Stiffness gradient sensitivity analysis method for evaluating the vibration reduction effect of complex variable-stiffness systems

doi:10.1007/s10483-025-3311-6

Abstract

Abstract:

An analytical method is proposed with the “stiffness gradient of the response” as a sensitivity metric, and the relationships between the vibration responses and stiffness changes are established. First, a 2-degree-of-freedom (DOF) system is used as an example to propose a stiffness gradient-based evaluation method, taking the effective control bandwidth ratio as a metric of effectiveness. The results show that there is an optimal mass ratio in both variable mass and variable stiffness cases. Then, a typical 16-DOF system is used to investigate the frequency domain characteristics of the stiffness gradient values in the complex system. The distributions of stiffness gradient values show multiple peak intervals corresponding to the sensitive regions for vibration control. By assigning random mass parameters, a significant exponential decay relationship between the subsystem’s mass and effective control is identified, emphasizing the importance of the optimal mass ratio. The finite-element simulation results of solid plate models with springs and oscillators further validate the theoretical results. In short, the gradient value of stiffness effectively quantifies the effects of subsystems on vibration control, providing an analytical tool for active control in complex systems. The identified exponential decay relationship offers meaningful guidance for implementation strategies.

Key words: stiffness gradient, sensitivity, optimal mass, stiffness regulation, vibration reduction

2010 MSC Number:

O328

Xingchi CAO, Xin FANG, Dianlong YU. Stiffness gradient sensitivity analysis method for evaluating the vibration reduction effect of complex variable-stiffness systems. Applied Mathematics and Mechanics (English Edition), 2025, 46(11): 2055-2074.

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Figures/Tables 15

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Fig.?4

Table?1

Fig.?5

Fig.?6

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Fig.?8

Table?2

Fig.?9

Fig.?10

Table?3

REAB of Subsystems 21, 33, and 42 across different frequency bands based on Model 1"

Frequency range/Hz	Subsystem 21	Subsystem 33	Subsystem 42
$[1, 100]$	0.07	0.38	0.48
$[100, 400]$	0.30	0.50	0.45
$[400, 500]$	0.34	0.43	0.30

Table?3

Fig.?11

Table?4

REAB of Subsystems 21, 33, and 42 across different frequency bands based on Model 2"

Frequency range/Hz	Subsystem 21	Subsystem 33	Subsystem 42
$[1, 100]$	0.01	0.48	0.82
$[100, 400]$	0.30	0.88	0.83
$[400, 500]$	0.80	1	0.76

Table?4

References 57

[13]	LI, Y. L. and XU, D. L. Vibration attenuation of high dimensional quasi-zero stiffness floating raft system. International Journal of Mechanical Sciences, 126, 186–195 (2017)
[14]	ZHAN, Q. C., CHEN, Y. L., ZHAO, Y. H., CHEN, M. F., and GUO, R. S. Vibration suppressing study of a simplified floating raft system by mixing using a nonlinear connecting intercalary plate and connecting nonlinear oscillators. Thin-Walled Structures, 206, 112686 (2025)
[15]	WANG, X. Z., RUI, S. T., YANG, S. K., ZHANG, W. Q., and MA, F. Y. A low-frequency pure metal metamaterial absorber with continuously tunable stiffness. Applied Mathematics and Mechanics (English Edition), 45(7), 1209–1224 (2024) https://doi.org/10.1007/s10483-024-3158-7
[16]	SUN, X. T., XU, J., and QI, Z. F. Mechanism properties of a bird-neck bionic rigid-flexible structure. Fundamental Research, 4(6), 1613–1624 (2024)
[17]	XIAO, C. D., TIAN, R. L., ZHANG, X. L., and LI, S. Variable stiffness and zero Poisson’s ratio of the butterfly-shaped mechanical metamaterial. Composites Communications, 49, 101958 (2024)
[18]	PAGLIARANI, N., ARLEO, L., DE LUCA, G., POZZI, J., and CIANCHETTI, M. Variable stiffness structure inspired by seashells. Smart Materials and Structures, 33(2), 025004 (2024)
[19]	HU, T., JIANG, L., PAN, L. Y., CHEN, B., GONG, N., YANG, J., GONG, X. L., and SUN, S. S. Development of a semi-active suspension using a compact magnetorheological damper with negative-stiffness components. Mechanical Systems and Signal Processing, 223, 111842 (2025)
[20]	XU, L. H., XIE, X. S., and LI, Z. X. Seismic performance and resilience of composite damping self-centering braced frame structures. Fundamental Research, 4(3), 603–610 (2024)
[21]	NABIL, T., BAKR, M., EL-DOMIATY, A., DAWOOD, M., and MANSOUR, T. M. Vibration damping by enhancing of magneto-rheological damper performance using novel smart fluid. Mechanics Based Design of Structures and Machines, 53(2), 1351–1367 (2025)
[22]	DEMIR, M. U. and YILMAZ, C. Realization of a wideband three-axis horizontal vibration isolator with adjustable stiffness and damping. Journal of Sound and Vibration, 600, 118876 (2025)
[23]	TIAN, R. L., WANG, M. H., ZHANG, Y. S., JING, X. J., and ZHANG, X. L. A concave X-shaped structure supported by variable pitch springs for low-frequency vibration isolation. Mechanical Systems and Signal Processing, 218, 111587 (2024)
[24]	CHAI, Z. Y., HAN, J. T., SONG, X. Y., ZANG, J., ZHANG, Y. W., and ZHANG, Z. Theoretical and experimental investigations on an X-shaped vibration isolator with active controlled variable stiffness. Applied Mathematics and Mechanics (English Edition), 45(8), 1371–1386 (2024) https://doi.org/10.1007/s10483-024-3135-6
[25]	FANG, X., WEN, J. H., CHENG, L., YU, D. L., ZHANG, H. J., and GUMBSCH, P. Programmable gear-based mechanical metamaterials. Nature Materials, 21(8), 869–876 (2022)
[26]	FANG, X., WEN, J., YU, D., GUMBSCH, P., and GAO, H. Gear-based metamaterials for extraordinary bandgap tunability. Applied Physics Letters, 127(11), 112201 (2025)
[27]	LIN, Y., WEN, G. L., LIU, C. X., HE, J. F., and LIU, J. A magnetorheological elastomer-based hybrid vibration isolation system with semi-active control and quasi-zero stiffness performance. International Journal of Non-Linear Mechanics, 174, 105063 (2025)
[28]	LI, Z. Y., WANG, K., CHEN, T. T., CHENG, L., XU, D. L., and ZHOU, J. X. Temperature controlled quasi-zero-stiffness metamaterial beam for broad-range low-frequency band tuning. International Journal of Mechanical Sciences, 259, 108593 (2023)
[29]	WEN, G. L., ZHANG, S. D., WANG, H. X., WANG, Z. P., HE, J. F., CHEN, Z. J., LIU, J., and XIE, Y. M. Origami-based acoustic metamaterial for tunable and broadband sound attenuation. International Journal of Mechanical Sciences, 239, 107872 (2023)
[30]	SONG, J., WANG, B. F., and HAO, X. H. Optimization algorithms and their applications and prospects in manufacturing engineering. Materials, 17(16), 4093 (2024)
[31]	MENG, X. L. An automatic frequency control system for transmission machinery based on back-propagation neural network algorithm in the Internet of Things environment. Journal of Testing and Evaluation, 52(3), 1376–1388 (2024)
[32]	ZHANG, T. S., YE, M., LI, X. F., BI, D. J., PENG, L. B., and XIE, Y. L. Fractional derivative kernel recursive generalized maximum correntropy for RUL prediction of rolling bearings. Mechanical Systems and Signal Processing, 217, 111527 (2024)
[33]	GONDZIO, J. and SOBRAL, F. N. C. Polynomial worst-case iteration complexity of quasi-Newton primal-dual interior point algorithms for linear programming. Computational Optimization and Applications, 91(2), 649–681 (2024)
[34]	LIU, Q., YU, Y. C., HAN, B. S., and ZHOU, W. A novel compact high-sensitivity fiber Bragg grating sensor for microvibration measurement of robot joints. IEEE Sensors Journal, 24(8), 12385–12399 (2024)
[35]	HASHEMIAN, F., YANG, H. Z., WANG, Y., DENG, X. M., KIM, S., and VAIDHYANATHAN, R. Parametric dynamic simulation and Bayesian design optimization of a front-loading washing machine. Journal of Vibration Engineering & Technologies, 12(1), 41–62 (2024)
[36]	KAMGAR, R., RAHMANI, F., and RAHGOZAR, R. Geometrical and material optimization of the functionally graded doubly-curved shell by metaheuristic optimization algorithms. Structures, 62, 106254 (2024)
[37]	ZHU, H. X., LI, S., ZHU, R. Y., GAO, F. Y., YIN, Z. Y., LIU, L. Q., and ZHENG, X. F. Residual vibration suppression of piezoelectric inkjet printing based on particle swarm optimization algorithm. Micromachines, 15(10), 1192 (2024)
[38]	XU, H. T., GAO, J., WEN, J. N., DU, J. S., and WANG, W. Data-driven simulation of aero-engine rotor assembly process optimization for unbalance based on fusion algorithm. Aerospace Science and Technology, 158, 109874 (2025)
[39]	QIU, Z. C., and LIU, Y. H. Visual feedback vibration control of flexible hinged plate system based on reinforcement learning algorithm. Mechanical Systems and Signal Processing, 224, 112005 (2025)
[40]	YIN, G. H., MA, M. L., JIA, P., and MA, X. X. Parameter optimization of friction pendulum bearings based on the adaptive genetic algorithm considering the overall evolutionary status. Buildings, 14(2), 435 (2024)
[41]	GUO, Y., XU, G. F., and DUAN, C. Y. Research on time-delayed vibration reduction control of 1/4 vehicle semi-active suspension system with three degrees of freedom. Advances in Mechanical Engineering, 16(9), 16878132241273541 (2024)
[42]	LIU, G., PAN, L., JIANG, W. Q., FAN, S., and BUHARI, A. Dynamic performance and optimization research for six-link mechanism considering the coupling effect of flexible structure and wear clearances. Nonlinear Dynamics, 112(6), 4299–4320 (2024)
[43]	LIN, Q., HU, J. X., ZHOU, Q., SHU, L. S., and ZHANG, A. F. A multi-fidelity Bayesian optimization approach for constrained multi-objective optimization problems. Journal of Mechanical Design, 146(7), 071702 (2024)
[1]	CHENG, J. W., BU, W. J., SHI, L., and FU, J. Q. A real-time shaft alignment monitoring method adapting to ship hull deformation for marine propulsion system. Mechanical Systems and Signal Processing, 197, 110366 (2023)
[2]	ZHANG, Y. T., SHEN, W. A., LONG, Z. T., ZHANG, Y. P., WANG, Z. C., ZHANG, Z. K., ZHU, S. Y., STOCCHINO, A., DENG, H. X., and ZHU, H. P. Active pendulation control of hoisting systems of ship-mounted cranes under ocean wave excitations: principle and experimental study. Mechanical Systems and Signal Processing, 222, 111802 (2025)
[3]	SHARMA, S. K., SHARMA, R. C., and LEE, J. Propelling precision of longitudinal vibration mitigation in ship propeller shafts through advanced nonlinear intelligent semi-active control leveraging adaptive neuro-fuzzy inference system with linear quadratic regulator. Journal of Vibration and Control, 31(7-8), 1472–1484 (2025)
[4]	WU, Y. W., DAI, Q. H., LIU, H. F., TANG, Y., and CHEN, X. C. Ship base vibration reduction design technology based on visualization of power flow and discrete optimization. Ocean Engineering, 309, 118494 (2024)
[5]	DAI, S. J., LIU, S. Y., JI, W. B., and LI, S. D. Vibration suppression in macro-micro grinding system of aeroengine blade based on impedance compensation prediction control strategy. The International Journal of Advanced Manufacturing Technology, 125(1), 793–807 (2023)
[6]	HU, B., FANG, X., WEN, J. H., and YU, D. L. Effectively reduce transient vibration of 2D wing with bi-stable metamaterial. International Journal of Mechanical Sciences, 272, 109172 (2024)
[7]	LAGEMANN, E., BRUNTON, S. L., SCHRÖEDER, W., and LAGEMANN, C. Towards extending the aircraft flight envelope by mitigating transonic airfoil buffet. Nature Communications, 15, 5020 (2024)
[8]	VAN DER VEEK, B., GUTIERREZ, H., WISE, B., KIRK, D., and VAN BARSCHOT, L. Vibration control of flexible launch vehicles using fiber Bragg grating sensor arrays. Sensors, 25(1), 204 (2025)
[9]	ZHOU, X., TONG, W. H., DAI, L., and WEI, B. Y. Vibration isolation and launch performance enhancement of the spacecraft in-orbit launch design using the nonlinear dynamic feature. Applied Sciences, 14(10), 4250 (2024)
[10]	KONG, C. Y., ZHAO, D. J., and LIANG, B. G. Vibration suppression of a flexible beam structure coupled with liquid sloshing via ADP control based on FBG strain measurement. Actuators, 12(12), 471 (2023)
[11]	SHENG, P., HU, B., FANG, X., and WEN, J. H. Random aeroelastic vibration of nonlinear metamaterial supersonic plates. International Journal of Mechanical Sciences, 297-298, 110371 (2025)
[44]	ORTIZ, R., MIRANDA-CHIQUITO, P., ENCALADA-DAVILA, A., MARQUEZ, L. E., TUTIVEN, C., CHATZI, E., and SILVA, C. E. An enhanced modeling framework for bearing fault simulation and machine learning-based identification with Bayesian-optimized hyperparameter tuning. Journal of Computing and Information Science in Engineering, 24(9), 091002 (2024)
[45]	WANG, X. G., MING, M. A., ZHAO, B., ZHANG, W. H., and SONG, S. Response surface model optimization algorithm for structural assessment based on vibration frequency. Shock and Vibration, 2025(1), 5446251 (2025)
[46]	MA, J. J., LIU, Z. T., WANG, C. S., GUO, Y., LIU, C. L., HAN, Y. W., and WANG, L. H. Multi-objective optimization research on nonlinear energy sink system of finite-length beam on elastic medium. Nonlinear Dynamics, 113(2), 1007–1024 (2025)
[47]	WU, Q. B., HE, J. J., CHEN, W. J., LI, Q. H., and LIU, S. T. Topology optimization of phononic crystal with prescribed band gaps. Computer Methods in Applied Mechanics and Engineering, 412, 116071 (2023)
[48]	ZHANG, H., DING, X. H., NI, W. Y., CHEN, Y. Y., ZHANG, X. P., and LI, H. Concurrent topology optimization of composite plates for minimum dynamic compliance. Materials, 15(2), 538 (2022)
[49]	ZHANG, H., TAKEZAWA, A., DING, X. H., GUO, H. H., NI, W. Y., and ZHANG, X. P. Topology optimization of composite macrostructures comprising multi-phase viscoelastic composite microstructures for enhanced structural damping. Composite Structures, 278, 114712 (2021)
[50]	HADIZADEH-BAZAZ, M., NAVARRO, I. J., and YEPES, V. Life-cycle cost assessment using the power spectral density function in a coastal concrete bridge. Journal of Marine Science and Engineering, 11(2), 433 (2023)
[51]	DURMAZGEZER, E., YUCEL, U., and OZCELIK, O. Damage identification of a reinforced concrete frame at increasing damage levels by sensitivity-based finite element model updating. Bulletin of Earthquake Engineering, 17(11), 6041–6060 (2019)
[52]	ZHAO, R. C., JIAO, Y. H., and QU, X. Q. Scaling design strategy for experimental rotor systems subjected to restricted support stiffness. Applied Mathematical Modelling, 109, 265–282 (2022)
[53]	WANG, X., LI, X., YUE, Z. S., YU, R. P., ZHANG, Q. C., DU, S. F., YANG, Z. K., HAN, B., and LU, T. J. Optimal design of metallic corrugated sandwich panels with polyurea-metal laminate face sheets for simultaneous vibration attenuation and structural stiffness. Composite Structures, 256, 112994 (2021)
[12]	LI, B. Y., SHUAI, C. G., and MA, J. G. Rolling stability analysis of high-static-low-dynamic stiffness floating raft vibration isolation systems. Journal of Vibration and Control, 29(21-22), 5161–5169 (2023)
[54]	VAN DEN WYNGAERT, J. C. E., SCHEVENELS, M., and REYNDERS, E. P. B. Broadband acoustic shape optimization of studs in double-leaf walls. Journal of Sound and Vibration, 485, 115562 (2020)
[55]	LEE, D. H., HWANG, W. S., and KIM, C. M. Design sensitivity analysis and optimization of an engine mount system using an FRF-based substructuring method. Journal of Sound and Vibration, 255(2), 383–397 (2002)
[56]	JUNG, J., HYUN, J., GOO, S., and WANG, S. An efficient design sensitivity analysis using element energies for topology optimization of a frequency response problem. Computer Methods in Applied Mechanics and Engineering, 296, 196–210 (2015)
[57]	GONI, S. A., MONDAL, S., and CHAKRABORTY, S. A new gradient based step size controlled inverse eigen sensitivity algorithm for identification of material and boundary parameters of plates. Journal of Vibration and Control, 23(16), 2619–2634 (2017)

Subsystem	Natural frequency/Hz	Frequency variation range/Hz
11	400	[200, 2 000]
12	66.7	[33, 333]
13	115.5	[57.7, 577]
21	400	[200, 2 000]
22	70.7	[35.4, 354]
23	89.4	[44.7, 447]
31	400	[200, 2 000]
32	66.7	[33.3, 333]
33	141.4	[70.7, 707]
41	400	[200, 2 000]
42	57.7	[28.9, 289]
43	81.6	[40.8, 408]

Subsystem	Initial mass	Random mass 1	Random mass 2	Random mass 3	Random mass 4
11	0.002 3	0.035 4	0.093 4	0.043 0	0.033 0
12	0.082 6	0.088 6	0.024 4	0.073 5	0.011 2
13	0.027 5	0.068 4	0.021 1	0.019 2	0.073 0
21	0.002 3	0.056 1	0.021 3	0.048 4	0.047 3
22	0.073 4	0.015 1	0.034 7	0.055 6	0.013 7
23	0.045 9	0.015 1	0.059 1	0.005 1	0.053 1
31	0.002 3	0.006 1	0.048 9	0.057 0	0.004 4
32	0.082 6	0.080 8	0.033 2	0.016 6	0.096 9
33	0.018 3	0.056 2	0.068 8	0.006 8	0.028 2
41	0.002 3	0.066 2	0.016 4	0.088 7	0.070 8
42	0.110 1	0.002 6	0.033 3	0.090 3	0.033 8
43	0.055 0	0.090 4	0.041 6	0.075 6	0.055 8