Applied Mathematics and Mechanics >
Natural vibration and critical velocity of translating Timoshenko beam with non-homogeneous boundaries
Received date: 2024-05-20
Online published: 2024-08-27
Supported by
the YEQISUN Joint Funds of the National Natural Science Foundation of China(U2341231);the National Natural Science Foundation of China(12172186);Project supported by the YEQISUN Joint Funds of the National Natural Science Foundation of China (No. U2341231) and the National Natural Science Foundation of China (No. 12172186)
Copyright
In most practical engineering applications, the translating belt wraps around two fixed wheels. The boundary conditions of the dynamic model are typically specified as simply supported or fixed boundaries. In this paper, non-homogeneous boundaries are introduced by the support wheels. Utilizing the translating belt as the mechanical prototype, the vibration characteristics of translating Timoshenko beam models with non-homogeneous boundaries are investigated for the first time. The governing equations of Timoshenko beam are deduced by employing the generalized Hamilton's principle. The effects of parameters such as the radius of wheel and the length of belt on vibration characteristics including the equilibrium deformations, critical velocities, natural frequencies, and modes, are numerically calculated and analyzed. The numerical results indicate that the beam experiences deformation characterized by varying curvatures near the wheels. The radii of the wheels play a pivotal role in determining the change in trend of the relative difference between two beam models. Comparing the results unearths that the relative difference in equilibrium deformations between the two beam models is more pronounced with smaller-sized wheels. When the two wheels are of equal size, the critical velocities of both beam models reach their respective minima. In addition, the relative difference in natural frequencies between the two beam models exhibits nonlinear variation and can easily exceed 50%. Furthermore, as the axial velocities increase, the impact of non-homogeneous boundaries on modal shape of translating beam becomes more significant. Although dealing with non-homogeneous boundaries is challenging, beam models with non-homogeneous boundaries are more sensitive to parameters, and the differences between the two types of beams undergo some interesting variations under the influence of non-homogeneous boundaries.
Yanan LI, Jieyu DING, Hu DING, Liqun CHEN . Natural vibration and critical velocity of translating Timoshenko beam with non-homogeneous boundaries[J]. Applied Mathematics and Mechanics, 2024 , 45(9) : 1523 -1538 . DOI: 10.1007/s10483-024-3148-7
| 1 | RAJ, S. K., SAHOO, B., NAYAK, A. R., and PANDA, L. N. Parametric analysis of an axially moving beam with time-dependent velocity, longitudinally varying tension and subjected to internal resonance. Archive of Applied Mechanics, 94 (1), 1- 20 (2024) |
| 2 | CHENG, Y., WU, Y. H., GUO, B. Z., and WU, Y. X. Stabilization and decay rate estimation for axially moving Kirchhoff-type beam with rotational inertia under nonlinear boundary feedback controls. Automatica, 163, 11597 (2024) |
| 3 | MOSLEMI, A., KHADEM, S. E., KHAZAEE, M., and DAVARPANAH, A. Nonlinear vibration and dynamic stability analysis of an axially moving beam with a nonlinear energy sink. Nonlinear Dynamics, 104 (3), 1955- 1972 (2021) |
| 4 | WANG, L. H., HU, Z. D., ZHONG, Z., and JU, J. W. Dynamic analysis of an axially translating viscoelastic beam with an arbitrarily varying length. Acta Mechanica, 214 (3-4), 225- 244 (2010) |
| 5 | LONG, S. B., ZHAO, X. Z., and SHANGGUAN, W. B. Method for estimating vibration responses of belt drive systems with a nonlinear tensioner. Nonlinear Dynamics, 100 (3), 2315- 2335 (2020) |
| 6 | SZE, K. Y., CHEN, S. H., and HUANG, J. L. The incremental harmonic balance method for nonlinear vibration of axially moving beams. Journal of Sound and Vibration, 281 (3-5), 611- 626 (2005) |
| 7 | HU, Y. D., and TIAN, Y. X. Primary parametric resonance, stability analysis and bifurcation characteristics of an axially moving ferromagnetic rectangular thin plate under the action of air-gap field. Nonlinear Dynamics, 112, 8889- 8920 (2024) |
| 8 | WU, Z. H., ZHANG, Y. M., and YAO, G. Natural frequency and stability analysis of axially moving functionally graded carbon nanotube-reinforced composite thin plates. Acta Mechanica, 234 (3), 1009- 1031 (2023) |
| 9 | YAO, G., XIE, Z. B., ZHU, L. S., and ZHANG, Y. M. Nonlinear vibrations of an axially moving plate in aero-thermal environment. Nonlinear Dynamics, 105 (4), 2921- 2933 (2021) |
| 10 | HU, Y. D., and CAO, T. X. Magnetoelastic primary resonance of an axially moving ferromagnetic plate in an air gap field. Applied Mathematical Modelling, 118, 370- 392 (2023) |
| 11 | KONG, L. Y., and PARKER, R. G. Vibration of an axially moving beam wrapping on fixed pulleys. Journal of Sound and Vibration, 280 (3-5), 1066- 1074 (2005) |
| 12 | ZHU, F., and PARKER, R. G. Influence of tensioner dry friction on the vibration of belt drives with belt bending stiffness. Journal of Vibration and Acoustics: Transactions of the ASME, 130 (1), 011002 (2008) |
| 13 | KONG, L. Y., and PARKER, R. G. Mechanics of serpentine belt drives with tensioner assemblies and belt bending stiffness. Journal of Mechanical Design, 127 (5), 957- 966 (2005) |
| 14 | DING, H. Periodic responses of a pulley-belt system with one-way clutch under inertia excitation. Journal of Sound and Vibration, 353, 308- 326 (2015) |
| 15 | AN, C., and SU, J. Dynamic response of axially moving Timoshenko beams: integral transform solution. Applied Mathematics and Mechanics (English Edition), 35 (11), 1421- 1436 (2014) |
| 16 | WANG, Y. Q., and ZU, J. W. Analytical analysis for vibration of longitudinally moving plate submerged in infinite liquid domain. Applied Mathematics and Mechanics (English Edition), 38 (5), 625- 646 (2017) |
| 17 | LI, Y. H., GAO, Q., JIAN, K. L., and YIN, X. G. Dynamic responses of viscoelastic axially moving belt. Applied Mathematics and Mechanics (English Edition), 24 (11), 1348- 1354 (2003) |
| 18 | MARYNOWSKI, K., and KAPITANIAK, T. Dynamics of axially moving continua. International Journal of Mechanical Sciences, 81, 26- 41 (2014) |
| 19 | PHAM, P. T., and HONG, K. S. Dynamic models of axially moving systems: a review. Nonlinear Dynamics, 100 (1), 315- 349 (2020) |
| 20 | ZHU, H., ZHU, W. D., and FAN, W. Dynamic modeling, simulation and experiment of power transmission belt drives: a systematic review. Journal of Sound and Vibration, 491, 115759 (2021) |
| 21 | ZHAO, Y. H., and DU, J. T. Nonlinear vibration analysis of a generally restrained double-beam structure coupled via an elastic connector of cubic nonlinearity. Nonlinear Dynamics, 109 (2), 563- 588 (2022) |
| 22 | MOJAHED, A., LIU, Y., BERGMAN, L. A., and VAKAKIS, A. F. Modal energy exchanges in an impulsively loaded beam with a geometrically nonlinear boundary condition: computation and experiment. Nonlinear Dynamics, 103 (4), 3443- 3463 (2021) |
| 23 | CHEN, L. Q., and YANG, X. D. Vibration and stability of an axially moving viscoelastic beam with hybrid supports. European Journal of Mechanics A/Solids, 25 (6), 996- 1008 (2006) |
| 24 | ZANG, J., CAO, R. Q., and ZHANG, Y. W. Steady-state response of a viscoelastic beam with asymmetric elastic supports coupled to a lever-type nonlinear energy sink. Nonlinear Dynamics, 105 (2), 1327- 1341 (2021) |
| 25 | LEE, U., and JANG, I. On the boundary conditions for axially moving beams. Journal of Sound and Vibration, 306 (3-5), 675- 690 (2007) |
| 26 | YURDDAS, A., ÖZKAYA, E., and BOYACI, H. Nonlinear vibrations and stability analysis of axially moving strings having nonideal mid-support conditions. Journal of Vibration and Control, 20 (4), 518- 534 (2014) |
| 27 | GAO, C. Y., DU, G. J., FENG, Y., and LI, J. X. Nonlinear vibration analysis of moving strip with inertial boundary condition. Mathematical Problems in Engineering, 2015, 1- 9 (2015) |
| 28 | LEE, J. Free vibration analysis of beams with non-ideal clamped boundary conditions. Journal of Mechanical Science and Technology, 27 (2), 297- 303 (2013) |
| 29 | BAGDATLI, S. M., and USLU, B. Free vibration analysis of axially moving beam under non-ideal conditions. Structural Engineering and Mechanics, 54 (3), 597- 605 (2015) |
| 30 | ATCI, D., and BAGDATLI, S. M. Vibrations of fluid conveying microbeams under non-ideal boundary conditions. Microsystem Technologies, 23 (10), 4741- 4752 (2017) |
| 31 | HERYUDONO, A. R. H., and LEE, J. Free vibration analysis of Euler-Bernoulli beams with non-ideal clamped boundary conditions by using Padé approximation. Journal of Mechanical Science and Technology, 33 (3), 1169- 1175 (2019) |
| 32 | WANG, Y. B., FANG, X. R., DING, H., and CHEN, L. Q. Quasi-periodic vibration of an axially moving beam under conveying harmonic varying mass. Applied Mathematical Modelling, 123, 644- 658 (2023) |
| 33 | DING, H. Equilibrium and forced vibration of an axially moving belt with belt-pulley contact boundaries. International Journal of Acoustics and Vibration, 24, 600- 607 (2019) |
| 34 | DING, H., LIM, C. W., and CHEN, L. Q. Nonlinear vibration of a traveling belt with non-homogeneous boundaries. Journal of Sound and Vibration, 424, 78- 93 (2018) |
| 35 | ZHANG, D. B., TANG, Y. Q., and CHEN, L. Q. Irregular instability boundaries of axially accelerating viscoelastic beams with 1:3 internal resonance. International Journal of Mechanical Sciences, 133, 535- 543 (2017) |
| 36 | CHEN, L. Q., TANG, Y. Q., and LIM, C. W. Dynamic stability in parametric resonance of axially accelerating viscoelastic Timoshenko beams. Journal of Sound and Vibration, 329, 547- 565 (2010) |
| 37 | RJOUB, Y. S., and HAMAD, A. G. Forced vibration of axially-loaded, multi-cracked Euler-Bernoulli and Timoshenko beams. Structures, 25, 370- 385 (2020) |
| 38 | YAO, L. Q., JI, C. J., SHEN, J. P., and LI, C. Free vibration and wave propagation of axially moving functionally graded Timoshenko microbeams. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 42 (3), 137 (2020) |
| 39 | ABBAS, W., BAKR, O. K., NASSAR, M. M., ABDEEN, M. A. M., and SHABRAWY, M. Analysis of tapered Timoshenko and Euler-Bernoulli beams on an elastic foundation with moving loads. Journal of Mathematics, 2021, 6616707 (2021) |
| 40 | CHEN, H. Y., WANG, Y. C., WANG, D., and XIE, K. Effect of axial load and thermal heating on dynamic characteristics of axially moving Timoshenko beam. International Journal of Structural Stability and Dynamics, 23 (20), 2350191 (2023) |
| 41 | DING, H., TAN, X., and DOWELL, E. H. Natural frequencies of a super-critical transporting Timoshenko beam. European Journal of Mechanics A/Solids, 66, 79- 93 (2017) |
| 42 | TAN, X., DING, H., and CHEN, L. Q. Nonlinear frequencies and forced responses of pipes conveying fluid via a coupled Timoshenko model. Journal of Sound and Vibration, 455, 241- 255 (2019) |
| 43 | KOÇ, M. A. Finite element and numerical vibration analysis of a Timoshenko and Euler-Bernoulli beams traversed by a moving high-speed train. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 43 (3), 165 (2021) |
| 44 | GHANNADIASL, A., and AJIRLOU, S. K. Dynamic analysis of multiple cracked Timoshenko beam under moving load-analytical method. Journal of Vibration and Control, 28 (3-4), 379- 395 (2022) |
| 45 | TANG, Y. Q., CHEN, L. Q., and YANG, X. D. Natural frequencies, modes and critical speeds of axially moving Timoshenko beams with different boundary conditions. International Journal of Mechanical Sciences, 50 (10-11), 1448- 1458 (2008) |
/
| 〈 |
|
〉 |
Email Alert
RSS