Transcript Aeroelastic effects - LSU Hurricane Engineering
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]