  For general nonlinear mechanical systems, we derive closed-form,
reduced-order models up to cubic order based on rigorous invariant manifold
results. For conservative systems, the reduction is based on Lyapunov Subcenter
Manifold (LSM) theory, whereas for damped-forced systems, we use Spectral
Submanifold (SSM) theory. To evaluate our explicit formulas for the reduced
model, no coordinate changes are required beyond an initial linear one. The
reduced-order models we derive are simple and depend only on physical and modal
parameters, allowing us to extract fundamental characteristics, such as
backbone curves and forced-response curves, of multi-degree-of-freedom
mechanical systems. To numerically verify the accuracy of the reduced models,
we test the reduction formulas on several mechanical systems, including a
higher-dimensional nonlinear Timoshenko beam.
