Parametric resonance of truncated conical shells rotating at periodically varying angular speed

• Parametric resonance of rotating conical shells with time-varying speed is studied.
• Equations of motion are derived based upon the thin shell theory and GDQ method.
• Instability regions for various modes and boundary conditions are obtained.
• Effects of speed and geometric parameters on instability regions are discussed.

Parametric resonance of a truncated conical shell rotating at periodically varying angular speed is studied in this paper. Based upon the Love׳s thin shell theory and generalized differential quadrature (GDQ) method, the equations of motion of a rotating conical shell are derived. The time-dependent rotating speed is assumed to be a small and sinusoidal perturbation superimposed upon a constant speed. Considering the periodically rotating speed, the conical shell system is a parametric excited system of the Mathieu–Hill type. The improved Hill׳s method is utilized for parametric instability analysis. Both the primary and combination instability regions for various natural modes and boundary conditions are obtained numerically. The effects of relative amplitude and constant part of periodically rotating speed and cone angle on the instability regions are discussed in detail. It is shown that for the natural mode with lower circumferential wavenumber, only the primary instability regions exist. With the increasing circumferential wavenumber, the instability widths are reduced significantly and the combination instability region might appear. The results for different boundary conditions are substantially similar. Increasing the constant rotating speed (or cone angle) all lead to the movements of instability regions and the appearance of combination instability region. The former will cause the instability width increasing, while the latter will reduce the instability width. The variation of length-to-radius ratio only causes the movements of instability regions.