Table of Content

31 July 2026, Volume 47 Issue 8

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Modular reconfigurable quasi-zero-stiffness isolators: design, analysis, and experiment

Kangfan YU, Yunwei CHEN, Chuanyun YU, Jianrun ZHANG, Xi LU, Xiaofei DU, Qidi FU

2026, 47(8):  1647-1668.  doi:10.1007/s10483-026-3417-7

Abstract ( 45 )   PDF (2364KB) ( 25 )  

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Quasi-zero-stiffness (QZS) isolators hold considerable promise for the development of low-frequency broadband vibration isolation platforms, particularly in aerospace, transportation, and civil engineering. However, stringent requirements for integration, lightweight design, reliability, and customization in high-end equipment limit the practical application of conventional QZS isolators. To address this challenge, this paper proposes a novel modular reconfigurable QZS isolator (MRQI) composed of elastic modules and complementary rigid modules, which is lightweight and exhibits programmable QZS characteristics. The elastic modules adopt a multi-section beam configuration with compliant joints to reduce stress concentration. Notably, this modular design not only endows the MRQI with reconfigurable and programmable properties but also simplifies fabrication from a three-dimensional (3D) additive manufacturing process to a two-dimensional (2D) planar process, thereby reducing manufacturing errors caused by spatial structural distribution. The results reveal that the MRQI achieves multi-stage QZS characteristics, enabling decoupled control of the QZS range and loading capacity, while each QZS range covers 80% of its corresponding stroke range. Moreover, the MRQI maintains robust low-frequency isolation performance against variations in excitation amplitude and loading mass under both displacement and force excitation conditions, provided that the displacement remains within the QZS ranges. Compared with linear isolators, the MRQI reduces the starting isolation frequency by more than 84%. Both the static and dynamic tests confirm the accuracy of the theoretical analysis. Overall, this work presents a novel approach for constructing scalable, integrated, lightweight, and customizable QZS structures, offering a promising framework for the practical engineering applications of QZS isolators.

A bistable liquid crystal elastomer self-oscillator with hysteretic optical occlusion

Xingui ZHOU, Zhuangzhuang ZHANG, Zuhao LI, Junjie CHEN, Kai LI

2026, 47(8):  1669-1690.  doi:10.1007/s10483-026-3415-9

Abstract ( 21 )   PDF (4231KB) ( 9 )  

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Self-oscillatory systems sustain continuous motion by harvesting energy from a steady environment through internal feedback. In contrast to conventional oscillators that depend on inertia and demand materials with rapid responsiveness to external stimuli, in this study, we experimentally design a bistable liquid crystal elastomer (LCE) self-excited oscillator operating under steady illumination. The oscillator consists of an LCE fiber, a spring, a rigid rod, a rope, and a hysteretic occluder. A quasi-static analysis under constant light reveals two distinct motion regimes: a steady regime and a self-oscillatory regime. Periodic motion arises from the contraction and relaxation of the LCE fiber, leading to alternating leftward inclination of the rigid rod under illumination and rightward inclination in darkness. Owing to the bistable configuration, asynchronous motion is generated, introducing a hysteresis-related delay in the optical feedback. This hysteretic effect relaxes the requirement for rapid response and enables sustained self-excited oscillations under steady illumination. Furthermore, the dependence of the critical contraction strain and oscillation period on the system parameters is systematically examined. Compared with the existing self-oscillating systems, the proposed design achieves reliable self-excited oscillation with reduced experimental complexity and fewer mechanical components, enabling its potential use in sensing, energy harvesting, and soft robotics.

Effect of the tetra-chiral auxetic cell and layer geometries on the thermomechanical vibration response of magneto-electro-elastic smart sandwich nanoplates

T. DAS, M. T. OZDEMIR, M. S. GUL, I. ESEN

2026, 47(8):  1691-1722.  doi:10.1007/s10483-026-3420-6

Abstract ( 28 )   PDF (1538KB) ( 9 )  

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This work investigates the thermomechanical vibration response of magneto-electro-elastic (MEE) smart sandwich nanoplates incorporating a tetra-chiral auxetic core. The analysis is performed within the framework of nonlocal strain-gradient elasticity combined with a four-variable refined shear deformation theory, enabling the simultaneous consideration of size-dependent effects, geometric chirality, and multi-field coupling. The studied configuration consists of a porous tetra-chiral auxetic core bonded to piezo-electro-magnetic face layers and subjected to thermal loading together with external electric and magnetic potentials. The effective elastic properties of the tetra-chiral auxetic core are evaluated by a unit-cell approach. The governing equations of motion are derived via Hamilton’s principle and solved analytically by Navier’s method. The model accuracy is verified through comparisons with the previously published results. A detailed parametric investigation is conducted to examine the effects of the auxetic geometric parameters, core-to-face thickness ratios, face-layer material composition, applied electric and magnetic fields, and nonlocal length-scale parameters on the fundamental vibration characteristics of the proposed smart sandwich nanoplates. The results indicate that the magnetic loading and strain-gradient effects enhance structural stiffness and stability, whereas the electric potential and nonlocal parameter introduce softening. Overall, the tetra-chiral auxetic cores provide an efficient mechanism for tuning and controlling the vibration behavior of smart sandwich nanoplates operating in coupled thermal and multi-field environments.

End-to-end analysis of vibration of time-varying mass systems

Kai WANG, Ao CHENG, Tingting CHEN, Jiaxi ZHOU, Zhuang LI, Shengtao ZHANG, Li CHENG

2026, 47(8):  1723-1746.  doi:10.1007/s10483-026-3414-8

Abstract ( 26 )   PDF (2948KB) ( 9 )  

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Vibration systems with time-varying mass are prevalent in engineering practice, exemplified by rockets with fuel depletion, vehicles with changing mass, and systems with cable-hoisted payloads. However, progress has been constrained by the lack of an end-to-end approach capable of integrating modeling, closed-form analysis, numerically stable calculations, and isolation design. Focusing on a typical system involving rocket fuel combustion with linear mass depletion, we first derive the equations of motion from the momentum theorem. A parameter transformation, constructed via the method of undetermined coefficients, converts the time-varying differential equation into a standard Bessel equation, yielding a closed-form analytical solution. To achieve reliable numerical solutions, a strategy combining variable upper-limit integration with grid-search-optimized lower bounds is deployed to replace indefinite integrals, thereby overcoming non-integrable products of Bessel functions. Benchmark comparisons with harmonic excitations show excellent agreement, validating the formulation and solution scheme. Building on the closed-form response, an end-to-end vibration isolation workflow is established via transmissibility and isolation failure-time-threshold (FTT) metrics, allowing for the selection of stiffness based on the instantaneous frequency ratio throughout the mass variation process. The resulting framework provides a general analytical tool for linear differential equations with time-varying mass and a practical pathway for vibration isolation design in mass-varying structures, with special relevance to aerospace applications.

A modular physics-informed neural network for nonlinear vibration isolators

Chenxu LIU, Yingjing QIAN, Guilan YU, Zhanli LIU, Xiaodong YANG

2026, 47(8):  1747-1768.  doi:10.1007/s10483-026-3413-7

Abstract ( 26 )   PDF (1077KB) ( 12 )  

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Nonlinear vibration isolation systems are of fundamental importance in safeguarding equipment, structures, and buildings against harmful vibrational excitations. However, existing intelligent methods are inadequate to fully capture their physical features, which limits the efficient prediction and design. In this study, we propose a novel modular physics-informed neural network (MPINN), enabling dynamic prediction and inverse design of harmonically excited single-degree-of-freedom nonlinear vibration isolators. This framework adopts a modular architecture consisting of a steady-state module informed by the harmonic balance network for predicting steady-state responses, a transient module that predicts decaying components through structured exponential and Fourier layers, and a restoring-force module that learns nonlinear stiffness in a polynomial form. These three modules are integrated through the governing equations and relevant physical quantities, thereby yielding a unified framework. By comparison with analytical solutions under linear conditions, the reliability of the MPINN is validated, and the steady-state and transient modules improve the accuracy by orders of magnitude compared with a specific multilayer perceptron (MLP)-based framework. Incorporating the energy-conservation loss significantly enhances the convergence of the MPINN for dynamic responses of Duffing-type isolators. The MPINN is further adopted to predict the responses of strongly nonlinear isolators, with the residual error reaching only 0.3% of that from the Runge-Kutta method. Finally, the inverse design of vibration isolators is achieved using the MPINN, generating desired results within only 0.52 s. This work proposes a novel modular physics-informed machine learning framework for the prediction and design of vibration isolators, providing new insights into investigating complex dynamic behaviors.

Dynamics modeling and attitude control of on-orbit refueling system

Yu LU, Xiaobing MA, Baozeng YUE

2026, 47(8):  1769-1788.  doi:10.1007/s10483-026-3421-7

Abstract ( 31 )   PDF (899KB) ( 4 )  

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Driven by the urgent demand for deep-space exploration and manned lunar landing, the on-orbit refueling technique has been widely studied. This approach involves delivering fuel to spacecraft via orbital refueling stations, which can extend the operational life of spacecraft. Additionally, by reducing the amount of fuel required at the launch stage, this strategy improves the launch transportation capacity and overall mission efficiency. This study investigates the dynamic characteristics and control methods of the on-orbit refueling system. First, based on Newton’s second law for variable-mass systems and combined with the mass conservation equation and initial conditions, the dynamic equations describing liquid level changes in the refueling and receiving tanks are derived. An analytical solution for this nonlinear ordinary differential equation is obtained. Second, the attitude dynamics equation of the spacecraft is established by considering the internal mass motion, liquid sloshing, and time-varying inertia, thus forming a complete dynamic model of the spacecraft during on-orbit refueling. Subsequently, through a specific case study, instability phenomena caused by changes in mass distribution and time-varying moment of inertia are analyzed, and an optimization strategy is proposed. Finally, through the discrete-form wave-based proportional-derivative (PD) control method, the attitude adjustment and fuel sloshing suppression of the liquid-filled spacecraft system are achieved.

Structural optimization for dynamic stability of a plane-grating monochromator

Guanghui HAN, Xinyu LIAN, Jixia YI, Huaxia DENG, Mengchao MA, Xiang ZHONG, Xinglong GONG

2026, 47(8):  1789-1810.  doi:10.1007/s10483-026-3416-6

Abstract ( 21 )   PDF (1326KB) ( 5 )  

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A structural optimization framework integrating the receptance method (RM) is developed to enhance the dynamic stability of a plane-grating monochromator (PGM). Three vertical geometric dimensions are selected as physically interpretable design variables. The plane-mirror angular vibration under ambient ground excitation is minimized directly in the geometric design space, so resonance and antiresonance features are modified implicitly, without prescribing modal targets. The meta-model of optimal prognosis (MOP) surrogates, selected using the coefficient of prognosis (CoP), enables global sensitivity analysis (SA) and surrogate-assisted multi-objective evolutionary optimization with a limited number of finite element (FE) evaluations. The SA identifies D₂ as the dominant driver of the mirror angular response, with D₃ providing secondary tuning and D₁ remaining nearly neutral within the investigated range. In the FE simulations, the optimized geometry reduces the peak angular vibration by 19.8% and the root mean square (RMS) angular vibration by 37.5%. Under identical ground microvibration excitations, laser Doppler vibrometer (LDV) measurements show a 22.8% reduction in the RMS. Receptance comparisons further confirm suppressed resonance peaks and a lower high-frequency response envelope after optimization. These results provide a practical and physically transparent method for improving the dynamic stability of complex optical systems.

Multi-physics coupling of piezoelectric laminated beam with temperature-dependent material property

Huirong ZHANG, Gantong CHEN, Bohao DUAN, Shengxi ZHOU

2026, 47(8):  1811-1834.  doi:10.1007/s10483-026-3418-8

Abstract ( 13 )   PDF (3181KB) ( 6 )  

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Piezoelectric laminated beams (PLBs) have attracted increasing attention because of their high energy density and ease of integration. However, the temperature sensitivity of their material parameters poses a critical challenge for accurately predicting their response under thermal and vibration excitations. To bridge this gap, we construct a coupled thermo-electro-elastic forced vibration model of the PLB configuration with a tip mass, characterized by a temperature-dependent material property, and derive its closed-form solutions. Furthermore, the dynamic responses induced by thermal and vibration excitations are decoupled and analyzed. The decoupling analysis indicates that the global displacement is predominantly governed by the displacement induced by the thermo-electric coupling effect under static and low-frequency thermal-vibration excitations, with the displacement amplitude on the order of micrometers. Due to the insignificant displacement, the thermal strain is excluded from the multi-physics coupling model, allowing for individual investigation of the effect of temperature-dependent material property on the dynamic response. The results show that the structural resonant frequency decreases with the increasing thermal source intensity, which is attributed to the deterioration of the flexural rigidity. The corresponding maximum voltage and power decrease by 4.5% and 9.7% with the increasing surface thermal source intensity from 0 kW/m² to 2.0 kW/m², respectively. Overall, this study is promising to promote the investigation of multi-physics coupling and provides critical insights into the performance degradation of piezoelectric structures under the combination of thermal and vibration excitations.

Semi-analytical contact modeling of finite-width functionally graded coatings

Xiang XU, Peilin FU, Xiaowei WANG, Jianming GONG, Jianping ZHAO, Qianhua KAN

2026, 47(8):  1835-1854.  doi:10.1007/s10483-026-3422-8

Abstract ( 22 )   PDF (1616KB) ( 16 )  

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Coatings widely applied on the surfaces of tribological components often exhibit depth-dependent material properties, and their optimal design requires the thorough understanding of contact mechanics for functionally graded (FG) coatings. However, most existing studies have adopted the assumption of an infinite-width coating, which differs significantly from the selective coating strategies commonly employed in engineering applications. Therefore, this paper develops a semi-analytical contact model for finite-width FG coatings. The coating modulus is allowed to vary arbitrarily along the depth direction, and no strict limitations are imposed on the coating width. The void zones flanking the coating are treated as zero-modulus coating segments, thereby extending the original coating into a fictitious infinitely wide layer. The modulus difference between coating and substrate enables the inclusion description of the coating, and the resulting disturbances are explicitly quantified through eigenstrains and related analytical solutions. In combination with the coupled relationship between the normal traction and the eigenstrain, the conjugate gradient (CG) method is used to robustly solve for the required normal traction. Parametric investigations based on the developed model demonstrate that increasing the coating modulus elevates structural stiffness and enhances contact stresses; the decreased distance from one coating edge to the initial contact point increases the normal traction and causes its profile to shift away from that edge, but the two edge effects vanish when the distance exceeds a certain threshold; deepening the FG coating intensifies the edge effects and amplifies the influence of the elastic dissimilarity between the coating and the substrate.

PINN-MG: a multigrid-inspired hybrid framework combining the iterative method and physics-informed neural networks

Daiwei DONG, Wei SUO, Jiaqing KOU, Weiwei ZHANG

2026, 47(8):  1855-1878.  doi:10.1007/s10483-026-3419-9

Abstract ( 19 )   PDF (4011KB) ( 5 )  

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Iterative methods are widely used for solving partial differential equations (PDEs). However, the difficulty in eliminating global low-frequency errors significantly limits their convergence speed. In recent years, neural networks (NNs) have emerged as a novel approach for solving PDEs, with studies showing that they exhibit faster convergence for low-frequency components. Building on these complementary frequency-convergence characteristics of iterative methods and NNs, and drawing inspiration from multigrid methods, we propose a hybrid solving framework consisting of a combination of iterative methods and NN-based solvers, termed physics-informed neural network multigrid (PINN-MG, abbreviated as PMG). In this framework, the iterative methods eliminate local high-frequency oscillation errors, while PINNs correct global low-frequency errors. Throughout the solving process, high- and low-frequency components alternately dominate the error, with each being addressed by the iterative methods and PINNs, respectively, thereby accelerating the convergence. We validate the proposed PMG framework on the linear Poisson equations and nonlinear Helmholtz equations. The results show significant acceleration of the PMG built on the Gauss-Seidel (GS), pseudo-time, and generalized minimal residual (GMRES) methods. A detailed analysis of the convergence process validates the rationality of the framework. To further evaluate the advantages of PMG, we apply it to indefinite Helmholtz equations, a class of problems in which traditional solvers often diverge due to the divergence of the low-frequency components. The PMG framework effectively overcomes this divergence, improving both the stability and the convergence speed. We propose the PMG framework as a data-free hybrid solver that does not rely on any pretraining and, more importantly, provides a unified mechanism to tightly couple the NN methods with classical iterative solvers, achieving an organic and iterative integration of the two paradigms.