  We investigate the role of dense environments in suppressing star formation
by studying $\rm \log_{10}(M_\star/M_\odot) > 9.7$ star-forming galaxies in
nine clusters from the Local Cluster Survey ($0.0137 < z < 0.0433$) and a large
comparison field sample drawn from the Sloan Digital Sky Survey. We compare the
star-formation rate (SFR) versus stellar mass relation as a function of
environment and morphology. After carefully controlling for mass, we find that
in all environments, the degree of SFR suppression increases with increasing
bulge-to-total (B/T) ratio. In addition, the SFRs of cluster and infall
galaxies at a fixed mass are more suppressed than their field counterparts at
all values of B/T. These results suggest a quenching mechanism that is linked
to bulge growth that operates in all environments and an additional mechanism
that further reduces the SFRs of galaxies in dense environments. We limit the
sample to $B/T < 0.3$ galaxies to control for the trends with morphology and
find that the excess population of cluster galaxies with suppressed SFRs
persists. We model the timescale associated with the decline of SFRs in dense
environments and find that the observed SFRs of the cluster core galaxies are
consistent with a range of models including: a mechanism that acts slowly and
continuously over a long (2-5 Gyr) timescale, and a more rapid ($<1$ Gyr)
quenching event that occurs after a delay period of 1-6 Gyr. Quenching may
therefore start immediately after galaxies enter clusters.
