System Analysis - Determining Dampening and Natural Frequency

[ConnorM]

Homework Statement

Problem and work attached.

It is mostly diagrams so I have just uploaded the pictures.

I think what I did to determine the transfer function was correct. I'm not quite sure how to do the last section of part (a) where I need to determine the dampening, static gain, and natural frequency as functions of k₁ and L.

I tried to set the general equation for a second order system equal to my transfer function and then I set the numerator and denominators equal to each other but I'm not sure where to go now.

Could someone verify that what I did for the transfer function is correct, and help me determine the functions to describe the dampening, natural frequency, and static gain?

 

[LvW]

I think, trhe last block diagram looks OK - but recalculate the corresponding transfer function.
I am not sure if it is correct.

 

[ConnorM]

Recalculating,

$H(s) = \frac{k}{s^2 + kLs + k + 2s}$

Does this look right?

So now I would have,

$2ζωs + ω^2 = kLs + 2s + k$

$k = Kω^2$

 

[LvW]

I think, the transfer functionis correct now.
You should put together both expressions with "s" and compare each factor in the denominator with the denominator of the general (normal) form.
This would result in w²=k which looks a bit uncommon to me because of the units.
I am an engineer and for this reason I propose to write from the beginning H1=k/(2+sT1) and H2=1/sT2 (later we set T1=T2=1sec).
Now, all the expressions will have the correct unit.

As a result, the transfer function is
H(s)=N(s)/D(s) with

N(s)=k/T1T2 and D(s)=s²+s(2+kL)/T1+k/T1T2

From this, it is easy to derive that

wn²=k/T1T2 and "theta"=(kL+2)/2wnT1=[(2+kL)*SQRT(T2/T1)]/2k

Now we can set T1=T2=1sec.

 

[donpacino]

I think, the transfer functionis correct now.
You should put together both expressions with "s" and compare each factor in the denominator with the denominator of the general (normal) form.
This would result in w²=k which looks a bit uncommon to me because of the units.
I am an engineer and for this reason I propose to write from the beginning H1=k/(2+sT1) and H2=1/sT2 (later we set T1=T2=1sec).
Now, all the expressions will have the correct unit.

As a result, the transfer function is
H(s)=N(s)/D(s) with

N(s)=k/T1T2 and D(s)=s²+s(2+kL)/T1+k/T1T2

From this, it is easy to derive that

wn²=k/T1T2 and "theta"=(kL+2)/2wnT1=[(2+kL)*SQRT(T2/T1)]/2k

Now we can set T1=T2=1sec.

why are you over-complicating the problem. You can't arbitrarily change the transfer functions. as a result those values you added will ALWAYS be 1.
it might be good to point out that the time constant of a pole is 1+sT, or write it in the form of (k/2)/(1+s/2), but even that is trivial.

2ζωs+ω2=kLs+2s+k

yes. now evaluate that equation for zeta and natural frequency

 

[LvW]

why are you over-complicating the problem. You can't arbitrarily change the transfer functions. as a result those values you added will ALWAYS be 1.
it might be good to point out that the time constant of a pole is 1+sT, or write it in the form of (k/2)/(1+s/2), but even that is trivial.

I am afraid, you didn`t get my target.
I feel and I think as an engineer (as I have mentioned) - that means: Every equation must be "correct", even as far as dimensions/units are concerned.
Do you know how the result would look like without the time constants T1 and T2? Look here: wn²=k
This automatically will create the question: Since k is a dimensionless constant, how can it be equal to wn² ?
Please consider that this is not a mathematical problem but a technical one!
And for such engineering problems it is not only helpful to control the correct units - it is even mandatory!

I didn`t change the transfer function; didn`t you really understand what I`ve done?
I only have written down the correct units with T1=T2=1sec.
And now we see that k is nothing else than a scaling constant.
Hence, I didn`t over-complicate the problem.
In contrary - I think, I have used a method that clearly shows the correct units and, thus, can help to avoid confusion and mis-interpretation.

 

[donpacino]

I am afraid, you didn get my target.
I feel and I think as an engineer (as I have mentioned) - that means: Every equation must be "correct", even as far as dimensions/units are concerned.
Do you know how the result would look like without the time constants T1 and T2? Look here: wn²=k
This automatically will create the question: Since k is a dimensionless constant, how can it be equal to wn² ?
Please consider that this is not a mathematical problem but a technical one!
And for such engineering problems it is not only helpful to control the correct units - it is even mandatory!

I didn`t change the transfer function; didn`t you really understand what I`ve done?
I only have written down the correct units with T1=T2=1sec.
And now we see that k is nothing else than a scaling constant.
Hence, I didn`t over-complicate the problem.
In contrary - I think, I have used a method that clearly shows the correct units and, thus, can help to avoid confusion and mis-interpretation.

that is 100% not necessary.
adding T does nothing but complicate the math.
As an engineer you should be able to know what the units are and solve these problems without adding further complication.
If you're doing dimensional analysis to check your work great.
there is zero value added by adding values like this, unless you are checking your work.

 

[LvW]

that is 100% not necessary.
adding T does nothing but complicate the math.
As an engineer you should be able to know what the units are and solve these problems without adding further complication.
If you're doing dimensional analysis to check your work great.
there is zero value added by adding values like this, unless you are checking your work.

Donpacino - I cannot see what your problem is.
* Please, can you explain why "it complicates the math"? Just by using a symbol with a value of unity with the aim to have the correct unit? Where is the problem?
* "You should be able to know what the units are": You are right, perhaps I am able to know this.
But I cannot always be sure that questioners in the forum are also able to interpret such symbols like "k" correctly. "k" is a constant value - and, suddenly, it turns out that it should have the unit (rad/s)² ? Don`t you see the possible source of error or misinterpretation?
* In this respect I can only say the following: For deriving formulas from circuit or block diagrams it is good (for my opinion: mandatory) not to suppress the units. This is a great help to check if the result is a plausible one.
* A transer function Vout/Vin has no dimension. And, therefore, in the denominator we must not have the sum of a dimensionless number ("1") and a frequency ("s" is given in rad/s). This is good engineering practice!
* I know nothing about your professional background. However, I can tell you that I have more than 25 years experinence in teaching electronics at a university.
Very often me and my collegues have seen why we have good reason to require not to supress the correct units during the process of deriving formulas from block or circuit diagrams. Believe me, I know what I am talking about!

Final comment: It would be interesting to read about the opinion of the questioner (ConnorM).

 

[rude man]

I haven't followed this squabble verbatim et litteratim but if LvW's point is that expressions should always be dimensionally consistent then I take his side 100%.
Writing F(s) = 1/(Ts + 1) is good.
Writing F(s) = 1/(s + 1) is really, really bad.
Substitute numerical values ONLY at the end. And maintain dimensional integrity ALWAYS. How many times does it have to be stressed that dimensions are de rigueur to check your work. ALWAYS.