In this paper, the stationary response of a multi-degree-of-freedom frictional stochastic system is studied. corresponding conservative system is approximately linear, while the damping effect induced by friction obtained using the generalized harmonic transformation. Based on the principle of stochastic averaging, properties of the corresponding conservative system are utilized, and the system is decoupled by modal analysis techniques, with the modal amplitudes treated as slow variables. The structure is excited by colored noises, only one vibration mode is activated. By leveraging the slow-varying effect of the modal amplitudes and modal transformation, the damping effect of friction can be expressed as a function of the modal amplitude. The original system is equivalently reduced to a quasi-linear single-degree-of-freedom system excited by white Gaussian noise. An analytical solution exists for this equivalent system. The response of each degree of freedom in original system is obtained through modal transformation. Three-degree-of-freedom stochastic systems with Dahl friction are studied as examples, and the effectiveness of proposed method is validated through comparison with Monte Carlo simulations.

The principle of stationary generalized action for non-conservative systems is mathematically elegant and physically interpretable, however, only for very few systems the generalized actions can be constructed theoretically. This work addresses this issue and proposes a data-driven method to automatically extract the generalized action for non-conservative systems. Here, the optimization objective is set by the mathematical implication of generalized action; solving conditions are derived by the arbitrariness of variation; and the generalized action with sparse form is determined by the iterative optimization with dimension increasing stepwise. Numerical investigation for four representative systems illustrates the efficacy of this method, and in the meanwhile, reveals the non-uniqueness of generalized action. Implemented within an integral-variational framework, this method exhibits strong robustness to data noise.

Effective control of randomly vibrating systems plays a critical role in enhancing the safety and performance of structures in various engineering applications. This paper proposes a data-driven stochastic optimal control methodology in which control laws are derived directly from random state data without requiring knowledge of the governing equations. The proposed approach consists of two stages. In the first stage, starting from random state data, the stochastic system is identified by estimating the drift and diffusion coefficients using the Kramers-Moyal expansion combined with sparse regression. In the second stage, the optimal control forces are determined by integrating genetic algorithms with sparse optimization, thereby reducing the control problem to a multidimensional optimization task. To compute the statistical moments involved in the control optimization, two complementary strategies are employed. When the chosen control-force structures render the controlled system analytically solvable, the statistical moments are obtained in closed form. Otherwise, they are estimated numerically using Monte Carlo simulations. While the analytical-based method is more efficient, the MCS-based method serves as a fallback when analytical solutions are not feasible. Representative examples demonstrate that the proposed method achieves effective control performance with good efficiency, particularly in handling strongly nonlinear regimes. Moreover, the control law derived from the analytical-based formulation remains effective despite structural restrictions on the control force.

The linearly evolving dynamics hidden in nonlinear systems provides crucial insights for understanding their diverse nonlinear behaviors; Existing studies, however, are predominantly confined to autonomous systems. This work addresses non-autonomous systems, automatically distilling linear evolving dynamics hidden in non-autonomous systems. It is formulated as an iterative optimization scheme which synchronously identifies the coordinate transformations (linear or nonlinear transformations of state variables) and the linearly evolving dynamics governing the transformed coordinates. Numerical examples demonstrate the efficacy of this method to non-autonomous linear/nonlinear systems, and the results identified possess acceptable accuracy across a rather wide parameter range. The proposed methodology distills the linearly evolving dynamics hidden in non-autonomous nonlinear systems, providing novel and instructive perspectives for analyzing such systems.

Predicting and assessing stochastic extremal processes is critical in many fields for tasks such as safety analysis, reliability-based control design, and extreme event forecasting. This paper proposes a novel data-driven method grounded in diffusion theory to predict the probability density of an extremal process directly from random state data. The approach consists of two main steps: first, a slowly varying process is identified from the random state data and approximated as a Markov diffusion process via stochastic averaging, and this process is augmented with its extremal process to form a Markov augmented vector. Second, using conditional moments from the Kramers-Moyal expansion and sparse regression algorithm, we extract the expressions for the drift and diffusion coefficients, leading to a Fokker-Planck-Kolmogorov (FPK) equation for the probability density of the Markov augmented vector. Solving the FPK equation and integrating the resultant yield the transient probability density of the extremal process. We demonstrated the method’s accuracy on three representative systems — a simple one-dimensional example, a Duffing oscillator, and a van der Pol system — each under Gaussian white noise excitation. The results show that the proposed diffusion-based data-driven approach enables effective long-term prediction of extremal process probability densities using only short-term historical state data.

Abstract Identifying explicit Lagrangians from discrete state data of dynamical systems is of both scientific significance and practical value. Existing methods are predominantly established within the second-kind Lagrangian framework, where the use of generalized coordinates reduces system dimensionality at the cost of increased functional complexity. In contrast, the first-kind Lagrangian framework formulated in physical coordinates involves higher dimensionality but features simpler functional structures. The intrinsic simplicity renders it particularly suitable for data-driven identification. This work addresses this problem by developing a data-driven method for identifying explicit Lagrangians in terms of physical coordinates. We first embed the first-kind Lagrangian framework into dynamical systems of concern, eliminate all Lagrange multipliers, expand the Lagrangian over a set of prespecified basis functions, and identify the explicit Lagrangian by sparse optimization. A well-posed problem is then constructed by combining the identified Lagrangian with the full set of constraint equations and initial conditions, enabling the prediction of state responses beyond the training data. Four representative examples are provided to demonstrate the application, effectiveness, and accuracy of the proposed method. In addition, we conduct a comparison with existing methods and discuss its robustness to data noise as well as the non-uniqueness and equivalence of Lagrangians identified.

Soft robots, with their compliant bodies, minimal environmental disturbance, and ability to withstand ambient pressures, offer promising solutions for deep-sea exploration. However, a common challenge of stiffening in soft materials impairs their effective actuation in harsh conditions. In this work, we integrated a liquid dielectric plasticizer within an electrohydraulic soft robot, serving dual critical functions as a softening agent to maintain the softness of the polymer shell and an electrohydraulic fluid for efficient actuation. In addition, by using the surrounding seawater as alternating electrodes, we prevented charge retention in dielectric layers, enabling sustained actuation performance. Field tests at depths of ~1360, 3176, and ~4071 meters confirmed the robot's ability to sense the environment, navigate complex trajectories, and withstand unsteady disturbances. Our work offers a generalized and straightforward framework for developing soft materials tailored for deep-sea applications, paving the way for soft robots to execute real-world missions.

Responsive liquid crystal elastomers (LCEs), being able to convert ambient energy into sustainable motions, have promoted the development of smart systems recently. However, the design of the LCE system and the corresponding nonlinear dynamics analysis remain a challenging task. In this paper, a novel opto-mechanical coupled nonlinear system composing a light-powered LCE fiber is proposed and its self-excited tristable oscillation is investigated. The LCE fiber is connected to a terminal mass, attached to two mechanical springs on each lateral side, where the springs are arranged as a “X” shape within a fixed frame. To obtain the nonlinear opto-mechanical governing equations, the elastic properties and the light stimuli response of the LCE fiber are combined and a piecewise dynamic coupled model is adopted. By using the iterative numerical method, the dynamic performance of the system is predicted. Once the energy of the illuminated light to the LCE exceeds the required critical threshold, a sustainable tristable oscillation will be triggered that enables the system to maintain a snap-through oscillation between its three equilibrium points and compensate the damping-induced energy loss. Furthermore, a comprehensive analysis of several crucial geometric and material factors that contribute to the behavior of the system is conducted, including energy-related parameters and the broke of symmetry by the gravitational acceleration. Compared to monostable and bistable light-driven LCE oscillators, the tristable one has more complicated motion types and tuning parameters. The investigation of this work can extend the knowledge about the nonlinear opto-mechanical systems having the light-responsive LCEs, which will be useful to the development of intelligent biosensors, soft robots, energy harvester, and smart actuators.

This work aims to establish a vibration isolation system with constant frequency property that is concise in form and easy in fabrication. Inspired by the insect leg mechanism a configuration space is preselected, then the optimal configuration with approximate exponential load-displacement relation is extracted by combining the energy method and genetic algorithm. Resorting to origami crafts, the isolation system is fabricated by three-step operation, that is cutting, folding and assembling: cutting the pattern in two-dimensional plane, folding to create the initial angles by plastic deformation, while assembling the system in three-dimensional space. Static experiments demonstrate that the load-displacement relation fits the exponential function with high precision; dynamic tests confirm its good vibration isolation performance, especially the excellent constant-frequency characteristics. The proposed design successfully combines bio-inspiration and origami-based fabrication to realize a practical constant-frequency isolation system. The intervention of origami crafts dramatically simplifies the fabrication process, and paves the way for the generalization from isolation cells to isolation metamaterials with constant frequency property.

Accurate identification of the dynamic system equations governing micro-electro-mechanical system (MEMS) resonators is critical for optimising their design and performance. This paper proposes a data-driven method based on the variational principle for dynamic system identification of MEMS resonators. Compared with traditional parametric modelling methods, this method does not rely on predefined models, but directly processes and analyses experimental data, which has higher flexibility and accuracy in identifying complex micro-scale systems. The research demonstrates the application of this method to two types of MEMS devices with distinct nonlinear characteristics. The results show that the method is not only suitable for theoretical research but also effective in identifying dynamic systems in practical applications. Furthermore, the system response under different excitation is predicted, which proves that it has good generalisation capabilities. This opens new avenues for dynamic analysis of MEMS resonators.

Extremely low dynamic stiffness is essential for the successful utilization of quasi-zero stiffness (QZS) structures for vibration isolation, energy harvesting, and micromechanical sensing. However, most conventional QZS structures are based on a parallel connection of positive and negative stiffness mechanisms, which poses challenges for stiffness matching, structure design, and fabrication. We propose a QZS structure based on a straight-arch-straight (SAS) tandem connection, in which the integrated design of the structure overcomes the problem of stiffness matching, and more importantly, the simple design of the structure greatly reduces the difficulty of fabrication and enables the design of a wide range of load-carrying capacity by only adjust the ratio of the length of the straight beam to the arch and the ratio of the height of the arch to the thickness. The symmetric deformation of the SAS structure extends the working range of the QZS structure, which can be further extended several times or divided into several different QZS intervals by connecting several identical or different SAS units in parallel. The experimental results show that the SAS structure has excellent vibration isolation performance in the low-frequency region, and more importantly, the low preload requirement also provides an important reference for structural design in the field of micromechanical sensing.

Synchronization underpins coherence in natural and engineered systems, unifying dynamics and countering noise while remaining vulnerable to disturbances threatening the stability and risking desynchronization. Here, we present a counterintuitive approach: harnessing noise to dilute the energy of unwanted fluctuations from external disturbances, thereby enhancing stability while preserving synchronization through the system's inherent noise suppression at the synchronized frequency. Through experiments with micromechanical oscillators and macroscale rotors, combined with stochastic averaging analysis, we show that this noise dilution effect improves synchronization efficiency, bolsters resistance to interference, and enhances long-term frequency stability. These findings position white noise of appropriate intensity as a dilution element for mitigating unwanted disturbances, providing previously unidentified insights into stability and resilience in complex synchronized systems.

Tracing back the past and predicting the future are of equal importance, while compared to the prediction, the backtracking is far from receiving the attention it deserves. With the explosive advances of the diffusion models, backtracking has undergone a complete renaissance, especially for stochastic systems. This work addresses this issue, tracing back transient information from near-stationary random data. Different from the diffusion models, we aim for statistical information but not sample information, and we only need near-stationary sample segments for identification, but not a large number of full-time samples for learning. The core idea of this data-driven method is: embedding Fokker-Planck equation (as a priori physical knowledge), which portrays the evolution of probability density of state, and then identifying and solving it to trace back the transient probability density. The efficacy of this method is demonstrated by three typical examples, namely, a one-dimensional linear system, a two-dimensional linear system, and the van der Pol system.

This work reports a high-quality and inexpensive vibration isolator with quasi-zero stiffness, which consists of two orthogonally aligned bands with appropriate cutting and folding. It is so concise that you can readily and immediately fabricate it anytime and anywhere if you want. Other advantages include, but not limited to, tunability, designability, and scalability: we can tune the static load-carrying capacity on-demand by intentionally adjusting the folding angle, design the static and dynamic properties by elaborately setting the geometry of cutting and folding, and fabricate small-scale devices by modern fabrication techniques. The operating principle and the isolating performance are demonstrated theoretically and experimentally; a universal scale law is derived analytically. Moreover, the relation between the geometry of the isolator and the mechanism of animals' legs is revealed, offering a novel perspective on understanding living bodies.

Design a multi-directional vibration isolator with concise configuration, low cost and high performance. A closed ring by bending an elastic strip inward serves as a positive stiffness component, while a buckled arch by bending an identical strip outward servers as a negative stiffness component. Aligning the closed ring and the buckled arch parallelly and orthogonally yields an isolator with high-static and low-dynamic stiffness property. Theoretical analyses derive the load–displacement relations of the closed ring and the buckled arch, along with the approximate formulas for positive and negative stiffness design. Numerical simulations validate the efficacy of the theoretical method. Experiments verify the outstanding isolation performance in the vertical direction and good isolation performance in horizontal plane by implementing sinusoidal sweep frequency and band-limited noise tests. It exhibits outstanding isolation performance in the vertical direction, and shows the anisotropy of isolation performance in horizontal plane. Furthermore, it possesses in-situ tunability of load-bearing capacity and isolation performance.

Stochastic averaging, as an effective technique for dimension reduction, is of great significance in stochastic dynamics and control. However, its practical applications in industrial and engineering fields are severely hindered by its dependence on governing equations and the complexity of mathematical operations. Herein, a data-driven method, named data-driven stochastic averaging, is developed to automatically discover the low-dimensional stochastic differential equations using only the random state data captured from the original high-dimensional dynamical systems. This method includes two successive steps, that is, extracting all slowly varying processes hidden in fast-varying state data and identifying drift and diffusion coefficients by their mathematical definitions. It automates dimension reduction and is especially suitable for cases with unavailable governing equations and excitation data. Its application, efficacy, and comparison with theory-based stochastic averaging are illustrated through several examples, numerical or experimental, with pure Gaussian white noise excitation or combined excitations.

Dimension reduction is one of the never-ending themes in investigating high-dimensional nonlinear random vibration systems. This work addresses this issue and establishes a theory-guided data-driven method to automatically distill a low-dimensional description of a high-dimensional presentation. It consists of four successive steps, just corresponding to four verbs, i.e., linearizing, truncating, constructing and compensating: linearizing a high-dimensional nonlinear system by deleting all nonlinear terms; truncating the insignificant modal shapes and building an approximate transformation between all physical coordinates and the modal coordinates retained; constructing a family of low-dimensional nonlinear systems by extending the low-dimensional linear system of retained modal coordinates; finally, compensating the adverse influences induced by the approximate transformation by picking out the low-dimensional equivalent system which balances the accuracy of prediction and the complexity of modelling. Through a toy example and a ten-degrees-of-freedom example with cubic nonlinearity, the application, efficacy, and tremendous potential of this method are illustrated in detail.

In view of the lack of an explicit expression for the stationary response probability density of generalized nonlinear systems subjected to combined harmonic and Gaussian white noise excitations, a data-driven method is proposed in this paper. The approach involves constructing an expansion expression with undetermined coefficients and determining these coefficients through solving an optimal problem. Initially, leveraging the principle of maximum entropy and the Buckingham Pi theorem, the stationary probability density of the system energy is represented in exponential form. The power of the exponential function is then expanded into a combination of basis functions of Pi groups with undetermined coefficients, constructed from system and excitation parameters, along with the system energy. Subsequently, the coefficients are determined by solving an optimal problem aimed at minimizing the residual between the expression and histogram-based estimations of the probability density of the system energy from random state data. Additionally, a sparse optimization algorithm is employed and then the explicit expression for the probability density of the system energy can be identified including system and excitation parameters. Two typical nonlinear systems, namely the Duffing oscillator and Coulomb friction system, are given to illustrate the effectiveness and accuracy of the proposed data-driven method. The identified expressions cover both resonant and non-resonant cases, showcasing the versatility and applicability of the proposed approach. Furthermore, the extensionality of the expression is thoroughly examined and discussed.

Multi-arm coordination has been receiving increasing attention in industry and our daily life. However, empowering arms to skillfully manipulate a common object within contact-rich tasks remains a daunting challenge. In this letter, we introduce a novel learning-based framework for arms to manipulate the object skillfully while maintaining tight coordination, which not only avoids dynamic modeling for keeping tight coordination but also eliminates the necessity to pre-program task-specific skills for handling complex contacts. To the best of our knowledge, our study is the first to concentrate on tightly coordinated arms in contact-rich tasks. Specifically, we improve and utilize value-decomposed multi-agent policy gradient methods to simultaneously learn tight coordination skills and task-specific skills. Contrasting with previous work on multi-arm skill learning, our approach efficiently learns skills without integrating human skills. Furthermore, we evaluate our approach through two challenging dual-arm assembly tasks and the performance exhibits promising robustness and generalizability.

Differing from the response control in civil and engineering applications, the excitation in a closed-loop sensing system is unknown and the control is designed to balance and measure the unknown excitation in real time. In this paper, a novel optimal force-balance control for a closed-loop sensing system with time delay is proposed. By introducing the errors between the control force and the unknown input as new variables, the force-balance control problem is transformed into a response minimization one with a time delay. Then, the discretization method is applied, and the continuous time system with time delay is converted into an extended discrete system without time delay, from which the analytical expression of the optimal force-balance control force is obtained. To illustrate the effectiveness of the proposed control, the optimal control of a practical quartz flexible force-balance accelerometer (FBA) is solved as an application. Numerical results show that the proposed control can realize the accurate measurement of wide-band random acceleration signals with a time delay of up to 10 times of control period. Besides, the proposed control can also significantly inhibit the vibration response of the FBA while guaranteeing measurement accuracy, which will benefit the obtaining of a large dynamic measurement range as a sensing application.