### Automatics

**Chapter 7 Oscillation Unit**

**Chapter 7.1 Introduction**

**The**

Fig. 7-1

Fig. 7-1

**oscillatory unit**and the previously known

**two-inertial**

**unit**are examples of transmittance

**G(s)**in which the numerator is a constant number

**d**, and the denominator is a

**binomial**of the

**second order**with parameters

**a, b, c**–>

**Fig. 7-1a**.

In

**Fig. 7-1b**parameters

**a,b,c,d**have specific values

**8,1,2,4.**

From the general form,

**Fig. 7-1**, it is not clear whether it is a

**two-inertial**or

**oscillatory**

**unit**.

Therefore,

**G(s)–>Fig7-1a**should be transformed into a

**normalized**form

**Fig. 7-2**

**Oscillatory**

**Unit-**Normalized form

And in it:

**k**–

**steady**state reinforcement

**q**–

**damping factor**– the higher it is, the stronger the oscillation damping

**T**–

**coefficient**of the oscillation period. I only emphasize. Factor

**T**, not the oscillation period itself

**Tosc**.

The

**theoretical period of oscillation**will be calculated according to the formula

**Tosc=2*π*T**. Moreover, it is an approximate model. The true oscillation period on the diagram will be accurate only for damping

**q=0**. Damping

**q**increases this

**Tosc**period.

Normalized

**G(s)**is a bit bizarre, but everything will be clear after examining the specific transmittance

**G(s)**from

**Fig. 7-1b**.

To begin with, let’s bring it to a

**normalized**form, i.e. as shown in

**Fig. 7-2**.

**Fig. 7-3**

And how to do it?

**1**Divide the numerator and denominator by

**2**, so that the free component of the expession (i.e. without

**s**) becomes

**1**

**2**…

**etc**

Of course

**L=P**and check it out.

In this way, we will get the normalized form on the

**right**. It shows that

**k=2**,

**T=2 sec**(i.e.

**Tosc=2*π*T=12.56 sec**) and

**q=0.25**. In the next sections, we will study this

**transmittance**and its modifications at various attenuations

**q**.

**Note**

The value of the damping coefficient

**q**results in:

**q=0**–> ideal oscillating unit – no damping –>

**Chapter. 7.5**

**0<q<1**real oscillating unit– with damping–>

**Chapter**.

**7.2 , 7.3 , 7.4 , 7.6**

**q>=1**two-inertial unit–>

**Chapter**.

**7.7**

It’s lab time!

**Chapter 7.2 k=2 T=2 sec q=0.25 with slider and bargraph**We start with the

**slider**and the bargraph. It is true that it is difficult to read the oscillation parameters by observing the bar graph, but you will feel the weight on the spring. It will come in handy in the pub brawl.

**Fig. 7-4**

**x(t)=0.5**

You will see oscillations. Wait a bit until it calms down and calculate the gain

**k**using steady-state digital meters. It should be

**k=2**. With the parameters

**T**and

**q**let’s leave it alone for now. We’ll come back to them using an oscilloscope.

**Chapter 7.3 k=2 T=2 sec q=0.25 with step and oscilloscope**

What will be the real oscillation period **Tosc** from the graph compared to the theoretical **Tosc=12.56** sec calculated in **Fig. 7-3**?

**Fig. 7-5**Gain

**k=2**agrees perfectly with the theory.

The real period of the oscillation

**Tosc=13 sec**on the oscilloscope is slightly larger than the

theoretical

**Tosc=12.56 sec***. And the damping

**q**? You can also calculate this parameter from the graph, but let’s spare it. It is enough for us to know that

**q**increases

**Tosc**and causes faster

**decay**of vibrations. We will see this in the next experiments.

***Tosc=2*Π*T=12.56 sec**

**Chapter 7.4 k=2 T=2 sec q=0.125 with step and oscilloscope**

How will **reducing** the **damping** twice to **q=0.125** affect the waveforms?

**Fig. 7-6**Oscillations last longer. and they didn’t even have time to calm down until

**60 seconds**. The oscillation period has decreased to

**Tosc=12.9 sec**but is still greater than ideal

**Tosc=12.56 sec**.

Gain

**k=2**is obvious. The first amplitude also increased.

What if we could completely get rid of damping by giving

**q=0**?

**Chapter 7.5 Ideal Oscillatory Unit k=2 T=2 sec q=0 with step and oscilloscope**

How will the lack of damping, i.e. **q=0**, affect the waveforms?

**Fig. 7-7**What’s up? There is no steady state

**y(t)=2**. Instead, there is a constant component

**2**around which a

**sine wave**with an amplitude also

**2**“flies”. And it will be like this until the end of the world. Notice that the real

**Tosc=12.56 sec**from the graph is equal to the theoretical one. This is a feature of an

**Ideal Oscillating Uni**t.

So far, we have been reducing the damping until we have reached the

**ideal oscillatory**where

**q=0**. Now let’s go the other way with damping and increase it to

**q=0.5**.

**Chapter 7.6 k=2 T=2 sec q=0.5 with step and oscilloscope**

**F****ig. 7-8**The output

**y(t)**seems to approach a

**two-inertial**one, although it is still an

**oscillatory unit**. The amplitude decreased and

**Tosc=14.8 sec**increased. The most deviates from the theoretical

**12.56**seconds now. So let’s go all out and give

**q>1**, e.g.

**q=1.5**.

**Chapter 7.7 “Oscillating”unit k=2 T=2 sec q=1.5 with step and oscilloscope**

The name** “Oscillating”** in quotes and **double-inertial** in the title suggest something. It’ll clear up in a moment.

**Fig. 7-9**Exactly. Typical response of the

**two-inertial**unit. It turns out that for

**q>1**the

**oscillatory**unit becomes

**two-inertial**!. Hence the quotation marks in the title of

**chapter 7.7**. Since it is

**two-inertial**, the transfer function from

**Fig. 7-9**can be reduced to the form from

**Fig. 6-1**from the previous

**chapter 6**.

**Chapter 7.8 “Oscillating” unit k=2 T=2 sec q=0.25 with Dirac and oscilloscope**

**Fig. 7-10**

**Dirac**shows what is most interesting in the oscillatory unit – only a variable component.

**Tosc=13 sec**is the same as in

**Fig. 7-5**.

**Chapter 7-9 Four with dirac and oscilloscope simultaneously**

As a summary, we will give a simultaneous **dirac pulse** to **4** dynamic units with different damping **q**. They all have **T=0.5 sec** and **k=1** We chose dirac as **x(t)**. Then the response does not have a constant component and it is easier to observe the effect of** damping** q on the time waveforms.**Fig. 7-11**The same short but strong signal

**x(t)**is fed to the inputs of successive terms with a decreasing damping factor

**q**.

This short

**x(t)**“hammer blow” is an approximation of the ideal

**Dirac pulse**..

**Fig. 7-12**There is a clear influence of

**q**damping. The greater the damping, the less the tendency to oscillate.

For the largest

**q=1.2**, there is no longer even an

**oscillating**unit but

**two-inertial**unit

**Chapter 7.10 Conclusions****1.** The transmittance of the **oscillating** unit for **0 <q<1** is shown in** Fig. 7-2**.**2.** For **q=0** it is an **ideal oscillating** unit in which the vibrations never die out.**3.** With increasing damping **q** the oscillations decrease. The oscillation period **Tosc** also increases.**4.** For **q>=1** we are dealing with a** two-inertial** ubit. Then the denominator **G(s)** can be presented as in** Fig. 6-**1 in **chapter 6**

The transmittance of the** oscillating** and **two-inertial** units can also be said differently**Fig. 7-13**The same can be expressed using

**complex numbers**

**Knowledge of**

Fig. 7-14

Fig. 7-14

**complex numbers**is not necessary, but if you want something more, click

**Courses**in the header and select the

**Complex Numbers**course.