4 DOF Model

[johnosb]

Hi all,

I've been working through the design volume and it's all starting to make some sense but I've hit a brick wall with the 4 DOF model. I've implemented it in Excel and using the model parameters from table 2.4-1 but it bears little resemblance to Fig 2.3-4.

There are a couple of items that I'm not sure that I've implemented correctly.
1/ The omega and omega squared value in the displacement denominator I assumed to be 2 Pi F where F is the model's input frequency.
2/ The denominator (Dt, Ds, Db and Da) is in the form z=x+jy and the solution is z=sqrt(x2+y2) where x = miwi2-miw2 and y=Riw

I hope that makes sense, it looks horrible without superscript and subscript.

The plot I get is below and I can upload an image of the sheet with the intermediate values if somebody has successfully tackled this using the same parameters and can compare them and point me in the right direction.

Any help would be greatly appreciated.

Os

[Trevor Gore]

Hi Os,

Well done for having a go at the 4-DOF model!

1/ The omega and omega squared value in the displacement denominator I assumed to be 2 Pi F where F is the model's input frequency.

Correct.

2/ The denominator (Dt, Ds, Db and Da) is in the form z=x+jy and the solution is z=sqrt(x2+y2) where x = miwi2-miw2 and y=Riw

Not sure where you're going with this one. I didn't go looking for a solution to the D's here, I just evaluated them for each value of ω, iirc.

Re: the plot, if you're getting what you have, you must be getting pretty close. It looks like you have an extra oscillator switched on, because of the 4th peak, or you have misplaced a decimal point, maybe on the side mass oscillator. If you check all the numbers (again!) and they seem OK, maybe you have another case of Excel ill-conditioning. The only way to check that is to re-code the spreadsheet breaking down cells with long formulas into smaller ones and see if you get the same answer.

[Dominic]

Hey Os, nice work on getting this far. First one to get the model working that I know of. I have my sights set on getting it up and working but have not read that bit enough yet.

But as guide to those of us behind you, could I suggest that when you ask questions, put down the page number or the equation number and use your questions to set up the contex a bit and Trevors answers to educate more broadly. It is OK to state the bleedingly obvious because there are always plenty of us who would benefit.
Alternatively, once you have it sorted, a brief guide to how you got it working and hints on the more obvious places where errors could be hiding.
So again, very cool you are working on this. I hope I can bounce ideas of you and see some results once you have it perfected.
Cheers
Dom

[johnosb]

Thanks for the reply guys.

Yep, sorry should have added more info.

The equations in question are in Appendix II, AII 2-25 through AII 2-28. Just below it states that:

"Letting the denominators of the four equations above = Dt, Ds, Db and Da respectively"

These equations are then inserted into the simultaneous equations and solved by the matrices. The denominators are complex and my question was are they in the form z=x+jy and are we interested in the magnitude z=sqrt(x2+y2) or, given that two of the w terms are in radians (2pif) are we after the angle theta=sqrt(y/x). The latter makes no sense to me so presumed that the former is true.

The values obtained for Di are then plugged into the equations AII 2-40 through AII 2-44 for the displacements.

I haven't had to worry about complex numbers for a lot of years and I'm stretching the memory so am likely totally wrong. I'll go through the equations again tonight and see if I can see anything obvious. In the interim, I'll attach a screen grab of the spreadsheet in the hope that something sticks out.

Thanks again,

Os.

[johnosb]

Another couple of hours going over the equations. I recoded the Dbar into several chunks as that was the longest calculation but no difference. I'm convinced it's my method and my misinterpretation of that denominator issue and how I'm handling equations AII 2-25 though AII 2-28. Oops, just looked at my previous post and realised I screwed up on the imaginary argument theta should have been theta = atan(y/x) not that I think it's used anyway.

Trevor I'm not sure I understand when you say you just solve for omega (how did you get the greek font in your post??), iirc - I don't know what iirc is. The equations on p2-35, 2.4-5 through 2.4-8 and Dbar all require a value for the Di variables and they are defined in Appendix, section 2 as the denominators of Aii 2-25 through AII 2-28. Am I missing something??


Os.

[simonm]

iirc= if I recall correctly

[johnosb]

Ah, I see. I thought it was i squared rc. I guess I'm jammed in math mode!!!

thanks for straightening that out.

Os

[Trevor Gore]

Os, sorry for not getting back earlier.

It's hard to see where you're going wrong (or where the book is incorrect!!). But to get a plot like you're getting, you must be pretty close.

So a few quick comments/questions:

1) the D's need to be calculated for each frequency increment (5 Hz, 10 Hz etc). I used increments of 0.67Hz because I was matching to a Visual Analyser tap response and that is what the frequency increment is
2) What happens when you change variables? If you increase e.g. soundboard mass, does everything move the way it should, e.g. the T(1,2) goes down and there is a related change in the the T(1,1)3 etc.
3) How are you handling the complex numbers in Excel? I used the IM functions (needs the Math add-in, but that should be on your MS Office disc if you can't load it directly)

Trevor I'm not sure I understand when you say you just solve for omega..

I said evaluate, not solve; (maybe just a language problem!!) If you use the IM functions you can manipulate the complex numbers just like real numbers (hate this real and imaginary nomenclature, but that's another story) and so you just type the formula into a cell and it evaluates it.

(how did you get the greek font in your post??)

Cut and paste from MSWord.

Let me know how you go and I'll have another look when my head feels a bit clearer. Got some bug at the moment and feel like I've been run over by a truck.

[kiwigeo]

Got some bug at the moment and feel like I've been run over by a truck.

This has been a commonly reported side effect from spending too much time staring at the equations in Trev and Gerards books

[johnosb]

Thanks Trevor,

There's no hurry on this. Sounds like a couple of double scotches and an afternoon in bed is what you really need. Come to think of it, I might do the same myself and I'm not even sick!!

I looked at using the IM functions but when I looked at the help on them they looked like they needed strings as arguments which was a bit confusing. I'll do a bit of research on them this evening and have a bit of a play. I just manually solved them for the magnitude (hypotenuse?):

Di=√((mt ωt^2 - mt ω^2 )+ (Rt ω))

I used the What_If function on a data table holding the Frequency values, just used a 5Hz resolution to start with.

I get a change by changing the top mass which I guess is as expected T(1,2) drops in frequency and the T(1,1)3 etc, drop slightly in frequency and are attenuated. Asoundhole radius change seems a bit odd 44mm radius sets T(1,1)2 to about 155 Hz but at 50mm it shoots up to about 180Hz. I had it set to 44.7mm to match your area and volume of the 'air pipe'.

Agree wholeheartedly about imaginary numbers, I've only remember using them in AC circuit theory and pretty well came close to failing that subject!!

Thanks again and please don't respond in a hurry, more important to get over the lurgy!!

Os

[Trevor Gore]

Asoundhole radius change seems a bit odd 44mm radius sets T(1,1)2 to about 155 Hz but at 50mm it shoots up to about 180Hz. I had it set to 44.7mm to match your area and volume of the 'air pipe'.

Hmmm. Missed out a 2 changing diameter to radius?

[Trevor Gore]

I'll do a bit of research on them this evening and have a bit of a play. I just manually solved them for the magnitude (hypotenuse?):

You need the phase angle as well, rather than just a conversion to a scalar. Not sure if you've got this in or not. The IM functions take care of this without having to think too hard about it.

[johnosb]

I worked out the method to pass the complex equation to the IM functions called COMPLEX funnily enough. I ran the IMABS function and got the same answers as the manually calculated cells (but you're right, a lot easier to manipulate). Which IM functions do you use? I assume from above that you use IMARGUMENT to get the angle so then we have the angle and modulus, then what?? I need to sit back and think about this a while I think, confusion is setting in!!

Os

[Dominic]

Hi Trevor, I have been chatting with Os about the model this week and I ended up sitting here at 4.00am this morning reading and playing with the spreadsheet. So the sooner I/we get this worked out the soon I will be able to go to bed.

Now, previously you mentioned calculating all the Di,s for each frequency increment. Easy enough, I end up with 4 columns of D's each calculated at .67 increments as my plots in VA. Force of habit perhaps but I want to see what's happening, I don't like black box calculations. At work we spend more time checking numbers than we do deriving them in the first place so transparency is ingrained. I want it long hand at first so I can look in the box- I assume this is doable?

Now with the 4 columns of Ds for each freq increment a value for sound pressure (p) is derived and then converted to Db and we can chart that as a frequency plot. But the equation for (p) is derived from the Y's so should these Y's be calculated using the table of Ds or do we just use the reference Ds derived from the reference frequency of 100 Hz? Do we need a ref freq? Am I close or are you thinking I stayed up too late last night and am talking gibberish.

Also, the imaginary number defined as j, what it seems to be from an economists pov is an indicator of a turning point if I understand the notes on page 1-97. 2j being a point where the Db stop falling and begin to rise or visa versa. Is that kind of in the right direction?

I am really savouring coding up all the stuff in the book but one day it will end and what will I do then??? Make more guitars I suppose and play them.

Cheers
Dom

PS, I hope my questions are clear or I am doomed

[Trevor Gore]

I worked out the method to pass the complex equation to the IM functions called COMPLEX funnily enough. I ran the IMABS function and got the same answers as the manually calculated cells (but you're right, a lot easier to manipulate). Which IM functions do you use? I assume from above that you use IMARGUMENT to get the angle so then we have the angle and modulus, then what?? I need to sit back and think about this a while I think, confusion is setting in!!

I just checked my spreadsheets: I used COMPLEX to get the D's, IMSUM to evaluate the determinants, IMDIV for the y's, IMPRODUCT for the p's; and there is a value for each of these parameters for each frequency value.

The phase angle relationships are intrinsically embedded in the complex numbers so don't need to be explicitly evaluated, but that information has to be carried through, which it automatically is if you use the IM functions. I wasn't sure that was happening as you seemed to be just evaluating the magnitude.

[Trevor Gore]

PS, I hope my questions are clear or I am doomed

Probably doomed!

I think my last post might have answered most of your questions. If it didn't, have another go!

[johnosb]

I think the penny finally dropped last night Trevor. I had assumed that the denominator issue was related to producing a real solution for Di and then plug that into the displacement calculations. I then gathered from what you were saying that the denominator stays in complex form and then each of the displacement formulae remain in complex notation and solved that way. I plugged them in last night and got a nice smooth plot with the 3 modes albeit very attenuated. The equations end up very messy with nested IM functions so there is a lot of scope for error typing them in and then there's the Excel issue you mentioned. I'll break them down into more user friendly chunks and see where that leads.

I hope your feeling better,

Os

[Trevor Gore]

I just checked my spreadsheets again because I couldn't recall any nested IM functions. I had a bunch of columns hidden (not noticed before, with current half speed brain!). So the determinants are a bunch of IMPRODUCTs in separate columns and then these are IMSUMed and that's how I avoided nesting things.

Sounds like you're on the home straight.

[Dominic]

There is a permanence about the word doomed that I refuse to concede to so I take it back. I’ll get this worked out one way or another. I was asking about producing columns of real numbers, one for each Da Db Dt Ds at each freq increment. Now I get that this is not what happens. People say economists are a bit loose with their assumptions but incorporating imaginary numbers in theory and then hearing that they are not really imaginary? Picturing the first part is hard enough but then trying imagine imaginary numbers that are not really imaginary. I am sure there is a good reason for it.
Cheers
Dom

[Trevor Gore]

Yes, history has not done us any favours with the nomenclature. Really, the i or j operator (depending on whether you are an electrical engineer or not) should be a sign like "+" or "-", at least for engineering applications. Anybody who teaches that j is the SQRT(-1) has no understanding of the concept. Just think of "complex" numbers as two dimensional numbers.

We could invent a new system:

Addition (a step forward) is "-"
Subtraction (not a step forward) is "+"
and j (a step sidewise) is "|"



(I'd better keep on taking the medicine and hope it kicks in some time soon...)

[johnosb]

I just nested the imsums and imsubs because I was trying save space to keep everything on the same page so that I could try to keep up with it. I broke it down today and broke them up into logical groups and put them on separate sheets. Found 3 or 4 oopsies in the process and it moves closer to resolution. See attached plot. Given that the resultant air pressure is still in a complex form (rectangular x+jy - for 100Hz it is about 3.3E-7 + j1.34E-6) so still the only way I can think of to to resolve it is sqrt(x^2+y^2) before feeding it into the sp(Db) calculation.


Os

[Dominic]

Thanks Trevor, along with Os' question of how to turn an imaginary number into something we can use, I just want to clarify the following.

When we calculate the Di's for each increment, do we also need to do the same with the Ys or are they in effect, constants associated with the top, back, sides and air irrespective of the frequency? Or perhaps I should say, are the Di's dynamic with respect to freq and the rest static?


(Please forgive my terminology. I look at the chart here and I can't help seeing a time series. I know its not but my brain has been hardwired after years of doing time series analysis.)

I hope this was better than my previous attempt to ask a question?
Cheers
Dom

[Dominic]

Yes, history has not done us any favours with the nomenclature. Really, the i or j operator (depending on whether you are an electrical engineer or not) should be a sign like "+" or "-", at least for engineering applications. Anybody who teaches that j is the SQRT(-1) has no understanding of the concept. Just think of "complex" numbers as two dimensional numbers.

We could invent a new system:

Addition (a step forward) is "-"
Subtraction (not a step forward) is "+"
and j (a step sidewise) is "|"



(I'd better keep on taking the medicine and hope it kicks in some time soon...)

I was just reading up on imaginary numbers and this site did not say j = sqrt(-1). But it did say the j is the answer to sqrt(-1). Credible? This stuff is hard enough without learning the wrong stuff at the same time. Its still cool though.
Cheers
Dom

[johnosb]

I was just reading up on imaginary numbers and this site did not say j = sqrt(-1). But it did say the j is the answer to sqrt(-1). Credible? This stuff is hard enough without learning the wrong stuff at the same time. Its still cool though.
Cheers
Dom

As I was taught j is the square root of a negative number, no two numbers squared can give a negative result as the result of negative numbers squared is positive (-2 X -2 = 4) it is imaginary. But j^2 = -1 so traditionally j = sqrt -1. interestingly, sqrt j9 for example can be either -3 or +3. Whole bunches of rules to learn! I think Euler is to blame or was it Gauss, maybe both had a finger in the pie. It's useful to electrical and electronic engineers in AC circuit analysis and I think I remember that there are some obscure polynomials that require imaginary numbers to reach a solution. For the rest of us it's just designed to confound and frustrate!

i is normally used to represent the imaginary component but j is used by engineers as i denotes current - just to add another layer of confusion. I passed some of my questions to a maths Phd (former colleague) and in his response he changed all the j's into i's with a snide comment about electronic engineers! I didn't get a straight answer but 3 pages of nonsensical derivations where I ended up with a real denominator but an imaginary numerator with a note saying "this should help", yeah right!

Os

[Trevor Gore]

I just nested the imsums and imsubs because I was trying save space to keep everything on the same page so that I could try to keep up with it.

I put it all on one sheet, but was out to column DG!

Given that the resultant air pressure is still in a complex form (rectangular x+jy - for 100Hz it is about 3.3E-7 + j1.34E-6) so still the only way I can think of to to resolve it is sqrt(x^2+y^2) before feeding it into the sp(Db) calculation.

Try that and post what you get.