turning forcing on and off
Often a push acts only for a while: a thruster fires for ten seconds, a heater runs until noon, a button is held then released. Turning forcing on and off is the practical skill of writing such finite-duration or delayed pushes as clean formulas — using Heaviside steps as gates — so the Laplace transform can swallow them whole.
The building blocks are simple. To start a forcing g(t) at time a, multiply by a step: g(t) u(t - a) switches it on at t = a and leaves it on. To confine it to a window from a to b, subtract a second step: g(t) [u(t - a) - u(t - b)] is g only between a and b and zero outside. A single brief pulse — on for a moment, off again — is just a narrow window. Stack several windows and you build a staircase, a burst pattern, anything that toggles. The key discipline, before transforming, is to rewrite each gated piece so its argument reads (t - a), matching the second shifting theorem, so each block transforms to e^(-as) times a table entry.
This is the everyday face of the whole field: the step function provides the gates, the second shifting theorem provides the transform of each delayed gate, and the result is that ANY on-again-off-again forcing reduces to a sum of exponential-tagged terms in the s-domain. From there the ODE solves algebraically, and the solution automatically knows when each push starts and stops.
A unit force acting only on 2 ≤ t < 5 is f(t) = u(t - 2) - u(t - 5), whose transform is (e^(-2s) - e^(-5s))/s — a single tidy expression for a force that turns on at 2 and off at 5.
Two steps gate a force into a finite window; each step contributes its own exponential tag.
When you gate g(t) with u(t - a), the transform needs g written as a function of (t - a), not t. For example u(t - a) g(t) and u(t - a) g(t - a) have DIFFERENT transforms; only the second fits the second shifting theorem directly.