AAE 556 Aeroelasticity Lecture 16 Dynamics and Vibrations Purdue Aeroelasticity 16-1 Aeroelasticity – the challenge the shapes to come New systems, vehicles, shapes, environments & challenges integrating aerodynamics, structures, controls and actuation to create unbeatable systems In the future • Future aircraft will be • • • • • Multi-purpose and robust Automated and robotic Semi-to-fully autonomous Able to change state Aircraft will be able to • Cope with environmental change, both manmade and due to nature • • Self-repair Use nontraditional propulsion …but the challenges loom large • Conceptual design of highly integrated systems with distributed power and actuation • Redefining aeroelastic stability concepts for structures that lock and unlock, move, stay fixed and then move again • • • Calculating loads and transient response Developing test plans for multi-dimensional structural configurations Assigning risk Low speed and high-speed flutter Gloster Grebe Handley Page O/400 X-15 Electra 16-5 Purdue Aeroelasticity Unusual flutter has become usual 16-6 Purdue Aeroelasticity Flutter is a dynamic instability it involves energy extraction 16-7 Purdue Aeroelasticity Understanding the origin - Typical section equations of motion - DOF x Plunge displacement h is positive downward & measured at the restrict to small angle h(t) θ(t) c.g shear center shear center xcg xθ h ( t ) = plunge freedom (bending ) θ ( t ) = pitch freedom ( twist ) measured at the shear center from static equilibrium position 16-8 Purdue Aeroelasticity Coupled Equations (EOM) are dynamically coupled but elastically uncoupled  m  mx  θ & mxθ   h&  Kh +  &   Iθ  θ& 0  h  0    =    KT  θ  0  x mg = weight restrict to small angle h(t) θ(t) c.g shear center xcg xθ xθ is called static unbalance and is the source of dynamic coupling 16-9 Purdue Aeroelasticity Prove it! Lagrange steps up to the plate z(t) is the downward displacement of a small potion of the airfoil at a position x located aft of the shear center z = h + x sin θ ≅ h + xθ x = xt T = ∫ ( ρ )(h&+ xθ&) dx x =− xl kinetic energy 1 U = K h h + KT θ 2 strain energy LaGrange's equations promise d  ∂ (T − U )  ∂ (T − U ) = Qi  ÷− dt  ∂η&i  ∂ηi 16-10 Purdue Aeroelasticity Kinetic energy integral simplifies x = xt T = ∫ ( ρ )(h&+ xθ&) dx x =− xl &2 & & & T = (mh + 2Sθ hθ + Iθ θ ) Sq is called the static unbalance m is the total mass m = ∫ ρ ( x )dx Sθ = mxθ = ∫ ρ ( x ) xdx Iq is called the airfoil mass moment of inertia – has parts Iθ = ∫ ρ ( x ) x dx = I o + mxθ2 16-11 Purdue Aeroelasticity Equations of motion for the unforced system (Qi = 0) ∂T = mh + mxθ θ ∂h ∂T  + I θ = mx h θ θ  ∂θ ∂U = Khh ∂h ∂U = KT θ ∂θ EOM in matrix form, as promised  m  mx  θ & mxθ   h&  Kh +  &   Iθ  θ& 0  h  0    =    KT  θ  0  16-12 Purdue Aeroelasticity Diff Eqn trial solution -separable Goal – frequencies and mode shapes  h  st h(t )    =  e θ (t ) θ  substitute into coupled differential eqns  m s   mxθ mxθ   h  st  K h  e +   Iθ  θ  0   h  st 0   e =    K h  θ  0  16-13 Purdue Aeroelasticity The eventual result Goal – frequencies and mode shapes  m s   mxθ ωθ2 mxθ   h  st  K h  e +   Iθ  θ  0 K = T Iθ ω h2   h  st 0   e =    K h  θ  0  Kh = m  Iθ  Iθ  2  Iθ  2 2  s = −  ÷( ω h + ωθ ) ±  ω h + ωθ )  −  ÷ω h ωθ (  Io   Io   Io  16-14 Purdue Aeroelasticity Natural frequencies change value when the c.g position changes  Iθ  Iθ  2  Iθ  2 2  ω = +  ÷( ω h + ωθ ) − or +  ω h + ωθ )  −  ÷ω h ωθ (  Io   Io   Io    = + a  x restrict to small angle h(t) θ(t) shear center xcg c.g 40 35 natural frequencies (rad./sec.) x Iθ = +  θ Io  ro Natural frequencies vs c.g offset 30 torsion frequency 25 20 fundamental (plunge) frequency 15 10 0.00 0.25 0.50 0.75 c.g offset c.g offset in semi-chords 16-15 Purdue Aeroelasticity 1.00 But – we have a derivation first collect terms into a single 2x2 matrix ( s m + K h ) 0 ( s mxθ )   h  st e =       2 ( s Iθ + KT ) θ   ( s mxθ ) 0 divide by exponential time term Matrix equations for free vibration ( s m + K h ) ( s mxθ )   h  0     =   2 ( s Iθ + KT )  θ  0   ( s mxθ ) 2 16-16 Purdue Aeroelasticity The time dependence term is factored out 2 (s m + K h ) ( s mxθ ) ( s mxθ ) ( s Iθ + KT ) Determinant of dynamic system matrix set determinant to zero (characteristic equation) ( s m+K ) ( s I 2 h θ + KT ) − ( s mxθ ) ( s mxθ ) = 2 16-17 Purdue Aeroelasticity nondimensionalize ( ) 2  mx  K h  KT   − s  s θ  = s +  s +   m  Iθ  I  θ   Define uncoupled frequency parameters ω h2 (s Kh = m + ω h2 )( ωθ2 KT = Iθ ) ( ) 2  mx s + ωθ2 − s  s θ  =   I θ     mxθ2  2  2 =  s 1 − ÷+ s ( ω h + ωθ ) + ω h ωθ ÷ ÷ I θ     16-18 Purdue Aeroelasticity Solution for natural frequencies ( )   Io  2  2  s   + s ω h + ωθ + ω h ωθ  =  I     θ  (as ) + bs + c = − b ± b − 4ac s = 2a 16-19 Purdue Aeroelasticity Solution for exponent s − s2 = ( ω h2 + ωθ2 )± ( ω h2 ) 2 + ωθ  Io  2 − 4 ω h ωθ  Iθ   Io  2   Iθ   Iθ  Iθ  2  Iθ  2 2  s = −  ÷( ω h + ωθ ) ±  ω h + ωθ )  −  ÷ω h ωθ (  Io   Io   Io  16-20 Purdue Aeroelasticity solutions for w are complex numbers Iθ = mrθ Iθ = Io Io = and I o + mxθ mro = Iθ = 1+ Io mro mro + mxθ mro xθ ro 16-21 Purdue Aeroelasticity Example configuration 2b=c xθ = 0.10c = 0.20b and xθ = 0.40 ro  xθ +   ro   = 1.16  ro = 0.25c = 0.5b and xθ = aro  xθ   ro   = 0.16  Iθ = 1+ a Io New terms – the radius of gyration 16-22 Purdue Aeroelasticity Natural frequencies change value when the c.g position changes  Iθ  Iθ  2  Iθ  2 2  ω = +  ÷( ω h + ωθ ) − or +  ω h + ωθ )  −  ÷ω h ωθ (  Io   Io   Io    = + a  x restrict to small angle h(t) θ(t) shear center xcg c.g 40 35 natural frequencies (rad./sec.) x Iθ = +  θ Io  ro Natural frequencies vs c.g offset 30 torsion frequency 25 20 fundamental (plunge) frequency 15 10 0.00 0.25 0.50 0.75 c.g offset c.g offset in semi-chords 16-23 Purdue Aeroelasticity 1.00