Mechatronic System Design: 2. signals and dynamic

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WB2414-Mechatronic System Design 2013-2014 1DelftUniversity ofTechnology

Mechatronic system design

Mechatronic system design wb2414‐2013/2014

Course part 2

Prof.ir. R.H.Munnig SchmidtMechatronic System Design

Signals and dynamic plots


Contents

•Signals

• Periodic, Harmonic signals

• Fourier analysis

•Dynamic plots

• Laplace & Fourier transform

• Time domain

• Frequency domain

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Signals

•A Signal is a function that conveys information about the behavior or attributes of some phenomenon.

•DC (Direct current) signal remains constant in a measured time period. It equals the average value of any signal.

•AC (Alternating current) has periodically alternating negative and positive values at a certain frequency (f )It’s average value is zero.

•Any signal can be described as a combination of different AC and DC signals, varying over time or space.

amount of events per time (Hz)

1Period: (s)

f

Tf


Special periodic function: Harmonic oscillations (sine-cosine)Angular frequency

A

ωt

A cos(ωt)

A sin(ωt)

x

y

t = 02 f

cos and sin have 90 degrees different phase angle

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Harmonic movement

22

2

( ) sin( )

d ( )( ) cos( )

d

d ( )( ) sin( )

d

x t A t

x tx t A t

t

x tx t A t

t

( ) ( )x x t

t 1A Derivative gives phase lead


Phase relation by means of complex number

A

Re

Im

( )Z

cos

sinj

( ) (cos sin )

( ) (cos sin )

j

j t

Z A j Ae

Z t A t j t Ae

Eulers formula

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Fourier decomposition of signals

•Periodic signals can be seen as a combination of harmonically related sinusoidal functions

• Fourier decomposition is based on the mathematicalfact that the multiplication of one sinusoidal function with another can only give a non‐zero average valueover time when they have an equal frequency and phase

1 2 1 21 2

cos( ) cos( )sin( )sin( )

2

t t t tt t


Triangle waveform

2 2 2

8 1 1ˆ( ) cos( ) cos(3 ) cos(5 ) .....3 5

f t t t t

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Square wave form

4 1 1ˆ( ) sin( ) sin(3 ) sin(5 ) .....3 5

f t t t t


Sawtooth waveform

2 1 1ˆ( ) sin( ) sin(2 ) sin(3 ) .....2 3

f t t t t

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Random signals


Fast Fourier Transform

Mind the window!

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Window functions


Contents

•Signals

• Periodic, Harmonic signals

• Fourier analysis

•Dynamic plots

• Laplace & Fourier transform

• Time domain

• Frequency domain

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Laplace transform, first step from the time into the frequency domain via the transfer function

0

0

d

i

d

( ) { ( )} ( )

( )F ( ) ( ) (0)

( )F ( ) ( )

d

d

st

t

f s f t e f t dt s j

x ts sx s x

t

x ss x t t

s


Laplace domain = complex plane with poles and zeros. Fourier transform gives frequency response

0

d

i

d ( )F ( ) ( )

d

( ) ( )F ( ) ( )d

t

x tj x

t

x xx t t j

j

s j

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With a phase change depending on the location of s

d

d

i

0

0

i

( )F ( ) ( ) (0)

d ( )F ( ) ( )

d

( )F ( ) ( )

d

d

d

( ) ( )F ( ) ( )d

t

t

x ts sx s x

t

x tj x

t

x ss x t t

s

x xx t t j

j

Multiply with s 90ᵒ phase lead, divide by s 90ᵒ phase lag

Derivative gives phase lead (+j), Integral gives phase lag (‐j)


Test and analysis stimuli of dynamic systems

•Frequency domain related responses

• Step and impulse

• Frequency sweep

• Noise

• Multi‐sines

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Stimuli in the frequency domain

• Step and impulse

• Need for limitation of force

• Frequency sweep

• Not too fast to enable stabilising of resonances (energy)

•Noise

• White noise, constant “Power Spectral Density”

•Multi‐sines

• Tuneable to frequency area of interest


Step response

Fourier analysis of a unit step gives wide spectrum with decreasing magnitude at higher frequenciesIt is a squarewave of almost 0Hz

4 1 1ˆ( ) sin( ) sin(3 ) sin(5 ) .....3 5

f t t t t

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Impulse time response

Fourier analysis of impulse is constant amplitude density over all frequencies (linear scale).


Bode plot

•Magnitude and phase

•Output versus input

•Output and input can be different (Force vs displacement)

•Double logarithmic scale loglog 1/x = straight line ‐1

•Horizontal frequency axis in Hz or rad/s

•Vertical axis magnitude abs or in dB

•Vertical axis phase in degrees

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Nyquist plot, the most important of all!

And don’t show the negativefrequencies!

Mechatronic System Design: 2. signals and dynamic

Technology

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