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# Mechatronic System Design: 2. signals and dynamic

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• 1. 1 WB2414-Mechatronic System Design 2013-2014 1Delft University of Technology Mechatronic system design Mechatronic system design wb2414‐2013/2014 Course part 2 Prof.ir. R.H.Munnig Schmidt Mechatronic System Design Signals and dynamic plots WB2414-Mechatronic System Design 2013-2014 2Delft University of Technology Contents •Signals • Periodic, Harmonic signals • Fourier analysis •Dynamic plots • Laplace & Fourier transform • Time domain • Frequency domain
• 2. 2 WB2414-Mechatronic System Design 2013-2014 3Delft University of Technology 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) 1 Period: (s) f T f   WB2414-Mechatronic System Design 2013-2014 4Delft University of Technology Special periodic function: Harmonic oscillations (sine-cosine) Angular frequency A ωt A cos(ωt) A sin(ωt) x y t = 0 2 f  cos and sin have 90 degrees different phase angle
• 3. 3 WB2414-Mechatronic System Design 2013-2014 5Delft University of Technology Harmonic movement 2 2 2 ( ) sin( ) d ( ) ( ) cos( ) d d ( ) ( ) sin( ) d x t A t x t x t A t t x t x t A t t              ( ) ( )x x t  t 1A   Derivative gives phase lead WB2414-Mechatronic System Design 2013-2014 6Delft University of Technology 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
• 4. 4 WB2414-Mechatronic System Design 2013-2014 7Delft University of Technology Fourier decomposition of signals •Periodic signals can be seen as a combination of  harmonically related sinusoidal functions •Fourier decomposition is based on the mathematical fact that the multiplication of one sinusoidal function  with another can only give a non‐zero average value over time when they have an equal frequency and  phase  1 2 1 2 1 2 cos( ) cos( ) sin( )sin( ) 2 t t t t t t            WB2414-Mechatronic System Design 2013-2014 8Delft University of Technology Triangle waveform 2 2 2 8 1 1ˆ( ) cos( ) cos(3 ) cos(5 ) ..... 3 5 f t t t t            
• 5. 5 WB2414-Mechatronic System Design 2013-2014 9Delft University of Technology Square wave form 4 1 1ˆ( ) sin( ) sin(3 ) sin(5 ) ..... 3 5 f t t t t             WB2414-Mechatronic System Design 2013-2014 10Delft University of Technology Sawtooth waveform 2 1 1ˆ( ) sin( ) sin(2 ) sin(3 ) ..... 2 3 f t t t t            
• 6. 6 WB2414-Mechatronic System Design 2013-2014 11Delft University of Technology Random signals WB2414-Mechatronic System Design 2013-2014 12Delft University of Technology Fast Fourier Transform Mind the window!
• 7. 7 WB2414-Mechatronic System Design 2013-2014 13Delft University of Technology Window functions WB2414-Mechatronic System Design 2013-2014 14Delft University of Technology Contents •Signals • Periodic, Harmonic signals • Fourier analysis •Dynamic plots • Laplace & Fourier transform • Time domain • Frequency domain
• 8. 8 WB2414-Mechatronic System Design 2013-2014 15Delft University of Technology 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 t s sx s x t x s s x t t s                              WB2414-Mechatronic System Design 2013-2014 16Delft University of Technology Laplace domain = complex plane with poles and zeros. Fourier transform gives frequency response 0 d i d ( ) F ( ) ( ) d ( ) ( ) F ( ) ( )d t x t j x t x x x t t j j                             s j  
• 9. 9 WB2414-Mechatronic System Design 2013-2014 17Delft University of Technology 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 t s sx s x t x t j x t x s s x t t s x x x 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 18Delft University of Technology Test and analysis stimuli of dynamic systems •Frequency domain related responses • Step and impulse • Frequency sweep • Noise • Multi‐sines
• 10. 10 WB2414-Mechatronic System Design 2013-2014 19Delft University of Technology 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 20Delft University of Technology Step response Fourier analysis of a unit step gives wide spectrum with decreasing magnitude at higher frequencies It is a squarewave of almost 0Hz 4 1 1ˆ( ) sin( ) sin(3 ) sin(5 ) ..... 3 5 f t t t t            
• 11. 11 WB2414-Mechatronic System Design 2013-2014 21Delft University of Technology Impulse time response Fourier analysis of impulse is constant amplitude density over  all frequencies (linear scale). WB2414-Mechatronic System Design 2013-2014 22Delft University of Technology 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
• 12. 12 WB2414-Mechatronic System Design 2013-2014 23Delft University of Technology Nyquist plot, the most important of all! And don’t show the negative frequencies!