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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.

Keywords elastic-support boundary      pre-twisted beam      semi-analytic method      modal strain energy ratio      torsional vibration     
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
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.
Mode Frequencies obtained from hammering test (benchmark)/Hz Frequencies obtained from sweep-frequency test/Hz Frequencies obtained from proposed model/Hz
Fixed support Elastic support
fn1 164.3 165.0 (0.426%) 160.4 (−2.374%) 159.6 (−2.861%)
fn2 578.5 580.5 (0.346%) 598.2 (3.405%) 580.9 (0.415%)
fn3 1149.0 1150.0 (0.087%) 1208.6 (5.187%) 1149.8 (0.070%)
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.
A Area of an arbitrary beam section
A1, A2, A3, A4, A5 Transfer matrices from oxyz to OXYZ
b0 Beam root width
C Coriolis matrix
E Young’s modulus
F External force vector
G Shear modulus
h0 Beam root thickness
Iy, Iz Second moment of area in the y- and z-axes, respectively
J Torsional moment of inertia
kx, ky, kz, krx, kry, krz Linear and angular support stiffness in odxdydzd
Ke, Kc, Ks, Kacc Structural, centrifugal stiffening, spin softening, and angular acceleration-induced stiffness matrices, respectively
L Beam length
M Mass matrix
n Rotating speed
N Modal truncation number
oxyz Local coordinate system located at the arbitrary beam section
odxdydzd Local coordinate system attached to the joint surface on the disk
OXYZ Global coordinate system
OrXrYrZr Rotating coordinate system
q, Q Displacement vectors in oxyz and odxdydzd, respectively
qr The rth right eigenvector
rp, r˙p Coordinate and velocity vectors, respectively
Rd Disk radius
t Time
Tb Kinetic energy
u, v, w; U, V, W Linear displacement components in oxyz and odxdydzd, respectively
Ub Potential energy
Ui(t), Vi(t), Wi(t) The ith-mode canonical coordinates related to U, V, and W
U', V', W' Linear displacements differentiating with respect to x
Xp, Yp, Zp; x, y, z Coordinates of an arbitrary point “p” on the beam section in OXYZ and oxyz, respectively
Q, F, Y Angular displacements odxdydzd
Q′, F′, Y Angular displacements differentiating with respect to x
Qi(t), Fi(t), Yi(t) The ith-mode canonical coordinates related to Q, F, Y
α, α˙ Angular displacement and angular velocity, respectively
β, β0 Setting angles of the arbitrary and root sections, respectively
βai, βfi, βci, βti Characteristic roots
γ, γ(L) Pre-twisted angles of the arbitrary and tip sections, respectively
δ Variation symbol
θ, φ, ψ Angular displacements in oxyz
ky, kz Shear factors along the y- and z-directions, respectively
xfi, xci Coefficients related to βfi and βci
ρ Density
th, tb Tapered ratio of thickness and width, respectively
ν Poisson’s ratio
φji Assumed mode shapes
MSER Modal strain energy ratio
  
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