# Frontiers of Mechanical Engineering

 Front. Mech. Eng.    2020, Vol. 15 Issue (3) : 374-389     https://doi.org/10.1007/s11465-019-0580-8
 RESEARCH ARTICLE
Dynamic modeling and coupling characteristics of rotating inclined beams with twisted-shape sections
Jin ZENG1, Chenguang ZHAO1, Hui MA1,2(), Bangchun WEN1
1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China
2. Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University, Shenyang 110819, China
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 Abstract In the existing literature, most studies investigated the free vibrations of a rotating pre-twisted cantilever beam; however, few considered the effect of the elastic-support boundary and the quantification of modal coupling degree among different vibration directions. In addition, Coriolis, spin softening, and centrifugal stiffening effects are not fully included in the derived equations of motion of a rotating beam in most literature, especially the centrifugal stiffening effect in torsional direction. Considering these deficiencies, this study established a coupled flapwise–chordwise–axial–torsional dynamic model of a rotating double-tapered, pre-twisted, and inclined Timoshenko beam with elastic supports based on the semi-analytic method. Then, the proposed model was verified with experiments and ANSYS models using Beam188 and Shell181 elements. Finally, the effects of setting and pre-twisted angles on the degree of coupling among flapwise, chordwise, and torsional directions were quantified via modal strain energy ratios. Results showed that 1) the appearance of torsional vibration originates from the combined effect of flapwise–torsional and chordwise–torsional couplings dependent on the Coriolis effect, and that 2) the flapwise–chordwise coupling caused by the pure pre-twisted angle is stronger than that caused by the pure setting angle. Corresponding Author(s): Hui MA Just Accepted Date: 02 April 2020   Online First Date: 29 April 2020    Issue Date: 03 September 2020
 Cite this article: Jin ZENG,Chenguang ZHAO,Hui MA, et al. Dynamic modeling and coupling characteristics of rotating inclined beams with twisted-shape sections[J]. Front. Mech. Eng., 2020, 15(3): 374-389. URL: http://journal.hep.com.cn/fme/EN/10.1007/s11465-019-0580-8 http://journal.hep.com.cn/fme/EN/Y2020/V15/I3/374
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 Fig.1  Pre-twisted and double-tapered beam: (a) Reference frames, (b) pre-twisted and setting angles, and (c) deformation modes of an arbitrary section. Fig.2  Test rig: (a) Hammer test, (b) sweep test, and (c) data acquisition system. Fig.3  Experimental results under γ(L) = 45°: (a) Frequency response function (FRF) of velocity obtained from hammering test and (b) spectrum cascades obtained from the sweep-frequency test. SFR: Sweep-frequency range; 1, 2, and 3: The first, the second, and the third sections of frequency sweep, respectively. Fig.4  Flow chart of the genetic algorithm. Tab.1  First three-order natural frequency obtained from the proposed model and experiment Fig.5  Finite element model of a pre-twisted beam with elastic-support boundary: (a) Beam188 and (b) Shell181. Fig.6  First five-order dynamic frequencies and modal shapes. Fig.7  Effects of β0 on yd, zd, and rotxd strain energy ratios varying with n: (a) fn1, (b) fn2, (c) fn3, (d) fn4, and (e) fn5. Note: The left column represents MSER in the yd direction; the middle column represents MSER in the zd direction; and the right column represents MSER in the rotxd direction. Fig.8  First five-order modal shapes under n = 0 r/min: (a) fn1, (b) fn2, (c) fn3, (d) fn4, and (e) fn5. Fig.9  Effects of γ(L) on yd, zd, and rotxd strain energy ratios varying with n: (a) fn1, (b) fn2, (c) fn3, (d) fn4, and (e) fn5. Note: The left column represents MSER in the yd direction; the middle column represents MSER in the zd direction; and the right column represents MSER in the rotxd direction.