Find the response of the mass m for the setup shown below Wh

Find the response of the mass m for the setup shown below: Where the force applied to the system could be shown as:

Solution

>> Firstly, as both springs are in parallel combination, so let\'s replace these two springs by a single one having equivalent stiffness as K = K1 + K2

=> K = 200 N/m

>> Now, in our system a mass , \"m\" is attached to a spring of spring constant, \"K\" = 200 N/m and a force, f(t) is acting on it.

>> So, considering the mass and writting its equation of motion,

m(d²x/dt²) + Kx = f(t) ..........(1).......

>> As, from graph,

f(t) = 3 , 0 <= t < 4

= 7 - t , 4 <= t < 7

As, m = 1 kg and K = 200 N/m

>> Between t = [0, 4]

(d²x/dt²) + 200x = 3

=> C.F. (Complementary Function) = A*sin(14.14*t) = sin(14.14*t)

>> P.I. (Partial Integral) = 3/200 = 0.015

>> So, x = sin(14.14*t) + 0.015

>> Between t = [4, 7]

(d²x/dt²) + 200x = 7 - t

=> C.F. (Complementary Function) = A*sin(14.14*t) = sin(14.14*t)

>> P.I. (Partial Integral) = (7 - t)/(D² + 200) = 0.035 - 0.005*t

=> x = sin(14.14*t) + 0.035 - 0.005*t

>> So, Required Response =>

x = sin(14.14*t) + 0.015    , 0 <= t < 4

= sin(14.14*t) + 0.035 - 0.005*t , 4 <= t < 7