Dynamic stiffness method for inplane free vibration of rotating beams including Coriolis effects

Abstract

a b s t r a c tThe paper addresses the in-plane free vibration analysis of rotating beams using an exact dynamic stiffness method. The analysis includes the Coriolis effects in the free vibratory motion as well as the effects of an arbitrary hub radius and an outboard force. The investigation focuses on the formulation of the frequency dependent dynamic stiffness matrix to perform exact modal analysis of rotating beams or beam assemblies. The governing differential equations of motion, derived from Hamilton's principle, are solved using the Frobenius method. Natural boundary conditions resulting from the Hamiltonian formulation enable expressions for nodal forces to be obtained in terms of arbitrary constants. The dynamic stiffness matrix is developed by relating the amplitudes of the nodal forces to those of the corresponding responses, thereby eliminating the arbitrary constants. Then the natural frequencies and mode shapes follow from the application of the Wittrick-Williams algorithm. Numerical results for an individual rotating beam for cantilever boundary condition are given and some results are validated. The influences of Coriolis effects, rotational speed and hub radius on the natural frequencies and mode shapes are illustrated.

Nonlinear free vibrations of centrifugally stiffened uniform beams at high angular velocity

“…This point corresponds to the critical rotating velocity, which is independent of the hub radius ratio δ h . These observations are also mentioned in [18,22], and showed graphically in other studies [9,40]. As shown in [9], this singular point occurs regardless of the Coriolis effect included or not (the singular point concerns the fundamental mode if the Coriolis effect is considered and the second mode if the Coriolis effect is not considered).…”

“…The critical point corresponds to Ω ⋆ ¼ βπ=2 as it is demonstrated in Eq. (18) and it is independent of the hub radius ratio δ h as it is showed in Fig. 4, where several curves are plotted for different hub radius ratios.…”

“…4), corresponding to Ω ⋆ ¼ 70π=2 ¼ 109:96. Another curve is presented in [18] (Fig. 6) corresponding to Ω ⋆ ¼ 100π=2 ¼ 157:08.…”

“…The determination of their dynamic characteristics (natural frequencies and mode shapes) are of great importance in design and control. A significant number of studies have been published on the bending vibrations of rotating beams, with the main objective of predicting the natural frequencies and associated mode shapes, as well as investigating their variations with angular velocity and other effects such as hub radius, taper and shear deformation [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. A review of several studies on rotating beams has been presented by Bazoun [19].…”

“…At high rotating speed, instability occurs at a certain critical value. It was explained in [18,22] that such static instability is explained by the fact that smalldeflection (linear) theory is used where large axial deformations induced by centrifugal force take place. The critical angular velocity is characterized by a single bifurcation point where the natural frequency becomes zero.…”