In this work, the governing equation of asymmetric continuous shaft in inertial frame of reference is studied. In particular, determination of the parameter ranges for the stability or instability of the shaft response is the focus of the present work. The governing equations are a fourth-order coupled partial differential equations containing time dependent coefficients. The equations are non-dimensionalized in terms of two parameters related to the average moment of inertia and the difference of moments of inertia about the principal axes. Using the latter as the asymptotic parameter and employing modal superposition, a formal methodology based on perturbation methods is developed to ascertain the stability and instability characteristics. The methodology is applicable to shafts subjected to some of the classical boundary conditions viz. simply supported, cantilever, and fixed–fixed. Similar stability curves are obtained for each mode for these different boundary conditions. The novel non-dimensionalization scheme chosen leads to the stability boundaries as well as the loci of varying speeds to be in the form of straight lines. The intersection of these lines determine the stable and unstable speed ranges of different asymmetric shafts. The results are generalized for different material and geometric properties of the shaft. © 2016 Elsevier Ltd