AAE 556 Aeroelasticity Lecture 23 Representing motion with complex numbers and arithmetic 1 Purdue Aeroelasticity

Our eigenvectors are expressed as complex numbers  What do you mean, complex amplitude?  Why are they complex?  What physical information is stored in these vectors?  The most important information is phase difference. Purdue Aeroelasticity 2

Phase definition lead or lag?  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. Purdue Aeroelasticity 3

Phase relationships lead and lag Cosine function “leads” sine function by 90 degrees - cosine reaches its max before sine does. Acceleration leads displacement by 180 degrees 4 Purdue Aeroelasticity

Motion is not purely sine or cosine functions Harmonic motion is represented as a rotating vector (a+ib) in a complex plane tt 5 Purdue Aeroelasticity

Flutter phenomena depend on motion phasing – lead and lag  System harmonic motion does not have the same sine or cosine function “phase” relationship Not only are the coefficients different, but the relative sizes (ratios) of the coefficients are different. 6 Purdue Aeroelasticity

How do we represent a sine or cosine function as a complex vector? 7 Purdue Aeroelasticity

The Real part of the complex function is the actual motion. The imaginary part is a by-product 8 Purdue Aeroelasticity

The phase angle for our example is negative real imaginary  9 Purdue Aeroelasticity

Aeroelastic vibration mode phasing is modeled with complex numbers and vectors 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. 10 Purdue Aeroelasticity

Summary At the merging point, the frequencies and the mode shapes become complex and one type of leads by 90 degrees while the other lags We have two different types of motion, pitch and plunge 11 Purdue Aeroelasticity

Jargon and derivatives 12 Purdue Aeroelasticity

The Real part of the complex function is the actual motion. The imaginary part is a by-product 13 Purdue Aeroelasticity Real Imag Plunge vector Torsion vector (and lift) Plunge velocity downward Plunge acceleration

Phased motion is the culprit for flutter  negative (torsion lags displacement) signals flutter Purdue Aeroelasticity 14

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) ,lift Lift, 90 degrees phase difference 15 Purdue Aeroelasticity

Stability re-visited (be careful of positive directions) Purdue Aeroelasticity 16

For the future Purdue Aeroelasticity 17

An example-forced response of a damped 1 DOF system 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. 18 Purdue Aeroelasticity

Solution approach using complex numbers – put complexity into the problem with a single complex equation real imaginary  Response To forcing Equilibrium 19 Purdue Aeroelasticity

Solution for complex amplitude real imaginary  20 Purdue Aeroelasticity

Solution real imaginary  response X(t) “lags” behind the harmonic force? 21 Purdue Aeroelasticity

Using complex numbers and doing complex arithmetic provides advantages  We use one complex arithmetic equation instead of two real equations to find the amplitudes of the motion 22 Purdue Aeroelasticity

The equation of motion solution can be represented as a vector relationship that closes kx 23 Purdue Aeroelasticity

Solution for resonant excitation 24 Purdue Aeroelasticity

Resonance definition velocity displacement acceleration 25 Purdue Aeroelasticity

Resonance with zero damping has a special solution Motion is not harmonic 26 Purdue Aeroelasticity