# Graphing decaying sinusoids by hand?

Discussion in 'Math' started by dewasiuk, Feb 26, 2011.

1. ### dewasiuk Thread Starter New Member

Feb 14, 2011
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How could this be roughly accomplished without making an extensive table of values?

For example if I use the laplace method of analyzing an RLC circuit with a pulse excitation, I know that I will arrive at a decaying sinusoidal current represented by the following equation:

I would like to be able to quickly graph the equation incase I didn't have access to software tools at that moment.

2. ### t_n_k AAC Fanatic!

Mar 6, 2009
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1. Draw the exponential envelope bounded by the curves

$f1(t)=e^{-\alpha t}$

and

$f2(t)=-e^{-\alpha t}$

2. Draw a sinusoid constrained in amplitude by the bounded region with damped period

$T=\frac{2\pi}{\omega}$

Since it is a sine function in this particular case the value at t=0 will be zero. If it were a simple cosine function the initial value would be 1. A phase displaced sine or cosine function would have an initial value determined by the value of the sine or cosine term at t=0 sec.

Last edited: Feb 27, 2011
3. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
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Here's an example sketch of a phase shifted exponentially decaying cosine function.

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4. ### dewasiuk Thread Starter New Member

Feb 14, 2011
24
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Oh I already know how to roughly sketch the actual shapes, but it's mostly about determining the amplitude of the first peak(for a sine wave) so I can draw it more accurately. I should have included that in my first post sorry.

5. ### Georacer Moderator

Nov 25, 2009
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1,266
Peaks and zero-crossings still occur in the multiples of pi/2, so these are points in time you look for.

For example, your first peak will be at the intersection of the envelope with the line x=pi/2 for a sinusoidal curve.
That is, of course, for zero time-shift. Shift your times accordingly for θ<>0.

Feb 14, 2011
24
0