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

Aeroelastic effects

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

Aeroelastic effects

• Example : Tacoma Narrows Bridge WA - 1940 • Example : ‘Galloping’ of iced-up transmission lines

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

Aeroelastic effects • Aerodynamic damping (along wind) : Drag force (per unit length) = D  C D 1 2 ρ a b( U   ) 2  C D 1 2 ρ a b U 2 (1  2 U )  C D 1 2 ρ a b U 2  C D ρ a b U  for small / U aerodynamic damping term transfer to left hand side of equation of motion : total damping term : m  x   c   kx  D(t) c   C D ρ a b U x  along-wind aerodynamic damping is

positive

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 : Δα   z / U

Aeroelastic effects • Galloping : Aerodynamic force per unit length in z direction (body axes) : F z = D sin  + L cos  = 1 2 ρ a U 2 b(C D sin α  C L cos  ) (Lecture 8) For  dF z dα  1 2 ρ a U 2 b(C D cosα  dC D dα sin α  C L sin α  dC L dα cos  ) = 0 : dF z dα  1 2 ρ a U 2 b(C D  dC L ) dα ΔF z  1 2 ρ a U 2 b(C D  dC L )Δ  dα

Aeroelastic effects • Galloping : Substituting, Δα   / U  F z  1 2 ρ a U 2 b(C D  dC L )(  dα U )   1 2 ρ a U b(C D  dC L )  dα (C D  dC dα L   F z is positive - acts in same direction as z 

negative

aerodynamic damping when transposed to left-hand side

Aeroelastic effects • Galloping : (C D  dC L )  0 dα den Hartog’s Criterion critical wind speed for galloping,  U crit , occurs when

total

damping is zero c   1 2  a U crit b(C D  dC L )  dα  0 U crit  2c ρ a b(C D  dC L ) dα U crit  8π  ρ a b(C D mn  1 dC L ) dα Since c = 2  (mk)=4  mn 1 (Figure 5.5 in book) m = mass per unit length n 1 = first mode natural frequency

Aeroelastic effects • 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 • Flutter : Consider a two dimensional body rotating with angular velocity Vertical velocity at leading edge :  d/2 Apparent change in angle of attack :   d/2 U Can generate a cross-wind force and a moment Aerodynamic instabilities involving rotation are called ‘flutter’

Aeroelastic effects • Flutter : General equations of motion for body free to rotate and translate : z   z   2η z ω z  ω 2 z z  F z (t) m  H 1   H 2   H 3 θ  θ   2η θ ω θ  ω 2 θ θ  M(t)  A 1 I  A 2  A 3 θ per unit mass per unit mass moment of inertia Flutter derivatives

Aeroelastic effects • Flutter : Types of instabilities :

Name

Galloping ‘Stall’ flutter ‘Classical’ flutter

Conditions

H 1 >0 A 2 >0 H 2 >0, A 1 >0

Type of motion

translational rotational coupled

Type of section

Square section Rectangle, H section Flat plate, airfoil

Aeroelastic effects • Flutter : 3 2 1 A 1 * 0 0 2 4 6 2 1 8 10 U/nd 12 0.4

A 2 * 0.3

0.2

0.1

0 -0.1

-0.2

unstable stable 2 1 0 0 -2 2 -4 -6 H 1 * 8 0 -2 6 4 2 H 2 * 4 6 8 10 12 2 1 2 A stable 1 1 2

Aeroelastic effects • 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

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

End of Lecture 14

John Holmes 225-405-3789 [email protected]