  We present a theoretical method to generate a highly accurate {\em
time-independent} Hamiltonian governing the finite-time behavior of a
time-periodic system. The method exploits infinitesimal unitary transformation
steps, from which renormalization group-like flow equations are derived to
produce the effective Hamiltonian. Our tractable method has a range of validity
reaching into frequency regimes that are usually inaccessible via high
frequency $\omega$ expansions in the parameter $h/\omega$, where $h$ is the
upper limit for the strength of local interactions. We demonstrate our approach
on both interacting and non-interacting many-body Hamiltonians where it offers
an improvement over the more well-known Magnus expansion and other high
frequency expansions. For the interacting models, we compare our approximate
results to those found via exact diagonalization. While the approximation
generally performs better globally than other high frequency approximations,
the improvement is especially pronounced in the regime of lower frequencies and
strong external driving. This regime is of special interest because of its
proximity to the resonant regime where the effect of a periodic drive is the
most dramatic. Our results open a new route towards identifying novel
non-equilibrium regimes and behaviors in driven quantum many-particle systems.
