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
WB2414-Mechatronic System Design 2013-2014 2DelftUniversity ofTechnology
Contents
•Signals
• Periodic, Harmonic signals
• Fourier analysis
•Dynamic plots
• Laplace & Fourier transform
• Time domain
• Frequency domain
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WB2414-Mechatronic System Design 2013-2014 3DelftUniversity ofTechnology
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
WB2414-Mechatronic System Design 2013-2014 4DelftUniversity ofTechnology
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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WB2414-Mechatronic System Design 2013-2014 5DelftUniversity ofTechnology
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
WB2414-Mechatronic System Design 2013-2014 6DelftUniversity ofTechnology
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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WB2414-Mechatronic System Design 2013-2014 7DelftUniversity ofTechnology
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
WB2414-Mechatronic System Design 2013-2014 8DelftUniversity ofTechnology
Triangle waveform
2 2 2
8 1 1ˆ( ) cos( ) cos(3 ) cos(5 ) .....3 5
f t t t t
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WB2414-Mechatronic System Design 2013-2014 9DelftUniversity ofTechnology
Square wave form
4 1 1ˆ( ) sin( ) sin(3 ) sin(5 ) .....3 5
f t t t t
WB2414-Mechatronic System Design 2013-2014 10DelftUniversity ofTechnology
Sawtooth waveform
2 1 1ˆ( ) sin( ) sin(2 ) sin(3 ) .....2 3
f t t t t
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WB2414-Mechatronic System Design 2013-2014 11DelftUniversity ofTechnology
Random signals
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Fast Fourier Transform
Mind the window!
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Window functions
WB2414-Mechatronic System Design 2013-2014 14DelftUniversity ofTechnology
Contents
•Signals
• Periodic, Harmonic signals
• Fourier analysis
•Dynamic plots
• Laplace & Fourier transform
• Time domain
• Frequency domain
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WB2414-Mechatronic System Design 2013-2014 15DelftUniversity ofTechnology
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
WB2414-Mechatronic System Design 2013-2014 16DelftUniversity ofTechnology
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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WB2414-Mechatronic System Design 2013-2014 17DelftUniversity ofTechnology
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)
WB2414-Mechatronic System Design 2013-2014 18DelftUniversity ofTechnology
Test and analysis stimuli of dynamic systems
•Frequency domain related responses
• Step and impulse
• Frequency sweep
• Noise
• Multi‐sines
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WB2414-Mechatronic System Design 2013-2014 19DelftUniversity ofTechnology
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
WB2414-Mechatronic System Design 2013-2014 20DelftUniversity ofTechnology
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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WB2414-Mechatronic System Design 2013-2014 21DelftUniversity ofTechnology
Impulse time response
Fourier analysis of impulse is constant amplitude density over all frequencies (linear scale).
WB2414-Mechatronic System Design 2013-2014 22DelftUniversity ofTechnology
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!
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