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This article concerns a class of beam equations with damping on rectangulartori. When the generators satisfy certain relationship, by excluding some valueof two model parameters, we prove that such models admit small amplitudequasi-periodic travelling wave solutions with two frequencies, which arecontinuations of two rotating wave solutions with one frequency. This resultholds not only for an isotropic torus, but also for an anisotropic torus. Theproof is mainly based on a Lyapunov--Schmidt reduction together with theimplicit function theorem.

Quasi-periodic travelling waves for a class of damped beams on rectangular tori