A mass of 1 slug is attached to a spring whose constant is 5 lb/ft. Initially, the mass is released 1 foot below the equilibrium position with a downward velocity of 5 ft/s, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2 times the instantaneous velocity. (a) Find the equation of motion if the mass is driven by an external force equal to f(t)

The equation of motion if the mass is driven by an external force is ;

   x(t) =

Given data :

mass ( m ) = 1 slug

constant ( k ) = 5 Ib/ft

Initial velocity ( x'(0) ) = 5 ft/s

Initial height ( x ) = 1 ft

Damping force ( β ) = 2

Determine the equation of motion if the mass is driven by an external force

f(t) =

 ( second derivative ) -------- ( 2 )

Insert the given values into equation ( 2 )

The auxiliary equation

m =  applying quadratic formula

m = -1 + 2i ,  and   - 1 - 2i  

Expressing the complementary solutions  

substituting these values

we will get ;

( a + 4b)cos2t  + ( b - 4a )sin2t  =  12cos2t + 3sin2t

comparing the coefficients on both sides

a + 4b = 12 ----- ( 3 )

- 4a + b  = 3  --- ( 4 )

resolving equations 3 and 4 simultaneously

a = 0

b = 3

∴ = 3sin2t

x'(0) = -A + 2B + 6

5 = -1 + 2B + 6 = 5 + 2B

∴ B = 0

Hence we can conclude that the equation of motion if the mass is driven by an external force is

x(t) =

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Explanation:

We are given that

Mass,m=1 slug

Spring constant,k=5lb/ft

x(0)=1 foot

Velocity,x'(0)=5ft/s

Substitute the values

Auxillary equation

By using quadratic formula

Complementary solution

Substitute the values

Comparing coefficient on both sides

....(1)

...(2)

Equation (1) multiply by 4 and then add to equation (2)

Substitute the value of b in equation (2)

Therefore,

General solution ,

Substitute the values

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