Exploring the Design, Modeling, and Identification of Beneficial Nonlinear Restoring Forces: A Review

Abstract

Exploring the design of beneficial nonlinear restoring force structures has become a highly popular topic due to their extensive applications in energy harvesting, actuation, energy absorption, robotics, etc. However, the current literature lacks a systematic review and classification that addresses the design, modeling, and parameter identification of nonlinear restoring forces. Thus, the present paper provides a thorough examination of the latest advancements in the design of nonlinear restoring forces, as well as modeling and parameter identification in contemporary beneficial nonlinear designs. The seven design methodologies, namely magnetic coupling, oblique spring linkages, static or dynamic preloading, metamaterials, bio-inspired, MEMS (Micro-Electromechanical Systems) manufacturing, and dry friction applied approaches, are classified. The polynomial, hysteretic, and piecewise linear models are summarized for nonlinear restoring force characterization. The system parameter identification methods covering restoring force surface, Hilbert transform, time-frequency analysis, nonlinear subspace identification, unscented Kalman filter, optimization algorithms, physics-informed neural networks, and data-driven sparse regression are reviewed. Moreover, possible enhancement strategies for nonlinear system identification of nonlinear restoring forces are presented. Finally, broader implications and future directions for the design, characterization, and identification of nonlinear restoring forces are discussed.

1. Introduction

To meet the increasing demand for structures and equipment with continuously advancing performance, academic researchers are attempting to utilize nonlinear phenomena to outperform linear designs. Over the past decades, beneficial nonlinear structures have been widely developed in energy harvesters, vibration isolators, actuators, robotics, morphing structures, etc., [1,2,3,4]. Typically, the intrinsic features of snap-through behavior and hardening stiffness effect can expand operational frequency bandwidth and maximize the output power of energy harvesters [5,6]. The ultra-low stiffness in dynamics and high stiffness in statics are widely adopted in quasi-zero stiffness vibration isolators [7,8]; the utilization of negative stiffness is frequently utilized to achieve swift transitions between various states of equilibrium, as required in the field of fast actuators and robotics [9,10]; and the dry friction-induced hysteretic type of nonlinear restoring force is widely encountered in wire rope isolators and connected joint design [11,12]. The essential mechanical property of structural design lies in its nonlinear restoring force, regardless of the approach used to achieve benefits. However, accurate modeling and identification of nonlinear restoring forces remain major challenges in practical applications.

There are numerous approaches to designing beneficial nonlinear restoring force structures, e.g., magnetic coupling [13,14], oblique spring mechanisms [15,16], static or dynamic preloading [17,18], mechanical metamaterial structures [19,20], bio-inspired designs [21,22], MEMS fabrication [23,24], dry friction structure [25,26], etc. The design of these nonlinear stiffness force structures has been mainly implemented in energy harvesting, vibration suppression, sensors, actuators, and connected joints [27,28]. For these nonlinear designs, mathematical models need to be used to characterize the nonlinear restoring forces for response prediction and dynamic control. Thus, polynomial-based [29], piecewise linear [30], and various hysteretic functions [31] are widely developed for model characterization. According to the design types and nonlinear restoring force models, it is imperative to provide a classification and overview in order to enhance future designs and modeling.

When a beneficial nonlinear engineering structure is designed, modeling is required to predict its responses, and accurate system parameter identification plays a crucial role in ensuring the precision of the model’s response predictions. Nonlinear system identification techniques have proven to be effective in accurately estimating the parameters of practical scenarios involving nonlinear vibrating structures. Extensive reviews on system identification methods for nonlinear structures have been conducted by Noël and Kerschen in previous decades [32]. It can be observed that among the established methods for nonlinear restoring force identification, the restoring force surface, the Hilbert transform–based method, time–frequency analysis, nonlinear subspace identification, and the evolutionary optimization method are the most commonly used. Moreover, the recently developed unscented Kalman filter, physics-informed neural networks, and sparse identification methods are widely employed in several active areas of nonlinear system identification. These methods have already been widely applied to the nonlinearity commonly encountered in engineering, such as piecewise linear [33,34,35], polynomial nonlinear stiffness [36,37,38], and friction nonlinearities [39,40,41].

With the above introduction, there are at least three topics that need to be discussed: design methodologies, characterization methods, and parameter identification techniques. Some interesting review papers on these topics are illustrated here. Yang et al. [42] conducted an extensive examination of the latest advancements in vibration attenuation and nonlinear energy harvesting from vibrations, encompassing applications and design configurations, nonlinear mechanisms, diverse optimization techniques, as well as prospects for future developments. The mechanical modulations for energy harvesting were comprehensively reviewed by Zou et al. [43], and the systems were categorized into three groups: transformations of excitation types, conversions to higher frequencies, and enhancements of force/motion. The study conducted by Fu et al. [44] provides a thorough analysis of the most recent progress in rotational energy research. Zhang et al. [45] provided a comprehensive review and guidelines on state-of-the-art self-powered methods for creating intelligent bearings, encompassing the elucidation of fundamental theories, modeling techniques, methodologies, and technologies. The study conducted by Quaranta et al. [46] offers a comprehensive overview of computational techniques within the field of artificial intelligence specifically designed for nonlinear dynamical system identification. In addition, both parametric and nonparametric identification problems are considered. Jing et al. [47] comprehensively examined nine distinct methodologies for nonlinear system identification, including restoring force surface analysis, Hilbert transform analysis, the direct quadrature method, the zero-crossing technique, the short-time Fourier transform approach, Gabor wavelet analysis, Morlet wavelet analysis, Morse wavelet analysis, and a neural network-based algorithm. These methods were evaluated in terms of their accuracy and subsequently validated using bolted connected joints. The comprehensive consideration of the entire loop encompassing nonlinear restoring force design, modeling, and identification has been overlooked despite some reviews summarizing recent advancements in methodologies for designing nonlinear restoring forces and identifying parameters. In addition, reviewing existing beneficial nonlinear restoring force design models and their system identification methods, the polynomial type of nonlinearity is a very typical and widely used model that needs to be paid specific attention. It must be noted that a hysteretic-type nonlinear restoring force may have more difficulty in system identification. The hysteresis model may be characterized by complex differential equation-based or operator models. The parameters may not be able to be measured directly, and multiple unknown parameters cause difficult convergence and low efficiency. Therefore, the classification and review of existing typical polynomials and hysteresis models, and their identification, is necessary.

Thus, in this paper, a state-of-the-art review of nonlinear restoring force design methodologies, as well as modeling and parameter identification in typical nonlinear designs, will be given. The eight kinds of design methodologies and three types of commonly used models will be detailed. The present nonlinear restoring force identification methods will be comprehensively reviewed, including restoring force surface, Hilbert transform, time-frequency analysis, nonlinear subspace identification, unscented Kalman filter, optimization algorithms, physics-informed neural networks, and sparse identification. Moreover, some enhancement strategies for nonlinear restoring force identification and a broader perspective on nonlinear restoring force design, characterization, and identification will be provided.

This paper is organized as follows: Section 2 and Section 3 review the design methodologies and modeling of beneficial nonlinear structures, respectively; Section 4 is dedicated to the review of eight kinds of identification methods; Section 5 presents possible enhanced strategies for nonlinear restoring force identification; and finally, Section 6 provides some conclusions and perspectives.

2. Design Methodologies and Their Applications

The classic linear vibration differential equation is modeled by Newton’s second law of motion. On the basis of linear dynamic equations, different dynamic response outcomes can be achieved by modifying nonlinear inertial forces, nonlinear stiffness forces, or nonlinear damping forces. If the resulting response proves beneficial to engineering structures, it is referred to as beneficial nonlinear design. Although the nonlinear inertia and damping can also adjust nonlinear dynamic responses, they will not be detailed in this paper, and some recent developments can be referred to these references [48,49]. Beneficial nonlinear structural designs can be achieved by altering the coupling nonlinear restoring forces through different structural configurations, thereby obtaining desired response outputs. The number of beneficial nonlinear stiffness designs far exceeds those involving nonlinear damping and nonlinear inertia. This may be attributed to the fact that nonlinear stiffness designs appear to be easier to implement. Thus, there are many engineering applications using nonlinear stiffness, such as vibration isolators, energy harvesters, etc. Specifically, the quasi-zero stiffness vibration isolation can be achieved by reducing the stiffness in a small displacement range around a stable equilibrium point; the broadband energy-harvesting performance of devices can be obtained by coupling bistable and tristable nonlinear stiffness forces; the significant reduction in resonance transmission rate can be achieved by increasing the damping ratio using hysteretic nonlinear restoring force isolators. Thus, the key step in exploiting benefits is designing a beneficial nonlinear restoring force.

In this subsection, the most recent developments in advanced nonlinear restoring force design techniques will be comprehensively reviewed. Figure 1 gives a schematic overview of techniques for achieving various beneficial nonlinear effects, including magnetic coupling, oblique spring linkage, mechanical or dynamical preloading, metamaterial structure designs, geometrically nonlinear bio-inspired designs, dry friction applications, etc.

2.1. Magnetic Coupling

Introducing magnetic interaction forces to modify nonlinear restoring forces represents the most prevalent approach. The hardening, softening, bistable, and multistable states can be easily achieved by adjusting the position and arrangement of magnets. There are various applications based on different arrangements of magnets.

The design of nonlinear restoring force involves three types of magnetic coupling methods based on the arrangement of magnets: linear, oblique or rotating, and circular arrangements. Figure 2 shows the very recent developments of magnet-coupled designs for different applications of nonlinear restoring forces. For a linear magnet arrangement, a tunable electromagnetic energy harvester based on a magnetic spring system for human motions was proposed by Wang et al. [58], and a nonlinear restoring force is quasi-static measured, as depicted in Figure 2a. A vibration harvester utilizing electromagnetic levitation was designed by Jenson et al. [59], incorporating three cylindrical permanent magnets arranged coaxially with two coils also arranged coaxially, as illustrated in Figure 2b. To accurately predict the system behavior, a fifth-order polynomial nonlinear restoring force approximation is essential for approximating the magnet–magnet force. Qian et al. [60] applied dynamic magnetic preloading to a piezoelectric stack energy transducer to increase the power output, as shown in Figure 2c. In addition, Wu et al. [61] developed a highly sensitive quasi-zero stiffness vibration sensor using a linear arrangement of circular magnets. Quasi-zero stiffness was achieved by Ma et al. [62] using an electromagnetic negative stiffness mechanism, and by Zhao et al. [63] using a magnetically modulated tetrahedral structure. Self-powered sensing through rotational energy harvesting and structural health monitoring is very popular nowadays [44,45]. The recently developed nonlinear restoring force energy harvesters using rotating magnet arrangements were developed by Fu et al., Narolia et al., Zou et al., Zhao et al., and Halim et al. [64,65,66,67,68], as illustrated in Figure 2d. Yan et al. [69] proposed a novel passive vari-stiffness nonlinear isolator using circumferentially rotatable magnets, and a cubic polynomial was adopted to describe the nonlinear restoring force, as shown in Figure 2e. The investigations conducted by Yao et al. [70] utilized magnets combined with negative stiffness to successfully achieve a multistable energy sink featuring a nonlinear behavior and stiffness characterized by piecewise linear properties, effectively suppressing the vibration of an unbalanced rotor system. Moreover, circular arrangement was commonly used for energy harvesters and sensors. Zhang et al. [71] invented a novel method for harnessing electromagnetic energy from the rotational movement of bearings using the Halbach circular configuration, as illustrated in Figure 2f. Wang et al. [72] employed a magnetic disk with a circular Halbach array for an energy harvester that self-adjusts its natural frequency. Zhao et al. [73] proposed a stacked magnetic modulated harvester with frequency up-conversion to enhance energy harvesting performance from swing motion, as shown in Figure 2g. Except for energy harvesters, the circular arrangement can also be used in sensors [74] and vibration isolators [75,76]. Above all, the utilization of magnetical coupling enables the introduction of a nonlinear opposing force that can effectively modify the overall nonlinear restoring force in quasi-zero stiffness, bistable, and multistable energy harvesters. In addition, the non-contact characteristics of different magnet arrangements make them well-suited for various conditions.

2.2. Oblique Spring Linkages

There are at least two types of oblique spring linkage-based nonlinear restoring force structures for designing energy harvesters and vibration isolators. The first type is only based on oblique springs, and the other type is usually coupled with a connecting rod, a cam roller, or gears. Yang et al. [51] introduced a novel tristable hybrid vibration energy harvester that utilizes a geometrically nonlinear spring mechanism to efficiently extract energy from ultra-low vibration sources, as shown in Figure 3a. Liu et al. [77] employed an oblique spring to devise a dual snap-through mechanism for enhancing the energy capture efficiency of a wave energy converter, resulting in significant improvements. Wen et al. [78] innovated a quasi-zero stiffness mechanism combining six oblique springs and a coil spring. The simulation results of the virtual prototype demonstrate that the proposed vibration isolator can effectively maintain exceptional vibration isolation performance even when subjected to significant amplitude vibrations under control. In addition, Lan et al. [79] developed a vibration isolator with the ability to effectively isolate various loads. The isolator consists of two inclined springs and one vertical spring, working together to achieve nearly zero stiffness at the balanced position, as illustrated in Figure 3b. Huang et al. [80] concerned the vibration isolation dynamics of an isolator employing two nonlinear Euler buckled beams as anti-stiffness correctors, as shown in Figure 3c. Except for this design, the quasi-zero stiffness can also be achieved based on the Stewatr mechanism [81], truss-spring-based stack Miura-ori [82], and bio-inspired limb-like spring structures [83] for ultra-low broadband vibration reduction. The oblique spring can be combined with other mechanisms to achieve a nonlinear restoring force. Zuo et al. [84] investigated a vibration isolator with quasi-zero stiffness using a parabolic cam-roller mechanism that can withstand excitation with large amplitude, as illustrated in Figure 3d. Thanh et al. [85] proposed quasi-zero stiffness driver seats based on horizontal springs, slide guide blocks, and oblique bars. Sun et al. [86] successfully achieved isolators with high static and low dynamic stiffness through the implementation of horizontal springs, rollers, and specifically designed curved surfaces that align with target force curves. In addition, the hybrid mechanisms for vibration isolators and energy harvesting backpacks were designed using cam–roller–spring–rod [87,88], rotating gear–spring [89], and adjustable gear–springs [90,91], as shown in Figure 3e,f.

2.3. Static or Dynamic Preloading

The application of static or dynamic preloading is a conventional approach for modulating stiffness. The natural frequency of a beam decreases under compressive stress, while tensile stress induces an increase in stiffness. Thus, the nonlinear restoring forces in actuators, energy harvesters, and vibration isolators can be adjusted to obtain broadband or low-frequency characteristics. The tunable-resonance vibration energy harvester designed and tested by Leland et al. [92] used a micro spiral head to apply axial compression on the piezoelectric bimorph, effectively reducing its resonance frequency, as shown in Figure 4a. A flex-compressive piezoelectric energy harvesting unit with a high-load capacity and customizable force transmission coefficient was created by Wang et al. [93], utilizing interchangeable components for assembly, as illustrated in Figure 4b. Other types of nonlinear energy harvesters using screw–spring mechanisms [94], diamond-shaped prestressed structures [95], and magnet-induced prestress [96] have also been developed. In piezoelectric actuators, Yang et al. [97] examined the key challenges linked to established methods for preloading within the framework of rapid nanopositioning systems utilizing piezoelectric stack actuators. The stick–slip piezoelectric actuators with flexure-hinge–induced preloading were designed by Yang et al. for higher speed [98]. The quasi-zero stiffness vibration isolator was designed using a preloaded buckled beam by Liu et al. and Lu et al. for better performance [99,100].

Unlike mechanical static preloading, dynamic loading is based on centrifugal or driving forces. Yang et al. [101] invented an energy-harvesting device utilizing compressive-mode piezoelectric technology, featuring exceptional efficiency. In this design, a duo of elastic beams applies a dynamic compressive load to the piezoelectric component. Zhao et al., Fang et al., and Zhou et al. investigated the centrifugal stiffening effect on magnetically coupled beam-type rotation piezoelectric energy harvesters [66,102,103], shown in Figure 4c. Wang et al. [104] proposed an energy harvester that involves utilizing pre-deformed springs to apply a tensile force, which effectively counteracts the constant centrifugal force at the target rotation speed, as shown in Figure 4d. The objective of this method is to optimize the performance of energy harvesters influenced by offset configurations. Qin et al. [105] proposed an adjustable magnetic-preloading piezoelectric traveling-wave ultrasonic millimeter-scale micromotor. The inclusion of an adjustable magnetic-preloading feature enhances torque production and enables accurate motion manipulation for the micromotor. Above all, mechanical and dynamic preloading can enhance the system performance by changing the nonlinear restoring force in energy harvesting, piezoelectric actuators, and vibration isolators.

Figure 4. Static or dynamic preloading to achieve nonlinear energy harvesting. (a) The micrometer head of axially compressing a piezoelectric bimorph [92]; (b) Flex-compressive piezoelectric energy harvesting cell [93]; (c,d) Magnetically coupled beam-type rotation piezoelectric energy harvesters [102,104].

Figure 4. Static or dynamic preloading to achieve nonlinear energy harvesting. (a) The micrometer head of axially compressing a piezoelectric bimorph [92]; (b) Flex-compressive piezoelectric energy harvesting cell [93]; (c,d) Magnetically coupled beam-type rotation piezoelectric energy harvesters [102,104].

2.4. Metamaterial Structure Design

In the past decade, mechanical metamaterial design has gained significant popularity in the field of vibration absorption, energy harvesting, soft robots, and actuators due to its exceptional nonlinear restoring force capabilities. For vibration isolation, Zhang et al. [106] proposed a class of customized mechanical metamaterials featuring programmable quasi-zero stiffness characteristics, where the underlying mechanism is solely derived from the structural geometry of the curved beams and is therefore independent of materials. Li et al. [107] employed shape-memory alloys to realize a temperature-controlled metamaterial beam with quasi-zero stiffness. The proposed method exhibits adjustable and low-frequency band gap characteristics by leveraging ambient temperature fluctuations, as shown in Figure 5a. In addition, the 3D-printed programmable graded cylindrical metamaterial structure [108], as shown in Figure 5b; pairs of folded and buckled beams [109], as shown in Figure 5c; a regulatory mechanism consisting of an electromagnetically charged coil and a magnetic ring [110]; and multi-step quasi-zero-stiffness metamaterial structures [111] are developed for broadband vibration isolation or energy absorption, as shown in Figure 5d. Zhai et al. [112] developed a mechanically engineered metamaterial inspired by origami, which exhibits deployability and collapsibility on demand, making it suitable for various applications, as shown in Figure 5e. Ma et al. [113] presented a pair of distinct two-dimensional auxetic metamaterials. One is characterized by a shuriken-shaped pattern that demonstrates four axes of symmetry, while the other features a triangular motif with six axes of symmetry, as shown in Figure 5f. In addition, Ma et al. [114] proposed a locally tunable resonance metamaterial featuring chiral-buckling structures for effective low-frequency vibration isolation, as shown in Figure 5g. For mechanical metamaterial-based energy-harvesting devices, the studies published on mechanical, acoustic, electromagnetic, and thermal applications using the appropriate mechanical metamaterials were briefly summarized by Tan et al. [115]. The solar energy harvester with an ultra-broadband metamaterial absorber was designed by Bagmanci [116], and a novel microwave energy harvester utilizing metamaterial absorbers featuring square split rings arranged in multiple layers was created by Karaaslan [117] for wireless communications. In soft robotics, Rafsanjani et al. [118] gave a brief review of programming soft robots with flexible mechanical metamaterials, e.g., beam-based structures and kirigami systems. Mark et al. [119] utilized mechanical metamaterials to achieve intrinsic synchronization between two passive clutches that make contact with their travel surface. The proposed design will facilitate the development of compliant and robust implementations that effectively adapt to the miniaturization of soft robot designs. For actuator design, the programmable soft-bending actuators with auxetic metamaterials were designed and fabricated by Qi et al. [120]. The computational approach proposed by Bonfanti et al. [121] offers an automated framework for the design of mechanical metamaterial actuators. Hu et al. [122] experimentally investigated intrinsic metamaterial properties and robotized methods, from fabrication to actuation, of a flexible origami polyhedra actuator. Overall, the essence of mechanical metamaterial design is still the design of the force deformation characteristics of metamaterial units.

2.5. Geometrically Nonlinear Bio-Inspired Design

Recently, various bio-inspired nonlinear structures have been designed for vibration energy harvesting, suppression, and actuators. Feng et al. [123] proposed a human body–inspired anti-vibration isolator, exploring its nonlinear stiffness and damping effect, as shown in Figure 6a. Inspired by the anatomical structure of the human middle ear, a V-shaped folded lever-type vibration isolator was designed by Kim et al. [124], as shown in Figure 6b. The ultra-subharmonics and subharmonics of a bio-inspired X-shaped structure were investigated by Wang et al. [125], demonstrating the possibility of adjusting the structural parameters to achieve either reduction or amplification of the subharmonic reaction. Inspired by a cat that safely landed after falling from a great height, Yan et al. [126] developed a bio-inspired polygonal skeleton structure that can release diverse stiffness, as shown in Figure 6c. Ling et al. [127] have developed a novel structure inspired by three different postures of a clicking beetle, which exhibits variable asymmetric stiffness characteristics to achieve effective vibration isolation, as shown in Figure 6d. In addition, the woodpecker-inspired shock isolator [128], bird multi-layer neck–inspired [129], jumping kangaroo–inspired, and pigeon leg–inspired [130] quasi-zero stiffness vibration suppression systems have been developed in the past five years. Moreover, Pan et al. [131] presented an analysis and design of a bio-inspired dynamics vibration sensor system in the shape of an X for quantifying the displacement of vibrations in an absolute manner. The track-based robot, which is furnished with an innovative suspension system inspired by biological mechanisms, has been systematically investigated by Sun et al. [132]. The exceptional suitability of this technology for diverse engineering assignments has been proven through its provision of a steady mobile platform with remarkable maneuverability, improved ability to navigate various terrains, and outstanding capacity for carrying heavy loads, as shown in Figure 6e. Jing et al. [133] invented an anti-vibration exoskeleton technology that integrates with hand-held jackhammers, effectively reducing vibration transmission while maintaining loading capacity.

For bio-inspired energy-harvesting structures, Qian et al. [134] et al. revealed a conceptual blueprint, executed initial experimental verification, and assessed the efficiency of an innovative bistable piezoelectric energy-capturing device inspired by the open state of the Venus flytrap, where the leaves exhibit curvature in two directions, as shown in Figure 6f. Drawing inspiration from the symbiotic interactions observed in plant species, Fu et al. [135] proposed a host–parasite vibration harvester for scavenging low-frequency random vibrations, as illustrated in Figure 6g. The string effect facilitates the effective conversion of low-frequency beam vibrations into high-frequency parasitic beam vibrations during resonance, thereby enhancing the efficient energy conversion of the broadband low-frequency motion collector. Drawing inspiration from the exercise of push-ups in human fitness training, Yang et al. [136] proposed a bio-inspired hexagonal skeleton structure for the design of wideband energy harvesters utilizing a nonlinear energy sink, as shown in Figure 6h. The harvester is capable of effectively converting the hybrid vibration into electricity, covering a wide low-frequency range.

The study by Llami et al. [137] provides a comprehensive review of notable advances in bio-inspired soft sensors and actuators (from 2017 to 2020), offering a foundational reference for material selection in the design of soft bio-inspired robots. Cai et al. [138] introduced a novel bistable gripper that draws inspiration from the closing motion observed in the mandible of a hummingbird. The robotic gripper, with its bistable characteristics, enables rapid object grasping without the need for continuous external force, as shown in Figure 6i. Drawing inspiration from the wings of mantas and eagles, Gu et al. [139] developed a bistable airfoil capable of undergoing morphological changes from symmetric to asymmetric shape, driven by pneumatic hinges, as depicted in Figure 6j. A three-finger soft gripper has been developed based on this actuator design, and an investigation into its performance in grasping objects of various shapes and sizes has been conducted. Ozaki et al. [140] introduced a piezoelectric direct-drive actuated flapping-wing micro aerial vehicle, inspired by biological mechanisms, that demonstrates exceptional lift capacity and minimal weight, as shown in Figure 6k.

Figure 6. Geometrically nonlinear bio-inspired designs. (a) Body-inspired anti-vibration isolator [123]; (b) Middle ear-inspired lever-type vibration isolator [124]; (c) Bio-inspired polygonal skeleton structure [126]; (d) Click-beetle-inspired structure [127]; (e) Bio-inspired vehicle suspension system [132]; (f) Bio-inspired bistable piezoelectric energy harvester [134]; (g) A host–parasite vibration harvester [135]; (h) A bio-inspired hexagonal skeleton structure [136]; (i) A robotic gripper [138]; (j) A bistable airfoil [139]; (k) Bio-inspired flapping-wing micro aerial vehicle [140].

Figure 6. Geometrically nonlinear bio-inspired designs. (a) Body-inspired anti-vibration isolator [123]; (b) Middle ear-inspired lever-type vibration isolator [124]; (c) Bio-inspired polygonal skeleton structure [126]; (d) Click-beetle-inspired structure [127]; (e) Bio-inspired vehicle suspension system [132]; (f) Bio-inspired bistable piezoelectric energy harvester [134]; (g) A host–parasite vibration harvester [135]; (h) A bio-inspired hexagonal skeleton structure [136]; (i) A robotic gripper [138]; (j) A bistable airfoil [139]; (k) Bio-inspired flapping-wing micro aerial vehicle [140].

2.6. MEMS Manufacturing

The self-powered wireless sensors are consistently reliant on MEMS-manufactured nonlinear energy harvesters. Therefore, a brief review of the MEMS-scale nonlinear energy harvester is briefly reviewed. Paul et al. [141] explored diverse tapering designs to attain the most favorable spring hardening nonlinearities for energy harvesting in MEMS devices, as shown in Figure 7a. The power density has been significantly enhanced relative to linear structures. Ando et al. [142] reported a novel bistable microelectromechanical system designed for energy-harvesting applications and successfully fabricated a micromachined SOI prototype. The electrostatic MEMS energy harvester investigated by Vysotskyi et al. [143] employs an innovative design featuring overlapping gaps in the sensor, as illustrated in Figure 7b. Nguyen et al. [144] presented the development, analysis, and simulation of a broad-spectrum MEMS electrostatic energy harvester integrating resilient nonlinear springs. The output power experienced a 68% increase in comparison to that of the linear model. Similarly, Liu et al. [145] proposed a unique electromagnetic energy-harvesting device for MEMS, which utilizes small suspension structures to achieve an innovative in-plane approximation. The spring hardening effect can optimize the frequency range of the device towards higher frequencies. Lu et al. [146] designed a wide-ranging frequency bandwidth electrostatic kinetic energy harvester utilizing silicon as the primary material, ranging from 1 Hz to 160 Hz, as shown in Figure 7c. For more designs of MEMS-manufactured nonlinear energy harvesters, the high figure of merit nonlinear micro electromagnetic harvester [147], multimodal electret-based MEMS energy harvester [148], magnetic tuning nonlinear MEMS electromagnetic [149], bandwidth-tolerant [150], and out-of-plane electret-based MEMS energy harvesters [151] have been investigated. In addition, Tian et al. [152,153] gave a brief review, and Todaro et al. provided an outlook on piezoelectric MEMS vibrational energy harvesters.

Except for energy harvesters, the MEMS-manufactured nonlinear structures can be used in sensors. Qiao et al. [154] presented an innovative MEMS bifurcation sensor that employs frequency unlocking caused by the internal resonance of 1:3 between two micro-resonators coupled electrostatically, as shown in Figure 7d. Zhang et al. [155] investigated the cubic nonlinearity of a MEMS mass sensor with resonant amplification through an auto-parametric effect. Fang et al. [156] introduced an innovative approach for multi-sensing utilizing nonlinear weakly linked resonators, including a bridge resonator and a cantilever resonator that are mechanically interconnected, as shown in Figure 7e. Liu et al. [157] explored mode localization in a system consisting of two coupled MEMS resonators. In the case of softening Duffing nonlinearity, the sensitivity of the amplitude ratio is greatly enhanced compared to the linear scenario, as shown in Figure 7f. Rhoads et al. and Bogue [158,159] gave a review of micro and nanoresonators’ nonlinear dynamics and recent developments in MEMS sensors.

Figure 7. Various nonlinear devices manufactured using MEMS. (a) Optimal spring hardening nonlinearities for MEMS energy harvesting [141]; (b) MEMS energy harvester with innovative gap-overlap transducer [143]; (c) A silicon-based electrostatic kinetic energy harvester [146]; (d) MEMS bifurcation sensor [154]; (e) New multi-sensing scheme [156]; (f) Two coupled MEMS resonators [157].

Figure 7. Various nonlinear devices manufactured using MEMS. (a) Optimal spring hardening nonlinearities for MEMS energy harvesting [141]; (b) MEMS energy harvester with innovative gap-overlap transducer [143]; (c) A silicon-based electrostatic kinetic energy harvester [146]; (d) MEMS bifurcation sensor [154]; (e) New multi-sensing scheme [156]; (f) Two coupled MEMS resonators [157].

2.7. Dry Friction Applied

Section 2.1, Section 2.2, Section 2.3, Section 2.4, Section 2.5 and Section 2.6 examine various design approaches of nonlinearities in real applications, where the nonlinear restoring force is always a smooth polynomial curve. However, the hysteresis type of nonlinear restoring force was widely encountered in engineering applications. Hysteresis is caused by materials or dry friction between joints. There are at least three types of dry friction dampers utilizing hysteresis restoring force that need to be investigated: wire rope, metal rubber, and bolted connected joints. For wire rope isolators, Balaji et al. [160] conducted a study on the impact of the number of coils, diameter of wire rope, and displacement ranges on the hysteretic properties exhibited by wire rope isolators under dynamic loading in both vertical and horizontal orientations. Carboni et al. [161] focused on a geometrically exact planar beam theory, where a nonlinear hysteretic beam was analyzed using a combination continuum extension of the Bouc–Wen model. The hysteresis behavior under different directions was also experimentally measured by Pellecchia et al., Rashidi et al. and Leblouba et al. [162,163,164], as shown in Figure 8a. In addition, the nonlinear dynamic analysis of the wire-rope isolator and Stockbridge damper has been investigated by Barbieri et al. [165], as shown in Figure 8b. Another type of dry friction–based vibration isolator is metal rubber. Ma et al. [166] designed a metal rubber particle damper based on an auxetic cellular configuration, achieving a maximum damping ratio of 0.5. Yang et al. [167] investigated the dependability characterization and damping capacity of the cyclic periodic metal rubber in the direction opposite to molding, as shown in Figure 8c. The combined stiffness characteristic of metal rubber materials was studied by Fu et al. [168]. The dynamic response of a satellite and carrier rocket system’s metal rubber isolators was investigated by Cao et al. [169]. In addition, the nonlinear modeling of metal rubber [170], semi-active vibration control of metal rubber [171], and hysteretic behavior of a novel metal rubber bridge bearing [172] were also studied. Except for the materials that cause hysteresis, the connected joints are very common structures that have a hysteresis-type of nonlinear restoring force. Lacayo et al. [173] conducted performance assessments of a comprehensive approach that considers the entire time-domain joint and a node-to-node analysis in the frequency domain of the Brake–Reuβ beam, as depicted in Figure 8d. Li et al. [174] developed a novel experimental setup for evaluating the friction hysteresis in fastened connections. The bolt preload and excitation amplitude on hysteresis were studied. The study conducted by Tan et al. [175] demonstrated that the utilization of a single bolt lap beam with threaded connections, along with the incorporation of a friction coefficient, can enhance the accuracy in predicting the hysteresis nonlinear restoring force of the joint. The beam–column connection in a prefabricated steel structure ensures exceptional seismic performance, thereby ensuring the overall structural stability. Thus, Yu et al. [176] proposed an innovative beam–column joint design that deviates from the conventional approach of utilizing bolts for connecting the steel beam. Instead, they opt for a butt-weld connection on the upper flange and a bolted connection on the lower flange, as shown in Figure 8e. The frictional nonlinearities present in the contact interfaces between blades and underplatform dampers in bladed disks were utilized to effectively attenuate vibrational energy. The reduction strategy developed by Quaegebeur et al. [177] aims to address large cyclically symmetric finite-element models in order to simulate complex nonlinear phenomena that may occur in the contact zones, such as stick, slip, and separation, as illustrated in Figure 8f. The Gaussian process nonlinear autoregressive with an exogenous model was employed by Teloli et al. [178] to accurately predict the hysteresis effects of bolted joint structures, and this prediction method was validated through accurate predictions of a bolted beam. For parameter identification of the hysteresis model, the harmonic balance method was employed by Miguel et al. [179] to identify the Bouc–Wen model, aiming to predict the nonlinear characteristics of bolted structures. The micro-slip model incorporates the spatial distribution features of contact pressure at the connection interface proposed by Li et al. [180], where parameters are determined through quasi-static experiments and optimization. From the above-mentioned, the main purpose of the hysteretic nonlinear restoring force is to model the wire rope, metal rubber, and connected interface more accurately. The accurate modeling and identification of this hysteresis is always a big challenge.

2.8. Others

The summary of the design methods for nonlinear restoring force described above is not exhaustive. There are also origami-inspired [181,182], track-designed [183,184], and magnetic-liquid-integrated [185,186] nonlinear restoring force designs for energy absorption, energy harvesting, and vibration isolation. The specific structural design will not be provided in detail here. In the following section, the modeling of various of these nonlinear restoring force structures will be illustrated.

3. Nonlinear Restoring Force Modeling

3.1. Various Polynomial-Based Nonlinear Models

Polynomial nonlinear restoring force models are widely used in structural dynamical analysis due to the fact that many analytical approaches can find closed analytic solutions. In addition, the polynomial-based models can describe many nonlinearities in real applications. In Figure 9, the various polynomial-based nonlinear models are plotted in recently published papers. The quasi-zero stiffness vibration isolator [187] has been widely developed in low-frequency vibration isolation systems and has the characteristics of high–static low–dynamic stiffness in the isolation region, as depicted in Figure 9a. When the isolated object oscillates around in the isolation region, it is very hard to excite due to a very low natural frequency. However, the nonlinear restoring force will gradually increase if the displacement becomes large. The hardening stiffness effect is a well-known nonlinearity widely encountered in engineering. Paul et al. [141] explored various tapering configurations to attain optimal nonlinearities in spring hardening for energy-harvesting purposes, as shown in Figure 9b. In addition, the softening stiffness characteristics can be seen in an adjustable device that adopts two pairs of linear oblique springs and gears [90], as depicted in Figure 9c. The nonlinear stiffness force gradually decreases with the increase in deformation. Figure 9d shows the bistable type of nonlinear restoring force, and there are three points of zero-force displacement [188]. The bistable nonlinear restoring force oscillator is the most popular design in energy harvesting and snap-through soft robots due to its inter-well motion characteristics. Moreover, the tristable and multistable systems can also be designed for different purposes, e.g., a multistable mechanical metamaterial for energy absorption [189]. The nonlinear stiffness can be seen in Figure 9e,f, and some review papers on this type of nonlinear stiffness force dynamic behaviors can be seen here [190,191,192,193]. For a multi-degree-of-freedom steel frame structure, the double Chebyshev polynomial [194] was used as a general nonparametric model for various nonlinear restoring forces, as shown in Figure 9g. To approximate the nonlinear spring and damper characteristics of suspension systems in motorcycles, Lauβ et al. [195] used cubic spline types of nonlinear restoring force, as depicted in Figure 9h, and the parameter is identified by efficient gradient computation using the adjoint variable approach. Another type of discontinuous polynomial-type bistable nonlinear stiffness force can be seen in a bio-inspired, prestressed double-buckling beam [134], as shown in Figure 9i. In addition, this type of nonlinear stiffness force can also be seen in bistable hybrid symmetric laminates in nonlinear energy sink structures [196].

3.2. Hysteretic Models

Hysteresis is widely encountered in engineering, e.g., dry friction isolators, connected joints, piezoelectric actuators, etc. Therefore, various hysteresis models have been developed for nonlinear restoring force characterization. In Figure 10, some representative hysteresis models are listed, although they are not exhaustive. The Bouc–Wen model, as shown in Figure 10a, is a first-order differential equation with memory characteristics. It is widely used in wire-rope, metal-rubber, and magneto-rheological dampers for vibration isolation [172,197,198]. Considering the limitations of the original Bouc–Wen model, the modified and improved Bouc–Wen model that can describe asymmetric and softening-hardening characteristics has also been proposed. For bolted-connected or other types of joints, the Iwan model is a very popular physical model, and the nonlinear restoring force can be seen in Figure 10b. It was originally proposed by Segalman in lap-type joints. Later, this model was developed by many scholars, such as the five-parameter Iwan model [199] and the six-parameter Iwan model [200]. Readers can refer to this book to find more detailed applications of the Iwan model [201]. The Prandtl–Ishlinskii model is applicable in various hysteresis materials and exploits stop-and-play operators, as depicted in Figure 10c. A comprehensive study on the Prandtl–Ishlinskii hysteresis model: an in-depth exploration of its inverse compensator was written by Janaideh [202]. In addition, there are significant investigations on the identification of the Prandtl–Ishlinskii model in piezoelectric actuators [203,204,205]. In terms of some mechanical, biomedical, and electrical systems [206,207,208], the Maxwell–Slip model was widely used. This model uses elastic sliding elements, which include massless linear springs and massless blocks that are easily affected by the Coulomb friction effect. The specific view of the model description can be seen in Figure 10d [209]. Another type of differential-based rate-independent hysteresis model is the Duhem model [210], as shown in Figure 10e. This type of model is often used in piezoelectric and magnetostrictive actuators for robust control [211,212]. The Jiles–Atherton model is specially used in ferromagnetic, magnetostrictive, and piezoelectric materials characterization, as depicted in Figure 10f. For instance, Jia et al. proposed a giant magnetostrictive material-based novel force sensor using the Jiles–Atherton model to analyze the relationship between force input and electromagnetic output [213]. For identification issues, genetic algorithms are commonly used methods to identify this type of model parameter [214].

3.3. Piecewise Linear Models

Except for the above polynomial and hysteresis-type nonlinear restoring forces, the piecewise linearity is also a typical nonlinear restoring force. There are at least three applications of piecewise linear nonlinear restoring force: nonlinear energy sink, gap or collision structure, and quasi-zero stiffness vibration isolators. The nonlinear restoring force of these three types can be seen in Figure 11. Yao et al. [217] developed a nonlinear energy sink using a piecewise linear stiffness spring, and the vibration attenuation performance is good under a moderate exciting force. In wing-to-payload mounting interfaces on F-16 aircraft, the piecewise linear nonlinear restoring force can be observed by frequency sweeping [218], and there is high stiffness in small displacement regions. For quasi-zero stiffness vibration isolators, zero stiffness in a small displacement region is often assumed in order to simplify dynamic analysis [219,220]. In addition, the piecewise linear can also be found in complicated dynamic response analysis in a multi-degree-of-freedom system with local stoppers [221]. For a review of piecewise-linear models, refer to these two articles [222,223].

4. Nonlinear Restoring Force Identification: Review

4.1. Typical Restoring Force Identification Applications

The nonlinear system identification strategies were comprehensively reviewed in 2016 by Noël and Kerschen [32]. In addition, Schoukens et al. [224] gave a user-oriented road map of nonlinear system identification and provided several examples with detailed mathematical explanations and formal proofs. Habibi et al. [225] investigated the nonlinear damping identification methods in structural dynamics. The techniques employed for the identification of nonlinear systems in free decay measurements were compared by Jing et al. [47] in the context of jointed structures. However, the systematic review of parameter identification methods for nonlinear restoring force models in Section 3 has not yet been classified.

Figure 12 gives a schematic overview of restoring force surface, Hilbert transforms, time-frequency analysis, nonlinear subspace identification, Kalman filter, optimization algorithms, physics-informed neural networks, and sparse identification methods. Among these nonlinear system identification methods, the restoring force surface identification method, Hilbert transform-based identification method, and time-frequency analysis method are nonparametric identification methods, which means the identified nonlinear restoring force needs to be fitted by a selected model. The other identification method is the parametric identification method. The identified nonlinear dynamical equation must be prior. It must be noted that the nonlinear restoring force function requires a certain transformation due to different types of nonlinear identification methods, utilizing different forms of dynamic equations, e.g., differential equations and state space equations.

4.2. Restoring Force Surface Method

The method of representing the stiffness and damping force characteristics of a nonlinear vibrator, known as the restoring force surface approach, utilizes graphical visualization based on displacement and velocity variables, and is directly incorporated into Newton’s second law of motion [232,233]. The nonlinear restoring force trajectory can be visualized by intersecting the restoring force surface with a vertical zero-velocity plane. It must be noted that the zero-velocity plane is not an absolute zero-velocity plane; it can be assigned to a reasonable region to keep identification accuracy. When the nonlinear restoring force is intercepted, the curve fitting is then utilized to obtain the final nonlinear restoring force function. Moreover, it is crucial to highlight that the simultaneous procurement of displacement, velocity, and acceleration is indispensable for implementing the restoring force surface technique.

Figure 13 illustrates the nonlinear restoring force identification of typical beneficial nonlinear structures. Allen et al. [234] applied the restoring force surface method to identify the force–velocity relationship for a microfabricated cantilever beam. The measurement results indicate the presence of a nonlinear oscillatory force exerted on the tip of the cantilever as its velocity approaches the maximum value in each cycle. Noël et al. [235] addressed the identification of a practical aerospace structure containing a highly nonlinear element featuring multiple mechanical stops and piecewise-linear stiffness. Liu et al. [236,237] utilized the restoring force surface method to identify the clearance value of the continuum structure, taking into account the dynamic properties of the nonlinearity in clearance. The effectiveness of the proposed method was experimentally validated through the design of a cantilever beam with adjustable clearances. Ma et al. [238] revealed the presence of nonlinear segmented restoring forces in hydrokinetic power conversion, which was achieved by utilizing flow-induced motions of a single cylinder. Dekeleme et al. [239] constructed the restoring force surface of a Duffing-type nonlinear energy sink structure. By employing mem-springs and mem dashpots in either series or parallel configurations, Pei et al. [240] employed the restoring force surface method to identify reconfigurable devices made of mild steel and SMA wire/wire rope. Liu et al. [241] comparatively investigated the numerical differentiation and integration method to identify quasi-zero stiffness, bistable, and tristable cantilever-beam structures. The results showed that the acceleration-measurement restoring force surface method cannot identify bistable or tristable structures. Pei et al. [242] combined restoring force surface with interpretable machine learning to find the nonlinear hysteresis in tube-like structures under shock impact. The restoring force surface of a double-well Duffing oscillator with position-dependent friction damping was constructed by Zhu et al. [243], and the parameters were identified using nonlinear subspace identification. Anastasio et al. [244] presented an experimental restoring force surface of a configuration of a spring in the shape of an X-shape, used as a mechanical oscillator. The findings contribute to the advancement of knowledge regarding the attributes of suspensions with nonlinear elastic properties and an X-shaped configuration exhibiting softening behavior and accommodating large displacements.

4.3. Hilbert Transform-Based Method

The method based on the Hilbert transform utilizes the concepts of envelope and instantaneous natural frequency to derive trajectories of nonlinear restoring forces, relying on the analytical signal representation of Newton’s second law of motion. The method can be directly applied to nonlinear monostable structures, such as those exhibiting hardening and softening nonlinearities, by utilizing a free vibration signal [35]. However, the free attenuation response contains rich harmonic components in strongly nonlinear structures that prevent the extraction of the envelope and instantaneous natural frequency [245]. Therefore, the Hilbert vibration decomposition was developed by Feldman to tackle this challenge [246]. By utilizing the Hilbert vibration decomposition method, it becomes possible to break down the free vibration of the strongly nonlinear structure into several gradually changing elements. This simplifies the process of determining a consistent envelope and corresponding natural frequency. Thus, strongly nonlinear structures can also be identified by Hilbert transform identification methods.

Figure 14 presents the most relevant Hilbert transform-based identification of typical nonlinear restoring force structures. Feldman et al. [247] discussed the experimental identification results of a nonlinear vibrating mechanical system that consists of a mass, elastic beam, actuator, and tension element. The piecewise linear restoring force can be extracted. Wu et al. [248] extended the Hilbert transform method to identify multi-degree-of-freedom structures with local piecewise nonlinearities, and experiments were performed on nonlinear hinges in typical folding wing structures. Feldman et al. [249] enhanced the Hilbert transform-based approach, enabling it to accurately capture the characteristics of nonlinear tangential stiffness and friction forces that arise during submicron motions. The Hilbert transform method was initially employed by Yuan et al. [250,251] to identify the presence of nonlinearity in a piezoelectric energy harvester utilizing circular laminated plates, which can exhibit either hardening or softening behavior. The novelty of the approaches lies in their utilization of both mechanical and electrical signals, rendering them applicable to a wide range of nonlinear piezoelectric mechanical systems. Harduf et al. [252] investigated the Hilbert transform method identification of an additively manufactured particle damper as a 2DOF frictional system, providing a comprehensive description of its dynamics.

4.4. Time-Frequency Analysis

The time-frequency analysis method may contain short-time Fourier transforms, early-time fast Fourier transforms, and wavelet transforms [253,254]. The intrinsic feature of time-frequency analysis is used to obtain the instantaneous amplitude and the instantaneous natural frequency. The parameters to be identified need to be interpolated. Thus, the time-frequency analysis method is also nonparametric.

Figure 15 illustrates the most relevant time-frequency analysis identification of typical nonlinear restoring force structures. Nagarajaiah et al. [255] proposed an algorithm for controlling wind-induced response in buildings that are equipped with variable stiffness-tuned mass dampers, utilizing the short-time Fourier transform. Avargel et al. [256] introduced a novel approach to enhance the identification of nonlinear systems in the domain of the short-time Fourier transform. Their research on quadratic nonlinear systems demonstrated that their method significantly enhances estimation accuracy while substantially reducing computational costs compared to time-domain Volterra models. Allen et al. [257] proposed a time-frequency signal processing technique to identify and characterize nonlinearity in transient response measurements of mass-spring systems with seven degrees of freedom and a nonlinear connection. The penalty function was incorporated by Wang et al. [258] into the ridges of the wavelet transform, which is continuous in order to accurately determine the instantaneous frequency of time-varying structures. A cable experiment was conducted with varying tension forces, and the proposed method allows for the identification of the instantaneous frequencies of the cable. Yang et al. [259] proposed a parameterized time-frequency transformation employing spline kernels to effectively identify systems exhibiting both linear and nonlinear time-varying characteristics. This approach was rigorously validated on a system featuring three distinct types of nonlinear time-varying stiffness, namely stiffness modulation with varying periods, segmented modulation of stiffness, and periodic modulation of nonlinear stiffness. A novel approach utilizing adaptive sparse time-frequency analysis was proposed by Bao et al. [260] to detect the changing cable tension of bridges. The findings indicate that, in comparison with the Hilbert-Huang transform technique, the adaptive sparse time-frequency analysis method offers a more precise estimation of the varying cable tension over time. Wang et al. [261] presented a discrete wavelet transform and substructure algorithm to track the sudden degradation of shear structures’ stiffness. Experiments conducted on a structure characterized by a three-story shear-type design that exhibits a sudden reduction in stiffness validate the effectiveness of this approach. Meng et al. [262] used a short-time Fourier transform to identify time-varying mesh stiffness of gear teeth with different crack lengths for fault characterization. Qu et al. [263] first introduced the adaptive wavelet transform analysis to the damage identification of the cable-stayed bridge, and a shake table test was carried out to verify the effectiveness. The extracted nonlinear stiffness and damping coefficients offer initial insights into the progression of damage during seismic input.

4.5. Nonlinear Subspace Identification

The nonlinear subspace identification method leverages robust numerical methods, such as orthogonal triangular decomposition and singular value decomposition, through the utilization of geometric tools [264]. The nonlinear restoring force in the dynamic equation can be considered as internal feedback forces that exhibit nonlinearity. The estimation of the extended frequency response function matrix is possible for both the dynamic system and the matrices related to input, output, and direct feedthrough.

Figure 16 shows the development of nonlinear subspace identification of typical nonlinear restoring force structures. The deterministic–stochastic subspace identification method was employed by Moaveni et al. [265] to characterize nonlinear structural systems adopting the time-varying and amplitude-dependent instantaneous modal parameters. The Giuffre–Menegotto–Pinto hysteretic model was utilized for verification. The investigation focused on the influence of false poles on nonlinear subspace identification by Marchesiello et al. [266], who also introduced several modal decoupling tools. The experiments conducted on a multi-degree-of-freedom system featuring a local hardening nonlinearity validate the substantial enhancements in estimation. Zhang et al. [36] developed a time-domain approach consisting of two stages that relies on the nonlinear subspace technique. The underlying linear system was identified prior to the local nonlinearities being integrated. Filippis et al. [267] employed frequency-domain nonlinear subspace identification and introduced the technique of spurious pole removal for enhancing the accuracy of identifying state-space matrices. The enhanced nonlinear subspace identification technique was validated using the Morane–Saulnier Paris aircraft. Liu et al. [268] devised an internet-based technique, known as correlation-subset-driven stochastic subspace identification, to oversee the state of suspension. Sun et al. [269] expanded the application of nonlinear subspace identification to vibrating structures with multiple degrees of freedom, incorporating freeplay. Experimental techniques were employed to estimate two types of freeplay, namely central freeplay and offset freeplay. Ma et al. [270] introduced an innovative technique for identifying nonlinear subspace in the time domain, specifically focusing on output-only data. This methodology has been successfully applied to three different systems, both numerical and experimental, which exhibit clearance nonlinearity. Liu et al. [271] first introduced the nonlinear subspace identification to an electromechanical coupling system, and a bistable piezoelectric energy-harvesting cantilever beam was utilized for verification. The results demonstrated that both the nonlinear restoring force and the electromechanical coupling factor can be simultaneously identified. Wei et al. [272] investigated a subspace identification method for nonlinear structures under oversampling. It combines the prediction error method with multi-frequency data to accurately estimate linear and nonlinear model parts, validated experimentally in a nonlinear energy sink structure. Yao et al. [273] suggested tackling the issue of stiffness identification in beam structures with elastic foundations by combining time-domain nonlinear subspace identification and global-mode analysis. Above all, the nonlinear subspace identification method and its improved versions have shown great potential in identifying various highly nonlinear systems.

4.6. Unscented Kalman Filter

The Kalman filter presents an alternative methodology for linearization. The extended Kalman filter employs analytical linearization to handle nonlinearity, whereas the unscented Kalman filter utilizes statistical linearization based on a predefined set of rules. The extended Kalman filter has emerged as a widely adopted technique in various nonlinear estimation and machine-learning applications [274].

Figure 17 depicts the development of unscented Kalman filter identification of typical nonlinear restoring force structures. Yuen et al. [275] proposed an advanced Bayesian algorithm for real-time system identification that incorporates state-of-the-art model-class selection components into the extended Kalman filter. This cutting-edge algorithm allows for real-time scenarios where model selection and parametric identification can be performed simultaneously. The presented examples demonstrate the application of noisy dynamic response measurement for damage detection in deteriorating structures. Jiang et al. [276] proposed a new approach for calibrating kinematics using an algorithm that combines extended Kalman and particle filters, which effectively enhances the positioning accuracy of robots. By utilizing the extended Kalman filter algorithm, the kinematic and error models of the robot were established, and its kinematic parameters were determined. Lei et al. [277] introduced a unique unscented Kalman filter that incorporates an unknown input, enabling the concurrent estimation of nonlinear structural systems and external excitations. The shear wall test was conducted, and the proposed method accurately enables the identification of the hysteretic force. Lund et al. [278] utilized an unscented Kalman filter to estimate the model parameters of an experimental nonlinear energy sink device, which exhibits geometric nonlinearity in its stiffness and hysteretic nonlinearity in its damping behavior. Xu et al. [279] proposed an EKF-based method employing a global iteration method with appropriate weighting for the identification of both the presence of a nonlinear restoring force and the inclusion of mass in structures simultaneously. The experimental validation of the algorithm is conducted through a dynamic test on a four-story frame structure equipped with a magnetorheological damper. Nguyen et al. [280] introduced an innovative approach to identifying adaptive parameters in a model consisting of three elements: a predicted model, a hysteresis observer, and algorithms that adapt. The validity of the approach was additionally confirmed for parameter estimation of a nonlinear vibration system that is equipped with an isolator. Paul et al. [281] explored various iterations of Kalman filter-based algorithms, including the extended Kalman filter, two-stage EKF, and the unscented Kalman filter. These algorithms were utilized to assess and compare their efficacy in determining the condition and characteristics of a nonlinear hysteretic model during the simulation of an unattached fiber-reinforced elastomeric isolator. Zhao et al. [282] developed an improved extended Kalman filter with unknown input using partial degree-of-freedom acceleration measurements. The experimental verification was performed on frame structures with multiple degrees of freedom, incorporating different parametric magnetorheological dampers to replicate a wide range of nonlinear characteristics.

4.7. Optimization Algorithms

The optimization techniques utilized in this scenario involve genetic algorithms, particle swarm optimization, differential evolution, and optimization based on artificial neural networks [46]. The dynamic equation of most nonlinear restoring force structures can be simplified as ordinary differential equations. The dynamic responses can be numerically calculated by the Euler method and the Runge–Kutta method. Once the dynamics equation can be simulated, the fitness function can be constructed, and then the identification problem becomes an optimization problem. Based on empirical knowledge, a reasonable parameter range can be given, which will greatly enhance the identification accuracy.

Figure 18 illustrates the development of the most recent optimization algorithm-based identification for typical nonlinear restoring force structures. Jiang et al. [283] proposed a novel approach to identify parameters in nonlinear hysteretic systems by leveraging sequence model optimization. The findings suggested that the identified numerical model is capable of accurately capturing the degradation in strength and stiffness, as well as the pinching effect observed in the structural system. Negash et al. [284] proposed a novel genetic algorithm for Bouc–Wen model parameters estimation for magnetorheological fluid dampers. The proposed novel genetic algorithm for Bouc–Wen model parameter identification can be applied to enhance the accuracy of control systems for magnetorheological fluid dampers. Nguyen et al. [285] introduced a combined approach of the adaptive differential evolution and Jaya algorithm for the detection of the Bouc–Wen hysteresis model in a piezoelectric actuator. The proposed novel algorithm can accurately and precisely identify the highly hysteretic nonlinearity of the piezoelectric actuator. The modified particle swarm optimization algorithm proposed by Feng et al. [286] effectively addressed the limitation of local optimization, thereby enhancing the accuracy in predicting high-precision actuating mechanisms and their output. Li et al. [287] proposed a multiobjective optimization technique known as NSGA-II for parameter identification of the Bouc–Wen–Baber–Noori hysteresis. The data on reinforced concrete column experiments were acquired from the database of the Pacific Earthquake Engineering Research Center, validating their efficacy. The Hammerstein-based hysteresis offline identification of a fraction-order polynomial-modified Prandtl–Ishlinskii model was presented by Yi et al. [288], enabling accurate identification of the rate/load/temperature hysteresis for smart material actuators. Liu et al. [289] developed an enhanced Iwan model to characterize softening-hardening hysteresis in a wire-rope isolator, and the particle swarm optimization–integrated hybrid identification was used for parameter estimation. A new approach was introduced by Li et al. [229] to identify multiple parameters of concrete dams, which combines the slime mold algorithm with polynomial chaos expansion methodology. The practicality of the proposed approach was demonstrated through its successful application to a sophisticated dam model with multiple variables, affirming its efficacy in addressing real-life engineering challenges. Zhou et al. [290] developed a novel approach combining the differential evolutionary algorithm and the sparrow search algorithm to detect a rate-dependent Prandtl–Ishlinskii model. The experiments on marine dampers demonstrate their superior efficiency and rapid convergence speed compared to other algorithms.

4.8. Physics-Informed Neural Networks

The introduction of the physics-informed neural network by Raissi et al. [291] aimed at addressing data-based solutions and uncovering partial differential equations through data analysis. The effectiveness of the proposed framework is convincingly showcased by a series of well-known challenges in fluid dynamics, quantum mechanics, reaction-diffusion systems, and the transmission of nonlinear shallow-water waves. They address the challenge of limited data availability in certain engineering systems, which undermines the robustness of most state-of-the-art machine learning techniques, rendering them ineffective in such scenarios [292]. The incorporation of prior knowledge regarding general physical laws serves as a regularization mechanism during the training process of neural networks, effectively constraining the solution space and enhancing the accuracy of function approximation.

Figure 19 presents the very recent development of physics-informed neural network identification for typical nonlinear restoring force structures. Li et al. [293] explored a physics-based neural network framework that incorporates an approximation function for boundary conditions, which aims to tackle the typical difficulties faced in the field of adaptable mechatronics and soft robotics. A boundary condition–embedded approximation function was employed in a biomechanical system to iteratively determine its dynamic parameters by leveraging the repetitive natural flexion of the foot. Lai et al. [294] exploited a novel approach to structural identification through the utilization of neural ordinary differential equations, with specific emphasis on incorporating domain knowledge such as structural dynamics. The method was verified on a structural system equipped with a negative stiffness hysteretic device. A general framework was proposed by Zhang et al. [295], which utilizes physics-informed neural networks to identify nonlinearities in unknown geometric and material parameters. Yang et al. [296] introduced a neural network that incorporates physics knowledge to forecast and detect the dynamics of collaborative robot joints. The methodology entails the creation of a dynamic model based on state-space principles, incorporating the system’s dynamics into a recurrent neural network through tailored Runge–Kutta cells. Subsequently, labeled training data is gathered to facilitate predictions on system responses and estimation of dynamic parameters. The training process of a physics-informed neural network has been enhanced by Guo et al. [297] through the incorporation of physical constraints, while finite element computation is integrated with the uniform design to generate an ideal collection of training data. The proposed physics-informed neural networks were utilized to estimate the nonlinear stiffness parameters of an actual frame structure. A novel framework combining model-based and data-driven approaches was proposed by Liu et al. [298] for identifying dynamic loads in interval structures. The proposed approach significantly enhances the precision, robustness, and generalization capabilities for identifying cantilever beam and wing structures.

4.9. Data-Driven Sparse Regression

The dynamics of most nonlinear dynamical systems are primarily defined by a limited number of significant terms, resulting in sparse governing equations within the basis function space. The sparse identification of a nonlinear dynamic system was originally proposed by Brunton et al. [231], and the algorithm was demonstrated across various problem domains, spanning basic canonical systems like nonlinear oscillators and the turbulent Lorenz system, to study fluid vortex shedding caused by obstacles. In the past five years, sparse identification and its enhanced iterations have gained significant traction in the realm of nonlinear system identification problems.

Figure 20 illustrates the very recent development of data-driven sparse identification for typical nonlinear restoring force structures. The recent sparse identification of nonlinear dynamics modeling procedures, as extended by Kaiser et al. [299], incorporates the effects of actuation and demonstrates the capability of these models to enhance the performance of model predictive control even with limited and noisy data. The work of Brunton et al. (2016) was extended by Lai et al. [300] to incorporate functions that enable the identification of significant nonlinearities, as well as hysteresis or inelastic behavior accompanied by permanent deformation. A novel structural system with nonlinearity is composed of a multi-degree-of-freedom structure and a negative stiffness device, demonstrating the remarkable capability of the proposed framework. Lin et al. [301] proposed an innovative method for identifying nonlinear dynamical systems, which combines the sparse regression algorithm with the separable least squares method. The utilization of Duhamel’s integral was applied to depict the dynamic correlation between the system’s input and output. Additionally, a novel approach based on reproducing kernel Hilbert space was introduced for non-parametric noise reduction in vibration displacement. This method aims to acquire velocity with minimized noise by denoising the displacement signal. Cheng et al. [302] proposed a two-stage sparse identification method, which was capable of localizing and characterizing local nonlinear structures containing nonlinear components, even when measurements are noisy. The proposed algorithm exhibits enhanced robustness to noise compared to one-stage sparse algorithms, as it can accurately identify local nonlinear structures even at lower signal-to-noise ratios. A sparse regression method incorporating physics knowledge was proposed by Novelli et al. [303], which was validated using a canonical spring-mass hopper and an electromagnetic energy harvester with non-continuous nonlinearities. Qian et al. [304] proposed a novel and advanced data-driven reconstruction method, which was based on the principle of data assembly and utilized a technique for determining the sparsification parameter. The bionic joint prototype with bistable characteristics had been developed, demonstrating both geometric and constitutive nonlinearity. The data-driven reconstruction method proves to be effective in addressing the challenges presented by non-ergodic data and undetermined sparsification parameters when reconstructing models of multistable systems. This approach holds potential for various applications in domains like robotics and deployable structures involving multistable nonlinear systems. Sun et al. [305] proposed a method of transcending the local attractors through the application of arbitrary perturbations on the time domain alongside an arbitrary initial condition in a compressive sensing process. The proposed generalized modeling reconstruction method enables continuous and compressive sensing of the dynamic behaviors and stability of multiple attractors in origami metastructures during parallel assembly and multistable cantilever-beam operation. The physics-encoded sparse identification of nonlinear dynamics, as introduced by Lathourakis et al. [306], expands upon the existing RK4-SINDy identification approach by integrating established principles of physics and domain expertise in three distinct ways. This proposed technique demonstrates the ability to ascertain the governing equations for dynamic systems that may involve non-smooth and discontinuous nonlinear stiffness forces, such as those arising from frictional contacts. Lu et al. [307] introduced a connected sparse least-square model, which incorporates an all-encompassing fused lasso regularization term to simultaneously detect governing equations of nonlinear dynamical systems using multiple state measurements corrupted by noise. The outcomes derived from the simulations and experiments indicate that their approach showcases a higher degree of precision in recognizing dynamic systems when compared to conventional sparse least squares. Chatterjee et al. [308] aimed to discover the governing laws for complex nonlinear structural dynamic systems by utilizing available noisy data and incorporating sparse Bayesian machine learning techniques. The proposed framework was successfully applied to actual datasets obtained from an experimental setup of a quasi-zero stiffness device, showcasing remarkable performance.

4.10. Comparisons

Based on these eight identification methods, their applicability to different nonlinear restoring force structures, noise resistance, capability for handling high-dimensional systems, and identification efficiency are summarized in four aspects. Since identification accuracy is easily affected by the combined influence of the other four factors, a comparison of identification accuracy is not considered here. The detailed summary is presented in

Table 1. Comparison of the nonlinear restoring force identification capability of eight identification methods.

Methods	Types of Restoring Force	Noise Resistance	Capability to Handle High-Dimensional Systems	Efficiency
Restoring force surface	Quasi-zero stiffness, bistable, multistable, hysteretic restoring force.	low	No	low
Hilbert transform	Piecewise linear/nonlinear, softening and hardening			low
Time-frequency analysis	Piecewise linear/nonlinear, softening and hardening	high		low
Nonlinear subspace identification	Quasi-zero stiffness, bistable, and multistable		Yes	low
Unscented Kalman filter	Any type of nonlinearity (the restoring force model must be defined a priori)		Yes	high
Optimization algorithms		low	No	low
Physics-informed neural networks			Yes	high
Sparse identification		high	Yes	high

.

The restoring force surface method and the Hilbert transform method are non-parametric identification approaches that do not require prior knowledge of the nonlinear restoring force model. They are primarily applied to single-degree-of-freedom structures with strong nonlinearity or to multi-degree-of-freedom systems with local single nonlinear oscillators. The restoring force surface method is suitable for quasi-zero stiffness (piecewise linear/nonlinear, softening, and hardening), bistable, multistable, as well as hysteretic restoring forces, whereas the Hilbert transform is not applicable for parameter identification of complex hysteretic restoring forces. Time-frequency analysis methods mainly focus on characterizing the variation in the system’s natural frequency with vibration attenuation time, reflecting stiffness and damping characteristics through changes in frequency and amplitude. This approach also belongs to non-parametric identification. Since piecewise linear systems may involve impacts that prevent observation of transient natural frequency variations with amplitude in time-frequency diagrams, it is only suitable for simple softening or hardening polynomial-type nonlinearities.

When applied to multi-degree-of-freedom nonlinear structures, the nonlinear subspace method relies critically on determining the model order and the type of nonlinearity. It is applicable to multi-degree-of-freedom nonlinear structures; however, no literature has reported its direct applicability for identifying parameters of complex hysteretic restoring forces. When identifying piecewise linear/nonlinear systems, it also requires pre-assumed model functions. Optimization algorithms, widely used across various disciplines, play a significant role in solving algebraic and differential equations. Their drawback lies in the time-consuming process of iterative optimization to minimize the error between reconstructed and actual responses, and they may fail to converge if assumptions are unreasonable. Theoretically, they are applicable to any nonlinearity.

The unscented Kalman filter, employed in control for real-time state estimation, offers advantages in speed, accuracy, and real-time capability. Its application in identifying nonlinear restoring forces in structural dynamics shows considerable promise and is suitable for arbitrarily assumed nonlinear types. Physics-informed neural networks and sparse identification methods have recently been applied to emerging research fields such as soft robotics and origami structures. The powerful fitting capability of neural networks provides an advantage for approximating arbitrary nonlinear restoring forces, while sparse identification can identify the primary contributing basis functions of the nonlinear restoring force, provided the library is constructed adequately. Although both methods require some prior knowledge, their improved versions, with faster iteration speeds, are gradually being adopted in engineering practice. The unscented Kalman filter, sparse identification, and physics-informed neural networks are expected to become mainstream methods for restoring force identification in nonlinear vibration control under complex working scenarios in the future.

Finally, it is important to note that using identification methods to obtain the dynamic restoring forces of vibration control components offers at least two distinct advantages over quasi-static measurements, highlighting the significance of dynamic parameter identification. First, dynamic testing can be performed under assembled conditions, relying only on limited sensor measurements, and can be conducted in real-time online to identify restoring forces in actual scenarios, which is particularly critical for real-time dynamic control and health monitoring. Second, quasi-static testing cannot capture the rate-dependent characteristics of nonlinear restoring forces (beyond the low-frequency range), whereas dynamic parameter identification can cover the mechanical properties of vibration control components under a broader range of frequency conditions.

5. Nonlinear Restoring Force Identification Enhancement Strategies

In Section 4, the development of eight different types of nonlinear system identification methods is reviewed. It can be found that the restoring force surface method, Hilbert transform-based method, and time-frequency analysis identification are nonparametric identification methods. It means that the identified nonlinear restoring force needs to be parameterized by curve-fitting techniques. Experience has shown that identification accuracy should be improved if the high level of noise is polluted. Thus, if the nonparametric identification methods are only used for parameter interval identification and followed by an optimization strategy, the identification accuracy will be greatly increased under high noise levels [226]. On the contrary, nonlinear subspace identification, unscented Kalman filter, optimization algorithms, and sparse identification methods are parameterized identification methods. Model selection plays an essential role in identification accuracy. Therefore, these four methods have their limitations and can be enhanced with the combination of contour variable-selection algorithms [309] and Bayesian model selection techniques [310].

Regarding nonparametric identification methods, constrained optimization may be a good way to achieve enhancement of identification accuracy. According to the present optimization methods, particle swarm optimization [311,312], differential evolution [313,314], and artificial neural network [315,316], etc., are widely adopted in optimization strategies for ordinary differential equations. However, optimization effectiveness and efficiency are highly dependent on prior knowledge. The nonparametric identification method is a good way to provide prior knowledge due to its natural physical meaning attribution. The combination of preliminary identification and optimization will enhance identification accuracy.

For the nonlinear subspace identification and sparse identification algorithm, a polynomial model is first assumed, and then the identification process can be successfully conducted. However, if the function of the nonlinear restoring force is preliminarily unknown or the polynomial order is unknown, the identification accuracy will significantly decrease [317]. Although sparse identification can give more candidate nonlinear functions in the library, the identification efficiency will greatly decrease. Moreover, more candidate basis functions may lead to overfitting problems. The nonparametric preliminary identification, such as restoring force surface and Hilbert transform-based method, may give a good prior knowledge before sparse regression. In a recently published article, Zhu et al. [318] introduced the nonlinear restoring force surface method into the nonlinear subspace method, which verified the ability of this hybrid strategy. Lai et al. [319] adopted the physics-informed idea in sparse identification and experimentally validated three-story structures equipped with a negative stiffness device. As is well known, physics-informed machine learning has been a hot topic in the last decade [320,321].

In addition to nonparametric preliminary identification, Bayesian model selection is now widely investigated in the system identification communities [322,323,324]. The most credible model, as commonly acknowledged, refers to the model with the highest probability of occurrence generated by Bayesian inference based on existing data. Assuming that there are N possible model candidates for the representation of a nonlinear restoring force. The occurrence probability of each model can be calculated quantitatively based on the probability analysis, and then the best model can be determined. Then, the corresponding dynamic response based on the selected model will be constructed. By implementing the above Bayesian inference, each model will give a prediction error. The final best model can be determined by the comparison probability of each model. Therefore, the Bayesian model selection is not only useful in the four identification algorithms studied in this paper but also for a wide applications of other modeling and identification methods. In fact, Bayesian model selection has already been widely developed in various areas [325,326].

6. Conclusions and Challenges

This paper presents a state-of-the-art review of nonlinear restoring force design methodologies, modeling, and parameter identification in typical beneficial nonlinear vibrating structures. The seven kinds of design methodologies and three types of commonly used models are classified. The system parameters identification methods covering restoring force surface, Hilbert transform, time-frequency analysis, nonlinear subspace identification, unscented Kalman filter, optimization algorithms, physics-informed neural networks, and data-driven sparse regression are reviewed. Moreover, some enhancement strategies based on the current nonlinear system identification of the nonlinear restoring force are presented.

It is believed that the provided review of nonlinear restoring force design methodologies and models will give some understanding to readers of beneficial nonlinear structural designs. Despite this evident progress, beneficial nonlinear structural design and modeling still have several important challenges ahead of us:

• Beneficial nonlinear structures are rarely applied in real engineering situations. For example, the nonlinear energy harvester should have MEMS-scale and high energy output simultaneously, which is a great challenge for nonlinear design and fabrication. The quasi-zero stiffness should be designed to simultaneously exhibit very low natural frequencies and high static load-bearing capacity.
• When considering the benefits brought by nonlinear design, additional negative impacts also need to be considered.

Moreover, although the system identification methods have been greatly developed in the past decades, the potential development of these and other identification algorithms could be explored further. New findings and several important challenges are still ahead of us:

• Several new applications of quasi-zero stiffness and bistable structures require additional attention and development. For instance, the metamaterial structures and bio-inspired smart structures. The high-dimensional, complicated hysteresis loops, multi-degree-of-freedom, multiscale, and non-lumped parameter modeling and identification may continue to be a great challenge.
• The lack of excitation measurement, the time-varying feature, and the strong noise disturbance are usually encountered in real situations. Therefore, when the identification algorithm is applied, the available datasets, the time-varying parameters, and the robustness of noise should be considered first.
• Many advanced machine learning methods have been used for system identification and control. The feasibility of obtaining a large amount of data needs to be considered. Balancing identification accuracy and computing efficiency is very important in data-driven identification methods. Moreover, the physical interpretation of data-driven methods needs further exploration.