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

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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 elasticsupport 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 pretwisted 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.

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Abbreviations

A :

Area of an arbitrary beam section

A1, A2, A3, A4, A5 :

Transfer matrices from oxyz to OXYZ

b 0 :

Beam root width

C :

Coriolis matrix

E :

Young’s modulus

F :

External force vector

G :

Shear modulus

h 0 :

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, ky, 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

o d x d y d z d :

Local coordinate system attached to the joint surface on the disk

OXYZ :

Global coordinate system

O r X r Y r Z r :

Rotating coordinate system

q, Q :

Displacement vectors in oxyz and odxdydzd, respectively

q r :

The rth right eigenvector

rp, ṙp :

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 odxdydzd, respectively

U b :

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

Θ, Φ, Ψ:

Angular displacements odxdydzd

Θ′, Φ′, Ψ′:

Angular displacements differentiating with respect to x

Θi(t), Φi(t), Ψi(t):

The ith-mode canonical coordinates related to Θ, Φ, Ψ

α, α̇:

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

κy, κz :

Shear factors along the y- and z-directions, respectively

ξfi, ξci :

Coefficients related to βfi and βci

ρ :

Density

τh, τb :

Tapered ratio of thickness and width, respectively

ν :

Poisson’s ratio

φ ji :

Assumed mode shapes

MSER:

Modal strain energy ratio

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Acknowledgements

This project was supported by the National Natural Science Foundation (Grant Nos. 11972112 and 11772089), the Fundamental Research Funds for the Central Universities (Grant Nos. N170308028, N170306004, N2003014, and N180708009), and Liaoning Revitalization Talents Program (Grant No. XLYC1807008).

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Authors and Affiliations

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

    Jin Zeng, Chenguang Zhao, Hui Ma & Bangchun Wen

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

    Hui Ma

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  1. Jin Zeng
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  2. Chenguang Zhao
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  3. Hui Ma
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Correspondence to Hui Ma.

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Zeng, J., Zhao, C., Ma, H. et al. Dynamic modeling and coupling characteristics of rotating inclined beams with twisted-shape sections. Front. Mech. Eng. 15, 374–389 (2020). https://doi.org/10.1007/s11465-019-0580-8

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