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Dynamic stiffness method for free vibration of an axially moving beam with generalized boundary conditions

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Abstract

Axially moving beams are often discussed with several classic boundary conditions, such as simply-supported ends, fixed ends, and free ends. Here, axially moving beams with generalized boundary conditions are discussed for the first time. The beam is supported by torsional springs and vertical springs at both ends. By modifying the stiffness of the springs, generalized boundaries can replace those classical boundaries. Dynamic stiffness matrices are, respectively, established for axially moving Timoshenko beams and Euler-Bernoulli (EB) beams with generalized boundaries. In order to verify the applicability of the EB model, the natural frequencies of the axially moving Timoshenko beam and EB beam are compared. Furthermore, the effects of constrained spring stiffness on the vibration frequencies of the axially moving beam are studied. Interestingly, it can be found that the critical speed of the axially moving beam does not change with the vertical spring stiffness. In addition, both the moving speed and elastic boundaries make the Timoshenko beam theory more needed. The validity of the dynamic stiffness method is demonstrated by using numerical simulation.

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Abbreviations

L :

length of the beam between two ends

A :

cross-sectional area

ρ :

beam density

X :

distance from the left end of the beam

T :

time coordinate

v(x, t) :

transverse vibration displacement

φ(x, t) :

angle of rotation of the cross-section

Γ:

axially constant speed

P 0 :

initial axial tension

E :

Young’s modulus of a uniform moving beam

J :

moment of inertial of a uniform moving beam

K :

shape factor

G :

shearing modulus

T :

kinetic energy

K L :

spring stiffness coefficient of the vertical elastic support at the left end

K R :

spring stiffness coefficient of the vertical elastic support at the right end

K t1 :

torsion spring stiffness coefficient at the left end

K t2 :

torsion spring stiffness coefficient at the right end

U f :

bending strain energy

U s :

shear strain energy

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Author information

Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, China

    Hu Ding, Minhui Zhu & Liqun Chen

  2. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai, 200072, China

    Hu Ding & Liqun Chen

  3. Department of Mechanics, Shanghai University, Shanghai, 200444, China

    Liqun Chen

Authors
  1. Hu Ding
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  2. Minhui Zhu
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  3. Liqun Chen
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Corresponding author

Correspondence to Hu Ding.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 11772181 and 11422214), the “Dawn” Program of Shanghai Education Commission (Nos. 17SG38 and 2019-01-07-00-09-E00018), the Key Research Project of Shanghai Science and Techno10gy Commission (No. 18010500100)

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Ding, H., Zhu, M. & Chen, L. Dynamic stiffness method for free vibration of an axially moving beam with generalized boundary conditions. Appl. Math. Mech.-Engl. Ed. 40, 911–924 (2019). https://doi.org/10.1007/s10483-019-2493-8

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  • DOI: https://doi.org/10.1007/s10483-019-2493-8

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