How do I write this interesting piecewise function?

[JosephK]

Homework Statement

[URL]http://www2.seminolestate.edu/lvosbury/images/VibSpringAnNS.gif[/URL]
Find the governing differential equation and position functions for a 32 pound object attached to the end of a spring with a spring constant of 1 and a forcing function that yields a constant velocity in the direction of motion. This velocity changes sign periodically. The forcing function is piecewise. The object is pulled down until the spring is stretched to 5 feet below its equilibrium position and then the object is released with an initial velocity of -1 ft/sec and the forcing function produces a constant velocity of -1 ft/sec. After the object has traveled 10 feet it is impeded and reverses direction with an initial velocity at that point of 1 ft/sec. and the forcing function changes to produce a constant velocity of 1 ft/sec. This behavior continues indefinitely.

Homework Equations

The Attempt at a Solution

I am interested in only finding the piecewise function.

I drew the graph.
[URL]http://assets.openstudy.com/updates/attachments/4dfd29cf0b8bbe4f12e6e1ca-joseph20111-1308436975669-piecewise.bmp[/URL]


I think I can use a calculator function such as frac(x) or int(x).

I wrote a piecewise (unfinished) function that I believe can be simplified.
[URL]http://assets.openstudy.com/updates/attachments/4dfd29cf0b8bbe4f12e6e1ca-joseph20111-1308447921585-pie.bmp[/URL]

-t represents line with negative 1 slope. t represents line with positive 1 slope.
I wrote (1)^n * t for alternating t.

To write inequalities, 5 + 10n.

 

[HallsofIvy]

Your forcing functio is periodic with period 20. The simplest thing to do would be to solve it as two separate problems for -5< t< 5 and then for 5< t< 15.

 

[LCKurtz]

Your piecewise description of your forcing function is not correct. For the period from -5 to 15 your formula would be:

f(x) = x, -5<x<5
f(x) = 10 - x, 5 < x < 15

You can write this portion of f(x) using the unit step function u(x):

f(x) = x + (10-2x)u(x-5) for -5 < x < 15

and call F(x) the periodic extension of f(x) with period 20. Then you could solve the DE using LaPlace transforms, making use of the expression for the LT of a periodic function.

 

[JosephK]

f(t) = t, -5<t<5
f(t) = 10 - t, 5 < t < 15

I think I can find the Laplace transform of this periodic function.

 

[LCKurtz]

f(t) = t, -5<t<5
f(t) = 10 - t, 5 < t < 15

I think I can find the Laplace transform of this periodic function.

Since your piecewise function is actually not defined for t < 0, you should use a formula on (0,20) to take the LaPlace transform. I think it is:

f(t) = t -10u(t-5) - 10u(t-15) on (0,20)

but you can check to be sure.

 

[chiro]

Homework Statement

[URL]http://www2.seminolestate.edu/lvosbury/images/VibSpringAnNS.gif[/URL]
Find the governing differential equation and position functions for a 32 pound object attached to the end of a spring with a spring constant of 1 and a forcing function that yields a constant velocity in the direction of motion. This velocity changes sign periodically. The forcing function is piecewise. The object is pulled down until the spring is stretched to 5 feet below its equilibrium position and then the object is released with an initial velocity of -1 ft/sec and the forcing function produces a constant velocity of -1 ft/sec. After the object has traveled 10 feet it is impeded and reverses direction with an initial velocity at that point of 1 ft/sec. and the forcing function changes to produce a constant velocity of 1 ft/sec. This behavior continues indefinitely.

Homework Equations

The Attempt at a Solution

I am interested in only finding the piecewise function.

I drew the graph.
[URL]http://assets.openstudy.com/updates/attachments/4dfd29cf0b8bbe4f12e6e1ca-joseph20111-1308436975669-piecewise.bmp[/URL]


I think I can use a calculator function such as frac(x) or int(x).

I wrote a piecewise (unfinished) function that I believe can be simplified.
[URL]http://assets.openstudy.com/updates/attachments/4dfd29cf0b8bbe4f12e6e1ca-joseph20111-1308447921585-pie.bmp[/URL]

-t represents line with negative 1 slope. t represents line with positive 1 slope.
I wrote (1)^n * t for alternating t.

To write inequalities, 5 + 10n.

One way of representing this that comes to mind is to use a x arcsin(sin(bx)) for constants a and b. From your drawing let a = A and b = pi/10 and hopefully that should give you what you need.