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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) = 6 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.) x(t) = ft

Respuesta :

Answer:

[tex]Y=(\dfrac{3}{16}+t \dfrac{3}{8})e^{-2t}-\dfrac{3}{16}cos 4t[/tex]

Explanation:

Given that m= 1 slug and given that spring stretches by 2 feet so we can find the spring constant K

mg=k x

1 x 32= k x 2

K=16

And also give that damping force is 8 times the velocity so damping constant C=8.

We know that equation for spring mass system

my''+Cy'+Ky=F

Now by putting the values

1 y"+8 y'+ 16y=6 cos 4 t ----(1)

The general solution of equation Y=CF+IP

Lets assume that at steady state the equation of y will be

y(IP)=A cos 4t+ B sin 4t

To find the constant A and B we have to compare this equation with equation 1.

Now find y' and y" (by differentiate with respect to t)

y'= -4A sin 4t+4B cos 4t

y"=-16A cos 4t-16B sin 4t

Now put the values of y" , y' and y in equation 1

1 (-16A cos 4t-16B sin 4t)+8( -4A sin 4t+4B cos 4t)+16(A cos 4t+ B sin 4t)=6sin4 t

So by comparing the coefficient both sides

-16A+32B+16A=0  So B=0

-16 B-32 A+16B=6  So A=-3/16

y=-3/16 cos 4t

Now to find the CF  of differential equation 1

y"+8 y'+ 16y=6 cos 4 t

Homogeneous version of above equation

[tex]m^2+8m+16=0[/tex]

So [tex]CF =(C_1+tC_2)e^{-2t}[/tex]

So the general equation

[tex]Y=(C_1+tC_2)e^{-2t}-3/16 cos 4t[/tex]

Given that t=0 Y=0 So

[tex]C_1=\dfrac{3}{16}[/tex]

t=0 Y'=0 So

[tex]C_2 =\dfrac{3}{8}[/tex]

[tex]Y=(\dfrac{3}{16}+t \dfrac{3}{8})e^{-2t}-\dfrac{3}{16}cos 4t[/tex]

The above equation is the general equation for motion.

batolisis

The equation of motion if the surrounding medium offers a damping force which is numerically equal to 8 times the instantaneous velocity is :

[tex]Y = (\frac{3}{16} + t\frac{3}{8} )e^{-2t} - \frac{3}{16} cos4t[/tex]

Determine the equation of motion

To determine the equation of motion we will consider the given variables

mass  = 1 slug

damping constant = 8

First step : calculate the spring constant

mg = kx

where ; m = 1,  g = 32 ft/s², x = 2 feet, k = ?

therefore ; x = mg / k = 1 * 32 / 2

                                   = 16

Next step : express the equation for spring mass constant

my''+Cy' + Ky = F ---- ( 1 )

where ; m = 1 slug, C = 8, K = 16,  F = 6 cos 4 t

input the values into the equation above

1 y"+8 y'+ 16y = 6 cos 4 t   ---- ( 2 )

General solution of equation : Y= CF + IP

Therefore for steady state

y(IP)=A cos 4t+ B sin 4t ----- ( 3 )

Next step : Derive the values of A and B by equating  ( 2 ) and ( 3 )

y'= -4A sin 4t + 4B cos 4t

y"= -16A cos 4t - 16B sin 4t

Input the values y' and y" into equation ( 2 ) then compare the coefficient on both sides of the equation

y = -3/16 cos 4t

The homogenous equation of equation ( 2 )

m² + 8m + 16 = 0

Therefore the CF value of equation ( 2 ) is

CF = ( C₁ + tC₂ )[tex]e^{-2t}[/tex]

Hence the general equation can be expressed as

Y = ( C₁ + tC₂ )[tex]e^{-2t}[/tex]  - 3/16 cos 4t

since ; y = 0, y' = 0, t = 0

C₁ = 3/16,   C₂ = 3/8

Therefore the equation of motion is [tex]Y = (\frac{3}{16} + t\frac{3}{8} )e^{-2t} - \frac{3}{16} cos4t[/tex]

Learn more about equation of motion : https://brainly.com/question/25951773

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