**Chapter 11.1 Introduction
**

**Fig. 11-1**

Oscillationand

Oscillation

**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.

**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**.

**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.

**Fig****. 11-4**

Click “Start”

**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

**Fig. 11-6**

Click “Start”

**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

**Fig. 11-8**

Click “Start”

**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

**Fig. 11-10**

Click “Start”

**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

**Fig. 11-12**

Click “Start”

**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

**Fig**

**. 11-14**

Click”Start”

**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

**Fig. 11-16**

Click “Start”

**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

**Fig. 11-18**

Click “Start”

**Fig**

**. 11-19**

The

**oscillation**unit changes to

**double inertial**unit when

**q=1.2**

**>1**

**.**

The lower is attenuation parameter

**q**–> 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. Idealunit when

2. Ideal

**Oscillation****q=0**

3. Attenuationrate

3. Attenuation

**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.

**Fig. 11-20**

The same but with so-called **complex numbers**

**Fig****. 11-21
**We will discuss

**complex numbers**later.

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