Problem 5

A system is tested on a moving platform

u

-moving base

The position of the mass relative to an inertial frame is y. The actual measurement of the mass

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.