Chapter 11.1 Introduction
11-1a
Fig. 11-1
Oscillation
and Double inertial units are G(s) examples where numerator is a constant number d and denumerator is a binomial with the parameters a, b, c–>Fig. 11-1a.
These parameters a,b,c,d are for example 8,2,2,4–>Fig. 11-1b.
You need to transform G(s)–>Rys. 11-1a to its standardized form. It’s this same G(s) of course but it’s possible to read some useful parameters.
11-2
Fig. 11-2
Oscillation unit- standardized form
k – steady state gain
q – attenuation  rate
T – oscillation rate.I emphasize. It isn’t oscillation period (as suggest the symbol T) but only  any rate. The real oscillation period is Tosc=2*Π*T. It’s  estimated period furthermore.
Standardized G(s) form is slightly bizarre , but all be clear at the moment. Let’s transform the Fig. 11-1b G(s) to the standardized G(s) as Fig. 11-2.

11-3a
Fig. 11-3
How to do it?
1 Divide numerator and denumerator by 2–>the free factor (without s) is 1 now.
2 … etc
Right=Left check it
Standarized form tell us that k=2, T=2 sec (i.e. Tosc=2*Π*T=12.56 sek) and q=0.25
We will test the standarized  G(s) with different q rates.
Let’s go to laboratory!

Chapter 11.2 k=2 T=2 sec q=0.25 with the potentiometer and bargraf
Call Desktop/PID/01_podstawowe_człony_dynamiczne/06_człon_oscylacyjny/01_oscylacyjny_bargraf.zcos.
11-4a
Fig. 11-4
Click “Start”
11-5a
Fig. 11-5
Play a little with this “weight and spring”. You observe the oscillations. Please count the gain k in steady state. You have to use digital meters here. It should be k=2.

Chapter 11.3 k=2 T=2 sec q=0.25 x(t) step type and oscillocope
What will be the real oscillation period Tosc beside theoretical period Tosc=12.56 as in Fig. 11-3 ?
Call desktop/PID/01_podstawowe_człony_dynamiczne/06_człon_oscylacyjny/02_oscylacyjny_skok_oscyloskop.zcos
11-6a
Fig. 11-6
Click “Start”
11-7
Fig. 11-7
Gain k=2 is acc. to the theory. The real Tosc=13 sec is a little more than Tosc=12.56 sec. And what about the attenuation  rate  q. It isn’t so easy read it from the Fig. 11-7 but it’s possible. Let’s poop out about it in this course.

Chapter 11.4 k=2 T=2 sec q=0.125 x(t) step type and oscillocope
Call Desktop/PID/01_podstawowe_człony_dynamiczne/06_człon_oscylacyjny/03_oscylacyjny_skok_oscyloskop.zco
11-8a
Fig. 11-8
Click “Start”
11-9
Fig. 11-9
Attenuation  rate q is minimised–>oscillations lasts longer. The first amplitude is bigger.
And what about q=0, no attenuation.

Chapter 11.5 k=2 T=2 sec q=0  x(t) step type and oscillocope
Call desktop/PID/01_podstawowe_człony_dynamiczne/06_człon_oscylacyjny/04_oscylacyjny_skok_oscyloskop.zcos
11-10a
Fig. 11-10
Click “Start”
11-11
RFig. 11-11
What is this? There is no steady state y(t)=2, but the constant component of the sinusoid y(t).
Note that real Tosc=12.56 sek is as theoretical! Let’s increase the attenuation  rate  up to q=0.5.

Chapter 11.6 k=2 T=2 sec q=0.5  x(t) step type and oscillocope
Call desktop/PID/01_podstawowe_człony_dynamiczne/06_człon_oscylacyjny/05_oscylacyjny_skok_oscyloskop.zcos
11-12a
Fig. 11-12
Click “Start”
11-13
Fig. 11-13
Draw conclusions  please. First amplitude minimized and Tosc=14.8 sec increased.
Let’s Make a Deal and q>1 for example q=1.5.

Chapter 11.7 “Oscillation unit” k=2 T=2 sec q=1.5  x(t) step type and oscillocope
Quotation marks suggest something.
Call Desktop/PID/01_podstawowe_człony_dynamiczne/06_dwuinercyjny_skok_oscyloskop.zcos

11-14a
Fig. 11-14
Click”Start”
11-15
Rys. 11-15
Typical double inertial response! When q>1 –>oscillation unit transforms to double inertal unit!

Chapter 11.8 k=2 T=2 sec q=0.25  x(t) Dirac type and oscilloscope
Call Desktop/PID/01_podstawowe_człony_dynamiczne/07_oscylacyjny_dirac_oscyloskop.zcos
11-16a
Fig. 11-16
Click “Start”
11-17a
Rys. 11-17
Dirac shows the most interesting attribute of the oscillation unit–>variable component.

Chapter 11.9   4 diracs simultaneously and oscilloscope
All the units are T=0.5 sec and k=1. You can observe the attenuation rate influence for the transient response.
Call Desktop/PID/01_podstawowe_człony_dynamiczne/08_4_na_raz _z_dirakiem_oscyloskop.zcos
11-18a
Fig. 11-18
Click “Start”
11-19
Fig. 11-19
The oscillation unit changes to double inertial unit when q=1.2>1.
The lower is attenuation parameter –> the the bigger are oscillations.
q=0–> infinity oscillations.
q=>1–>no oscillations. It’s double inertial unit, not a oscillation unit!

Chapter 11.9 Conclusions
1. Oscillation unit when 0 <q<1 –> Fig. 11-2
2. Ideal Oscillation
unit when q=0
3. Attenuation
rate  increases–>oscillations decreases and Tosc increases
4. When q>=1–>oscillation unit changes to double inertia unit

The other look for oscillation and double inertia unit.
11-20a
Fig. 11-20

The same but with so-called complex numbers
11-21a
Fig. 11-21
We will discuss complex numbers later.

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