1. Aeroelastic effects Wind loading and structural response Lecture 14 Dr. J.D. Holmes

2. Aeroelastic effects Very flexible dynamically wind-sensitive structures Motion of the structure generates aerodynamic forces Positive aerodynamic damping : reduces vibrations - steel lattice towers if forces act in direction to increase the motion : aerodynamic instability

3. Aeroelastic effects Example : Tacoma Narrows Bridge WA - 1940 Example : ‘Galloping’ of iced-up transmission lines

4. Aeroelastic effects Aerodynamic damping (along wind) : Relative velocity of air with respect to body = Consider a body moving with velocity in a flow of speed  U

5. Aeroelastic effects Aerodynamic damping (along wind) : Drag force (per unit length) = for small aerodynamic damping term total damping term : along-wind aerodynamic damping is positive transfer to left hand side of equation of motion :

6. Aeroelastic effects Galloping : galloping is a form of aerodynamic instability caused by negative aerodynamic damping in the cross wind direction Motion of body in z direction will generate an apparent reduction in angle of attack,  From vector diagram :

7. Aeroelastic effects Galloping : Aerodynamic force per unit length in z direction (body axes) : F z = D sin  + L cos  = (Lecture 8) For  = 0 :

8. Aeroelastic effects Galloping : Substituting, For,  F z is positive - acts in same direction as negative aerodynamic damping when transposed to left-hand side

9. Aeroelastic effects Galloping : den Hartog’s Criterion critical wind speed for galloping,  U crit, occurs when total damping is zero Since c = 2  (mk)=4  mn 1 (Figure 5.5 in book) m = mass per unit length n 1 = first mode natural frequency

10. Galloping : Cross sections prone to galloping : Square section (zero angle of attack) D-shaped cross section iced-up transmission line or guy cable Aeroelastic effects

11. Flutter : Consider a two dimensional body rotating with angular velocity Vertical velocity at leading edge : Apparent change in angle of attack : Can generate a cross-wind force and a moment Aerodynamic instabilities involving rotation are called ‘flutter’

12. Aeroelastic effects Flutter : General equations of motion for body free to rotate and translate : per unit mass per unit mass moment of inertia Flutter derivatives

13. Aeroelastic effects Flutter : Types of instabilities :

14. Flutter : 1 2 -0.1 -0.2 A2*A2* 0.1 0 1 2 6 4 2 0 8 1 2 H2*H2* -2 A 0.4 0.3 0.2 unstable stable Flutter derivatives for two bridge deck sections : A1*A1* 3 2 1 0 0 2 46 8 10 12 1 2 -6 -4 -2 0 0 H1*H1* 2 4 6 8 10 12 1 2 U/nd Aeroelastic effects

15. Flutter : Determination of critical flutter speed for long-span bridges: Empirical formula (e.g. Selberg) Experimental determination (wind-tunnel model) Theoretical analysis using flutter derivatives obtained experimentally

16. Aeroelastic effects Lock - in : Motion-induced forces during vibration caused by vortex shedding Frequency ‘locks-in’ to frequency of vibration Strength of forces and correlation length increased

17. End of Lecture 14 John Holmes 225-405-3789 JHolmes@lsu.edu