Study on vibration reduction of two-scale system coupled with dynamic vibration absorber

doi:10.1007/s10483-024-3138-9

摘要/Abstract

Abstract:

The dynamic vibration absorber with inerter and grounded stiffness (IG-DVA) is used to control a two-scale system subject to a weak periodic perturbation. The vibration suppression effect is remarkable. The amplitude of the main system coupled with absorber is significantly reduced, and the high frequency vibration completely disappears. First, through the slow-fast analysis and stability theory, it is found that the stability of the autonomous system exerts a notable regulating effect on the vibration response of the non-autonomous system. After adding the dynamic vibrator absorber, the center in the autonomous system changes to an asymptotically stable focus, consequently suppressing the vibration in the non-autonomous system. Further research reveals that the parameters of the absorber affect the real parts of the eigenvalues of the autonomous system, thereby regulating the stability of the system. Transitioning from a qualitative standpoint to a quantitative approach, a comparison of the solutions before and after the introduction of the dynamic absorber reveals that, when the grounded stiffness ratio and the mass ratio of the dynamic absorber are not equal, the high-frequency part in the analytical solution disappears. As a result, this leads to a reduction in the amplitude of the trajectory, achieving a vibration reduction effect.

Key words: two-scale system, dynamic vibration absorber, vibration control, inerter

中图分类号:

34E13

. [J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(8): 1335-1352.

Honglin WAN, Xianghong LI, Yongjun SHEN. Study on vibration reduction of two-scale system coupled with dynamic vibration absorber[J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(8): 1335-1352.

图/表 19

参考文献 31

1	REN, W. Analysis of control measures for metro vibration and its impact on museum building. Museum, 2, 124- 129 (2021)
2	LI, Y. Research on Design of Low Frequency Vibration and Noise Reduction of Aircraft Panel based on Acoustic Metamaterial (in Chinese), Ph. D. dissertation, National University of Defense Technology (2018)
3	LIU, Y. P., LU, Z. M., YAN, X. X., LIU, Z. J., and TANG, L. Q. Measurement and modelling of the vibration induced by working equipment on an offshore platform. Ocean Engineering, 219, 108354 (2021)
4	SONG, X. Y., CHAI, Z. Y., ZHANG, Y. W., ZANG, J., and XU, K. F. Nonlinear vibration isolation via an innovative active bionic variable stiffness adapter (ABVSA). Nonlinear Dynamics, 109 (2), 353- 370 (2022)
5	CHAI, Y. Y., and JING, X. J. Low-frequency multi-direction vibration isolation via a new arrangement of the X-shaped linkage mechanism. Nonlinear Dynamics, 109 (4), 2383- 2421 (2022)
6	LIU, J. G., LI, Y. M., ZHANG, Y., GAO, Q., and ZUO, B. Dynamics and control of a parallel mechanism for active vibration isolation in space station. Nonlinear Dynamics, 76 (3), 1737- 1751 (2014)
7	WANG, L., LI, J. Z., YANG, Y. Y., WANG, J., and YUAN, J. B. Active control of low-frequency vibrations in ultra-precision machining with blended infinite and zero stiffness. International Journal of Machine Tools and Manufacture, 139, 64- 74 (2019)
8	SONG, A. L., WANG, X. P., CHEN, T. N., JIANG, P., and KAI, B. Low-frequency bandgaps of two-dimensional phononic crystal plate composed of asymmetric double-sided cylinder stubs. International Journal of Modern Physics B, 30, 1650029 (2016)
9	ZHAO, W. J., WANG, Y. T., ZHU, R., HU, G. K., and HU, H. Y. Isolating low-frequency vibration via lightweight embedded metastructures. Scientia Sinica Physica, Mechanica & Astronomica, 50 (9), 162- 175 (2020)
10	GARDONIO, P., and ZILLETTI, M. Sweeping tuneable vibration absorbers for low-mid frequencies vibration control. Journal of Sound and Vibration, 354, 1- 12 (2015)
11	EHAB, B., MEHDI, G., and SAMIR, E. Flutter control and mitigation of limit cycle oscillations in aircraft wings using distributed vibration absorbers. Nonlinear Dynamics, 106, 1975- 2003 (2021)
12	FRAHM, H. Device for Damping Vibrations of Bodies, U. S. Patent, 089958 (1909)
13	ORMONDROYD, J., and DEN HARTOG, J. P. The theory of the dynamic vibration absorber. Transactions of the American Society of Mechanical Engineers, 49-50 (2), 021007 (1928)
14	ACAR, M. A., and YILMAZ, C. Design of an adaptive-passive dynamic vibration absorber composed of a string-mass system equipped with negative stiffness tension adjusting mechanism. Journal of Sound and Vibration, 332, 231- 245 (2013)
15	YAO, H. L., CHEN, Z. D., and WEN, B. C. Dynamic vibration absorber with negative stiffness for rotor system. Shock and Vibration, 2016, 5231704 (2016)
16	PENG, H. B., SHEN, Y. J., and YANG, S. P. Parameters optimization of a new type of dynamic vibration absorber with negative stiffness (in Chinese). Chinese Journal of Theoretical and Applied Mechanics, 47 (2), 320- 327 (2015)
17	SHEN, Y. J., XING, Z. Y., YANG, S. P., and SUN, J. Q. Parameters optimization for a novel dynamic vibration absorber. Mechanical Systems and Signal Processing, 133, 106282 (2019)
18	CHEN, M. Z. Q., HU, Y. L., HUANG, L. X., and CHEN, G. R. Influence of inerter on natural frequencies of vibration systems. Journal of Sound and Vibration, 333, 1874- 1887 (2014)
19	HU, Y. L., and CHEN, M. Z. Q. Performance evaluation for inerter-based dynamic vibration absorbers. International Journal of Mechanical Sciences, 99, 297- 307 (2015)
20	AGATHOKLIS, G., and FRANCESCO, P. Wind-induced vibration mitigation in tall buildings using the tuned mass-damper-inerter. Journal of Structural Engineering, 143 (9), 04017127 (2017)
21	WEI, W., ZHU, S. Y., ZHAI, W. M., and ZHANG, Q. L. Low frequency vibration reduction method for floating slab tracks based on tuned viscous mass damper. Scientia Sinica Technologica, 51, 1391- 1400 (2021)
22	BADUIDANA, M., and AURELIEN, K. J. Optimum design for a novel inerter-based vibration absorber with an amplified inertance and grounded stiffness for enhanced vibration control. Journal of Vibration and Control, 28, 2502- 2518 (2022)
23	CHEN, J., SUN, W. G., WU, Y. J., ZHANG, L. M., and HE, X. L. Minimization of beam response using inerter-based dynamic vibration absorber with negative stiffness. Journal of Vibration and Shock, 39 (8), 15- 22 (2020)
24	SUI, P., SHEN, Y. J., and YANG, S. P. Parameters optimization of a dynamic vibration absorber with inerter and grounded stiffness. Chinese Journal of Theoretical and Applied Mechanics, 53 (5), 1412- 1422 (2021)
25	FAN, S. T., and SHEN, Y. J. Parameter optimization of viscoelastic dynamic vibration absorber with inerter and grounded stiffness. Journal of Vibration Engineering, 35 (4), 814- 825 (2022)
26	RINZEL, J., and LEE, Y. S. Dissection of a model for neuronal parabolic bursting. Journal of Mathematical Biology, 25 (6), 653- 675 (1987)
27	MENG, P., LU, Q. S., and WANG, Q. Y. Dynamical analysis of bursting oscillations in the Chay-Keizer model with three time scales. Science China Technological Sciences, 54 (8), 2024- 2032 (2011)
28	BI, Q. S., ZHANG, R., and ZHANG, Z. D. Bifurcation mechanism of bursting oscillations in parametrically excited dynamical system. Applied Mathematics and Computation, 243, 482- 491 (2014)
29	QIAN, Y. H., and YAN, D. M. Fast-slow dynamics analysis of a coupled Duffing system with periodic excitation. International Journal of Bifurcation and Chaos, 28 (12), 1850148 (2018)
30	NGAMSA TEGNITSAP, J. V., FOTSIN, H. B., KAMDOUM TAMBA, V., and MEGAM NGOUONKADI, E. B. Dynamical study of VDPCL oscillator: antimonotonicity. bursting oscillations, coexisting attractors and hardware experiments. European Physical Journal Plus, 135, 591 (2020)
31	ZHANG, X. F., ZHANG, B., HAN, X. J., and BI, Q. S. On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified van der Pol-Duffing system with slow-varying periodic excitation. Nonlinear Dynamics, 108 (3), 2097- 2114 (2022)

[1]	Zeqi LU, Ke LI, Hu DING, Liqun CHEN. Nonlinear energy harvesting based on a modified snap-through mechanism[J]. Applied Mathematics and Mechanics (English Edition), 2019, 40(1): 167-180.
[2]	邓明香冯永平. Two-scale finite element method for piezoelectric problem in periodic structure[J]. Applied Mathematics and Mechanics (English Edition), 2011, 32(12): 1525-1540.
[3]	Xianghong LI, Jianhua TANG, Yanli WANG, Yongjun SHEN. Approximate analytical solution in slow-fast system based on modified multi-scale method[J]. Applied Mathematics and Mechanics (English Edition), 2020, 41(4): 605-622.
[4]	E. GHAVANLOO, S. EL-BORGI. Nonlinear wave dispersion in monoatomic chains with lumped and distributed masses: discrete and continuum models[J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(4): 633-648.