A mass of 1 slug when attached to a spring stretches it 2 fe

A mass of 1 slug, when attached to a spring, stretches it 2 feet and then comes to rest in the equilibrium position. Starting at t = 0, an external force equal to f(t) = 10 sin 4t is applied to the system. Find the equation of motion if the surrounding medium offers a damping force that is numerically equal to 8 times the instantaneous velocity. (Use g = 32 ft/s² for the acceleration due to gravity.)

Solution

Given that

m = 1
1 slug = 32 lb

f = kx
32 = k(2)
k = 16

c = 8 ( 8 times the instantaneous velocity)

mx\'\' + cx\' + kx = 10sin4t
x\'\' + 8x\' + 16x = 10sin(4t)

first we find the complimentary solution xh
The characteristic equation is r² + 8r + 16 = 0
(r + 4)² = 0
r = -4, -4 (real and repeated roots)

xh = ce^(-4t) + cte^(-4t)

Now we find the particular solution xp
xp = Acos(4t) + Bsin(4t)

xp\' = -4Asin(4t) + 4Bcos(4t)
xp\'\' = -16Acos(4t) - 16Bsin(4t)

x\'\' + 8x\' + 16x = 10sin(4t)
-16Acos(4t) - 16Bsin(4t) + 8[ -4Asin(4t) + 4Bcos(4t) ] + 16 [ Acos(4t) + Bsin(4t) ] = 10sin(4t)
-16Acos(4t) - 16Bsin(4t) - 32Asin(4t) + 32Bcos(4t) + 16Acos(4t) + 16Bsin(4t) ] = 10sin(4t)
-32Asin(4t) + 32Bcos(4t) = 10sin(4t)
-16Asin(4t) + 16Bcos(4t) = 5sin(4t)

solve for A and B
16Bcos(4t) = 0
B = 0

-16Asin(4t) = 5sin(4t)
A = - 5/16

The particular solution is xp = - 5 /16 cos(4t)

Therefore the general solution is
x(t) = xh + xp
x(t) = ce^(-4t) + cte^(-4t) - 5/16 cos(4t)
Now at t = 0 it starts from rest that is initial velocity = 0
x\'(0) = 0

at t = 0 it starts from equilibrium
x(0) = 0

x(t) = ce^(-4t) + cte^(-4t) - ¼cos(4t)
0 = c + c(0) - 5/16 cos(0)

c₁ = 5/16.

and we have from the general solution x(t) = ce^(-4t) + cte^(-4t) - 5/16 cos(4t)

Then x\' (t) = - 4ce^(-4t) - 4cte^(-4t) +c₂ e^(-4t)+ 5/4 sin(4t)

Now use x\'(0) = 0

x\' (0) = - 4ce^(-4t) - 4cte^(-4t) +c₂ e^(-4t)+ 5/4 sin(4t)
0 = - 4c + c e^(0) + 5/4 sin(0)
0 = -4c₁ + c
c = -4c₁
c = 5/4

Therefore , the equation of motion is

x(t) = (5/16)e^(-4t) + (5/4) te^(-4t) - 5/16 cos(4t)