Your company is developing a product that measures oscillatory motion using a sp

Question

Your company is developing a product that measures oscillatory motion using a springmass
system. The mechanism is housed in a container that only provides information to a
person via a read-out screen on the container (of course, there is a computer board inside
the container that analyzes the motion of the spring-mass system used). The container
also is filled with a substance that causes a damping force to the motion of the springmass
system. This damping force has been measured to be equal to the instantaneous
velocity of the spring-mass system when it is in oscillatory motion.
You are about to run a test of the system. You know that an object weighing ten pounds
causes a free spring that is normally five feet long to stretch to a length of seven feet.
Now, for testing purposes, you attach, to the spring, an object that weighs eight pounds
and place it into the container so that it is held 3 feet above the equilibrium point of the
system. The system experiences a phenomenon which causes the mass of the springmass
system to have an initial downward velocity of 10 feet per second.

1. Find the equation of motion for the spring-mass system.

2. After computation by the computer, the container’s read-out screen will show the
phase angle, the damped amplitude, the quasi period, and the quasi frequency of the
system. What values will the screen show for this test?

Explanation / Answer

Alright, to set up your D.E. you need the constants for mass(m) dampening(gamma) and the spring(K). There is no external force so this will be homogeneous.
M= weight / Gravity = 8/32 = 1/4
Gamma= 1 because instant velocity is u'(t)
and K = weight/ length = 10/2= 5
So the D.E. = (1/4)u''(t) + u'(t) + 5u(t) =0
and the initial conditions are u(0)= -3 and u'(0)= 10
now solve this equation.
Chac. eqn. = r2 + 4r + 20 =0 complete the square to get (r+2)2 = -16 ==> r=-2+/- 4i

Which has soln's in the form of u= C1e-2t Cos( 4t) + C2e-2t Sin(4t)

use the initial conditions to find C1 and C2

C1 = -3 and C2= 4

So the equation of motion is u(t) = -3e-2tCos(4t) + 4e-2t Sin(4t)

Part 2

amplitude is sqrt( 42 +32 ) = sqrt( 25) = 5

phase angle= arctan(4/-3) approx= 0.927295 + pi=2.2143 add pi because it is negative

quasi freq= 4

quasi period = 2/4 = /2

I hope that helps

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