Nonlinear coupling mechanism of quasi-zero-stiffness units in multi-level structures

doi:10.1007/s10483-026-3392-6

Abstract

Abstract:

Multilayer structures composed of quasi-zero-stiffness (QZS) units exhibit mechanical characteristics distinct from those of a single unit, and their behaviors are governed by the coupling mechanism between the QZS units. This paper introduces the coupling coefficient to quantitatively describe this mechanism, classifying the system into strongly coupled and weakly coupled states. Through theoretical analysis, numerical simulation, and experimental testing, the static and dynamic responses under different coupling states are comparatively investigated. The results show that in the strongly coupled system, the deformation behavior of each QZS unit shows high consistency, leading to a wider QZS region, weaker nonlinear characteristics, and stronger dynamic response. In the weakly coupled systems, the low degree of deformation coordinations among the units results in different QZS regions, enabling low-frequency vibration isolation under varying loads. The analytical approach of the coupling mechanisms and the static and dynamic response behaviors generated by the two coupling mechanisms provide guidance for the structural design of multifunctional and highly adaptable multi-level QZS metamaterials.

Key words: multi-level structure, coupling mechanism, strong coupling, weak coupling, deformation coordination

2010 MSC Number:

O328

Shaokun YANG, Xingxing SHI, Xingzhong WANG, Jiuhui WU, Fuyin MA. Nonlinear coupling mechanism of quasi-zero-stiffness units in multi-level structures. Applied Mathematics and Mechanics (English Edition), 2026, 47(6): 1215-1240.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3392-6

https://www.amm.shu.edu.cn/EN/Y2026/V47/I6/1215

Figures/Tables 18

Fig. 1

Fig. 2

Fig. 3

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Fig. 5

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Fig. 8

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Fig. 13

Table 1

Parameters at different positions on the force-displacement curve"

State	x/mm	F/N	m_i/kg	$k i, 0$ /(N·mm^-1)
Position 1	5.00	1.29	0.131 6	0.827 7
Position 2	13.14	2.01	0.205 2	0.026 7
Position 3	20.00	2.31	0.304 1	0.467 1
Position 4	29.15	2.98	0.304 1	0.055 8
Position 5	35.05	3.26	0.332 6	0.750 2

Table 1

Fig. 14

Table 2

Fig. 15

Fig. A1

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Load	M/kg	State	Frequency/Hz
M₁	4.0	Out of QZS region	10.4
M₂	4.8	First QZS region	6.5
M₃	7.80	Second QZS region	7
M₄	12.2	Third QZS region	6