A nonlinear damping absorber for broadband and multi-directional vibration suppression

doi:10.1007/s10483-026-3395-9

Abstract

Abstract:

Most existing vibration absorbers are limited by the need for precise tuning, and are difficult to suppress broadband vibration. A nonlinear damping absorber (NDA) is proposed in this paper to control the vibrations of structures in multiple modes, providing nonlinear damping in multiple directions and enabling broadband and multi-directional vibration suppression. A pipe is selected as the research object to analyze its dynamic behaviors and vibration suppression mechanisms. The vibration reduction performance of the NDA on the in-plane vibration and out-of-plane vibration of the pipe is studied through experimentation and theory. First, the configuration of the NDA is designed, and its mechanical model is established. An experimental platform is built for the parameter identification and multi-directional vibration reduction test of the absorber. Second, based on the generalized Hamilton’s principle, the mechanical model describing the in-plane vibration and out-of-plane vibration of the pipe with the NDAs is derived. The approximate solution of the nonlinear response is derived and numerically validated. The multi-directional and multi-modal vibration reduction efficiency of the NDAs for the structures is analyzed. The research results show that the proposed absorber can significantly control the multi-directional vibration. Nonlinear damping effectively broadens the vibration reduction bandwidth, and improves the damping efficiency, with the cubic term playing the dominant role. Finally, a concept of multi-modal weighted optimization is proposed. The absorber parameters are optimized through the particle swarm optimization (PSO) algorithm. This paper provides a new type of absorbers for the vibration control of multiple directions in broadband.

Key words: absorber, nonlinear damping, multi-directional vibration, vibration control

2010 MSC Number:

O322

Haiting ZHENG, Hu DING, J. C. JI. A nonlinear damping absorber for broadband and multi-directional vibration suppression. Applied Mathematics and Mechanics (English Edition), 2026, 47(6): 1301-1322.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3395-9

https://www.amm.shu.edu.cn/EN/Y2026/V47/I6/1301

Figures/Tables 19

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Table 1

Identified dynamic parameters of the NDA1"

Parameter	Symbol	Value	Unit
Mass of the oscillator	m	0.3	kg
Linear stiffness	k₁	$1.14 × 105$	N/m
Linear damping	c₁	6.6	N·s/m
Square nonlinear damping	c₂	$− 453.3$	N·s²/m²
Cubic nonlinear damping	c₃	$− 4.8 × 105$	N·s³/m³

Table 1

Fig. 5

Table 2

Identified dynamic parameters of the NDA2"

Parameter	Symbol	Value	Unit
Mass of the oscillator	m	0.3	kg
Linear stiffness	K₁	$1.40 × 106$	N/m
Linear damping	C₁	7.62	N·s/m
Square nonlinear damping	C₂	$− 434.9$	N·s²/m²
Cubic nonlinear damping	C₃	$1.44 × 105$	N·s³/m³

Table 2

Fig. 6

Fig. 7

Table 3

Parameters of the pipe and external excitation"

Parameter	Symbol	Value	Unit
Pipe length	L	2.2	m
Inner diameter	d	$5 × 10 − 2$	m
Outer diameter	D	$6 × 10 − 2$	m
Pipe density	ρ	7 850	kg/m³
Young’s modulus	E	210	GPa
Damping coefficient of in-plane vibration	c_w	5	N·s/m
Damping coefficient of out-of-plane vibration	c_v	5	N·s/m
Spring stiffness	K	$1 × 106$	N/m
Acceleration amplitude	A_a	1.6	m/s²
Angle of inclination	θ	45	°

Table 3

Fig. 8

Fig. 9

Table 4

Dynamic parameters of LDVA1"

Parameter	Symbol	Value	Unit
Mass of the oscillator	m	0.3	kg
Linear stiffness	k₁	$1.14 × 105$	N/m
Linear damping	c₁	6.6	N·s/m
Square nonlinear damping	c₂	0	N·s²/m²
Cubic nonlinear damping	c₃	0	N·s³/m³

Table 4

Table 5

Dynamic parameters of LDVA2"

Parameter	Symbol	Value	Unit
Mass of the oscillator	m	0.3	kg
Linear stiffness	K₁	$1.40 × 106$	N/m
Linear damping	C₁	7.62	N·s/m
Square nonlinear damping	C₂	0	N·s²/m²
Cubic nonlinear damping	C₃	0	N·s³/m³

Table 5

Fig. 10

Fig. 11

Table 6

The initial parameters of the PSO"

Parameter	Symbol	Value
Population size	M	40
Maximum iteration	$n iter, max$	70
Maximum inertia weight	$ω max$	0.9
Minimum inertia weight	$ω min$	0.4
Individual learning factor	$γ 1$	1.5
Group learning factor	$γ 2$	1.7
Penalty coefficient	α	1

Table 6

Table 7

The optimal parameters of NDAs"

Parameter	Symbol	Value	Unit
Linear stiffness of NDA1	k₁	$8.32 × 104$	N/m
Linear damping of NDA1	c₁	7.17	N·s/m
Square nonlinear damping of NDA1	c₂	297.46	N·s²/m²
Cubic nonlinear damping of NDA1	c₃	$5.81 × 105$	N·s³/m³
Linear stiffness of NDA2	K₁	$1.46 × 106$	N/m
Linear damping of NDA2	C₁	8.93	N·s/m
Square nonlinear damping of NDA2	C₂	500.72	N·s²/m²
Cubic nonlinear damping of NDA2	C₃	$1.11 × 105$	N·s³/m³

Table 7

Fig. 12

References 51

[1]	SABAHI, M. A. and SAIDI, A. R. An analytical investigation on nonlinear vibrations and stability of Timoshenko pipes conveying two-phase flow. Thin-Walled Structures, 198, 111749 (2024)
[2]	ATTIA, S., MOHAREB, M., MARTENS, M., and ADEEB, S. Finite element analysis for free vibration of pipes conveying fluids — physical significance of complex mode shapes. Thin-Walled Structures, 200, 111894 (2024)
[3]	CZERWIŃKI, A. and ŁUCZKO, J. Experimental and numerical study on vibrations of a helical pipe with fluid flow. Journal of Sound and Vibration, 535, 117116 (2022)
[4]	EBRAHIMI, R. and ZIAEI-RAD, S. Nonplanar vibration and flutter analysis of vertically spinning cantilevered piezoelectric pipes conveying fluid. Ocean Engineering, 261, 112180 (2022)
[5]	XIE, W. D., LI, Z., WANG, L., JIANG, Z. Y., and LIANG, Z. L. Stability and vibrations of a flexible pipe under stochastic parametric excitation. International Journal of Mechanical Sciences, 307, 110895 (2025)
[6]	YUAN, J. R. and DING, H. An out-of-plane vibration model for in-plane curved pipes conveying fluid. Ocean Engineering, 271, 113747 (2023)
[7]	ZHU, B., GUO, Y., LI, Y. D., and WANG, Y. Q. Three-dimensional nonlinear vibrations of slightly curved cantilevered pipes conveying fluid. Journal of Fluids and Structures, 123, 104018 (2023)
[8]	ZHU, B., ZHANG, X. L., and ZHAO, T. Y. Nonlinear planar and non-planar vibrations of viscoelastic fluid-conveying pipes with external and internal resonances. Journal of Sound and Vibration, 548, 117558 (2023)
[9]	PICKLES, D. J., HUNT, G., ELLIOTT, A. J., CAMMARANO, A., and FALCONE, G. An experimental investigation into the effect two-phase flow induced vibrations have on a J-shaped flexible pipe. Journal of Fluids and Structures, 125, 104057 (2024)
[10]	JIANG, T. L., ZHANG, L. B., GUO, Z. L., YAN, H., DAI, H. L., and WANG, L. Three-dimensional dynamics and synchronization of two coupled fluid-conveying pipes with intermediate springs. Communications in Nonlinear Science and Numerical Simulation, 115, 106777 (2022)
[11]	HESHMATI, M., DANESHMAND, F., and AMINI, Y. Corrugated pipes conveying fluid: vibration and instability analysis. Ocean Engineering, 271, 113507 (2023)
[12]	DEHROUYEH-SEMNANI, A. M. Nonlinear geometrically exact dynamics of fluid-conveying cantilevered hard magnetic soft pipe with uniform and nonuniform magnetizations. Mechanical Systems and Signal Processing, 188, 110016 (2023)
[13]	GUO, Y. C. and DING, H. Theoretical and experimental study on dynamic characteristics of L-shaped fluid-conveying pipes. Applied Mathematical Modelling, 129, 232–249 (2024)
[14]	JING, J., MAO, X. Y., DING, H., and CHEN, L. Q. Parametric resonance of axially functionally graded pipes conveying pulsating fluid. Applied Mathematics and Mechanics (English Edition), 45(2), 239–260 (2024) https://doi.org/10.1007/s10483-024-3083-6
[15]	WANG, D. L., JIANG, T. L., DAI, H. L., and WANG, L. Non-planar vibration characteristics and buckling behaviors of two fluid-conveying pipes coupled with an intermediate spring. Applied Mathematics and Mechanics (English Edition), 46(10), 1829–1850 (2025) https://doi.org/10.1007/s10483-025-3306-9
[16]	DAI, H. L., XING, H. R., HE, Y. X., and WANG, L. Parametric resonance and suppression for L-shaped pipe conveying pulsating fluid. Nuclear Engineering and Technology, 57(4), 103318 (2025)
[17]	IQBAL, M., KUMAR, A., JAYA, M. M., and BURSI, O. S. Vibration control of periodically supported pipes employing optimally designed dampers. International Journal of Mechanical Sciences, 234, 107684 (2022)
[18]	WANG, W. W., MA, H., LIU, S. H., and GUO, X. M. Vibration suppression of spatial pipes with varied clamp configurations: dynamic modeling and experimental validation. Mechanical Systems and Signal Processing, 241, 113437 (2025)
[19]	DENG, T. C., DING, H., KITIPORNCHAI, S., and YANG, J. Machine learning-based design strategy for weak vibration pipes conveying fluid. Applied Mathematics and Mechanics (English Edition), 46(7), 1215–1236 (2025) https://doi.org/10.1007/s10483-025-3276-7
[20]	HAROUNI, P., ATTARI, N. K. A., and ROFOOEI, F. R. Evaluation of a robust dynamic vibration absorber based on negative stiffness and internal resonance against seismic excitation. International Journal of Non-Linear Mechanics, 146, 104130 (2022)
[21]	BARREDO, E., MENDOZA LARIOS, J. G., COLÍN, J., MAYÉN, J., FLORES-HERNÁNDEZ, A. A., and ARIAS-MONTIEL, M. A novel high-performance passive non-traditional inerter-based dynamic vibration absorber. Journal of Sound and Vibration, 485, 115583 (2020)
[22]	SAEED, N. A., HOU, L., YI, H. M., SHUKUR, A. A., ALAMRY, S. M., and EL-SHOURBAGY, S. M. On a broadband vibration isolator with tunable stiffness: from quasi-zero-stiffness to zero-stiffness behavior. Applied Mathematics and Mechanics (English Edition), 47(2), 255–282 (2026) https://doi.org/10.1007/s10483-026-3351-9
[23]	HUANG, X. B., WANG, B., HUANG, Z. W., HUA, X. G., and CHEN, Z. Q. A theoretical model for a low-frequency two-stage hybrid vibration isolator with a nonlinear energy sink and a negative stiffness spring. Applied Mathematical Modelling, 142, 115948 (2025)
[24]	ZHANG, X., HUANG, X. B., and WANG, B. A quad-stable nonlinear piezoelectric energy harvester with piecewise stiffness for broadband energy harvesting. Nonlinear Dynamics, 112(22), 19633–19652 (2024)
[25]	HUANG, X. B. Exploiting multi-stiffness combination inspired absorbers for simultaneous energy harvesting and vibration mitigation. Applied Energy, 364, 123124 (2024)
[26]	QIAN, D. Y., CONG, Y. Y., KANG, H. J., and SU, X. Y. Dynamics on vortex-induced vibration suppression of a flexible cable by tuned mass damper. Communications in Nonlinear Science and Numerical Simulation, 151, 109129 (2025)
[27]	NEŠIĆ, N., KARLIČIĆ, D., CAJIĆ, M. L., SIMONOVIĆ, J., and ADHIKARI, S. Vibration suppression of a platform by a fractional type electromagnetic damper and inerter-based nonlinear energy sink. Applied Mathematical Modelling, 137, 115651 (2025)
[28]	BASTA, E., GUPTA, S. K., and BARRY, O. Frequency lock-in control and mitigation of nonlinear vortex-induced vibrations of an airfoil structure using a conserved-mass linear vibration absorber. Nonlinear Dynamics, 112(11), 8789–8809 (2024)
[29]	ZHANG, H., WEN, B., TIAN, X., LI, X., LI, X., and PENG, Z. Vibration reduction of floating offshore wind turbine with nonlinear vibration absorber: concept, numerical analysis and experimental tests. Ocean Engineering, 331, 121196 (2025)
[30]	MA, K., DU, J. T., LIU, Y., and CHEN, X. M. A study on torsional vibration attenuation of closed-loop crankshaft using nonlinear energy sink under abnormal engine operating condition. Communications in Nonlinear Science and Numerical Simulation, 143, 108537 (2025)
[31]	LIU, H. L., XIE, S. C., CHEN, J., LI, F. M., and ZHOU, W. Reduction of moving-load induced vibrations of graphene-reinforced composite beams with general boundary conditions via a nonlinear energy sink. Applied Mathematics and Mechanics (English Edition), 46(11), 2075–2094 (2025) https://doi.org/10.1007/s10483-025-3318-9
[32]	SHI, D. J., LIU, P. Z., ZHANG, X. H., ZHAO, C. Y., NG, B. F., and WANG, P. An in-depth investigation into the modeling and nonlinear vibration behavior of a coupled particle damper-nonlinear energy sink system. Nonlinear Dynamics, 113(22), 30973–31007 (2025)
[33]	CHEN, H. Y., ZENG, Y. C., DING, H., LAI, S., and CHEN, L. Q. Dynamics and vibration reduction performance of asymmetric tristable nonlinear energy sink. Applied Mathematics and Mechanics (English Edition), 45(3), 389–406 (2024) https://doi.org/10.1007/s10483-024-3095-9
[34]	LI, M. and DING, H. A vertical track nonlinear energy sink. Applied Mathematics and Mechanics (English Edition), 45(6), 931–946 (2024) https://doi.org/10.1007/s10483-024-3127-6
[35]	ZENG, Y. C., JI, J. C., and DING, H. Multistable nonlinear energy sinks: a state-of-the-art review. International Journal of Dynamics and Control, 13(11), 390 (2025)
[36]	KANG, H. J., WANG, Y. F., and ZHAO, Y. Y. Nonlinear vibration analysis of a double-cable beam structure with nonlinear energy sinks. Communications in Nonlinear Science and Numerical Simulation, 142, 108529 (2025)
[37]	LUO, Y. F., GAO, J. B., PENG, J., SUN, H. X., XUE, S. W., LUO, Z. Y., and ZHANG, M. Strongly modulated response and bifurcation characteristics of a lever-type nonlinear energy sink. Nonlinear Dynamics, 113(23), 32139–32166 (2025)
[38]	ZENG, Y. C., DING, H., and JI, J. C. An origami-inspired nonlinear energy sink: design, modeling, and analysis. Applied Mathematics and Mechanics (English Edition), 46(4), 601–616 (2025) https://doi.org/10.1007/s10483-025-3239-6
[39]	DEVARAJAN, K. and SANTHOSH, B. Inertial amplification as a performance enhancement method for snap-through vibration energy harvester. Applied Mathematical Modelling, 137, 115734 (2025)
[40]	KHASAWNEH, M. A. and DAQAQ, M. F. Response behavior of bi-stable point wave energy absorbers under harmonic wave excitations. Nonlinear Dynamics, 109(2), 371–391 (2022)
[41]	ZHAO, G. F., YANG, Z. Y., MA, Y. H., WANG, J. J., CHEN, S. P., and YANG, H. Eddy-current nonlinear energy sink for seismic response mitigation: numerical and experimental study. Structures, 82, 110434 (2025)
[42]	PHILIP, R., SANTHOSH, B., BALARAM, B., and AWREJCEWICZ, J. Vibration control in fluid conveying pipes using NES with nonlinear damping. Mechanical Systems and Signal Processing, 194, 110250 (2023)
[43]	LI, S. B., DING, H., and JING, X. J. Identification and influence of cubic nonlinear damping of vertical track nonlinear energy sinks. Mechanical Systems and Signal Processing, 238, 113164 (2025)
[44]	LUO, C. Q., ZHU, Z. H., GUO, Y. Q., WANG, J. Q., and JING, X. J. Multi-direction vibration isolation with tunable QZS performance via novel X-mechanism design. Communications in Nonlinear Science and Numerical Simulation, 137, 108140 (2024)
[45]	HE, F. F., DU, J. T., and LIU, Y. The bio-inspired spider web dynamic vibration absorber applied to the transverse-torsional-axial vibration in rotor systems. Mechanical Systems and Signal Processing, 237, 112993 (2025)
[46]	GENG, X. F., DING, H., JING, X. J., MAO, X. Y., WEI, K. X., and CHEN, L. Q. Dynamic design of a magnetic-enhanced nonlinear energy sink. Mechanical Systems and Signal Processing, 185, 109813 (2023)
[47]	LI, Y. N., DING, J. Y., DING, H., and CHEN, L. Q. Natural vibration and critical velocity of translating Timoshenko beam with non-homogeneous boundaries. Applied Mathematics and Mechanics (English Edition), 45(9), 1523–1538 (2024) https://doi.org/10.1007/s10483-024-3148-7
[48]	ZHENG, H. T. and DING, H. Softening-hardening nonlinear dynamics of a beam with concentrated mass and axially elastic boundaries. Nonlinear Dynamics, 113(20), 27085–27105 (2025)
[49]	WANG, J., FENG, S. D., ZHANG, K., and DING, H. Transverse vibration suppression of an inclined beam with a nonlinear energy sink. Engineering Structures, 327, 119658 (2025)
[50]	ZHENG, H. T., MAO, X. Y., DING, H., and CHEN, L. Q. Distributed control of a plate platform by NES-cells. Mechanical Systems and Signal Processing, 209, 111128 (2024)
[51]	GENG, X. F., DING, H., MAO, X. Y., and CHEN, L. Q. Nonlinear energy sink with limited vibration amplitude. Mechanical Systems and Signal Processing, 156, 107625 (2021)