Free flapewise vibration analysis of rotating double-tapered Timoshenko beams

Abstract

Free vibration analysis of a rotating double-tapered Timoshenko beam undergoing flapwise transverse vibration is presented. Using an assumed mode method, the governing equations of motion are derived from the kinetic and potential energy expressions which are derived from a set of hybrid deformation variables. These equations of motion are then transformed into dimensionless forms using a set of dimensionless parameters, such as the hub radius ratio, the dimensionless angular speed ratio, the slenderness ratio, and the height and width taper ratios, etc. The natural frequencies and mode shapes are then determined from these dimensionless equations of motion. The effects of the dimensionless parameters on the natural frequencies and modal characteristics of a rotating double-tapered Timoshenko beam are numerically studied through numerical examples. The tuned angular speed of the rotating double-tapered Timoshenko beam is then investigated. IntroductionThe modal characteristics, i.e., the natural frequencies and the corresponding mode shapes, of rotating beams, especially tapered ones, are critical to the design and analysis of rotating structures, such as the rotating machinery, helicopter blades, wind turbine blades, etc., since most rotating beams in engineering applications are tapered. The modal characteristics of rotating beams often vary significantly from those of non-rotating beams due to the influence of centrifugal inertia force. Due to this significant variation of modal characteristics resulted from rotation, the modal characteristic of rotating beams has been widely investigated.Study of the modal characteristics of rotating beams originated from the pioneering work of Southwell and Gough [1]. Based on the Rayleigh energy theorem, an equation, known as the Southwell equation, that relates the natural frequency to the rotating frequency of a uniform beam was developed. This study was further extended by Liebers [2], Theodorsen [3] and Schilhansl [4], to obtain more accurate natural frequencies of rotating beams. However, due to the large amount of calculation and lack of computational devices, the mode shapes were not available in these investigations. Later, using digital computers, investigations in mode shapes and more accurate natural frequencies were published for uniform rotating and non-rotating beams [5][6][7][8][9][10][11][12][13].The modal characteristics of rotating and non-rotating tapered beams have also been widely investigated. Hodges and Dowell [14] derived the non-linear equations of motion of twisted non-uniform rotor blades based on the Hamilton's Principle and the Newtonian method. Klein [15] analyzed the vibration of a non-rotating tapered beam with a method which combined the advantages of a finite element approach and of a RayleighRitz analysis. Downs [16] applied a dynamic discretization technique to calculate the natural frequencies of a

Size-dependent vibration of functionally graded rotating nanobeams with different boundary conditions based on nonlocal elasticity theory

“…In the present study, the arc length s will be introduced. For both EBT and TBT nanobeams, the geometric relation between the arc length stretch s and the Cartesian variables is given by 10,11,13 uðx;tÞ ¼ sðx;tÞ � 1 2…”

“…Obviously, rotating motion can cause significant changes in structural performance. [10][11][12][13][14] With the development of micro-manufacturing technology in the past two decades, structural elements have become more and more miniaturized, even at the micro/nano scale, for example, micro-turbines, 15 micro-grippers, 16 micro-scanners, 17 micro-actuators, 18,19 etc. The applications of these micromechanisms are widely involved in high-tech fields such as micro/nano electromechanical systems (MEMS/NEMS), biomedical equipment, rocket propulsion systems, etc.…”

Free vibration analysis of rotating tapered Timoshenko beams via variational iteration method

“…The example of rotating tapered beam (E/kG=3.059, γ=0.08, ν=0, δ=0) with linearly varying height and width is taken from Downs (1977), where natural frequencies are extracted via dynamic discretization technique (DDT). The same example was re-analyzed by Ozdemir Ozgumus and Kaya (2008) using differential transformation method (DTM), by Zhu (2012) using Rayleigh-Ritz method (RRM) and by Huang et al. (2013) using power series solution (PPS).…”

Nonlinear forced vibration analysis of a rotating three-dimensional tapered cantilever beam

“…Bulut (2013) investigated the dynamic stability of rotating tapered beams with varying rotating velocity by applying the Floquet theory. Zhu (2012) analyzed the flapwise vibration of rotating double-tapered Timoshenko beams by introducing a set of hybrid deformation variables. Chen et al (2017) developed a variational iteration method to obtain accurate natural frequencies and mode shapes for rotating tapered Timoshenko beams.…”