Question

Consider a damped forced mass-spring system with m = 1, γ = 2, and k = 26, under the influence of an external force F(t) = 82 cos(4t).

a) (8 points) Find the position u(t) of the mass at any time t, if u(0) = 6 and u 0 (0) = 0.

b) (4 points) Find the transient solution uc(t) and the steady state solution U(t). How would you characterize these two solutions in terms of their behavior in time?

c) (4 points) Find the amplitude R and the phase angle δ for this motion and express U(t) as a single trigonometric term: U(t) = R cos(ωt − δ).

d) (4 points) Justify the following description of what happens: “the transient motion uc(t) dies out with the passage of time, leaving only the steady state periodic motion U(t)

Answer 1

Any doubt in aamy step then comment below.. i will explain you..

In part b .. we see clearly that both solution have oscillatory motion but in transient solutionn, amplitude of motion decrease exponentially with time...

In part d) ... Transient solution have exponential term with negative exponents... so thts why it decreases with time and tends to 0 as t tends to infinity...so we have only steady solution...