~ × & Σ …

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~ × & Σ …

Post by WebPilot » Thu Feb 19, 2009 12:24 am

Let's start with a simple sinusoid …

As you all know (or should by now) a sinusoid can be expressed as

y = A · sin (ω·t)

where A is a constant, ω is its frequency expressed in radians/sec and t is time expressed in secs. This is all well and good.

If we define φ,

φ ≡ ω/ωn

as some driving frequency ratio, where ωn is defined as a resonant frequency of the system in which we are interested (and constant for purposes of this discussion), then, substituting this into the sinusoid expression, one obtains

y = A · sin (φ·τ)

where τ, tau, is defined as dimensionless time with units of [1]. It can be calculated using the expression,

τ ≡ ωn · t
Image

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Re: ~ × & Σ …

Post by WebPilot » Fri Feb 20, 2009 7:22 am

Here we have the simple sine curve.

y = A ‧ sin(x)

The abscissa, x, is given in radians not degrees, the amplitude is 1.0 and the period is 2Π or 6.283185308…

Image

The reader should realize, that in the previous post, τ which is given by ωn · t, is also expressed in radians. This can be easily shown by looking at the units of each parameter.

ωn is given in radians/sec, and t is in sec, so multiplying them together we obtain radians/sec × sec ⟿ radians. A radian is a "dimensionless" parameter.
Image

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Re: ~ × & Σ …

Post by Mike Everman » Fri Feb 20, 2009 6:51 pm

Got it so far...
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Re: ~ × & Σ …

Post by WebPilot » Sat Feb 21, 2009 3:03 am

happy to read that …

Image

Here is a plot of sin(x) and cos(x). Note that cos(x) = sin(x+Π/2) where Π/2 is the phase angle. The waveform, cos(x) leads sin(x) in time (dimensionless or not).
Image

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Re: ~ × & Σ …

Post by WebPilot » Sun Feb 22, 2009 3:32 am

A further peculiarity of sinusoids.

If a particle were to move, having its position (in 2D) vary harmonically with time, given by
  1. x(t)=cos(t) and
  2. y(t)=sin(t)
a plot of y(t) versus x(t) yields

Image

The movement is circular. Any point on the curve represents the position of the particle at some time, t. If the movement goes forward in time, the particle position on the curve proceeds in a counter-clockwise manner; starting at the coordinates, (1,0).

This representation is known as a parametric plot.
Image

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Re: ~ × & Σ …

Post by WebPilot » Mon Feb 23, 2009 1:56 am

If we let x(t)=cos(t), then, the derivative with respect to time is dx(t)/dt=-sin(t). This means, if x(t) denotes the position of a particle, as it oscillates to and fro, dx/dt is its velocity.

If we plot velocity versus postion, we obtain again a circular plot.

Image

However, at t=0 the motion now starts at coordinates (1,0) and as time progresses, say Π/2, the coordinates have become (0,-1). IOW, the direction of the path followed is now CW.
Image

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Re: ~ × & Σ …

Post by WebPilot » Sat Mar 07, 2009 7:01 am

I am going to return to this thread, but first some lateral thinking …
Figure 1.png
Parametric plot in 3d:
x=cos(3*t)
y=sin(3*t)
for 0 ≤ t ≤ 2π
Figure 1.png (18.83 KiB) Viewed 4523 times
Image

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Re: ~ × & Σ …

Post by WebPilot » Sun Mar 08, 2009 5:00 am

changing ω from 3 to 1,
Fig 1.png
Fig 1.png (8.92 KiB) Viewed 4439 times
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Re: ~ × & Σ …

Post by WebPilot » Mon Mar 09, 2009 12:44 am

To summarize,
Fig1.png
Fig1.png (10.99 KiB) Viewed 4407 times
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Re: ~ × & Σ …

Post by WebPilot » Wed Mar 11, 2009 1:55 am

some lateral drifting
someMeshes.png
3 types
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Re: ~ × & Σ …

Post by WebPilot » Thu Mar 19, 2009 2:35 am

One sinusoid in a random (noisy) environment can be extracted …

Image
Image

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Re: ~ × & Σ …

Post by larry cottrill » Thu Mar 19, 2009 11:39 am

WebPilot wrote:One sinusoid in a random (noisy) environment can be extracted …
Interestingly, I witnessed a demonstration on National Public Radio of the ability of the human ear/brain to perform this exact kind of extraction. A high-pitched sine wave below the volume of normal perception was delivered with white noise superimposed on it. With the noise level low, there was no perception of the tone. Amazingly, as the sound level of the superimposed noise increased, without increasing the amplitude of the tone, the tone became audible. The higher the noise level, the more apparent the tone became (at least, up to a point)! This phenomenon was referred to as 'stochastic resonance'.

L Cottrill

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Re: ~ × & Σ …

Post by WebPilot » Fri Mar 20, 2009 3:50 am

Interesting, but strictly speaking, stochastic resonance occurs in bistable systems, …

continuing
  1. I numerically created the above waveform (noise and sine);
  2. I then exported it to a .wav format (binary) file;
  3. I then reloaded this waveform and analyzed it.
Here is the saved .wav file which is playable on the reader's computer system

and here is the analysis:

Image
REM A little clipping and reduction in amplitude is observed, but the end result is still here; a 100 Hz signal is found embedded in the noise.
Image

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Re: ~ × & Σ …

Post by WebPilot » Mon Mar 23, 2009 3:27 am

I may need this capabilility in the not too distant future …

Image
Vector Field
Image

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Re: ~ × & Σ …

Post by Mike Everman » Thu Mar 26, 2009 3:01 am

Thanks for this. Very fun to watch, Forrest.
Mike
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