Arch Appl Mech (2013) 83:765–781 DOI 10.1007/s00419-012-0716-3 ORIGINAL Mostafa Talebitooti Three-dimensional free vibration analysis of rotating laminated conical shells: layerwise differential quadrature (LW-DQ) method Received: 3 May 2012 / Accepted: 1 November 2012 / Published online: 21 November 2012 © Springer-Verlag Berlin Heidelberg 2012 Abstract This paper focuses on the free vibration analysis of thick, rotating laminated composite conical shells with different boundary conditions based on the three-dimensional theory, using the layerwise differential quadrature method (LW-DQM). The equations of motion are derived applying the Hamilton’s principle. In order to accurately account for the thickness effects, the layerwise theory is used to discretize the equations of motion and the related boundary conditions through the thickness of the shells. Then, the equations of motion as well as the boundary condition equations are transformed into a set of algebraic equation applying the DQM in the meridional direction. This study demonstrates the applicability, accuracy, stability and the fast rate of convergence of the present method, for free vibration analyses of rotating thick laminated conical shells. The presented results are compared with those of other shell theories obtained using conventional methods and a special case where the angle of the conical shell approaches zero, that is, a cylindrical shell and excellent agreements are achieved. Keywords Rotating conical shell · Layerwise theory · Natural frequency · Differential quadrature method 1 Introduction Rotating laminated conical shells are increasingly being used in many engineering applications such as the drive shafts of gas turbines, high-speed centrifugal separators, motors and rotor systems because of their strength to weight ratio. Hence, it is of a great importance to understand the vibration behavior of rotating laminated conical shells for the design of aforementioned structures. Most studies were restricted to the vibration analysis of non-rotating and rotating conical shells based on classical laminated shell theory (CLT). They include the works by Sivadas [1] and Lam and Hua [2] on the rotating conical shells with simply supported boundary condition as well as a discussion on influence of boundary condition on rotating conical shell by Lam and Hua [3] and Ng et al. [4]. For non-rotating conical shell, many papers are involved to discuss the influence of boundary conditions on the frequency characteristics, particularly the works done by Sivadas and Ganesan [5], Thambiratnam and Zhuge [6] and Tong [7, 8]. This theory cannot be used for thick shells and even thin shells when the number of circumferential waves increases as a result of neglecting shear deformation and rotary inertia effects in CLT. The refinement of thin-shell theories has resulted in a number of the so-called first- and higher-order shear deformation theories to include the effects of transverse shear deformation [9–13]. These theories behave much more accurately than the thin-shell theories for the analysis of slightly thick shells but are still inadequate for the analysis of thick shells. In order to analyze the thick conical shells precisely, the three- dimensional theory should be used to account all the transverse stress and strain components. The literature shows that the dynamic analysis of rotating conical shells based on 3-D theory is complex and the common M. Talebitooti (B ) Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, 16846-13114 Tehran, Iran E-mail: mtalebi@iust.ac.ir