A method for the numerical analysis of nonlinear normal modes of forced vibrations in strongly non-linear systems with piecewise linear elastic characteristics is proposed. The approach is based on the combination of the Shaw–Pierre method of nonlinear normal modes with the Rauscher technique. As a result of application of this approach, the nonautonomous piecewise linear system is transformed into an autonomous system. For this system, we determine the Shaw–Pierre nonlinear normal modes. We also study the nonlinear torsional vibrations of the power transmission in a three-cylinder transport engine.