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Sagot :
Answer:
Equation of motion is x(t) = [tex]-te^{-4t}[/tex] + [tex]\frac{1}{4}[/tex] sin(4t)
Explanation:
P.S - The exact question is -
Given - 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 [tex]f(t) = 8 cos(4t)[/tex] is applied to the system.
To find - Find the equation of motion if the surrounding medium offers a damping force that is numerically equal to 8 times the instantaneous velocity.
Proof -
Given that,
Mass = 1 slug
We know that, 1 slug = 32 lb
Now,
Force, f = kx
⇒32 = k(2)
⇒k = 16
Now,
Given that, C = 8 ( 8 times the instantaneous velocity)
Now,
The differential equation of motion is equals to
mx'' + Cx' + kx = 8 cos(4t)
⇒x'' + 8x' + 16x = 8 cos(4t) ...........(1)
Let the General solution of equation (1) be
x(t) = x(c) + x(p)
Now,
The auxiliary equation is
m² + 8m + 16 = 0
m² + 4m + 4m + 16 = 0
m (m+4) + 4 (m+4) = 0
⇒(m+4)(m+4) = 0
⇒m = -4, -4
So,
The Complimentary equation becomes
x(c) = [tex]Ae^{-4t} + Bte^{-4t}[/tex] ...........(2)
Now,
Let the particular solution be
x(p) = C cos(4t) + D sin(4t)
x'(p) = -4C sin(4t) + 4D cos(4t)
x''(p) = -16C cos(4t) - 16D sin(4t)
It also satisfy equation (1)
Equation (1) becomes
-16C cos(4t) - 16D sin(4t) + 8 [ -4C sin(4t) + 4D cos(4t) ] + 16 [ C cos(4t) + D sin(4t) ] = 8 cos(4t)
⇒-16C cos(4t) - 16D sin(4t) - 32C sin(4t) + 32D cos(4t) ] + 16C cos(4t) + 16D sin(4t) ] = 8 cos(4t)
⇒-4C sin(4t) + 4B cos(4t) = cos(4t)
By comparing, we get
4B = 1 , A = 0
⇒ B = [tex]\frac{1}{4}[/tex] , A = 0
So, The particular solution becomes
x(p) = [tex]\frac{1}{4}[/tex] sin(4t)
Now,
The General solution becomes
x(t) = [tex]Ae^{-4t} + Bte^{-4t}[/tex] + [tex]\frac{1}{4}[/tex] sin(4t) .......(3)
Now,
Given that, At t = 0, initial velocity is zero and the system starts equilibrium
⇒x(0) = 0, x'(0) = 0
By putting t = 0 in equation (3) , we get
A = 0
Now,
Differentiate equation (3), we get
x'(t) = [tex]-4Ae^{-4t} + Be^{-4t} - 4Bte^{-4t}[/tex] + [tex]\frac{1}{4}[/tex] *4 cos(4t)
Put t = 0, we get
0 = -4A + B + 1
⇒B = -1
∴ we get
The general solution becomes
x(t) = [tex]-te^{-4t}[/tex] + [tex]\frac{1}{4}[/tex] sin(4t)
Equation of motion is x(t) = [tex]-te^{-4t}[/tex] + [tex]\frac{1}{4}[/tex] sin(4t)
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