AAE 556 Aeroelasticity Lecture 23 Im Representing motion with complex numbers and arithmetic iωt f(t)=(a+ib)e plunge velocity +work b φ a Real V plunge velocity Our eigenvectors are expressed as complex numbers i  h  iωt  e θ  i i i What you mean, complex amplitude? Why are they complex? What physical information is stored in these vectors? The most important information is phase difference Phase definition lead or lag? i i i i Phase is the difference in time between two events such as the zero crossing of two waveforms, or the time between a reference and the peak of a waveform In our case the waveform is a sine wave or a cosine wave The phase is expressed in degrees It is also the time between two events divided by the period (also a time), times 360 degrees Phase relationships lead and lag Cosine function “leads” sine function by 90 degrees - cosine reaches its max before sine does h(t ) = h1 sin ωt θ (t ) = θ1 cos ωt Acceleration leads displacement by 180 degrees Motion is not purely sine or cosine functions Harmonic motion is represented as a rotating vector (a+ib) in a complex plane e iωt = cos ωt + i sin ωt f = a + ib = re ωt f (t ) = ( a + ib ) e i ( ωt ) iφ iωt = re e = re i ( ωt +φ ) iφ Flutter phenomena depend on motion phasing – lead and lag i System harmonic motion does not have the same sine or cosine function “phase” relationship h ( t ) = h1 sin ( ω t ) + h2 cos ( ω t ) θ ( t ) = θ1 sin ( ω t ) + θ cos ( ω t ) Not only are the coefficients different, but the relative sizes (ratios) of the coefficients are different How we represent a sine or cosine function as a complex vector? f ( t ) = ( a + ib ) ( eiωt ) = ( a + ib ) ( cos ω t + i sin ω t ) f ( t ) = ( a cos ω t − b sin ω t ) + i ( b cos ω t + a sin ω t ) h ( t ) = h1 sin ( ω t ) + h2 cos ( ω t ) The Real part of the complex function is the actual motion The imaginary part is a by-product h ( t ) = h1 sin ( ω t ) + h2 cos ( ω t ) f real ( t ) = ( a cos ωt − b sin ωt ) a = h2 b = − h1 h ( t ) = Re ( h ) = Re ( h2 − ih1 ) e iω t The phase angle for our example is negative h2 − ih1 = ho e = ho ( cos φ + i sin φ ) iφ h1 = −ho sin φ h2 = ho cos φ imaginary φ real − h2 tan φ = h1 h ( t ) = ho eiφ eiωt = ho e ( iωt +φ ) Aeroelastic vibration mode phasing is modeled with complex numbers and vectors h  cos ωt + i sin ωt    iωt   i ω t  b e =  e =   θ + i θ θ + i θ cos ω t + i sin ω t ( ) Imag  Imag ) ( Real   Real  θ  h  cos ωt sin ωt       i ω t b  e =  +i  ( θ Real cos ωt − θ Imag sin ωt )  ( θ Real sin ωt + θ Imag cos ωt )   θ  1) The plunge and the twist motions are not “in phase.” 2) The Real part of the complex function gives us the expression for the actual motion Jargon and derivatives dh  i ( ωt +φ ) = h = iωho e dt ( h( t ) = Re iωh ei ( ωt +φ ) o velocity ) Imaginary Real acceleration The Real part of the complex function is the actual motion The imaginary part is a by-product Imag Plunge velocity downward Plunge vector Real Plunge acceleration Torsion vector (and lift) Phased motion is the culprit for flutter Wair dh = − ∫ L dt = −qSCL π hoθ o sin φ α dt π φ =± plunge velocity +work V plunge velocity φ negative (torsion lags displacemen t) signals flutter Flutter occurs when the frequency becomes complex quasi-steady flutter mode shape allows lift - which depends on pitch (twist) to be in phase with the plunge (bending) Im θ,lift acceleration plunge velocity displacement Lift, 90 degrees phase difference Real Stability re-visited (be careful of positive directions) For the future An example-forced response of a damped DOF system mx + cx + kx = Po cos ω t Motion is harmonic The solution for x(t) is a sine-cosine combination that has a phase relative to the force Two equations are necessary x(t ) = x1 cos ω t + x2 sin ω t Solution approach using complex numbers – put complexity into the problem with a single complex equation Response ( x( t ) = Re Xe To forcing iωt ) imaginary φ real ( Po cos ωt = Re Po e iωt ) Equilibrium ( −mω X + iω cX + kX ) e iω t = Po e iω t Solution for complex amplitude Po X= − mω + iωc + k ( ) imaginary Po X= − mω + k + iωc ([ ] φ real ) Po X= ( m −ω + k + iω c m m ) Solution Po X= ( ( −ω m ) m ) + (ω c m)  + ω o2 − iω c  −ω +ω  o   −ω c  m  φ = tan −1  − ω + ωo2    k ω = m o imaginary φ real response X(t) “lags” behind the harmonic force? Using complex numbers and doing complex arithmetic provides advantages i We use one complex arithmetic equation instead of two real equations to find the amplitudes of the motion The equation of motion solution can be represented as a vector relationship that closes mx + cx + kx = Po cos ω t Imaginary P Real acceleration kx x velocity Solution for resonant excitation mx + cx + kx = Po cos ωt Po X= m ( −ω + ω − iω c 2 o ( m ) )  −ω + ω 2 + ω c  ( ÷ o ) m   Resonance definition Po X=    ( ( −ω m o ) m ) + (ω c m)  + ω o2 − iω c 2 − ωo + ωo Im acceleration X = −i Po ωc Im velocity P velocity acceleration displacement Real damping force Real P X displacement Resonance with zero damping has a special solution X = −i Po ωc x( t ) = X ot sin ωt Motion is not harmonic