AAE 556 Aeroelasticity Lecture 17 Typical section vibration Purdue Aeroelasticity 17-1 Understanding the origins of flutter 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 17-2 Purdue Aeroelasticity A peek ahead at the final result coupled equations of motion 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 17-3 Purdue Aeroelasticity Lagrange and analytical dynamics an alternative to FBD’s and Isaac Newton z(t) is the downward displacement of a small portion of the airfoil at a position x located downstream of the shear center d  ∂ (T − U )  ∂ (T − U ) = Qi  ÷− dt  ∂η&i  ∂ηi z = h + x sin θ ≅ h + xθ x kinetic energy restrict to small angle h(t) θ(t) c.g x = xt T = ∫ ( ρ )(h&+ xθ&) dx x =− xl shear center strain energy xcg 1 U = K h h + KT θ 2 17-4 Purdue Aeroelasticity Expanding the kinetic energy integral ( ) x = xt x = xt & & T = ∫ ( ρ )(h + xθ ) dx = ∫ ρ h&2 + ρ xh& θ&+ ρ x 2θ&2 dx x =− xl x =− xl m = ∫ ρ ( x )dx m is the total mass Sθ = mxθ = ∫ ρ ( x ) xdx Sq is called the static unbalance Iθ = ∫ ρ x dx = I o + mxθ2 Iq is called the airfoil mass moment of inertia – it has parts &2 & & & T = (mh + 2Sθ hθ + Iθ θ ) 17-5 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  17-6 Purdue Aeroelasticity Differential equation a trial solution Goal – frequencies and mode shapes  h  st h(t )    =  e θ (t ) θ  Substitute this into differential equations  m s   mxθ mxθ   h  st  K h  e +   Iθ  θ  0   h  st 0   e =    K h  θ  0  17-7 Purdue Aeroelasticity There is a characteristic equation here  m s   mxθ mxθ   h  st  K h  e +   Iθ  θ  0 ( s m + K h )   ( s mxθ )    h  st 0   e =    K h  θ  0  s ( mxθ )   h  st 0 e =    s ( Iθ + K h )  θ  0 17-8 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 17-9 Purdue Aeroelasticity Nondimensionalize by dividing by m and Iθ ( ) 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     17-10 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 17-11 Purdue Aeroelasticity Solutions for exponent s These are complex numbers − 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  e =e st ± iωt 17-12 Purdue Aeroelasticity solutions for s are complex numbers Iθ = mrθ Iθ = Io Io = and I o + mxθ mro = Iθ = 1+ Io mro mro + mxθ mro xθ ro 17-13 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 17-14 Purdue Aeroelasticity Natural frequencies change when the wing c.g or EA positions change  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 17-15 Purdue Aeroelasticity 1.00 Summary? 17-16 Purdue Aeroelasticity