  We propose a reformulation for the integral equations approach of Jain,
Breunung \& Haller [Nonlinear Dyn. 97, 313--341 (2019)] to steady-state
response computation for periodically forced nonlinear mechanical systems. This
reformulation results in additional speed-up and better convergence. We show
that the solutions of the reformulated equations are in one-to-one
correspondence with those of the original integral equations and derive
conditions under which a collocation type approximation converges to the exact
solution in the reformulated setting. Furthermore, we observe that model
reduction using a selected set of vibration modes of the linearized system
substantially enhances the computational performance. Finally, we discuss an
open-source implementation of this approach and demonstrate the gains in
computational performance using three examples that also include nonlinear
finite-element models.
