This study focuses on the steady-state periodic response of supercritically transporting viscoelastic beams. In the supercritical speed range, forced vibrations are investigated for traveling beams via the multiscale analysis with a numerical confirmation. The forced vibration is excited by the spatially uniform and temporally harmonic vibration of the supporting foundation. A nonlinear integro-partial-differential equation is used to determine steady responses. The straight equilibrium configuration bifurcates in multiple equilibrium positions at supercritical translating speeds. The equation is cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for nontrivial equilibrium configuration. The natural frequencies and modes of the supercritically traveling beams are analyzed via the Galerkin method for the linear standard form with space-dependent coefficients under the simply supported boundary conditions. Based on the natural frequencies and modes, the method of multiple scales is applied to the governing equation to determine steady-state responses. To confirm results via the method of multiple scales, a finite difference scheme is developed to calculate steady-state response numerically. Quantitative comparisons demonstrate that the approximate analytical results have rather high precision. Numerical results are also presented to show the contributions of foundation vibration amplitude, viscoelastic damping, and nonlinearity to the response amplitude for the first and the second mode.

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