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Coupled flapwise-chordwise-axial-torsional dynamic responses of rotating pre-twisted and inclined cantilever beams subject to the base excitation

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Abstract

A rotating pre-twisted and inclined cantilever beam model (RPICBM) with the flapwise-chordwise-axial-torsional coupling is established with the Hamilton principle and the finite element (FE) method. The effectiveness of the model is verified via comparisons with the literatures and the FE models in ANSYS. The effects of the setting and pre-twisted angles on the dynamic responses of the RPICBM are analyzed. The results show that: (i) the increase in the setting or pre-twisted angle results in the increases in the first-order flapwise and torsional frequencies while the decrease in the first-order chordwise frequency under rotating conditions; (ii) a positive/negative setting angle leads to a positive/negative constant component, while a positive/negative pre-twisted angle leads to a negative/positive constant component; (iii) when the rotation speed is non-zero, the pre-twisted angle or non-zero setting angle will result in the coupled flapwise-chordwise-axial-torsional vibration of the RPICBM under axial base excitation.

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Abbreviations

A :

cross-sectional area

A 0 :

amplitude of the base excitation

OXYZ :

global coordinate system

OX r Y r Z r :

rotating coordinate system

o b x b y b z b :

local coordinate system

o e x e y e z e :

element coordinate system

A i :

coordinate-transformation matrix

b :

beam width at the arbitrary section

b 0 :

beam width at the root section

c :

a constant

C :

the Rayleigh damping

E :

Young’s modulus

F b :

base excitation force in the x-direction

F c :

centrifugal force

F e :

element nodal force vector in oexeyeze

\(F_{\rm{r}}^{\rm{e}}\) :

element nodal force vector in OXrYrZr

F :

global nodal force vector

f e :

excitation frequency

G :

shear modulus

f i :

concerned end frequency

G e :

element Coriolis matrix in oexeyeze

\(G_{\rm{r}}^{\rm{e}}\) :

element Coriolis matrix in OXrYrZr

G :

global Coriolis matrix

h :

beam thickness at the arbitrary section

h 0 :

beam thickness at the root section

I x :

area moment of the cross-section inertia about ox3

I y :

area moment of the cross-section inertia about oy3

I z :

area moment of the cross-section inertia about oz3

J :

torsional moment of the inertia of a pre-twisted rectangular beam

J s :

torsional moment of the inertia of a straight rectangular beam

J a :

additional torsional moment of the inertia caused by γ(L)

\(K_{\rm{e}}^{\rm{e}}\) :

element structural stiffness matrix in oexeyeze

\(K_{\rm{c}}^{\rm{e}}\) :

element centrifugal stiffening matrix in oexeyeze when \(\dot{\alpha}=1 \rm{rad}\cdot{s}^-1\)

\(K_{\rm{s}}^{\rm{e}}\) :

spin softening matrix in oexeyeze when \(\dot{\alpha}=1 \rm{rad}\cdot{s}^-1\)

\(K_{\rm{e,r}}^{\rm{e}}\) :

element structural stiffness matrix in OXrYrZr

\(K_{\rm{c,r}}^{\rm{e}}\) :

element centrifugal stiffening matrix in OXrYrZr when \(\dot{\alpha}=1 \rm{rad}\cdot{s}^-1\)

\(K_{\rm{s,r}}^{\rm{e}}\) :

element spin softening matrix in OXrYrZr when \(\dot \alpha=1\;{\rm{rad}}\cdot{{\rm{s}}^{ - 1}}\)

K e :

global structural stiffness matrix

K c :

global centrifugal stiffening matrix

K s :

global spin softening matrix

\(K_{\rm{acc}}^{\rm{e}}\) :

stiffness matrix related to \(\ddot{\alpha}\) in oexeyeze

\(K_{\rm{acc,r}}^{\rm{e}}\) :

stiffness matrix related to \(\ddot{\alpha}\) in OXrYrZr

K acc :

global stiffness matrix related to \(\ddot \alpha \)

L :

beam length

l k :

kth element length

M e :

element mass matrix in oexeyeze

\(M_{\rm{r}}^{\rm{e}}\) :

element mass matrix in OXrYrZr

M :

global mass matrix

N :

total number of the beam elements

n :

rotation speed

R d :

disk radius

R k :

x coordinate of oe in OXrYrZr

r P :

coordinate vector of the arbitrary point P in OXYZ

ṙ P :

velocity vector of the arbitrary point P in OXYZ

T k :

kinetic energy of the kth rotating Timoshenko beam element

T :

transformation matrix from OXrYrZr to oexeyeze

t 1 :

starting moment of the variation

t 2 :

ending moment of the variation

U k :

potential energy of the fcth rotating Timoshenko beam element; linear displacements of the centroid of an arbitrary beam cross-section along the positive xe-, ye-, and ze-axes

\(\dot{u}, \dot{v}, \dot{w}\) :

velocities of the centroid of an arbitrary beam cross-section along the positive xe-, ye-, and ze-axes

\(\ddot{u}, \ddot{v}, \ddot{w}\) :

accelerations of the centroid of an arbitrary beam cross-section along the positive xe-, ye- and ze-axes

u′, v′, w′:

the derivative of u, v, and w versus the coordinate x

\(X_u^{\rm{r}}\) :

base excitation along the Xr-direction

x k :

x coordinate of oe in obxbybzb

(X P, Y P, Z P):

coordinate components of Point P along the X-, Y-, and Z-directions

(x, y, z):

coordinate components of Point P along the x3-, y3-, and z3-directions

Y I :

first-order flapwise frequency

Y II :

second-order flapwise frequency

Z I :

first-order chordwise frequency

Z II :

second-order chordwise frequency

α :

rotating angle

\(\dot \alpha \) :

angular velocity

\(\ddot \alpha \) :

angular acceleration

α 1, β 1 :

Rayleigh damping coefficients

β 0 :

initial setting angle

β k :

setting angle of the kth Timoshenko beam element

γ k :

pre-twisted angle of the fcth Timoshenko beam element

γ(L):

pre-twisted angle at L

δ :

variational symbol

δ e :

element nodal displacement vector in oexeyeze

\(\delta_{\rm{r}}^{\rm{e}}\) :

element nodal displacement vector in OXrYrZr

ζ :

correction factor

θ :

angular displacement with respect to the xe-axis

\(\dot\theta\) :

velocity with respect to the xe-axis

\(\ddot\theta\) :

acceleration with respect to the xe-axis

θ′:

the derivative of θ versus the coordinate x

θ I :

the first-order torsional frequency

κ y :

shear coefficient in the y-direction

κ z :

shear coefficient in the z-direction

ξ 1, ξ 2 :

modal damping ratios

ρ :

density

τ b :

breadth taper

T h :

thickness taper

υ :

Poisson’s ratio

φ :

angular displacement with respect to the ze-axis

\(\dot\varphi\) :

velocity with respect to the ze-axis

\(\ddot\varphi\) :

acceleration with respect to the ze axis

φ′:

the derivative of φ versus the coordinate x

χ :

a symbol representing global matrices

\(\chi_k^{\rm{e}}\) :

a symbol representing element matrices

ϕ :

angular displacement with respect to the ye-axis

\(\dot\phi\) :

velocity with respect to the ye-axis

\(\ddot\phi\) :

acceleration with respect to the ye-axis

ϕ′:

derivative of ϕ versus the coordinate x

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Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, China

    Jin Zeng, Hui Ma, Kun Yu, Zhitao Xu & Bangchun Wen

  2. Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University, Shenyang, 110819, China

    Hui Ma

Authors
  1. Jin Zeng
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  2. Hui Ma
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  3. Kun Yu
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  4. Zhitao Xu
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  5. Bangchun Wen
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Correspondence to Hui Ma.

Additional information

Citation: ZENG, J., MA, H., YU, K., XU, Z. T., and WEN, B. C. Coupled flapwise-chordwiseaxial-torsional dynamic responses of rotating pre-twisted and inclined cantilever beams subject to the base excitation. Applied Mathematics and Mechanics (English Edition) 40(8), 1053–1082 (2019) https://doi.org/10.1007/s10483-019-2506-6

Project supported by the National Natural Science Foundation of China (No. 11772089), the Fundamental Research Funds for the Central Universities of China (Nos. N170308028 and N170306004), the Program for the Innovative Talents of Higher Learning Institutions of Liaoning of China (No. LR2017035), and the LiaoNing Revitalization Talents Program of China (No. XLYC1807008)

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Zeng, J., Ma, H., Yu, K. et al. Coupled flapwise-chordwise-axial-torsional dynamic responses of rotating pre-twisted and inclined cantilever beams subject to the base excitation. Appl. Math. Mech.-Engl. Ed. 40, 1053–1082 (2019). https://doi.org/10.1007/s10483-019-2506-6

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  • DOI: https://doi.org/10.1007/s10483-019-2506-6

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