AAE556 lecture 23 Representing motion with complex numbers and arithmetic

AAE 556 Aeroelasticity Lecture 23Representing motion with complex numbers and arithmetic a b Im plunge velocity1Purdue Aeroelasticity... i In our case the waveform is a sine wave or a c

AAE 556 Aeroelasticity Lecture 23

Representing motion with complex

numbers and arithmetic

a

b Im

plunge velocity1Purdue Aeroelasticity

Our eigenvectors are expressed

in these vectors?

i The most important information is phase difference.

Phase definition

lead or lag?

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

i In our case the waveform is a sine wave or a cosine wave

i The phase is expressed in degrees

i It is also the time between two events divided by the period (also a time), times 360

degrees.

Purdue Aeroelasticity

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Phase relationships

lead and lag

t t

t h

t

h

ω θ

θ

ω

cos )

(

sin )

4Purdue Aeroelasticity

Motion is not purely sine or cosine functions Harmonic motion is represented as a rotating

vector (a+ib) in a complex plane

Flutter phenomena depend on motion

phasing – lead and lag

i System harmonic motion does not have the same sine or cosine function

How do we represent a sine or cosine

function as a complex vector?

The Real part of the complex function is the

The phase angle for our

i

oe e h e h

t h

9Purdue Aeroelasticity

Aeroelastic vibration mode phasing is modeled with complex numbers and

vectors

Re

Real Imag al Imag

cos sin 1

2 1

We have two different types of motion, pitch and plunge

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Purdue Aeroelasticity

Jargon and derivatives

h dt

h

( ) = ( ω i ( ω t + φ ) )

o e h

i Re t

h

Real

Imaginary velocity

acceleration

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Purdue Aeroelasticity

The Real part of the complex function is the

Plunge velocity downward

Plunge acceleration

Phased motion is the culprit

for flutter

V

+work

plunge velocity

plunge velocity

2

π

φ = ±

φ negative (torsion lags displacemen t) signals

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)

θ ,lift Lift, 90 degrees

phase difference

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Stability re-visited (be careful of positive directions)

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For the future

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Two equations are necessary.

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Solution approach using complex numbers – put

complexity into the problem with a single

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Solution for complex amplitude

P

+ +

−

=

ω ω

2

2 o 2

2 o

2 o

m c

m

c i

m

P X

ω ω

ω

ω ω

1tan

o

m

c

ω ω

ω φ

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Purdue Aeroelasticity

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

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The equation of motion solution can be represented as a vector relationship that

closes

RealImaginary

velocity

accelerationP

Solution for resonant

P kx

x c x

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displacement

Im

Real damping force acceleration

displacement acceleration

−

2 o

2 o

2 o

2 o o

m c

m

c i

m

P X

ω ω

ω

ω ω

ω

c

P i

Resonance with zero damping

has a special solution