

g K (73) [ ] [ ] [ ] [ ]ΦΦ= Τ

dg CCg (74)

[ ] [ ] [ ] [ ] [ ] [ ]H2A0ΤG

2g ΨΦ=Α ΡT, (75)

[ ] [ ] [ ] [ ] [ ] [ ]H1A0ΤG

1g ΨΦ=Α ΡT, (76)

[ ] [ ] [ ] [ ] [ ] [ ]H0A0ΤG

0g ΨΦ=Α ΡT, (77)

[ ]0M is discussed in Appendix B

In the present analysis [ ]gC will be neglected and the structural damping will be introduced as a fictitious damping proportional to the stiffness matrix [4].

29

)i21(MK s2sgsgs ξ+ω= , s=1,S (78)

where sω , sξ are the natural frequency of the sth mode and the structural damping ratio of that mode, respectively.

Thus the generalized stiffness matrix, [ ]gK , becomes a diagonal matrix of size S x S that consists of the elements given by Eq.(78) and can be expressed in following manner:

[ ]

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

ξ+ω

ξ+ω

ξ+ω

=

)i21(M0000

00)i21(M00

0000)i21(M

K

S2SSg

s2ssg

1211g

Gg

(79)

Assuming the solution for Eq. (71) is of the form:

ti

0S

0s

01

e

q

q

q

λ

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=q , or ti0 e λ= qq (80)

and taking into account that qq λ= i , qq 2λ−= , the flutter eigenvalue problem can be written as:

[ ] [ ] [ ] [ ] [ ] 0q 0=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛Κ+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛Α⎟⎟

⎠

⎞⎜⎜⎝

⎛σ

+Α⎟⎟⎠

⎞⎜⎜⎝

⎛σ

+Α−Μλ− gSxS0g

2

750SxS1g

750SxS2gg

2

Vb

Vbi ,

.,

., (81)

where:

Vb 75.0λ

=σ

If the following notation is used:

30

[ ] [ ] [ ] [ ]SxS0g

2750

SxS1g750

SxS2gg Vb

Vb

iA ,.

,.

, Α⎟⎟⎠

⎞⎜⎜⎝

⎛σ

+Α⎟⎟⎠

⎞⎜⎜⎝

⎛σ

+Α−=

Then the flutter eigenvalue problem can be written as:

[ ] [ ]( ) [ ]( ) 0q 0=Κ+Α+Μλ− ggg2 (82)

Dividing expression (82) by a referenced frequency, 20λ , after rearranging terms, Eq.

(82) can be written as an eigenvalue problem in following standard form:

[ ] [ ]( ) [ ] 0q 0=⎟⎟⎠

⎞⎜⎜⎝

⎛Κ

λ+Α+Μ

λ

λ− g2

0gg2

0

2 1 (83)

Ω=λ

λ ~20

2

; Ω=λλ ~ii

0

; (84)

[ ] [ ][ ] 0q0 =Ω−Π B~ (85)

where

[ ] [ ]g20

K1λ

=Π (86)

[ ] [ ] [ ]ggMB Α+= (87)

It should be noted that this is the nonlinear eigenvalue problem since the matrix [ ]gΑ is a function of λ . The solution of the above eigenvalue problem (85) results in S complex eigenvalues:

ν+µ=⎟⎟⎠

⎞⎜⎜⎝

⎛λλ ii

0

; 00 ii λν+λµ=λ (88)

The real part of the eigenvalue ( )µ represents the damping ratio, and the imaginary

part ( )ν represents the damped frequency; flutter occurs if 0≥µ for at least one of the eigenvalues.

The matrix [ ]gΑ includes, in the general case, complex terms.

31

5. Computer Code

The flutter analysis of propellers is based on the normal modes method. The flutter part of the code that uses two-dimensional subsonic unsteady aerodynamics will be used in the present investigation. The code is written in CI and is presented schematically in the form of a flow chart in Fig. 2. The input to the computer code and the steps involved of solving the flutter problem are shown in the flow chart. The input consists of the blade geometry described by the grid point's coordinates using the XYZ system (Fig.1) and the modal information - frequencies, mode shapes, and generalized masses. The main step, which is step No. 3 in Fig. 2, includes the calculation of the generalized aerodynamic matrix [ ]gΑ , using normal modes and an aerodynamic strip representation of the blade. The blade is divided into a series of discrete aerodynamic strips, each strip having constant properties. Each strip experiences two motions: plunging, h , and pitching,α , about an arbitrary reference line. h and α are expressed in terms of the normal modes and normal coordinates by using Eq.(D.26). The lift and moment acting on the nth section are obtained by using Eq. (C.16). For calculating modal data, the equivalent material properties for each element are generated by using a preprocessor code of the FEM software. After the stiffnesses, masses, and actual or equivalent nodal loads for the elements are generated, they are used to form the system matrices using the direct stiffness method [12]. In this approach the contributions of all the elements are added to obtain stiffnesses, masses, and nodal loads for the entire system. Thus, by a summation along the chord one obtains:

[ ] [ ]∑=

=c

cc

M

1mmn KK , [ ] [ ]∑

=

=c

cc

M

1mmnM M , ∑

=

=c

cc

M

1mmnQ Q (89)

where CM is the number of elements in the chordwise direction, mc=1, 2, 3,…, CM . Note, that for the operations of Eqs. (89), is necessary the matrices or vectors on the right side are extended in size (using zeros) so that they will have the same size as the matrix or vector on the left side.

In Eqs. (89) the symbols [ ]nK , [ ]nM and nQ represent the stiffness, mass matrix and the nodal loads vector, respectively, for the nth strip of the blade. Then the undamped equations of motion, Eqs. (38), for the assembled system, become

[ ] [ ] ggg QKM =+ qq (90)

Equation (90) gives the system equations of motion for all nodal displacements. The frequencies, mode shapes and generalized masses are obtained as a result of a solution of the homogeneous Eq. (90).

32

Figure 2: Flow Chart

33

5.1. Calculation of the Mode Shapes:

The finite element method (FEM) to be used in this study involves assumptions concerning the displacement shape functions (see Appendix D) within each shell element. These functions give approximate results when the shell element is of a finite size and exact results at infinitesimal size. The shape functions define the displacements at any point as functions of the nodal displacements.

From FEM we obtain, for each node, three linear displacements 1w~~ , 2w

~~ , 3w~~ and

three rotary displacements 4w~~ , 5w~~ , 6w~~ corresponding to the X,Y,Z system of coordinates and to the frequency of natural oscillations. If we are interested in calculating the heave, h , and pitch,α (see Appendix D):

For bending oscillations

03

s,n02

s,ns,n cosv~sinv~h β∆+β∆= (91)

2s,nv~∆ and 3

snv~∆ are defined as:

∑=

=∆c

c

c

M

1m

2s,m,n

c

2s,n w~

M1v~ (92)

∑=

=∆c

c

c

M

1m

3s,m,n

c

3s,n w~

M1v~ (93)

s=1, 2, 3… number of modes, n=1, 2,…, Ns are number of strips along the blade, mc=1, 2,…, Mc are number of finite element along the chord.

∑=

=∆=αc

c

c

M

1m

4s,m,n

c

4s,ns,n w~

M1v~ (94)

5.2. Calculation of the Generalized Mass From ANSYS we obtain, for the nth strip of the blade, for its motion in the sth mode, the following value of kinetic energy, ( nE )s,

( ) ( )∑=

=m

1esesn EE (95)

E e is the kinetic energy of eth finite element, m is the number of finite element , m=1, 2, 3…. SC MM × s , where CM and SM are the number of elements in chordwise and spanwise directions, respectively.

34

Note that in what follows the number of elements in the spanwise direction will be equal to the number of strips in same direction, namely SS NM = . In ANSYS the kinetic energy, by definition, is:

( ) [ ]( )snnΤ

nsn uMu21E = , (n=1, 2, 3,…, Ns, s=1, 2, 3,..., S) (96)

If one uses the notations:

[ ] qu nn Φ= ; [ ]ΤΤΤ Φ= nn qu (97)

the kinetic energy for the nth strip of the blade, for its sth mode, can be written as:

( ) [ ] [ ] [ ]( )sn

2ssn M

21E nn ΦΦω= Τ (98)

Thus the generalized mass of the strip is

[ ]( ) [ ] [ ] [ ]( )snnnsng M ΦΦ=Μ Τ (99)

Hence

[ ]( ) ( )2s

sn

sng

E2ω

=Μ (100)

5.3. Calculation of the Lift-Curve Slope

A computer program which is based on NASA-TM-85696 [17] has been written. This code provides a comprehensive data base for the NACA 16-series airfoils. The geometry covered in the program is limited to cambers representing design-lift coefficients from 0.0 to 0.7 and thickness ratios from 4 to 21 percent. The data includes Mach numbers from 0.3 to 1.6, angles of attack from -4 to 8 degrees, and lift coefficients from 0.0 to 0.8. Extrapolation is used in order to obtain data for Mach numbers, angles of attack, and lift coefficients that are outside the above mentioned region. An additional subroutine was introduced [17] in order to account for the nonlinear behavior of lift coefficient beyond the stall angle. Essentially, the lift coefficient becomes a function of the angle of attack as well as of the Mach number, Reynolds number, and geometry. The drag coefficient is also calculated using this subroutine. After obtaining the angle of attack, the lift-curve slope is calculated.

35

6. Results and Discussions

For an initial verification of the aeroelastic code the SR2-EC propeller is chosen. This SR2 propeller has highly loaded thin blades that experience deflections that are larger than in the case of conventional propellers. There is an experimental data of this propeller, which can be compared with the numerical results

6.1. The SR2-EC propeller

The SR2-EC propeller was designed to maintain high propulsive efficiencies at high cruise speeds, up to Mach 0.8. The SR2-EC propeller has eight straight blades, each blade has a large twist with thin airfoil sections, to minimize compressibility losses [1]. EC in the title (SR2-EC), stands for Equivalent Composite.

The SR2-EC was designed, fabricated and tested by Hamilton Standard (USA), under the NASA Advanced Turboprop Program (ATP). The propeller disk diameter is 2 ft (0.622m) with a design blade tip speed of 800 ft/sec (243 m/sec). The global coordinate system of the SR2 blade is shown in Fig. 3. The geometry of the blade is shown in Fig. 4 and is summarized in Table 1. A computer code was developed to construct the blade geometry and FEM, based on the above data. Table 2 presents the equivalent mechanical properties of the material, which are used during the analysis.

`The blade FEM model includes 128 ANSYS SHELL 93 elements and 433 nodes as is shown in Ref. 1 (see Fig. 3.2). The boundary conditions at the blade root are chosen such that it is fully fixed, namely the six degrees of freedom along the blade root are restrained in the FEM, (see Fig. 3.2 in Ref. 1)

36

Figure 3: Global coordinate system of the SR2 blade.

37

Table1.The geometry of a blade of the SR2-EC propeller

(the cross sections are NACA16 airfoils )

r/R c/d t0 /c β0 [deg] CLd

0.235 0.1440 0.2200 21.119 -0.165 0.25 0.1450 0.1400 20.334 -0.132 0.30 0.1460 0.1020 17.785 -0.035 0.35 0.1470 0.0700 15.340 0.039 0.40 0.1490 0.0580 13.000 0.096 0.45 0.1495 0.0500 10.764 0.135 0.50 0.1500 0.0430 8.6330 0.159 0.55 0.1493 0.0360 6.6055 0.170 0.60 0.1490 0.0330 4.6823 0.170 0.65 0.1480 0.0250 2.8633 0.160 0.70 0.1460 0.0220 1.1486 0.143 0.75 0.1450 0.0210 -0.4617 0.120 0.80 0.1410 0.0200 -1.9679 0.095 0.85 0.1360 0.0190 -3.3698 0.068 0.90 0.1260 0.0188 -4.6674 0.041 0.95 0.1100 0.0186 -5.8608 0.017 1.0 0.0580 0.0185 -6.9499 -0.001

Table 2: Equivalent material properties of the SR2-EC blade

Material Name

Density ρ0, [kg/m3]

Young’s modulus Ε0, [GPa]

Poisson’s ratio µ0

Equivalent Composite

1540

64

0.33

38

Figure 4: The geometry of a blade of the SR2-EC propeller 6.2. Results and Discussions

In what follows initial results of using the code are presented. Figure 5 shows the mode shapes used in the study, as calculated by ANSYS for

Ω=6000 rpm, β0.75R =55.4 deg. It can be seen from Fig.5 that the first mode is primarily bending, the second mode is bending-torsion and the third mode is primarily torsion. The presence of a torsion component in the first mode and a flexural component in the third mode is negligible small. At the same time the flexural and torsion components in the second mode are of similar importance

The variations of the angle of attack,α , the lift curve slope, αddCL , along the blade are presented in Figs.6 and 7, for β0.75R =55.4 deg, Mach number M=0 and rotational speeds that vary in the range, Ω=2000 -10000 rpm. As shown in Fig. 6, the angles of attack are larger than 20 degrees for all the cross-sections of the blade for the entire indicates range. This indicates a stalled flow over the entire blade. The lift-curve slope along the blade is calculated in this case using NASA Code, [17], and is presented in Fig. 7. From Fig. 7 it can be seen that the slope increase sharply from 0 to 1.6 for angles of attack in the range .deg35deg50 ≥α≥ The slope increase from 1.6 to 2.25 for angles of attack the range .deg20deg35 ≥α≥

Figure 8 shows the variation of the damping and the frequency vs. the rotational speed. The calculations were carried out using NASA Code [17] to describe stall. The pitch angle equals 55.4 deg and the structural damping ξ=0.002 for all the three modes. The mode damping is presented as damping ratio which is:

39

100dampingcritical

damping×

, where the critical damping is equal to 0.002

From Fig. 8 it can be seen that the frequencies of oscillations of the first and second modes increase with an increase of the rotational speed. At the same time the frequency of the torsion oscillations, namely the third mode, decrease vs. the increase of the rotational speed. The damping ratio decrease significantly for the first bending and torsion modes and decrease slightly for second bending mode with increase of rotational speed. The damping ratio of the first bending mode is larger than the damping ratio for the second bending mode, but the damping ratio of the third mode (the first torsion mode) is larger than the damping ratio of the first bending mode.

The effect of structure damping on the damping ratio and the frequency is presented in Figs. 9-11. As expected, the increase of structure damping leads to the increase of the damping ratio for all modes. Moreover, the damping ratios of the first bending and torsion modes get close by the increase of structural damping. The dependence of the frequencies on the structure damping is smaller than the dependence of the damping ratio.

The variations of the angle of attack,α , the lift curve slope, αddCL , along the blade are presented in Figs.12 and 13 for β0.75R =55.4 deg, rotational speed, Ω=6000 rpm and various Mach numbers in the range 0.1M0 ≤≤ . As shown in Fig. 12,.the angles of attack are larger than 10 degrees for all the cross-sections for velocities in the range 2.0M0 ≤≤ . This indicates a stalled flow over the entire blade. The lift-curve slope along the blade is presented in Fig. 13. From Fig. 13, it can be seen that the slope is less than 5 for angles of attack in the range .deg10≥α and Mach number M<0.4. This indicates a state stall.

Figure 14 shows the variation of the damping ratio and the frequency vs. the Mach number. The results of the calculations are presented for the following case: β0.75R =55.4 deg., ξ=0. It can be seen that the damping for all modes decrease as the Mach number increases. At the same time the frequencies of the bending modes are almost invariable, but the frequency of first torsion mode is decreasing. The sequence of modes in this case is remained the same.

40

0.0 0.2 0.4 0.6 0.8 1.0r/R

-0.010

0.005

0.020

0.035

0.050

TorsionFlap

Mode 1

0.0 0.2 0.4 0.6 0.8 1.0r/R

-0.015

-0.010

-0.005

0.000

0.005

Mode 2

0.0 0.2 0.4 0.6 0.8 1.0r/R

-0.1

0.2

0.5

0.8

1.1

Mode 3

Figure 5.Mode shapes calculated with respect to the blade coordinates system: β0.75R =55.4 deg, Ω=6000 rpm.

41

0.0 0.2 0.4 0.6 0.8 1.0r/R

20

25

30

35

40

RPM = 2000RPM = 4000RPM = 6000RPM = 8000RPM = 10000

α

Figure 6.Variation of the angle of attack along the blade for various rotational speeds: β0.75R =55.4 deg, Mach number M=0.

42

0.0 0.2 0.4 0.6 0.8 1.0r/R

0.5

1.0

1.5

2.0

2.5

dCL/

dα

RPM = 2000RPM = 4000RPM = 6000RPM = 8000RPM = 10000

Figure 7.Variation of the lift-curve slope along the blade for various rotational speeds: β0.75R =55.4 deg, Mach number M=0.

43

Figure 8.Variation of damping and frequency vs. the rotational speed: β0.75R =55.4 deg., ξ=0,

44

Figure 9. Variation of damping and frequency vs. the rotational speed: β0.75R =55.4 deg., ξ=0,002.

45

Figure 10. Variation of damping and frequency vs. the rotational speed: β0.75R =55.4 deg., ξ=0,02,

46

Figure 11. Variation of damping and frequency vs. the rotational speed: β0.75R =55.4 deg., ξ=0,2.

47

0.0 0.2 0.4 0.6 0.8 1.0r/R

-10

0

10

20

30

40

M = 0M = 0.2M = 0.4M = 0.6M = 0.8M = 1.0

α

Figure 12.Variation of the angle of attack along the blade for various Mach numbers: β0.75R =55.4 deg, Ω=6000 rpm

48

0.0 0.2 0.4 0.6 0.8 1.0r/R

0

1

2

3

4

5

6

7

8

dCL/

dα

M = 0M = 0.2M = 0.4M = 0.6M = 0.8M = 1.0

Figure 13.Variation of the lift-curve slope along the blade for various Mach

numbers: β0.75R =55.4 deg, Ω=6000 rpm.

49

Figure 14.Variation of damping and frequency vs. the free-stream Mach number: β0.75R =55.4 deg., Ω=6000 rpm, ξ=0.2.

50

7. Conclusions

The model is based on a finite element structural model of the blade and a two-dimensional unsteady aerodynamic model of a cross-section of the blade, based on Theodorsen's model. The coupling between the two models, structural and aerodynamic, is described in the report.

A computer code, based on the model has been assembled. The basic state of the blade is calculated using nonlinear aerodynamic data of the blade's cross-sections.

Initial results of the code are presented in the report and include frequency and damping of the various modes.

51

References 1. Yadykin, Y., Tenetov, V., Weissberg, I., and Rosen, A.; ” Aeroelastic Analysis of

Propellers, Part I - “ Static ” Analysis “ Technion-Israel Institute of Technology, TAE No.932, 2004

2. Mikkelson, D.C., Mitchell, G.A., and Bober, L.J., "Summary of Recent NASA Propeller Research," NASA TM-83733, 1984.

3. Kaza, K.R.V., Mehmed, O., Narayanan, G.V. and Murthy, D.V. "Analytical Flutter Investigation of a Composite Propfan Model," Journal of Aircraft, Vol. 26, No. 8, Aug. 1989, pp. 772-780

4. Reddy, T.S.R, Srivastava, R., and Mehmed, O. " ASTROP2-LE: A Mistuned Aeroelastic Analysis System Based on a Two Dimensional Linearized Euler Solver", NASA/TM-2002-211499, May 2002.

5. Elchuri, V., and Smith, G.C.C., "Flutter Analysis of Advanced Turbopropellers," AIAA Journal, Vol. 22, No. 6, June 1984, pp.801-802.

6. Turnburg, J.E., "Classical Flutter Stability of Swept Propellers, " AIAA Paper 83-0847, May 1983.

7. Mehmed, D., and Kaza, K.R.V., "Experimental Classical Flutter Results of a Composite Advanced Turboprop Model," NASA TM-88972, 1986.

8. Theodorsen, T., General Theory of Aerodynamic Instability and the Mechanism of Flutter, N.A.C.A. Report 496, 1935.

9. Gupta, K. K., Development of a Unified Numerical Procedure for Free Vibration Analysis of Structures, International Journal for Numerical Methods in Engineering, Vol. 17, 1981, p.p.187-198.

10. Kosmatka, J. B. and Friedmann, P. P., Structural Dynamic Modeling of Advanced Composite Propellers by the Finite Element Method, AIAA Paper 87-0740, 1987, pp. 111-124.

11. Loewy, R.G., Rosen, A., Mathew, M. B., Application of the Principal Curvature Transformation to Nonlinear Rotor Blade Analysis, AIAA/ASME/ASCE/AHS, 27th Structures, Structural Dynamics and Materials Conference, Part 2, A Collection of Technical Papers, San Antonio, Texas, May 19-21, 1986, Paper 86-0843, pp. 55-76.

12. Weaver, J.R., Timoshenko, S.P., Young, D.H., Vibration Problems in Engineering, Fifth Edition, 1990

13. Foss, K.A., "Coordinates which uncouple the equations of motion of damped linear dynamic systems.," Jour. Appl. Mech., ASME, 25, 1958, pp.361-364.

14. Srivastava, R., Reddy, T.S.R. and Stefko, G.L., A Numerical Aeroelastic Stability Analysis of a Ducted-Fan Configuration, AIAA Paper 96-2671, 1996, pp.1-8

15. Ie, C.A. and Kosmatka, J.B., "Formulation of a Nonlinear Theory for Spinning Anisotropic Beams", Recent Advances in the Structural Dynamic Modeling of Composite Rotor Blades and Thick Composites, AD-Vol. 30, ASME 1992, pp.41-57.

16. Bisplinghoff, R.L., Ashley, H., and Halfman, R., Aeroelasticity, Dover Publications, Inc. Mineola, New York, 1996.

17. Maksymiuk,C.M., and Watson, S., A., A Computer Program for Estimating the Aerodynamic Characteristics of NASA 16-Series Airfoils, NASA TM-85696, Sep. 1983.

18. Yadykin, Y., Tenetov, V., and Rosen, A., “Aerodynamic Modeling of Propellers, Part 1” Technion-Israel Institute of Technology, TAE No.888, 2002.

52

Appendix A: Explanation of Nonlinearities in the Finite Element Analysis A.1. Introduction to Nonlinearities

This Appendix discusses the different geometrically nonlinear options within the ANSYS program including large strain, large deflection, stress stiffening, and spin softening. Only elements with displacements degrees of freedom (DOF's) are applicable. Geometric nonlinearities refer to the nonlinearities in the structure or component due to the changing geometry as it deflects. The stiffness [ ]K is a function of the displacements u . The stiffness changes because the shape changes and/or the rotations. The program can account for four types of geometric nonlinearities: 1 Large strain- Assumes that the strains are no longer infinitesimal (they are finite). Shape changes (e.g. area, thickness, etc.) are accounted for. Deflections and rotations may be arbitrarily large. 2. Large rotation- Assumes that the rotations are large but the strains (those that cause stresses) are evaluated using linearized expressions. The change in the shape of the structure is neglected, except for rigid motions. The elements refer to the original configuration. 3. Stress stiffening- Assumes that both, strains and rotations, are small. A 1st order approximation to the rotations is used to capture a few nonlinear rotation effects. 4. Spin softening- Also assumes that both, strains and rotations, are small. This option accounts for the radial motion of a structural mass as it is subjected to an angular velocity. Hence it is a type of large deflection but small rotation approximation. In what follows the effects of stress stiffening and spin softening are further

explained: A.2. Stress Stiffening Stress stiffening is the stiffening (or weakening) of a structure due to its stress state.

This stiffening effect normally needs to be considered for thin structures with small bending stiffness compared to their axial stiffness, such as thin beams. The effect couples the in-plane and transverse displacements. This effect also augments the regular nonlinear stiffness matrix produced by large strain or large deflection effects. The effect of stress stiffening is accounted for by generating and then using an additional stiffness matrix, hereinafter called the "stress stiffness matrix". The stress stiffness matrix is added to the regular stiffness matrix in order to give the total stiffness. Stress stiffening may be used for static or transient analyses. The stress stiffness matrix is computed based on the stress state of the previous equilibrium iteration. Thus, to generate a valid stress-stiffened problem, at least two iterations are normally required, with the first iteration being used to determine the stress state that will be used to generate the stress stiffness matrix of the second iteration. If this additional stiffness affects the stresses, more iterations need to be done to obtain a converged solution In some linear analyses, the static (or initial) stress state may be large enough that

the additional stiffness effects must be included for accuracy. Modal analysis is linear analysis for which the prestressing effect can be included. Note, that in this case the

53

stress stiffness matrix is constant, since that the stresses computed in the analysis is assumed small compared to the prestress stress A.3. Spin Softening

The vibration of a spinning blade will cause relative circumferential motions, which will change the direction of the centrifugal load which, in turn, will tend to stabilize or destabilize the structure. As a small deflection analysis cannot directly account for changes in geometry, the effect can be accounted for by an adjustment of the stiffness matrix, called spin (or centrifugal) softening. Spin softening is used during the modal analysis.

54

Appendix B: Matrices The ANSYS code calculates the stiffness and mass matrices. Yet, in what follows

expressions of these matrices, that were used for verification, are presented. Matrices [ ]0M , [ ]CFK , CFF , GRF , [ ]0T , and [ ]0T∆ , that are presented in

Eq.(15), are defined as follows [ 15 ]: [ ] ( ) dzdymmmmmmM

A

T33

T22

T110 ∫∫ ++ρ= (B.1)

[ ] [ ] [ ]( )Τ+= SSCF KK21K (B.2)

[ ]( )( )( )

dzdy

mm2mm

mm2mm

mm2mm

KA T

21yxT

332Y

2x

T31zx

T22

2z

2x

T32zy

T11

2z

2y

s ∫∫⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

ΩΩ−Ω+Ω+

+ΩΩ−Ω+Ω+

+ΩΩ−Ω+Ω

ρ= (B 3)

( )( ) ( )( )( ) ( )( )( ) ( )

∫∫⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

+−++

++−++

++−+

ρ=A

3c

yt

xz3n2

Y2x

2t

xn

zy2c2

z2x

1n

zc

yx1t2

z2y

CF dzdy

m)(hΩ)(hΩΩmhΩΩ

m)(hΩ)(hΩΩmhΩΩ

m)(hΩ)(hΩΩmhΩΩ

F (B.4)

[ ]( ) (

) ( ) dzdymmmmmm

mmmmmm2F

AT

12T

21zT

31

T13y

T23

T32x

GR ∫∫⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−Ω+−

−Ω+−Ωρ−= (B.5)

Here

[ ]y,z,0,0,0,1m T1 −= , [ ]0,0,z,0,1,0m T

2 −= , [ ]0,0,y,1,0,0m T3 = (B.6)

The components of the rotational velocity vector are [15]:

( ) ( ) [ ]T0zyx 00 T,,,, Ω=ΩΩΩ (B.7)

Hence the matrices can be written a follows: [ ] [ ]( ) ( ) dzdymmmmTK

A

T22

T11

2T0s ∫∫ −Ωρ= (B.8)

[ ] [ ]( ) ( ) dzdym)h(m)h(TΩρF

A2

c1

t2T0CF ∫∫ += (B.9)

[ ] [ ] ( ) dzdymmmmT2F

A

T12

T21

T0GR ∫∫ −Ωρ−= (B.10)

55

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ββ−ββ=

00

000

00

001

cossinsincosT , (B.11)

0β is the pitch angle of the cross-section of the blade.

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−=

ααααT)(sinβ)(cosβ0

)(cosβ)(sinβ0001

∆

00

000 , (B.12)

In two-dimensional case [ ] [ ]30 TT =

[ ]⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−≡

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

LDM

cossinsincos

LDM

TFF

M

0

Z

Y

X

00

00

ββ0ββ0

001 (B.13)

56

Appendix C: Aerodynamic Forces and Moments Acting on an Oscillating Airfoil The problem of calculating the aerodynamic forces acting on a two-dimensional

airfoil moving in simple harmonic motion about an equilibrium position (of incoming uniform flow-fixed wing), was analyzed by Theodorsen [8]. In [8] it was assumed that the lift and moment per unit span, due to blade motion, are linearly related to the displacements and their derivatives with respect to time.

In the present analysis the influence of the drag force will be neglected, namely 0=∆ D . In the future it will be possible to replace it by more complicated models.

Following Theodorsen, we consider a chordwise-rigid airfoil. The airfoil motion is described by a vertical translation )(th and rotation about axis OX at abY = , through an angle )(tα . The positive direction of these variables is indicated in Fig. 2.

Theodorsen determined the forces and moments acting on an oscillating airfoil section, in an incompressible flow. The expressions for the lift, L∆ , and moment ,

M∆ , are derived using the airfoil system of coordinates. With the sign convention of both, the plunging and leading edge displacements due

to pitching, positive upward (Fig. 1), the lift and moment, at the cross-section r of the blade, for a specified frequency, using the notation of Ref. 8, are:

( ) ( )[ ]⎭⎬⎫

⎩⎨⎧

+−++−+−= ασ)(FbV2α)σ(Fa2Vαabh)σ(F

bV2hbρπtr,L

2

11∆ 2 (C.1)

( ) ( ) +α⎟⎠⎞

⎜⎝⎛ −+

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−++=∆

21213 aVαa

81bh

bV)σ(Fahabρπtr,M 2

( )⎭⎬⎫

⎥⎦

⎤⎢⎣

⎡+−⎟

⎠⎞

⎜⎝⎛ ++ ασ)(F

bVαVσ)(Faa

2

2121 (C.2)

Here, a is the nondimensional distance between the mid-chord point and the point where h is measured.. Vbλ=σ is the reduced frequency, λ is the circular frequency of oscillations, b is half chord of the blade cross-section and V is the velocity of incoming air flow The quantities h and α represent the plunging and pitching displacements of the blade cross-section and are functions of time.

The lift deficiency function, ( )σF , is a function of the reduced frequency,σ , and is expressed by Hankel functions, or separated into real and imaginary parts and expressed in terms of Bessel functions of the first and second kind:

( ) )σ(Gi)σ(ΘσF += , (C.3)

where

( ) ( ) ( )( ) ( )2

012

01

011011

JYYJJYYYJJ

−++−++

=σΘ (C.4)

and

57

( )( ) ( )2

012

01

0101

JYYJJJYY

−+++

=σG (C.5)

The function ( )σF is worth a short discussion, since it plays a roll in the analysis

of not only harmonic oscillations but also other unsteady cases [16]. The argument of all the Bessel functions is the reduced frequency, σ , which is regarded as the best measure of the “unsteadiness” of oscillations in an incompressible flow. An idea of the roll of ( )σF comes from an examination of the simplified lift and moment expressions used by aeronautical engineers when dealing with low-reduced-frequency unsteady motions.

In the simplified case it is assumed that all aerodynamic loads are obtained from steady-state formulas except that the angle of attack α is replaced by the instantaneous inclination between the resultant velocity vector and the chordline. In the case of h and α motions, this procedure would lead to [16]:

( )( ) ⎪⎭

⎪⎬⎫

+−≅

+−≅

a0.5bLM

αVhVbρπ2L

00

0 (C.6)

Note, that Eq. (C.6) can be rearranged as:

⎪⎪

⎭

⎪⎪

⎬

⎫

⎥⎦

⎤⎢⎣

⎡α+⎟

⎠⎞

⎜⎝⎛ +≅

⎥⎦

⎤⎢⎣

⎡α+−≅

)(V2

1b

)(V

b

L2

L

Chπ2aVρM

Chπ2VρL

20

20

(C.7)

where, according to the linear theory:

απ=α 2CL )( (C.8)

The expression (2π ) in equations (C.7)-(C.8) is the theoretical value of the lift curve-slope, αddCL , [16]:

Using the lift curve-slope, αddCL , equations (C.1) and (C.2) may be expressed as:

( )

( ) ( ) ( )

( )⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

αα

π+

⎥⎦

⎤⎢⎣

⎡α

απ

−++−

αα

π+

−=

ασ)(Fb

V

αV)σ(Fa2αabh)σ(FVhbρπtr,L

ddC

ddC

211

ddC

b∆

L2

LL

2 (C.9)

58

( ) ( ) ( )α⎟

⎠⎞

⎜⎝⎛ −+

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−

αα

π+

+=∆21

ddC

21 L3 aVαa

81bh

bV)σ(Fa2habρπtr,M 2 +

( ) ( ) ( )⎪⎭

⎪⎬⎫

⎥⎦

⎤⎢⎣

⎡

αα

π+

αα

π−

⎟⎠⎞

⎜⎝⎛ ++ ασ)(F

ddC

bV

21αVσ)(F

ddC

2a21a

21 L

2L (C.10)

Expressions (C.9) and (C.10) can be written, in a matrix form, as:

( )( ) [ ] [ ] [ ]

⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

∆∆

αh

CαhC

αhC

tr,Mtr,L

321 (C.11)

Where, the aerodynamic matrices [ ]1C , [ ]2C and [ ]3C are:

[ ] ( )⎥⎦⎤

⎢⎣

⎡

+−−

= 243

32

1 a0.125bρπabρπabρπbρπ

C (C.12)

[ ]

( ) ( ) ( )

( ) ( ) ( ) ( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡π−

+−+

⎥⎦

⎤⎢⎣

⎡−

π−

−=

)σ(F)σ(F

)σ(F2

)σ(Fb1

2

dααdCa0.250.5abπa0.5

dααdC

1dααdC1a2π

dααdC

VbρCL

2L

LL

2 (C.13)

[ ] ( )( )⎥⎦

⎤⎢⎣

⎡+

−=

a0.500

αdαCdC L

3 b1

σ)(FbVρ 2 C14)

During the present aerodynamic analysis the blade is divided into sN strips in the spanwise direction (see Fig.D.2). Each strip has two "aerodynamic" degrees of freedom: a plunging displacement, nh , and a pitching displacement, nα . Since the motion is harmonic, a circular frequency λ is defined. The plunging and pitching motions of the nth cross-section ( sNn1 ≤≤ ) can be expressed as:

ti

n

n

n

n eh

αh λ

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

α=

⎭⎬⎫

⎩⎨⎧

0

0

(C.15)

00nn ,h α are the plunging and pitching amplitudes at cross-section n.

After substituting Eq.(C.15) into Eqs. (C.11), the lift and moment acting on nth oscillating strip of the blade, are given by:

59

[ ] [ ] [ ]( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

αΨ+Ψλ+Ψλ−=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∆

∆

n

n0n1n2n

2

n

n hi ,,,M

L (C.16)

The matrices 2n ,Ψ , 1n ,Ψ , 0n ,Ψ are complex square matrices of order 2 x 2, defined as:

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡=Ψ

22

22

n2nn,n,

n,n,

, SR

QPbπ (C.17)

nlnn,2 bρP −= , albρQ nnn,22= , albρR nnn,2

2= , ⎟⎠⎞

⎜⎝⎛ +−=

81albρS 2

nnn,23 (C.18)

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡=Ψ

11

11

n1nn,n,

n,n,

, SR

QPbπ (C.19)

( ) ( )nn

nnn,1 σFdααdC

bVlbρP L

22

π−= (C.20)

( ) ( ) ( )⎥

⎦

⎤⎢⎣

⎡π

−+−= n

nnnn,1 σF

dααdC2a11

bVlbρQ L2

2 (C.21)

( ) ( ) ( )n

nnnn,1 σF

dααdCa21

bVlbρR L2

2π+

= (C.22)

( ) ( ) ( ) ⎥

⎦

⎤⎢⎣

⎡π

+−⎟

⎠⎞

⎜⎝⎛ −= n

nnnn,1 σF

dααdCa211

21a

bVlbρS L3

2 (C.23)

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡=Ψ

00

00

n0nn,n,

n,n,

, SR

QPbπ (C.24)

00n =,P ( ) ( )n

2

nn,0 σFdααdCVlρQ

π−=

00n =,R ( ) ( )n

2

nnn,0 σFdααdCa

21VlbρS ⎟

⎠⎞

⎜⎝⎛ +

π= (C.25)

Here, nb , nσ , nl , ρ are, respectively: half chord, reduced frequency (based on nb ), length of the strip, and air density.

60

The aerodynamic matrices, [ ]2Ψ , [ ]1Ψ and [ ]0Ψ for the entire blade, are a diagonal matrices of order ss N3N3 × , given by:

[ ]

[ ]

[ ]

[ ] ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡Ψ

⎥⎦

⎤⎢⎣

⎡Ψ

⎥⎦

⎤⎢⎣

⎡Ψ

=Ψ

2N

2n

21

N3xN32

s

ss

000

000

000

,

,

,

(C.26)

[ ]

[ ]

[ ]

[ ] ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡Ψ

⎥⎦

⎤⎢⎣

⎡Ψ

⎥⎦

⎤⎢⎣

⎡Ψ

=Ψ

1N

1n

11

N3xN31

s

ss

000

000

000

,

,

,

(C.27)

[ ]

[ ]

[ ]

[ ] ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡Ψ

⎥⎦

⎤⎢⎣

⎡Ψ

⎥⎦

⎤⎢⎣

⎡Ψ

=Ψ

0N

0n

01

N3xN30

s

ss

000

000

000

,

,

,

(C.28)

The aerodynamic matrices 2n ,Ψ , 1n ,Ψ and 0n ,Ψ are defined by Eqs.(C.17), (C.19), and (C.24) respectively.

61

Appendix D: Calculating the matrix [ ]H

According to Eq. (25) of the main text:

( ) [ ] ( )tt qu Φ=∆ (D.1)

u∆ is the displacement vector. If we use a finite element code, then ku∆ denotes the kth displacement (displacement related to the kth degree of freedom) of the th node. Thus the vector u∆ becomes:

KL

kL

1L

K1kk1k1k1

21

11 000

uu,u,u,u,u,u,u,u,u,u ∆∆∆∆∆∆∆∆∆∆∆=∆ +−Τu (D.2)

where:

0L1 ≤≤ , Kk1 ≤≤ (D.3)

0L is the total number of nodes, K is the number of degrees of freedom at each node.

q is the vector of generalized coordinates, defined as follows:

Ss21 q,,q,q,q ……=Τq (D.4) [ ]Φ is the matrix of eigenvectors, the dimension of which is SP × , where S is the number of modes used in the analysis:

[ ]

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

φφφ

φφφ

φφφ

=Φ ×

S,Ps,P1,P

S,is,i1,i

S,1s,11,1

SP (D.5)

and:

KLP 0 ×= , Pi1 ≤≤ (D.6)

If ks,w~ denotes the kth displacement (displacement related to the kth degree of

freedom) of the th node, of the ths mode shape, then:

( )[ ] s,kK1k

s,w~ +−φ= (D.7)

where Kk1S,s1,L1 0 ≤≤≤≤≤≤ (D.8)

The blade is divided into sN strips. In each strip there are Mc elements. The element (n, 1) is the leading edge element of strip n, while the element (n, Mc) is the trailing edge element of that strip. Each element has eI nodes. The perturbation of the

62

displacement which is related to the kth degree of freedom, of node i, of element mc of strip n, is denoted: k

i,m,n cw~~ (D.9)

eccs Ii1;Mm1;Nn1;Kk1 ≤≤≤≤≤≤≤≤

It is convenient now to define a vector cm,nw of dimensions eN where:

KIN ee ⋅= (D.10)

The vector

cm,nw describes the displacements at the nodes of the element (n, mc) and is defined as follows:

KI,m,n

ki,m,n

2I,m,n

21,m,n

1I,m,n

12,m,n

11,m,nmn, ecceccecccc

w~~,,w~~,,w~~,,w~~,w~~,,w~~,w~~=Τw

(D.11) For further derivations it is convenient do define a vector w of the perturbation displacements of all the elements, the dimensions of which is N0, where:

ec0 NMN ⋅= (D.12) The vector w is defined as:

TM,n

Tm,n

T2,n

T1,n cc

,,,, wwwwwn =Τ (D.13)

It is possible, based on the finite element scheme, to write down the following relation:

[ ] )t(PN0uAwn ∆= × (D.14)

The elements of the matrix [ ]A are either one (1) or zero (0), depending on the finite element structure of the blade.

For a three-dimensional shell element as shown in Fig. D.1, k=1, 2, 3 correspond to the linear displacements in the directions X, Y, Z, (respectively) of the global system of coordinates, k=4, 5, 6 refer to the rotations about the axes X, Y, Z, respectively, K is equal to 6.

63

Figure D1. 3D quadratic shell element Let us return to the element (n, mc). The kth displacement over the element depends on the kth displacements at its nodes and is given as a function of the element coordinates r, s and t see Fig. D.1. Thus

ki,m,n

I

1ii

km,n c

e

cw~~)t,s,r(Nw~ ∑

=

⋅= (D.15)

where )t,s,r(Ni are the element shape functions. In the present analysis Shell 93 element is used (see Fig.D.1). In this case 8Ie = and the functions )t,s,r(Ni are reduced to )t,s(Ni where:

( )( )( )1tst1s141N1 −−−−−= , ( )( )( )1tst1s1

41N 2 −−−+= ,

( )( )( )1tst1s141N3 −+++= , ( )( )( )1tst1s1

41N4 −+−+−= ,

( )( )t1s121N 2

5 −−= , ( )( )26 t1s1

21N −+= ,

( )( )t1s121N 2

7 +−= , ( )( )28 t1s1

21N −−= . (D.16)

The indeces 1, 2, 3, 4, 5, 6, 7, 8 correspond to the nodes I, J, K, L, M, N, O, P of the finite element (Fig. D.1 ) At the point s = t = 0, namely the middle point of the element Shell 93, the values of the shape functions are:

41NNNN 4321 −==== ,

21NNNN 8765 ==== (D.17)

The representative kth displacements of element (n, mc), k

m,n cw~~ , will be the

displacements of the middle point of the element. For the element Shell 93, it is given by:

64

⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

⋅⎥⎦⎤

⎢⎣⎡ −−−−=

k8,m,n

17,m,n

k6,m,n

15,m,n

k4,m,n

13,m,n

k2,m,n

k1,m,n

km,n

c

c

c

c

c

c

c

c

c

w~~w~~w~~w~~w~~w~~w~~w~~

21

21

21

21

41

41

41

41w~ (D.18)

For aerodynamic calculations the representative values of the displacements of the

thn strip are required. knv~∆ is the representative thk component of the displacement

of the thn strip. It will be chosen as the average of all the km,n c

~w over the strip, namely:

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=∆

kMn,

kmn,

k2n,

k1n,

c

kn

c

c

w~

w~

w~w~

1,,1,1M1v~ (D.19)

It is convenient now to define a vector nv of order K as follows:

Kn

kn

2n

1nn v~∆,v~∆,,v~,∆v~∆=Τv (D.20)

Based on Eqs. (D.11), (D.13), (D.14), (D.18), (D.19) and (D.20) it is possible to write, for the thn strip of the blade, the following expression:

[ ] nn wv nB= (D.21)

where the dimensions of the matrix [ ]nB are 0NK × .

The displacements which are important for the present aerodynamic calculations are:

1) The plunging displacement, h, (motion perpendicular to chord). It is measured along the reference line

2) The pitching displacement, α . It indicates the rotation of the cross-section about the reference line.

65

The numerical steps for computing the plunging displacement, nh , of the nth strip and the rotation, nα , are listed below:

(1) The tangent unit vector t , that is a radial vector at the middle chord (tangent to the reference line), is computed .

(2) The chord unit vectors of the strips are computed next. The cross-section plane of the strip is defined, perpendicular to the tangent unit vector at the middle chord point. The points of intersection of this plane and the blade leading and trailing edges define the chord. A chord unit vector c is drawn along the line that joins these points.

(3) The normal unit vectors are computed next. The chord unit vector c is translated to the middle chord point. The normal unit vector n at this point is the vector cross product of the tangent and the chord unit vectors at the middle chord point:

tcn ˆˆˆ ×= (D.22)

(4) The plunging displacements in the direction of the normal vector at a middle chord point, denoted nh , is the dot product of the normal vector and the corresponding displacement vector:

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

∆

∆

∆

= •3n

2n

1n

nzyxn

v~v~v~

n,n,nh (D.23)

(5) The pitching displacements about the tangent vector, at the middle chord point, represented by nα , is the dot product of the tangent vector and the corresponding rotation vector

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

∆

∆

∆

= •6n

5n

4n

nzyxn

v

v

v

t,t,tα (D.24)

Based on the geometry of the problem it is possible to write:

3n

3n

2n

2n

1n

1nn v~Cv~Cv~Ch ∆+∆+∆= (D.25)

where nh is the plunging motion of the nth blade's strip. ,C,C 2n

1n and 3

nC are coefficients of the transformation.

In a similar manner to Eq. (D.25) it is also possible to write:

66

6n

6n

5n

5n

4n

4nn v~Cv~Cv~C ∆+∆+∆=α (D.26)

where nα is the pitching motion of the nth blade's strip and ,C,C 5n

4n and 6

nC are coefficients of the transformation.

Based on Eqs. (D.25) and (D.26) one can write:

[ ] nvnn

n Ch

=⎭⎬⎫

⎩⎨⎧

α (D.27)

[ ]nC is a matrix of the coefficients of the transformation, defined as follows:

[ ] ⎥⎦

⎤⎢⎣

⎡= 6

n5n

4n

3n

2n

1n

n CCC000000CCC

C (D.28)

Substitution of Eqs. (D.1), (D.14) into Eq. (D.21) and then into Eq. (D.27), result in:

[ ]qnn

n Hαh

=⎭⎬⎫

⎩⎨⎧

(D.29)

where

[ ] [ ] [ ] [ ] [ ] SPPNnNKnK2nS2n 00ABCH ××××× Φ= (D.30)

67

Figure D.2: Matrix [ ]Φ Components.

68

Appendix E: Calculating the matrix [ ]AP The matrix [ ]AP is the matrix that describes the distribution of the resultant cross-

sectional aerodynamic loads over the blade. In the present analysis of it is assumed that the aerodynamic loads can be calculated using steady-state formulas while the angle of attack α is given by the instantaneous angle between the resultant velocity vector and the chordline. If the cosines of small angles are taken equal to unity, then the lift and pitching moment acted on the nth strip of the blade are:

( ) ( )⎪⎪

⎭

⎪⎪

⎬

⎫

+−=−∆=

Γρ=γρ=∆−=

∫

∫ ∫

−

− −

a50bLdyabypM

VdyVdypL

n

b

bn

b

b

b

bnn

. (E.1)

after using the flat-plate chordwise loading [16]:

ybybV2

+−

α=γ (E.2)

the lift force can be described as follows:

αVρπb2dyybybV2L 2

b

b

2n =

+−

αρ= ∫−

(E.3)

where ϕ= cosby and:

( ) bd1bdyybyb

0

b

b

π=⎟⎟⎠

⎞⎜⎜⎝

⎛ϕϕ−=

+−

∫∫π

−

cos (E.4)

Dividing the interval of integration by a number of segments that is equal to number of chordwise finite elements, Mc, (see Fig. E.1), the integral (E.7) is written as:

( ) ( ) ( )

( ) ( ) ( ) bdyyfdyyfdyyf

dyyfdyyfdyyfdyybyb

b2

M

b12

M

b12

M

b22

M

b

0

0

b

b22

M

b12

M

b12

M

b2

M

b2

M

b2

M

c

c

c

c

c

c

c

c

c

c

π=+++

++=+−

∫∫∫

∫∫∫∫

∆

∆⎟⎠

⎞⎜⎝

⎛ −

∆⎟⎠

⎞⎜⎝

⎛ −

∆⎟⎠

⎞⎜⎝

⎛ −

∆

∆−

∆⎟⎠⎞

⎜⎝⎛ −−

∆⎟⎠⎞

⎜⎝⎛ −−

∆⎟⎠⎞

⎜⎝⎛ −−

∆−

∆

∆−

(E.5)

where ( )ybybyf

+−

= describes the distribution of the aerodynamic forces,

bb2

Mc ±=∆± are the limits of integration and b∆ is the chordwise length of the FE.

69

If it is assumed that Eq. (E.5) describes also the unsteady aerodynamic loads, then:

( ) ( )

( )

( ) ( ) ( )⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

αα

π+⎥

⎦

⎤⎢⎣

⎡α

απ

−+

+−α

απ

+

−= ∫− ασ)(F

ddC

bVαV)σ(F

ddC

2a211

αabh)σ(Fd

dCb

Vh

dyyfbρtr,L∆L

2L

L∆b

2M

∆b2

M

c

c

(E.6)

( ) ( ) ( ) ( )+

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−

αα

π+

+=∆ ∫∆

∆−

αa81bh

bV)σ(F

ddC

2a21hadyyfbρtr,M 2L

b2

M

b2

M

2

c

c

+

( ) ( ) ( )

⎪⎭

⎪⎬⎫

⎥⎦

⎤⎢⎣

⎡

αα

π+

αα

π−

⎟⎠⎞

⎜⎝⎛ ++α⎟

⎠⎞

⎜⎝⎛ −+ ασ)(F

ddC

bV

21αVσ)(F

ddC

2a21a

21

21aV L

2L (E.7)

Figure E.1. The scheme of the numbering of the nodes of the finite element

The matrix [ ]AP can be described as follows:

[ ]

[ ]

[ ]

[ ] ⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=×

ss

s

N,N

,

1,1

N3

P

P

P

P

M

nnM

M

PA

00

00

00

(E.8)

[ ]AP is of order sN3P × , where scm NMNKP ⋅⋅⋅= . In accordance with equation

(E.5), the matrix [ ] nnMP , (which is of order 3MNK cm ⋅⋅⋅ ) can be presented as follows:

70

[ ]( )

( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

π=

∫

∫

∫

∫

∫

∫

∆

∆⎟⎠

⎞⎜⎝

⎛ −

∆

∆⎟⎠

⎞⎜⎝

⎛ −

∆⎟⎠⎞

⎜⎝⎛ −−

∆−

∆⎟⎠⎞

⎜⎝⎛ −−

∆−

∆⎟⎠

⎞⎜⎝

⎛ −−

∆−

∆⎟⎠

⎞⎜⎝

⎛ −−

∆−

×⋅

000000

Adyyf00

0Adyyf0

000000

000000

Adyyf00

0Adyyf0

000000

000000

Adyyf00

0dyyf0

000000

b1P

c

nc

nc

c

nc

nc

c

nc

nc

c

nc

nc

nc

nc

nc

nc

mC

M

b2

M

b12

M

M

b2

M

b12

M

m

b12

M

b2

M

m

b12

M

b2

M

1

b12

M

b2

M

b12

M

b2

M

n3NM6n,nM

(E.9)

The coefficient cmA are defined as follows

( )cc mmm N1A = (E.10)

( )cmmN1 is a parameter representing a uniform distribution of the aerodynamic loads

over the mN nodes (which are defined in Fig.E.1) of the cm th finite element

71

Appendix F: Calculating the matrix [ ]0T

The transformation matrix, [ ]0T , for the entire blade, is a diagonal matrix of order PP × :

[ ]

[ ]

[ ]

[ ] ⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=×

ss NNM

nnM

11M

PP0

T

T

T

T

,

,

,

, scm NMNKP ⋅⋅⋅= (F.1)

[ ]( )

[ ]

[ ]

[ ] ⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=⋅⋅×⋅⋅

cc

c,ccmcm

M,MF

m,mF

1,1F

MNKMNKn,nM

T

T

T

T (F.2)

[ ]( )

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡

=⋅×⋅

mm

mmmmc,c

N,N

n,n

1,1

NKNKm,mF

T00T

T00T

T00T

T (F.3)

[ ]T is obtained by using the Euler angles 012 ,, βββ . [ ] [ ] [ ] [ ]321 TTTT = (F.4)

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ββ−ββ=

00

001

00

001

cossinsincosT , [ ]

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ββ

β−β=

11

11

2

0010

0

cossin

sincosT , [ ]

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ββ−ββ

=10000

22

22

3 cossinsincos

T

(F.5)

72

Finally the matrix [ ]mm n,nT is:

[ ]

mm

mm

n,n10

20

210

20

210

1020

210

20

210

12121

n,n

coscoscossin

sinsincossinsin

cossincos

cossincoscos

sinsinsinsincos

cossinsin

sinsincoscoscos

T

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

ββββ−

βββββ+

βββ

ββββ+

βββββ−

βββ

β−ββββ

= (F.6)

where mm Nn1 ≤≤ . Note, that for the two-dimensional case 021 =β=β , [ ] [ ]1TT = , then:

mm

mm

n,n00

00

00

00

n,n

cossin0sincos0

0010

0cossin0sincos0

001

T00T

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

ββ−ββ

ββ−ββ

=⎥⎦

⎤⎢⎣

⎡ (F.7)