  We discuss an integral equation approach that enables fast computation of the
response of nonlinear multi-degree-of-freedom mechanical systems under periodic
and quasi-periodic external excitation. The kernel of this integral equation is
a Green's function that we compute explicitly for general mechanical systems.
We derive conditions under which the integral equation can be solved by a
simple and fast Picard iteration even for non-smooth mechanical systems. The
convergence of this iteration cannot be guaranteed for near-resonant forcing,
for which we employ a Newton--Raphson iteration instead, obtaining robust
convergence. We further show that this integral-equation approach can be
appended with standard continuation schemes to achieve an additional,
significant performance increase over common approaches to computing
steady-state response.
