motion, however, is its acceleration, ÿ, which is provided by an accelerometer attached to the mass. The mass, spring rate, and damping rate are 1, 4 and 8, respectively, so the ODE is ÿ + 4y + 8y = 4ů + 8u. 5 1. Solve the unforced (u= 0) IVP for y with y(0) = -1, g(0) = 1, by computing the characteristic roots A₁ and ₂ and letting y(t) = Aet + Bet, t≥ 0. The parameters A and B are determined by enforcing the initial conditions. Once y(t) is determined (it is a real-valued signal), compute the mass acceleration by differentiation (note: there are no discontinuities in y or any of its time derivatives in a neighborhood of t = 0). 2. Now use the unilateral Laplace transform to find the acceleration due to these initial con- ditions (u= 0 still) by following these steps: 1) apply the transform to the ODE, 2) iso- late the expression for ŷ, 3) use the Derivative Theorem again for the mass acceleration, ỹ = s²ŷ - sy(0) - (0), 4) "invert" the expression for to determine ÿ(t) for, t≥ 0. Compare to the result from Part 1. Note that when differentiating a dependent variable it is always necessary to apply the Derivative Theorem to account for possible non-zero initial conditions associated with it. 3. Now consider a forced IVP in which the ICs are the same as Part 1, however, u(t) = tµ(t), t20. Use the unilateral Laplace transform to determine ÿ, t≥ 0. Note that "external" inputs are always considered "abrupt", so u(0) = 0, ú(0) = 0, etc. Hint: develop the expression for first, then apply the Derivative Theorem to determine , then reverse engineer to time-domain signals. Make sure to distinguish which signals in ÿ are abrupt, versus those that are not.
Fig: 1