Dynamic modeling and coupling characteristics of rotating inclined beams with twisted-shape sections

doi:10.1007/s11465-019-0580-8

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 MA^1,²(

), 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:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-019-0580-8
https://academic.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.

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 y^d, z^d, and rotx^d strain energy ratios varying with n: (a) f_n1, (b) f_n2, (c) f_n3, (d) f_n4, and (e) f_n5. Note: The left column represents MSER in the y^d direction; the middle column represents MSER in the z^d direction; and the right column represents MSER in the rotx^d direction.

Fig.8 First five-order modal shapes under n = 0 r/min: (a) f_n1, (b) f_n2, (c) f_n3, (d) f_n4, and (e) f_n5.

Fig.9 Effects of γ(L) on y^d, z^d, and rotx^d strain energy ratios varying with n: (a) f_n1, (b) f_n2, (c) f_n3, (d) f_n4, and (e) f_n5. Note: The left column represents MSER in the y^d direction; the middle column represents MSER in the z^d direction; and the right column represents MSER in the rotx^d 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
I_y, I_z	Second moment of area in the y- and z-axes, respectively
J	Torsional moment of inertia
k_x, k_y, k_z, k_r_x, k_r_y, k_r_z	Linear and angular support stiffness in o^dx^dy^dz^d
K_e, K_c, K_s, K_acc	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
o^dx^dy^dz^d	Local coordinate system attached to the joint surface on the disk
OXYZ	Global coordinate system
O^rX^rY^rZ^r	Rotating coordinate system
q, Q	Displacement vectors in oxyz and o^dx^dy^dz^d, respectively
q_r	The rth right eigenvector
$r p$ , $r ˙ p$	Coordinate and velocity vectors, respectively
R_d	Disk radius
t	Time
T_b	Kinetic energy
u, v, w; U, V, W	Linear displacement components in oxyz and o^dx^dy^dz^d, respectively
U_b	Potential energy
U_i(t), V_i(t), W_i(t)	The ith-mode canonical coordinates related to U, V, and W
U', V', W'	Linear displacements differentiating with respect to x
X^p, Y^p, Z^p; x, y, z	Coordinates of an arbitrary point “p” on the beam section in OXYZ and oxyz, respectively
Q, F, Y	Angular displacements o^dx^dy^dz^d
Q′, F′, Y′	Angular displacements differentiating with respect to x
Q_i(t), F_i(t), Y_i(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
β_a_i, β_f_i, β_c_i, β_t_i	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
k_y, k_z	Shear factors along the y- and z-directions, respectively
x_f_i, x_c_i	Coefficients related to β_f_i and β_c_i
ρ	Density
t_h, t_b	Tapered ratio of thickness and width, respectively
$ν$	Poisson’s ratio
$φ j i$	Assumed mode shapes
MSER	Modal strain energy ratio

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