Mechatronic System Design: 2. signals and dynamic

  • 108 views
Uploaded on

 

More in: Technology , Education
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Be the first to comment
    Be the first to like this
No Downloads

Views

Total Views
108
On Slideshare
0
From Embeds
0
Number of Embeds
1

Actions

Shares
Downloads
0
Comments
0
Likes
0

Embeds 0

No embeds

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
    No notes for slide

Transcript

  • 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!