AC THEORY

[__dan]

At theta = pi, the spring is compressed to its maximum value of displacement x, velocity = 0, acceleration = max (decel at that point, max negative accel).

So you can say it is at position x = -1. But the radius, or the triangle hypoteneuse representing total energy, is constant (1 for the unit circle) and at max compression the spring potential energy, is also a positive value (PE = 1, KE = 0). Summing the right hand side using Pythagoras gives the correct value of e^i(pi) = 1.

The book is wrong.

 

[Carultch]

At theta = pi, the spring is compressed to its maximum value of displacement x, velocity = 0, acceleration = max (decel at that point, max negative accel).

So you can say it is at position x = -1. But the radius, or the triangle hypoteneuse representing total energy, is constant (1 for the unit circle) and at max compression the spring potential energy, is also a positive value (PE = 1, KE = 0). Summing the right hand side using Pythagoras gives the correct value of e^i(pi) = 1.

The book is wrong.

Both PE and KE depend on squaring a number to get them, and you lose information when you square numbers. Multiple numbers square to the same result, such as -1 and +1 both square to +1. Take the square root, and you can't tell which number was originally squared to get +1, and it is only by convention that we default to considering +1 as *the* square root.

e^(i*pi) is not +1. You are making the same false conclusion, as you'd make to conclude that -1 = +1, just because (-1)^2 = (+1)^2.

 

[__dan]

Like this.

 

[Carultch]

Like this.

You keep having the same misunderstandings.

This is true.
TE = 1/2*m*v^2 + 1/2*k*x^2

This is where you've introduced terms without any reason, and begun to make your false conclusion:
TE = 1/2*m*v^2*cos(theta) + 1/2*k*x^2*i*sin(theta)

Neither component of total energy should ever go negative in this situation. KE can't be negative, since rest is the slowest it can move. By convention of letting PE equal zero at x=0, the lowest PE can also be is zero. So it doesn't make sense to just multiply them by cos(theta) and i*sin(theta) like you have done.

This is how you really should approach what you are trying to do.
x(t) = A*sin(w*t)
v(t) = A*w*cos(w*t)

The known general solutions for x as a function of time, and its derivative, v as a function of time. A is amplitude and w is angular frequency. You can make A & w both equal to 1 for simplicity, but I'll keep them for the general case.

Take these equations, and substitute them in to the equation for each form of energy:
PE = 1/2*k*x^2
PE = 1/2*k*(A*sin(w*t))^2
PE = 1/2*k*A^2*sin(w*t)^2
sin(w*t)^2 = 1/2 - 1/2*cos(2*w*t)
PE = 1/4*k*A^2*(1 + cos(2*w*t))

KE = 1/2*m*v^2
KE = 1/2*m*(A*w*cos(w*t))^2
KE = 1/2*m*A^2*w*cos(w*t)^2
cos(w*t)^2 = 1/2 + 1/2*cos(2*w*t)
KE = 1/4*m*A^2*w^2*(1 + cos(2*w*t)

Using the standard known solution that w^2 = k/m, we can see that:
KE = 1/4*k*A^2*(1 + cos(2*w*t))

Neither of these energy values are ever negative, or imaginary. These both are based on cosine functions of time that always add up to the same total energy (TE = 1/2*k*A^2), but not at the same frequency as the original functions x(t) and v(t), and they are not centered on zero.

 

[drcampbell]

... That's what DiffEQ is all about ...

I took those classes: Ordinary Difficult Equations and Partial Difficult Equations.

 

[__dan]

I took those classes: Ordinary Difficult Equations and Partial Difficult Equations.

Yes my point exactly.

I would love it if they would teach the way that nature actually works. What they do is teach auto mechanics. Flunk Diff Eq and have to go make a living at Jiffy Q. The first step in what they teach is to separate the model based on what is observable in nature, and follow a recipe that reads like 'take your socks off and throw those in too'.

In one cycle of a sinewave there are more than one million instances of this instant time t. so of course they have you solving for everything is time t, x(t), v(t), u(t), (they love u substitutions, probably get a nickel for each one). Solving sinewave phenomena where the problems are set up as instantaneous time t is a waste of time. In the complex plane t is there, but in a way that is subtle and not dominating. x = r * theta and solving in terms of theta is a lot easier. Too easy. They don't want people actually knowing how nature works. Or they want more auto mechanics, same result.

I am going to clean up the earlier post but not today.

 

[Carultch]

I am going to clean up the earlier post but not today.

I have a correction to make to mine as well.

The equation PE = 1/4*k*A^2*(1 + cos(2*w*t)), should really be:
PE = 1/4*k*A^2*(1 - cos(2*w*t))

 

[junkhound]

re:
weep along with you about how hard it is to solve

Had to grin at this, my first big raise and promotion at the airplane company was to write a set of Bernoulli equations and reduce to simple cubic expressions to balance the settings for dozens of orifices on electronic equipment racks (back in mostly vac tube days). Prior practice was a mockup where a few techicians spent a few days setting orifices by trial and error 'cause the math was too hard for even engineers

Now of course, nearly all electronics are sililcon, and nearly all equippment has its own cooling system based on ambient temp ranges.

Ya dont need much theory for residential wiring, == the more theory you do know the more ridiculous some aspects of the code (aka AF, dual ground rods, etc) have become.

 

[junkhound]

One more aside.
DO not think I have written out a calculus or diffyQ equation or Laplace since 1995 or so. About that time all the FEA programs reached early maturity and became the 'goto' tools for just about everything. PSpice for circuits, Maxwell for fields and thermal, etc. etc.

 

[Carultch]

One more aside.
DO not think I have written out a calculus or diffyQ equation or Laplace since 1995 or so. About that time all the FEA programs reached early maturity and became the 'goto' tools for just about everything. PSpice for circuits, Maxwell for fields and thermal, etc. etc.

That's a good point too, in that a lot of computer programs do the heavy lifting that engineers from 50 years ago had to do manually.

The important value of knowing the theory today, is that it helps you understand what results you are generally expecting from these computer programs. It can help you develop an intuition of the trends in typical engineering problems, so you can do a reality check on your results. For the select few problems you can solve in closed-form, knowing how to do so can allow you to use it to guide your design decisions. Computer programs like FEA, are computationally intensive to run. While knowing theory is useful for knowing where to start with your design.

 

[drcampbell]

... The important value of knowing the theory today, is that it helps you understand ...

typo: The essential value of knowing the theory ...

If you don't have a solid comprehension of the theory -- developed by doing numerous examples by hand -- you should not be using the software.
see, for example, Henry Petroski's essay, "Should the Computer be Registered?" (might not be verbatim)

 

[Carultch]

typo: The essential value of knowing the theory ...

True. It still means approximately the same thing as I had in mind, but I agree, essential is a much better word.

 

[Sberry]

I have help when it requires a mind greater than my own. As one bud said when I was trying to learn some electronics, you will never be that good you will still need to ask, get better at something you are already good at. I can drill holes.

 

[PaulMmn]

Serious question. How often or when really would you ever need to know most things taught in AC theory levels 1 through 4 out in the field? To me it all seems unnecessary

I've spent my career in the computer world. At assorted training seminars I loved to attend sessions by one of the guys who -really- knew how the system was put together. He talked about things like how the computer responded to different things.
It was stuff that had no use in the 'real world.' But it added to your knowledge of how things were -supposed- to work. When things went awry, you'd look at the problem and say, "But that's not how it's supposed to work!" Those little pieces of information were lurking inside my head, and popped up at opportune moments.
As Scotty once said, "Ach! It's not logical, and makes no sense, but the ship doesn't -=feel=- right!"
Information is never wasted, and it's surprising how often 'useless' knowledge turns out to be useful!

 

[LarryFine]

Serious question. How often or when really would you ever need to know most things taught in AC theory levels 1 through 4 out in the field? To me it all seems unnecessary

While AC vs DC may not be important, understanding the relationships of electrical values certainly is.

I'm referring to Ohm's and Watt's Laws and other simple related calculations.

 

[PaulMmn]

View attachment 2565472

No, that's almost like

 

[PaulMmn]

Why are words i didnt write in my post?!
Anyway, you knew me when i did band instrument repair? Bach didnt use harmonics either, maybe stravinsky.

Violinists use harmonics to play high notes. They stretch their pinky and lightly touch the node of a harmonic to force the string to emphasize the harmonic they want-- usually a multiple of the 'base note.'

 

[PaulMmn]

You teach one of them the 3 4 5 rule to get things square, and they can't comprehend that the units don't matter

There are other things like that-- like the % of a grade-- it's Rise/Run * 100 = %. It can be feet, miles, inches... and the division 'cancels out' the units, and you're left with pure numbers!

 

[rambojoe]

Violinists use harmonics to play high notes. They stretch their pinky and lightly touch the node of a harmonic to force the string to emphasize the harmonic they want-- usually a multiple of the 'base note.'

I know what harmonics are... But if it makes you feel better....
P.s., any finger can produce it, its a thing called postion just so ya know...
Just about any instrument can produce harmonics, btw, not just a violin.... Including wind and brass instruments. Saying "violin" is like saying "i googled it and educated somebody"..

 

[rambojoe]

I guess next is a clip of "eruption"...