# B Linear amplitude decay function

1. Aug 4, 2016

### houlahound

I collected data on a periodic mechanical motion. The amplitude is damped linearly. What mathematical function models linear amplitude decay. All I can find is exponential decay of a sine wave.

2. Aug 4, 2016

### Staff: Mentor

Instead of multiplying the sine with an exponential, you can multiply it with a linear function.

3. Aug 7, 2016

### houlahound

hi all. I tried multiplying by a linear function ie;

-0.1x*sin(20x)

to get;

in comparison an exponential multiplying factor gives this with the envelope of the amplitude decreasing exponentially;

my data looks like this with a linear envelope;

the linear multiplier clearly does not work to model this data. this should be a simple model but my math is not working.

any tips to general models to account for the data above appreciated.

4. Aug 7, 2016

### nasu

Your amplitude decreases so you need to multiply by a decreasing linear function, something like (1-x) rather than x.

Actually your decay may be exponential but the time constant is much larger than the period of the periodic function.
So you will have something like
$e^{ \frac{-t}{\tau}} sin(\omega t)$
If t is much smaller than $\tau$ this can be approximated by
$(1-\frac{t}{\tau})sin(\omega t)$
and will look like a linear decay of the sin wave.

5. Aug 7, 2016

### houlahound

I might have to calculate by hand but my plotting tool for this input as you suggest;

gives the following output;

6. Aug 8, 2016

### Staff: Mentor

Look at your linear part: It starts at 1, then goes to zero for x=2. You plot it up to x=100 where it is increasing in magnitude again.

Try something like 1-0.01*x if you want an x-range up to 100.

It looks like your sine function takes degrees as input, which is quite odd in the context of those problems.

7. Aug 8, 2016

### nasu

Your data plot has no unit so it is impossible to guess reasonable values of parameters.
You need to adjust the parameters in the function to fit your actual plot.

If you extend the plot beyond 10 s the amplitude it will increase.

Here is an example of parameters showing the desired behavior:
f[t] = (1 - t/10) Sin[2 t]

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Last edited: Aug 8, 2016
8. Aug 8, 2016

### houlahound

I wish I could like your post more than once, you nailed it.

will put up some data next chance I get.

cheers