Redacted for privacy

Transcription

1 AN ABSTAT OF THE THESIS OF Somsak haiyapinunt for the degree of Doctor of Philosophy in Mechanical Engineering presented on December 7, Title: Linearized Model for Wind Turbines in Yaw edacted for privacy Abstract approved: Konem t. wiison The analysis of rigid hub, three-bladed horizontal-axis, axisymmetric wind turbines in yaw is made using a linearized, four-degree-of-freedom model. The linearized equations of motion of rotor and nacelle are developed using quasi-steady blade element theory and Lagrange's equations. The yaw behavior of the system is studied from coefficients of the equations of motion. presented and studied. Analytical results for two wind turbines are The study shows that yaw tracking error is primarily caused by tower shadow. The contribution of the nacelle to the yaw stability is proved to be a destabilizing one. The yaw stability of a wind turbine in a reverse position is investigated and used for verification of the analysis. The characteristics of yaw static stability are primarily determined by the characteristics of the in-plane force coefficient and the location of the yaw axis relative to the rotor plane. A sensitivity study of the terms in the yaw stiffness coefficient is

2 made. The yaw static stability is strongly affected by a change in the coning angle. Two Fortran computer programs are developed to compute the numerical values of coefficients of the equations of motion. Program listings and sample outputs are included.

3 Linearized Model for Wind Turbines in Yaw by Somsak haiyapinunt A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy ompleted December 7, 1983 ommencement June 1984

4 APPOVED: edacted for privacy Professor of meditiattawneering in charge of major edacted for privacy Head of-department/o Prebianitil Engineerin Dean of Graduate edacted for privacy 7 ool Date thesis is presented December 7, 1983 Typed by Express Typing Service for Somsak haiyapinunt

5 AKNOWLEGMENTS I would like to express my sincere gratitude to Dr. obert E. Wilson, my major professor, for his guidance, encouragement, criticism, and financial support. I am also grateful to Dr. James. Welty, the department head, for providing me teaching assistantships during the course of my graduate studies. I also wish to thank Dr. harles E. Smith for his advice and useful discussions about the dynamics of wind turbines, and ockwell International orporation, ocky Flats Plant for its financial support from September 1981 to March Finally, I would like to express my deepest gratitude to my parents, Surin and Suwannee haiyapinunt, for their love, encouragement, and untiring support.

6 TABLE OF ONTENTS 1. INTODUTION 1 2. ANALYSIS 3 Development of the Equations of Motion 3 3. PHYSIAL HAATEISTIS OF TEST ASES 9 Grumman WS33 9 Enertech ESULTS AND DISUSSION 17 Analytical esults for the Grumman WS33 18 Static Pitch Angle 18 Static Flapwise Deflection 18 oefficients of the Equation of Motion in Yaw 19 Yaw Forcing Function 22 Yaw Tracking Error 29 Verification of the Analysis with the Test Data 31 Yaw Stiffness oefficient 34 Sensitivity Study 47 Analytical esults for the Enertech ONLUSION 70 BIBLIOGAPHY 75 APPENDIES APPENDIX I KINEMATIS 77 APPENDIX II OTO AEODYNAMIS 90 II.1 elative Velocity Aerodynamic Forces and Moments Linearized Aerodynamic Forces Aixal Induction Factor na" Variation of Axial Induction with Generalized oordinates Tip Loss Model Power and Thrust oefficient 116 APPENDIX III DEIVATION OF GOVENING EQUATION 117 III.1 Kinetic and Potential Energy Virtual Work Nonconservative Forces Nacelle, Gravity Tower Shadow 128

7 APPENDIX IV IV.1 LINEAIZED EQUATIONS OF MOTION Linearization APPENDIX V YAW STIFFNESS OEFFIIENT 157 V.1 Yaw Stiffness oefficient 157 V.2 In-Plane Force and Out-of-Plane Force 159 APPENDIX VI OMPUTE ODES 161 VI.1 POP ode 161 VI.2 AEO ode 165 VI.3 Sample ases 174 VI.4 omputer Listings 177

8 LIST OF FIGUES Figure Page 3.1 Airfoil sectional data for NAA from reference Lift and drag coefficient for the NAA 4415 airfoil section (ef. 9) The Grumman WS33 and the Enertech Variations of yaw moment per shadow width for selected locations of blade shear center, the Grumman WS Variations of yaw moment per shadow width based on PM for selected locations of blade shear center, the Grumman WS Variations of yaw forcing function based on PM for selected values of velocity deficit, the Grumman WS Velocity diagrams for a turbine blade section in a conventional position and in a reverse position Force coefficients for the NAA airfoil section in the airfoil's coordinates Force coefficients for the reverse NAA airfoil section in the airfoil's coordinates Spanwise distribution of yaw stiffness coefficient for the Grumman WS Spanwise distribution of yaw stiffness coefficient for the Grumman WS Spanwise distribution of yaw stiffness coefficient for the Grumman WS33 in a reverse position Spanwise distribution of yaw stiffness coefficient for the Grumman WS33 in a reverse position Force coefficients for the Grumman WS33's blade section in the rotor's coordinates versus angle of attack Force coefficients for the Grumman WS33's blade section, in a reverse position, in the rotor's coordinates versus angle of attack. 45

9 Figure Page 4.13 Spanwise distribution of components of yaw stiffness coefficient for the Grumman WS33 due to in-plane force and out-of-plane force Effect of pitch angle on yaw stiffness coefficient for the Grumman WS Effect of modulus of elasticity on static flapwise tip deflection for the Grumman WS Effect of modulus of elasticity on yaw stiffness coefficient for the Grumman WS Effect of rotor speed on yaw stiffness coefficient for the Grumman WS Effect of rotor speed on yaw stiffness coefficient based on the same PM for the Grumman WS Effect of the location of the blade cross section's shear center on yaw stiffness coefficient for the Grumman WS Effect of the location of the yaw axis relative to the rotor plane on yaw stiffness coefficient for the Grumman WS Effect of coning angle on yaw stiffness coefficient for the Grumman WS Effect of the location of the yaw axis relative to the rotor plane on yaw stiffness coefficient of the Grumman WS33's nacelle Effect of the location of the yaw axis relative to the rotor plane on yaw stiffness coefficient of the nacelle and rotor for the Grumman WS Yaw tracking errors versus tip speed ratio for the Enertech Yaw stiffness coefficient versus velocity ratio for the Enertech otor system. 78 ^^^ 1.2 oordinate systems XYZ, xyz, and xyz oordinate systems xyi, x y z, and xpypzp. 83

10 Figure Page 1.4 oordinate systems xoff3z, xeyeze, and xix2x Transformation matrices. 85 II.1.1 Velocity diagram at blade cross section Velocity diagram at blade cross section Velocity diagram at blade cross section evaluated at nominal value Windmill brake state performance Nacelle geometry Tower shadow. 129 V.1 egions of operation for momentum calculations. 168

11 LIST OF TABLES Table 3.1 Page Physical and operating characteristics of the rotor of the Grumman WS Physical and operating characteristics of the blade of the Grumman WS Nacelle properties of the Grumman WS33. Physical and operating characteristics of the rotor of the Enertech The moment inertia of the blade cross section of the Enertech Mass distribution of the blade of the Enertech Nacelle properties of the Enertech Static tip pitch angles at B = Static tip deflections at B = oefficients of the equation of motion in yaw from AEO code oefficients of the equation of motion in yaw for the combined rotor and nacelle system from AEO code oefficients of the equation of motion in yaw from POP code oefficients of the equation of motion in yaw for the combined rotor and nacelle system from POP code oefficients for harmonic terms (g, Dg, Eg, Fg), stiffness coefficient, and forcing function of a single-blade equation of motion in pitch Yaw tracking error for the Grumman WS33 with B = 4, el/ = 0.25, 40 shadow width, and 50% velocity deficit oefficients of the equation of motion in yaw for the reverse turbine, static tip deflection, and power coefficient Static pitch angle under normal operating condition. 63

12 Table Page 4.11 Static flapwise tip deflection under normal operating condition from both computer codes oefficients of the equation of motion in yaw from AEO code oefficients of the equation of motion in yaw for the combined rotor and nacelle system from AEO code oefficients of the equation of motion in yaw from POP code oefficients of the equation of motion in yaw for the combined rotor and nacelle system from POP code Yaw forcing function with 20 shadow width and velocity deficit = 50% Some integration values. 131

13 NOMENLATUE a A B c D L F n r t n P axial induction factor area number of blades blade chord drag coefficient lift coefficient force coefficient in the direction normal to the rotor force coefficient in the direction tangential to the rotor normal force coefficient power coefficient c() t T torque coefficient tangential force coefficient thrust coefficient drag force e distance from mass center to shear center of the blade cross section el distance from 1/4 blade chord to shear center of the blade cross section e2 distance from 3/4 blade chord to shear center of the blade cross section e3 distance from mid-blade chord to shear center of the blade cross section E modulus of elasticity

14 f 1 f 2 f 3 f 4 F F 1 F 2 F 3 F 4 g G G 1 G 2 G 3 G 4 h mode shape of the blade twisting mode shape of the blade deflection mode shape of the lead-lag deflection mode shape of the yaw displacement tip loss factor linearized aerodynamic force term linearized aerodynamic force term linearized aerodynamic force term linearized aerodynamic force term gravitational force shear modulus of rigidity linearized aerodynamic force term linearized aerodynamic force term linearized aerodynamic force term linearized aerodynamic force term a function defined in gravitational force I moment of inertia I 1 mass moment of inertia in nl direction 12 mass moment of inertia in n2 direction 13 mass moment of inertia in n3 direction jn J J1 J2 J 3 k n Tin Glauert coefficient polar moment of inertia moment of inertia in nl direction moment of inertia in n2 direction moment of inertia in n3 direction Glauert coefficient stiffness coefficient of the system

15 distance from nacelle yaw axis to rotor center L lift force nn mass coefficient of the system M n n N n P qm q1 q2 q3 moment unit vector linearized normal force power dynamic pressure T,pwV2 1 generalized coordinate of pitch angle generalized coordinate of flap deflection generalized coordinate of the variation of azimuth angle q4 qs r rn rs H S S t H n u u n U v n generalized coordinate of yaw angle static tip deflection local blade radius local blade radius in the rotor plane distance of the local blade radius to blade root blade radius hub radius distance from blade tip to blade root cross-sectional area of nacelle time linearized tangential force axial velocity at the rotor radial displacement strain energy normal displacement

16 V. wind velocity V w W W e wind velocity at reference point flap deflection relative velocity relative velocity excluding the pitching velocity at 3/4 blade chord W n W t x X xi,x2,x3 normal relative velocity tangential relative velocity local tip speed ratio tip speed ratio coordinate system on the blade cross section after blade pitching xe,ye,ze coordinate system on the blade cross section after blade flapping xa,ya,za coordinate system on the blade cross section after accounting for the pretwist angle x,y 2z P P P coordinate system at rotor center accounting for the coning angle x,y,z coordinate system fixed to the blade at azimuth angle 1) x,y,z coordinate system with its origin is at the rotor center and the system is fixed to the nacelle x,y,z coordinate system fixed to the nacelle and its origin located at nacelle yaw axis X,Y,Z coordinate system located on top of the tower

17 Greek Symbols blade pitch angle x variation of azimuth angle azimuth angle (ct + x) yaw angle p coning angle density dummy variable rotor angular velocity a wl w2 w3 a ae pretwist angle blade angular velocity in xl direction blade angular velocity in x2 direction blade angular velocity in x3 direction angle of attack angle of attack plus the effect of pitching velocity. integral term for the variation of axial induction factor

18 Linearized Model for Wind Turbines in Yaw 1. INTODUTION Wind-powered machines can be classified into two types according to the orientation of the axis of rotation: turbines and vertical-axis wind turbines. horizontal-axis wind For a horizontal-axis wind turbine, the system can be further distinguished as either a downwind rotor or an upwind rotor system. When the rotor is upwind of the tower, the system usually has a yaw controller to force the wind turbine to track the wind mechanically. There is often no need for a yaw controller in the downwind rotor case. When the rotor is downwind of the tower, the wind turbine will typically track the wind. Most of the downwind turbines are free-yaw systems. Unfortunately, in many free-yaw systems, a yaw instability can occur: instead of tracking with the wind, the turbine yaws away from the wind. Little work has been done on wind turbines in yaw. Most of the previous work has been on the structural dynamics and control of wind turbine systems. The cause of the yaw instability has still not been fully understood. The technology and methodology used to develop present-day wind turbines are adapted from the fixed and rotating wing aircraft technology. ibner [16] has done an analysis of induction velocity and side force in terms of the shape of the blade when the propeller is yawing. For wind turbines, Miller [13] looked into the static stability characteristics of horizontal-axis wind turbines with a

19 2 free-yaw system operating only in a low wind condition (i.e., no stall model). Hirschbein [7] analyzed the dynamics and control of large horizontal-axis axisymmetric wind turbines in his Ph.D. thesis. He modeled the blade motion by considering the blades to be composed of an inboard series of massless, rigid links restrained by linear springs and dampers with a much larger, massive blade attached to the outermost link. In this dissertation, yaw of wind turbines will be studied by using a four-degree-of-freedom system to represent the axisymmetric wind turbine system. The study will focus on the cause of yaw tracking errors and the characteristics of the yaw static stability. This will be done by developing the equations of motion of the system and then studying from the coefficients of the equations. The equations of motion of the rotor are developed using the Lagrange method, and the aerodynamic forces and moments are developed using the quasi-static blade element theory. The aerodynamics of wind turbines in the stall region is also considered. The analysis is primarily developed for a three-bladed horizontal axis wind turbine, but it also can easily be applied to a turbine with any number of blades greater than three. The contribution of the nacelle to the rotor system is also examined.

20 3 2. ANALYSIS Development of the Equations of Motion A four-degree-of-freedom system is chosen to model the axisymmetric turbine system. The degrees of freedom are blade pitch deflection, blade flap, nacelle yaw, and rotor speed. These degrees of freedom are defined as follows: Blade pitch is defined as the rotation of the blade cross section around the control axis; Blade flap is defined as the deflection of the blade in the direction perpendicular to the blade chord; Nacelle yaw angle is defined as the angle of the nacelle around the yaw axis with respect to the wind; and otor speed variation is defined as the variation of the rotor speed from the nominal value. The equations of motions are developed using the Lagrange method. Since each of the variables is a function of time and radial distance, the partial derivatives of these variables will be encountered during the development of the equations of motions. To avoid dealing with partial differential equations, the assumed modes method [11] is used in this study. The purpose of this method is to eliminate the spatial dependence from the dependent variable by discretizing the spatial variable. Thus each of the system's degrees of freedom is expressed as the product of the displacement function (assumed mode shape), which is the function of the spatial coordinate, and the time-dependent generalized coordinate. By this method, the equations of motion of the system will be developed in ordinary rather than partial differential forms.

21 4 In order to attack the problem, the kinematics of the rotor are first developed. Then, the kinetic energy is obtained from the expression of the kinematics. The potential energy expression is developed from the strain energy of the rotor system. With quasi-steady blade element theory, the aerodynamic forces and moments are developed. Then, the nonconservative forces in Lagrange's equation are derived from the virtual work of the aerodynamic forces and moments. Finally, when the Lagrangian functions and nonconservative forces are substituted back into Lagrange's equation, a set of nonlinear equations of motion of the rotor system is obtained. The rotor extracts energy from the wind, converting it into mechanical energy. Since the energy is extracted from the airstream, the velocity of the wake will be decreased. To represent the reduction of the wind velocity at the rotor and in the wake, the axial induction factor "a" [20, 22] is introduced. In this study the nonrotating wake model is used. The local value of the axial induction factor can be calculated by equating the windwise force developed by using momentum theory, and the same force developed by using blade element theory. The Glauert empirical relationship [4] is used instead of the momentum theory when the axial induction factor is greater than The tip loss model is used to account for the flow at the tip of the turbine blade. The development of the axial induction factor and the tip loss model is presented in Appendix II. The above steps lead to a set of nonlinear equations. If the ranges of values of the dependent variables can be retricted, the

22 5 system may be well-approximated as linear. In this study, the system is analyzed in the linear range. In the process of equation linearization, the variation of the axial induction factor with yaw, pitch, flap, and rotational speed will be encountered. Linearized aerodynamic forces and moments are developed. Let us define the variation of the induction factor with the dependent variables as the summation of two terms: 1) the product of a coefficient and the distance along the yaw axis of the rotor, and 2) the product of a coefficient and the distance along the rotor pitch axis. 9a = jn cost') + kn sr lily Here, * is the blade azimuth angle. The value of the two coefficients, jn and kn, can be calculated by equating the derivative of yaw or pitch moment developed by the momentum theorem to the derivative of yaw or pitch moment developed by the blade element theory. With the known values of the coefficients, jri and kn, the variation of the axial induction factor can be determined. The result shows that the variation of the axial induction factor exists only for the yaw and yaw rate variables in the uniform flow case. The linearization of the aerodynamic forces and the variation of the axial inducation factor are presented in Appendix II.

23 6 The linearized rotor equations of motion are expressed in matrix form [M]fili) + + [K]fqi) = fg) where {qi} is a four-dimensional generalized coordinate column vector representing the system's degrees of freedom; fg) is a four-dimensional forcing function column vector; [M], [], and [K] are the four dimensional square mass, damping, and stiffness coefficient matrices, respectively. For a large wind turbine system, gravity loads are very important to dynamic and structural analyses. To make the analytical model for the turbine system applicable regardless of the size of the system, gravity loads are included in this study. The gravitational force is added to the system by means of a potential function. For a downwind system, the rotor is located behind the nacelle and tower. The effect of the nacelle and tower shadow on the system was studied. The nacelle is considered as a slender body. The shape of the nacelle is assumed to he a cylinder with hemispheres on both ends. The equation of motion of the nacelle will be developed by using the Lagrange method. The nacelle will be considered as a rigid body rotating around its yaw axis when the kinetic and potential energy are calculated. The nonconservative force on the nacelle is derived from the virtual work of the nacelle. The forces on the nacelle are

24 7 calculated by using slender body theory with forces generated only from the forebody part of the nacelle. The tower shadow is modeled as a velocity deficit from the rotor axial velocity value over a selected region of the rotor disk. The system's equations of motion are developed with the tower shadow. Throughout this analysis the wind turbine is modeled with a three-bladed rotor. The turbine blades are elastic. The hub, nacelle, and tower are rigid. The nacelle is allowed to yaw freely. The center of mass of the nacelle and rotor is located over the central axis of the tower. This axis will be referred to as the nacelle yaw axis. The absolute motion of the turbine blade is determined by the motion of blade deflection relative to the hub, the motion due to rotor rotation, and the motion of the nacelle and tower. Since in this analysis no movement of the tower is allowed, the tower is considered as an inertial reference frame. A series of coordinate systems is used to describe a point on the blade. A series of transformation matrices is then used to transform the coordinate systems that describe motion of a point on the blade in its original reference frame into the inertial reference frame. Two computer codes have been written to handle the numerical computations which yield the coefficients for the equations of motion. One of them has a simplified lift and drag curve in the stall region. The other was developed later and was necessary because of the need for a more accurate model for lift and drag in the stall region. This second computer code only emphasizes the yaw equation, since it was

25 8 found from the analytical results that there is no coupling between yaw and the other degrees of freedom. The inputs to the computer code are the geometry of the rotor, the wind condition, and the operating conditions. Both computer codes will calculate the axial induction factor along the blade at a particular tip speed ratio. At the same time, they also calculate the integral terms for variation of the axial induction factor with yaw and yaw rate. Finally, the programs will calculate the coefficients in the equations of motion (mass, damping, stiffness, and forcing function). Besides the coefficients of the equations of motion, the codes also calculate the thrust and power coefficients for the rotor.

26 9 3. PHYSIAL HAATEISTIS OF TEST ASES Two test cases are chosen to verify the analysis. The test cases are the Grumman WS33 and the Enertech Both of these machines are three-bladed horizontal axis downwind wind turbines. Grumman WS33 The Grumman WS33 is a three-bladed, downwind machine designed to interface directly with an electrical utility network. The machine is rated at 15 kw at 24 mph and peak power of 18 kw at 35 mph. Utility compatible electrical power is generated in winds between a cut-in speed of 9 mph (4.0 m/s) and a cut-out speed of 50 mph (22 m/s) by using torque characteristics of the unit's induction generator combined with rotor aerodynamics to maintain essentially constant speed. The rotor's diameter is 33.3 feet. The blades are extruded aluminum alloy with a modified NAA airfoil section. The blades are adjusted in pitch by a redundant pitch actuator system controlled by a solid state programmable logic controller, the Microcomputer ontrol Unit. The overall weight of the nacelle and rotor assembly including the blades is 2,589 pounds. The blades weighed 185 pounds each. Power is generated by a 240/280 volt, 30,60Hz induction generator with a 20kw capacity. The generator operates between 1800 and 1835 rpm. The gear box with 25.1 to 1 ratio will cut in at about 72 rpm with full power being generated at a little above 74 rpm.

27 10 The physical and operating characteristics of the rotor, the blade, and the nacelle are given in tables 3.1, 3.2, and 3.3, respectively. The information describing the Grumman WS33 was obtained from reference 1. The airfoil lift and drag coefficient of the Grumman WS33 are plotted as functions of eynolds number and angle of attack in Figure 3.1. These data are obtained from reference 19. Table 3.1 Physical and operating characteristics of the rotor of the Grumman WS33. otor diameter Blade chord oot cut-out Airfoil type otor speed otor coning angle Number of blades ft 1.5 ft ft NAA (modified) 74.1 rpm Table 3.2 Physical and operating characteristics of the blade of the Grumman WS33. Blade density Mass per unit length Moment of inertia in flapwise direction Moment of inertia in chordwise direction Moment of inertia in radial direction Modulus of elasticity Shear modulus slug/ft lug /ft in in in 4 10x1Ou psi 3.8x106 psi Table 3.3 Nacelle properties of the Grumman WS33. Distance of the nacelle yaw axis to blade hub Length of the nacelle ross section area of the forebody end ross section area of the nacelle Mass moment of inertia of the nacelle and hub assembly around the yaw axis ft ft ft ft slug-ft2

28 L 0.4 O ANGLE OF ATTAK O (DEGEES) D ANGLE OF ATTAK c (DEGEES) 15 Figure 3.1 Airfoil sectional data for NAA from reference 19.

29 12 Enertech 1500 The Enertech 1500 is a downwind system with a three-bladed rotor. The geometry and material properties of a blade of an Enertech 1500 wind turbine were measured. The blade has linear twist with slight linear taper over the outer 22 percent of the rotor blade and the blade's thickness is varied from root to tip. For the calculation of the aerodynamic forces and moments, the blade profile section was represented by the NAA 4415 airfoil section. The airfoil lift and drag coefficients are plotted as functions of eynolds number and angle of attack in Figure 3.2. These data are obtained from reference 9. This rotor is designed to operate at tip speed of 117 fps (170 rpm). The physical characteristics of the rotor are presented in Table 3.4. Table 3.4 Physical and operating characteristics of the rotor of the Enertech otor diameter ft Blade chord 6.8 in. from root to r/ = linear taper to 6.1 in. at r/ = 1.0 Airfoil type NAA 4415* PM 170 Tip speed 117 fps Number of blades 3 oot cut-out 0.84 ft Twist 5 from root linear to 1 at blade tip Precone 0 *Used as representative airfoil section. The profile of the blade cross sections were measured at six stations along the blade. The weight of the blade was measured. By knowing the weight and the profile of each cross section, properties

30 , 1.0 U z 0.8 U U ANGLE OF ATTAK. a (DEG) ANGLE OF ATTAK. a (DEG) 15 Figure 3.2 Lift and drag coefficient for the NAA 4415 airfoil section (ef. 9).

31 14 of the blade were calculated. The expressions for the moment of inertia and mass distribution per unit length of the blade were written as a function of the distance along the blade. These expressions are given in tables 3.5 and 3.6. Table 3.5 The moment inertia of the blade cross section of the Enertech r/ J2 (in 4) J3 (in 4) exp ( r/) exp ( r/) exp ( r/) exp (-0.76 r/) exp ( r/) (r/) exp ( r/) (r/) Here J.'s are the moment of inertias of the blade cross section at the mass center in xi direction and J1 = J2 + J3. Table 3.6 Mass distribution of the blade of the Enertech r/ p (slug/ft) exp ( r/) exp ( r/) (r/)4' Since the blade is made of wood (orthotropic material), its material properties depend on the orientation of wood grain. It is difficult to find the mechanical properties of a nonuniform orthotropic beam by experiment. Thus, it was decided to treat the

32 15 blade as an isotropic material and use the values obtained from the U.S. Forest Products Laboratory on Sitka Spruce with 10 percent moisture content. The values are EL = 1.84 x 106 psi, G L X 105 Psi, \I = 0.25 For simplicity, the elastic axis was assumed to be a straight line which is parallel to the trailing edge of the blade. The location of the elastic axis on the blade cross section was chosen arbitrarily. The location of the axis then is varied to see the effect on the system. The nacelle of the Enertech 1500 has a cylindrical shape with a hemisphere on each end. The properties and geometry of the nacelle are given in Table 3.7. Table 3.7 Nacelle properties of the Enertech Distance of the nacelle yaw axis to the blade hub Length of the nacelle adius of the nacelle cross section Mass moment inertia of the nacelle around the yaw axis 2.46 ft ft 0.84 ft slug-ft2 The generator for the Enertech 1500 is a single-phase induction motor connected to a gearbox having a measured to 1 ratio.

33 16 The Grumman Windstream 33 PP r The Enertech 1500 Figure 3.3 The Grumman,WS33 and the Enertech 1500.

34 17 4. ESULTS AND DISUSSION The Enertech 1500 and the Grumman WS33 were used as the test cases. Two computer codes, AEO code and POP code, were used to generate the numerical values of the analytical results. The AEO code uses a simplified lift and drag curve to calculate the axial induction factor and its variation. The AEO code also generates the coefficients of the equation of motion for a four-degree-of-freedom system. The revised version of POP code [21] uses the actual lift curve to calculate the axial induction factor and its variation. The POP code also gives more accurate results for the static tip deflection and the coefficients of the equations of motion in yaw. More detail regarding these two computer codes is shown in Appendix VI. It was found that the POP code is preferred because of the accuracy of the lift and drag models in the stall region. The yaw response of wind turbines will be examined from the numerical values of coefficients in the equations of motion generated from the computer codes. The analytical results of the Grumman WS33 is examined first. The cause of yaw tracking error is obtained from studying the coefficients in the yaw equation. The tower shadow effect is included in this analysis. The verification of the analysis is obtained from the calculated yaw stability of the Grumman WS33 in the upwind position. The analytical results for the Enertech 1500 are presented and discussed in brief.

35 18 Analytical esults for the Grumman WS33 With the numerical values of the coefficients in the equations of motion, the static pitch angle and the static flapwise deflection are first examined. Then, the sources of yaw forcing function, which are wind shear, gravity, blade cyclic force, blade flap, and tower shadow, are investigated. Static Pitch Angle The equilibrium pitch angle can be obtained from the pitch stiffness coefficient and the pitch forcing function. The equilibrium pitch angles that differed from the assumed zero value for different tip speed ratios are given in Table 4.1. These angles are so small that they have negligible effect on the system. Table 4.1 Static tip pitch angles at = 4. X est (degree) Static Flapwise Deflection The flapwise deflection (coning due to blade load) is examined. The static tip deflections for the Grumman machine from the AEO code and POP code are given in Table 4.2.

36 19 Table 4.2 Static tip deflections at 13= 4. AEO X ds (ft) POP ds (ft) The difference between the values obtained from the AEO and POP codes is small except at the tip speed ratio 3 and 4. Those are the conditions under which most of the blade is operating in the stall region, and the AEO code does not have an accurate model in the stall region. The results in Table 4.2 show that the blades exhibit a negative coning effect under low wind. The blade tips are deflected upwind because the centrifugal deflection overcomes the aerodynamic deflection. oefficients of the Equation of Motion in Yaw For a uniform wind condition, there is no coupling between the yaw angle and the other three variables explicitly on the equations of motion. For the nacelle, the equation of motion is in the form of undamped second order system in yaw. The stiffness coefficient of the nacelle is not dependent on the tip speed ratio. The development of the nacelle is given in Appendix III.

37 20 Because of the linearity of the system, the nacelle's equation of motion can be added directly to the rotor equation of motion in yaw. The nacelle destablized the system in yaw. The coefficients for the equation of motion in yaw from both computer codes are given in tables 4.3, 4.4, 4.5, and 4.6. Table 4.3 X oefficients of the equation of motion in yaw from AEO code. m44 44 k44 m44n* k44n *n refers to the nacelle Table 4.4 X oefficients of the equation of motion in yaw for the combined rotor and nacelle system from AEO code. m44t 44T k44t

38 21 Table 4.5 X oefficients of the equation of motion in yaw from POP code. m44 44 k44 m44n k44n Table 4.6 X oefficients of the equation of motion in yaw for the combined rotor and nacelle system from POP code. m44t 44 T k44 T There is some doubt in the accuracy of the stiffness coefficient for the Grumman WS33's nacelle. The analytical model of the nacelle in this analysis is based on slender body theory. Unfortunately, the shape of the Grumman WS33's nacelle and most other wind turbines' nacelles are not slender. There are some correction factors suggested to use with the slender body theory by reference 2 and 8 for a noncircular body and a finenesi ratio effect. However, the experimental results for a fineness ratio effect on the aerodynamic characteristics of bodies of revolution in reference 6 does not agree with the correction factors given in references 2 and 8. There is no

39 22 accurate model for the noncircular body with blunt nose. More work is needed in order to obtain an accurate model. The values in tables 4.3 and 4.5 are given without the correction factors. These correction factors, given in Appendix III, are always less than one. Therefore, the values in tables 4.3 and 4.5 are the most destablizing condition for the nacelle effect based on the slender body theory. According to the POP code, the system is stable for all of the tip speeds considered. However, a negative stiffness coefficient is encountered for the AEO code's result at the tip speed ratio equal to 3.. The expression for the stiffness coefficient consists of the derivative of the aerodynamic force and its moment arm. The derivative of the aerodynamic force is dependent on the slope of the lift and drag curve versus the angle of attack. Therefore, an accurate model of the lift and drag curve is needed to obtain an accurate result. Thus, the POP code is preferred to the AEO code, which uses the simplified lift and drag curve in the stall region. Yaw Forcing Function Possible candidates for a yaw forcing function are wind shear, blade pitch, blade flap, and tower shadow. The in-plane yaw force on the rotor is related nonlinearly to the wind speed; therefore, the difference in the axial velocity on the rotor due to wind shear would result in a yaw moment created by the net in-plane yaw force. However, there was no wind shear in the test data obtained from the ocky Flat esearch Energy enter for the Enertech 1500 or the Grumman WS33.

40 23 Gravity and blade cyclic pitch. The gravitational force is added to the system equations of motion by means of a potential function. The analysis shows that there is no effect of gravity appearing in the yaw equation. Therefore, the gravity is not a source for the yaw forcing function. The gravity effect also appears as a cyclic moment in the pitch equation for a single-bladed rotor. However, this moment cancels out for an axisymmetric three-bladed rotor. The effect of the blade cyclic force on the cyclic pitch angle is also examined by considering the gravitational force on a single blade. The cyclic moment due to gravity appears in the stiffness coefficient and forcing function in pitch. These cyclic moment terms are given in the following forms: (g cos4)+ Dg sinflqi = (Eg cos* + Fg sinip) The coefficients g, Dg, Eg, and Fg are given in Table 4.7. The stiffness coefficient and forcing function also are given in Table 4.7. Table 4.7 oefficients for harmonic terms (g, Dg, Eg, Fg), stiffness coefficient, and forcing function of a single-blade equation of motion in pitch. X k 11 GO1 gx103 Dgx102 Egx103 Fgx

41 24 For the static condition, the cyclic pitch angle is given as the following form: ql = G 01/3 + (Eg cost + Eg sinq)) s k11/3 (g cost + Dg sin1p) It can be seen from the magnitude of the coefficients given in Table 4.7 that the blade cyclic force has a negligible effect on the static pitch angle. Blade flap. The flapwise displacement appears implicitly and explicitly in most of the coefficient terms. The flapwise deflection definitely affects the yaw behavior, but it is not a source of the yaw forcing function. The effect of blade flap will be discussed in a later section. Tower shadow. The tower shadow is modeled as a velocity deficit from the axial velocity over a sector of the rotor disk, centered about the tower centerline. The development for the equations of motion with the tower shadow is given in Appendix III. Because of the difference of the axial velocity on the rotor inside and outside of the tower shadow region (i.e., when the blade is in the 6 o'clock position and the 12 o'clock position), there will be different values of aerodynamic forces. The difference of the in-plane force (force that is tangential to the rotor plane) in the

42 25 tower shadow produces a net yaw moment around yaw axis, creating a static yaw angle. This yaw moment turns out to be the yaw forcing function that is needed. onsider the yaw moment inside the tower shadow. The expression of this yaw moment can be simplified as follows: My =f f Ft + sin p)t: f Nofi dr 4. H 14_,o Ar4 B 2 sin 2' H H H 27 2 (1) Here Ft is the in-plane force, t is the distance from the rotor to the yaw axis, p is the coning angle, No and Ho are the forces normal and tangent to the blade chord, el is the distance from the shear center to the blade 1/4 chord, and wo is the static flapwise deflection. The full expression of this yaw moment is given in Appendix IV. The first term in the equation (1) primarily depends on the in-plane force. This in-plane force is directly related to the power output. Therefore, the power output response would be similar to the in-plane force. The third term is the yaw moment due to the tangential force and flapwise deflection acting in the same direction as the moment in the first term. The second term is the yaw moment due to the normal force and the offset distance of the shear center (e1). This yaw moment acts in the opposite direction of the other two terms in the equation (1).

43 26 1 A DISTANE FOM BLADE LEADING EDGE TO SHEA ENTE I MY o ), 51 IT N Figure 4.1 Variations of yaw moment per shadow width for selected locations of blade shear center, the Grumman WS33.

44 DISTANE FOM BLADE LEADING EDGE TO SHEA ENTE.001 M BSIN- IT Voo n. Figure 4.2 Variations of yaw moment per shadow width based on PM for selected locations of blade shear center, the Grumman WS33.

45 28 The yaw forcing function due to tower shadow is obtained by subtracting the yaw moment created inside the tower shadow from the one created outside the tower shadow (i.e., 6 o'clock position and 12 o'clock position). The yaw moment created by forces inside the tower shadow is shown in Figure 4.1. The effect of the shear center position on the yaw moment is that if the shear center is behind the 1/4 blade chord position, the yaw moment is decreased by the effect of this positive offset distance. Vice versa, the yaw moment will be increased if the shear center is ahead of the 1/4 blade chord. For the purpose of illustration, the yaw moment normalized by PM is considered. The yaw forcing function is given by G04 = Mql Mq2 (2) Here, mci = -M 1 Y11)(12 mq2 = +My2422 Subscript 1 refers to the condition at the rotor outside the tower shadow in the opposite direction of the tower shadow region. Subscript 2 refers to the condition inside the tower shadow. The plot of Mq versus the advance ratio (V./Q) for different shear center positions are shown in Figure 4.2. The yaw forcing is dependent upon: 1) the width of the tower shadow; 2) the velocity deficit in the tower shadow; 3) the shear center position; and 4) the power output of the rotor.

46 29 For zero offset distance, if the power increases continuously with wind speed, the in-plane force also is expected to increase continuously with wind speed. For such a situation the net yaw force produced by the tower shadow would always have the same sign since the wind speed in the tower shadow is less than in the free stream, then the in-place force in the tower shadow would be less than the in-plane force in the free stream. If the power peaked and then decreased with the wind speed, then the yaw force would change sign as the velocity increased. This effect has been observed on the Enertech 1500 and the Grumman WS33. As the peak power is achieved, the rotor yaw changes sign. The yaw forcing function due to tower shadow with 40 shadow width, 50% velocity deficit, and 80% velocity deficit is shown in Figure 4.3. Yaw Tracking Error A yaw tracking error is defined as an angle which the rotor yaws away from the wind in a static condition. The static yaw angle of a downwind turbine is obtained by dividing the yaw forcing function with the stiffness coefficient of the rotor and nacelle. The yaw angles for the Grumman WS33 with a 40 shadow width and a 50% velocity deficit are given in Table 4.8. The predicted static yaw angle must be small according to the linear analysis. Therefore, any large predicted static yaw angle would indicate that the result is not accurate. This is shown in Table 4.8 at tip speed ratio equal to 3 and 4.

47 GO4 x X VELOITY DEFE I T 80% VELOITY DEFE IT ,1 0, n. Figure 4.3 Variations of yaw forcing function based on PM for selected values of velocity deficit, the Grumman WS33.

48 31 Table 4.8 Yaw tracking error for the Grumman WS33 with 13 = 4, el/ = 0.25, 40 shadow width, and 50% velocity deficit. i ( degree ) It can be seen that for this analysis the yaw angles depend on many variables. These variables are tower shadow width, velocity deficit, nacelle stiffness coefficient, the position of the shear center, and the power output of the rotor. With many uncertainties in these variables, especially the stiffness coefficient of the nacelle, there is no exact solution for the yaw tracking error. Thus, the values given in Table 4.8 are for the preliminary study of the yaw behavior of the system rather than the prediction of the magnitude of yaw tracking error. Verification of the Analysis With the Test Data The test data from the ocky Flats esearch Energy enter indicates that the Grumman WS33 has a yaw instability near start-up. The machine will rotate about the yaw axis from a downwind position to an upwind position. Although the analysis indicated that the Grumman WS33 in a downwind position is stable in yaw, it is possible that with a more accurate model of the nacelle, the system could be unstable in yaw near start-up. Thus, the focus of the analysis is directed to the

49 32 yaw stability of a reverse turbine (a downwind turbine rotating to an upwind position). If the analytical result indicates that there is a region of stability for the reverse turbine, this result would verify the analysis. everse turbine. When a downwind turbine is rotating to the upwind position, the upper surface of the blade cross section will be facing the wind instead of the lower surface of the blade cross section. This reflection of the blade cross section will result in a negative angle of attack and pitch angle. The geometry of a reverse turbine blade and a conventional one are shown in Figure 4.4. The shape of the hub (nose cone) of the Grumman WS33 is a hemisphere of in. radius. The hub is considered as the forebody part of the nacelle for the turbine in an upwind condition. The coefficients of the equation of motion in yaw for the reverse turbine including the hub effect are given in Table 4.9. Table 4.9 X oefficients of the equation of motion in yaw for the reverse turbine, static tip deflection, and power coefficient. m44t 44 k44 T ds (ft) p

50 (a) conventional position (b) reverse position Figure 4.4 Velocity diagrams for a turbine blade section in a conventional position and in a reverse position.

51 34 As shown in Table 4.9, the system is unstable in yaw because of the negative yaw stiffness coefficient, except at the tip speed ratio equal to 5. This result indicates that for a certain, operating condition if the Grumman WS33 rotates to an upwind position, it could operate as a stable upwind turbine. This yaw stability verifies the analysis. The static flapwise tip deflections for the reverse turbine also are given in Table 4.9. When a downwind turbine with a coning angle (coning away from the wind) rotates to an upwind position, the blade will be coning to the wind. This negative coning yields a positive flapwise deflection (deflect in the wind direction) because the bending moments created by the aerodynamic force and the centrifugal force are in the same direction. Yaw Stiffness oefficient The numerical value of the yaw stiffness coefficient for a small blade element is obtained and its distribution along the blade is examined to see what causes the destablizing effect. The yaw stiffness coefficient is the linearized variation of the yaw moment from its nominal value with respect to the yaw angle. This linearized variation of the yaw moment can be expressed as the product of the derivative of the aerodynamic force and its moment arm around the nacelle's yaw axis. Therefore, the stiffness coefficient is primarily dependent on the derivative of the aerodynamic force instead of the force itself. The behavior of the aerodynamic force in the coordinates of the airfoil, normal and tangential to the blade chord, is observed. Let n and t be the force coefficients expressed in the normal and the

52 35 tangential directions to the blade chord. Figures 4.5 and 4.6 show the plots of n and t versus the angle of attack a for the Grumman blade section in a conventional position (downwind turbine) and in a reverse position (reverse turbine). Note should be made that for the reverse turbine, although the blade is operating with a negative angle of attack, the sign convention of the analysis makes the aerodynamic forces on the blade result in a positive value. Figures 4.7 and 4.8 show the distribution of the yaw stiffness coefficient along the blade at different tip speed ratios for the Grumman turbine in a downwind position. The destablizing effect (a negative value) of the yaw stiffness coefficient appears in figures 4.7 and 4.8 as a reverse hump curve starting at the blade tip at the tip speed ratio equal to 2 and moving in board as the tip speed ratio is increased. The further the reverse hump curve moves in board, the smaller the magnitude of the hump curve becomes. This hump curve starts at the portion of the blade which is experiencing a 14 angle of attack and stops at the portion of the blade which is experiencing an angle of attack greater than 21. These two angles are the angles which the slope of the tangential force for a Grumman rotor blade changes sign. For the Grumman turbine with a reverse position (staying upwind), the distribution of the yaw stiffness coefficient is shown in figures 4.9 and It can be seen that the shape of the distribution of the yaw stiffness for a reverse turbine has the overall shape similar to the shape of the distribution of the yaw stiffness coefficient for a downwind turbine except it is upside down. The destabilizing effect becomes the stabilizing effect. Now the hump curves in figures 4.9

53 n 1.0 I -10 o Figure 4.5 Force coefficients for the NAA airfoil section in the airfoil's coordinates.

54 ni.0 0' I 500 0: loo' 0.3 I I t I I 50* a 100' Figure 4.6 Force coefficients for the reverse NAA airfoil section in the airfoil's coordinates.

55 k 44 = dk dk X = r 1 I r k 44 = # dk 44 0 dk X = t I r 1.0 Figure 4.7 Spanwise distribution of yaw stiffness coefficient for the Grumman WS33.

56 dk a5 r 1.0 Figure 4.8 Spanwise distribution of yaw stiffness coefficient for the Grumman WS33.

57 .0015 i k 44 = k 44 = dk 44 0 dk X = X = r r Figure 4.9 Spanwise distribution of yaw stiffness coefficient for the Grumman WS33 in a reverse position.

58 dk r I dk dk r 1.0 Figure 4.10 Spanwise distribution of yaw stiffness coefficient for the Grumman WS33 in a reverse position.

59 42 and 4.10 represent the stabilizing effect and this curve also is governed by the slope of the tangential force in Figure 4.6. It would be difficult to analyze the yaw stiffness coefficient explicitly from its lengthy expression (see expression of k44 in Appendix IV). The purpose of this section is to analyze the yaw stiffness coefficient in more detail. To accomplish this task, a simple model of the rotor geometry is chosen and studied. The expression of the yaw stiffness coefficient can be expressed into three terms according to the sine of the coning angle: 1) terms with zero order of the sine of the coning angle; 2) terms with first order of the sine of the coning angle; and 3) terms with second order of the sine of the coning angle. They are k44 k440 k441 sin p k442 sin2p (3) The expression of k440,, k441, and k442 are given in Appendix V. The expression of the stiffness coefficient is then simplified by assuming 1) the coning angle is zero; 2) the variation of the axial induction factor with yaw and yaw rate is zero; and 3) the shear center is at the 1/4 blade chord position (i.e., el = 0). From this simple model, the stiffness coefficient can be expressed into two components: the component of the linear equation of the yaw moment due to an in-plane force (force in the rotor plane) and the one due to an out-of-plane force (force normal to the rotor plane). These two terms can be expressed as

60 43 k 3 G k f 2 dr+ 3 F rf 2 dr 44 _ 2 H T 4 A 4 2 H (4) where GT is the derivative of the in-plane force and FA is the derivative of the out-of-plane force. The expressions for the GT and FA are given in Appendix V. From equation (4), it can be seen that the stiffness coefficient is dependent on the derivative of the in-plane and the derivative of the out-of-plane force with respect to the angle of attack. Furthermore, the sign of the distance from the yaw axis to the rotor, A,, is a factor which controls the in-plane force term to be either the stablizing term or destabilizing term. Figures 4.11 and 4.12 show the plots of the in-plane and the out-of-plane force coefficient versus the angle of attack for the Grumman turbine blade in a downwind position and in an upwind position. The distribution of the first term and second term in equation (4) along the blade is superimposed on each other for a reverse turbine in Figure The effect of the flapwise deflection on the stiffness coefficient is investigated from Figure Since this expression of the derivative of the out-of-plane force FA consists of the sine of the local slope of flapwise deflection, the flapwise deflection is therefore essential for the out-of-plane force contribution to the stiffness coefficient. According to Figure 4.13, it can be seen that

61 N 1.0, I * Oc, O O 100 Figure 4.11 Force coefficients for the Grumman WS33's blade section in the rotor's coordinates versus angle of attack.

62 T I 1 N I.0 I 0 - o o Figure 4.12 Force coefficients for the Grumman WS33's blade section, in a reverse position, in the rotor's coordinates versus angle of attack.

63 in-plane force out-of-plane force dk r in-plane for ---- out-of-plane force in-plane force force dk 44 dk r Figure Spanwise distribution of components of yaw stiffness coefficient for the Grumman WS33 due to in-plane force and out-of-plane force. 1.0

64 47 though the magnitude of the flapwise deflection is small, the magnitude of the out-of-plane force term is almost the same order of the in-plane force term. Thus, the effect of the flapwise deflection on the stiffness coefficient is that it adds the contribution of the out-of-plane force to the stiffness coefficient. The effect of the coning angle on the yaw stability can be investigated from the expression of the stiffness coefficient in equation (3). For small coning angle, the numerical value of the third term (term involved with square of sine of the coning angle) in equation (3) is negligible and k441 turns out to be a positive definite number. Therefore, the sign of the coning angle is the indication whether k441 sin p will act as a stabilizing term or a destabilizing term. In conclusion, the yaw stability of the system can be improved by adding a positive coning angle to the system. The numerical values of the yaw stiffness coefficient for different values of tip speed ratio are given in the sensitivity study section. Sensitivity Study The yaw stability of the wind turbine system in this analysis is primarily determined by the sign and magnitude of the yaw stiffness coefficient of the rotor and the nacelle. Thus, the sensitivity of the yaw stiffness coefficient to the selected input parameters is studied. In the previous sections, the coefficients for the equation of motion are given by

65 48 k = stiffness coefficient 1/2 POD Voo23 However, the turbine is operating at a constant rotor speed in the test condition. Therefore, for the purpose of comparison, the coefficents normalized by PM and the advance ratio are used. The coefficient based on PM is related to the coefficient normalized by the dynamic pressure by K nn = stiffness coefficient = knn 1/2 pop N23 X2 The advance ratio is seen to be the reciprocal of the tip speed ratio. J =1 X Because of the linearized system, the stiffness coefficient of the rotor and the nacelle can be separately studied. otor stiffness coefficient. The sensitivity of the rotor stiffness coefficient in yaw normalized by PM to the selected input parameters for the Grumman WS33 is examined. Torsional stiffness. The effect of the blade's torsional stiffness (blade's shear modulus G) on the yaw stiffness coefficient is examined. The only effect of the blade's torsional stiffness appears in the pitch equation (of the four-degree-of-freedom system).

66 49 Unless the static pitch deflection is significantly different from the zero value, there will not be any effect on the yaw equation. Pitch angle. Figure 4.14 shows the plots of the yaw stiffness coefficient versus the advance ratio for different values of pitch angle. First, let us consider the curve of the yaw stiffness coefficient for pitch angle equal to 4 as the typical case. The stiffness coefficient increases as the wind velocity is increasing to a certain value. Then the stiffness coefficient starts to decrease when the wind velocity is further increased because the destabilizing effect due to the negative slope of in-place force is starting to dominate (i.e., the blade is experiencing the angle of attack from 14 to 20 near the tip). When the wind velocity is further increased, the stiffness coefficient then starts to increase again because the stabilizing effect due to the stall region (a positive slope of in-plane force versus angle of attack) dominates the destablizing effect. Finally, when the whole blade is stalled, the stiffness coefficient is controlled by the stabilizing term due to a positive slope of the forces (in-plane force and out-of-plane force) versus angle of attack. This results in a large magnitude of the stiffness coefficient in high wind. With the understanding of the nature of the stiffness coefficient related to the wind velocity, the effect of the pitch angle on the stiffness coefficient is investigated. Increasing pitch angle lowers the angle of attack. Thus, increasing the pitch angle delays the destabilizing effect in the stiffness coefficient at the same wind condition. For (3 = 6 and 10, the stiffness coefficient is increased

67 50 K VDO D, Figure 4.14 Effect of pitch angle on yaw stiffness coefficient for the Grumman WS33.

68 51 at a low wind condition and decreased at a high wind condition. For the negative pitch angle, the opposite is true. That is, the curve is shifted to the left-hand side. The stiffness coefficient is decreased at the advance ratio to 0.25 and then increased again. The blade experiences the destabilizing effect at lower wind than the one with a positive pitch angle. Modulus of elasticity and flapwise deflection. The flapwise deflection depends on the blade stiffness (modulus of elasticity E), the centrifugal force, and the aerodynamic load. The static tip flapwise deflections for the Grumman machine with E = 10x106 psi and E = 20x106 psi are given in Figure The stiffer blade has a smaller deflection. According to Figure 4.16, the stiffer blade causes the stiffness coefficient in yaw to increase except at the advance ratio equal to 0.5. However, under the values of blade stiffness considered, the increasing of this stiffness coefficient in yaw is negligibly small. Although the stiffness coefficient in yaw is less sensitive to the changes of flapwise deflection, the flapwise deflection itself is an essential part of the stiffness coefficient for the contribution of the out-of-plane force. Speed. The effect of a change in rotor speed on the yaw stiffness coefficient is considered. Figure 4.17 shows the curves of the yaw stiffness coefficient for rotor speeds of 60, 74, and 90 rpm. Increasing the rotor speed slightly increases the nondimensional yaw stiffness coefficient at a high wind condition but slightly decreases the nondimensional yaw stiffness coefficient at a low wind condition.

69 ds (ft) X Figure 4.15 Effect of modulus of elasticity on static flapwise tip deflection for the Grumman WS33.

70 VD3 2 Figure 4.16 Effect of modulus of elasticity on yaw stiffness coefficient for the Grumman WS33.

71 54 K SA Figure 4.17 Effect of rotor speed on yaw stiffness coefficient for the Grumman WS33.

72 55 Note should be made that the different curves of the nondimensionalized yaw stiffness coefficient are based on the different rotor speeds. In order to compare the net difference in the yaw stiffness coefficient due to change in rotor speed, the nondimensional yaw stiffness coefficients in Figure 4.17 are corrected so that they are based on the same rotor speed, 74 rpm. These conditions are shown in Figure The relative values of these yaw stiffness coefficients are large. Shear center position. The stiffness coefficient in yaw with shear center positions at 10%, 25%, 50%, and 75% of the blade chord, measured from the leading edge of the turbine blade, are shown in Figure The stiffness coefficient is increased by moving the shear center closer to the trailing edge. This effect is significant at a higher wind condition. Distance from the rotor to the nacelle yaw axis. The distance from the rotor to the nacelle yaw axis is varied to see its effect on the stiffness coefficient in yaw. Figure 4.20 shows the curves of the stiffness coefficient versus the advance ratio for I = 0.1, 0.176, 0.25, and 0.5. The effect of this parameter is small at a low wind condition and rather significant at a high wind condition. For the advance ratio less than 0.16, the stiffness coefficient is increased when t/ is increased. For the advance ratio greater than 0.16, the effect is reversed: increasing 9/ decreases the stiffness coefficient. oning angle. From the previous section, it was found that the coning angle is one of the important parameters in determining the

73 56 K Voo Figure 4.18 Effect of rotor speed on yaw stiffness coefficient based on the same PM for the Grumman WS33.

74 DISTANE FOM BLADE LEADING EDGE TO SHEA ENTE K V, D. Figure 4.19 Effect of the location of the blade cross section's shear center on yaw stiffness coefficient for the Grumman WS33.

75 K ?, Figure 4.20 Effect of the location of the yaw axis relative to the rotor plane on yaw stiffness coefficient for the Grumman WS33.

76 59 yaw behavior. The positive coning angle adds the stablizing effect to the stiffness coefficient in yaw. The vice versa is true: the negative coning angle causes the destabilizing effect to the system. This coning angle effect is illustrated in Figure Figure 4.21 shows the system is unstable for a negative coning angle (p = -3.5"), partially stable for a zero coning angle, and stable for a positive coning angle (P = 3.5, 10'). Stiffness coefficient of the nacelle. The nacelle plays an important role in the yaw stability of the system because of its largest negative value in the stiffness coefficient. So, reducing the negative value of its stiffness coefficient would mean improving the yaw stability. The effect of the distance from the rotor to the nacelle yaw axis on the stiffness coefficient of the nacelle is examined. The stiffness coefficients of the nacelle with different values of the distance from the rotor to the nacelle yaw axis are given in Figure Increasing 9/ decreases the yaw stiffness coefficient of the nacelle. So, the yaw stability can be improved by increasing the distance from the rotor to nacelle yaw axis. However, for a given nacelle, the distance from the yaw axis to the rotor is limited by the space necessary to install the generator unit. In addition, the effect of the 2/ on the nacelle will be dominated by the rotor for the combined system (rotor and nacelle), especially for a high wind condition (low tip speed ratio). These effects are shown in Figure 4.23 for the stiffness coefficient for the rotor and nacelle.

77 60 Figure 4.21 V. 2 Effect of coning angle on yaw stiffness coefficient for the Grumman WS33.

78 K44N KJL Figure 4.22 Effect of the location of the yaw axis relative to the rotor plane on yaw stiffness coefficient of the Grumman WS33's nacelle.

79 62 Figure 4.23 Effect of the location of the yaw axis relative to the rotor plane on yaw stiffness coefficient of the nacelle and rotor for the Grumman WS33.

80 63 Analytical esults for the Enertech 1500 The Enertech 1500 was used as a test case in this section. The Enertech 1500 and the Grumman WS33 are basically the same, according to the type of wind turbine: axis downwind wind turbines. both of them are three-bladed horizontal The differences between these two turbines are the geometry and physical properties of the rotor and the nacelle. Therefore, the general trend of the Enertech 1500's yaw behavior would be similar to the Grumman WS33's. The discussion of the yaw behavior of the Enertech 1500 would be qualitatively the same as the previous section (i.e., the discussion of the Grumman WS33). Thus, the purpose of this section is to show the analytical results of another wind turbine rather than to discuss or verify the results in detail. The tower shadow is also the source of the yaw forcing function for the Enertech The static pitch angle, static flapwise deflection, and coefficients in the equation of motion in yaw are given in tables 4.10, 4.11, 4.12, 4.13, 4.14, and Table 4.10 Static pitch angle under normal operating condition. X 0 st (degree)

81 64 Table 4.11 X Static flapwise tip deflection under normal operating condition from both computer codes. AEO ds (ft) POP ds (ft) Table 4.12 X oefficients of the equation of motion in yaw from AEO code. m44 44 k44n m44n* k44n *n refers to the nacelle Table 4.13 X oefficients of the equation of motion in yaw for the combined rotor and nacelle system from AEO code. m44t 44T k44t

82 65 Table 4.14 X oefficients of the equation of motion in yaw from POP code. m44 44 k44n m44n k44n Table 4.15 X oefficients of the equation of motion in yaw for the combined rotor and nacelle system from POP code. m44t 44T k44t It can be seen from Table 4.10 that the magnitudes of the static pitch angle are so small that they have negligible effect on the system. As shown in Table 4.11, the blade exhibits a positive flapwise deflection for all the tip speed ratios considered. The difference in the static tip deflections between the ones obtained from the AEO code and the POP code is significant at the tip speed ratio equal to 3 and 4. The cause of this difference is primarily due to the simplified lift and drag curve in the AEO. The coefficients in tables 4.13 and 4.15 show that the system is unstable in yaw due to the negative stiffness coefficient. The

83 66 primary causes of this yaw instability are proved to be the nacelle and lack of blade coning. The yaw forcing function due to tower shadow with 20 shadow width and velocity deficit equal to 50% is given in Table Table 4.16 Yaw forcing function with 20 shadow width and velocity deficit = 50%. G One of the reasons that the Enertech 1500 was chosen as a test case is the availability of the data for yaw tracking error to verify the analysis. These test results were obtained from the ocky Flats Wind Energy esearch enter. The test procedure is explained in reference 23. This yaw tracking error is shown in Figure Since the analysis used the linear approximation method, the analytical results are valid only in a small region around zero yaw angle. Therefore, the analytical results for the Enertech 1500 should represent the linear part around the tip speed ratio equal to 3 and 4 or the linear part around the tip speed ratio equal to 9. A sign change of yaw angle in the analytical results would confirm the analysis.

84 Figure 4.24 Yaw tracking errors versus tip speed ratio for the Enertech 1500.

85 68 Unfortunately, according to the analysis the system is unstable in yaw. This contradicts the test data that the system is stable and operating with the, static yaw angle. The explanation for this contradiction may be the use of single-section data (e.g., 3/4 radius) to represent the aerodynamics for the entire variable thickness blade. The analysis used airfoil NAA 4415 to represent the variable thickness blade of the Enertech 1500 for calculating the aerodynamic forces and moments. One positive thing about the analytical results of the Enertech 1500 is the nacelle. The Enertech 1500's nacelle has a cylindrical shape with a hemisphere on each end. And the forces and moments calculated from the nacelle in this analysis are based on the slender body theory. Thus, with the Enertech 1500's nacelle shape, the model agrees well with the theory. The predicted forces and moments on a cylindrical body with a hemisphere at the end, which yaws at a small angle to the wind, agrees quite well to the experimental result. This can be seen by comparing the theoretical result to the experimental result in reference 14. Finally, the yaw stiffness coefficient normalized by PM for the Enertech 1500 with different wind condition is shown in Figure 4.25.

86 K Voo Figure 4.25 Yaw stiffness coefficient versus velocity ratio for the Enertech 1500.

87 70 5. ONLUSION The yaw behavior of horizontal-axis wind turbines was examined in this dissertation. The study of the yaw behavior undertook to find the cause of poor yaw tracking, investigate the parameters that control yaw behavior, and perform the sensitivity study of those parameters. The yaw behavior of wind turbines was analyzed by studying the linearized equations of motion around the zero yaw angle. Two computer codes, AEO and POP, were developed to handle the numerical values of coefficients of the equations of motion. The POP code was perferred because of its accuracy in the stall region. esults were obtained for the Grumman WS33 and the Enertech 1500: three-bladed horizontal-axis wind turbines with free yawed system. Between these two test cases,the Grumman WS33 was preferred to the Enertech 1500 because of the geometry and material properities of its blade. The Grumman WS33's blade is made of aluminum (isotopic material) and has a uniform cross section. The Enertech 1500's blade is made of wood (orthotropic material) and has a variable thickness and chord. The study showed that the yaw tracking error of a downwind wind turbine without wind shear was primarily caused by tower shadow. The effect of the tower shadow appears as the yaw forcing function in the yaw equation. This yaw forcing function is dependent on 1) the width of the tower shadow, 2) the velocity deficit in the tower shadow,

88 71 3) the position of the blade's shear center, and 4) the power output of the rotor. The yaw forcing function for zero offset distance (the blade's shear center is at 1/4 chord) will always have the same sign if the power output of the rotor increases with the wind speed. If the power peaked and then decreased with wind speed, then the yaw forcing function would change sign as the velocity increased. the case with the Grumman WS33 and the Enertech This is exactly As the peak power is achieved, the rotor yaw changes sign. The presence of the nacelle in a downwind wind turbine destablizes the system in yaw in the form of a negative stiffness coefficient. The analytical model of the nacelle was developed using the slender body theory. However, the nacelle of the Grumman WS33 is not a slender body; therefore, the predicted yaw behavior from its nacelle contains some uncertainty. Thus, the predicted yaw tracking error of the turbine system (rotor and nacelle) for the Grumman WS33 should be viewed as the qualitative behavior rather than the exact magnitude. In order to obtain accurate predictions of yaw behavior, an accurate nacelle model must be available. The yaw stability of the Grumman WS33 in an upwind position was used to verify the analysis. The analytical results indicated that the Grumman WS33 in the upwind position is stable in yaw at tip speed ratio equal to 5. The sign and the magnitude of the yaw stiffness coefficient were found to be the indicators of yaw stability for the Grumman WS33.

89 72 Therefore, the characteristic of the yaw stiffness was studied to gain more understanding about the yaw behavior. The yaw stiffness coefficient is the linear variation of the yaw moment around the zero yaw angle. This yaw moment consists of the moment due to the in-plane force and the moment due to the out-of-plane force. The yaw moment due to the out-of-plane forces exists only when the static flapwise deflection exists. Because the derivative of force was encountered in calculating the yaw stiffness coefficient rather than the force itself, the negative derivative of the in-plane force (negative slope of the in-plane force versus the angle of attack) and its position on the blade length were therefore the important factors in calculating the yaw stiffness coefficient. If this negative derivative force appears near the blade tip, its contribution would be large. The further the negative derivative force moves in-board, the smaller its contribution becomes. This contribution will act as a stabilizing or destabilizing term depending on the sign of the distance from the rotor to the yaw axis (i.e., downwind or upwind turbine). It will act to destabilize for a downwind turbine and act to stabilize for an upwind turbine. The sensitivity of the yaw stiffness coefficient to the selected input parameters was studied. The coning angle was found to be the most sensitive parameter. Increasing the coning angle increases the rotor yaw stiffness coefficient. Decreasing the coning angle (i.e., negative coning angle) decreases the rotor yaw stiffness coefficient. The next parameters to which yaw stiffness coefficient is sensitive are the position of the blade's shear center and the

90 73 distance from the rotor to the yaw axis. The effect of these parameters on the yaw stiffness coefficient is small at low wind conditions and is increased as the wind speed increases. Moving the shear center closer to the blade trailing edge increases the yaw stiffness coefficient. Increasing the distance from the rotor to the yaw axis increases the yaw stiffness coefficient at a low wind condition. But as the wind increases, the effect is reverse: increasing the distance from the rotor to the yaw axis decreases the yaw stiffness coefficient. The effect of this distance on the nacelle is reverse to the effect on the rotor. That is, increasing the distance from the rotor to the yaw axis increases the stiffness coefficient to the nacelle. However, for a given nacelle, this distance can be varied only slightly because of the space necessary to install a generator unit. For the Grumman WS33, the effect of this distance on the rotor yaw stiffness coefficient is dominant over the effect of this distance on the nacelle yaw stiffness coefficient. Increasing blade pitch angle increases the yaw stiffness coefficient at a low wind condition and decreases the yaw stiffness coefficient at a high wind condition. The yaw stiffness coefficient is slightly increased by decreasing blade stiffness. Increasing the rotor speed increases the yaw stiffness coefficient but the nondimensional yaw stiffness is hardly affected by the changes in rotor speed. Finally, the analytical results for the Enertech 1500 were studied. It was found that the theory developed for the Grumman WS33

91 74 can be applied to the Enertech 1500 since they are the same type of wind turbine: three-bladed horizontal-axis wind turbine. The study indicated that the Enertech 1500 was unstable in yaw. The nacelle and lack of blade coning are the primary causes of the system instability. The yaw prediction for the Enertech 1500 is in contradiction with test data. This contradiction could have resulted from 1) using single airfoil-section data (e.g., 3/4 radius), NAA 4415, to represent a variable thickness blade and 2) using the analysis for isotropic material to predict the yaw behavior of a rotor made of orthotropic material. In order to obtain more accurate results, more accurate models of the rotor and the nacelle are needed.

92 75 BIBLIOGAPHY 1. Adler, F. M., P. Henton, and P. W. King. "Grumman WS33 Wind System, Prototype onstruction and Testing, Phase II Technical eport," eport under Subcontract No. PF71787-F, prepared for ockwell International orporation, November 1, Munk, Max. M. "Aerodynamics of Airships," in W. F. Durand (Ed.), Aerodynamic Theory, Vol. 6, Division Q, Julius Springer, Berlin, Fung, Y.. An Introduction to the Theory of Aeroelasticity, John Wiley, Glauert, H. "The Analysis of Experimental esults in the Windmill Brake and Vortex ing States of an Airscrew," eports and Memoranda, No. AE.222, February Goldstein, S. "On the Vortex Theory of Screw Propellers," Proc. oyal Soc., A123, 440, Hayes, William., and William P. Henderson. "Some Effects of Nose Bluntness and Fineness atio of the Static Longitudinal Aerodynamic haracteristics of Bodies of evolution at Subsonic Speeds," NASA Technical Note D Hirschbein, M. S. "Dynamics and ontrol of Large Horizontal- Axis Axisymmetric Wind Turbines," Ph.D. thesis, University of Delaware, Hoerner, Sighard F., and Henry V. Borst. Fluid-Dynamic Lift, published by Mrs. Hoerner, New Jersey, 19/5, hapter Jacobs, E. N., and A. Sherman. "Airfoil Section haracteristics as Affected by Variations of eynolds Number," NAA T586, Kane, Thomas. Dynamics, Stanford University, 3rd edition, Meirovitch, L. Analytical Methods in Vibrations, Macmillan and o., Miley, S. J. "A atalog of Low eynolds Number Airfoil Data for Wind Turbine Applications," eport under Subcontract No. PFY12781-W, prepared for ockwell International orporation, February Miller,. H. "On the Weatherizing of Wind Turbines," AIAA Journal of Energy, Vol. 3, No. 5, September-October 1979.

93 Polhamus, Edward. "Effect of Nose Shape on Subsonic Aerodynamic haracteristics of a Body of evolution Having a Fineness atio of 10.94," NAA esearch Memorandum M L57 F25, August Prandtl, L. Gottinger Nachr., p. 193, Appendix, ibner, H. S. "Propellers in Yaw," NAA Technical eport No Satrou, D., and M. H. Snyder. "Two-Dimensional Test of GA(W)-1 and GA(W)-2 Airfoils at Angles-of-Attack from 0 to 360 Degrees," WE-1, Wind Energy Laboratory, Wichita State University, Wichita, Kansas, Sheldahl, obert E., and Paul. Klimas. "Aerodynamic haracteristics of Seven Symmetrical Airfoil Sections Through 180-Degree Angle of Attack for Use in Aerodynamic Analysis of Vertical Axis Wind Turbines," SAND , Sandia National Laboratory, Albuquerque, New Mexico, Sweeney, T. E., and W. B. Nixon. "Two Dimensional Lift and Drag haracteristics of Grumman Windmill Blade Section," Department of Aerospace and Mechanical Sciences, Princeton University, February Wilson,. E., and P. B. S. Lissaman. Applied Aerodynamics of Wind Power Machines, Oregon State University, May Wilson,. E., and S. N. Walker. "A Fortran Program for the Determination of Performance, Loads, and Stability Derivatives of Wind Turbines," Oregon State University, 25 October Wilson,. E., P. B. S. Lissaman, and S. N. Walker. Aerodynamic Performance of Wind Turbines, Oregon State University, June 119/b. 23. Wilson,. E., and E. M. Patton. "Performance Prediction and omparison with Test esults for Horizontal Axis Wind Turbines," eport under Subcontract No. PFN-13318W, prepared for ockwell International orporation, February 1982.

94 APPENDIES

95 77 APPENDIX I KINEMATIS A four-degree-of-freedom wind turbine system is illustrated in Figure I.1. The degrees of freedom of the axisymmetric rotor system are blade pitch deflection, blade flap, speed variation, and yaw angle. In developing the mathematical model for the turbine system, we use assumed mode shapes and generalized coordinates to represent the dependent variables. By this method we can derive the governing equations in ordinary differential form rather than partial differential form. Each degree of freedom is expressed as the product of the displacement function (assumed mode shape) and the generalized coordinate. These relations are given as: e(r,t) = fl()g1(t) (blade pitch) (1) w(r,t) = sf2(i)(q2(t) + qs) (blade flap) (2) i(r,t) = f3()g3(t) (speed variation) (3) y(r,t) = f4()g4(t) (yaw angle) (4) where rotor. is the distance from the tip of the blade to the hub of the The qi(t) terms are the generalized coordinates of the rotor system and the f ( ) terms are the assumed mode shapes. The mode i shapes are expressed as:

96 f i (pre *11 otor system. 78

97 79 2r, r, fl(112 ) s ) f2(rigrrs)2 A(rS)3 (rs)4 "1. ) (S) s Here rs is the radial distance from the local point on the blade to the blade root. The mode shape in Eq. (5) is for a uniform cantilever beam in static equilibrium with applied torque at the open end. The mode shape in Eq. (6) is for a uniform cantilever beam in static equilibrium with uniform forces applied on the beam. The mode shapes in Eqs. (7) and (8) are those of a rigid body. For a turbine blade with blade stub, the mode shapes of blade pitch and blade flap differ from the ones in Eqs. (5) and (6). They are expressed as follows: for r > H f tgj) = 24H )tr) + 2bS - (7S ) 2 "S "."'"H "S S

98 80 for 0 < r < H H rs EI (H, ( f1(i) `ET) ' 4( ) 2( )) H S S s 9( H) GJ ( ( S)( S)) f2(i) `` (12 1 `Ft S ww IH H S r where = r - H Ko,HN2 EI ( 44L1)) 21-s,/ (E)H',H, EI ( K1 = 12qT (E0H ) Here H is the length of blade stub, measured from the blade section to the rotor center. (EI) H and (GJ) H are flapwise stiffness and torsional stiffness of the blade stub. Having defined the degrees of freedom in terms of generalized coordinates, we are now ready to develop the kinematics of the rotor system. The absolute motion of the turbine blade is determined by the motion of blade deflection relative to the hub, the motion due to rotor rotation, plus the motion of the nacelle and tower. Since in this analysis no movement of the tower is allowed, we consider the reference frame fixed to the tower as the inertial reference frame. onsider the motion of a point on the blade whose absolute position is represented by a series of relative position vectors. A series of coordinate systems is used to describe these vectors. located on the top of the tower. Let the coordinate system X,Y,Z be The coordinate system x,y,z is fixed

99 81 on the nacelle and its origin is at the same point as the coordinate system X,Y,Z.. A A The coordinate system x,y,z is the same as the coordinate system x,y,z except its origin is moved to the center of the rotor. The coordinate system x,/,z is obtained by rotating the coordinate sys- A A A tem x,y,z by the magnitude of angle * (where p = at + x). The coordinate system x,y,z P P P around the /axis by the angle p. coordinate system x,y,z a a a is obtained by rotating the coordinate system x,y,z Then, at position r on the blade, the represents the effect of the pretwist angle, a. The coordinate system xa,ya,ze is obtained by moving the origin of the coordinate system xa,ya,za in the z3 direction over the distance "w" and rotating it around the ya axis by the angle w' (aw/ar). Finally, the coordinate system xi,x2,x3 is located on the shear center of the blade cross section and differs from the coordinates xe,ya,za by the amount of the pitch angle 8. These coordinate systems are shown in order from the inertial reference frame to the final reference frame that is fixed on a point on the blade in Figures 1.2, 1.3, and 1.4. A series of transformation matrices is used to transform from one coordinate system to the others. These transformation matrices are shown in Figure 1.5. Another variable that we will deal with is the radial displacement of the blade. This displacement occurs during the blade deflection when the assumption of an inextensible blade is made. This radial displacement is defined as (9)

100 82 x X Z i AA A Figure 1.2 oordinate systems XYZ, xyz, and xyz.

101 Figure 1.3 oordinate systems xyz, x y z, x y P z. 83

102 84 Xs Figure 1.4 oordinate systems xy13 z x6 ye z 8, and x 1 x 2 x3.

103 85 XI X 2 ( x ) = [T] (x') 0 0 [T] = 0 OS 71 Sin 71 0 sin 771 cosni / 77 ( x ) = [T] cos Sin 772 X3 [1] = 0 I 0 -sin cos 772 ( x ) [T] ( x') / cos773-sin773 0 [T] sin 173 cos Figure 1.5 Transformation matrices.

104 8,6 where vc is the deflection of the blade in the direction that is perpendicular to the axial line (flapwise direction). The velocity of a point on the blade is found by using the kinematic relation [10] x T c a a (10) where ic is the velocity of point c in a reference frame. a is the velocity of point c in a reference frame a.. a is the angular velocity of the body that the reference frame a is fixed to, observed from the reference frame. T. is the position vector of point c. For the angular velocity, we have ya ut ea where is the angular velocity of the body that a reference n frame is fixed to, observed from a reference frame n. The absolute motion of a point on the blade can be found by using the transformation matrices and Eqs. (10) and (11). The blade velocity and blade angular velocity measured at the center of mass of the blade cross section are: = V Xp Xp + V YP nyp + V Zp Zp (12)

105 LS ajalim A du = A + UA U + mua + LIA u = z`px pue xa = - I.ScISODAT hu JXA = A mx = xa a = Dn - i.s4kt4 dsoogu + - i.stia esoogi4s))a, OS00 rlisodgulsdin.$)as - dsopla esoogsop) s,msoogsop - eu 4kui.s( (444S9SOD + msop, 1.s (gin.sou + du.ska 0S00gS00) + MSOD S gu t.s (11S00( A A. = - (iisopk7 A 14'4A = = J) + ISM gu dso341(dn - Ls4hM gsopdu!.s,msooesopea gu - + du!_sitlsop!..('n - disoogsopdso3).44 Ma LS, 1.s,mtu!.seu gu A ak s] du i.$) esoogu -,msoogsoo au!.s ( - i.s,mt4sdsop [Ou dsod]dlsopi:a esoogu - el4s,msoogsop ( + sdu!.s mu, reu 217A sior sdu (11u JZA za = = + (144s):(3n gs0dtn + I.Schm + MSO3eSO39a LS, gu tsgu du + 1.SdS0)44 filsodgu - Ma LS, mu!.sau Osoo, + du es0d9sod) + 914seui.s,msoo ( aza a = + dsopka osoogsop) - i.sdsopka mu 4u!.seui.s, + I.s,msoo gui.seu 41sop(

106 88 where e = distance from mass center to the shear center of the blade cross section. = distance from the rotor plane to the nacelle's yaw axis. w' ar g W' = raw) at Lar) 1.1) = + wfi w 21'2 + w3r-3 (13) where w. 1 = w 81 w Wi + + w yi i = 1,2,3 and Wel = 5 w wl = 0 w *1 = (sinpcosw' + cospsinwicoss) w yl = ;L(cospcosw' - sinpsinwicosa)cos* - sinw'sinssimpj '482 ww2 = - W'cose colp2 = - 11/(sinpsinw'sine + cospsinocose - cospcosw'sinecos8)

107 6 Vs-co = - eui.simui.sdsoo)a. 9S SdU MSOOdU gsodou IttS00( OSODEISOD)P: + msod 1S, gu1seu S( "cr) = 0 EtAm = M LS, GU - esodmin.sdul.$)41 - eu!.sikil.sdsod - (gsodesoo,mso3dso0 Ekcn - esoo,mul.scisop)1, + el4s 41.q.sdui.s + (Isoo(gsopesop,teisoodui.s + eu!.sgsoo)) - 41u!.s(gL4sesop,misop

108 90 APPENDIX II OTO AEODYNAMIS II.1 elative Velocity The relative velocity that the blade element experiences at the rotor is defined as the vector sum of the blade element velocity at midchord and the wind velocity at the rotor. = w B (1) Here /w is the wind velocity at the rotor and 12, is the blade element velocity at mid-chord, it does not include pitch velocity (5). The wind velocity at the rotor is given by w = V w ; z - av w n, (2) where "a" is the axial induction factor. The development of the axial induction factor will be explained in a later section. In the strip theory method (2-D assumption), the relative velocity in the spanwise direction does not produce lift force or drag force. The velocity to be considered in evaluation of the aerodynamic forces and moments is the relative velocity in the plane of the blade cross section. Thus the relative velocity is expressed as e Ow -%).-13/Xlye Ow (3)

109 91 By using the unit vectors en and ;t, we obtain e = Wnen - Wt et (4) where W n = (V w - B Wt Ow 9 en = Z8 e. = ye The expression for VB can be obtained by following the same procedure used in Appendix I. Substituting the value of -13. and Vw into Eqs. (5) and (6) we obtain the normal and tangential relative velocities as W n = awn + V Bn W t = cwt + V Bt

110 92 V wn = - Vw(cosy - a)(sinpsinw' - cospcosvecos$) - V siny[(cospsinw' + sinpcosw'cowsin* - cosw'sinscos*3 - Zi[(sinpcosecosw' + sinw'cosp)sinp - sinecoswicosto] + d sinw' - (r + u d );coswicospsin$ - (r + ud)y(cos8cosw'sin* - sinpsin8cosw'cos*) V Bn - Wcosw' - wi,sinvecospsin$ - w4(sinw'sinpsinacos* - sinw'cos$sin*) + e3icose(sinpcosw' + cospsinwicos8) + e3 -.cose[(cospcosw' - sinpsinwicoscos* - sinw'sin$sinlo] V wt = {- Vw(cosy - a)cospsin$ - V siny(sinpsin0sin* + cos8cos*)} - Zy(cos0cos* sinpsinesin*) + (r + ud);cospcos$ - (r + ud)-7((sinbsin) + sinpcosacos*) V Bt = - wlpsinp - vicospcos* + e3isine(sinpcosw' + cospsinw'cos$) + e 4sine[(cospcoswl 3 - sinpsinw'cose)cos* - sinw'sinsin*] where e3 is the distance from the mid-chord to the shear center of the blade cross section. The velocity diagram of the relative velocity at the blade cross section is shown in Figure II Aerodynamic Forces and Moments Figure shows a blade profile section at radius r with the relevant velocities and forces. The air flow gives rise to a lift force L and a drag force D whose resultant can be resolved into components of normal force dfn and tangential force dft.

111 Figure Velocity diagram at blade cross section. 93

112 94 From the geometry we have df n = dl cos. + dd sin. (7 ) df t = dl sin. - dd cos. (8) The expression for the normal force and tangential force can also be expressed as 2 dfn = 1 0 w 2 -. e c dr n 1 df = p W 2 c dr t e t where h = L cos. + D sin. t = L sin. - D cos. The aerodynamic moment at 1/4 chord can be expressed as dm n = pwc dr n c/4 1 7 e M c/4 1 and according to Fung [3] c/4 IT = - OSa 5 8 (12) Substituting the expression of mc/4 back into Eq. (11), we obtain

113 95 3 dm = - pmwecosa Trc 5 dr (13) c/ Linearized Aerodynamic Forces In this study the linearized aerodynamic forces will be developed. These functions will consist of the nominal terms plus the linear variations of the aerodynamic forces with the dependent variables. Let us first consider the aerodynamic forces. Figure shows the blade profile section at radius r with the relevant velocities and forces. The components of the aerodynamic forces are expressed as dfn =, -7p. We2cndr (14) df t 1 = p W 2 c t dr e (15) where n = L(aE)OSO + p(ae)sin0 t = L(aE) sino - o(ae) coso We = Wn + Wt ae is the effective angle of attack measured at 3/4 chord when including the effect of the pitching velocity at that point. Normalizing Eqs. (14) and (15) by dividing through with 1 -p V yields W 2 c d F ('A) n 721 n (16)

114 Figure Velocity diagram at blade cross section. 96

115 97 w e )2 c, F (7,17 't t dr (17) The derivative of the normal force with respect to the dependent variables is defined as N dr ---- ac F an N = (2 Wn a Wn 2W, z _ an w c We a 17t )] +(Tr)2 3n n (18) The derivative of n with respect to h becomes a n a n aae a n an acte an ao an (19) The velocity of the fluid that accounts for the pitching velocity at 3/4 chord is expressed as VP = e 2 ecose; n - e 2 esineet (20) and W2 = (Wn + e 2 5cose) 2 + (Wt + e 2 esi ne) 2 From the velocity diagram in Figure , the tangent and cosine of the effective angle are expressed as tance Wn + e 2 5cose Wt + e 2 esine (21)

116 98 cos(0e Wt + e 2 8sine (22) where (I)E e From trigonometric relations we obtain an (tan (1,E) = sec 2 0 E 4E aa E 2 a (,_ = - cos (0E LanoEj an an (23) By substituting Eqs. (21) and (22) into Eq. (23), we obtain the expresaae sion of as ace E 12 r w + e nn 2 an ))(Wt + e2esi ne - (Wn + e28cose)(wt + e2 (esine))] an (24) n where Wn a Wn an W t aw t an In the same way, the expression of 21- can be expressed as an

117 q 99 = [WnnW - WnWtn] an w2 t "e (25) Substituting Eqs. (19), (24), and (25) back into Eq. (18) we then evaluate all the dependent variables at nominal values. The derivative of the normal force can be expressed as N n W W n t = F 1V n+fi 2 VG, for n * ql,q1 Wn Wt q1 N = F + F --:- + F q1 1 v-1,-o, 2 v-m-os 4 (26) Wn cl1 N = F 1 V +. (1 1 co Wt. F qi F 1 i 4. 3 a vco where W in n r t) = t c + 1 V n n V V =, 2= n-nv F = V F3 e Wt 2 f.< n V V 1 F4 = ft, (AZ en fi co a. Figure shows the velocity diagram of the blade evaluated at the nominal value. The relation of lift and drag at the nominal value can be expressed as

118 100 Figure Velocity diagram at blade cross section evaluated at nominal value.

119 101 n = L cosa + D sina n = L cosa + sina - D t v ae ae t = sina - D cosa t = L sina - cosa + D n ae ae n = L cosa + D sina a ae ae t + L sina - 0 cosa a ae E The variation of the tangential force with the dependent variables can be found in the same way. The derivative of the tangential force is defined as dr n 3F an t and W W n t H = G n 1 n+gn 2 V-- for n * ql,1 W n Wt ql q1 H = G + G, + G 1 V. 2 V 4 (27) W n. Wt. ql H = G + G 1 2 -V-- 4- G3 f7:-

120 102 where 2W \T n { co ct} v 2W W c t n G2 2 V t t V co V co G = e W t 3 Tt V V 1 a OD OD f We c 2 G = () 4 V t co ce 11.4 Axial Induction Factor "a" In this analysis, the nonrotating wake model is used. We can calculate the local value of the axial induction factor by equating the windwise force developed on the blade to the momentum flux in an annular ring of radius r. Applying the momentum theorem to the flow in the annulus "dr" one obtains an expression for the windwise force as dt = po,(2wrdr)u(vw - V2) = p. V2 (1 - a)2a271.rdr (28) Defining a local thrust coefficient by dt (T)L 1 2 p.vda Equation (28) becomes (T)L = 4a(1 - a) (29)

121 103 The local thrust coefficient based on the blade force in the windwise direction is developed using the blade element theory dt = 1 p W 2 Bc dr e (30) Using the definition of (T)L, we obtain (T)L W (Bc)(ve)2 (31) With a given value of L, the local axial induction factor can be found by equating Eqs. (29) and (31). The simple momentum theory approach leads to the result that the induction factor "a" cannot be greater than 0.5 as this would yield zero downstream velocity. However, increasing thrust coefficient values are obtained for a > 0.5. When the axial induction factor "a" is greater than acritical, the critical, Glauert relationship [4 ] has been used instead of the simple momentum theorem. The Glauert relationship is shown in Figure This empirical relationship can be approximated by a straight line with good accuracy using wind tunnel test data. The straight line approximation used in this analysis for a > ac is (T)L = 4ac(1 - ac) + 4(1-2a)(a - ac) (32) where ac = 0.38.

122 I I NAA TN 221-8=4 O NAA TN 221-8=2 A BM 885-uv A &M 885-cv T A ) r GLAUET'S HA. /._s GL A UET'S UVE EMPIIAL FOMULA 4,2 6O " MOMENTUM THEOY Axial Induction Factor, a 1.0 Figure Windmill brake state performance.

123 Variation of Axial Induction Factor with Generalized oordinates In the process of linearizing the aeroforces, the variation of the axial induction factor with the dependent variables is encountered. We can calculate the local variation of the axial induction factor by equating the derivative of the moments developed by the blade force to the derivative of the moments developed by the momentum flux. Defining the variation of the axial induction factor as en r = k n sin + j n cos* (33) Substituting the expression for the variation of the axial induction factor back into the linearized aerodynamic forces terms, we now have two new coefficients to solve for, kn and jn. The coefficient k n can be calculated by equating the derivative of the yaw moment developed by the momentum theorem to the yaw moment derivative developed by the blade element theory. In the same way, the coefficient jn can be calculated by equating the derivative of the pitching moment developed by the momentum theorem to the pitching moment derivative obtained from the blade element theory. onsidering the segment "rndrnde of the annulus "dr", we obtain the expression of the moment as the cross product of the rn vector and the windwise force of that segment. da x d1 (34) N where

124 106 rn = (r + um)cosp - wsinp (35) dt = p. V2 (cosy - a)2a rndrnd* (36) A local moment coefficient is defined as dm d M p.v7r 7 (37) Substituting Eqs. (35) and (36) into Eq. (37), we obtain the expression of the yaw moment as the component of the vector "dm" in the nx direction and the pitching moment in the ny direction. The expression for the yaw moment is 1 r N d M = 4a(cosy - a) --- sin* x dr N (38) The expression for the pitching moment is d M 1 Tr 2 rn = 4a(cosy - a) IT- cos* dr N (39) By taking the derivative of the yaw moment and the pitching moment with respect to the dependent variables then integrating over the whole rotor, we obtain the expression a a 2 27r, x r r P la rn 3.n IT aa o o a an at 27r rn y 1 L aa N 7r of I I dr N Si NI) 4) -17 " " costod* (40) (41)

125 107 where TL = 4a(1 - a) Substituting the expression of H_ from Eq. (33) into Eqs. (40) and (41), we obtain am x_ knit 1 an am = j an n 1 (42) (43) where a T r 3 dr L N N ni of as 3 (44) Now we will look into the same yaw moment and the same pitching moment but they will be developed by blade force instead of momentum flux. onsidering the small element of blade "dr", the moment created by the aeroforces and aeromoments are expressed as = rm x dt + da c (45) where rm = (r + u Xp + W zo + e 1 2 dr, df n a + df t ye

126 108 da c = dm c the expression for the yaw moment is obtained from the component of dtm in the n x direction dm = (TL1) d + (TL2) d x (46) where we 2 N = cn c W e)2 c H = -- e 1 ---cose[(cospcosw' - sinpsinwlcos8)cos* - sinw'sinbsimpj TL1= (r+u ) cosw' sinw'j(sinpsinocos* - cos8sin*) (r+u ) - [ sinw' - cosw' sine] [ (sinw'sinasinip) el + (cospcosw' - sinpsinw'cosa)costp] TL2 = (r+um) [ cosw' + sinws][(cospsinw' + sinpcosw'cosa)cos* + cosw'simsinild Now we take the derivative of this moment with respect to the dependent variables. Then we add the effect which accounts for the "B" turbine

127 109 blades in the system. The expression of the average yaw moment derivative is given as 3 Mx B 27r arr f f N (TLI) dip f f H ( TL2) r H 0 n Zir H 0 (47) where N = n 3 N an H = n a H an By substituting the expression of H. from Eq. (33) into Nn and Hn terms, the derivative of the yaw moment is expressed in terms of kn and jn. The expression of the pitching moment developed by the blade force is expressed as the component of dt in Eq. (45) in the n direction. M 3m Then the derivative of the pitching moment any is obtained in the same 3m way as it is done in anx. Now we can equate the derivative of the yaw moment developed by momentum flux to the one developed by blade force and the derivative of pitching moment developed by momentum flux to the one developed by blade force. The analysis results in two equations and two unknowns (kn and The result of this linearized analysis shows that the variation of the axial induction factor exists only for the yaw and yaw rate variables 3a an n * q4 and q4 (48)

128 110 The coefficients k n and j for yaw and yaw rate are given by k = ("4 "5 "5 "7)( ) "2(1112 "13 "14 "15) (49) 2 2 q4 ("1 "3) "2 ia '4 1(114 "5 "5 "7)112 ("1 n3)(1112 "13 ("1 "3) " )} (50) k (118 " "11)("1 113) 112( ) q4 (ni - n3) 2 + n2 (51) -I (118 "9 "lo 1111)112+ Or "3)( )1 q4 where ni's are the integral terms. ( ) 2 + (52) These integral terms are given as follows: N a T dr r L (rn)3 NO N f as ) el 3 r 57 f (Ni) (N1) e(cospcosw(i) - sinpsinw(')cose) n H (r+u ) w r 3 m N dr -.7; 2 f (Ni) ( cosw(' + sinw(;)sinpsine T T if 2 H (r+u ) 3 m +--- f (H1)( 2 r H sinw' wo coswc;)(cospcosw' - sinpsinw(1 coss) r N dr 3-27r f (r+u ) w m o, (H1)( --- coswo + TE-sinwo)(cospsinwc1) H dr + sinpcoswc;cosa) N

129 ITT f H8 H a I (IN) gu Ls?mui.s j N JP 8 u 8 ) (-ri+j w -a- f (TN) ) H H8 - IL, ---- H ) (-ri+j w f )(1H) H H8?msoo s?mu!. J OM N JP ElsoD(?mu!.s um 1.s flu -1, (?msoo 1.s?mu NJ JP -I ---ti f He ) TH ) (n+j?ms + om LS MU (? i.s?msod gu N JP u -3- =f4- f Ha 8 te -3-](zN) 54s00ds0o) - S S ( EISOD(,)MU ) (-n+j -77ji--),)MSOD + o M!.s mu (0, LS [9u!.sdu Su = -=-Q- f H8 8 to --1)(EN) I.s gu 1.s?mu ) ) (-n+.1 w H m?msoo + o a i.s Jp sod(?mu (g =- f b H 21 )(al) n+j (,mul.s o - OM 8 msop,msoodsop)(,o o - Jp I.sdu!.s mu 17j.(gso3, - f H8 8 ) )(ZH ) 11 w (?MS00 + om S (54U OSOD)?MU + msodu!.s (gscn, t7 Jp

130 = -27 f (H3)[( cosw(13 + H ( r+um) (r+um) o, sinwo)coswolsi n8 + si two' wo dr - coswo')sinwo'si no] f 4 il = f (N4)[}-rel (cospcoswo H - si npsi nwo'cos(3) ( r+u ) w dr ( m coswo + o sinw '0 )sinpsin8jf4 ir el ( r+u ) 3, w. o., dr Tr-r 05) [-ir sinw'sin$ ( rn cosw' + s nw )cosej f o 4 H 3 ( r+um) wo -7 f (H4)( sinwo' - cosw' ) (cospcosw - 0 H - Si npsinwo'cosa)f4 dr 1110 = 3 r arr H ( r+u m ) (H4)( coswo + sinwl)(cospsinw' + si npcoswolcoso) f4 dr 1111 = 3 f " 5 ) ( r+u ) w m o cosw' + [( o si nw')cosw'si ns (r+um) wo., dr + ( s nw - cosw')sinw'sine]f o 4 e (r+ um) _r d n12 = f (N2)(11-- sinw'sine. ( coswo' + T sinwo')cosa)f 4 H 1113 " H ) f (N3) (Tr- (sinpsinw(icose-cospcoswo.)+(( coswo. + sinw 1)si npsi nb) f dr 4

131 1114 = 3y f H ( r+ u (H2)[( ) wo m sinw' - coswol)sinwossi nf (r+ um) cosw o dr + o sinw )cosw si ne.]f 4 3 ( r+ um) wo f (H3)( H sinw' - coswo )(cospcosw' o - si npsi nwo'cos8) f4 d r ( r+ um) 1115 = + f (H3)( H cosw' + o s i nwo) (cospsi nw ' + Si npcoswo'cos8) f 4 d r II e ( r+um) = o., -27. f (N4)[7 nw Isi ( cosw' + si nw )cos o o f 4 " dr = - T e f (N5)[T ḻ (cospcoswo' - sinpsinwo'cos8) I H (r+um) wo ( coswo si + nw1) 7-- si npsi ne]f dr o 4 ( r+u ) w 3 m 1118 = f ( H4 ) [ ( si nw' - -Fro coswo)si nwo'si n8 77 H ( r+u ) + ( coswo' + w si nwo' )cosw' si n8] f 4 dr

132 114 3 f (H5)( (r+u m ) sinw' o o cosw1)(cospcosw( o 13 - sinpsinw('cosa) f4 dr f (r+u ) m 0-15)(-7 coswc; + 0., sinw )(cospsinw' o "71 H + sinpcosw(;cosa) f4 dr where N1 = (sinpsinwt; - cospcosw(;cosa)f1 - cospsinaf2 N2 = cosw'sinaf 1 - cos8f 2 N3 = (cospsinw'd + sinpcosw('cos$)f1 + sinpsin$f2 V (r+u w di o sinacosw ' + sinpsinacosw' + sinw'sinpsina)f 1 V o V N4 = e ( cospcoswl - sinpsinw'cosa)f, O (r+u di w -(2- toss + sinpcosa + o cosp)f2 V... (2-\T- (sinpcosacosw ' + sinwcosp)+ ' (r+u d ) v cosacosw' N5 = V +, sinw o cosa +- 3 sinw'sin$)f 1 V +(vsinpsino + (r+ud) V slna)f 2

133 115 The expressions for H1, H2, H3, H4, and H5 are the same as N1, N2, N3, N4, and N5, respectively, except F1 and F2 in Ni terms are replaced by G 1 and G 2 in Hi terms Tip Loss Model In order to account for nonuniform flow in the wake of a wind turbine, flow models have been adapted from the propeller theory. Physically, the tip correction accounts for the fact that the maximum change in axial velocity, 2aV,,, in the wake occurs only at the vortex sheets and the average velocity change in the wake is 2aV.F, where F is the tip loss factor. "Tip losses" have been treated in a variety of different manners in the propeller and helicopter industries. The simplest method is to reduce-the maximum rotor radius by some fraction of the actual radius, which in helicopter studies is of the order of A more detailed analysis was done by Prandtl [15] as a method for estimation of lightly loaded propeller tip losses. Later Goldstein [5 J developed a more rigorous analysis. But due to the ease of use and the fact that the available experimental data are not sufficiently accurate to resolve the differences predicted by various approaches, only the Prandtl method will be considered. Prandtl's factor is defined as F = erc cos e -f where f = B -r sin$t B /r-1 sing)

134 116 The expression for f can be suitably approximated by writing rsino in place of sino.r. Here B represents the number of blades; 0/. is the angle of the helical surface with the slipstream boundary Power and Thrust oefficient From the blade elementary theory, the windwise force and torque at the nominal value are given as dt = 1 dr p 2 BW 2 c (53).eN dq = 1 dr p.bwe2 ctr (54) Power is defined as tfte product of torque and angular speed dp = adq (55) 1 Normalizing Eqs. (53) and (55) with - rp.y./ 2 1 T and p.v3 I1 2, respectively and making use of the relationship of the relative velocities and angles at the blade cross section, one obtains 3 xtin clor: p ;i7s.,/ (1;3.2 ) [(1-a)L - xyx2dx (56) x hub 3 x tip Bc 1-a 2 T f [xl + (1-a)D]xdx /x x hub (57)

135 117 APPENDIX III DEIVATION OF GOVENING EQUATIONS In order to develop the equations of motion, the Lagrange method is used. The expression of kinetic and potential energy of the system will be developed. Then, by using the virtual work concept an expression for the nonconservative forces can be obtained. Lagrange's equation is used to develop the equations of motion. The Lagrange equation is given as d (at.,a L.= al aqi aqi Qi where L Qi = Lagrangian function = KE-PE = nonconservative force qi = generalized coordinate With the expression of KE, PE and Qi substituted back into Lagrange's equations, we obtain the equations of motion. III.1 Kinetic and Potential Energy In order to obtain the expression for kinetic energy of the rotor system, the velocity and angular velocity of the blade element are first developed. With known values of mass and mass moment of inertia of the blade element, the kinetic energy is expressed as (1)

136 118 Here Vc is the velocity of the blade element of length dr, wi's are angular velocities of the blade element in the direction normal and tangent to the blade, dm is the mass of the blade element, and dii's are the mass moment of inertias of the blade element at mass center in the same direction as the wi's. The total kinetic energy of the blade system is obtained by integrating over the blade length and adding the contributions of each blade B B B B KE = E f V2dm + z f wl 2 di, + z f 4dI2 + z f 4dI3 i=1 H i=1 H i=1 h i=1 h (2) where B is the number of blades. The additional kinetic energy due to the hub mass and generator are considered. The additional kinetic energy terms are expressed as. 1 KE = I * 2 + I (N tp) T G (3) G Here I H is the mass moment of inertia of the hub around the rotor shaft, IG is the mass moment of inertia of generator around the rotor shaft, and NG is the step-up gearing ratio between the turbine and the generator. An expression for the potential energy of the rotor system can be derived from the strain energy due to the blade deflection and blade twisting. The expression for the strain energy of an element of a blade is first developed, then integrating along the blade span and adding the contribution of each blade to get the total potential energy. Thus, we obtain

137 119,2w dr 2 ar i=1 H ar i=1 H U = 2 f EI(r)(2--r dr + E f GJ(r)(----) (4) Virtual Work The virtual work principle can be stated as, "If a system of forces is in equilibrium, the work done by the externally applied forces through virtual displacements compatible with the constraint of the system is zero," [11 n SW = E Pi. = 0 i=1 where = external force dri = virtual displacement Virtual displacement is defined as infinitesimal arbitrary changes in the coordinates of a system. These are small variations from the true position of the system and must be compatible with the constraints of the system. The total virtual work of the system can be expressed as the summation of the virtual work of conservative forces and the virtual work of nonconservative forces. The conservative forces are the forces that do depend on position and can be derived from a potential function. onservative forces are the inertia forces, the contact forces, and body forces. The nonconservative forces are energy-dissipating forces, such as friction forces and forces imparting energy to the system, such as external forces. Nonconservative forces are forces that do not depend on position alone and cannot be derived from a potential function.

138 120 In this analysis we will consider the virtual work of the nonconservative forces alone. The nonconservative forces in our case are the aerodynamic forces and moments Nonconservative Forces First, let us redefine the virtual displacement and virtual angular displacement (virtual rotation) of the system. In this analysis, we assume that the aeroforces and moments act.at 1/4 chord position of the blade cross section. The virtual displacement and virtual angular displacement are defined as [10 av d 91 Sa 3w. agi qi where a, d the partial rate of change of position with respect to aqi 3w = is the partial rate of change with respect to qi of orientaaqi qi at the 1/4 blade chord in the inertial reference frame. tion of the blade in the inertial reference frame. The virtual work is defined as the summation of the inner product of the aerodynamic force and the virtual displacement and the inner product of the aerodynamic torque or couple and the virtual angular displacement

139 121 sw = Ps-P + A.stt (7) The aerodynamic force and couple at 1/4 chord are defined as P = Fnen + Ft e t (8) A mit, 1 Substituting Eqs. (7) and (8) into Eqs. (6) the expression for the virtual work becomes 64 Q16q1 Q26q2 Q36(13 + Q4"4 where Qi represents the nonconservative force relevant for the right hand side of the Lagrange's equation (.d) + A (a4: 3q1 ov (A + A. (222) aq2 42 = p a d) A (aw av, Q4 = p (77'') A (14 3'44 3q4 Now we have the expression for the Lagrangian function and the nonconservative forces. Substituting these expressions back into Lagrange's equation, we obtain four equations of motion. These equa-

140 122 tions can be written in matrix form as [M] {qi} + [] {qi} = {G(q1,...q4, qi,...q4,t)} (13) where [M] = nonlinear mass coefficient matrix [] = nonlinear damping coefficient matrix from -Ed f (al 7H ig1 = a vector consisting of nonlinear terms from "i + Oi Nacelle, Gravity Nacelle In this analysis we will consider the nacelle as a slender body. The shape of the nacelle is assumed to be a cylinder with a hemisphere on the forebody and afterbody. Figure shows a picture of the nacelle. Since we assume that the nacelle acts like a rigid body and the only movement it is allowed is rotation around the yaw axis, the kinetic energy and potential energy can be expressed as KE = I y n PE = 0 where I n is the nacelle's mass moment of inertia around the yaw axis KE = Ifa 2 n 4.4 (14)

141 123 OTO PLANE WIND PA z Figure Nacelle geometry.

142 124 For the nonconservative force, the forces on the nacelle are calculated by using the slender body theorem. The forces on the body can be expressed as ds df y = 2q co dz dz (15) u 2 zl d(z))21 j1; dfz = - q.(i + (,z) + ( dz dz ) dz (16) where s = the cross section area of the body (z) = the radius of the body cross section The virtual displacement of the nacelle is expressed as 6-P = zf4bg4 ny (17) The virtual work of the nacelle system is given by d(sw) = dfy sp = (2qwz dz f24q4)sq4 (18) dz The nonconservative force for the nacelle is expressed as ds dqn = 2qmz dz f 2 4 q4 az (19)

143 125 The force on the nacelle exists only at the hemispheres at both ends of ds the nacelle (-ay* 0). The afterbody of the nacelle is in the hub area. flow would separate before it reaches the afterbody. In real flow, the Only the forebody part of the nacelle is considered. The equation of motion of the nacelle is developed by substituting the expression for kinetic energy and the nonconservative force in Lagrange's equation. The nondimensionalized equation of motion is given by m 44 q 4 + k 44 q 4 = 0 (20) where I n 2 m44 n -7 ' 4 2 rn ds 2 k44 f4 dz n -7 J dz -(n-m) n = distance from the nacelle's yaw axis to the forebody end of the nacelle M = radius of the hemisphere on forebody and afterbody of the nacelle. The correction factor for the nacelle with a non-circular cross section is obtained from reference 2. From the analysis of airships, Munk[2] defined the inertia factor for the cross section effect on the

144 126 lateral forces developed by the slender theory as for common circular cross section =. 1 for other cross section b2 4S where S denotes the area of the cross section and b is its largest width or height at right angles to the motion considered. Hoerner [8] also suggested the correction factor for'the effect of fineness ratio on the slender body as P = (1 - d/1) where 1 is the length of the body and d is the diameter of the body. With these correction factors, the stiffness coefficient of the nacelle becomes 23 in z 41 f24 dz (0) 44 n where z 1 is the distance from the forebody end of the nacelle to the point where ds = 0. Gravity Effect For a larger wind turbine system, the effect of gravity is very important in dynamic and structural analysis. Although the Enertech

145 is a small wind turbine system, the gravity effect will be included in the analysis to make the analysis applicable to any size turbine system. The gravity effect will be added to the system by means of a potential function. The gravitational force of the blade element dr is defined as -gdm nx (21) The potential function for the gravitational force is given by dp = ghdm (22) where h is a function of ql, q4, and t, whose absolute value is equal to the distance between the mass center of the blade element cross section and any fixed horizontal plane H. We are dealing with the expression for the derivative of the poten- tial function instead of the potential function itself when we 3q i develop the equations of motion by using Lagrange's equation. Therefore we take the derivative of the potential function in Eq. (22) with respect to the generalized coordinate a(dp) aqi ah aq. (23) The velocity of the blade element "dr" measured at the mass center can be expressed as

146 128 c dh + dt nx 4 ah ah + = - ( E q. + + aqi at x 1=1 (24) The expression ah be found by dotting Eq. (24) with the unit vecaqi ah for nx and assuming that -57. equals zero. ah avc + aqi x aq (25) Substituting the expression ah in Eq. (25) back into Eq. (23), we have aq. P) the expression a(d accounting for the gravity effect to be put into aq Lagrange's equation aid.el av, c gdm. aqi --.n x a7qi (26) Tower Shadow When a rotor is downwind of the tower, the blades pass through the wind shadow cast by the tower. The performance of the wind turbine will be affected by this tower shadow. In this study, the tower shadow is modeled as the velocity deficit from the rotor axial velocity value over a selected region of the rotor disk, centered about the tower center line. For the simplicity of analysis, the width of the tower shadow is assumed as a segment of the rotor area. The width and the velocity deficit of the tower shadow are dependent on the geometry of the tower. This tower shadow model is shown in Figure

147 129 Figure Tower shadow.

148 130 To account for the tower shadow effect on the equations of motion of the system, the width and the velocity deficit are arbitrarily chosen. Then for this linear system, the superposition method is used. The average forces on the rotor with the tower shadow will be the average forces on the rotor without the tower shadow, plus the difference of average forces in the shadow region between the one with and the one without the velocity deficit due to the tower shadow. The coefficients of equations of motion will be recalculated for the shadow region. Many terms in the expression for forces and moments that depend on the azimuth angle, which are usually balanced out in the 3-bladed rotor case, will remain in the tower shadow case. The average forces and moments in the shadow region are given by F shadow B 27; X 7-7 A F f H (df)d* (27) shadow f (it x dr)dip 1T - 7 H (28) where df = the force on the blade element X = the shadow width. The flow conditions in the tower shadow are developed from a uniform flow model. Thus flow conditions in the tower shadow vary only with velocity deficit and tip speed ratio. Table gives the values of the integrations from the lower limit of w - 7to +. X X

149 131 Table Some integration values. I A. X IT -, 7 X - 7 si n 2 4, chp A - sinx 2 1T '7 7 A Tr 2 cos 2 tiglip (x - sinx) 2 Tr + 2 1T A 2 si nlocostpd4) = 0 Tr + A - 7 A 2 = -2 sl n 2 2 si mod* = 0'

150 132 APPENDIX IV LINEAIZED EQUATIONS OF MOTION IV.1 Linearization eal systems contain some nonlinearity. If the ranges of values of the dependent variables are sufficiently restricted, the system may be well approximated as linear. In this study we will treat the system in the linear range. The first thing we need in linearization is the equilibrium value of each dependent variable. Because of the complexity of this rotor system's mathematical model, the equilibrium values have been chosen as e o = 0 (1) w f ( f.)(1 (2) o s2s xo = 0 (3) Yo (4) where qs is the static tip deflection and the subscript 0 indicates that the values are evaluated at nominal values. We now define the dependent variable as the nominal (equilibrium) term plus a small variation term. qi(t) = qi0 + dqi(t) (5)

151 133 Substituting the value of the generalized coordinate shown in Eq. (5) into the equations of motion and developing a Taylor's Series for the nonlinear function of the generalized coordinates and their derivatives yields the relation given below f(qi'qi'qi) f(qi 'qi 'qi ) + O 0 o. " af(qi 'qi 'qi ) q. aq i 1 - af(q.,q. ),q. af(qi 'qi 'qi ) 0 0 o o o o 6q + 6q + (6) i 4.1 aq. Neglecting higher order terms, we obtain the linearized equation of motion as - af(qi oi oi ) " " f(qi 'cli 'qi ) f(qi 'qi 'cli ) + aq. O q1. af(cli,qi,qi ) af(qi,qi i ) 6q + 6q = 0 (7) aqi aqi Linearized Equation of Motion With the known values of k n and j n' the expression for an in the linearized aeroforce is defined. Then, the linearized equations of motion of the system are expressed in the matrix form as [M*] {6qi} + [*]{ aii} + [K*] {6q.11 = { 6} where M* = linearized mass coefficient matrix * = linearized damping coefficient matrix

152 134 K* = linearized stiffness coefficient matrix G = linearized forcing function vector The components of the matrices M*, *, K* and the vector G are: Mass matrix of the rotor m cl 2 dr 3 e 2f2 dr 6 11c1 e. dr o. r "H pr.1 k qe. r p(.t.-=) fl - f sinwori T co 3 O 3 f 2 4r2. TZ 3, d uc2 e,4nw,f dr 1 cl c2 r f-7- ' 11112= m21 q I / mh e s + 3 u cosw o' '1 O H 7dr u w 3 dr - f o si nocosp + -e cospcos3)f3 '10. H 3 e 2 dr + f )1( ) sinwo'cospcosafif3 T- qm H 7 ml3= m31 (r+u ) e c e f ur coswo isi nocosp + ( ) coswisi np]f f o 'co H d 1 3 r w 3 e o dr + f 4 si nwlsi nf3cospf if 3 T- o qc H I 3 1 dr + f (si npcosw + OSpOS8Si nw 1) f f cic,, o 1 3 m14= m41 0 rric2)2 dr 3 (5)2 r f2 dr i 2 f2 dr m22 f 11Y-17) + F1: qr) 'I 2 IT- q 2 17 O H H O H

153 M23= m32 =. uc2 wo 3 -, f u (7 Si ni3osp + -t. H 4 (r+u ) 3 e. + 7 f u(.0, H OS cosa)f P 3 Sint3OSp + FT sing),71. f f I2 + :2-- f cospcosat3 dr go H x dr _dr ' m24= = 0 w 2 3 o e H f 4(--2- (sin a + sin pos a) + -7 (cos a + sin2psin2a) ft" (r+u ) 2 c dr cos p) f3 r (r +u) (r+u ) w e c o 6 q I w(--rṟ sincospsin8 ce H sjipcospcos8 m33 = w e + o sinacosacos2p)f2 dr f -2 (si npcosw:3 + cospcosasi nwc; Qm H 2 dr dr + j 5- cos psin a f 3 (1... " H f (si npsi nw,'0 - cospcoswo'cosa) f2 qm H V' 1 (I H 2 T G) f 2, `3 m34= m43 = 0

154 q r H 1.1 w 2 2pcos20) (Te(2 + cos ) + sin 2sOS2p) (r+u ) + 2c (1 + si n2p)} f 42 SI!: w 3 2 o + r+uc) LL 1. 'FT OSP OS6 S np - e cospsi na] f 2 (r+u ) w (r+u ) 3 r 0 c si npcospcoss si npcospsi ns m44 = wo - e IT- Si 2 2 dr si nbcosa(1 - n pnf4 + 3 I J. 7 (cos peos + Si n psi n wolcos B + si n 2 2s wo'si n H - 2 si npcospsi nwo'coswo'coss) f 2 dr f 7 (sin psi n + cos 2 2 dr A)f 4 q. H 2q f (cos2psi n 2 wo' + si n 2 pcos 2 w(')cos 28 + cos 2 wo'si n 4 r- dr + 2si nw 'cosw 'si npcospcoss) f 2 4 where = blade's mass per unit length 1:1 c c = evaluated at nominal value 3q1

155 137 arj c 7.1c2 = evaluated at nominal value 4 2 I H = mass moment of inertia of the hub I G = mass moment of inertia of the generator unit and gear box Damping coefficient matrix of the rotor q. f H c aq (-17-0 Si nbcosp + cospcose) dr al..' 1 w 3 e dr 77- j ua -- cospcosw'si naf Tr- H c 11 au _I. 2. c dr w 3 e 0 2 dr j a cosw'sinbcospf - I pa rf /7-- si npf 1-7- aqi o 1 q.h 3 r, (r+uc) W e 2 dr 31r r.t. _(..) 2 c dr cos8cosp f IT- + T3 j vce) Vee 1 H H + ---q i 4" W e e c Llim1.,. dr f1 Si nwojfi f 1r 17 n Tc: (Tco a H

156 138 3 auc2 w f us2 (-2- s i n$cosp H ql + /7-z cospcosf3 dr 3 c2 e dr + J ut cospcosw'si n$f 1 K dr + f 1.62 IT Tr Si nasi nwoicospfif2 - m., H jr (r+u) 3 c, dr (71 sinwisinbcosp flf2 12 = < 3 r dr + ) 2si nosi nw o co J H w 3 o e o - --, f ut2 cosw' ( sinscosp + cospcoss)flf Tz 4e H 2 (12-43) 3 dr 9. H Fi LId2, + 3 f 7-!-- (ecoswo'f2 - (---7)si nwo)(-7-- H ', a(si npsi nw ' -cospcosw icos8)flf -ro o el f il dr Ill' sinwi) dr 1 o

157 139 6 "co H e u t 1 ( ) 2 (si npcospsi nbsi nw - si nacoseosw (1-Si n 2p) )f 1 f 3 dr K 6 e 2{, Ili/ 7 coswosi n 13+si n pos f3) 'co e H dr + Si npcospcosf3si nwio )f 1 f 3 (r+u ) 6 f -- dr + i pa (O S 2pSi nw ' +si npcos8cosposw 1)f f 7 o o 1 3 ". H 13 = a uc (r+uc) 2 + f us1 ( cos p 'co H - w o cos8 - Si n8) Si npcosp)f3 7-- d r 14= 0 ( f -- racospsi dr 2-13) nf3(si npsi nw ' -co spcosw 'cos6) f if3 - o o '1. H ( r+u ) w e d riml., dr V o V o 1 o f ((--coswi+ 2si nw 1)cospsi ni3)f (-1 si nw ) f H co 3. e 3. ii f ir (si npsi nw'l-cospsi nw Icosa) F ( f1 dr ITI'' si nw if H "co o o 1 1 o 3 wo (r+u ) d el + 3 f (3 Si np - dr cospcosb)f f f si nwo)f3 T- H co co

158 e qm r au w usz Si 118OSp OSPOSB) "dr j H r ci ss dr sinscospf2a 3 H --7Tr 140 dr f un (-p2sinpsinwoifif--- + (r+u c ) f si nbsi nwq'fif2 4. H dr + q dr f 1.62 I Si flf3osf3si npcoswo'fif2 7- H 21 =.1 3 H au c e dr un aq 2 cosw 'si nbcospf w - f A cosw1( o si nocosp + cospcosa)f if2-4 ' H dr 3 s e dr - f i2 Si nw'si nbcospf f 1 2 K H 3 + f 4 c2(si npsi nwl-cospcosw 'o cose)f 1f2-03 H dr Wt e,, r e,, s DJ "-v H f `' cosw' m 2 dr si nw ' )f o o 1 4. H ail w e dr (72. si necosp + 7 cospcosa) = r t'ic2 dr + j usz si nf3cospf - ico H F 1 HL2)si + 3 f ( cosw'f - - Si nw 1) s o 2 o 2 o o Um2 dr

159 141 6 f H wo us1 (sin a+sln pos f2f3 Ks dr 6 q r " s si nscosa(1-si n2p)f2f3 r H auc (r+u,) w usta 1( `" cos 2p - (7- o O - si ne.)si npcosp)f3 dr (r+u,) e 6 + f " s npco f 2f3 gm H dr ( 10-13) 23 = < - f ci[sinpcospcosecos2w0' gee H 2 + si nw 'cosw 1(cos 2pOS2 - si n 2p)] f, 2f dr FT + 3 IH (r+u ) [ d s o. cosw'cospsi + si nw, coscos] f2cosw(') m2 sinw,)f o 3 dr e wo - 31 [v3 (si npsi nw01+cospsi nwocosa)f1-(7 si np H (r+ud) V O s Irri2 cospcos8)f21(7- f2cosw - si nw )f o 3 dr 24 = 0

160 Wt e9 ( r+u ) [ cosw c osp si ns 3 f a H "co n co 142 o dr + -- si nwocospsi nfi] f if = t c r V r r e 2 H m n V a co el (Si npcosw +cospsi nw 'cosa)f 1f3 dr Wt c e2 f 7T t ((r cospsinf si np) fif T- H H W a co 14T f 1)2-7c (si ncosw+cossi nwlcs)f f3 - dr H co wo dr F 1 3 f (scoswo'f2 - H (r+um) sinw0)( cosw 'cospsina w dr + si nwo'cospsi ne) f f F 1 V H ca cosw if -1:1 sinw') (sinpcoswl+cospsinw 'cosf3 s 2 d2 dr G1 +) wo (r 3 f v--(scoswof2- tjd2sinwo)( "' cospsin8 - sinpj H ca r ( r+um) w 3 5 N6( coswolcospsint3 + o H 2r dr + 1 (si npcosw +cospsi nw 'cosi3))f3 sinwo'cospsin$ ( r+um) 3 f H6( cospsi ne - rw si np) H

161 , = 0 41 = 42 = 43 = 0 where { (r+u d ) v o coswicospsina + --sinw cospsinalf/ V o N6 = e3 w (r+u d ) - --(sinpsinw(;+cospsinw(;cosa)f, +(vsinp - cospcosa)f2 V (r+u d ) v w coswh cospsina + sinwlcospsina)g 1 V H6 w (r+ud) o sinpsinwci+ cospsinwcosa)g1+ (i7snp -, v. cospcosa)g2 G = Slip rate

162 ( r+u m) f (N4)(. sin0cosw' + sinpsinacosw' H wo,., 2 dr + T- si nwosi npsi no) r 4 K 3 dr 7 f (N4) if (sinpsinwicosa - cospcoswi)f., 2 Tr- 4 H 3 f (N5)(T (sinpcosocoswt;+sinw('cosp) + " H ( r+u m) cos$cosw' wo 2 dr sinw(')cosa)f4 e 1 f (N5) 7- sinw f2 dr 3 7 o i 4 H ( r+u ) w m. dr + 3 f (H4) (S-' cos$ + slnpcosa + o T- cosp)f 2 4 H (r+um) 2 d r - f (H5)(t sinpsin$ + sinf J (Ni)( (N1)(7 sinficosw' + q4 H (r) um sinpsin$cosw' + s inw'sinpsin$).7( N f4 d r e 1 rn d 0 o o 4 q4 -H + 3- i f (Ni) ir (sinpsinw'cosa-cospcosw) f4 3 t + j. f (H1)(- OS$ + q4 H (r+u m ) wo rn sinpcosa + -4 cosp).-- K dr f4-7 3 ( ) 1. k. 5 (N1)(-FT (sinpcosacoswl+sinw'cosp)+ cosacosw' o o q H 4 + w 0 r sinwocosa).-7- N K f4 dr

163 e 3 r + k f l (Ni) sinw 'sins f dr 2 o IT N 4 q4 H + 7 k f (H1) 3 2. q4 H (r+um) sinpsino (Fsins) + sins) TK Stiffness coefficient matrix of the rotor.: dr f " f usz ---py OSW(13 (Si n 8+sin pos $)fi - " dr 3 ri 2(e ) (cos e. +Si n2 n psi n 8)f2 dr q j - 6 e 2 2 dr pa 2 (7) si npcos 2 pos(3si nw icoswo if, 'ice H - f pa4(r)`cos`psi n2wo'ff ;Ls 9ce H l + 7, 3 -- f 2 '1. H w o 2 dr T-z si nocoso(1-si n 2 p)f1 7 f 2 e Tr (r+uc) 2 dr 3 + sinpcospsinf3f1 9 ' H 2 2 auc 2 dr 3 - f pa cos p(7,--) '1 H dc 11 kll 4 ^ f Ua ( 9*2 H 2 e '-. O S c'p.-1si riposp0sf3+ tillpospsi n6) aci4 l< 1 (r+u,) 1 w, 02-13) 2 2 dr - f a (sinpsinw:)-cospcoswo'cosa) f1 7-- '1. H (I2-I..,) s 22,42,42 dr,. 3 r GJ.F,2 dr - - f 7-- if lovj P3111 pli ir, j zr li Tri. H I" gee H a uc_clr e3 el "Iml.,..- dr - 3 f.t F2(si npcosw'o+cospsi nwo'cos8)(7 f1 si nw op-1 H H c

164 au d s2-3 f v () (-coswo'cospsin8 F1+ cospcoso F2)(7. H '11 e l f m1. dr 7- - H 4 nw1) 3 f F o el dr Si nw10) ariml +3 N sinw ' +3 f D o o, Ho fl H dr el 2 dr k 12 = k 12a + k 12b 3 2 f us22 (7) ( s i rlf3osbsi nwg 1-s i n 2p -Si r1pospsi nbcosw(;)f H, dr f 2 e usl s (coswgsi n 2 B+si n 2 pos 28) H + si npcospcosbsi nw0')flf2 dr 3 r 2 e j wo pa 7-- (si nw (si n 8+si n pos f3) k 12a = < - si npcospcosacosw0')fif, dr 3 2 e - f (r+u c) slco H (si npcospcosf3si nw-cos 2 pcosw0)f dr 2 a uc (r+uc) w I US/ 17OS P cosy - Si n8)si npcoss) dr -- O H 1 2 3Lic 3Uc 2 dr q f aq 1 aq 2 cos p oz. H 3 (12-13) f cospsi ns(si npcosw01+ cospsi nwocoss)f -ice H dr

165 f H (N7) (Tc f - 1. dr nwoj k 12b 3 f. ml dr o aq2 o N si nw' + j H dr c o s w f T - where (r+u i ((si npcosw +cospsi nw 'o cosa) (1-a )f d -, o si nwocospsi nof v + and i coswicospsina FT )c V l 2 p 1 N7 = Qwo (ti s si nwo'cospsi ne,f + cosw 'cospsi nfif ' L V 2 ne + V 3 (sinpsinwi-cospcoswicosf2)fi O a s u auri + sinpf cospcos8)f 2 2 V. aq 2 2 c No = [ (7We) ni evaluated at nominal value O Ho W e) c evaluated at nominal value k 13 = k 14 = 0 k21 = k 12a + k21b

166 148 Qe 3-3 f (sinpcoswl+ cospsinw'coso)f cosw'f H "co, dr sinw jf 1 o e o 2 17 o 2 k2lb = Q, au d v 7-,)(coswo'si n8f/- cosof2)cosp H co "1 s cosw(v2 1:1 m2., dr nw o j ar, + 3 f N sinw' EIL - 3 f F 4 ( -= coswo'f2 o ag2 o H H e sinwil dr - 3 iii.22(s)2(si n28 + cos2bsi n2p)f 22 -k-- dr g H a 3" f uq2cos2p(.)2 dr H 3c12 -r 32u (r+u,) 9 w - dr ust [ cos-p - (e,= cosy sinf3)sinpcosp] slco H 3'12 3 (11-13) + I a `cos2w,;(si n2p-cos2pcos23)f '22 dr sic= H k 22 = 1 + cl2 (I143) 2 sinpcospsinw'coswicos' 2 dr c= H ` o 0 o 2 K H EI f2 dr 3 " 7 2 lr 1.1 (N7) (.-F f2coswo - m2 sinwl) dr o 7-- e + 3 f NO e sinwtof2 f 2 dr IT f 31.1m2 dr N si nwo - ati f No m` coswof H dr

167 149 k 23 = k 24 = 0 ae (r+urri) 3-3 f 77 (sinpcosw1+ cospsinw 'cosa)f coswicospsins 2 H "ca wo l dr +i snw'cospsins + npcosw ' + cospsi nw Icosan f f f - s7e 3 (si V= H (r+um) npcoswl+cospsi nw 'cosa)g2( cospsi na wo. dr - np)fif3 k 31= au w si d - 3 f [-v- coswosi nsfi+ cosaf2)cosp+f.4+ ](o si nw 'cospsi ns H co '11 (_ar+u cosw'cospsins + --(sinpcoswl+ cospsinwlcosa))f 3 dr au 3 [v_( ad )(-coswo'sinsgi+ cossg2)cosp H co ql (r+urn) + G4] OSpSine - 0 sinp)f3 r au au m dr No aq o 3 K H 1 H q 1-3 f N cosw'cospsinaf 77-3 f Ho --/cm ospcoss r + 3 f H Ho el (si npcoswl+cospsi nw lcoss)f if dr _F.

168 150 ( r+um) 3 J (N7) ( H coswicospsine, + wo si nw cospsino)f dr el + 3 J (N7) le- (si npcoswol+cospsi nwolcose)f H dr /T k32 =4 ( r+um) + 3 f (H7)( (--- cospsino - sinp)f3 dr --- H - 3 f N H No au.02 (1 ( r+um) s OSW0IOSpSi nef3 + 3 f No(-17 v f2 - f2)sinwo'cospsin8f3 H dr dr w - 3 f No T9-9- coswocospsi n8f 2f3 H dr e 1..,, dr 3 f N - (si npsi nwo-cospcoswo'cosa)f2f3 - H 1/4 - aum s 3 f dr Ho ag cospcosaf3 T.+ 3 f H sinpf2f3 dr H H k 33 = k 34 = 0 k 41 = k 42 = k 43 = 0

169 151 - H r+u ) (N2)(7 Si si nacosw'+ m o si npsi nacosw + si nw Isi npsi na)f 4 2 dr 3 2- H (N2) el (si npsinw 'cosa-cospcoswl)f 2 dr o 4 3 f H um) (N3)(fi-(sinpcosacosw01+sinwo'cosp)+ (r+-- cosacosw' wo 2 dr 3 + sinwo'cosa)f4 7 f H 2 dr (N3) FT- sinwlsinaf o H (H3q- sinpsina+ ( r+um) si na)f4 2 dr + 3 f (H21- cosa + H (r+urn) sinpcosa w 2 + OSp)f4 dr + in f (N1)1- sinacoswo'+ '4 H ( r+u m ) sinpsinacoswo + sinw' si npsi,rn dr f 4 el rn dr f (Ni) ( npsi nwocosa-cospcoswo)-- f4r 3., jq4 H 3 7 jq 4 j (H1)( cosa+ H (r+u w rn m) sinpcosa+ o cosp) f4 I.< 4 3 k q f (N1)(7k (sinpcosacosw'+sinw'cosp) H (r+uni) cosacosw' + el rn o dr sinwocosa- sinw'sina), f44 17 ( r+um) rn dr + k (H1)(- sinpsina+ sina)-f- 2 f4 (14 H

170 152 Forcing function vector 3 1.)3 H 2 f ( e 2 2 usl (si nacosacosw4(1-si n p) -si npcospsi nisi nw 'o )f dr ) e f us2 4 ' H w o (cosw 1(si n 2 a+si n 2 pos 20) + si npcospcosasi nwo) fi dr 3 ( r+u ) 2 e c (cos 2 psi nw '+si npcospcosacosw1)f o 1 H - f uil sice dr G 01 au ( r+u ) c + pa 2 c Kaqi ( e H 4. 2 OS p - w o S i ripospos$ dr + si npcospsi na) - ( 9 + f 2 s' "cospsi na(si npsi nw 1-cospcosw 'cosa)f dr cle,, 2 o 1 e jml, dr + 3 f N f - si nw ) o 1 o H wo 2 e s --, f pa 7 s (Tr( sin 2 a + sin 2 pos 2 dr $)+ 7 r-si nacos8cos2p) f 2 1. H 3 -c-i co I 1412 (r+uc si i npcosaf 2 dr G 02 au ( r+u 1 w+ 3 2 c c' 2 -Si aq 'leoh 2 --, f usl --(--OS p - S rtpospos(3+ Si ripospsi nf3) 3-13) + f S1 (Si rip OSp0Sf3OS2W 0 qco H 2 dr + si nwo'coswol(cos2pos 2 a - si n p)) f r- s EI,2 dr f2 Ico H H - -, f qs + 3 f N0(7 f2cosw(;- m2 si nw(;) 1(4'

171 'o 153 (r+u m ) 3 N { coswicospsina + -2sinwicospsina o H G 03 e + l (si npcosw' + cospsinw'coso)}f o o 3 dr (r+u ) w m o dr + 3 I H ( cospsino sinp)f3 IT- - nntc o w 'ci H where T G = generator torque at nominal speed 0 n G = gear box ratio G 04 0 Yaw moment due to tower shadow 3 (r+u m ) 17 I (N0){4+ sinp)sinacoswo ' + H My e dr - 7-4cospcosw' - sinpsinwcosa)1(2sin f 2 4 (r+u ) w 3, m. - dr f th,){ F OS0+ SlnpOS$ + os p 1(2 sint) A f4 - " v where M is the yaw moment in the tower shadow sector normalized by dynamic pressure. Note: the computer codes in this dissertation will calculate QO instead of My, where QO = - M 3 y / {-27 (2514)}.

172 154 Summary of symbols used in this section u c u d u m = blade's radial displacement at center of mass = blade's radial displacement at mid-chord = blade's radial displacement at 1/4 blade chord au ; n - at au ; - evaluated at nominal value where aqi n = c,d,n i = 1,2,3,4 e = distance from mass center to shear center of blade cross section el = distance from 1/4 blade chord to shear center of blade cross section e2 = distance from 3/4 blade chord to shear center of blade cross section e3 = distance from mid blade chord to shear center of blade cross section = blade's mass per unit length m c -o ) n evaluated at nominal values Ho, = (:14 We r c Lv ) evaluated at nominal values N1 = (sinpsinw()-cospcosw(1coso)f1 - cospsin$f2 N2 = cosw'sinef 1 -cos8f 2 N3 = (cospsinw+sinpcoswcos3)f1 +sinpsin0f2

173 155 (r+u 1 w di.. o.,.. (1 sinbcoswi + sinpsinbcosw' sinwosinpsinB)Fi V o V o 02 O O N4 = e 3 + (cospcosw1-sinpsinw 'cosb)fi V - ( ( r+ud) wo toss + sinpcosb + cosp) F2 V 02 ( r+u d) w o ( (sinpcosbcosw+ ' sinw 'cosp)+ cosscosw1+ sinw'cos0 V V V N5 = 3, + V3 O + (-- V sinpsin8 + ( r+ud) V si n8) F2 N6 = (r+ud) w, ( cosw icospsi nr3 + yo sinw no) Fi o ocospsi co e wo ( r+u ) 3 d - sinpsinw '+cospsinw 'cosb)fi + (-sinp cospcosB)F2 o V. "00 '0. (si npcosw '0+ cospsi nw scos8) (1 -a )f' - ( r+u d) si nwo'cospsi nsfi au d + a cosw'cospsinb)f Vce aq2 1 N7 = awo + ( s Si nw 'cospsi nef + coswo'cospsi n0f2 V O + ste 3 V O (sinpsinw -cospcoswicos8)f ')F, 2 + (4 sinpf au cospcoso)f d co 2

174 156 The expressions for H1, H2, H3, H4, H5, H6, and H7 are the same as N1, N2, N3, N4, N5, N6, and N7, respectively, except F1 and F2 in Ni terms are replaced by G1 and G2 in Hi terms.

175 157 APPENDIX V YAW STIFFNEST OEFFIIENT V.1 Yaw Stiffness oefficient The expression for the yaw stiffness coefficient can be expressed into three terms according to the sine of the coning angle. They are k 44 = k44 + k 44 sinp + k 44 sin 2 p ( 1) where 3,2 dr 3 7 I (N2)(LL1)r - fj cospsinw1f (LL2)f2 dr ' H H w 3.2, "y I (H2)(T cos + " H 2 2 dr -cosp)f (r+um) dr I cospsinwo G sin6f 1 4 k = 44 o rn (N8)OSp(LL1)w- K dr f4 w,rn dr - 4.j f (H8)cosp(icosa + FTo cosp)-r- f4 c14 H 3 rn k I (N8)OSp(LL2)-K 7 (14 H dr f4 3 (r+um), rn dr k f (H8)cosp sins -- K 4 ` c14 H

176 21 ti 21 I Hli tb z 7) J., --,z? {gul.s 9 - imsoo5soo mt. Al f 4s 1 a b 17 b, T, o H tb, IT j. Nu w 21 ID + ( 11) I Jlimi4s!- -E '4.1 ( n+.4) 21 4 ou,s(8h) - I?msoogs00(8N)} -E H o tb T P Na /i ul.s to - (z-n)ij}(3,mul.s n+.4) H, u vb t N J. dsoo LI{EluLs(8H) - imsoof1soo(8n)} )1 -AP 'a t -7-{(dS00) )19 + j}?mui.s (111)1 UP NU vt4 tb 21, vb, z b 17j. _LI dsoo{9soo it, (8H) +.( 11)(8N)} r N.1 (n+j) 21 H Hb 21 -,,-til 173j guts w b (8H) I -,. - --=-1-il,,J(Z11)(8N) I ( n4,4) ` -r " el T l- 17,1?Mill.SdS00-11{9411.SID -?msoogs0d'j} f ap Z 'a' 21 E H21 7 Hli 7 4- tzl5soo w b I (ZH) -`, + j-in, i /7.4( 11)(ZN) ( n+,,t) b c ''''' c' 851

177 159 and N2 = cosw'sin$f 1 - cos$f 2 N8 = cosw'o cos$f 1 + sin$f 2 H2 = cosw'o sin0g 1 - cos$g 2 H8 = cosw'o cos$g 1 + sin$g 2 LL1 = T sinacosw3 - ff 1 coswcosp (r+u ) w el LL2 = t- cospsinwo + --T -T cosacosw(1) + esinw(1)cos$ - esinvq)sin$ (r+u ) w e1 m LL3 = sin0coswl + -P sinosinw' + o o o V.2 In-Plane Force and Out-of-Plane Force The in-plane force is referred to the force tangential to the rotor plane and the out-of-plane force is referred to the force normal to the rotor plane. The yaw stiffness coefficient developed in this dissertation is based on the forces and moments in the airfoil's coordinates. However, the components of the force in the airfoil's coordinates can be related to the components of the force in the rotor's coordinates by = G A n x + G B 7+1 F D = F D -ri z where G A = Ncosw'sin$ + Hcos$ = -N(cospsinw' + sinpcosw'cos$) + Hsinpsin$

178 160 F D = N(-sinpsinw' + cospcosw'cosa) - Hcospsin8 Here N and H are the forces expressed in the airfoil's coordinates: normal to and tangential to the chord line. GD and FD are the in-plane force and the out-of-plane force, respectively. nx, nx, and nz are the unit vectors in the coordinate system defined in Appendix I. The expression for the yaw stiffness coefficient can be rewritten in terms of the in-plane force and the out-of-plane force using Eqs. (2) and (3). Let us consider a simple case: a rotor with no conning angle, no variation of the axial induction factor with yaw, no offset distance from the shear center (e1=0), and no explicit contribution from the flapwise deflection. The expression for the yaw stiffness coefficient becomes k44 = dr,3 r r c2 dr G f 2 T 17'4 2 H (4) where GT = {(cosw'osinafi - cosaf2)coswlosina +(coswslosinagi - cosag2)cosal {(sinw'f 1 )sinw'} F A = {(sinwji)coswcosa - (sinw(1302)sina) Here GT is a component of the in-plane force coefficient and FA is a component of the out-of-plane force coefficient.

179 161 APPENDIX VI OMPUTE ODES Two FOTAN computer programs, POP code and AEO code, are developed to handle the numerical values of the coefficients of the equations of motion. The AEO code uses a simplified lift and drag curve for a four-degree-of-freedom system while the POP code uses an actual lift curve and only emphasizes on the yaw equation. Both codes will calculate the axial induction factor along the blade at a particular tip speed ratio. At the same time, they also calculate the integral terms for the variations of the axial induction factor with yaw and yaw rate. Finally, the codes calculate the coefficients in the equations of motion ( mass, damping, stiffness coefficients, and forcing function ). VI.1 POP ode The POP code in this dissertation is a modified version of the original POP code developed by Wilson [21]. The original POP code uses the modified strip theory solving for the axial induction factor, thrust coefficient, and power coefficient. Besides these features from the original code, the new POP code also has the following features: 1) it is written in the FOTAN V computer program language. 2) it uses the Glauert relationship [4] instead of the momentum theory to calculate the axial induction factor when its value exceeds the critical value. 3) it calculates the variations of the axial induction factor with yaw and yaw rate. 4) it uses the iteration method to calculate the static flapwise

180 162 deflection. 5) it calculates the coefficients of the equation of motion in yaw. Input Data The input data for the program consists of two parts. The first part of the data is stored in a computer data file (Tape 60). The second part of the data is inputed to the program through the terminal. The parameters to be inputed in a data file are HB D radius of blade, ft hub radius, ft incremental percentage (percent of radius for integration incremental) THETP B H HH GO pitch angle, degrees number of blades altitude of hub above sea level, ft altitude of hub above ground level, ft tip loss model controller 0 Prandtl 1 Goldstein 2 no tip loss model 3 Mostab tip loss model HL hub loss model controller 0 none 1 Prandtl APP angular interference model code 0 angular interference factor calculated 1 angular interference factor set equal to 0 XETA velocity power law exponent

181 163 TH ALO AMOD blade maximum thickness/chord angle of attack for zero lift, degrees axial interference model code 0 standard 1 Wilson NF NFS number of input stations for blade geometry number of data points on lift and drag curve NPOF NAA profile or profile subroutine 4415 NAA NAA data inserted in tabular form 9999 NAA (I) I(I) THIT(I) AAT(I) LT(I) DT(I) percent radius for stations chord for stations, ft twist angle for stations, degrees angle of attack, degrees coefficient of lift data coefficient of drag data The parameters to be inputed through the terminal are pitch angle (degrees), tip speed ratio: minimum, maximum, and increment, rotor speed (rad/sec), conning angle (degrees), blade shear center position given as the ratio of the distance from the blade leading edge to the shear center and blade chord (esc), position of center of mass for the blade cross section given as the ratio of the blade leading edge to center of mass and blade chord (xcg), location of the yaw axis (YL, ft), modulus of elasticity (AE, psi), and modulus of shear (AG, psi).

182 164 Output The output for the program is on a tape file entitled TAPE 61. On this file are written both the program operating conditions and program output. P A L D The following are the output quantities: local distance on the blade, r/ axial induction factor lift coefficient drag coefficient PHI summation of angle of attack and twist angle, a + ALPHA F E NO T P QS angle of attack, degrees tip loss factor eynolds number thrust coefficient for a rigid rotor power coefficient for a rigid rotor non-dimensional static tip deflection (final value after the iterations) The coefficients of the equation of motion in yaw are ZMDD ZDD ZKDD QO mass coefficient damping coefficient stiffness coefficient yaw moment coefficient due to the tower shadow per unit shadow width SKDEL coefficient accounting for the variation of the axial induction factor with yaw, k SJDEL coefficient accounting for the variation of the axial induction factor with yaw, jy

183 165 SKDEL coefficient accounting for the variation of the axial induction factor with yaw rate, k. SJDEL coefficient accounting for the variation of the axial induction factor with yaw rate, j,;, PAA power coefficient for a rotor with flexible blades (accounting for the flapwise deflection effect) Given the values of the shadow width and the velocity deficit, the yaw forcing function can be obtained from the relation G 04 2{Q01x (SF) 2 OOlx } 2 sing 2 where X = width of the shadow sector (degrees) number of blades SF = correction factor due to non-dimensionalized value at different tip speed values f =1-(%velocity deficit) /100} x 1 = tip speed ratio considered x 2 tip speed ratio of the shadow sector : x2 = (SF)x1 VI.2 AEO ode The AEO code uses a simplified lift and drag curves to calculate the aerodynamic loads. The lift curve is approximated and can be described in a simple yet fairly accurate form by six parameters.. The curve consists of four straight line segments as follows: L 27msin(a+a o ) a < a L max L = Lmax a < a < ab max L a B < a < a stall L fl at

184 166 sin(i - a) L Lflat sin(2 - a ) -stall' The six parameters are a > a stall - lift curve slope divided by 27 0 a L - zero lift angle of attack - maximum lift coefficient ab max - angle at which drops to, L "flat L - an approximate to the average L on the far side of the L flat curve, this can be adjusted up and down depending upon L the characteristics of the airfoil astall - angle at which L begin to decrease The drag coefficient curve is also in multiple sections. Below a, the drag is given by the following: L max D = D(1 + a ( a )n) a o L max where a, n. and are constants determined by the airfoil character- D 0 istics. If a > a the drag coefficient can be represented by a L max single curve fit or a series of curve fits. The axial induction factor "a" is calculated by equating momentum flux to blade force. There are six possible intersections of blade force and momentum relations due to two regions on momentum relations and three regions on blade force. Two-regions on momentum relations are the region of parabolic curve when and the "a" "acritical" straight line when "a" ". Three regions on blade force are > "acritical the linear slope curve where the angle of attack is less than the angle at the maximum lift force, the flat part of lift curve (, an d Lmax L flat)' and the lift curve in the stall region. Once the particular region is

185 167 identified, the solution is a straightforward procedure of finding where the momentum and blade element curves intersect. These intersections of blade force and momentum relations are shown in Figure V.1. A subroutine and two functions are developed to handle the inner integral term of the double integration. The inner integral terms are terms involving the derivative of flapwise deflection (radial displacement and its derivative). The composite Simpson's rule method is used for the numerical integrations in the code. Input Data The input data for the program consists of the physical characteristics of the wind turbine itself. They consist of physical airfoil data and operation variables. The physical airfoil data and operation variables are B B EM D XMIN XMAX DBX D ZEO L FLAT chord to radius ratio at blade root (Bc/) number of blades slope of linear portion of lift curve/2w dr/ tip speed ratio to start program last tip speed ratio - used to end the program the increment of tip speed ratio minimum lift coefficient lift coefficient on the horizontal portion of the lift curve L MAX ALPHA BEAK maximum lift coefficient angle of attack where the lift curve changes values, from the maximum value to L FLAT, degrees

186 168 case 1 case 2 case 2 T T a a case 3 case 4 T T a a 1. case a a Figure V.1 egions of operation for momentum calculations.

187 169 ALO AST SI PITH BETA OOT DBETA T angle of attack at zero lift, degrees stall angle of attack, degrees coning angle, degrees prepitch angle, degrees pretwist angle at blade root, degrees (aroot - $tip); twist angle change, degrees local radius at twist angle change from linear to constant twist DND (c/ at chord change - c/ at tip), chord change ratio local radius at chord change from linear taper to constant chord H ES hub radius blade shear center position given as the ratio of the distance from the blade leading edge to the shear center and blade chord (e s /c) XG position of blade cross-section's center of mass given as the ratio of distance from blade leading edge to center of mass and blade chord (xcg/c) E G OMEGA HO YL modulus of elasticity, psi modulus of shear, psi rotor speed, rad/sec air density, slug/ft3 distance from the nacelle yaw axis to the center of the rotor, ft M blade tip radius, ft number of integration steps in subroutine

188 170 Output The output for the program is on a tape file entitled TAPE 1. On this file are written both the program operating conditions and program output. QS P T Mnn The following are the output quantities: nondimensional static tip deflection power coefficient thrust coefficient rotor mass coefficient where nn is the indication of which variables it represents nn Knn HP HF SKDEL rotor damping coefficient rotor stiffness coefficient rotor forcing function of pitch equation rotor forcing function of flap equation coefficient accounting for the variation of the axial induction factor with yaw, ky SJDEL coefficient accounting for the variation of the axial induction factor with yaw, ji SKDEL coefficient accounting for the variation of the axial induction factor with yaw rate, k. SJDEL coefficient accounting for the variation of the axial induction factor with yaw rate, j71, For nn parameters P F generalized coordinate in pitch generalized coordinate in flap 0 generalized coordinate in speed generalized coordinate in yaw

189 171 If the code is not suppressed, additional output quantities are printed on the output list. P A PHI BETA ALPHA L D B PB TB They are as follows: local distance on the blade, r/ axial induction factor summation of angle of attack and pretwist angle pretwist angle angle of attack lift coefficient drag coefficient local chord to radius ratio, Bc/ power coefficient thrust coefficient For the tower shadow part in yaw equations, the additional quantities on the output list are: SMnn mass coefficient in shadow region per unit shadow width/(b/2w) Gnn damping coefficient in shadow region per unit shadow width/(b/2w) 5Knn stiffness coefficient in shadow region per unit shadow width/(b/2w) QO forcing function per unit shadow width generated from the shadow /(B /2n) The quantities from the tower shadow effect will be calculated when the magnitude of the velocity deficit is given. For example, if the velocity deficit value is 50%, the forcing function at tip speed ratio of 2 due to tower shadow is given as

190 172 27r ["Ix=2 (SF) 2 O 01 x.4 ]2 sin 2 where j B x = width of the shadow segment (degree) = number of blades = tip speed ratio SF = correction factor due to nondimensionalized value at different tip speed value [= 1 - (% velocity deficit)/100] For the gravity effect, the gravity forces on the pitch equation for a single blade are listed as the components of sine and cosine of the azimuth angle. GNOS osine component of the forcing function due to gravity GNSIN GPUS Sine component of the forcing function due to gravity osine component of the k11 of a single blade due to gravity GPSIN Sine componet of the k11 of the single blade due to gravity GFOS osine component of the k12 of a single blade due to gravity GFSIN Sine component of the k12 of a single blade due to gravity GOOS osine component of the k13 of single blade due to gravity GOSIN Sine component of the k13 of a single blade due to gravity

191 173 The properties of the blade and shear center position are also listed in the output E El E2 E3 el e 2 e 3 AE Modulus of elasticity, psi AG Shear modulus of rigidity, psi The integration step sizes are shown as N Number of integration step sizes used in the main program Number of integration step sizes used in subroutine (double integral) Note The code does not calculate some of the terms in the expression of the coefficient of rotor equations of motion. These terms have to be calculated by hand then added to the results of the computer code. These terms are G in 33 G T G in G 03 o q 1 22 q (LT H -", GTIG) In 1 m33 33

192 174 VI.3 Sample ases The Grumman WS33 and the Enertech 1500 are used as test cases. The physical characteristics of both rotors are needed as inputs. Some simplifications of these data must be made to use in the computer codes. Some parameters for the simplification schemes are presented in this chapter. AEO ode With the lift curve defined in Appendix VI.2, the lift curve's parameters of the Grumman WS33 and the Enertech 1500 are given as for the Grumman WS L max L flat al o ab astall for the Enertech L = max L Lfl at al o ab astall

193 175 The drag curves of the Grumman WS33 and the Enertech 1500 can be approximated in a series of curve fits. These curve fits are shown as follows: for the Grumman WS33 D = D ( (a - ad) 2 ) a < al 0 D = D + 0.6(a - al) al<a<a2 0 D = (a - a2) a2<a<a3 D = 5 sin 2 a 4/ 7r + sina a3<a<a4 D = 4.5 sin 2 a a4<a 4/7r + sina where D = a.'s are given in radians; By converting radians into degrees, the values of ails are given as follows: ad = 2, al = 10, a2 = 14, a3 = 20, and a4 = 28. for the Enertech 1500 D = D ( a 2 ) ciz12 0 D = 3.36 D + (tana - tan12 ) 12 <a<15 D = (tana) 2.15 Lflat 15 <a<a2 D = tana Lflat a2 <a"/4 D = D2 sin 2 a 1 + sina where D = D2 = a2 = arctan {(41 Lflat ).87}

194 176 For the AEO code, the combination of the effective radius and Prandtl method is used for the calculation of tip loss factor. The effective radius is given by and eff B2 /3x 1/2 B 2/3 x eff B 2/3 x 1/2 ' 8 2/3 x for x < 3 which was obtained from an empirical relation which expressed the maximum power coefficient of wind turbines. The tip loss factor is expressed as 2 arc cos (ef) where 77 (1 - r/)/(1 - eff/) ef = {cos(0t--)1 POP ode POP code uses the actual lift and drag curves of the blade section to calculate the aerodynamic loads. The airfoil sectional data of the NAA in a reverse position are needed for analyzing the Grumman WS33 in a reverse position. Unfortunately, these data are not available. But by studying the behavior and trend of the aerodynamic characteristics of airfoil sections through degree angle of attack from references 9, 17, and 18, the lift and drag curves of the reverse NAA can be expressed as

195 177 L = (a - ao) a < a1 L = a a a a a 0 L = a a a a a 0 al< a < a2 a2< a < a3 n2a L = si/7 + sina a > a3 for drag curve D = D (1 + 20a 2 ) 0 D = (a - a4) a < a4 a4 a < a5 D = (a - a5) D = 2 i a /11. + sina a5< a < a6 a > a6 Here a.'s are angles of attack in radians measured from the reverse side of the airfoil (i.e., negative angle of attack of the conventional airfoil). By converting radians into degrees, the values of i's are as follows: al = 4, a2 = 17.7, a3 = 20, a4 = 10, a5= 14, and a6 = 20. VI.4 omputer Listings The listings of the POP code, the POP code's output, the AEO code, and the AEO code's output are given as follows:

196 FOGAN POPIINFOTOUIPUT,TAL64.14.E/ APP -- ANGULA INTEHFEEN;E 103K3UT 61 VESION MAH ANGULA INTEFEENE FATO ALULATED 64 MAIN PEGAN ANGULA INTEFEENE FATO SET EQUAL TO S. POP ALULATES THE THEOETIAL PEFOMANE PAAMETES Of A 3 POPELLE TAPE WIND tuine. IT UTILIZES A SIMPSONtS..ULE 6...HIIIPE3ENT ADIUS FO STATIONS II 3...HITHOD FO STATIONS - FT 69 METHOD / THEE PASS TEHNIQUE OF NUMEIAL INTEGATION. 9 DIMENSION 12511, THEMES., AMMO, LI $ E TUITTTWIST ANGLE FO STATIONS 0.3EES FE DIMENSION STA X ,ALPN11001,F OMMON ild x.thei.ahod.h.s/40.0mega.ho.vis.hl.p1x.w H-MAX THIKNESS/HO ATIO 73 ENPOF,APP.A T TEST.WEII.HH.4LO,A i 74 OMMON XGASO1E.AG...ALOANGLE OF ATTAK FO 2E43 LIFT DEGEES TS OMMON.IONETIIINF ELEM...OE,. OF LIFT DATA 77 INPUT PAANETESI AADIUS OF PIA DE - FT IT OEF. OF DAG DATA MU- -HUB ADIUS FT 19 ANGLE OF ATTAK DE;EES e--IML4EmEmIAL PEENTAGE 22 EAD INPUT DATA THEIPITH ANGLE - DEGEES 24 EA016439)ODNET.O.H.HP 25 E NL0APP.XETA.TH.6L O -- HUNKE 26 EA INFOFS.NPOF..4.--WIND VELOITY - MPH 27 EAD III.1111,THETTITI.1.10F1 28 IF INPOF.NE GO TO I A 92..,.k ETA- -VELOITY POWE LAW EXPONENT 29 E I 114L NEST 34 I ONTINUE A 95...OMEGA -- ANGULA VELOITY OF POP PN 31 PI A PINT 53 3 INTEFEENE MODEL 100E 13 EAD.OMETP 0 -- STANDAD 3% PINT WILSON IT EAD 11XMINOLMAX.DX 36 PINT 55...M -- ALTITUDE OF NU ABOVE SEA LEVEL Ft TT EAD'ONEGA 38 PINT 54 EAD.41...NH -- ALTITUDE OF HUB MOVE GOUND LEVEL FT PINT TI 3...DI -- ONING ANGLE DEGEES 44 EAOIESIPX;GOL 42 PINT 59 EAO.,AE,AG...NF-- NUMBE OF INPUTED STATIONS GEOMETY 4$ 44...NPOF--NAA PPOFILE O POFILE SUBOUTINE 4$ IL.YL/ 4415 NAGA XXMIN NAA V..ONEGOVX A DATA INSETED IN FAULA FOM POFILE UVEFIT TO E USED TN NAAXX 49 PINTINPUT AND TITLES FO OUTPUT A 99 SO A ALL TITLES I41140ETI.NF.SOLDI A 10/...GO - -TIP LOSS MODEL ONTOLLE I AMOIL E4P1-.797N210u00.1 A GOLDSTEIN SI INITIALIZATION AND ONSTANT PAAMETE ALULATIONS A NO TIP LOSS MODEL S4 3 MOSTA TIP MODEL 65 A w S6 A 105 I=Il...ML - -HUT LOSS MODEL ONTOLLE 57 0 NONE $ d TUOS.TITS.0. I PALOIL S9 DI.D t A MOSTA0 TIP LOSS MODEL PAAMETE I AI. 0.0 A 113 TO. 8.0 A 144

197 2 3 A 0800 AMYX = 0.0 A U05.305=0. OX = 0.0 A=.38 IX = 0.0 FXX01 = 0.0 FIXP fy = 0.0 PY = 0.0 %MU = AMP 2.0 A. 1.0 AP = 0.0 ONTOL = 3.0 FX6 = O. F.O. SI = 51.1,180. EF..OSISII INEIF THEIPP1/180. AL0=AL0PI/100. ALP4A=-1110 HO EXPI / VIS = , NN.I-081/ X = L8 = X DP = 1X-IBI0SISII DO O = OSISII NO 11405ISIO L... THEE PASS NUMEIAL INIETATION FOM TIP TO NUB AI = 1. IF IGO.E0.2.1 AT 2. U = 1. IF LFA 0.0 IF GO TO 2 ALL SEAHILA.ItINETIgNigGgfNEf.SM00E1 LOLL AL IL.:1THET.FXXPIOTXPI.X5XX.XITX.OXoTXtEePHIL.00.X. $00,A.AP.XL.AK.ALPNA.F.LFA.AT.AATeall.00I.NFS.501.DON.SMODE.U0S. $0051. STAIII.U AXIIII.A ALPNIII.ALPNA FOIGIII=F A. 0.0 AT 0.0 DO N TIP.4. IF IIL GO TO 3 AMP. ASI IF 1AST0P.GE.2.1 GO TO 9 D NO1 IF GO TO 5 IF IONtOL GO TO S IIP = L -0P IF ITIPAT.EF1 GO TO 4 IF 1ONTOL GO /0 5 A 117 A A A A A A A A A A A A A 131 A $ S5 A 157 A A 1E0 A 161 A 162 A 163 A 164 A 165 A 166 A 16? A r UN = 141 -LF1 169 LIO (PEI-UPI/ANL-1M 170 LF.5LTO 171 :1.11TOL GO TO S 173 LF /16.OSIS , X L L -O2 177 IF 1OUTOL LF I. 178 IF 1ONtOL.E LF ILF0.1.1 / AK ALL SEANILA4.1*INETIgNF E1 ALL AL IL.INET.FXXP2.FTXPIL.X11XXPI.XMIXPLOI0P1.TXPleE. 182 S 21.16,2L.o,cr.cr.6.1p0m...111,64.F.cir a.T $ , ,00511./.1 STAIKI.L, AXIIKI.A ALPHIKI=ALPNA FOIGIK1.F L L -D12 A 185 If 1ONIOL LF IF IONI LF 1.0 A 188 AK 0.8 A 199 ALL SEAHIL.FeIONETIINF..INET.SMOOEI ALL AL IL..INETOTXT.FTI.XNXXP.XNIXPDXP.TXP.E.PHIeLID. A X.V.A0AP.X1..AK.ALPHA.F.LF.ATIoAAT.LI.0010NFS.S010.14SMOOE. $ W KK=1011 SIATIKI.1f AXIIKK1.A 111.PNIKKIALPNA FNIGIKKI.F FX6 FX6.0161FXXP1.4.FXXP2IFX111 11TX 016.1QX14.O1PI.OXPI Or 01,10110 TO I TX.4..TXPI.TXP/ Plf PI.OMEGAOIX 1U0S.ITUIS U0S.4.UT/ XNXT XMXVI0T61011XX.4.XNXXP11111XXPO ONTO X4,100161XMVX.4.X111XPIOXI1xPI IF IONIOL.E0.2.1 ONTOL. 0.0 IF 1ONTOL.E GO TO 6 IF III...f/PI.E GO TO 6 IF IONIOL O. EF -TIP 1r IONI01.10./.1 ONTOL = 2. SO TO 7 00 a OO ONTINUE OX = OXP IX TOP 011X1 = (HOOP ONYX XMIAP FAXPIIXT U0S.U0S2 OUS T/ TY/1.5NOV21.1X21 PT PV/1.5PNU113P1X2) IP PY/ A 194 A 195 A A 190 A 199 A 200, A 201 A 202 A 203 A 204 A 205 A 206 A 207 A 200 A 209 A 210 A 211 A 212 A 213

198 PHI = phi.188./p1 A FOMAI 1/61H TOQUE VAIATION 61/E TO SHEA SEOND DEIVATIVE A A 133 ALPHA = ALPHI2180./PI A FOIAITYELEIYPI INPUT SEI10111/114PUT PIT:11 ANGLE.) P 2 UIX.OSIS111 A FOMAT). TIP SPEED ATIOS IINI.IMAXI.IINTEMENTI./ A 2IA 55 FOMATI.OIU AD PE SE') PINT OUTPUT A FOMAIIIKONING ANGLE') S? FOMATIWOIXOPEA., A?..SAOA.$ Astifil.. P2WL/IX.OSISIII 141..ALPHAY MOE NO WIT P.A.AP.LIO.PHIU.ALPHA.F.E.IT.PT A F INPUT tsc.xcc.rt 1 8 ONTINUE fomaii. INPUT AE,AG 1 9 HP. TP A FONAIII1o IN NOMALIZED STATI TIP OEFLEtION IS t.g13.61 WITE A FOMAT( /.' OS PEVIOUS OS NO OF WIIE PT S 11E62110N OS.INIS/TDIS if fomaii/s7a zdo,oll?..f100',11x000./.2a.616/ WITE OS A A FOMA11/.700 SKDELY.IIMOSJOELY.1140SKDEL SIDEL2/.24. PISTIL P A SYS $ PIESIEL = PI A FOM/11/.1140 PAA 2 '.113.6) OP = 13E14.M.1222/11412PH0124META.PISTEL.IXETAI.I.PIESTEL128/4. A 250 Ol FOMAT( /.71.' TANGENTIAL FOE PE BLADE IN OTO PLANE PAV = PT.DP A 251 WITE PAS A 252 ENO WITE 16108/ cur A 269 WITE PISTIL A SUBOUTINE TITLES I.IONEtleNF.SOLDI WITE / PI2STEL 2 WITE TITLES PINTS GUT INPUT DATA ON A DESIPTIVE 3' ALL ITEA1113S.ALFM.AXI.STA.POS001.1(11 FOM. AND PINTS OESIPIIONS OF SYMBOLS/IITLES F OUTPUT. WITE IS.POS0KL 5 ALL YAIOS.ALPH.AXI.STA.FBIG.MITOLgSKDEL,SJOEL.SKOEL.S.IDEL. DIMENSION I251. I4251. IHETI M00.1D13.2KOD.000,240102KO.ZOW.W100.PAA.LIKI.OII ONNO.D V.A.THETP,A GO.ONEGA.NO.VISIMIL.PI.1.W. WITE MO.I00.21( NPOF.APP TEST.ETI.M WITE1E1.791 SEL.SJOEL.SKDEL.S./40EL 9 WITE KIIE PAA 10 WITE WIIE 161,141 It OMEGA/1 WIII ) V 12 D*DI WITE META 13 MB.HO/OSISII WItE HS ,1 WITE N 15 THETP2THEIP.180./PI 16 MIME ) OMEGA /P1 17 WITE /14 IA WITE 161,211 THEIP IFIX.GT.AMAMI GO TO A WITE SI SI GO 10 IGO 21 SI SI2P1/ ONTINUE 22 L.752 FUMAIS FO INPUT 4NO OUTPUT STATEMENTS 212 ALL SEANIL.II.INETIOF.G.TWET.SMODEI SI SI 274 BANG INET21011./P1.THETP WITE BANG STOP WITE SI WITE 'WIE ( FONAT13F WITE 161, FDMATIIA.F4.3.2X.F6.4.3A.F F4.2.3%.F5.3.2A.F6.3.2X.F6.3.20, 31 WITE A F II FOMAT 1///.214 PEFOMANE SL KOTIP SPEED ATIO.1E8.31 A 305 ALL SOLIDI (.101F.9..P FOMAI 1/ H TOTAL POWE OEFFIIENt WITE ) SOLO FOMAT H AVEAGE POWE OEFFIIENT WITH GADIENT 2.F ALL ATIVI I.I.NF.40.P1sAT) A 310 WITE 161,291 AF SI 34 FOMAT H 701AL MOGI OEFFIIENT.F7.41 A 311 WITE NFOF WITE ) OF 19 FOMAt17F WITE FOMATI3F10.41 WITE 461, I FOMAT FOMAT 1/584 IDFOUE VAIATION OUE to AEODYNAMI SEOND DEIVATIVE A 330 WIT( 161,341 IIII NET1( F1 41 A 331 WITE 461,351 2.F15.41

199 IA /I it WITE D IF IAPP.E GO TO I WITE IF 1AM GO TO 2 WITE rat.it, GO 10 3 WIIE IF GO 10 4 IF GO TO 6 IF GO 10 5 IF IGO.E GO 10 6 IF GO 10 7 GO TO 9 IF 100.(0.2.0) GO TO 6 IF GO TO 7 GO 10 6 WITE GO IL 9 NITc 161,421 GO TO 9 WITE 161,431 GO TO 9 NAM ) IF IME.E GO TO ,451 GO TO 11 WITE ONTINUE WITE WITE WITE ETUN FOMAT FOMAT 157N IHEOETIAL PEFOMANE OF A POPELLE TYPE WIND TUBI!NEI FOMAT 1/// OPEATING ONOITIONSII FOMAI 1//g150.23H W140 VELOITY - FPS.FT.41 FOMAI 1/ H WIND VELOITI GADIENT EXPONENT.rr.as FOMAI HU9 HEIGHT AOVE GOUND LEVEL - FT- OF12.41 FOMAT 1/.15X,401 ALTITUDE OF HUB ABOVE set LEVEL - Ft. of19.41 FOMAI 1/016X.30H ANGULA VELOITY AO/SE.F10.41 FOMAT (/ N TIP SPEED ATIO FOMAI I/.15X04414 PITH ANGLE FOM NOMINAL TWIST - DEGEES.F10. $4) FOMAT ,401 PITH ANGLE (I 0.75 ADIUS DEGEES.F10.41 FOMAI 1/ OHENS ANGLE - DEGEES.F10.41 FOMAT I/H.50.14H BLADE DESIGNS) FOMAT 1// H NUMBE OF BLADES FOMAT 1/.150./9H TIP ADIOS FT.07.4) FOMAT 1/ H HUB ADIUS - FT 0F7.41 FOMAI 1/ H SOLIDTY.F12.51 FOMAT 1/ H ATIVITY FATO,F12.51 FOMAI 1/ H NAA POFILE.141 FOMAI 1/ NUNBE OF STATIONS ALONG SPAN 0141 FOMAT 11/.10X.29M HOD AND TWIST OISTI3UTIONI FOMAT 17/.16X.1 EH PEENT AOIUS.50.8HH0O0.F1) NIST.DEGO FOMAI 1/.200J5.1.11X.F X1F10.51 FOMAI (///15X,301 POGAM OPEATING conorrtonss) FOMAT 1// INEMENTAL PEENTAGE 0E7.41 FOMAI 1// ANGULA INTEFEENE FATO, AP 0.01 FOMAT 1//1, WILSON AXIAL INTEFEENE METN00 USED) az as / ? TO TT Is Al S I S FONA1 I1/ SIANDANO AXIAL INTEFEENE USED) FOMAT 1//.15X.3914 TIP LOSSES MODELED SY POMOTES FOMULA) FOMAT 1/ H IIF LOSSES M00EEE0 DY cotosurms FOMULA) FOMAT 1// H NO TIP LOSS MODEL) FOMAT 1// TIP LOSS MODEL -- NASA VESION) FOMAI 1// H NO HUBLOSS MODEL USED) FOHAT 1//g HUBLOSSES MODELED BY POMOTE) FOMAT FOMAI 11/ PEFOMANE ANALYSIS', FOMA11///40..P..6X0***.50..APTOX.*ET06X000O0.PHI..40. $4ALPHAT.50.*F..$0..E NOVOIX..1*.AX.TP*1 END 0 suricourtne SEAHIL.A.1.1HE1IOW..IHET.SMODEI 2 SEAH DETEMINES SHE HOD AHO SHE TWIST ANGLE AT 3 A GIVEN ADIUS ALONG THE SPAN. IT UTILIZES A LINEA 4 INTEPOLATION 1EHNIQUE. 6 DIMENSION It1251. I1251. INETII251 T OMMON e v.a.the/p.an ,60.04ega,400,visolidpi.x ,W 9 PI/ E/IXOSISIII* IF IV.EO.II/O GO TO 4 11 IF IV.GE.IIII GO TO 2 12 IF II.EQ.NFI GO ONTINUE 14 J I.1 15 PE s IV-IJ111/IIJ-21.IJ PE IIJ-111* Al THEE PEITHEII1J-21.40E *THET GO IINFI 20 THEO EMINE) 21 GO TO THEY THE THEO THEIOI/ E.(L.110I/IP.HB) SHODEXMODIXOL$ ETUN END SUBOUTINE ALL IL..THE101F.ITTF.AMFXFONFVF.W.IF.E.PHI0L,0 $40.V.A.AP.ALgAX.ALPHA.F.EF.AT.AAT.LI.:DT.NFS.SOLOON.SMODEt 11ILIOSOOSI AL DETEMINES the AXIAL AND ANGULA INTEFEENE FATOS AT A GIVEN ADIUS AND DETEMINES FUNTIONS DEPENDENT UPON THESE PAAMETES. DIMENSION AA11251, LTI251, 01(25) OMMON X.THETP.AH GOOMEGAIIMO.VISAL.P INPOF.APP,T102,13,T T7.14.IEST.XETO.HH.ALO.A OMMON AG.ES.AE.AG XL N HO MO J A DEL IA AP PHI ATAN111..A1LOSIS11/111.01P1XLII PHIAA s ABSIPHII IS

200 3Ao*FsiNIP.111**2. SINPNI.SINIPMIAAI *AI.8AD/V8p.0.1 IFISINPHI.E SINPHI =11.-0A0.11*AiV491 %XL LOSIPHIAAI/SINPHI 20 AMN '1 XALO 21 XXL./PL 22 IFIANN.L ANN=0.0 PHI * PHI A= SOMAMMI ALPHA = PHI-IHET-IHEIP ONTINUE ODI = ATAN111.-A1/1X APIII-AtA4111.-Al/ALI IFIAA.E0.0.1 THEN DA1 ODI/4. 0A /19.1.ISOLD.THI/m1/411./x10.201L/X1**21 ELSE DALPHA A2 27 AP VA/IF*SINIPNIIOSIPHII-VAI 80 ALPHA = ALPHA-DALPHA 24 ENDIF ALPHA = ALpHA IF IAPP.E0.1.1 AP 0.0 el 9 P.LAIAOSISIII ALULATION OF SET INAL LIFT ANO OAL OEFFIIENTS 31 DAMPENING OF AXIAL AND ANGULA INTEFEENE FATO a ITEATIONS. 84 IF INPOF GO TO 1 33 OS IF INPPOF.E GO TO IF ,12,10 IF INPOF O TO IF 1410/ 13,12,11 47 IF INPOF O TO 4 36 /I IF 1.k MITE A. 1/1.8E ALL NAA44 IL010310ALPHAI,L.OgAL0141 AP = IAPIOELTA0..S 90 ALL NA444 IL.X.S10ALPNAO.LOADDIALOoN1 01 ONTINUE 91 GO TO GALL NAA00 IALPHAAL UST FO ONVEGENE 93 ALL NAA00 IALPNA AL SO TO 5 44 IF IAPPA0.1.1 GO TO IN 95 3 ALL NA ALPHA,L.0011A18.4.AAI.LIIIOTOFS,S0101 4S IF IAOSIIAPOELTAIYAPI.LE GO TO ALL NAATI IL LPHAOLO.000.ALS.N.AAT.LT.OTINFS$ GO TO 15 $ I 14 IF IAOSIIA-0ETAIYA GO GO i ALL MAAU( ,ALPNA.L ONTINUE :ALL NAU% ,SI,ALPNA AL IF IAW.GE.1.1 GO S ONTINUE SO IF GO TO 6 S/ 16 ONTINUE FO PEATION OF VOTEX ING STATE EPLAE THE FOLLOWING ADS 102 L. cunt 52 P IS :to. LF.LO F GO 10 7 SS IF IJAT.401 GO TO WI1E 161, ALULATION OF TIP AND NU LOSSES S7 17 ONTINUE IF IAT.E F ALULATION OF FUNTIONS OEPENOENI UPON AXIAL AND 116 IF IAI.E GO TO 7 60 ANGULA INTEFEENE FATOS. 117 ALL TIPLOS F01.601,41.PlooL,PHOHI ONSUME M = L.SINIPHII-O*OSIPHII 63 M 121 :I u OOSIPHII.D.SINIPHII 64 SoIII11.-WVOSISIII**20,111.*API.OONEGAI..21 E = HO.W.AVIS 121 x%.. L.SINIPHIO 65 ONST 10.5.HO* LOOSIPHII 66 FAF = LONST*x 121 SIG 10.1/IPPLO 67 FYI. ONST'T 124 IF GO TO O 6$ XmFAF = FAF.IL-1031/OSISIO 125 VO SIG.MITIOSISII.!211/1$141P411"21 69 XmfVF FVF.I*1-1031/OS1STI 126 VA SIGTXXO/IF.SINIPHIITOSIPHI11 70 II s 10.5.HO.O ' AN = F.F04.VO.F OF. II.L.% 128 IF IAN.L AN IF II.V.OSISII 129 A. 12..V8W-SOITAN$1712.1V0,FTF/1 73 DP s D/12..XI 1311 AP VA/111.A.F1711.AIVA/ 74. Gill 131 If 1APP AP SO TO 9 76 LA ILO-LI/0.001 OA = tcoo-cot.o.00t G.1V.IOSISII OP a LA.SINIPHII.OA.OSIPHIIy 114 VA MXT 78 GYP = LA.:OSIPHIItEM.SINIPHII-A /12.SINIPH11..24VOI N.L.OSIALPHAILO*SINIALPHAI IFIA.LE.41 GO TO 12

201 19 ONTINUE IF INPOF.E THEN ALL POPEEA6.ES.X..NIT..AE.AGOMASS.411.Al2.A13.E.EloE2. 1EI.FF.FFP.FFPP.FP.FPP0L.E1.611 ELSE ALL POPEITAG.ESOX.S.411..AE.AG.AMASS.A11.Al2gAII.E.E1.E2. 6 E3.ff.FFP.FFPPOP.FPPOL.E1.611 ENDIF INTOT/ S.1.H OTNA.6./IHO.V.V1 SOLVE FG QS ASIATI TIP DEFLEtIONO I 2 AMM = 1. AM * 4.0 IF GO 10 1 If 160.E GO 20 2 IF IGO.E GO TO 4 IF IGO.E GO 10 3 IF 160.E GO TO 3 ONTINUE TV /12..LSTINTISINIPHIT IFITI THEN FI. ELSE F.12./PII.AOSIEXP VVVVV ENDIF BEETA.INET.THEIP GO TO 5 E 20 3 F 1.0 E 21 B.OSIBEETAI GO WO 5 E 22 SO.SINIBEETAI 4 IF 11ABSISINIPHIIII.LT GO 13 2 E 23 SI=OSISII ALL Gaps! Iu.uo.F.4,60.etotom,p41.sun2,sum.AK.Ammoks1 E 24 3 HUBLOSS ALULATIONS E 25 c.cix E 26 I=U 5 IF EHL.ED.1.11/ GO TO 6 E 27 WTOMEGA Fl * 1.0 E 26 IF *OF THEN GO TO ALL SUHOWIL.N.S.E.E1.(3.0BO.HM.E1.61.X1 6 FI 42./PII. AOSTEXP410.1LHO1/12..APSOTISINIPHIE E 38 ELSE - $001 E 31 ALL SUMONLIL.O.S.E.E1.E N.EI.G.1.71 f f. f*ft E 32 ENDIF ETUN 1 33 BOONA=SI X1.SI.S911E3SSI END F 14 BTANG=SI.SB11...A1X.LASI.00 NNOM.19NOMA.SNOMATBIANG*9IANG1 SUBOUTINE MESSES. 11.1*ATO F 1 OSE OTNA AM4SS*OH*OH IE SBB11.55I.SSIOIS$11111f BESSEL ALULATES F 3 F 2 BESSEL FUNTIONS FO THE GOLDSTEIN OS2.OTNA.441t N.SSI.SI.O*FFP 3 TIP LOSS MODEL F )04ON*FF.N. F S SINE euc.ssucbtas F 6 prin -00ft AK a 8.8 F 7 OSSI.OTNA.ITHASS.OH.OM ILMSMSTE.SS1351.$51PAUL I. F 8 0S52.OYNA*AMASS.ON SBISS FFOFF 00 K.1.16 F 9 OSST.OYNA*EI.FFPPFFPP *AK f 10 U0S.051/ TEAK F IL BOS=OSSIOSS2tOSS3 I/ I OSIPH11.2) SINIPHIIOSIPH OSIPHII 151 A. F K F 13 SLA / f ITK.LE.0.01 GO TO 2 F 14 T SINIPH11"21/OSIPHII.LO.SINIPHII.LA.SINIPHIII *TKP F F VO 1 F 17 ETUN I56 2 p f Al 3 ist = 8/1E15 F FOMAT 493H TOU HAVE SPEIFIED A NAGA POFILE NOT STOED IN THE P 199 K. AK.l. F 28 TOGAN. THE POGAM ILL USE NAGA * AK F IT 27 FOMAT 1,.31(.4911 ATTENTION NO ONVEGENE AT THIS STATION) ON 1NUE F AI *VOS F 23 END ETUN F 24 1 ENO F 25 SUBOUTINE TIPLOS 2 3 TIPLOS DETEMINES THE TIP AND HINT LOSSES 3 SUBOUTINE NAA00 TALPHA t BASED UPON GOLOSTEIN.S THEOY. O PANOTLAS THEOT. 5 G NAA DETEMINES THE OEFFIIENTS OF LIFT ANO DAG 6 3 O FO THE ASE OF NO LOSSES. AT A GIVLN ANGLE OF ATTAK. ALPNAI FO A NAGA 8212 AIFOIL. 6 fo EQUATIONS HEE OBTAINED OT A OTHOGNAL POLYNOMIAL S SOME 0.0 OVETIT OF NAGA DATA PUBLISHEO IN NAGA E'OT NO. 669.PAGE SUN AK. 1. E E E f E E E E Al ,ES 16 17

202 1 NOT PEVIOUSLY STOED. UE MUST INSET UVE FIT EQUATIONS 4 AO = 5.73 G 9 A2 = 7.A4 3 FO SETIONAL LIFT ANO DAG OEFFIIENTS 13 A FUNTION OF 5 6 AS SOO ANGLE OF ATTAK IN DEGEES SDI = G 12 SO2 =.13 EQUATIONS FO LIFT AND DAG OEFFIIENTS FO 6 G 15 NAA AIFOIL HAVE BEEN INSETED. 9 S SD4 = 0.0G A S ALPHA180./ G AMAX = A ALPH /160. ADD UVE FIT POGAM FO L AND D 13 G 16 IF IA.GT.ANATO GO TO I PI L = 10A 6 19 ONVI4.1/ O. SOOPIS GO TO 3 AL0.1.1ONVI 6 22 I L AAMAX G 23 IF IL.GE.0.01 GO TO , L = E.G B O. S0I.SD411..AMA A-4M1/ IF ILO.LE.1.01 GO O = 1.0 O G 20 3 ETUN ENO S SUBOUTINE NAA44 IL.AgS1.ALPIA...D.AL PI ALI.ALPNA LFL ALIYALI AMAX.P172. AL3.4L2ALI AST=45P1/ AL3.ALI L ONV=PI/180. 1E11.6E44.1 L.00.0/.1LE.BPAL L4 DO F ,.71 L.00K G24123ALIS4AL4 1E114E021.1 L.2.25.SIN121L11114./PlYSINIAL ABSA.AOSIALPHAI ALI. ALPHA D AL21 ALE.AALI IFIA.GE.10.1 D IALI ALI.AL2*ALL 1E11.6E L1..14.ONVII AOILTHA1130./P1 IFIA.GE.20.I D.4.5ISIN..2./14./01PSINIEL E14.1E4.4.1 THEN ETUN I 33- l ENO AL21 ELSE II.LE.14..AND.A.GT.4.1 THEM SUBOUTINE NEGATE ILOTAgSIOILP41.L.O.AL1,11.11T.LT.OT.NFS.SOLO J J 2 L ALI Si J 3 IFIA.LE.10.0 THEN 3 O /4611L140ONVI HAW? IS AN INTEPOLATING SUBOUTINE TO INTEPOLATE J 4 ELSE AIFOIL DATA IMPUTED IN TABLE FOM. J 5 J i L1-10..ONV/ J 7 ENLIF DIMENSION AATI261. LI ELSE IF IA TNEN A ALPHA J 4 J 9 L L AL2 DO I 1.10NFS IFIA.LE.16.1 THEN EF O TO 4 J 10 D A ONV3 IF II.LE.1 lllll I GO TO 2 J Al ELSE IF II.EQ.NFSI GO TO 3 J 12 J 13 D= ONVI 1 ONTINUE 2 ENO IF J lot J 14 ELSE PE. IA AATIJ J 21 A J 15 L.1.2S1N12.ALII L J 16 O A1116ONVI O = PE.1:01(J /1 J 17 ENDIF GO I 5 J 16 3 ETUN LL LTINFSI J /9 I 311 ENO D OTINFSI J 20 GO 10 5 J 21 I I SUBOUTINE MAIAM IL,X.GTIALPHA,L,O.AL L J D. OMO J 23 NAAXX IS AN EMPTY SUJOUTINE FO USE FO A POFILE I 3 5 O = IE /3.471 J 24

203 J 25 ETUN J 26- END K 1 SUBOUTINE SOLIDI I.1,6F0B0,PI,SOLDI K 2 3 SOLIDI DETEMINES THE TOTAL SOLIDITY OF THE K. 3 K 4 WINO FUYINE DESIGN. K 5 K 6 DIMENSION 11125). I1251 K 1 HEX = NF -1 K A 51 = 0, 1E110*(0.11 THEN ELSE K 9 DO 1 I=1,NFA K II SOL ,/II11,2.11III-II.111/100. K 11 St = SIASOL K 12 1 ONTINUE END IF K 13 SOLO 951/1PIlt.20 K 14 ETUN K /5- END t 1 SUBOUTINE ATIVI I.I.AF.001.PI.AF1 L AM! - OEtENINES INE AIIVIII FATO OF THE L 3 L WINO TUBINE DESIGN. L 3 t 6 DIMENSION I1251 L 1 HEX. NFI t 6 SI 0. t /./INFA t /6 I / K111/1/2. t I/ O IIIIIII4111,2..1/.111/100. L 12 OO = (14111,100. L 13 FA 131/ DO L SWAG L 15 1 ONTINUE L 16 AE '16. I IT ETUN L 16. END SUBOUTINE GOLOST 111.O.F ,OL.PHIISUNI,SUM,AK.ANN01111 GOLOST DEIENIAES THE GOLOSTEN TIP LOSS FATOS FO THE 2 BLADED DESIGN ASE. FO DESIGN ASES WITH BLADE NUNES OTHE IAN TWO THE PAIIOTL MODEL IS USED. THIS AN 9E HANGED IF NEESSAY, ANO THE GOLOSTEN MODEL FO ANT NUMBE OF BLADES NAV SE SUBSTITUTED FO THIS OUTINE. DO 5 H.1,3 V 12.*IN.1.1 IA - U V ALL BESSEL SALL BESSEL IF 12.GE.3.51 GO TO 1 A = D = 8.'8. N 22 N 23 tivz a 22/1A /11A ,122,31/11A-01sto-021 N V /11A-V2010-1, N 25 0I102 1V.P1410/12.*S/N1.5.V.P111IIVE N 26 SO TO 2 N = IWUT/(1..U U.U11.U.Ulf111..UU1.41 N U.U.21.*U440U61/111.4UU1*71 II 30 I U.U 603..U" *U636.U N 31 I 161/ N 32 3T111 fat2/62,14/ /16231 N 33 FU 1UUlt11.1.UU1TI0E N 34 SUN SUN.FVU/V2 N 35 IF IAM.NE.6.01 GO TO 3 N 16 E /1U IF IAN.NE.1.61 GO TO 4 N 37 N 3$ E 6.531i1U0o If IAN.II.I.01 E 0.0 N 39 N 40 SUN? SUM2411UO.UO.ANNI/11.4UOU01IAI/41.61 N At AM ANTI. N 42 AK AKI/12..ANI N 43 5 ANN AK/ N 44 G fueulitt..tput-te./iptploisul N 45 I 6-12aPtsSun2 N 46 f 411.4,001111PUIPTI 47 ETUN N 46 END FUNTION KNOOIPO DIMENSION O I102*(.161/ , , SUN.AH000. DO 1 IJ016 SW /1 AMOD.AN PI 4 ONTINUE ENOD.XMODJSUH ETUN END SUBOUTINE POPE! f=6,,esoix,.n84,4e.a;.anass.aii,ei2sai3io ,E2.E.I.Ff.rEP,EFEU.FP.FPPOILAIOJI N I O./X N 2 E.LA N 3 H.110/ N 4 E.TESAGI N 5 E1.ILS.25O N ESI N 7 E ESI N I N 9. POPETIES OF THE BLADE N IB N It 20 AFA /144. N 12 IFIL.LE.1..ANO.LeGI /!NEN N II EVPI-3.3I3'LDI N E2Ff=2.0236SLOI N IS ANASS16041FATOLIPI 'LI N 16 ELSE IFIL.LE AND.L.GI T4EN N 17 B EXPI-1.802LI N EAP10.26.LI N 19 ANASS.9.65AFAIO.EXPf-O.E4IT N 20 ELSE IFIL.LE NO.PL.GI TVEN 11 21

204 3 812=4.5679*EXPI4.000A*II IFIL.G =2.111EXPI =L/ BII= =L** AMASS=5.243*AFATO=I** ELSE AMASS=10.88*AFATO ENDIF *B13 EI=S12AE J=BIPAG HANGE UNIT TO FEET AII=AFATO011/1144.*X=XI AII.AFAIO*012/1144.X=XI A13cAFAIO *X.OXI EI=E1/1144.PX*.A.1 G.I.GJ/1144..X"4., Al2=LASI512 AIT=OINSI.BIT EI=AE*012* =AG*1111=144. AII=A11/1XX FIX*XI AII.A13/IX*XI EI=E1/111X GJ/IX*.0 J0=7.2047EOS EI0=1.1577E06 210=EI0 IIM*0 GJ0.G.10FIX*4.1 F0=1. FO.I. S.11.-HI ZP=IL-HI/S ZP2=ZPIP ZPI=IP2ZP IPA=ZPIP FF=6../P24.=IPI*IP4 FFP=12.M12.ZP2*A..IPI FFPP= P2P.12.=ZP2 FP=2.ZP-ZPIP IPP.2.*11,ZPI ooetion OF MODE SHAPE THAT ASED ON LENGTH OF THE BLADE I HO NOT ON THE ADIUS OF THE OTO I OETION FO FF=FFS MASSOAMPIN.STIFFNESS MATIES I FFOP.FFPP/S EPP:FN./S F0.1. F0=1. S.41..ANO IPA.KH/S ZP.IL-HI/S IPI.ZPIP IPT=IPE*IP ms16.1.1p AKJ0=2.*ZPAGJ/G12 AKKO.2..EZIEWIPOZPA*13.t4.*ZPAI AKK1.12.PEI/EIS*IPAP11.*IPAI FF.IIKKA*AKKIIP6.IP2..4.*/PIZP4 FFP=AKK10120I0-1201P24.*IPI FFPP=12,24. P*12.*ZP2 FP.A P.W.ZP OETION OF MODE SHAPE THAT BASED ON LENGTH OF THE BLADE I HO NOT ON THE ADIUS OF THE OTO OETION FO ASSOAMPING.SILFFNESS MATIES 1 ETUN FF=FF*S ENO FFPP=FFPP/S IPP=IPP/S E1UN SUBOUTINE ITEATIOSIIALPN.AXII,SI40051,KKIIKLI DIPENSIUN IESI THE AATI2SIs LT 1251, OTI2S1 END OIMENSION stattoollmmodttetatoso SUBOUTINE POPE IYG.ESIX..HB,OIE,AG.ANASS.A11.Al2sAII. OMMON eogno.o.v.m.inetp,amoolona1s6o,onga.ho.vison,pi0na.n. SE.FI.E20E3,FF.FrP,FFIPP.FP.FPP.L.EI.011 ANPOF.APP.T1012.I7.I EST0XETIGHH.ALOsA OMMN XG.ES.AEIA F.IX OMMON,I.THETI.NF LI2=L/ EXTENAL TOM, TIH H=HA/ KL.I E=IESXGI. MMAAG gt.itsc-.251ne i2 ALEFI=IGHT.O. E ESI* DO KK EI.14.S-IS)* NNALF=1/2 2 1 POPETIES OF THE BLADE L.1.1 NTEST=1-2.NNALF L.STAIII 012= A=AXIIII 011= ALPHA=ALPHIII 1111=8II*0II ALPHAD=ALFHA*0.001 DENSI= L.L' ANASS= ALL SEAHILoIAOHETI.NA.aEisSMOOEI AII=LENSIPATI

205 IFIMPOP.E THEM ALL POPEIXG.ESX..MB.oAEOG.ANASS.411.Al2.4130E.ElvE24, SE3,TT,FFP,FFPP.FPrEPPLETyI1 ALL HAAAXIL.X.SI.ALPHAsL,OIDALOOTO ALL NAAXXEL.X.SI,ALPHAO.LG.0001rAL0.4) ELSE ALL POPEITXGIrES.X,OB,.AE.Al.AHASS,AII.Al2.A11,EP.E1.12. SEWF.FTP.FFPPaPIPPPIILIII.GJI ALL NAA XoSItALPHA.I.00sALO.MI ALL NAA44112LIIX*SI.ALPNAO.L.000.ALB.M1 ENOIF H.HOJ W=SIFOS NP.FFPBS W=OSINPI SIT.SININP) IN=W.M-SW.S4 BEETA.THEIINETP O.OSEBEETAI SB.SINI BEEIAI SI=OSISTI SS1=i1NISTI =/X OH.OHEGA ITIL.LE.XXI THEN O.STA111.-STAILI ELSE O=STAILLI..STATIT EMOTE GH.OSIALPHAI.ISINTALPHAT HOSINTALPMAT..OOSIALPHAT X.OMX/V ITINPOF.EQ HEN ALL SUMSUMITOMoTINAIIE.E1..E3OOUIIBU.AAU.ABUOMU.IUNA*0 1M.ABUM.AUO$MMASII0H.EI.G.I0AT ELSE ALL SUNSUMITTOM,TIM.I.E,ElfE3oU.AU.B1.AAUeABUOBU0AUNIAA IUM.ABUNAUTTOMASeHATAJATO MIT U0Uti.U UP0*AU umpouc UOTO.UMFO=BU UMPO.AUM 8( U3 AUPTI.AAU AUMPO=AAUM BUP0=AUFO.ABUE BUMPD.AUMFO=A00m BUF0=BUNFO=BBU BOUMOBU DI5mIALfUTS1-,M.SSI SOME OFATOS PATI.SSI'SM-SPOI.O PAE12=SSIWOSI.SM.A TAILizILAUMI.M.SI*S94N.SHSI.SIDET.PIT12 TAILI.IL.UNI*SIASB-NSSI IA Iflis:ST Il.*Al.XEILAU01GS1.3-MSSIT 101.-PAT1.11,41-XIIPLH101.M*.IST'S.PI'SW.SI.SO-E3.PAE121 NE2.104.MN.MtMT E2 0 NNO.MEN MASS MATIX OVEl=06SHOV V OINA.13./OVELT NES=1A11-AIWOM*OWISSI.SIO*2H.SWWITSI.001*.Z.-SSI.SSIII HiSsI.WNO.IFF.NAINFO.SNI TEI.ESB.B.E1.-SSI.SSIIILFAUIASSI.A IFI.BUG IILAUISISIAE.SSSI.SOI Ii4=AMASS.OH.W SBATSSI*0121.PPIF.SSI.SI..BUT TES.11.FFPA.TIPP OALETIOTNAlaFOPSOO OIGHI.UTUAEAMASS.OWOM11F1oTP,IT21.HTSPrp1,MTS DIGHT.OIGHTO FEB=2.13. IffNEESNELO FEO.4./3. IF I.ETTAK1 ree.t.,:. STAT THE SIMPSON INTEGATION ALEF1=ALEFT.FEBOMETT IGNI.IGNIE0pEGNI ONSTUE POS.HIS OS.IGHT/ALETT TFOOS.E O TO IA HE;X=ABS110S.POS11051 IFI:HEK TO 555 TIFTKLAT.101 GO TO ASS GO TO 22 lb NITE II FomATI,. US OSS ONTINUE ETUN ENO SUBOUTINE SUHOMAILONOSsE6E1..E GJOIT BOUS.O. ALON*A. AHIGH.ILII.HT/PS EWEN OS.1AHIH-ALOMI/N O SIA05 /3. LS=ALON ION,* IHALE.1/2 ITES1-I-2IHALF IPALS 2P2=ZPolP ZP3.2P2IP TPA:IP32P FFP.12.IP-12.EPEt4.Ep3 FEPP ZP4, FPA2mIP..ZP2 FPP.2.'11,1P1 O BOUL.4TP.FFPPS F evi. IFTITEST.E0.01 FEN*4.

206 le IF EU.K1 FEB=1. BU=aBO41,1=00H1OS3 LS=ISIIOS ONTINUE ETUN ENO SUBOUTINE SUNOWILOTHOTS,EIElsElyBOUINNEloGJ.AI OBU=0. GJO=7.2647E05 El0=1.7033E06 GJO=GJ0 /TX=4.1 EIO=E10/1A.4.) ALON=O. AHIGH=L/S 1PA=H/S N=NH DS=TAHIGH-ALOTIN OS3=DS/3. LS=-IFA K= I=101 IHALF=1/2 ITES1=1.4THALF IF=LS 11.2=ZP=OP ZPI=ZP2ZP IpysIPI=IP IFIELS THEN AKJ0=2.IFAGJ/GJO AKK3=2.EI/E10ZPAZPA13.04.IPAT AKK1 =12.EI/E10=ZPA 41,01.1PAI FF=ANKO=AKKOZ.6.IP2-4..ZPI.Z=4 IFF=AKK1.12.1P-12.TP2.40,11.3 FFP0= P.12.*P2 FP=A *IP.IP=IP FFP=2.11.-ZPI ELSE FF=2..11FA.ZPI= *IPA2.FIBI FFP.12.12PAIIIP101.o0p0-tpi fifts p1 FP.2..ZPA41102P/SPAI FPP=2. ENOIF FFPFFPS FE0.2. IFTITES FEB=4. FE9s1. BOU=BOU=FEBOBBU=DSI LS=LSOS 10 ONTINUE ETUN ENO SUBOUTINE SUMSUH111014TINOLgEggE1 lol3eu.au,bu,aau.abuobus 1AUM,AAUN.AOUN,AU0sH11, Elo3JoX, UG.AU0=0A1.400= AUM.AOUN=AU =0. S.1.-PH ANIG0=4111-HI/S N=MN DS=TAHIGH-ALON1/N DS3=OS/1. le 1SrALOW K.N =1.K IHALF=1/2 ITEST= ALF SP.US POP 1P3.ZP211. ZP P FF.6.2P2-4../PUZ04 FFP.12.ZP-124ZP2.4'.fP] FFPP= IP2 FP.Z.ZP*ZP2 FPPs ,1 N.11SPS WP=FFPOS IMPuFFPPOS N=OSINPI SN.SININPI OU=-0.514MPNP TONO=TONTE4N.SIT,FP.FPB.NFBI tont=tomal,cw.sw.fp,rppopp, EEl -E3 TONI=TOMIEE3oN.SW,FF,FO.NPPI OAU=..NP10110 OAUH=-11PT0H1 0AU0=-NPt0H3 OAAU=.40M0TOHO/S OAAUN=TON1=T0MI/S 11$0=ITHIE.N.SN,FP.IPP,UPP.FFoFFPO TINI=IIIIITEL.oMFPO.NPP*FFP.FFPPI OABUG=NP=TINO-FFPTOMO =NP=T1H1..FFPTOMI OBBU=...FFPFFP=S FEB= EST ree.%. FE=1. uc=ucortimuc.os3 AU=AUFLIAUDS3 AUN=AUEIOAUH=OS3 AUO=AUEB0AU0=OS3 AAU=AAUTFE*OAAU=OS3 AAUN=AAUN4EAAUWOS0 ABU=ABUWEITABUOS1 0111/0=A0UN4(0.000UN.OS3 00U.00UE0000UOS7 OU.O0UOS LS=ISONS ONTINUE ETUN ENO SUBOUTINE SUNSUNITON.fIV.1a,E1.E.10.1.AUAUO1AU.A0UOBU.A AUG,M1,0S,IIH.EI.G.I.X1 U=AU=AAU=ABU=BU=AUN=AAUH=AOUN=AU0= =7.204TEO5 E11=1.7013E06 Ele=i104.1 GJO=6J0 /(111(4.1 ALOW=0.

207 s=1.-h AHIGH=L/S ZPA.H/S N=HM OS.IAHIGH-ALOWT/N OSI=OS/S. LS=-ZPA K=H I=1.1( IHALF=1/2 ITEST=1-2.1HALF ZP=LS zpzp 21,3=21,22P 21,4=21,3.2P IFILS.GE.0.1 THEN AKJ0=2..ZPAGJ/6.10 AK0.2.EI/EIO.ZPZPA ,2PAT A10(1.12.E1/E102PA.11..ZPA1 FF.AKAA.AKAI*2P.6..IP2-4..IPI.ZA4 FFP=AKK1.12..ZA-12..ZP2.4..2P3 IFPP= ZP.12..ZP2 FP=4KJ0.2..ZP-21..IP FPP tP1 ELSE FFP=120(2P4.2P1.11..ZPA-7P1 FFPP=l /P) FF=2..ZPA.t1..ZP/ZPAI FPP=2. ENOIF H=FFOS.S wp.ffpos IIPP=FFPP.OS H.:OSUMI SN.SINIOPT 00=-0.5.SHP.NP 10/101041E.N.SW.FP.FPPOIP1 TONI.TOMIEI.W.SN.FPOPP.PPI EE3.-E3 tw.tomuo.cw.swo.p.rep.wm DAU.-WP10110 DAUN=-HP.IONI DAU0s-ITP.TONI OAAU=-TONO.TOMB/S OAAUm.-10M11010IS T1110=TINTE,W.SH,FPOPPAPP.FfP.FAPPI TIHI=TIMIE FPPO.P.FFP.FFPPI OABU=WP.TINO-FFP.10He OABUM=WPIIHI-FFP.TON1 THIBM=-FFP.FFP.S FEB2. IFTITEST.E0.01 FEB4. IFTI.E0.1.0.I.E0.K1 FE1.1. U.UL.FEBOU.OS1 AU.Au.FE9.DAUOS3 AuH=AUM.FEBDAUMOSI AUO=AUO.FEW,DAUO.OSI AAU.AAU.FEB.OAAUOSI AAUM.AAUM.FEMAAUM.OS3 ABU=ABO.FE110AUOS] le AiHni=AlluMFE00ABOS3 BU=OBUFEB.000UOS3 Bu=OBU.OS LS=LSOS ONTINUE ETUN ENO FUNTION ,0.E.F1 10H.A.B.E-A.F.0 ETUN END FUNTION TINIA*D*o00E,F0G* A.E.S*A.F A ETUN END SUBOUTINE VAMIOS*ALPH*AXI*STA*F0131,KKOL01KOELoSJOLL*5K0EL. 1SJDEL*ZMODeZOOpiK00*O0gINDLeZKO* ,PAA,E DIMENSION I251* I1251* THET11251 AAT1211* LI1251* ODIUM DIMENSION STAII011*AX111001*ALPH11011,F OMMON ODOD0B.V.X.INETF*AN G9ONEGAANO*VISOLO.IIX*Nt SNPOF*APP lgt21,1304*t est*xetion*alega OMMON XGAS*AE*AG OMMON oionetiof EXTENAL 10/10111 IN./ A KNALF.N/2 IFIIKK.2.KNALF IN. SLUATA AI IL3 F01* F0.1* / /1/ Z I19.1* THOD.ZDT.201.1D2=2KOZ.ZKO1.2K02.N180.PAA L D N2.10N3=OLl.OI*0* ZZO ZZO23=0* I.XX L.I1 NNALF.IFE NTEST=1.2.NNAO L.STAIII A=AXIII1 ALPNA.ALPM111 FLOSS=F ALPHAD=ALPNA1G*001 L.L.X.OSISI/ ALL SEAHILO*IIINETIOF.oINET*SHODEI IFINPOF THEN ALL FOPEIXGIDESOX*OD,*AE*A6*AMASS*A11,Al2*A13*E.E1*EE* SE30FF*FFP*FFPPOP*FPPAL*E1s621 ALL NAAXXILOX*SI*ALPHA*L.000AL0ON ALL NAAXXILOXeS1*ALPNA *AL001 ELSE ALL POPEIIX60ESOIX EgAGgANASSI,A11*Al2*A13*E*E1vE2* SEI*FF*FFP*FFPF*FP.FPP*L.E14.11 ALL NAA441L.XIISI*ALPNA$LIIOgAL001 ALL NAA441LOX*SI*ALPFAD.L0*000A10*N1 ENDIF LA.ILO-.L1/0400/ OA / N.140/1X.:05ISIII A 11 A It A 13

208 S=1.-M N=SFFOS NP=FFPIQS N=OSINPO SN=SININPI 2N=NW.SISN BEETA=TNETAINETP D=OSIDEETAI $8=SINIDEETAI SI=OSISIO SSI=SINISII =/A ON=OMEGA IFIL.LE.KKI THEN O=STAIII..STAILI ELSE D=STAILLI-S lllll END IF N=LOSIALPNAItD.SINIALPHAI O=:LSINIALPHAI..00OSIALPMAI A=ON.X/V IF INPOF.EQ THEN ALL SUMSUNITONstIN.LIpEwEl.E3gUIpAUoOU.AAUeAUOIOU.AUNAAU IN,ADUM.AUOOMASOINgEloGJ.AI ELSE ALL SUNSUNIITON,TINOLA41470,AUONIAAU.ASUOSU.AUM.AA IUM.AOUN,AUD.NNOS.N.ElloGJeX1 END IF UO=UM=U UPD=AU UFO=OU UDFD=UMFD=OUG UMPO=AUM BUD=OUN=9U AUP=AAU AUNPO=AAUM BUAO=AUED=ADU BUNPU=AUNFO=ABUN OUF=OUNFD=ODU DUM=OBU SEMI OFATOS =X PAII=SSISW.SI.leO PAT2=SI.SO PAT3=ESIIISSI.SNAO PAT4=IEtUNIPN,NSII PAT5=412L,UMIPSW-NN PAT6=SISNeSSINO PAI7 =/V.M.SO*01*TLIUDIPSSItSB*NINSWPSSI!SO.E3.PATIO PATO=/VITLOILOUDISSI.O*NSIt PAT9=/VITL.PAI6.IL,U0148GIONSN.9tE3SWS01 PATI0 =/V.IILPSSIPSIUIL.U01.01 PATII=N.SD.SIPESI.O PA1I2=S5PNISISND PAT13.NSSI.SOIEMS.SSI ILFI=ITOSISNfILIUMI SSI SO.SN-NNS1ISB..E1PAT6IFFP ILFI=TLFISN*SSIS0FFOUVPSSI.SB.3N PALI=ILoUI.SIMISSIWIESSIPS9 patt.n.motouclassi.wp.cst.co-efsmst 3 TAILIaILPUMI.N.SISIN.SN.:SI.SOIEI.PATI2 TA/L2=ILIUMISIPSO..NISSI 0AIL3=YlSON iol.uhi SSIS49,111.wsw.SWSO-E1ePAT3 TAIL,3=IIA U SWISISO-EPAI3 tall4.pat6.1lioumi.o.wou.sh.o-wilf.s8 tails=wowitiltuomsio.wa.si 71I155=ILOUMI*SSI.O.NSST TAIL6=YLPSSISOIILIUMI* pAWIi.-41-xIIIA.U01W.WSO,WSwSI.S0-E7PWITO w (2.14N.ON.141ot Nt0=142O NNO=WE2N. APPIONHIMEI SIoO WSBO10.L51.391x3aPI HA.1.4.OSIALPHAIKOASINIALPHAI TA.GlA.SINIALPHAO-OA.OSIALPH41 OP=NA-O OP.IA.N fl=:12..1on.opoti f2..12.nof-op.oni GI=.42.ONN.OP.NTI G2= OPNNI F4=NET.NA.EP G4=.NE2IAFP HELP2=AU01-onSSO.GtSIO.G21 IIPP.X.EPPATIVFP NFIIAA.ILAUOI.SN*SI.SIPLFFP-X.SN :SI.SIPFE-APNN.SISTIeFFP NFIS=.11E3'PATIFFP...XOUINSIPSA NFI=NFIttNFIPINFIS Oft.-X.ISSPIFF-OWSI.OI N01.IILOW01.IPSISO.NO.SW.SIMO-WISSISW.SI.VOO11'IMo/V w021101ssi-ilipw01.si.op..,0/11 N0I.IVOSO.W.OL.U01SSMO.W.WPWSSMO.E3'PAI31.0/ =/VPITLDIAL.U0IPSSIPDINSII mo7.civflirt.cssiececm.sims11.41mou01necutwa.sece.e3.wsei w 04=pe1 rt.ss1osetilfuoi.sell N NOI=PAT/FI-PAT2F2 NTOI=PAIP6I-PAT2G2 NN02=PAT6'FI,SSIASBF2 WNDI=NSO.FID.F2 NTO2=PAT6.61,1SSIS0.G2 NT03 =M.S8.G1-.O.G2 N N002=NO3FItN04.F2 NNOD3 =N0IF11402F2 N IDO2=110361*N04.62 N TOD3 =NOIPI FIND THE INTEGAL OEFFIIENTS DF TAN AND TAN ATE SAM=II.S/PII OTOA ,A = f1035 SI IFIAAT.AO GO IGLUAT/DIDAI A0I12=IPAEIFi-PIT2 '111.4E1oP4513-pAT4eSSPSOI e cl12.4paiigi-patvg21 'IPAWPAT3-PAAPAT61 OZI2SAM*IA0II24,00ZI210ISOP 110I13 4PATI F1..PATF211.4/.SlSIPAT ,..IPA1161..PAIG21 IPATS.SBIPAA14.GNSDI DIII=SAN.IADZI3.OU1I31.01SO

209 DZIE=SAMFIHTWFI-TFZiTIEITPATI-PATETSSITSBITFOFD DZIT=SANTIPATVFITSSI.SD.FEITIEITSNTSS-PAT ATOITFO.OA DZIEutSANTINTSPG1-01%2ITIPAISTPAI3-PAI4PATEITEDTO OTIF.#SANFIPATEFGIFSSITSOFGETTIFAIEMSOTPAITTSNTSBITFOTDA D II0zSAMTIPATFTFI-PAT3 F2IFIETTPANII-PAT4SSITSOITFOAO DZI9.-SAIPAT9 F1oPAIITEIIPAT4T9-WSOITFOTDA 0211E..winTIPAT761-PATIM21TIPAISTPAII-PAIETPAIGITFOTO D ZIIL=FSAMTIPAT9TGITPATIOTGFITIPATETNTSDTPATSFSNTSOIAFOTD O 2112=SAN IIITSOFFI-OFEIIEISNSD-PAT4'OITFOO OZIIJ.SANTIPAIOTIFSSI.SOFFZITA-EITPATJTPAIEFSSITSDIFFOTO DZI14 SAM IN S8 61-OTG2ITIPAI5TSNSWAEETNSBIFOO 0III5=-SANTIFAIETGIT5SITSBG2IFIPATWAII-PATETPATEITF0FO 02116=SANTIPAI7FFI-PATAFFEATIET.SNTSD-PATETOOFFDTD D ZIII=-SAIPAIE FI FATIOFF2IIEITPATI-PATISSITSOIFFOTO O 2118=tsAH IPAT/ GI-PATATG21.1PAI5 SNSTEPATINTS810FOTO DII19.-SAHTIPAI9FGETPAIISTGEMPATE5FPAES-PATVPAt4IFFDPO PASS MATIX OVEL=.SHOTVE D INA=iTsom, 001=2.TVLELT4.TYLTN$SP0.4.TYLFIETuIFssi-4.TYLTETSIFS6 002=NTE11..IcSITITT2.ItE.ETII.AISO.SIITT =ILATUIn.T11. SSI SSII-2.FEFNFSSHT11.-SSIFSSII O 04=2.TIL.uIeWNFSSI.SITB-2..IEWITETSSITSIoSIO 005=ISIFNI.T2..ISSITSNF SWTSOIA*2.-2.*SSITSITSNFNTH SSIS8IT2..BOIAI2 o cir=gcsl.sw.2..(ssi.cm.cwe.flawsw.z..2.oswycw.ssmstne WHOo.OVNATIAmASSTIO01.0D2.003,0041TAIITOOSEDDEFAITTOOFITFOFIPO DAMPING OEFFIIENTS MATIX FI-NOT F2ITTAIL3FtwoIFFND4FF2IFTAILEIFOFF0 0D2=14IND1.61-NO2.621TAIL5-IN0IT GZITTAIL6ITFOTFD ozcoz.i.s*icool.coowtom DIOI=IPATIFFE-PAI2 MFIAILITIPATV1-PAT2T2ITTAILS D IOL= STFOTO 02O2=IPAIEFFL-PAT2' FEI IAIL4-IPATIGE-AT2-'GEITTAIL6 D io2.1.5toio2disafoto STIFFNESS OEFFIIENTS NATIX 2001=IIN SO FI-9 F2IFTAILUIPATEFITSSITSBFEITTAIL4IEDF IPIIOWI-en2IETAILS-IPATETGITSSITSBEITTAILEIFFOTF0 0ZK0Z.E.5IIDDIAZ0021DO O TE01.11FATIFFI-PATETFEITtAILITIPATITGI-TATE.G2IFIAIL5 DINDI.I DISFODE 02KOZ.IPATITFI-PAT2TFEITIAILE-IPATit-.AT2'GEITTAIL6 DZKO2=1.5TDZKOVOISTFOTOO Z0D11a SI-9TF2IFIAIL3'FDFOO IDDIT.I.STIPAIEFISSITSOFEITIEILETFOFOTDIt ZODEI.I.STWSEGIFIAILSTFOFFOO FVFOO PATEITSSITS0G2ITTAILEFOFOTO STIFFNESS IN TAN D IDDLI*IINTSBTFI-OFF2ITSOTNTINTSIITGI-9FG2ITPITFOFO DZOOL2.1PAITI SSITSOTFEITISSITBTNASNT35IIFOTF IPAT6 'GITSSITSOTEITSSITSIPFDFO OZOOL4.14PAUFFI-PAIEFF21SO.NEIPAIIT1-PA TOITOISTF0 0700LS.-IPATI.F1-PAT2TF2IFISSITNTSIISIIOISFD DZOOLE,IPATITGI-PATVGLITSSITSOTOISTFO DZOLI=1.5.10ZDOLITOZOOL2OZOOL3IFD DIDL2.1.5OZOOLNTO LEADZODLEITO 0100N1.11NTSTFI-S.F2,FIWSWFSSIFSOTINTSOFGE-O.OFIFEFS/1 AFT ***** DiDONL=011,0141FDF0 OZODI2.IPATEFFITSSISOTF2IWETSN8TFOFU O PATIFFI-PAT2'F2ITATSMTISITSEPOIS.F0 ONON4.IPAti61-PAt SIOTO O 2001(5.-IPAATIFFI-PAI2TF2ITHTSN30DISTFO DIONI.I.STIOZOONITOZOOEITD 0201( I00N3.OZOONEIO DIDNI=1.510IUDESITO DIDOI.ISOSO.F1-WFZITTIL.UNITSSITSO.NTEITISSITSWO-SITHII OIDDI.OIDOI.FOTF0 OIDD2*IPATETFIeSSI.SOTF2ITIILAUNIT9TN-EleSNTSSITFOTFO DIDO3.IWTSIITGI-DAG2IFIILAUNITSSI.OPTWO OZ0DE.-IPAT6' GIISSISOFGEITIILIOUNISOIFOFF0 DIDD5 ILAUNIASSIS9IDEPISSIFSNT9-SIFNI DIDDI15 010OSTIPATIFFI-PAT2FF2OT0IS DIDD6.1PAII1-PAIZTE2IFIILAUNITSSIT9'DISiTF0 DZOO7.-IPAEI.FL-PAIETFEITIILAUNITOTN-EITSNTSWOISF PAEITGI-FATETGFIILATUMITSOTOISFD IOZOOITUDOETOZDOP3OIDO41.OT OZD2=1.5.IDZOOEFOIDDEITO DID3.I.5TIDIDD7OZDOSIDIT EFFET OF TONE SHADoN 000=-INN6IAILITNEETAIETIFFOTO DOLI.-INNOSOTNOITOITWO DO1.-INNOTTAIL33ANTATIAIL55ITFDO IFIIII.EQ.11 THEN FE IFII.EO.I,O.l. E0.41 FE0=1./8. IFI IN=A ELSE FE02./T. IFINIEST.EDJI FE5.4./3. IFII.E0.1.0.I.E0.KKI FESsial. ENO IF 3 STAT THE SINPSON INTEGATION IFFE8TOZII 2I2II2OFEBI I3.ZIEFFES ZIT/ILOTIMIE E II6ZIEFEOFIDZIEI ztr.tir.rtuntr, ZIO.2I6.FEIDII ES* itio.ziteeortim ZI M itiuttis.rommtso ni4=illwle.lortt =21150E Z116WIDZI ,0E ills=z11114e010i11111 ittuzitweitozus1 ZMDD=Z000.FEOI IOZ./OIFEOTIOIOZI IDI.ZOIFFED.IDI011 2O2sZO24E0.10ZO21 11(01.ZKOIJFEADZKOI 11(02=ZKO24EVIDZKO21

210 ..,., NM NM 0 O Pal 114 O ;..1. rim s. /4 1,1. woe w M. rft 1. in. M. N. N or ,./ 0, VNn.w... : t :r.rr-!,: Z :: :::?. 2dtdd or. b...or m2222 ZW.N.N. 0 ft.. N N N 2! t22 n: i!eee. O...,.....= e,tass,......lpes2181't tg Z : tt 1:::: , www. t A :::t Ting_ YJ WUJYA rtria......Oo - 4L... :2:2 :.::;:t..., : t :1 om.p,)..0, '47 :-:T7774 Ntgag z z w...on.1.1.j., u. Z?1:.414A.IZA4:::=2:.41..t: tl itt.gde.dtttttaa!&=zg 5'gdttgggagg: "-"'" z ' -c'=?9,9411=1% =2,1 "t.,,-, WOO e 1.:t...!21.. tzvw

211 193 POP 's Output 'THEOETIAL PEFOMANE OF A POPELLE TYPE WIND TUBINE OPEATING ONDITIONS: WIND VELOITY - FPS WIND VELOITY GADIENT EXPONENT HUB HEIGHT ABOVE GOUND LEVEL - FT- ALTITUDE OF HUB.ABOVE SEA LEVEL - FT ANGULA VELOITY - AD/SE TIP SPEED ATIO PITH ANGLE FOM NOMINAL TWIST - DEGEES PITH ANGLE AT 0.75 ADIUS - DEGEES ONING ANGLE - DEGEES BLADE DESIGN: NUMBE OF BLADES 3. TIP ADIUS - FT HUB ADIUS - FT SOLIDTY ATIVITY FATO NAA POFILE 9999 NUMBE OF STATIONS ALONG SPAN 1 HOD AND TWIST DISTIBUTION PEENT ADIUS HOD-FT TWIST-DEG POGAM OPEATING ONDITIONS: INEMENTAL PEENTAGE ANGULA INTEFEENE FATO. AP STANDAD AXIAL INTEFEENE METHOD USED TIP LOSSES MODELED BY PANDTLS FOMULA NO HUBLOSS MODEL USED

212 NAINNNNW asaanmmmt nnww441.aavia0.1.,,,00'00..n.p.ou 4'41):41':124:14:.=8.18::=,144:1818,-;81N88,t884T.V48,18W' 8288i ' V225:r4;8.00,...832:3n..a..-8Fa58.?-34,1,1!882!zWi.L4 V;;It88;8'41):1M-n8,1VV:27;;;:pa.38822U12,gi:::::::::G "..:4;42.484rae, VPP.40 00:00 rn"?!7-0:4-, Nr.84'848V4,14.V.4:841t- 03,. O PP NNOU(V PN N0 V :142:, tom: a, r.,

213 4 PPoGFAH AEPo TINPOTOUTPUT.TAPtil OWIU HI10,1 /STIFF/ Al 1,E10, AGJ.GJO.N.OS 150E PI NOEST VANIA PLE Wu! SETION EXTENAL N TININ=SEON011 THIS POOFAI ALULATES 7HE POWE OEFFIIENT AND NIHOWISL PINTS FOE GIFFIIENT FO A HOIZONTAL AXIS NINO ft./pupil PINTZ AND GENEATES TNT Z.ofFFIIENTS OF N OF NOTION kead.0.11,eh.of,anin,xm4x,09x or TH.. TU=9ITE SYSTEM PINT 3 EAO;D6.LN.LFL.AO PINT 65 PEAD.M.AST.SI.PITH.OTOITA PINT 66 EAD.FT,OcNo,.H OEGON STATE UNIVESITY PAINT 67 MAH FOPATI INPUT SFLA ENTE POSITION ES/c AGi Wink or MASS L EA0,ES.KG PAINT FILE AssINPENTs FoNATI INPUT I AND G 1 IA0.AE.AG INpul FO WEPT OF INDEPENDENT VAFJABLES PAHT SI OUTPUT FOP POGAN THE USE 51 FOMAT I. INPUT NIA: I TAP( I FO A LISTING OF THE P5OA4 OUTPUT IAD.ON.PNOOL.A.M4 PINT 68 TADS.SUPp VAPIAPU:S INPUT FiON TELETVPI 66 FOMAIISIPPESS INTENED. OUTPUT? IVI'l 9 5UH5EF OF iancs 9c DIHENSIONLESS HOD TO ADIUS A110 AT BLADE OOT HEADINGS AND ALULATIOH OF ONSTANTS TM SLOPI OF LIFT OI FFIIENT UVE O INFUNINT4L INTEGATION STEP ALONG THE BLADE OO HININO4 OPAL, OEFFIIENT writcli.cco LM MAXIMUM LIFT OEFFIIENT NITEI1t1i1 60PONO.F.T.OTH.9T.OTA. PITH PITH ANGLE -I11 DEGEES NFITEII;61LMiIFL.IN.ASTiA6PiA FT TWIST ANGLE AT PLDE OOT-IN LOSETS NITE11,91 AmIN.VMAX,O.S1 OBTA IfiEHENAL TWIST ANGLE HANGE -IN [AWES ONTS.PI/I88. AtO ANLI OF ATTAK FO ZEO LIFT AE..S8 AST STALL ANGLE OF ATTAK raf.i. SI ONING AfiGLE IL=IL/ T ADIAL PSITION AT WHIH TWIST ANGLE HANES SISINV1 ONO INEMENTAL HOD HANGE ATIo SI=OSISII P ADIAL POSITION AT WHIH DLA01 HOD HANES SSI.SINISIT fin N119 FAOIPL POSITION DOTA=DDIAONVI Pig! LIFT BEAK ANGLE OF ATTAK AB=ADONVT ALO=ALOONVT AST=ASTNVT OTHE vafiaple ASSIGN1IATS AHAX=P1/2. ZLOT=1. A AIIAL INDU:TION FATO OE=LFL/MOT P PWE OEFFIIENT O2=1LFL/ZLOTI 11.SINTASTIT/I.ESIN12.ASTIT T WINOWISE FOE OEFFIIENT OSE=ATAN11.41/LFLI.471 XL LAL TIP SPEED ATIO EHT=LFL/ISINIANAX-ASTII SETS LAL TWIST ANGLE. 200 NITE11.211PIH L LAL LIFT OEFFIIENT xotmlo co LAL DAG OE FFIIENT ANG=85,1ONVT 9 LOAL DIFENSIONLISS HOD TO ADIUS ATIO I=T/F PHI LOAL ANGLE OF ELATIVE vuocxtr WITH OTO PLANE =/F ALPHA LAL A1GLE OF ATTAY PH=N/ 27 ont1nuf KK=0 PI= E356 V=04/X

214 KLA1 IN=S nsa.23 oxaora DAEIFFE ThumAI-pi/o NUNATNu4 NANur IFINNN.M.TiluNt N=NUnft KL=1( A1I3A/14.715=716=7I7A7le=1I9=7110=7111=7112A7II3= AlliE1117AZII5A7I15=0. thf=zh7f=zipo=?nff=i4f0a7400=7he0 =lciws=7ff=icp0w1cfpaicff=0. ZEFOrIOAI0F.7(00AZO/A7DIA2co2AIKH=IKFAIK0iIKIA2MFFoL. 7KF0A7K0FAIK0FA7KOOAZKOZ=IK0IAZKO2AtHA/HFAmAe. ccoashopasijfa5100.v00.5cop=scofasocasc01=5oja5c0ka0. SFDPAsKIFAIK00=5vo1 AsKoJASKOKAO. scpolasm72=sf0ia1fo2aso0ias skfol=skfr2askidiask/02=sk001a5k V..52=53.551=552=5534. chcosagnsin=gf05=0,51n=gicas=gesinagocos=sosin=0. GA32.2 GwAG/t IuL1=IOL2AML3A10IIA1D2A70.13A70w1 Azow2.1oN3=0. IAO. PAO. AlAAPAATOAO. APO = -I0EIsc-7ONol 7FA cOSIP1/2.IFI iasnftiii /ix8.e KEwSOFT18,7.L666/ E4.04$ lutpra ai KL=I to.ah JUAI HEK To SE( IF N OF INTEVAL IS IAN VEN t/3 FULL 000-3/0 PULE ON Mk TIME INUI.KALS KHALFAN/2 IFTIN-2KHALFI.E.1I 'M=E IFIKL.E9.10 GG I IFISuPP.MiNyi GO TO 555 NFI1F IFIKL.E0.11 THEN NHALFAJ11 /2 NTESTAJU-2NHALF (ES NHALF=1/2 NTE5T=1-24HALF EhniF LAML/K I2AELK71( KL23A0LF2*co KI7AELE ALcULATIEN OF ELM_ TWIST ANGLE ANo LOAL HOD TO AaluS atio E=XL/X EAA(1.-PII,(1.-LI EFSHLAEA FpIGAE./PIAcSIEF PcIAIG I. AF IFIPL.GE.cI 7c09c4-DNoIL-cI/II.-I If IFB /G.NE.0.1 9cIAc/F0IG IAB/ EPAIES-KGIcil EIAIES-.251T E2 =10:75-EScI0 E3A10.5-EsI VA/V SITAAANG IFIGL.GE.IT ETA=ANG-OTAIL-TI/11.-TI BETAABETAPTHoNoT EHEIENIFXSI/I2.4IT StELwO/015/01 AHAAsINIUN/12.71,1EnlI-ALO ALULATION OF AXIAL INDUTION FATO IFILA0.1.1 THIN LAALHAAO. PHIAETA A=I. ELSE GA4.II.-2.AI NA4.A*11.-AI-A ANAIF.).ENSIIOSIIETA-ALOI-ILSINIETA-ALIII-H AAAP/I0KFxENSIOSILIA-AL010 PHIAATANIti.-Al/KLI ALPHAANI,MIA IFIA.G1.AI GO lc 50 IA7.OIAH.XcSE.OSEOLIA-AL01 I*OKF Ew KS:1:7SIdTA-ALOT-XLSINIEhTA SIT III/A. PHI=ATANIII..-41/XLI ALPHA=PHIETA 30 LA2.EISIHIALHA.AL51 IFIASILIAT.LNI GO TO 507 LAIABSIALPHAT/ALHAIcLH IFIeSIALFHoI.GT.A41 L7fALF0A/AISIALLIILIL 1NANF*I.Loc51/12..PII 02AIG.HEN7721/2-I1121 OAIIII211.AL21-H01/1E2-N IFI000N.L1..61 A=3. IFT0HA3..01 GG TO 32 AA-52*5IET102.ZII 32 INTINUE FHT=AtANIII.-Al/KLI ALPHA=PHI-7110 IFIASIALPHAI LAtA1S(ALHAI/ALHHAIGL1 IFIAGSI4LHAI.G1.:11 LAIALFHA/AnsiAL.NAII.LIL

215 ...7, um t. ; tr-. J. I Z... r &A.- T r... :..: x,..i'..- Vii %.% ti 2:e ;F.r z N i. 0. l... 0 I.41, o "a. IIcs2y..6 1 ID.4.. W I I N.7. ;-..7; LI ZZ..., n iii...'.., 0.;.,:vsL'.'t:tz.:4... :;.. -..: :."..7 =..,... Sen...4.o , 0 ;1 1 Sfe. OW 0 ley , I..,... Y O..- I I 0 ' : f. :.: ; (4) WW.00, Sr Q. PM'?.7. eag 7,7 can: (.14, t g TTF , ii;i::e;ni5if.;..; & t 04.,,.2,, VI I.... a ohl. 40 Alr.. i 4 WV. NI *I W...sO Ply....,.... w,,, II.A..., Ply, 5 OW) N P O 1[W.L. 0 4Jr...^..7.12,g0 #...QU W F7W N...N " *0.JIII.J u OMeO.O.e...M......w. 0 0.,.., , o...m.d. NI10 sr.. IA..,. in... =...,... 44FL=...,0A1;Ig f.1=4.44"..'la:=ata=44:2g:t;t10 :..ZA.':Vf.g z...m.. um.... =m..a.m....m...m,...m...x all.a.l.aya04.0.r talW.W.w474,WW.K4. u0ww =.7. l' 4 er..,., :: lir : ::;', : F. : ie. i'. :-.!. - If.m...,.J..1 1,.. =...v. x...., =... _,..,. Z..-, Oa, 0 0 I WW!...; 4.: Y".. i 6J -0 A *"... N.0 10, 4J4 OUWW, -IX. I ,. o 6) I.. I, :.1181":::.:.,D * 1,f..7g...S24/ ,h...SS ftwft...s.j.400,.l.w..m..10m,,...ft".jj J. ILI mx u x../...,z0.., I."0= _ -gx-7ezi:i- V'tT:t17.1.1EI:oot.?-t..1',... m.e IP ^.. :.,... ". 3.w..t J...2.J..,>...mw...m..cmw.r..z...m.,...=f.:vm:Itig EE;i:Legt,t4fia' W V 2 = = 701 = = J N.1.11[ I.m.= m us 7:2/2:: t 2:17:...2. faat , NW I, "... 6.= lu 0. J...IN E I, ,0 a mown. OA......OW ;..1.m. 5.:4.17..; t'vet? tr:'; :417AAgg.Lt Og;;g cau.o... o...t...,.,..., ID "::... z.an m ,.....!..!..te...=:n,.., j....., oow...., &...4..mmw. ".66.NM Ot.w W. N,. 6J-WW W1W.7M. "N.w.A...O.O.., N.N.....O.,0..",,. N0.0o.0N. t, ::L/?:: "6 LNJV N.UP..H ONWNW Ow 6UNNO OW...n. trat7ae... E:t5:-.W.t.tVi.ZALEZ=5f,

216 . OvEL.0.5Y.PAIP F0.1. FO.E. 7PYILF-TH5I/FS ZP2=IPIP 7P3=TP27P ZA./p37p F1=6.7P3-4.IP3TIP4 FFP.12.7P-12./2y47p3 FIPPY: A3p2 FP=2.Zp-IPIP FPP= P1 FpHpy.p. SUmE STATI MODE SHAPL EQUAL DYNAMI NODE SHAPE FFS=FT FISF=FIF FFSpp=FFp FrmLyTH FFHS.FFHF rfusepaffhpp wv.ffso53 Ne.EFSTOS NE.FFSPOS H.OSIMP1 SwYSINIPPI ED.PW-SISN P.OSIDLIAT SB=SINTBETAT EITANG=SIS511.-AIAALFSie NNOM.IDNMAY5NMAynTAIWPTANSI O SI=DYNA AHASS DN OM TE.T SO B 11.-SSISSI1-ITSSI01FFD DSI.DYN TAII-III1iOHD4SSISIMFFPO OSS.I.FITNOFFYLNYYOK SINE BO=ODUOS PAUL= -nnuc DSSIYOYNAAmASSON IL SISIAESSISISITTPAUL3A D SSE=-DYNA ATIASS SBSDAISSIBI2.1FFFFOT DSSIi111.11AFilliFFPEFFO,O IFI1M.E0.11 THEN FID9./8. IFIJU.E4.1.0.JU.AM.1 FFMPT.fey IFIJU.GE.41 IM=0 ELSE FEB.2./3. IFINIEST.E.0.01 FLB=4./3. IFEJU.E0.1.0.JUAM.NUT FEem1./3. ENDIF STAT 1K INTEGATION SEIS1IFEOSI 5252/FES2 SI.S3KODII SSI.SSI1FEEIDSSI SS2=SSEEFEWOSS2 SS3.SSSFEBOSSI EliEEDX AYBL/B IFIAAI.HT THEM K=KI OETION OF 430 SHAPE THAT BASED ON LENGTH OF THE BLADE XL.S14B IF -PHI NOT ON THE ADIUS OF THE OTO 1 OETION FO DEYTTNUP-HUH10k ASS.0:11P1hG.STIFFNESS MATIES 1 ELSE ENDIF FF=FFYPS JUPJUTI FrPP=FFPFAS IFEJU.LE.NNI GO TO 555 FP.Fpp/FS ELSE ENOIF ALL SUPFIT11.11M.K..E.EI.MU.AUOU:.AEU.AK.EIU.AUM.AAU IFIAL.EO.11 THEN IH.ABlim,AUOHT UOYUM.U OS.ISILS2S3I/ISSEASSLASS31 IwPDFAU NV11E OS UcFYU ot.10.4i urro.voro.nuc KEPI UAPO..UP OFOE/A AuDYBUP.BU IFTIN-APALFT.E0.01 THin AuPDAAAO IHY0 AUmPo.A,AuH ELSE DPPD=AUF0mAK: I11.1 BUMFD=AurFO=Amul ENDIF UFOYDUPFDYalU GO ONUm.IU ELSE SO ENDIF 3=OSIPLEAT SOE OFATOS OISYIALGYUISI-w2SsI PATI.SSISH-SImel DYNA.6.,1Am0PVI PAI2=5ISi PATI=SIM-SSISIEl SOLVE rco OS (cum TIP OEFATIomb PeT4PIALF.UNTO.44Sw PATS=TELF.UMISw-N4 IFIAL.LO.11 THEN PATETISWSSIN0 DhOPMAP-c AI.XALSIS3-XE3SSI PAIE.,11.1FE.S SSIS9.N.K.S.SSI.SEL3AVIII

217 POE=1 1401W0eF211FFP-UMPI.SYT 070,0=1::.*OTNAIpmASS.01.4col.P02.POScF040scP5110.POGIo FPT=OucP3.1,AFTIA4NGPO'SOSI.FFfEtrEmfUT.ST.SN.FFp.FP F2=EF.IFS5I.SWFFPFP.F.S9.ooSSIM.FF.FP-EOuc.cm.SO.SIFr F3.1.F.TN.PAhT11.rFP.FP-I.SM.SOSI.fP.FF FP4=AI1. OM.PAIIFFFP FS.I..NT.F*40,22.IFF.4-UAFO.SwlFP /V 7F.TOTNA.TANAES Om IFPIFFP21erp31FP4IIPSToF 07FF.DYNA.TAMASS.04.(9UF041TI1FOSISIFFIT 07FF=IOTFF 3.FI/V 1- U0FO SN.wFF11FF.4-UmFlSWITO F01.m.TSVSNtISSOAI"Z.T4EA,S9,0,1i.-SSIoSSIi-TI.iUT4SSI4O r02.,1ilfut SI SI-OFSSI5I4.EFSSSISAU F01,-AAI SSI*SIIP2m4SIMM.II:5I SSI.SSITI cf04.3.*two1.fi*no2.f21iffcw-unfo.sni 07F0=1-Z,,OYNATAIASS.ON.IFOPFF.cF21FOKFOS.FFP.FOITF OP=3.*NME.TIV.IH4.TAILI.TA.TAIL21.P1/8...oPATI2I/V OZOP.OIOP.FPFJ'A DIOF=3./V*I-F,UILTO.Swow.FF A11.14SI.TAIL21.F0.0Fk OI0.3.*I1mo1fF141402FI1TAILIFIWOOGI1w0I.G21.1A1121.F0D 001=IINI.F1-143i.F21TAIlltINOT.FI,NL.F:161:41.41,WFO p02.14mg:-402 G21 TAIL5-1W03*GINwocilTAIL61DF0 07OT=1.SIODIKJWO 02OI.IPAT FI-FATZ.F21 TAIL341PATIGI-PAIG214TAILS OZO1= FOOFN D7D2.1PA2TI FI-PAATE F21 TA/L4-IrATI.G1-PAF12G21TAIL6 OIO2=1.5OZO2V1iFOO STIFFNESS cofffiients MATIX NTL.Auol-W.SIoSiFl.SPO.F21 rp1=-eeiisos441ssi issisw2.10 P2=-LF EF SISPIE.SSI8Slcw$WSMI PT.[Ty%F.S3*9.l1.-SSI.SSII,EIkth1UTSSI.SI.SO rp4=-au.iu.si.si-aao.iie.uisi.si-s3psi.in-l*s811 TS=GP.FPFFPA. P6.3.*1-x.miLP-Mr2.F2-Wk2.P.NA.FI.IIIF-11NP0.SNI Pr7.1.01kN0+5wAO11.7.Nwri.F.FP! OIFT=TOTA.(A4ASS.ON01.11PPI.P2oPP31.F.F.F.P4ItP5I) og.taie-ai SSISW-SIm.001,.EFPFP or3 IA12-A ,SIoSIS3'S9.Fr.Fr 07KPP=EIXFPFOYNAtroPPP91 07KPP=107KrPOP6APP71aF prize EFeSO I9 Sm 11.-SSISSIT-SSISINIFrPFP pf2= io ASSIlI2.1.55I.SIOSWI.FF.FF PF3=kr.NAI5TOISB.S1.1SSI OSSPSIOTFFP.FP F4=-IIPL UT ISSISIS4-05ISIm)FFpFp F5=-AUEU.SI.SI-AOUIfFEWI.SPSI-SSI.SPI,O-ESOI1 PF6=-1Al2-AI.21.0WOVSI.SVPATI7,Fip.T FF7.-3 1wFfi1UFF211EIFm-Um.0sm) PFe=3,WNOTOOMPOim.0.0.0N.FFP1 OIXFFAIDYNA84ANASS.ON.34*(PrI4F2.PF3.PF4.0F5IOF61.PF7tPF81.0 ZFP1=3.1-X.1'EEP-mV.F21IFF*w-UNFaSMIA3;AuMFONNOSW IFP1=IFPI-7 NE2.T11AFPIFFW-UNFSNI Obar.IOYNIAmASS OH*04.1pFl F2oF3sPF4,,PF51FF611,7FPO IFF1=-ISP SO IO SSII2.0FrFF-SISIUBU IFF2=-1111OIESI*SI-SS10SI.Im0-0.18IT03U IFF3.11II-AIII.OV34.2M.ISSISSI-ISI* FFPFF IFF4.401A1I-A SSI.SISm.MFF.FF IFFS=EIFrrPFFIV IFFE=-3,. IINFTF11NFF21IFFw-uMFO.SNII IFF7.7..0,0*ISmFFFFP.SUMF7VNFOMFF.0 OIKFF.MA.gamASs.04,0447FF1.7FrZioIFF3.7FF4.7FFST 07XFF=17xFF.7FFE2FFTIO OPT= -NPVFOIFZTAILAG mNoAUNwSISOF0 OPI=OPI-IFTAILI.G4TAI121F0 0m TAUNSI.7F0-PATIFPF =I-8.NELP.IAILI-AAUO.I-SI.S.GIK51'8.52V.TAIL21.F0 5700i3.I0P1i ,OF OF1=-INFIFleNFVF21.TAILIFO-INFOGI.wF2'SITTAI(2.F0 OF2=MNO.SI.S13.(SW.ThEIWWFAP-OUNW-ok.W.PFP-FF.SNI.F0 OF3=NNOWPAI FFP-F UMSI4-SFFt OZKOF=3.10FIT0171'0F3IfO ZO01.11 S FI-.F21.1AtE3FTPAT6FI.SSISF21141l40.FOFO 7002*ITIWITPG1;..V.G2181AILF-IPITE.FITS$1451.G FD.F0 07K07 = OZKOI.IPAIA FI-PATVF2ITAII).(rAQT161-PArT2621.1AIL S.07x FT.OP 02KD2=IPATI.FI-FAt2 F21 TAIL4-1PAIA.G1-F4PTG211ATE6 070( XD2'GIS.F0yoF OING FUNTIONS HP2.E.FIN ISO.7 ISSI7I2.1.SSISIO.SNIF MPS'-461.(tg4uwospcm.suisspcsvictocuiere NP4.AUIIEUTSI SI-M.SSISIO.ESSISISOT HP5=1412-AIIIONOMSISPAPilFP NP6.3.*NO.III.FP-ONPO.5WI 0Z10.1OrNA IANASs INPI.min.NP3.1041O51EIk NFI=NISOSIOISSI.81"/.1AESO*0.11.-SSISSII-ILAIUcloSSI. NFEAUTIFE4UIFSI41I-WISSI.SI.oiESSI.SIS =1A11.-AI31.0WON.ISSI.SI.1.2W.Sw.cN.11SI SSI.SSITT N F4=-EFFPPFFTOS NFS.3.mNO.IFFW-UNFoSNI OZNF.lOYNATANASS04.0 INFIFF.mF21NFFFPNr41NF51ocr *INN0.TAILI-NTOTAIL21FON EFFET OF TONE SNAOu D00.-IWN0TAILI.NIO.TAIL5IF01FP OISNPIA-UO.FATII.AN.S1.1/1*(E.UISSITF-E.L.mSI.FP OZSMOmIANASSOZENPI-A11.PAI3'FPI.FO.OT/OVEl OISOF1.-UFO.PAHTI1.SN.IYL.IFI.UT.SSII.FF-E.MFF O 7SMOFAI1HASS07SNFITAIVSSI.S1AFFP1or0.0F/OVIL DISNOImIANASseAEI.PAL2-AIIFATIVPArTI-AIVSISSIWS (00.I07SP0I.APATI.PA761.FPMEL ISHOI=ITOSI-m 4tSPI2ITLSSITE.UTIE IpAT13..2.PALV$I.PAGT SN03.Al1ISTU.S9 I2AI2'9OTAITIwS412 7SN04=,IAIPP AI2'ISS165110,24573 A111"2) O 7SHOO=IAMASS 17EMOIUS1021M03.1$4041FDF0n/oVEL ZSP1.-.2 P mtf.iultail3tclatail51.f0 /v ZS2=-PI/7.*WIF001.FPPATI.F0 OSDP.IZSPAtn7P21.04k 7SP3=TIU.SI-E PATEO.FI,P3L24AIAU*SSI-LPAFTFP11 ZSP4m1A13-A SWISISI-SSISSITA2.*SSI.SIMWS.cel.F ISP5=-NA.0O3.1E1FP-UNP)S01F5 OS02-14N0101.7(ii.FP-UrPl.Smk.O OSOI.755A0motd4aSS./SP3.ZS41.FD/OVELTOFA DSDF.IN.UDFOSN-W.FF141/V.IFI.TA113.WI11151.F F2=ANISS OSI-SSI.1.FrloPAL2.PALI.IOVSSIIIA:SI.AFFII I50F3.-IAII-A13104.cOFF ZSOF4.-mN003.IFF.m-U4Fa'SwIeF0.015 OSF02.-FH01.OIS.(FF.M-W,F.P'SMS.1 OSF01.7SOF4.1-75OF2.75UF31FOO/9trEl

218 ANT0=/VolY J O.N.SII P1I9=/V.IYL.PAkT6IL cw.N.SN.c0.( $ PAPTI0=/V.IW.SSI.S.II.UDISI AT SItfySI. PAT12=5NSI391 P0113.0fSSI TIFI.III.28*S SSI*59.,Sw-Nk.4.SSI*594.1PAFT61.FFP TLF1=1IFI-Sw20129.Fr-IUN.SSI*50.04 PAct.IFIPWI..s1-4.SSI.00.1.SS =y1.141.F.uI.S3I.114.SI.00-FoS9oSI 11111=1FLF UNI w451sasnsistiotip4iti2 It112=IFT.F,AP11I N.SSI T111_3=Y1.E0,0444cafilml.SSI.SWW.N.SW.SSISO-WPAT3 TAIL4=11 PA-d NIflw.4PSii0-flSNS0 1A115=VL9tILPiu414551iWOSI IIt6=YL.V.11S9.0-1.m41.S0 vt.csi.s a141.11l i*9-nk.ssii 01,-poTI.( (.11LIT.O01NoSI.S9owSNST.S8-(3.PAP.Ti2I Nt2=N11NNNTwi F WNO.NLZT. FI.Ik.NNNKNPNTI r2=12.040t-upni G1.12.c9m11c0PNil 12.F.I OP.1011 l..wei.afp c4.nr2t0fp 11ELP2.AU0f SI4621 wp2=xe3pati2fp WrI1=-LF FF0 NT L.U01505I.S0FFP-450.SI.S13.Fr-X.w.GN.SInSoFFP VF13=-X3PAftffP-K9U0 otosi.s9 wft4wfil.wfi2.4f13 wr2.-x1551ff-auocscal mc1.olpir.un.cwcsi.59.ms.swcsi smcsifswce11mfo/v 102=INS51-IL.1.101SI91F0 /V Nct.irl.51.c.1.111,..unloSSI.S0.w4w E30PAT3IM/V NO2.0,11ITO,IIFI4U0).S.ST.IIINF.T511 NGT./VoITOISSI.9,,0*.i4"S11.1F.UOI.901NPSN, /Volvl.S5P,SOW001m ="ou1.fi.pt120f2 W101=PATI11-PAFtG2 Whly..pAFTEF1g,551.39F E.?FI-9F7 4107=PA1f.G NT9I.N =003*F1.004.F F1-102F = G2 NT003=NG1G1-NOG2 FIN() TNT INTEGAL OEFFIIENT: OF TA 000 TAN.011 SAN.11.5/PII OTDA.4.I Oill.00IA,01501s01002PPAIGSI IFIPAT.AI GG I GluAT/3IGAI A02I2.11,1;11.F1-PATI2*T21.1E18,A,13-1,014.S51, m.IPA11Gt-PATG21IPAT5PaTI-PA114PAT61 07I2.SAH ,0712I.3IS.P A0713.1PAFT F1-0ATF Pt140I 00213=-IPAT1 GI-P412G21IPA5T5SNS9$PAT4cWS M S.W., 0714.SWI.50.F1-11.F21.IET,PAT3-PA FO.D9 DZI5ISA8,IPANT,F F.04I1SNS0-PAT4.00I.F0 02I6*SAP.10450G1-00 GilIPATSPAP13-PAT4PAT61F0.DF SAN IPAN16.GINSSI SA.G21 IPA T5Sw.S91FO.O D IIII=SA IPAFT7 11-PAT9 F2IIPATI-.0FT4SSI501FGk O 719.-SAWIAFT9FI.PAT10*.21.1PAT kF D IT10.41SWIPAI7.GILPATO421.1AT5oPA13-PAT4.eAK161.F00.:N D ZI11=43A1,+10A19.011PAT10.621,1AT4.N50PAF.T5SliSOiFO*Of4 0III2=SAN.IN.S0*F1-00.FEINIE P014..:01F0. 0iIi3ssANIPATWF1.SSI S0 F21.1-E1PATZWASSI.501F0.0 ( AP.110, IAT5.SNeS0.1,0I4.1.S01.F.POh O 7/10=-SAI.IPAT4.G1.SSI.STI*G2I.IPAT50013-AT4ePAFT6I.F =SAM IPAPT, F1-PATF21E19050-PAT491FDD O P41.19tiiPAi110TF21.1!1PAT3-PIFT4fSSPS01TO.O 07I10..SWIPAIT.G1-PATP,G21.1PA PAT4 4.SalFOD SAP.IPAPI9oGt.PATIOG2IfIPAT5PAFT3-0AT(ongtvioF0.0h MASS MATIX nycl= ov.v /OVELI O INPP101U *UOwESIWPTEEFP.F1,1ANASS.AII.FPFP O 7NPPOYNAOZNPPIO OZNFF.OYNAWASS IUF0UFaFFFFI.412FF01T N.N.ISIPS E.SEI3.0.15S LITAII..2.*SI.SI*2.*IPLWI.SSI.51.(F.S..Ng *E.S.00.SI.SI 04=111153MMiSI.9.541* ,51.SI50eil..113.PATI..i. D / IANASS( F0F0.OPP 001=2.Il.ft.4..YL.NF SI 9.4 1,1IL.UIS51-4..Il.ITSP50 02=010.10*11.i1SI.91s EP11..IS UI82011!OSSIoS L*N.S0o011.-SSI.SSII O 04.2.*IL.UTNSSI SI0-2.1IWI.ESSSSO OSEIS ;TISSTISO81.42:TSIOSI S5VSI!Sw*Gw*0 006m(ISSIS9I SI ISSIW.31..?.01W,S01"2.12.*SN.wS3I.SP0 O 7400=VNA.11010$ ATMWFWOO OTNPF.DTKPANA3S10PWUFD-UF0.E.Sw.FP*E.NFF.FPIO P01.PAT11.1-UP.E.S4fF014N.II.uI*SI.OLT.SSII6FP p02.litinstocw4csl.cnswofp TWIANA55*P010.02).F0.09 ZFOT.-UF0PAPT1IMPLA.00IoS0*SI.t*SSIIFf TF02.AISIIFFp =OTN481ANASS.IF0I*U021.F0.00 AMPING OEFFIIENTS NAMIX PPT=OUPDPATIIUP0 SIIFP-AUESe.51FP e1.2.- w SSPF0FP.EILOISIF..r0 PP3.3. NT4 FP IN2 (EIFP-Um.0101.P//8.FPAleV 02PP.10Y04VIANAS5OWIPP1PPZII.PP!1f? FFI.AUF.PAFT11.UF1.iFiPSN.FPWT SIFr.44, P2=1.IPLW FF.Fr*ELF.SSIeSN.FF0.FP PF3*-(V.IPPAr.T11.,FFP.FP PF1, ,0P.PATWFP.PP 0F / F054leTr FP-Omoo.7w, 07PF=f0TIA.I111ASS.04.1PFIpF2.cOFII-PT41.FSIs3Fii POI=.T.ISSI.SI E030-$ SSI.SSITTFP T02.-TN * ESTSI.1.50).FP P03=E.TI.0.UI1231., I.SI..0.wIF0 P04*AU.IILF4UT.SI.fT-NFSSISI.94,.551..S1901 O WSI.SPPIPTIorr

219 OSD0=-IIMI.E1twO2.171.TAI13,1401GTH02.G21.TAII : S.I.IF..I.11YL.HF1-00'S9T ATIlfIIF.UclPATIII40 OSol.EIATI-AIII.ESN.m.SP.3ST-vW.SI.STII-AI2.SISHoITo OS T4IIIHI003.T41121.F0F.W OsO2.-( IIIIW1 O1'TAII21,01IFTO.I OS00I=DS044(ANASS.OSOZOS031.0P,F0.0AP/OVEL wHow.TAIL4*N40103TAIII*HT002.TAIEitwTOWTAIISIFED.E0FO 0503).114NVTAII!f4t /0FO.DIS.O OSOOF=114NOEFIA AIL61.31SOFO OSKPI=IwTOILI PATI Fo.AUM,SSIOIiHNOiAUNiSSI.SONI,FO OSKZ,(1WP2.FEWHEL.F,IFTATII*IPVGZWNELF2641TAIL510EDFO OF02=SP2.HE2SFFP.IHA.TAILIKTA.TAII5I.F0FO OSEDP=ESFPIIOSKPE OSKFI.E-W110 TIFI4WO,9UISSI.IFOFDI, OFK12.111WIFT1.612Fil.TAIETAINFT.GI.I TOO OSKF=ISETiGSKTE OSK00=1-NtsOF1AIL4.NT0FTAIL6IFE0II DFA01 =I-HhO IAII4.wNO3' IAII34AT0TAIL6.NT03.TAIL51.TOF0.O 051(001=IHN0ITAII3.4TU1TAIL5Io1SFOO OSK0.1NNOI.TA114-TOI.TAIL6101SFF0FO OFAP1=-$,NO3.I11F2-UMPOSNI,E0FO =-0N01DISYIEIFFPUHOS41,D OS (FE.w-UNE3.SWIF S6102.-kAJ1DISIFFN-UMFDos41040 O5K001.-IHNI1TAIII*WTO1'TAIL21FO.FOok OSKOOZ-IwN)1.TAILI.ODI.TAII0.0ISmF0.EIE GAVITY TENS OGNO5.ANASSG.I1UP0-ESW.M.SI-ToSSI.O 01.FP1.04/OYLL OGNSIN.-AHASSG.tiNSBO/OVII. PGFOS.ANASS.G.IAUcP0.SI-FPSSI.SOFF.FiFDPMVEL DESIN.EVASS.GE.FFF.FP.OPOVEL OFEOS.AHASS.G1100cPD-E.W.FEPIcsI1EFSSIFS.SN.FP.EPI*O oarccs=ocrcosinvel OF5IN.ANA5S.4EF'SH.S3.FE.F.001/OVIL TIGOOS=-MASS.FEIHFSBFF.FOO/OVEL OGOSIN.ANASS.G IE.SSINFP-UPOSTIFFOOK/OVEL IFIIM.E0.11 THEN FIS.9./0. IT(1.i FEI.E./A. IFII.GE =0 ELSE FE0=2.n. IFINTEST.EQ.01 F11.4./3. TE NN) FE.1./3. LNOIE STAT THE slepsob INTEGATION 712.2i2uE0.( FEn.(0713) 2I4=II44E.107I StEEB TE7TOZIGI ZI ='7I7.TEIOZI7I 7I6.7I0IFEB107I01 719=719.FE =7110IEE9'(07I =7I12.FEI*I E113.2II3'FE III4=ZI14'FEE4.1071J6) III5=III5'FIO.I =7 116.FE8.I = 2Ii7i1EO'fJ7U 7) 2111.Z119fFEWA ZUPP.IIIPWWWWPO impt=empf.fef10/hei zmpo.mosfee( irriti.p102rirri ZP00.FI.INN301 MOO=2N00'FE.E IHO.ZITO.FEOF IPP.TPP.FEEIFIDFPI IP=IPT4FE.IDUPF1 2P0 =2P0IF1 0, FP.IEP.FEIOZFPI IEFFIEEETOPIDEFI EOFIFOoTEO.107F01 7ENP,.20F4TE8.1DIOP) ZOF.ZOF0FE Z0007OOTTE TE I01=201,FE IO2.1O2.TE8102)21 TKPP.ZKPPoTEEPIOTKPPI IFF.EPF.TEVIIKFI 21(10FI6E0 OE EFT.ZIEFF.FEFIOZFFFI miz=tkozsree(orkvi ZKOP.IKOP.FE.OlicoP ItOraIKOFfEE071coF ZK0IDMI.FEEIFI ' NPFEH.FEEMOIHPI ZHE,INT.FEBIOZHEI OO.OWEBOOO SND.SNPEFE9.02SH)P SNOF.SPOF.FEOP SH00=SHDOTFEPOPSH)0 SN00.S1100.FE9075hli0 SO.SDPFES.OSEIP SOF=SOFIFEVOSHE 5E00=SOO.1ENeosm scoirscolefeosodi SEDJ=SOJOIMS001 SOKFSO.FET.0300( SKO.SOPTEEEVOST. SOF=SX0E.F F SFOG*SK000FE9'OSvO3 SOI.SKOI$FEEVOSKOl SKOJ=SKILITFEEPOSIMJ SKOK.SKOKITEPOSKOK SPOI=SPOTIFEPS.01 SP02=SP32.FE9FSO2 SFOI.SF01*FT9'SFOI SIDE.SFUITE9.OSFP2 scon.sccotavoscaol S002=SO02.FEI.ES5052 SPOI=SKPEI.FI.DS<DI SXF.02=SE2tit9PS<P02 SVEDI.SKFEI.TEAT,SFDI

220 WITE SVF02=SKF2101,5002 WITE P,T SV001.SVIIFT9S(001 WITE S002=50G2/FE9.GS<OD I11PF,IMPF.711POTHEFaMFO M00 ONOS=ONSTFE9.04OS NFIIETT.E061 GIKSIN=GNEINIFFI.ONSIN IMITE PPaPF,UP0,7FPOFF.7F0,IOP.70Fg GP3E.GPOSIFIl.DG POS GPSI4tOPEINIFTWOGPSIN HF/TEI IPP.7KPFZAFP,IFFOKOP.IOF.2A00 GEOS=GISIFLO.OUS NFTIE11,5081 GSIO.GESIN.F110GFSIN W11E HP,ZHE GOOS=GS.F11G005 NIII0II L.SOU.SKOL.S.PWIL ysti,g.1-1n.14.0gosia IMITE H0t7140.FE0107H01 N OO5.GNSINtGPOSIGPSIN.GFOSeGFSINt6005,GOSIN IIIUPP.10.1HTI GO TO 41 ETE SHOP.SMOF.SMDO.SHOO WI0E SOP.IOF,SOO.SDO IFIGE.EO.HT G E11,517) SKOPSKOF,SV IFIAAT.51 GO NITE11;3111 scoo.scrnscod 554 FITT11.16)FIoliOPHLOOTT.DAL.E.O.00IP3.TO wit1ii (P.SKFO.SA00 KK.0 NFitE OO 43 ON11hUE AMK*1 DP=0A/X AL.Xl-DY 2=X10)01 VBX I=AL/A I=Ifi IFIT.GTOTPAYI GO TO 40 GO FO 27 LF241.1.M41 IM:4 40 ONTINUE 11=H7 ta05.ans DP=ITUM-NUMIOF IFI0ANS.65.1HY1 GO ELSE MINI 6 IOW EA0.PIH IFIT.E.ONT GO T 555 GO TO 200 ALULATE THE YAW ANO TA ATE VAIATION IFIE THIN FOMAI STATEMENTS SOIL.SJOSEtSKOIlsSADEL.O. ELSE FOMAT. FOES INPUT SEQUENE.) E 73 OIN Z.Z12 2 FOMATE. WITE M,EltOIAHIN.XMAA,00.1 SAUL )/OEN 69 FOMAT.' WITE ALE,ASI.SI.IIcH,os.TA F001,01E1A.1 SJOEL= M /0EM 66 FOMAT)' WITE T.30001H*1 SJOE=-SJEL 3 FOMAT VIlt O IEOi L 4A1, L FLAT, ALPHA OIAI SAOEL=1718/ , I FOI4AII YOU ANT ANOINI PITN ANGLE? 111'1 SAo6L.I$K3EL-71217I atisini4it/oiNo 5 FOMAIIAIT SJ I,MITIO0I FOMAWINPUT PITH ANG(.1 SJ0El...fSODEL ) /0NO 0 FOHATIIEWAEOOVNAHI 04111*//.6XII.LM..EVIo'LF0.41,,4.4,40,'ASI' HOU 1.11WAO6.?Tit'ALtt /7F10.1/1 Z3 O.LIOI15)OEL701..SKFDEL*7O02 9 FOMIATITSWOPEATIONAL VAIAPEES.//54../NIN.61,0Y1AA.,73.O.. 2V E31..SVOIL XWS1./4F10.3/1 S00.,SOIISTDELSOJIS5EDELSO 15 FO3ATII5WOHYSIAL AIFOIL OATA.//1731..O..7W44'.61WOO SV ITSJOEL.S3JT5KOILSVOK 1142.*111'.0WE' 5WN., I4*/8,10.3/1 SPO.SP01.3JTOEL*SP12 60 FOHATI////25A0F3S111 OPEATING ONO/fIONS'/I SED.S0I/S3OWSF02 21 F01HA1I///253..POGAH FOES OUTPUT AT PITH t'.f7.3.. OEGMES.I SOO=S0DI1SJ6DELS FONATIM/A1.' PV..60.'0.01WPWOMOOETA*. SKP3 =SKPOIISJOELSKPD2 153s.AEPMA A..D.OX,.9*03..P4',73.14.) SKED=SKFOITSJOSKFZ 16 FOMAT F SVOO=SODITSJDZSKON 100 FONATI//153.1.THIS IM USED'oF1.3.3A'SENDS*) OETION FO HASS OIF. DUE TO TOWE 5H0004 SO 17 IS IN 407 FOMATI/ 33. SDIV.123..SJ7W.120SVO.O.123..SJPotE'/. FOM Of IHIJ-HIJS1 405 VAINN;113: sww.-shop 406 FOIA G12.01 SmnF.-SmoF 400 FOHATI3140.G/2.610 SN00.-SHOO 402 FOHATI t III =..111 SHOD = -SHOO 505 OFITE11,70011.E M.AO FOPMA11//.61WMPP..121WMPF*.027, WHF',120,.MFO..123

221 1.'900'.122..M FOMATI/7.6WPF',1111..PF'.11X..P0.,11X,=FP...11X...FF..11K, 1' 70=.11WO OF'1117.' WDO FOAP/1//.6,..KPP'.12WKPF*.127,=KFP'.17X11.K7".12X..KOP.,12Kt 1=KOF K FONAT17/.6WHP142X,'NF',12Xio'NO=1 509 F P=. e.f10,5,811,' I...F FO4411/,11.' OS s = FOMAT177.2X..GOO5',97(0.NTIN',93,.P *GPSIWOWGFOS =GF51h=1, W1 514 FO9A11811X FO0A1/.44..SHOP = '.012.6q. 59OF = ' SM00 = ' ' ' FO98117,4X.=5OP = ' SOF '.012.6,' SDO = t SDO FOMATI/.4X,'SKOF = ' ' SKDF ' ' SKDO = ' ' 15K00 = '0 FOML117,0OO = FOD8117.4X0'SP0 = '.612.6,' SFO = '.612.6,' 5000 = " FO4ATI/.48.5KPO scfn = ' ' $1110 ' FO9804/.80e. ED '.900'22".9K.F3.4;94o=42'.9WAG' 11,25.611l , 300 TIDOUT.SIONII-I/OTN NFITF TIMOUT STOP ND SUPOOTINT SUMKOMITOO.TIMATO AUOUgAAUgABU009UIA 1OD,SPUNIPEUMOU00441 ONDON 1, D00 /TUFF/ EI ,JOOTO.01 U.AU=8A0O.APUITU..AUN.AAUM=A9UM.AUD.O. ALOW.PHP AplGosli ,05 NOV 075.(ANIG0-8L =OFT K= T.I.K IDATF.I/2 ITT TF OAU.-NP*TONO DIUD...4P1091 0A00=-MP70113 O AO M0/43 Ilt18".4170tXIrt NJP.FPPINPP.FFP,FFPP) TIN ,511.FPaPPOIPP,FFP.FFPPI 0A8U.UP FFP.(0110 D A911N.NP FFP.IONI DETOU=aFFF.FFP.45 FIlF:gEST Ft9.4. trit.co.c.oc.t.c000 FE9=1. U.U6 TFEPOU=0453 AU.AUTF00.DAU=OS3 AUN.AUWEB=0009=01233 AUD.AUDIFEB.D8U AU.FAUIFE908AU0851 AAUD=AAUNTFEO ADU.AOU=F013.08OU= PU11.AOUNTFEPOAOUI BOUTFES.030U.DITS3 OU100U.05 ES ONTINUE :TKO FUNTION TON O.E.F1 OMPON lifas =EsAn*F.D WETUN FUNTION TIOIAO..0asFe011 OMMON 1o$ P4I,6 4PF,0, O.M :LOU" P./P 713=7P2=2P IP ' FF '7P31P4 FF ,77.0 FFPO P112.0p2 FP.2.*ZP...7P2 FFP s7P1 N.FF= WP.FFF=O5 NFP.FFPP=OS HOSIDP) SO DU=..0.5=WP 11P=55 10M0.TOKII.P.38,FP,FPPO5511 TONT= ,FP,FPP.OPPI 213= ,FP,FPp.wPpl

222 AEO' s Output POGAM 0EATIN6 ONDITIONS PHYSIAL AIFOIL DATA 8 B DNO T H AT DATA AEODYNAMI 0.14 GIN LFL M AST ALO OO OPEATIONAL VAIABLES OIN o SI POGAM FOES OUTPUT AT PITH OS E-02 PA DEGEES P11 A PHI BETA ALPHA L D O PO ie E ! E 21 t2 63 AE AG E E E N 68 N 40 P T HPP HPF NPO OFF 0F0 HOO E E E E E PP PF PO FP FF FO OP OF OO E E E E E-41 n454314e LOD KPP KPF KFP off OOP KOF KOo E E-02 NP HF E-14 HO E-01

223 SKOEL SJOEL SKDEL 5J110E E E KO N GN05 G8518 GPOS GPSIN GFOS GFSIN GOOS E E E E E E-05 5NOP E E-03 SMOO n E-01 SOP E-03 SOF E-02 SOO E-03 SOD E P E-01 SKOF E E E-02 5P E-02 SFO SOD WO E-02 SKFO OI E-01