aState Key Laboratory of New Ceramics and Fine Processing, School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China
bDepartment of Engineering Mechanics & Center for Nano and Micro Mechanics & Key Laboratory of Applied Mechanics, Tsinghua University, Beijing 100084, China
cState Key Laboratory for Manufacturing System Engineering, Xi’an Jiaotong University, Xi’an 710049, China
dInstitute of Machinery Manufacturing Technology, China Academy of Engineering Physics, Mianyang 621900, China
Quasi-zero-stiffness (QZS) metamaterials have attracted significant interest for application in low-frequency vibration isolation. However, previous work has been limited by the design mechanism of QZS metamaterials, as it is still difficult to achieve a simplified structure suitable for practical engineering applications. Here, we introduce a class of programmable QZS metamaterials and a novel design mechanism that address this long-standing difficulty. The proposed QZS metamaterials are formed by an array of representative unit cells (RUCs) with the expected QZS features, where the QZS features of the RUC are tailored by means of a structural bionic mechanism. In our experiments, we validate the QZS features exhibited by the RUCs, the programmable QZS behavior, and the potential promising applications of these programmable QZS metamaterials in low-frequency vibration isolation. The obtained results could inspire a new class of programmable QZS metamaterials for low-frequency vibration isolation in current and future mechanical and other engineering applications.
Mechanical metamaterials are artificial materials with unusual physical properties developed through the rational design of microstructures [1], [2], [3], [4]. Some examples of these unusual properties include a designable Poisson’s ratio [5], negative stiffness [6], and a negative coefficient of thermal expansion [7]. Among such properties, the negative stiffness exhibited by mechanical metamaterials has attracted increasing attention, particularly with the emergence of quasi-zero-stiffness (QZS) metamaterials for low-frequency vibration isolation.
The concept of QZS has attracted the attention of many scholars since it was first proposed [8], and QZS isolators have been continuously studied for their high static stiffness and low dynamic stiffness [9]. Thus far, QZS-based vibration isolation has been applied in various fields, such as transport engineering [10], civil engineering [11], and medicine [12], to name just a few. In such fields, QZS-based vibration isolation is usually achieved by means of a stiffness-combined mechanism; that is, a negative stiffness element is connected to a positive stiffness element in parallel to produce zero stiffness. In this regard, a great deal of research has been done by many scholars, which can be found in Refs. [13], [14], [15], [16], [17], [18], [19] and the references therein. However, an extensive amount of research has found that the current stiffness-combined mechanism limits the widespread use of QZS-based vibration isolation. Structures using this mechanism are typically complex and insufficiently compact for precision instruments or microdevices with very small dimensions. Furthermore, the QZS-based vibration will be limited in such contexts due to the narrow working range and unique payload.
To overcome these limitations, researchers have made some valuable attempts to design new QZS metamaterials by introducing advanced theories and techniques. For example, Cai et al. [20] and Zhang et al. [21] studied the wave attenuation properties and programmable characteristics of QZS metamaterials using optimization and compliant mechanisms, respectively. Jiang et al. [22] and Deng et al. [23] introduced a bioinspired approach to designing novel QZS metamaterials, thus investigating the properties of high static stiffness and low dynamic stiffness at large vibration amplitudes, as well as the low-frequency characteristics under micro-vibrations. In addition, origami (or kirigami)-based structures have recently attracted increasing attention due to their specific features, which include bistable states [24], self-deployability [25], and nonlinear force–displacement relationships [26], [27]. Despite research works on QZS metamaterials that have been proposed in recent years, the development of a simplified structure and design mechanism suitable for practical engineering applications remains a challenge. Therefore, this field still has room for new and creative innovations.
In this context, we exploit the negative stiffness of mechanical metamaterials, using inclined beams as representative unit cells (RUCs) [28], [29] and taking advantage of a control mechanism based on a structural bionic design method [30], we design and realize novel mechanical metamaterials with strong programmable QZS features in terms of overall mechanical behavior, which we name programmable QZS metamaterials. “Programmability” is defined here in terms of the targeted QZS characteristics of the QZS metamaterials that can be realized by programming a combination pattern of unit cells with predetermined QZS features, which is similar to the definition of programmability used in Refs. [31], [32], [33], [34]. Compared with previous works, the innovation of the current work is reflected in two main aspects: The design mechanism of the RUCs with QZS features stems from a novel structural bionic mechanism, which simplifies the structure of the RUC; and the simplified RUC structure can be effectively combined with a manufacturing process for traditional engineering materials, facilitating the application of QZS metamaterials for vibration isolation in practical engineering. To thoroughly demonstrate the above description, the structural bionic mechanism and QZS metamaterials are described in detail in Section 2. In Section 3, metamaterial samples are designed and fabricated, and numerical simulations and experimental tests are presented through finite-element analysis (FEA) and experiments. The conclusion and future work are summarized in Section 4.
2. Design mechanism and QZS metamaterials
2.1. Structural bionic mechanism
Structural function formation is a transition from disorder to order, and can be found in areas ranging from crystal growth [35] and biological structures [36] to complex bionic structures [37], [38]. From these works, it can be seen that—regardless of the research field in which the control of structural function is applied—the bionic morphological features determine the function of the structures, in what is called the “structural bionic mechanism.” Here, we take a bar under stress as an example, as shown in Fig. 1(a). A simple bar exists in different states, from a one-dimensional (1D) structure to a three-dimensional (3D) structure, as shown on the horizontal line. Remarkably, the force–displacement relationship corresponding to each deformation state is different, as shown below the horizontal line. Following this mechanism, we exploit the negative stiffness of mechanical metamaterials and take the inclined beams as RUCs [28], [29], as shown in Fig. 1(b), in order to design and realize a novel RUC for a mechanical metamaterial with QZS features, as shown in Fig. 1(c).
To thoroughly demonstrate the description above, we start by demonstrating the QZS features of the RUC. Since the RUC is mirror-symmetric to the centerline, only half of the RUC (simply referred to as a “building block”) and its key constituent element, the folded beam, is illustrated in Fig. 2(a). The building blocks include auxiliary structures and folded beams, where the folded beam is rotated to form a symmetrical structure while keeping the basic topology unchanged (Fig. 2(a), right panel). This topology is formed by splicing five sections of beams and adjacent beams vertically.
For the RUC, when a vertical displacement d occurs, the resultant force f1 of the folded beams is , where α is the inclined angle and (h0d)/L; L1 and h0 are the initial length and height of the folded beam, respectively; L is the compressed length of the folded beam, and ; l is the half span of the horizontal constraint ends; and k1 is the equivalent stiffness of the folded beam. Assuming that the folded beam reaches the equilibrium position when it is compressed to the horizontal position, and taking this as the initial position, the displacement in the vertical direction is X = d − h0. We then have the force f1 of the folded beam, as follows:
By differentiating X with Eq. (1), the system stiffness K of the RUC can be obtained:
When X = 0 is the static equilibrium position, then the system stiffness K is as follows:
Since the equivalent stiffness can be regarded as a combination of the stiffness brought by the inclined beam and the length c (Fig. 2(a)), and therefore using the energy method [39], Eq. (3) can be expressed as follows:
where E is the Young’s modulus of the material from which the folded beam is made; A is the cross-sectional area and , where b is the out-of-plane thickness of the folded beam and h is the thickness of the folded beam; I is the moment of inertia of the section, where I = bh3/12; and a2 and c are the length parameters of the folded beam, as shown in Fig. 2(a). Details on the derivation of Eqs. (1), (2), (3), (4) are provided in Appendix A.
As shown in Fig. 2(a), if the length parameter c = 0, then the folded beam becomes an inclined beam and the RUC becomes a unit cell composed of two inclined beams. In this case, the term involving the length parameter c in Eq. (4) is 0; that is, . With , K is always negative at the static equilibrium position, as shown in Fig. 2(b) (details are provided in Appendix A). The system stiffness curves of the RUC under different values of c, when , are shown in Fig. 2(c). Based on the system stiffness curves of the RUC in Figs. 2(b) and (c), because of the existence of the length parameter c, the inclined beam is transformed into a folded beam. Correspondingly, the negative stiffness of the inclined beam is offset by the positive stiffness brought by the existence of the length parameter c, which is consistent with the theoretical analysis given in Eq. (3). Thus, the proposed structural bionic mechanism is effectively demonstrated.
Based on the description above, an RUC sample was fabricated using 3D printing (ultraviolet-curing technology), with photosensitive resin as the parent material, as shown in Fig. 2(d) (details provided in Appendix A). The force–displacement relationship of the RUC under vertical compression was obtained, as shown in Fig. 2(e). The material properties used in the simulation were derived from the real stress–strain relationship of 3D-printed dog-bone specimens, as shown in Fig. 2(f). In summary, we verified the structural bionic mechanism and obtained an RUC with the expected QZS features.
2.2. Design of programmable QZS metamaterials
Having described in the previous sections the design mechanism and how an RUC with QZS features was obtained, we introduce in this section the programmable QZS behavior of the proposed QZS metamaterials. Fig. 3(a) shows the QZS metamaterials composed of RUCs through a periodic linear array. Here, the two important parameters of the QZS metamaterials—that is, the numbers of the unit cells in series and in parallel ( and )—should be noted. These two parameters confer programmable QZS behavior to the QZS metamaterial. More specifically, the characteristic exhibited by these two parameters (m and n) can be interpreted by the analogue of connecting Hooke’s springs in series or parallel, respectively. The parameter m determines the QZS displacement range d0—which means that a force can cause the same displacement of the springs in series, and the parameter n influences the QZS payload F0, that is, the total load under parallel conditions is equal to the sum of all the loads. To make it easier to study the roles of m and n, a matrix composed of displacements and loads is used to characterize the QZS features and is denoted as . For example, when two unit cells are connected in series, the matrix “” is obtained (Fig. 3(b-i)); when two groups of unit cells, corresponding to the matrix , are connected in parallel, the matrix “” is obtained (Fig. 3(b-ii)); and, when three unit cells are connected in parallel, the matrix “” is obtained (Fig. 3(b-iii)). Therefore, the proposed QZS metamaterials are obtained by using multiple RUCs in series or parallel, as illustrated in Fig. 3(c-i). Accordingly, a more complex matrix can be obtained, which corresponds to force–displacement relationships that contain multiple payloads, as illustrated in Fig. 3(c-ii). In addition, the proposed QZS metamaterials still inherit and extend the advantages of the vibration isolation application of QZS structures; that is, full-band vibration isolation can be achieved as long as the mass is consistent with the payload of the QZS metamaterial (Fig. 3(d)).
3. Results and discussion
3.1. Compression experiments
As described above, a programmable QZS metamaterial (Fig. 4(a)) can be obtained by assembling RUCs with QZS characteristics. To validate the programmable QZS behavior, four metamaterial samples were assembled using RUCs (Fig. 4(b)). The associated matrix representations of the assembled metamaterials are shown in the right panel of Fig. 4(b). Quasi-static compression tests were performed on these metamaterial samples utilizing a universal testing machine (AG-IC, Shimadzu, Japan). (Details on the quasi-static compression test are provided in Appendix A.) The experimentally measured force–displacement curves corresponding to the four metamaterial samples are given in Fig. 4(c). The highlighted area denotes the QZS region, and the annotation of d0 is the QZS displacement range, which is consistent with d0 on the force–displacement relationship in Fig. 2(e). As shown in Fig. 4(c), the measured QZS features of the metamaterial samples compare quite well with the expected QZS features (the matrix under the force–displacement relationships). For example, as presented in Fig. 4(c-i), the matrix expression of the measured QZS features of the type I sample is , while the expected matrix expression is ; obviously, the two matrices are almost identical.
In addition, as shown in Fig. 4(a), the QZS metamaterials can be discretized into multiple series and parallel groups of RUCs with QZS features; conversely, modular structures with different QZS characteristics can be assembled into QZS metamaterials with programmable QZS features. This property makes up for the limitations of previous QZS metamaterials in terms of their narrow work ranges and single QZS payload. For example, the type III metamaterial sample shown in Fig. 4 can be considered to be formed by two type II samples in parallel; therefore, the QZS payload of type III should be double that of type II. This can be effectively verified by the “plateau” feature of the force–displacement relationship, as shown in Figs. 4(c-ii) and (c-iii). Of course, other complex mechanical metamaterials can also be obtained through a modular design, such as the type IV metamaterial sample in the figure. Correspondingly, force–displacement relationships that contain multiple payloads can be obtained, as shown in Fig. 4(c-iv). As mentioned above, full-band vibration isolation can be achieved as long as the mass is consistent with the payload of the QZS metamaterial; therefore, the proposed programmable QZS metamaterials can provide excellent vibration isolation performance under different loadings.
3.2. Experimental vibration tests
To better understand the vibration isolation performance of the proposed programmable QZS metamaterials, experimental tests for vibration isolation were carried out. Fig. 5(a) shows a schematic diagram of the metamaterial sample and the test installation used for the experiment. More specifically, one metamaterial sample, characterized by an assembly matrix (Fig. 5(b)), was fixed to a pressure plate using hot melt adhesive, and the pressure plate was installed on an electromechanical shaker (HEV-50, CHENG TEC, China). A mass corresponding to the payload of the QZS metamaterial sample was fixed to the support plate using hot melt adhesive. Two accelerators (LC0135T and CT1010L, CHENG TEC) were used to collect input and output signals. Afterward, a sinusoidal wave was used as the output signal and was applied to the pressure plate via the shaker in a random sweep manner, using a low-frequency range (0–30 Hz). The experimental setup is shown in Fig. 5(b). By performing experiments, we obtained the transmittance curves under five typical support masses using , where is the acceleration at the payload and is the acceleration generated by the shaker, as shown in Fig. 5(c).
The experimental results showed that the masses corresponding to the QZS payloads could be effectively isolated in the low-frequency range, which was consistent with the QZS “plateau” characteristics predicted for the mechanical metamaterials. Two masses that did not correspond to QZS metamaterial payloads were also tested, and the measured resonance peaks in the measured frequency range further explained the vibration isolation performance of the proposed QZS metamaterials. While there were some differences in the results, such as the experimentally obtained QZS payloads 4.13F0, 6.05F0, and 8.11F0, they were not integer multiples of the payloads of the RUC (i.e., [4F0 6F0 8F0]) marked in Fig. 5(c), where F0 = 2 N comes from Fig. 2(e). The isolated support mass did not correspond exactly to the QZS payload, owing to inevitable errors, such as thickness differences in the 3D-printing process, deviations in the assembly process, and measurement error. It should also be noted that the experimental vibration testing of programmable mechanical metamaterials that have undergone multiple compression experiments will result in some differences in the results, which can be explained by the viscous effect of the photosensitive resin itself causing a delay in restoring the original shape after the structure is deformed. Furthermore, maintaining the overall structural stability of the QZS metamaterials composed of multiple RUCs during vibration testing is key in ensuring effective testing. In general, it is possible to achieve the stability of the QZS metamaterials by increasing the out-of-plane thickness of the RUCs (i.e., the parameter b in Fig. 2(d)). At the same time, it is necessary to ensure the symmetry of the designed QZS metamaterial as much as possible in practical applications (see Movies S1–S3 in Appendix A).
To further validate the advantages of the proposed design mechanism and the application of these QZS metamaterials in practical engineering, and to reflect the differences between the QZS metamaterials proposed in this work and those in previous studies, another case is provided in Appendix A. In that case, the material used to manufacture the QZS metamaterial was changed from a photosensitive resin to an engineering material (stainless steel); the fabrication process of simple RUCs from stainless steel is provided in Appendix A. The experimental results showed that the masses corresponding to the QZS payloads could be effectively isolated in the low-frequency range (Fig. S3 and Movie S4 in Appendix A). Overall, the results further verified that the proposed QZS metamaterials can achieve full-band vibration isolation in the low-frequency range when the mass corresponds to the QZS load.
3.3. Effects of the design parameters
As mentioned above, during the design of the RUCs, changing the geometric parameters will affect the QZS features they exhibit. To further guide RUC design, the effects of the design parameters of the RUC were comprehensively examined using the simulation model. The stiffness in the QZS zone (i.e., the QZS displacement range d0 in Fig. 2(e)) was used as the performance index. The analysis results are shown in Fig. 6, where parts (a)–(d) show the effect of single-parameter variation on the stiffness, and the interactive influence of two-parameter changes on the stiffness is shown in parts (e) and (f).
From the results shown in Fig. 6, three conclusions can be drawn: QZS characteristics can be realized only when the inclined angle α and the length parameter c have a specific value range; changes in the in-plane and out-of-plane thickness of the folded beams will not cause a qualitative change in the QZS characteristics, but the thicker the value of the in-plane and out-of-plane thickness, the greater the QZS payload of the QZS characteristics; and the length parameter a2 and the length parameters c and a1 have a significant interactive influence on the QZS characteristics. For example, in Fig. 6(e), on the premise of achieving QZS characteristics, the larger the value of length parameter a2, the smaller the value of length parameter c. These conclusions can provide theoretical guidance for the practical application of QZS metamaterials in engineering to a certain extent.
4. Conclusions and future works
We have designed and demonstrated new QZS metamaterials formed by an array of RUCs that possess strong programmable QZS features in terms of their overall mechanical behavior. The design mechanism of the RUC originates from a structural bionic mechanism, rather than the stiffness-combined mechanism mentioned in the literature, simplifying the structure of the RUC. The RUC of the proposed metamaterials includes two folded beams; these RUCs with a simple topology and reasonable design parameters can be effectively combined using manufacturing processes for traditional engineering materials, facilitating the vibration isolation application of QZS metamaterial in practical engineering.
We designed and fabricated metamaterial samples and validated them through FEA and quasi-static compression tests. Furthermore, experimental tests were carried out in two cases to show that the proposed programmable QZS metamaterials have excellent vibration isolation effects in the low-frequency range. Given the metamaterials’ characteristics and merits, we believe that the proposed structural bionic mechanism could inspire a new class of programmable QZS metamaterials for low-frequency vibration isolation in current and future mechanical and other engineering applications. As one example of many possible directions, based on the research mechanism of 1D programmable QZS metamaterials, future studies can overcome the 1D limitation and explore the vibration isolation of two-dimensional (2D) external excitations, or even the design and realization of 3D multi-degree-of-freedom and multi-stage QZS vibration isolation systems. In addition, compared with the QZS metamaterials proposed in this work, which are limited by the application of small deformation, QZS metamaterials with the characteristics of large deformation are worthy of further study. In addition, future exploration of integrated smart materials that combine QZS metamaterials with active control is recommended, with four-dimensional (4D) printing being expected to advance this development.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (52332006), the National Key Research and Development Program of China (2022YFB380600 and 2023YFB3811401), the China Postdoctoral Science Foundation (2022M721850), and Southwest United Graduate School Research Program (202302AO370008).
We thank Quan Zhang (National University of Ireland, Galway) for his valuable discussion of the manuscript and his suggestions for the preparation of figures in this manuscript.
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