List of figures

1.1 Aeroelastic (Collar’s) triangle of forces 2

1.2 Principle of airfoil torsional divergence 5

1.3 Principle of control surface reversion 6

1.4 Harmonic motion of airfoil with single DOF – torsion 8

1.5 Harmonic motion of airfoil with single DOF – bending 8

1.6 Harmonic motion of airfoil with 2 DOFs (bending / torsion – ‘in-phase’) 9

1.7 Harmonic motion of airfoil with 2 DOFs (bending / torsion – ‘out-of-phase’ – shift π/2) 9

1.8 Harmonic motion of airfoil with 2 DOFs (bending / aileron – ‘in-phase’) 10

1.9 Harmonic motion of airfoil with 2 DOFs (bending / aileron – ‘out-of-phase’ – shift π/2) 10

2.1 Gyroscopic system with propeller 15

2.2 Independent mode shapes ((a) pitch, (b) yaw) 15

2.3 Backward (a) and forward (b) whirl mode 16

2.4 Stable (a) and unstable (b) state of gyroscopic vibrations for backward flutter mode 16

2.5 Aerodynamic forces due to pitching deflection (angle Θ) 17

2.6 Aerodynamic forces due to the yawing velocity ˙ (movement around vertical axis) 18

2.7 Aerodynamic forces due to pitching angular velocity ˙ (movement around lateral axis) 19

2.8 Kinematical scheme of the gyroscopic system 20

2.9 Influence of the propeller advance ratio (V∞/(ΩR)) on the stability of an undamped gyroscopic system 24

2.10 Influences of structural damping and propeller – pivot point distance on whirl flutter stability 25

2.11 Static divergence of the gyroscopic system 26

2.12 Whirl flutter boundaries (Ω = const.) 27

2.13 Whirl flutter boundaries (KΘ = const.; KΨ = const.) 28

2.14 Whirl flutter boundaries (Jo = const.) 28

2.15 Influence of inflow angle to whirl flutter boundaries 29

3.1 Lockheed L-188 C Electra II aircraft 34

3.2 Beechcraft 1900C aircraft 36

4.1 NACA wind tunnel model of wing and nacelle 43

4.2 Propeller wind tunnel model 43

4.3 Propeller simple wind tunnel model 44

4.4 Experimental whirl flutter boundaries 45

4.5 NASA propeller wind tunnel model 45

4.6 Comparison of experimental aerodynamic derivatives with theory 46

4.7 Effect of thrust on whirl flutter stability 47

4.8 Hinged blades propeller wind tunnel model 48

4.9 Effect of blades flapping on whirl flutter stability 48

4.10 NAL hinged blade propeller model 49

4.11 Effect of blades flapping on whirl flutter stability 50

4.12 Semispan wing / engine component model (NASA Langley) 51

4.13 L-188 C Electra II aeroelastic model WT measurement (NASA Langley) 53

4.14 L-188 C Electra II aeroelastic model WT measurement (NASA Langley) 53

4.15 Effect of stiffness reduction on the whirl flutter boundary for the starboard outboard engine, (KΘ/KΨ) = 1.0; g = 0.014 54

4.16 Effect of stiffness and damping reduction (KΘ – 67%; g – 35% of nominal) on the whirl flutter boundary for various starboard engines, (KΘ/KΨ) = 1.0 or 1.5 55

4.17 Effect of structural damping on the whirl flutter boundary, KΘ = 3.6e3 [in-lb/rad]; (KΘ/KΨ) = 1.8 55

4.18 Effect of starboard inner propeller overspeed on the whirl flutter boundary (others at nominal rpm) 56

4.19 Flapping blades prop-rotor wind tunnel model 58

4.20 Flapping blades prop-rotor – results summary 58

4.21 Flapping blades prop-rotor results – influence of blade flapping hinge 59

4.22 Flapping blades prop-rotor results – influence of stiffness ratio 60

4.23 Simple table-top prop-rotor model 60

4.24 Prop-rotor model 61

4.25 Four-blade prop-rotor model 62

4.26 Three-blade prop-rotor model 63

4.27 WRATS tilt-rotor aircraft component model 64

4.28 WRATS measurement results – influence of WT medium 65

4.29 L-610 commuter aircraft 66

4.30 L-610 complete aeroelastic model at TsAGI T-104 wind tunnel test section 67

4.31 L-610 aeroelastic model starboard wing/engine component 67

4.32 L-610 aeroelastic model aileron actuation 68

4.33 Aeroelastic model engine nonlinear attachment (1 – engine mass; 2 – nonlinear attachment; 3 – pitch 1st attachment 4 – yaw attachment; 5 – pitch 2nd and 3rd attachments; 6 – hinge) 69

4.34 W-WING demonstrator nacelle design drawing (1 – motor and gearbox; 2 – wing spar; 3 – pitch attachment; 4 – yaw attachment; 5 – mass balancing weight) 70

4.35 W-WING demonstrator, uncoated nacelle 70

4.36 W-WING demonstrator, uncoated nacelle integrated into wing structure 71

4.37 W-WING demonstrator wing and coated nacelle 71

4.38 W-WING demonstrator wing – strain gauges in half-span section 72

4.39 Tool for the propeller blade adjustment 73

4.40 W-WING demonstrator FE model (structural) 75

4.41 W-WING demonstrator FE model (aerodynamic) 75

4.42 Example of W-WING analytical results – required stiffness for neutral stability (Ω = 2000 rpm, light blades), parameter: flow velocity 76

4.43 Example of W-WING analytical results – required stiffness for neutral stability (V = 20 m·s−1, Ω = 2000 rpm), parameter: IP (light/heavy blades) 76

4.44 Example of W-WING analytical results – required stiffness for neutral stability (V = 20 m·s−1, heavy blades), parameter: propeller revolutions 77

5.1 Effective quasi-steady angles 84

5.2 Whirl flutter critical dimensionless frequency and damping (G = (γΨ/γΘ) = 1.0 and γ2 = (KΨ/KΨ) = 1.4) 87

5.3 Influence of the propeller hub distance on the backward whirl mode critical damping (G = (γΨ/γΘ) = 1.0 and γ2 = (KΨ/KΘ) = 1.0) 89

5.4 Stability boundaries – assessment of the stiffness asymmetry via KRMS (γΨ = γΘ = 0.03; Ω = 1020 rpm)...