Natural frequency analysis of gear system with interval shaft misalignment in multidimensions

Abstract

Due to the existing tolerances, the shaft misalignment of gear system is inevitably uncertain and shows interval property, which will cause the natural frequency to fluctuate in an interval range, therefore the traditional dynamics analysis that takes it as a deterministic parameter cannot reflect the interval characteristics of natural frequency. To overcome this conundrum, an improved interval eigenvalue algorithm based on matrix non-negative factorization and the Chebyshev inclusion function was proposed to find the natural frequency range of gear system under such uncertainty. First, the average meshing stiffness range of the gears with interval shaft misalignment is obtained by the Chebyshev interval approach. Then combined with the consideration of the remaining excitation parameter deviation coefficients, the global interval stiffness matrix and mass matrix of a single-stage gear system are obtained. Afterwards, the non-negative factorizations of these two interval matrices are carried out and parameter vertex theory is finally employed to obtain the upper and lower bounds of the eigenvalues. The feasibility of the applied algorithm is verified through comparison with the scanning method. Numerical results show that the uncertainty of shaft misalignment will enormously widen the system's resonance bandwidth and intensify oscillation, these will provide a new and valuable perspective for the tolerance allocation in gearbox design.