Nonlinear dynamic behavior of a bionic quasi-zero stiffness isolation system inspired by cobweb with the amplification effect

Abstract

Inspired by the hunting strategy of spiders utilizing cobwebs, this study proposes a novel bionic cobweb structure (BCS) that facilitates the layer-by-layer nonlinear amplification of physical attributes for its embedded isolation components. By integrating a conventional quasi-zero stiffness (QZS) isolator within the BCS, a bionic QZS isolation system is achieved, exhibiting amplified negative stiffness and damping characteristics. The influences of BCS parameters on negative stiffness and damping are systematically investigated. A dynamic equation of the isolation system is formulated, incorporating the cubic trinomial stiffness and cubic quadrinomial damping terms. The displacement transmissibility is derived by the average method and verified via the Runge–Kutta method. The results manifest the effectiveness of the BCS nonlinear amplification effect on the enhancement of vibration isolation performance, accompanied by good coupling between the amplified negative stiffness and damping. Increasing the number of BCS layers, enlarging the initial angle θ_i of the i-layer BCS, augmenting the pre-compression δ₂₀ of the negative stiffness springs, and shortening the connecting rod L_s, synergistically contribute to a superior amplification of negative stiffness for better counterbalancing the substantial positive stiffness encountered in heavy-load scenarios. Furthermore, the amplified damping exhibits an anti-resonant characteristic, effectively mitigating the hardening nonlinearity without compromising high-frequency performance. The constructed bionic QZS isolation system with the BCS outperforms the initial QZS system in terms of resonance frequency, peak transmissibility, and isolation frequency band. Moreover, the proposed BCS has the prospect of emerging as a usual platform without modifying the current isolation elements. The intrinsic amplification effect and design logic can offer heuristic insights for future research.

1 Introduction

Unexpected low-frequency vibration is ubiquitous in many mechanical structures and instruments and difficult to attenuate, which leads to performance degradation and service life reduction. Furthermore, low-frequency vibrations are commonly generated by heavy equipment, thereby significantly augmenting the intricacy of vibration isolation with respect to loading capacity limitation and installation difficulty. It is demonstrated that when encountering low-frequency vibration, a lower natural frequency is beneficial for reducing the beginning vibration isolation frequency and widening the isolation frequency band [1]. Hence, low positive stiffness and large isolation object mass are preferred. However, the high positive stiffness required for structural stability under the high-loading requirement conflicts with the above requirements [2, 3]. Low-frequency vibration isolation should jump out of the scope of traditional linear vibration isolation.

Various research efforts focused on incorporating specific nonlinear stiffness into the vibration isolation for improved performance. A beneficial stiffness called high static and low dynamic stiffness (HSLDS) embodies high loading capacity under small static deformation, alongside a diminished dynamic stiffness that facilitates a reduced resonant frequency. [4]. As a representative HSLDS vibration isolator, the quasi-zero stiffness vibration isolator (QZS-VI) is usually constructed by paralleling the positive and negative stiffness elements. According to the implementation method of negative stiffness, the QZS-VI can be divided into semi-active, active, and passive isolators. The active and semi-active isolators can more accurately and fast adjust parameters for better isolation performance and enhance the robustness of the system. However, active and semi-active types usually need external energy to input current for electromagnetic force through electromagnetic effect. Due to the lags and delays of signal acquisition, processing, transmission, and the potential deviations between the theoretical and actual electromagnetic forces because of the hysteresis and heat during the high-speed operation, the generated negative stiffness and isolation performance would be affected in practice. The above can increase the complexity and cost of the system.

Passive isolators may not have these shortcomings and attenuate vibration through material deformation or structural displacement. The corresponding negative stiffness element contains oblique-spring type [5, 6], spring-rod type [7, 8], cam-roller-spring type[9,10,11], bucked-beam type[12, 13], disk-spring type [14, 15], origami type [16, 17], and permanent magnet type[18,19,20]. Hao et al. [21] and Margielewicz et al. [22] formulated a more precise dynamic equation for the oblique-spring QZS-VI, extensively analyzing its characteristics to furnish practical implementation guidelines. Bouna et al. [23] explored the efficacy of QZS-VI in controlling multi-span bridge vibrations, highlighting its superiority in suppressing low-frequency vibration. Liu et al. [24] incorporated dual mass blocks into the spring-rod QZS-VI, inducing nonlinear inerter behavior, contributing to weakening the right bend of the resonance peak and improving isolation performance. Ye et al. [25] presented a structurally optimized cam-roller-spring QZS-VI to adapt different loads. By introducing the four piezoelectric buckled beams, Liu et al. [26] endowed the bucked-beam QZS- VI with the capacity to harvest energy. Sadeghi et al. [27] developed a QZS-VI with a unique pressurized fluidic origami cellular structure. With less number of magnets, Carrella et al. Li et al. [28], Zheng et al.[29], Zhou et al. [30], and Yuan et al. [31] proposed the negative stiffness element consisting of a pair of axially-implemented permanent magnetic rings with various magnetized directions. Araki et al.[32] constructed a QZS-VI using super-elastic Cu-Al-Mn SMA bars to simplify the mechanism while maintaining high loading capacity and large stroke length. The above magnetic QZS-VIs all exhibit good performance in reducing the resonant frequency, widening the isolation frequency range, and weakening the peak transmissibility.

However, the negative stiffness force of existing QZS-VIs originates from stored pre-compressed elastic or magnetic force. Due to a considerable portion of elastic or magnetic force being wasted by mutual cancellation, the negative stiffness force is yielded as a changing bilateral elastic or magnetic force difference. The essence of negative stiffness is the derivative of the bilateral force difference with respect to deformation. When the deformation caused by excitation is constant and relatively small, the pre-compressed or magnetic forces cannot even be fully converted into the negative stiffness force, thereby restricting the negative stiffness.

As illustrated in Fig. 1, the insufficient negative stiffness results in inadequate counteraction against the high positive stiffness in heavy-loading and low-frequency occasions with the restriction of space and weight, such as ships[33], aerospace[34], vehicles[35], etc. To achieve higher negative stiffness, conventional research has concentrated on costly and space-occupying approaches, including special high-stiffness materials [32, 36], increased driving force mechanisms [37,38,39], and intricate magnet arrays [40, 41]. In this paper, a cost-effective method is suggested for higher negative stiffness, disrupting the traditional identical relationship between the excitation displacement and deformation of pre-stored energy components. Small excitation displacement elicits substantial deformation, thereby amplifying negative stiffness forces. Consequently, negative stiffness intensifies as the derivative of the augmented force with respect to small displacement.

Fig. 1

The insufficient negative stiffness inadequately counteracts the high positive stiffness in heavy-loading occasions

The integration of bionics with vibration isolation has garnered significant attention due to its potential to provide unique and high-quality stiffness properties. Specifically, various biological structures can be extracted as the improvements and extensions of X-shaped structures. Wu et al. proposed a bio-inspired limb-like structure with adjustable loading capacity and advantageous nonlinear stiffness [42]. Sun et al. [43] introduced a bio-inspired multi-layer structure, modeled on avian necks, achieving effective external excitation isolation with potential applications in diverse fields. Jiang et al. [44] systematically investigated a bio-mimetic multi-joint leg-like vibration isolation structure, which exhibits superior HSLDS properties and exceptional low-frequency isolation capabilities. Yan et al. [45] developed a bio-inspired polygonal skeleton (BIPS), which was inspired by the safe landing of a cat and exhibited nonlinear vibration suppression properties through adjustable stiffness. In essence, the aforementioned isolation attributes stem from the nonlinear interplay between the deformation of elastic components and excitation. This insight motivates our quest for optimal biomimetic prototypes, leading to the design of a targeted bionic cobweb structure (BCS).

The intricate cobweb architecture enables spiders to subdue struggled prey with small exertion, aligning with our quest for excitation displacement and deformation in elastic components. The intricate interplay between cobweb, spider, and hunting behavior spurs our pursuit of elucidating and investigating potential analogies with envisioned bionic structures, isolation components, and vibration mitigation mechanisms. Structurally, the bionic cobweb structure (BCS) exhibits a multi-layer rhombic configuration, comprising rods interconnected by hinges. A systematic exploration reveals the geometric properties and amplification effect of the BCS. Notably, it facilitates a layer-by-layer augmentation of the physical attributes of the embedded negative stiffness and damping elements. The bionic QZS isolation systems employing BCSs with different layers (BQZS-BCSs) are modeled, and their amplified negative stiffness and damping coefficient are systematically analyzed, considering the effects of structural parameters. Then, the dynamic equation for BQZS-BCS is formulated, solved utilizing the average method, and verified through the Runge–Kutta method. Some discussions considering the influences of structural parameters demonstrate that the amplification effect of the BCS effectively enhances the isolation performance of the original QZS system. The integration of the BCS with the initial QZS system yields an enhanced anti-resonant behavior, characterized by a reduced resonance frequency, diminished resonant peak, and expanded vibration isolation band, all attributable to the synergistic effect of amplified negative stiffness and damping coefficient. The evolution and application of biomimetic thinking in QZS vibration isolation are elaborated in Fig. 2.

Fig. 2

The evolution and application of Biomimetic thinking in QZS vibration isolation inspired from cobweb

The paper is organized as follows. The bionic design philosophy is described in Sect. 2. Section 3 exhibits the configuration of the proposed BCS, and reveals the unique amplification effect for negative stiffness and damping. Section 4 systematically investigates the static characteristics of the amplified stiffness and damping caused by the BCSs with different layers. In Sect. 5, the dynamic equation of BQZS-BCS is established and resolved. Some analysis of the isolation performance is given. Section 6 concludes the study.

2 Bionic design philosophy

2.1 Design target

When the base is stationary, the deformation of the negative stiffness element typically aligns with the excitation displacement. The negative stiffness is derived from the negative stiffness force's derivative with respect to excitation displacement. Irrespective of material type, the negative stiffness force exists as a bilateral force difference, significantly influenced by deformation. To achieve high negative stiffness with constant excitation, an inverse deformation-excitation relationship must be established, resulting in large deformations from small excitations to compensate for the bilateral force inefficiency.

Furthermore, nonlinear damping has been proven to enhance vibration isolation when coupled with nonlinear cubic springs [46, 47]. To mitigate resonant peaks and reduce jumping phenomena, a heightened equivalent damping coefficient is required, necessitating an augmentation in the relative displacement between the ends of the damping element per unit of time.

Consequently, the design structure ought to possess the capacity to exhibit substantial deformation in response to small excitation displacement, thereby optimizing the performance of the negative stiffness and damping elements.

2.2 Bionic inspiration

Amidst the localized vibrations emanating from the struggling prey, the immediate milieu of the spider remains impervious to such disturbances. Upon the exhaustion of the prey, the spider astutely harnesses the intricate geometry of its web to enwrap it with small action. This adverse deformation relationship between prey and spider, linked to the web architecture, inspires the design of a bionic cobweb structure (BCS) that enables significant deformation with small excitation.

As depicted in Fig. 3, salient parallels exist between the spider's hunting strategy and bionic QZS vibration isolation, enlightens the development of BQZS-BCS. The objective of the vibration isolation is to attenuate the isolation object response displacement, analogous to weakening the struggle of prey. The bionic cobweb structure (BCS) imitates the cobweb, with the installed positive springs and damping elements mimicking its elasticity and damping. The negative stiffness element functions as the controller of vibration isolation, resembling the spider in hunting behavior.

Fig. 3

The parallel connections between spider’s hunting and bionic QZS vibration isolation

However, it is crucial to recognize the inverse relationship inherent in these analogies. The spider’s movement is relatively small in comparison with the prey’s violent struggle, while the excitation displacement is a small quantity when compared to the expected deformation. Considering the unique deforming relationship of the cobweb in the spider-hunting strategy, the targeted relationship can be achieved by investigating the cobweb’s characteristics and then doing the opposite.

3 The bionic cobweb structure (BCS) and its amplification effect

This section exhibits the models of the BCSs, elaborates on the mechanisms of its one-layer amplification effect for the embedded isolation elements, and formulates a comprehensive design principle for the BCS. Based on that, the BQZS-BCS are proposed, and the multi-layer amplification effects for the negative stiffness and damping are explicated.

3.1 The mechanism of the amplification effect

As shown in Fig. 4a, the negative stiffness element is embedded in the core of the one-layer BCS. The projection lengths of the movable rod 1 in the horizontal and vertical directions are is L_x and L_y, respectively. θ₁ stands for the angle between the movable rod 1 and the base. Assuming the exciting force exerted on the loading plate is F_ny, and the resulting displacement is Y. The force generated by the negative stiffness element is F_nx, and the resulting displacement is X. A geometric relationship can be achieved that L_x² + L_y² = (L_x + X/2)² + (L_y − Y/2)². When neglecting the high-order terms and little variables, it gives X/Y = L_y/L_x = tanθ₁. Based on the principle of virtual displacements, the equation F_ny⋅Y = F_nx ·X can be attained. Thus, F_ny = F_nx·tanθ₁. tanθ₁ determines whether F_ny gets enlarged or not in comparison with F_nx. When θ₁ varies within (π/4, π/2), the BCS has an amplification effect on F_nx, and a small change in Y could induce a substantial variation in X. While, it can lead to an opposite effect when deforms within (0, π/4). Further, the initial negative stiffness can be derived as $d(F_{nx} )/dX$, and the equivalent negative stiffness with the BCS can be given as $d(F_{ny} )/dY$, which can be expressed as $\tan^2 \theta_1 \frac{{d(F_{nx} )}}{dX}$. When θ₁ keeps within (π/4, π/2), the equivalent negative stiffness with the BCS gets amplified in comparison with the initial negative stiffness. Here, $\tan^2 \theta_1$ performs as a decisive parameter and is named as the amplifying coefficient of the negative stiffness.

Fig. 4

The model of the BCS embedded with the isolation elements for manifesting the amplification effect. a with the negative stiffness element; b with the damping element

Similarly, the one-layer BCS implemented with the damping element is described in Fig. 2b. The equivalent damping force can be expressed as F_cy (F_cy = F_cx·X/Y = F_cx·tanθ₁ = C $\dot{X}$ tanθ₁ = C $\dot{Y}$ tan²θ₁, C denotes damping coefficient), and its amplification effect occurs when θ₁ varies within (π/4, π/2), Compared with the initial damping force C·$\dot{Y}$, the amplifying coefficient of the damping force is also tan²θ₁.

Following the above, the amplification coefficient needs to be multiplied by tan²θ_i (i = 1,2,3…) for each additional layer of BCS. The approximate amplification coefficient of i-layer BCS is ∏tan²θ_i (i = 1,2,3…).

3.2 The design principle of the BCS

Based on the amplification effect, the BCS can be extended to multi-layer, and the related design principle is developed as follows. (1) sublayers can be nested in the outer layer of the BCS layer by layer to construct more compact configurations for more great amplification; (2) the initial angles between the movable rod and the direction perpendicular to the sublayer excitation has a decisive influence on the amplification effect, which is effective within (π/4, π/2); (3) the negative stiffness and damping elements that need to be amplified are implemented in the center of the located sublayer, the positive springs are installed between the loading plate and the base. The two terminals of the negative stiffness or damping elements are connected with the located layer hinges, and their moving direction is perpendicular to the located sublayer excitation.

3.3 Amplification effects for the isolation elements in the BCSs with different layers

For the isolation system with the i-layer bionic cobweb structure (BCS), ${\text{x}}_i$(i = 1, 2, 3) stands for the horizontal displacement. L_i (i = 1, 2, 3) is the length of the movable rod i for the i-layer BCS. The angle between the rod and the base in its initial position is θ_i, while the deformation-induced angle is φ_i (where i = 1, 2, 3). L_s denotes the length of the rod for connecting the negative stiffness springs. The angle between the connecting rod and the deformation direction of the negative stiffness springs is taken as α_i (where i = 0, 1, 2, 3. For the initial QZS system, i = 0). The original, equilibrium-position, and current lengths of the springs (whose stiffness is k_l) for providing the negative stiffness are S₂, S₂₀, and $S_2^{\prime}$($S_2^{\prime} { = }S_{20} + L_s - \sqrt {{L_s^2 - {\text{x}}_i^2 }}$). Its pre-compression and current compression are δ₂₀ and δ₂, respectively. Here,$\delta_{20} = S_2 - S_{20}$, $\delta_2 = S_2 - \left( {S_{20} + L_s - \sqrt {L_s^2 - x_i^2 } } \right) = \delta_{20} + \sqrt {L_s^2 - x_i^2 } - L_s$. The stiffness and pre-compression of the spring for providing load is k_p and δ₁₀, respectively.

3.3.1 Initial QZS structure

Figure 5 depicts a typical three-spring QZS isolation system, whose negative stiffness force is formulated in Eq. (1). For the convenience of subsequent calculations, the negative stiffness force and the corresponding negative stiffness can be approximated by the Taylor series expansion in Eq. (2) as a polynomial about y. Figure 6 shows that the results of the negative stiffness force F_kn0 calculated by Eqs. (1) and (2) agree well. Besides, Eq. (3) gives the expression of the damping force.

$$ F_{kn0} = - 2k_l \left( {\delta_{20} - L_s + \sqrt {L_s^2 - y^2 } } \right)\tan \alpha_0 ,\;\;\;\tan \alpha_0 = \frac{y}{{\sqrt {L_s^2 - y^2 } }}. $$

(1)

$$ \left\{ {\begin{array}{*{20}l} {F_{kn0} \approx \left( {kn_{01} y + kn_{03} y^3 } \right){ = } - \frac{{2k_l \delta_{20} }}{L_s }y - \left( {\frac{{k_l \delta_{20} }}{L_s^3 } - \frac{k_l }{{L_s^2 }}} \right)y^3 } \hfill \\ {K_{kn0} { = }\frac{{\partial F_{kn0} }}{\partial y} \approx - \frac{{2k_l \delta_{20} }}{L_s } - 3\left( {\frac{{k_l \delta_{20} }}{L_s^3 } - \frac{k_l }{{L_s^2 }}} \right)y^2 } \hfill \\ \end{array} } \right.. $$

(2)

$$ F_{{\text{c}}0} = f_{c0} \dot{y} = c\dot{y}. $$

(3)

where f_c0 is denoted as the damping coefficient of the initial QZS isolation system.

Fig. 5

The schematic diagram of the initial QZS isolation: (a), plan drawing; (b), 3-D drawing

Fig. 6

Comparison of the negative stiffness force for the initial QZS isolation system between original equation and Taylor expansion when k_l = k_p = 5e4 N/m, L_s = 0.05 m, δ₂₀ = 0.025 m

3.3.2 Amplification effect with the one-layer bionic cobweb structure

Figure 7 describes the bionic QZS isolation system with a single-layer BCS, comprising four identical movable rods arranged in a rhombus via hinges. Embedded in the BCS are parallel implemented negative stiffness and damping elements. The movable rod, with a length of L₁, is inclined at an angle θ₁ with respect to the base. Vertical displacement of the isolated object deforms the structure horizontally and vertically, causing displacements x₁ and y. The rod connecting pre-compressed springs shifts vertically by y/2 and horizontally by x/2. This results in an angle increase φ₁ between rod 1 and the base, with α₁ representing the angle between the rod and the deformation direction of the springs. The geometric relationship between deformation (x₁ and y) and initial conditions (L₁ and θ₁) can be deduced as Eq. (4).

$$ x_1 = 2L_1 \cos \theta_1 - 2\sqrt {{L_1^2 - \left( {L_1 \sin \theta_1 + \frac{y}{2}} \right)^2 }} . $$

(4)

Fig. 7

The schematic diagram of the QZS isolation with the one-layer BCS: a plan drawing; b 3-D drawing

After the deformation, the angle (θ₁ + φ₁) can be expressed by Eq. (5).

$$ \tan \left( {\theta_1 + \varphi_1 } \right) = \frac{{L_1 \sin \theta_1 + \frac{y}{2}}}{{L_1 \cos \theta_1 - \frac{x_1 }{2}}} = \frac{{L_1 \sin \theta_1 + \frac{y}{2}}}{{\sqrt {{L_1^2 - \left( {L_1 \sin \theta_1 + \frac{y}{2}} \right)^2 }} }},\;\;\;\;\varphi_1 = {\text{arctan}}\frac{2L_1 \sin \theta_1 + y}{{2L_1 \cos \theta_1 - x_1 }} - \theta_1 . $$

(5)

So, the one-layer amplified negative stiffness force can be achieved as

$$ F_{kn1} = - 2k_l \left( {\delta_{20} - L_s + \sqrt {L_s^2 - x_1^2 } } \right)\tan \alpha_1 \tan \left( {\theta_1 + \varphi_1 } \right),\;\;\;\;x_1 \in \left( { - L_s ,L_s } \right), $$

(6)

where $\tan \alpha_1 = \frac{x_1 }{{\sqrt {L_s^2 - x_1^2 } }}$, $y \in \left( { - 2L_1 (1 + \sin \theta_1 ),2L_1 (1 - \sin \theta_1 )} \right)$.

To simplify with the Taylor series expansion, the above expression can be transformed as

$$ F_{kn1} \approx \left( {kn_{11} y{ + }kn_{12} y^2 + kn_{13} y^3 } \right), $$

(7)

where $kn_{11} { = } - \frac{{2k_l \delta_{20} \tan^2 \theta_1 }}{L_s }$, $kn_{12} { = } - \frac{{3k_l \delta_{20} \tan \theta_1 }}{2L_s L_1 \cos^3 \theta_1 }$, $kn_{13} { = } - \left[ {\frac{{k_l \delta_{20} \left( {5\tan^2 \theta_1 + 1} \right)}}{4L_s L_1^2 \cos^4 \theta_1 } + \frac{{k_l \delta_{20} \tan^4 \theta_1 }}{{L_s^3 }} - \frac{k_l \tan^4 \theta_1 }{{L_s^2 }}} \right]$.

As demonstrated in Fig. 8, the outcomes computed via the initial expression exhibit good concurrence with those obtained through Taylor series expansion.

Fig. 8

Comparison of the negative stiffness force for the QZS isolation system with the one-layer BCS between the original equation and Taylor expansion when k_p = 5e4 N/m, k_l = 2e4 N/m, L_s = 0.08 m, δ₂₀ = 0.04 m, L₁ = 0.2 m, θ₁ = 55°

Accordingly, the one-layer amplified negative stiffness can be taken as

$$ K_{n1} = \frac{{\partial F_{kn1} }}{\partial y} \approx kn_{11} { + 2}kn_{12} y + 3kn_{13} y^2 . $$

(8)

Similar to Eq. (6), the one-layer amplified damping force can be expressed as

$$ F_{c1} = c_1 \dot{x}_1 \tan \left( {\theta_1 + \varphi_1 } \right) = c_1 \frac{\partial x_1 }{{\partial y}}\frac{{L_1 \sin \theta_1 + \frac{y}{2}}}{{\sqrt {{L_1^2 - \left( {L_1 \sin \theta_1 + \frac{y}{2}} \right)^2 }} }}\dot{y} = c_1 \frac{{\left( {L_1 \sin \theta_1 + \frac{y}{2}} \right)^2 }}{{L_1^2 - \left( {L_1 \sin \theta_1 + \frac{y}{2}} \right)^2 }}\dot{y} $$

(9)

Further, Eq. (9) can be fitted as Eq. (10) with the Taylor series expansion. Figure 9 illustrates a good accordance between the fitted and the exact results for the one-layer amplified damping coefficient.

$$ F_{c1} \approx \left( {c_{13} y^3 + c_{12} y^2 + c_{11} y + c_{10} } \right)\dot{y}{ = }f_{c1} \dot{y}, $$

(10)

where $c_{13} { = }\frac{{\left( {2\tan^3 \theta_1 + \tan \theta_1 } \right)}}{2L_1^3 \cos^5 \theta_1 }c$, $c_{12} { = }\frac{{\left( {4\tan^2 \theta_1 + 1} \right)}}{4L_1^2 \cos^4 \theta_1 }c$, $c_{11} { = }\frac{\tan \theta_1 }{{L_1 \cos^3 \theta_1 }}c$, $c_{10} { = }\tan^2 \theta_1 c$. f_c1 is taken as the one-layer amplified damping coefficient.

Fig. 9

Comparison of the damping coefficient for the QZS isolation system with the one-layer BCS between the original equation and Taylor expansion when kp = 5e4 N/m, k_l = 2e4 N/m, L_s = 0.08 m, δ₂₀ = 0.04 m, L₁ = 0.2 m, θ₁ = 55°

3.3.3 Amplification effect with the double-layer bionic cobweb structure

Increasing the number of BCS layers has the potential for enhancing isolation performance. Figure 10 illustrates a double-layer BCS, effectively a 90-degree rotation and nesting of the single-layer design within an enlarged counterpart. Geometric parameters and excitation displacement (y) of the inner layer remain consistent with the single-layer configuration. Movable rod 2 has a length L₂ and a vertical angle θ₂. Deformation-induced angles φ₂ and α₂ represent the inclinations of rod 2 and the connecting rod from the initial position to the current position, respectively. Analogous to x₁, the outer layer exhibits a symmetric horizontal displacement x₂.

Fig. 10

The schematic diagram of the QZS isolation with the double-layer BCS: a, plan drawing; b, 3-D drawing

By extending the relationships of geometric parameters and displacements in the one-layer BCS, the corresponding equations for the double-layer are derived in Eqs. (11) and (12).

$$ x_2 = 2L_2 \cos \theta_2 - 2\sqrt {{L_2^2 - \left( {L_2 \sin \theta_2 + \frac{x_1 }{2}} \right)^2 }} ,\;\;\;x_1 = 2L_1 \cos \theta_1 - 2\sqrt {{L_1^2 - \left( {L_1 \sin \theta_1 + \frac{y}{2}} \right)^2 }} $$

(11)

$$ \tan \left( {\theta_2 + \varphi_2 } \right) = \frac{{L_2 \sin \theta_2 + \frac{x_1 }{2}}}{{L_2 \cos \theta_2 - \frac{x_2 }{2}}}{ = }\frac{{L_2 \sin \theta_2 + L_1 \cos \theta_1 { - }\sqrt {{L_1^2 - \left( {L_1^{\,} \sin \theta_1 + \frac{y}{2}} \right)^2 }} }}{{\sqrt {{L_2^2 - \left( {L_2^{\,} \sin \theta_2 + \frac{x_1 }{2}} \right)^2 }} }},\;\;\;\tan \left( {\theta_1 + \varphi_1 } \right) = \frac{{L_1 \sin \theta_1 + \frac{y}{2}}}{{L_1 \cos \theta_1 - \frac{x_1 }{2}}}. $$

(12)

Equations (13) and (14) exhibit the explicit and approximated expressions of double-layer amplified negative stiffness force, respectively. The two results coincide well with each other in Fig. 11. Equation (15) describes the corresponding equivalent negative stiffness.

$$ F_{kn2} = - 2k_l \left( {\delta_{20} - L_s + \sqrt {L_s^2 - x_2^2 } } \right)\tan \alpha_2 \tan \left( {\theta_2 + \varphi_2 } \right)\tan \left( {\theta_1 + \varphi_1 } \right), $$

(13)

where $\tan \alpha_2 = \frac{x_2 }{{\sqrt {L_s^2 - x_2^2 } }}$, $y \in \left( { - 2L_1 (1 + \sin \theta_1 ),2L_1 (1 - \sin \theta_1 )} \right)$, $x_1 \in \left( { - 2L_2 (1 + \sin \theta_2 ),2L_2 (1 - \sin \theta_2 )} \right)$, $x_2 \in \left( { - L_s ,L_s } \right)$.

$$ F_{kn2} \approx kn_{21} y{ + }kn_{22} y^2 + kn_{23} y^3 , $$

(14)

where $kn_{21} { = } - \frac{{2k_l \delta_{20} \tan^2 \theta_1 \tan^2 \theta_2 }}{L_s }$, $kn_{22} { = } - \left( {\frac{{3k_l \delta_{20} \tan \theta_1 \tan^2 \theta_2 }}{2L_s L_1 \cos^3 \theta_1 } + \frac{{3k_l \delta_{20} \tan^3 \theta_1 \tan \theta_2 }}{2L_s L_2 \cos^3 \theta_2 }} \right)$,$kn_{23} { = } - \left[ {\frac{{k_l \delta_{20} \tan^2 \theta_2 (5\tan^2 \theta_1 + 1)}}{4L_s L_1^2 \cos^4 \theta_1 } + \frac{{k_l \delta_{20} \tan^4 \theta_1 (5\tan^2 \theta_2 + 1)}}{4L_s L_2^2 \cos^4 \theta_2 } + \frac{{3k_l \delta_{20} \tan^2 \theta_1 \tan \theta_2 }}{2L_s L_1 L_2 \cos^3 \theta_1 \cos^3 \theta_2 } + \frac{{k_l \delta_{20} \tan^4 \theta_1 \tan^4 \theta_2 }}{L_s^3 } - \frac{k_l \tan^4 \theta_1 \tan^4 \theta_2 }{{L_s^2 }}} \right]$.

$$ K_{n2} = \frac{{\partial F_{kn2} }}{\partial y} = kn_{21} { + 2}kn_{22} y + 3kn_{23} y^2 . $$

(15)

Fig. 11

Comparison of the negative stiffness force for the QZS isolation system with the double-layer BCS between initial expression and Taylor expansion when k_p = 5e4N/m, k_l = 2e4 N/m, L_s = 0.08 m, δ₂₀ = 0.04 m, L₁ = 0.2 m, L₂ = 0.15 m, θ₁ = 55°, θ₂ = 55°

The double-layer amplified damping force can be formulated as

$$ \begin{aligned} F_{c2} & = c\dot{x}_2 \tan \left( {\theta_2 + \varphi_2 } \right)\tan \left( {\theta_1 + \varphi_1 } \right) = c\frac{\partial x_2 }{{\partial x_1 }}\frac{\partial x_1 }{{\partial \hat{y}}}\frac{{L_2 \sin \theta_2 + \frac{x_1 }{2}}}{{\sqrt {{L_2^2 - \left( {L_2^{\,} \sin \theta_2 + \frac{x_1 }{2}} \right)^2 }} }}\frac{{L_1 \sin \theta_1 + \frac{y}{2}}}{{\sqrt {{L_1^2 - \left( {L_1 \sin \theta_1 + \frac{y}{2}} \right)^2 }} }}\dot{y} \\ & = c\frac{{\left( {L_2 \sin \theta_2 + \frac{x_1 }{2}} \right)^2 }}{{L_2^2 - \left( {L_2^{\,} \sin \theta_2 + \frac{x_1 }{2}} \right)^2 }}\frac{{\left( {L_1 \sin \theta_1 + \frac{y}{2}} \right)^2 }}{{L_1^2 - \left( {L_1 \sin \theta_1 + \frac{y}{2}} \right)^2 }}\dot{y} = f_{c2} \dot{y}. \\ \end{aligned} $$

(16)

Then, F_c2 can be fitted with the Taylor series expansion as Eq. (17). Figure 12 illustrates that f_c2 calculated by Eqs. (16) and (17) exhibit a high degree of agreement.

$$ F_{c2} \approx \left( {c_{23} y^3 + c_{22} y^2 + c_{21} y^{\,} + c_{20} } \right)\dot{y} = f_{c2} \dot{y} $$

(17)

where $c_{23} = \left[ {\frac{{3\tan^3 \theta_1 \left( {4\tan^2 \theta_2 { + }1} \right)}}{8L_1 L_2^2 \cos^3 \theta_1 \cos^4 \theta_2 }{ + }\frac{{\tan \theta_1 \tan \theta_2 \left( {11\tan^2 \theta_1 { + 4}} \right)}}{8L_1^2 L_2 \cos^4 \theta_1 \cos^3 \theta_2 }{ + }\frac{{\tan \theta_1 \tan^2 \theta_2 \left( {2\tan^2 \theta_1 + 1} \right)}}{2L_1^3 \cos^5 \theta_1 } + \frac{{\tan^5 \theta_1 \tan \theta_2 \left( {2\tan^2 \theta_2 + 1} \right)}}{2L_2^3 \cos^5 \theta_2 }} \right]c$,

Fig. 12

Comparison of the damping coefficient for the QZS isolation system with the double-layer BCS between the original equation and Taylor expansion when k_p = 5e4N/m, k_l = 2e4N/m, L_s = 0.08m, δ₂₀ = 0.04m, L₁ = 0.2m, L₂ = 0.15m, θ₁ = 55°, θ₂ = 55°

$c_{22} = \left[ \begin{gathered} \frac{(4\tan^2 \theta_1 + 1)\tan^2 \theta_2 }{{4L_1^2 \cos^4 \theta_1 }} + \frac{\tan^4 \theta_1 (4\tan^2 \theta_2 + 1)}{{4L_2^2 \cos^4 \theta_2 }} \hfill \\ + \frac{5\tan^2 \theta_1 \tan \theta_2 }{{4L_1 L_2 \cos^3 \theta_1 \cos^3 \theta_2 }} \hfill \\ \end{gathered} \right]c$, $c_{21} = \left[ {\frac{\tan \theta_1 \tan^2 \theta_2 }{{L_1 \cos^3 \theta_1 }} + \frac{\tan^3 \theta_1 \tan \theta_2 }{{L_2 \cos^3 \theta_2 }}} \right]c$,$c_{20} = \tan^2 \theta_1 \tan^2 \theta_2 c$,

f_c2 represents the damping coefficient of the bionic QZS isolation system with the double-layer BCS.

3.3.4 Amplification effect with the triple-layer bionic cobweb structure

The triple-layer BCS is constructed and shown in Fig. 13. Structurally, the triple-layer BCS is constructed by nesting the double-layer BCS in one-layer BCS inside. The length of the movable rod 3 is L₃, and the angle from the vertical direction is θ₃. The changing angle between the movable rod 3 and the horizontal direction is φ₃. The angle formed by that rod and the direction of the negative stiffness springs' deformation is α₃. A symmetrical horizontal displacement of x₃ exists in the inner layer.

Fig. 13

The schematic diagram of the QZS isolation with the triple-layer BCS: a, plan drawing; b, 3-D drawing

The relationships between the geometric parameters and the displacements for the triple-layer are given in Eqs. (18) and (19).

$$ x_3 = 2L_3 \cos \theta_3 - 2\sqrt {{L_3^2 - \left( {L_3 \sin \theta_3 + \frac{x_2 }{2}} \right)^2 }} ,\;\;\;x_2 = 2L_2 \cos \theta_2 - 2\sqrt {{L_2^2 - \left( {L_2 \sin \theta_2 + \frac{x_1 }{2}} \right)^2 }} ,\;\;\;x_1 = 2L_1 \cos \theta_1 - 2\sqrt {{L_1^2 - \left( {L_1 \sin \theta_1 + \frac{y}{2}} \right)^2 }} , $$

(18)

$$ \begin{aligned} \tan \left( {\theta_3 + \varphi_3 } \right) & = \frac{{L_3 \sin \theta_3 + \frac{x_2 }{2}}}{{L_3 \cos \theta_3 - \frac{x_3 }{2}}}{ = }\frac{{L_3 \sin \theta_3 + L_2 \cos \theta_2 - \sqrt {{L_2^2 - \left( {L_2^{\,} \sin \theta_2 + \frac{x_1 }{2}} \right)^2 }} }}{{\sqrt {{L_3^2 - \left( {L_3^{\,} \sin \theta_3 + \frac{x_2 }{2}} \right)^2 }} }}, \\ \tan \left( {\theta_2 + \varphi_2 } \right) & = \frac{{L_2 \sin \theta_2 + \frac{x_1 }{2}}}{{L_2 \cos \theta_2 - \frac{x_2 }{2}}}{ = }\frac{{L_2 \sin \theta_2 + L_1 \cos \theta_1 { - }\sqrt {{L_1^2 - \left( {L_1^{\,} \sin \theta_1 + \frac{y}{2}} \right)^2 }} }}{{\sqrt {{L_2^2 - \left( {L_2^{\,} \sin \theta_2 + \frac{x_1 }{2}} \right)^2 }} }},\begin{array}{*{20}c} {\,} \\ \end{array} \tan \left( {\theta_1 + \varphi_1 } \right) = \frac{{L_1 \sin \theta_1 + \frac{y}{2}}}{{L_1 \cos \theta_1 - \frac{x_1 }{2}}}. \\ \end{aligned} $$

(19)

Equations (20) and (23) give the triple-layer amplified negative stiffness force and damping force, respectively. Their approximations fitted with the Taylor series expansion are expressed in Eqs. (21) and (24), which keep in a good agreement with their respective explicit results in Figs. 14 and 15. By taking the derivative of Eq. (21) with respect to y, the resulting negative stiffness is obtained in Eq. (22).

$$ F_{kn3} = 2k_l \left( {\delta_{20} - L_s + \sqrt {L_s^2 - x_3^2 } } \right)\tan \alpha_3 \tan \left( {\theta_3 + \varphi_3 } \right)\tan \left( {\theta_2 + \varphi_2 } \right)\tan \left( {\theta_1 + \varphi_1 } \right), $$

(20)

where $\tan \alpha_3 = \frac{x_3 }{{\sqrt {L_s^2 - x_3^2 } }}$,$y \in \left( { - 2L_1 (1 + \sin \theta_1 ),2L_1 (1 - \sin \theta_1 )} \right)$, $x_1 \in \left( { - 2L_2 (1 + \sin \theta_2 ),2L_2 (1 - \sin \theta_2 )} \right)$, $x_2 \in \left( { - 2L_3 (1 + \sin \theta_3 ),2L_3 (1 - \sin \theta_3 )} \right)$, $x_3 \in \left( { - L_s ,L_s } \right)$.

$$ F_{kn3} \approx \left( {kn_{31} y{ + }kn_{32} y^2 + kn_{33} y^3 } \right), $$

(21)

where $kn_{31} { = } - \frac{{2k_l \delta_{20} \tan^2 \theta_1 \tan^2 \theta_2 \tan^2 \theta_3 }}{L_s }$, $kn_{32} { = } - \left( {\frac{{3k_l \delta_{20} \tan \theta_1 \tan^2 \theta_2 \tan^2 \theta_3 }}{2L_s L_1 \cos^3 \theta_1 } + \frac{{3k_l \delta_{20} \tan^3 \theta_1 \tan \theta_2 \tan^2 \theta_{3} }}{2L_s L_2 \cos^3 \theta_2 } + \frac{{3k_l \delta_{20} \tan^3 \theta_1 \tan^3 \theta_{2} \tan \theta_3 }}{2L_s L_3 \cos^3 \theta_3 }} \right)$ $kn_{33} { = } - \left[ \begin{gathered} \frac{{3k_l \delta_{20} \tan^2 \theta_1 \tan^3 \theta_2 \tan \theta_3 }}{2L_s L_1 L_3 \cos^3 \theta_1 \cos^3 \theta_3 } + \frac{{3k_l \delta_{20} \tan^4 \theta_1 \tan^2 \theta_2 \tan \theta_3 }}{2L_s L_2 L_3 \cos^3 \theta_2 \cos^3 \theta_3 } + \frac{{3k_l \delta_{20} \tan^2 \theta_1 \tan \theta_2 \tan^2 \theta_3 }}{2L_s L_1 L_2 \cos^3 \theta_1 \cos^3 \theta_2 } \hfill \\ + \frac{{k_l \delta_{20} (5\tan^4 \theta_1 + 6\tan^2 \theta_1 { + }1)\tan^2 \theta_2 \tan^2 \theta_3 }}{4L_s L_1^2 \cos^2 \theta_1 } + \frac{{k_l \delta_{20} (5\tan^4 \theta_2 + 6\tan^2 \theta_2 { + }1)\tan^4 \theta_1 \tan^2 \theta_3 }}{4L_s L_2^2 \cos^2 \theta_2 } \hfill \\ + \frac{{k_l \delta_{20} ({5}\tan^4 \theta_3 + 6\tan^2 \theta_3 { + }1)\tan^4 \theta_1 \tan^4 \theta_2 }}{4L_s L_3^2 \cos^2 \theta_3 } + \frac{{k_l \delta_{20} \tan^4 \theta_1 \tan^4 \theta_2 \tan^4 \theta_3 }}{L_s^3 } \hfill \\ - \frac{k_l \tan^4 \theta_1 \tan^4 \theta_2 \tan^4 \theta_3 }{{L_s^2 }} \hfill \\ \end{gathered} \right]$.

$$ K_{n3} = \frac{{\partial F_{kn3} }}{\partial y} = kn_{31} { + 2}kn_{32} y^{\,} + 3kn_{33} y^2 . $$

(22)

$$ \begin{aligned} F_{c3} & = c\dot{x}_3 \tan \left( {\theta_3 + \varphi_3 } \right)\tan \left( {\theta_2 + \varphi_2 } \right)\tan \left( {\theta_1 + \varphi_1 } \right) \\ & = c\frac{\partial x_3 }{{\partial x_2 }}\frac{\partial x_2 }{{\partial x_1 }}\frac{\partial x_1 }{{\partial y}}\tan \left( {\theta_3 + \varphi_3 } \right)\tan \left( {\theta_2 + \varphi_2 } \right)\tan \left( {\theta_1 + \varphi_1 } \right)\dot{y} \\ & = c\frac{{L_3 \sin \theta_3 + \frac{x_2 }{2}}}{{L_3^2 - \left( {L_3^{\,} \sin \theta_3 + \frac{x_2 }{2}} \right)^2 }}\frac{{L_2 \sin \theta_2 + \frac{x_1 }{2}}}{{L_2^2 - \left( {L_2^{\,} \sin \theta_2 + \frac{x_1 }{2}} \right)^2 }}\frac{{L_1 \sin \theta_1 + \frac{y}{2}}}{{L_1^2 - \left( {L_1 \sin \theta_1 + \frac{y}{2}} \right)^2 }}\dot{y}. \\ \end{aligned} $$

(23)

$$ F_{c3} \approx \left( {c_{33} y^3 + c_{32} y^2 + c_{31} y^{\,} + c_{30} } \right)\dot{y} = f_{c3} c\dot{y}. $$

(24)

where $c_{33} = \left[ \begin{gathered} \frac{{\tan^3 \theta_1 \tan^2 \theta_3 \left( {12\tan^2 \theta_2 { + 3}} \right)}}{8L_1 L_2^2 \cos^3 \theta_1 \cos^4 \theta_2 }{ + }\frac{{\tan^3 \theta_1 \tan^4 \theta_2 \left( {12\tan^2 \theta_3 { + 3}} \right)}}{8L_1 L_3^2 \cos^3 \theta_1 \cos^4 \theta_3 }{ + }\frac{{\tan^5 \theta_1 \tan^3 \theta_2 \left( {12\tan^2 \theta_3 { + 3}} \right)}}{8L_2 L_3^2 \cos^3 \theta_2 \cos^4 \theta_3 }{ + }\frac{{\tan \theta_1 \tan \theta_2 \tan^2 \theta_3 \left( {11\tan^2 \theta_1 { + 4}} \right)}}{8L_1^2 L_2 \cos^4 \theta_1 \cos^3 \theta_2 } \hfill \\ { + }\frac{{\tan \theta_1 \tan^3 \theta_2 \tan \theta_3 \left( {11\tan^2 \theta_1 { + 4}} \right)}}{8L_1^2 L_3 \cos^4 \theta_1 \cos^3 \theta_3 }{ + }\frac{{\tan^5 \theta_1 \tan \theta_2 \tan \theta_3 \left( {11\tan^2 \theta_2 { + 4}} \right)}}{8L_2^2 L_3 \cos^4 \theta_2 \cos^3 \theta_3 }{ + }\frac{{\tan \theta_1 \tan^2 \theta_2 \tan^2 \theta_3 \left( {2\tan^2 \theta_1 + 1} \right)}}{2L_1^3 \cos^5 \theta_1 } \hfill \\ + \frac{{\tan^5 \theta_1 \tan \theta_2 \tan^2 \theta_3 \left( {2\tan^2 \theta_2 + 1} \right)}}{2L_2^3 \cos^5 \theta_2 }{ + }\frac{{\tan^5 \theta_1 \tan^5 \theta_2 \tan \theta_3 \left( {2\tan^2 \theta_3 + 1} \right)}}{2L_3^3 \cos^5 \theta_3 }{ + }\frac{15\tan^3 \theta_1 \tan^2 \theta_2 \tan \theta_3 }{{8L_1 L_2 L_3 \cos^3 \theta_1 \cos^3 \theta_2 \cos^3 \theta_3 }} \hfill \\ \end{gathered} \right]c$,

Fig. 14

Comparison of the negative stiffness force for the QZS isolation system with the triple-layer BCS between original equation and Taylor expansion when k_p = 5e4N/m, k_l = 2e4N/m, L_s = 0.08m, δ₂₀ = 0.04m, L₁ = 0.2m, L₂ = 0.15m, L₃ = 0.12m, θ₁ = 55°, θ₂ = 55°,θ₃ = 55°

Fig. 15

Comparison of the damping coefficient of the QZS isolation system with the triple-layer bionic cobweb structure between initial expression and Taylor expansion when k_p = 5e4N/m, k_l = 2e4N/m, L_s = 0.08m, δ₂₀ = 0.04m, L₁ = 0.2m, L₂ = 0.185m, L₃ = 0.17m, θ₁ = 55°, θ₂ = 55°, θ₃ = 55°

$c_{32} = \left[ \begin{gathered} \frac{\tan^2 \theta_2 \tan^2 \theta_3 (4\tan^2 \theta_1 + 1)}{{4L_1^2 \cos^4 \theta_1 }} + \frac{\tan^4 \theta_1 \tan^2 \theta_3 (4\tan^2 \theta_2 + 1)}{{4L_2^2 \cos^4 \theta_2 }} + \frac{\tan^4 \theta_1 \tan^4 \theta_2 (4\tan^2 \theta_3 + 1)}{{4L_3^2 \cos^4 \theta_3 }} \hfill \\ { + }\frac{5\tan^2 \theta_1 \tan \theta_2 \tan^2 \theta_3 }{{4L_1 L_2 \cos^3 \theta_1 \cos^3 \theta_2 }}{ + }\frac{5\tan^2 \theta_1 \tan^3 \theta_2 \tan \theta_3 }{{4L_1 L_3 \cos^3 \theta_1 \cos^3 \theta_3 }} + \frac{5\tan^4 \theta_1 \tan^2 \theta_2 \tan \theta_3 }{{4L_2 L_3 \cos^3 \theta_2 \cos^3 \theta_3 }} \hfill \\ \end{gathered} \right]c$,

$c_{31} = \left[ {\frac{\tan \theta_1 \tan^2 \theta_2 \tan^2 \theta_3 }{{L_1 \cos^3 \theta_1 }} + \frac{\tan^3 \theta_1 \tan \theta_2 \tan^2 \theta_3 }{{L_2 \cos^3 \theta_2 }}{ + }\frac{\tan^3 \theta_1 \tan^3 \theta_2 \tan \theta_3 }{{L_3 \cos^3 \theta_3 }}} \right]c$, $c_{30} = \tan^2 \theta_1 \tan^2 \theta_2 \tan^2 \theta_3 c$, f_c3 is noted as the damping coefficient of the bionic QZS isolation system with the triple-layer BCS.

For extension, the geometric relationship between deformation (x_i and y_i) and initial conditions (L_i and θ_i) of the QZS isolation system with the i-layer BCS can be summarized as Eqs. (25) and (26). The i-layer amplified negative stiffness and damping force are expressed in Eqs. (27) and (28), respectively.

$$ x_i = 2L_i \cos \theta_i - 2\sqrt {{L_i^2 - \left( {L_i^2 \sin \theta_i + \frac{{x_{i - 1} }}{2}} \right)^2 }} . $$

(25)

$$ \tan \left( {\theta_i + \varphi_i } \right) = \frac{{L_i \sin \theta_i + \frac{{x_{i - 1} }}{2}}}{{L_i \cos \theta_i - \frac{x_i }{2}}}{ = }\frac{{L_i \sin \theta_i + L_{i - 1} \cos \theta_{i - 1} - \sqrt {{L_{i - 1}^2 - \left( {L_{i - 1}^2 \sin \theta_{i - 1} + \frac{{x_{i - 2} }}{2}} \right)^2 }} }}{{\sqrt {{L_i^2 - \left( {L_i^2 \sin \theta_i + \frac{{x_{i - 1} }}{2}} \right)^2 }} }} $$

(26)

$$ \begin{gathered} F_{kni} = 2k_l \left( {\delta_{20} - L_s + \sqrt {L_s^2 - x_i^2 } } \right)\tan \alpha_i \tan \left( {\theta_i + \varphi_i } \right)\tan \left( {\theta_{i - 1} + \varphi_{i - 1} } \right) \cdots \tan \left( {\theta_1 + \varphi_1 } \right) \hfill \\ \begin{array}{*{20}c} {\,} \\ \end{array} \begin{array}{*{20}c} {\,} \\ \end{array} = 2k_l \left( {\delta_{20} - L_s + \sqrt {L_s^2 - x_i^2 } } \right)\frac{x_i }{{\sqrt {L_s^2 - x_i^2 } }}\frac{{L_i \sin \theta_i + \frac{{x_{i - 1} }}{2}}}{{\sqrt {{L_i^2 - \left( {L_i^{\,} \sin \theta_i + \frac{{x_{i - 1} }}{2}} \right)^2 }} }}\frac{{L_{i - 1} \sin \theta_{i - 1} + \frac{{x_{i - 2} }}{2}}}{{\sqrt {{L_{i - 1}^2 - \left( {L_{i - 1}^{\,} \sin \theta_{i - 1} + \frac{{x_{i - 2} }}{2}} \right)^2 }} }} \cdots \frac{{L_1 \sin \theta_1 + \frac{y}{2}}}{{\sqrt {{L_1^2 - \left( {L_1 \sin \theta_1 + \frac{y}{2}} \right)^2 }} }} \hfill \\ \hfill \\ \end{gathered} $$

(27)

$$ \begin{aligned} F_{cn} & = c\dot{x}_n \tan \left( {\theta_n + \varphi_n } \right)\tan \left( {\theta_{n - 1} + \varphi_{n - 1} } \right) \cdots \tan \left( {\theta_1 + \varphi_1 } \right) \\ & = c\frac{{L_n \sin \theta_n + \frac{{x_{n - 1} }}{2}}}{{L_n^2 - \left( {L_n^{\,} \sin \theta_n + \frac{{x_{n - 1} }}{2}} \right)^2 }}\frac{{L_{n - 1} \sin \theta_{n - 1} + \frac{{x_{n - 2} }}{2}}}{{L_n^2 - \left( {L_n^{\,} \sin \theta_n + \frac{{x_{n - 2} }}{2}} \right)^2 }} \cdots \frac{{L_1 \sin \theta_1 + \frac{y}{2}}}{{L_1^2 - \left( {L_1 \sin \theta_1 + \frac{y}{2}} \right)^2 }}\dot{y}. \\ \end{aligned} $$

(28)

4 The static characteristics of the amplified stiffness and damping coefficient

To minimize the number of variables to be investigated, the approximated formulations of stiffness and damping are nondimensionalized in "Appendix A" for further calculating simplification and intuitionistic results.

4.1 The amplified negative stiffness and damping formulas

The nondimensionalized negative stiffness force $\hat{F}_{kni} \left( {i = 0,1,2,3} \right)$ and the corresponding negative stiffness $\hat{K}_{ni} \left( {i = 0,1,2,3} \right)$ for the initial QZS-VI and the BQZS-BCS are exhibited in Eq. (A.1) of "Appendix A". Eq. (A.2) presents the combined restoring force $\hat{F}_{ki} \left( {i = 0,1,2,3} \right)$ and the resulting combined stiffness $\hat{K}_i \left( {i = 0,1,2,3} \right)$ for the initial QZS-VI and the BQZS-BCS.

$\hat{K}_{n0}$ and $\hat{K}_0$ of the initial QZS-VI show bilateral symmetry on the coordinate axis. As displayed in Eq. (A.3), the nonlinearity of the initial QZS-VI can be obtained as $\left| {\hat{F}_{kn03} \hat{y}^3 /(\lambda_1 \lambda_2 + \hat{F}_{kn01} )\hat{y}} \right|$.

Compared with $\hat{K}_{n0}$, the minimum value of $\hat{K}_{ni}$ of the BQZS-BCS has a right offset by $- \hat{F}_{kni2} /3\hat{F}_{kni3}$, suggesting that the peak negative stiffness for the bionic QZS isolation system does not exist at the equilibrium position. The offset $- \hat{F}_{kni2} /3\hat{F}_{kni3}$ depends on the geometrical parameters (θ_i, λ_i). The BCS intensifies the nonlinearity to a value of $\left| {(\hat{F}_{kni2} \hat{y}^2 + \hat{F}_{kni3} \hat{y}^3 )/(\lambda_1 \lambda_2 + \hat{F}_{kni1} )\hat{y}} \right|$, which manifests that the implementation of the BCS can significantly aggravate the nonlinearity of the isolation system in comparison with the initial one.

Equations (A.4) and (A.5) express the general formulation of the amplified damping coefficient and its nonlinearity, respectively. When compared to the initial QZS-VI damping coefficient, the amplified damping coefficient demonstrates substantial augmentation in magnitude and nonlinearity.

4.2 The influences of θ ₁, θ ₂, θ ₃, λ ₁ on the stiffness

With certain commonalities, Fig. 16, 17, 18, 19, 20 and Figs. 33, 34 in "Appendix A" illustrate the influences of θ₁, θ₂, θ₃ on the negative stiffness and combined stiffness of the different QZS isolation systems. Figures 16, 17, 18, 19, 20 describe the comparison between the initial and one-layer amplified stiffnesses, as well as the contrasting assessment between the triple-layer amplified and double-layer amplified stiffnesses. Analogous to the parametric influence mechanisms manifested by the comparison between the one-layer amplified and initial amplified BSC stiffnesses, the comparison chart for the double-layer amplified and one-layer amplified stiffnesses is exhibited in "Appendix B".

Fig. 16

The effects of θ₁ on the non-dimensional amplified stiffnesses with different BCSs. a, b The comparison between the one-layer amplified and the initial when λ₂ = 4 (k_p = 20e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m), θ₁ = 45°, 50°, 55°, 60°, 65°, c = 80N⋅s/m, z₀ = 0.5e−3m; c, d The comparison between the triple-layer amplified and the double-layer amplified when λ₂ = 14 (k_p = 60e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m, L₃ = 0.8m), θ₁ = 50°, 55°, 60°, 65°, θ₂ = 55°, θ₃ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m. a The negative stiffness curves; b The combined stiffness curves

Fig. 17

The effects of θ₁ on the nonlinearities of the non-dimensional combined restoring forces with different BCSs. a, The comparison between the one-layer BCS and the initial when λ₂ = 4 (k_p = 20e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m), θ₁ = 45°, 50°, 55°, 60°, 65°, c = 80N⋅s/m, z₀ = 0.5e−3m. b The comparison between the triple-layer BCS and the double-layer BCS when λ₂ = 14 (k_p = 60e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m, L₃ = 0.8m), θ₁ = 50°, 55°, 60°, 65°, θ₂ = 55°, θ₃ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m

Fig. 18

The effects of θ₂ on the non-dimensional amplified stiffnesses with different BCSs. a, b, The comparison between the triple-layer amplified and double-layer amplified when λ₂ = 12 (k_p = 60e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m, L₃ = 0.8m), θ₁ = 55°, θ₂ = 50°, 55°, 60°, 65°, θ₃ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m. a The negative stiffness curves. b The combined stiffness curves

Fig. 19

The effects of θ₂ on the nonlinearities of the combined restoring forces with the triple-layer and double-layer BCSs when λ₂ = 12 (k_p = 60e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m, L₃ = 0.8m), θ₁ = 55°, θ₂ = 50°, 55°, 60°, 65°, θ₃ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m

Fig. 20

The effects of θ₃ on the non-dimensional stiffnesses with triple-layer BCS and the corresponding nonlinearity of the combined restoring force when λ₂ = 12 (k_p = 60e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m, L₃ = 0.8m), θ₁ = 55°, θ₂ = 55°, θ₃ = 50°, 55°, 60°, 65°, c = 80N⋅s/m, z₀ = 0.5e−3m. a, The negative stiffness curves; b The combined stiffness curves. c The nonlinearity curves

Compared to the initial QZS isolation system, the incorporation of the BCS prominently reduces the equivalent negative and combined stiffnesses for the BQZS-BCS, especially the negative stiffness at the equilibrium position. For the initial QZS isolation system and the bionic QZS isolation systems with one-layer, double-layer, and triple-layer BCS, the negative stiffness at the equilibrium position equals to − 2, − 2tan²θ₁, − 2tan²θ₁tan²θ₂, − 2tan²θ₁tan²θ₂ tan²θ₃, individually, which complies with the mechanism of the amplification effect mentioned in Sect. 3.1.

However, the incorporation of the multi-layer BCS introduces a right offset ($- \hat{F}_{kni2} /3\hat{F}_{kni3}$), causing the minimum negative stiffness to deviate from the equilibrium position (zero point) to the right side of y axis. More layers of the BCS can result in a much greater reduction of the negative and combined stiffnesses, while the right offset decreases with the number of BCS layers. An enlightening finding is deduced that the stiffnesses have conspicuous right offset after the one-layer and double-layer amplification, while the stiffness curves appear almost symmetric after the triple-layer amplification. For the BCS with the same layer, the negative and combined stiffnesses can be reduced with θ_i (i = 1,2,3), and the right offset decreases with θ_i. Moreover, increasing the number of structural layers and θ_i results in an increment in the nonlinearity of the amplified combined restoring force.

Figures 21 and 22 exhibit the influences of λ₁ on the negative and combined stiffnesses for different QZS isolation systems. Although λ₁ indicates the relationship between δ₂₀ and L_s, it refers to the pre-compressed elastic energy that has been stored when L_s remains constant.

Fig. 21

The effects of λ₁ on the non-dimensional amplified stiffnesses with different BCSs. a, b, The comparison between the one-layer amplified and the initial when λ₂ = 4 (k_p = 20e4N/m, k_l = 5e4N/m), λ₁ = 3.3, 5 (L_s = 0.05m, δ₂₀ = 0.015m, 0.01m), λ₃ = 0.25 (L₁ = 0.2m), θ₁ = 60°, c = 80N⋅s/m, z₀ = 0.5e−3m. c, d, The comparison between the triple-layer amplified and the double-layer amplified when λ₂ = 9 (k_p = 45e4N/m, k_l = 5e4N/m), λ₁ = 0.67, 1, 2, 3.3, 5, λ₃ = 0.25, λ₄ = 0.125, λ₅ = 0.0625 (L_s = 0.05m, δ₂₀ = 0.075m, 0.05m, 0.025m, 0.015m, 0.01m, L₁ = 0.2m, L₂ = 0.4m, L₃ = 0.8m), θ₁ = 55°, θ₂ = 55°, θ₃ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m. a, c, The negative stiffness curves. b, d, The combined stiffness curves

Fig. 22

The effects of λ₁ on the nonlinearities of the non-dimensional combined restoring forces with different BCSs. a The comparison between one-layer BCS and the initial when λ₂ = 4 (k_p = 20e4N/m, k_l = 5e4N/m), λ₁ = 3.3, 5 (L_s = 0.05m, δ₂₀ = 0.015m, 0.01m), λ₃ = 0.25 (L₁ = 0.2m), θ₁ = 60°, c = 80N⋅s/m, z₀ = 0.5e−3m. b The comparison between the triple-layer BCS and the double-layer BCS when λ₂ = 9 (k_p = 45e4N/m, k_l = 5e4N/m), λ₁ = 0.67, 1, 2, 3.3, 5, λ₃ = 0.25, λ₄ = 0.125, λ₅ = 0.0625 (L_s = 0.05m, δ₂₀ = 0.075m, 0.05m, 0.025m, 0.015m, 0.01m, L₁ = 0.2m, L₂ = 0.4m, L₃ = 0.8m), θ₁ = 55°, θ₂ = 55°, θ₃ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m

For the initial and one-layer QZS isolation systems, the negative and combined stiffnesses transform from concave to convex with the decrease of λ₁. The stiffnesses transition from hardening to softening at λ₁ < 1 for the initial system and λ₁ < 2 for the one-layer system due to convexity. λ₁ marks this critical point, and introducing the one-layer BCS raises this threshold. That is due to the second-order term of the initial negative stiffness force, which is $- 3\left( {1 - \lambda_1 } \right)$. While the second-order term of the one-layer amplified negative stiffness is $- 3\tan^4 \theta_1 \left[ {\lambda_3^2 \left( {5\tan^2 \theta_1 + 1} \right)/\left( {4\sin^4 \theta_1 } \right) + (1 - \lambda_1 )} \right]$. Since the latter contains an additional nonlinear positive buffer term compared to the former, a higher λ₁ is required to change the second-order term's sign. The one-layer BCS amplifies the initial negative stiffness, prominently. Especially, the initial negative stiffness at the equilibrium position is $\lambda_1 \lambda_2 - 2$, while the one-layer amplified one is $\lambda_1 \lambda_2 - 2\tan^2 \theta_1$. A smaller λ₁ can result in a higher negative stiffness at the equilibrium position. The parallel implementation of the positive and negative stiffnesses scatters the combined stiffness curves, and its magnitude at the equilibrium position differs. The combined stiffness ought to be non-negative, with the utilization of BCS constraining the effective range to a narrower spectrum encompassed by the value of λ₁. In essence, a decreased L_s or increased δ₂₀ optimizes negative stiffness performance with hardening characteristics. The one-layer restoring force exhibits enhanced nonlinearity compared to the initial QZS system, and its nonlinearity is inversely correlating with λ₁.

Moreover, the nondimensionalized negative and combined stiffnesses at the equilibrium position keep decreasing with the increase of nesting layer for the BCS. Specifically, the initial, one-layer amplified, double-layer amplified, triple-layer amplified negative stiffness at the equilibrium position are − 2, $- 2\tan^2 \theta_1$, $- 2\tan^2 \theta_1 \tan^2 \theta_2$, $- 2\tan^2 \theta_1 \tan^2 \theta_2 \tan^2 \theta_3$. The initial, one-layer amplified, double-layer amplified triple-layer amplified combined stiffnesses are $\lambda_1 \lambda_2 - 2$, $\lambda_1 \lambda_2 - 2\tan^2 \theta_1$, $\lambda_1 \lambda_2 - 2\tan^2 \theta_1 \tan^2 \theta_2$, $\lambda_1 \lambda_2 - 2\tan^2 \theta_1 \tan^2 \theta_2 \tan^2 \theta_3$. For the BCS with the same layer, a small λ₁ can produce a lower combined stiffness. The nonlinearity initially rises and then diminishes with increasing λ₁. Notably, the combined stiffness around equilibrium must be non-negative for effective vibration isolation, limiting λ₁ to (3.3, 5) for single-layer and (2, 5) for double-layer and triple-layer. Within this range, λ1 positively correlates with combined stiffness and nonlinearity.

4.3 The influences of θ ₁, θ ₂, θ ₃, on damping coefficient

The effects of θ₁, θ₂, θ₃ on the amplified damping coefficient are comparable and are shown in Figs. 23, 24, 25, 26, and 35, 36. The results can be summarized as follows. For the BCS with the same layer, a larger θ_i (i = 1, 2, 3) can result in a larger damping coefficient and a more severe nonlinearity within most of the ranges. The number of BCS’s layers and the damping coefficient are positively correlated with the same θ_i. While, for the BCSs with different layers, the nonlinearity increases with the number of the BCS and then decreases. When the BCS changes from the initial to the double-layer, the nonlinearity monotonically increases, but it rapidly drops once it reaches the triple-layer. In terms of the amplification efficiency and the variations of nonlinearity, the difference between the double-layer amplified and one-layer amplified damping coefficients is less evident than the difference between the one-layer amplified and initial damping coefficients, as well as that between the triple-layer amplified and double-layer amplified damping coefficients. The comparison between the double-layer amplified and the one-layer amplified damping coefficients is presented in "Appendix C".

Fig. 23

The effects of θ₁ on the non-dimensional amplified damping coefficients with different BCSs. a The comparison between the one-layer amplified and the initial when λ₁ = 3.3, λ₃ = 0.25 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m), θ₁ = 50°,55°,60°,65°, c = 80N⋅s/m, z₀ = 0.5e−3m; b The comparison between the triple-layer amplified and the double-layer amplified when λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m), θ₁ = 50°, 55°, 60°, 65°, θ₂ = 55°, θ₃ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m

Fig. 24

The effects of θ₁ on the nonlinearities of the non-dimensional damping coefficient with different BCSs. a The comparison between the one-layer BCS and the initial when λ₁ = 3.3, λ₃ = 0.25 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m), θ₁ = 50°, 55°, 60°, 65°, c = 80N⋅s/m, z₀ = 0.5e−3m. b The comparison between the triple-layer BCS and the double-layer BCS when λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m), θ₁ = 50°, 55°, 60°, 65°, θ₂ = 55°, θ₃ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m

Fig. 25

The effects of θ₂ on the non-dimensional amplified damping coefficients and their nonlinearities with the triple-layer and the double-layer BCSs when λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m), θ₁ = 55°, θ₃ = 55°, θ₂ = 50°, 55°, 60°, 65°, c = 80N⋅s/m, z₀ = 0.5e−3m. a The damping coefficient. b The nonlinearity

Fig. 26

The effects of θ₃ on the non-dimensional amplified damping coefficient and its nonlinearity with the triple-layer BCS when λ₂ = 14 (k_p = 70e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m), θ₁ = 55°, θ₂ = 55°, θ₃ = 50°, 55°, 60°, 65°, c = 80N⋅s/m, z₀ = 0.5e−3m. a The damping coefficient. b The nonlinearity

5 Dynamic results and analysis

5.1 Dynamic modeling and resolving of the bionic QZS isolation system with the BCS

The dynamic models of the BQZS-BCSs are exhibited as Fig. 27, in which the excitation displacements exerted on the base and the response displacements of the isolation object are z (z = Z₀coswt) and w, respectively. y is the relative displacement between the loading plate and the base, which equals to w-z.

Fig. 27

The schematic diagram and 3-D model of the QZS isolation systems using the BCSs with different layers

The fitted formulation of amplified negative stiffness and damping forces are introduced to describe the isolation system dynamic behavior.

$$ m\ddot{y} + k_p y + F_{kni} + F_{ci} = - m\ddot{z},\;\;\;i = 0,1,2,3 \ldots . $$

(29)

Specifically,

$$ m\ddot{y} + k_p y + \left( {kn_{i1} y{ + }kn_{i2} y^2 + kn_{i3} y^3 } \right) + \left( {c_{i0} + c_{i1} y + c_{i2} y^2 + c_{i3} y^3 } \right)\dot{y} = - m\ddot{z}, $$

(30)

where, “⋅⋅” and “⋅” denote the second and first derivatives of variables with respect to time t, respectively.

For calculation simplification, the nondimensionalized dynamic equation is given by Eq. (31).

$$ y^{\prime \prime} + \gamma_1 y{ + }\gamma_2 y^2 { + }\gamma_3 y^3 { + }\left( {\zeta_0 + \zeta_1 y + \zeta_2 y^2 + \zeta_3 y^3 } \right)y^{\prime} { = }Z_0 \Omega^2 \cos (\Omega \tau ), $$

(31)

where $\omega_0 = \sqrt {{\frac{k_p }{m}}}$,$\tau = \omega_0 t$,$\gamma_1 = \frac{{k_{ni1} }}{k_p }{ + }1$,$\gamma_2 = \frac{{k_{ni2} }}{k_p }$,$\gamma_3 = \frac{{k_{ni3} }}{k_p }$,$\zeta_0 = c_{i0} /\sqrt {k_p m}$,$\zeta_1 = c_{i1} /\sqrt {k_p m}$,$\zeta_2 = c_{i2} /\sqrt {k_p m}$,$\zeta_3 = c_{i3} /\sqrt {k_p m}$, $\Omega = \frac{\omega }{\omega_0 }$. “ ′′ ” and “ ′ ” denote the second and first derivatives of variables with respect to nondimensional time τ, respectively.

Then, the above can be transformed as follows with the assumption that $y = \rho \cos \left( {\omega t + \beta } \right) = \rho \cos \left( {\Omega \tau + \beta } \right) = \rho \cos \varphi$. Besides, the tuning parameter $\sigma_1 = \Omega^2 - \gamma_1$ is introduced to reflect the proximity of the natural frequency and excitation frequency in Eq. (32).

$$ y^{^{\prime\prime}} + \Omega^2 y = \sigma_1 y - \gamma_2 y^2 - \gamma_3 y^3 - \left( {\zeta_0 + \zeta_1 y + \zeta_2 y^2 + \zeta_3 y^3 } \right)y^{\prime} + Z_0 \Omega^2 \cos (\Omega \tau ). $$

(32)

Adopting the average method to solve the above equation under the assumption of first order vibration, the amplitude and phase of the response displacement are as follows

$$ \left\{ {\begin{array}{*{20}l} {\rho^{\prime}} = - \frac{\varepsilon }{2\pi \Omega }\int_0^{2\pi } {\left[ {\sigma_1 y - \gamma_2 y^2 - \gamma_3 y^3 - \left( {\zeta_0 + \zeta_1 y + \zeta_2 y^2 + \zeta_3 y^3 } \right)y^{\prime} + Z_0 \Omega^2 \cos (\varphi - \beta )} \right]} \sin \varphi d\varphi \hfill \\ {\beta^{\prime} = - \frac{\varepsilon }{2\pi \Omega \rho }\int_0^{2\pi } {\left[ {\sigma_1 y - \gamma_2 y^2 - \gamma_3 y^3 - \left( {\zeta_0 + \zeta_1 y + \zeta_2 y^2 + \zeta_3 y^3 } \right)y^{\prime} + Z_0 \Omega^2 \cos (\varphi - \beta )} \right]} \cos \varphi d\varphi } \hfill \\\end{array} } \right. . $$

(33)

After simplification, it is taken as

$$ \left\{ {\begin{array}{*{20}l} {\rho^{\prime} = - \frac{1}{2\Omega }\left( {\zeta_0 \rho \Omega + \frac{\zeta_2 \Omega \rho^3 }{4} + Z_0 \Omega^2 \sin \beta } \right) = - \frac{\zeta_0 \rho }{2} - \frac{\zeta_2 \rho^3 }{8} - \frac{Z_0 \Omega \sin \beta }{2}} \hfill \\ {\beta^{\prime} = \frac{1}{2\Omega \rho }\left( {\sigma_1 \rho + \frac{3\gamma_3 \rho^3 }{4} - Z_0 \Omega^2 \cos \beta } \right){ = }\frac{\gamma_1 }{{2\Omega }} - \frac{\Omega }{2} + \frac{3\gamma_3 \rho^2 }{{8\Omega }} - \frac{Z_0 \Omega \cos \beta }{{2\rho }}} \hfill \\ \end{array} } \right.. $$

(34)

Since the average method assumes the magnitude changes of ρ and β to be relatively minimal in a period of nondimensional time τ and can be treated as constants, the right ends of the following expressions equal zero.

Under the consideration that sin²β + cos²β = 1, Eq. (34) can be obtained as

$$ \left( {\frac{4\zeta_0 \rho + \zeta_2 \rho^3 }{{4Z_0 \Omega }}} \right)^2 + \left( {\frac{3\gamma_3 \rho^3 - 4\sigma_1 \rho }{{4Z_0 \Omega^2 }}} \right)^2 = 1. $$

(35)

It can be also illustrated as

$$ W\left( {\rho ,\Omega } \right) = \Phi^2 \left( {\rho ,\Omega } \right) + \Psi^2 \left( {\rho ,\Omega } \right) - Z_0^2 \Omega^4 = 0, $$

(36)

where $\Phi \left( {\rho ,\Omega } \right) = \zeta_0 \rho \Omega + \frac{\zeta_2 \Omega \rho^3 }{4}$, $\Psi \left( {\rho ,\Omega } \right) = \gamma_1 \rho - \Omega^2 \rho + \frac{3\gamma_3 \rho^3 }{4}$.

The absolute response displacement of the isolation object can be expressed as

$$ w = z + y = Z_0 \cos \left( {\omega t} \right) + \rho \cos \left( {\omega t + \beta } \right) = A\cos (\omega t + \Phi_1 ), $$

(37)

where $A = \sqrt {\rho^2 + 2\rho Z_0 \cos \beta + Z_0^2 } ,\begin{array}{*{20}c} {\,} \\ \end{array} \Phi_1 = \arctan \frac{\rho \sin \beta }{{\rho \cos \beta + Z_0 }} = \arctan \left( { - \frac{\Phi }{\Psi }} \right)$,

The transmissibility of the BQZS-BCS is derived as

$$ T_{\,} = \frac{A}{Z_0 } = \sqrt {{\frac{\rho^2 }{{Z_0^2 }} + \frac{{\rho^2 \left[ {3\gamma_3 \rho^2 - 4\left( {\Omega^2 - \gamma_1 } \right)} \right]}}{2Z_0^2 \Omega^2 } + 1}} . $$

(38)

Equation (35) can be transformed into the equation with the variable Ω² as

$$ 16(\rho^2 - Z_0^2 )\left( {\Omega^2 } \right)^2 + \left[ {(8\zeta_0 \zeta_2 - 24\gamma_3 )\rho^4 + (16\zeta_0^2 - 32\gamma_1 )\rho^2 + \rho^6 \zeta_2^2 } \right]\Omega^2 + 16\gamma_1^2 \rho^2 + 24\gamma_1 \gamma_3 \rho^4 + 9\gamma_3^2 \rho^6 { = }0 $$

(39)

Solve Eq. (39), it gives

$$ \Omega^2 = \frac{{8(3\gamma_3 \rho^4 + 4\gamma_1 \rho^2 ) - \left( {4\zeta_0 \rho + \zeta_2 \rho^3 } \right)^2 }}{{32\left( {\rho^2 - z_0^2 } \right)}} \pm \frac{{\rho \sqrt {\Delta } }}{{32\left( {\rho^2 - z_0^2 } \right)}}, $$

(40)

where $\begin{aligned} \Delta & = \zeta_2^4 \left( {\rho^2 } \right)^5 + 16\zeta_2^2 \left( {\zeta_0 \zeta_2 - 3\gamma_3 } \right)\left( {\rho^2 } \right)^4 + \left( {96\zeta_0^2 \zeta_2^2 - 384\zeta_0 \zeta_2 \gamma_3 - 64\gamma_1 \zeta_2^2 } \right)\left( {\rho^2 } \right)^3 \\ & \;\;\;\; + (256\zeta_0^3 \zeta_2 + 576\gamma_3^2 z_0^2 - 768\gamma_3 \zeta_0^2 - 512\zeta_0 \zeta_2 \gamma_1 )\left( {\rho^2 } \right)^2 \\ & \;\;\;\; + (256\zeta_0^4 + 1536\gamma_1 \gamma_3 z_0^2 - 1024\zeta_0^2 \gamma_1 )\left( {\rho^2 } \right) + 1024\gamma_1^2 z_0^2 . \\ \end{aligned}$.

The solution of the equation Δ = 0 corresponds to the resonant amplitude ρ_max. The resonant frequency Ω_max and T_max of the bionic QZS isolation system with the amplified negative stiffness and amplified damping coefficient (BQZS-AA) can be achieved as

$$ \left\{ {\begin{array}{*{20}l} {\rho_{\max } = \sqrt {{Root\left( {\Delta = 0} \right)}} } \hfill \\ {\Omega_{\max } = \sqrt {{\frac{{8(3\gamma_3 \rho_{\max }^4 + 4\gamma_1 \rho_{\max }^2 ) - \left( {4\zeta_0 \rho_{\max } + \zeta_2 \rho_{\max }^3 } \right)^2 }}{{32\left( {\rho_{\max }^2 - z_0^2 } \right)}}}} } \hfill \\ {T_{\max } = \sqrt {{\frac{{\rho_{\max }^2 }}{Z_0^2 } + \frac{{\rho_{\max }^2 \left[ {3\gamma_3 \rho_{\max }^2 - 4\left( {\Omega_{\max }^2 - \gamma_1 } \right)} \right]}}{{2Z_0^2 \Omega_{\max }^2 }} + 1}} } \hfill \\ \end{array} } \right.. $$

(41)

For the bionic QZS isolation system with the amplified negative stiffness and the initial damping coefficient (BQZS-AI), ζ₂ = 0, and it gives

$$ \Delta = 64\left[ {(9\gamma_3^2 z_0^2 - 12\gamma_3 \zeta_0^2 )\left( {\rho^2 } \right)^2 + (4\zeta_0^4 + 24\gamma_1 \gamma_3 z_0^2 - 16\zeta_0^2 \gamma_1 )\left( {\rho^2 } \right) + 16\gamma_1^2 z_0^2 } \right], $$

(42)

$$ \rho_{\max } = \sqrt {{\frac{{ - b_1 - \sqrt {b_1^2 - 4a_1 c_1 } }}{2a_1 }}} ,\;\;\Omega_{\max } = \sqrt {{\frac{{8(3\gamma_3 \rho_{\max }^4 + 4\gamma_1 \rho_{\max }^2 ) - \left( {4\zeta_0 \rho_{\max } + \zeta_2 \rho_{\max }^3 } \right)^2 }}{{32\left( {\rho_{\max }^2 - z_0^2 } \right)}}}} , $$

(43)

where $a_1 = (9\gamma_3^2 z_0^2 - 12\gamma_3 \zeta_0^2 )$, $b_1 = (4\zeta_0^4 + 24\gamma_1 \gamma_3 z_0^2 - 16\zeta_0^2 \gamma_1 )$, $c_1 = 16\gamma_1^2 z_0^2$.

Assuming that (ρ_s, β_s), singularities in the dynamic phase plane, represent the stable response of the system, Eq. (34) can be achieved based on Eq. (44).

$$ \left\{ {\begin{array}{*{20}l} { - \frac{1}{2\Omega }\left( {\zeta_0 \rho_s \Omega + \frac{\zeta_2 \Omega \rho_s^3 }{4} + Z_0 \Omega^2 \sin \beta_s } \right){ = }0} \hfill \\ {\frac{1}{2\Omega \rho_s }\left( {\gamma_1 \rho_s - \Omega^2 \rho_s + \frac{3\gamma_3 \rho_s^3 }{4} - Z_0 \Omega^2 \cos \beta_s } \right) = 0} \hfill \\ \end{array} } \right.. $$

(44)

Besides, W (ρ_s, Ω) equals to zero too.

By introducing the perturbation variables h (h = ρ-ρ_s) and k (k = β-β_s), the first order approximation to the system in equations near the singularity can be written as

$$ \left\{ {\begin{array}{*{20}l} {h^{\prime} = \left( { - \frac{\zeta_0 }{2} - \frac{3\zeta_2 \rho_s^2 }{8}} \right)h - \frac{Z_0 \Omega }{2}\cos \beta_s k} \hfill \\ {k^{\prime} = \left( {\frac{3\gamma_3 \rho_s }{{4\Omega }} + \frac{Z_0 \Omega }{2}\frac{\cos \beta_s }{{\rho_s^2 }}} \right)h + \frac{Z_0 \Omega }{2}\frac{\sin \beta_s }{{\rho_s }}k} \hfill \\ \end{array} } \right.. $$

(45)

Combing Eq. (44), the above linear perturbation equation can be transformed as

$$ \left\{ {\begin{array}{*{20}l} {2\Omega h^{\prime} { + }\left( {\Omega \zeta_0 { + }\frac{3\Omega \zeta_2 \rho_s^2 }{4}} \right)h{ + }\left( {\gamma_1 \rho_s - \Omega^2 \rho_s + \frac{3\gamma_3 \rho_s^3 }{4}} \right)k = 0} \hfill \\ {2\Omega k^{\prime} - \frac{1}{\rho_s }\left( {\gamma_1 - \Omega^2 { + }\frac{9\gamma_3 \rho_s^2 }{4}} \right)h{ + }\left( {\Omega \zeta_0 { + }\frac{\Omega \zeta_2 \rho_s^2 }{4}} \right)k = 0} \hfill \\ \end{array} } \right.. $$

(46)

The eigenequation of this linear perturbation equation can be derived as

$$ \left| {\begin{array}{*{20}c} {2\Omega \lambda { + }\left( {\Omega \zeta_0 { + }\frac{3\Omega \zeta_2 \rho_s^2 }{4}} \right)} & {\left( {\gamma_1 \rho_s - \Omega^2 \rho_s + \frac{3\gamma_3 \rho_s^3 }{4}} \right)} \\ { - \frac{1}{\rho_s }\left( {\gamma_1 - \Omega^2 { + }\frac{9\gamma_3 \rho_s^2 }{4}} \right)} & {2\Omega \lambda { + }\left( {\Omega \zeta_0 { + }\frac{\Omega \zeta_2 \rho_s^2 }{4}} \right)} \\ \end{array} } \right|{ = }4\Omega^2 \left( {\lambda^2 {\text{ + A}}\lambda {\text{ + B}}} \right){ = }0, $$

(47)

where

$$ \left\{ {\begin{array}{*{20}l} {{\text{A = }}\frac{1}{2\Omega \rho_s }\left[ {\frac{{\partial \left( {\rho \Phi } \right)}}{\partial \rho }} \right]_s { = }\zeta_0 { + }\frac{\zeta_2 \rho_s^2 }{2}} \hfill \\ {{\text{B = }}\frac{1}{8\Omega^2 \rho_s }\left[ {\frac{\partial W}{{\partial \rho }}} \right]_s { = }\left( {\frac{\zeta_0 }{2} + \frac{3\zeta_2 \rho_s^2 }{8}} \right)\left( {\frac{\zeta_0 }{2} + \frac{\zeta_2 \rho_s^2 }{8}} \right) + \left( {\frac{\gamma_1 }{{2\Omega }} - \frac{\Omega }{2}{ + }\frac{9\gamma_3 \rho_s^2 }{{8\Omega }}} \right)\left( {\frac{\gamma_1 }{{2\Omega }} - \frac{\Omega }{2}{ + }\frac{3\gamma_3 \rho_s^2 }{{8\Omega }}} \right)} \hfill \\ \end{array} } \right.. $$

According to Liapunov's first order approximation stability theory A > 0 and B > 0 are sufficient conditions for the asymptotic stability of the singularity (ρ_s, β_s). It indicates that the singularity is unstable when B < 0. The boundary between the stable and unstable regions is determined by B = 0, which refers to the point where the slope of the magnitude-frequency curve with respect to its vertical axis is 0.

5.2 Verification

To ensure the correctness of the derived solutions, it is essential to conduct a comparison between the results calculated by analytical and numerical methods.

In this section, the analytical results of the initial QZS isolation system and the bionic QZS isolation system with one-layer BCS are attained by solving the transcendental equation of amplitude-frequency relationship in Eq. (35), and the numerical results are achieved by using Runge–Kutta method to forward and backward sweep. From Fig. 28, it can be concluded that the two kinds of results keep a good agreement.

Fig. 28

The comparisons between the results calculated by analytical and numerical methods for the QZS isolation systems with one-layer cobweb structure or not when λ₂ = 4 (kp = 20e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m), θ₁ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m

5.3 Bionic QZS isolation systems using the BCS with different layers (BQZS-BCSs)

To elucidate the benefits and coupling mechanism of the amplified negative stiffness and damping coefficient, the two BQZS-BCSs are divided into types: one with the amplified negative stiffness and the initial damping coefficient (BQZS-AI), another with the amplified negative stiffness and damping coefficient (BQZS-AA). The influences of various geometric parameters on the isolation performances of the two types are analyzed and compared, to demonstrate the indispensability of both amplified negative stiffness and damping coefficients. The double-layer amplified negative stiffness and damping characteristics are both presented in the "Appendices B and C", the dynamic properties of the bionic isolation system with the double-layer BCS are exhibited in the "Appendix D" for clarity and conciseness.

5.3.1 Bionic QZS isolation system with the one-layer BCS

It is known from Sects. 4.2 and 4.3 that a larger θ₁, λ₃ or smaller λ₁ can cause a smaller combined stiffness and a larger damping coefficient. From the common sense, the amplified negative stiffness can counteract the positive spring much more for a lower resonant frequency. The unstable region may decrease or even disappear, broadening the effective isolation frequency band. However, the result in Fig. 29 manifests a noticeable rightward distortion in the resonance curves of the BQZS-AI. The severity of distortion increases with θ₁ and λ₃, even surpassing the level of initial QZS system.

Fig. 29

The effects of parameters on the BQZS-AA and BQZS-AI with the one-layer BCS. a and b describe the effects of θ₁ when λ₂ = 4 (k_p = 20e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m), θ₁ = 50°, 55°, 60°, 65°, c = 80N⋅s/m, z₀ = 0.5e−3m; c and d describe the effects of λ₁ when λ₂ = 4 (k_p = 20e4N/m, k_l = 5e4N/m), λ₁ = 3.3, 5 (L_s = 0.05m, δ₂₀ = 0.015m, 0.01m), λ₃ = 0.25 (L₁ = 0.2m), θ₁ = 60°, c = 80N⋅s/m, z₀ = 0.5e−3m; a, c represent the amplitude-frequency curves; b, d represent the transmissibility curves

With the decrement of λ₁, the distortion in the BQZS-AI diminishes, yet it maintains more severity than the initial QZS-VI across most ranges. The much higher jump-down frequency leads to a broadened unstable region and an enlarged resonant frequency. This is due to the substantial rise in the nonlinearity of the hardening stiffness as the displacement departs from the equilibrium position.

These outcomes contravene the intended objectives of QZS technology. The degradation of the BQZS-AI isolation performance renders the incorporation of the BCS redundant. The adverse effects of nonlinearity brought by the specific structure undermine the benefits of amplified negative stiffness.

The integration of a damping element at the core of BCS introduces a pivotal shift. The amplified negative stiffness and damping coefficient are well coupled in the BQZS-AA. As illustrated in Fig. 30, the amplified negative stiffness lowers the resonant frequency, while the amplified damping coefficient alleviates the nonlinear effects, rectifying the right-distorted resonance curves to the left, thus reducing the unstable region and the resonance amplitude. The resonant frequency, resonance amplitude, and peak transmissibility of the BQZS-AA all reduce with the increase of θ₁ or the decrease of λ₁, and are much lower than the initial ones. The isolation performance, including that in the high-frequency field, of the BQZS-AA with the one-layer BCS gets greatly improved with a smaller transmissibility when compared to the initial QZS-VI.

Fig. 30

The effects of parameters on the resonant frequency and response amplitude of the BQZS-AA with the one-layer BCS. a Describe the effects of θ₁ when λ₂ = 4 (k_p = 20e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m), θ₁ ∈ (50°,65°), c = 80N⋅s/m, z₀ = 0.5e−3m; b describe the effects of λ₁ when λ₂ = 4 (k_p = 20e4N/m, k_l = 5e4N/m), λ₁ ∈ (3.3,5) (L_s = 0.05m, δ₂₀ ∈ (0.01m, 0.015m)), λ₃ = 0.25 (L₁ = 0.2m), θ₁ = 60°, c = 80N⋅s/m, z₀ = 0.5e−3m

5.3.2 Bionic QZS isolation system with the triple-layer BCS

With an increasing number of nested layers, the integration of amplified negative stiffness and damping elements within the triple-layer BCS significantly enhances the isolation performance. Sections 4.2 and 4.3 demonstrate the advantages of the triple-amplified attributes over other-layer amplified ones, which can be corroborated by their dynamic isolation performance in Figs. 31 and 32. With identical structural parameters except for the outmost layer, the BQZS-AA with the triple-layer BCS exhibits outperform its double-layer counterpart.

Fig. 31

The effects of parameters on the BQZS-AA and BQZS-AI with the triple-layer BCS. a and b describe the effects of θ₁ when λ₂ = 14 (k_p = 60e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m), θ₁ = 50°,55°,60°,65°,θ₂ = 55°, θ₃ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m; c and d describe the effects of θ₂ when λ₂ = 14 (k_p = 70e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125, λ₅ = 0.0625 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m, L₃ = 0.8m), θ₁ = 55°, θ₂ = 50°,55°,60°,65°, θ₃ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m; e and f describe the effects of θ₃ when λ₂ = 14 (k_p = 70e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125, λ₅ = 0.0625 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m, L₃ = 0.8m), θ₁ = 55°,θ₂ = 55°, θ₃ = 50°,55°,60°,65°, c = 80N⋅s/m, z₀ = 0.5e−3m; g and h describe the effects of λ₁ when λ₂ = 9 (k_p = 45e4N/m, k_l = 5e4N/m), λ₁ = 2, 3.3, 5, λ₃ = 0.25, λ₄ = 0.125, λ₅ = 0.0625 (L_s = 0.05m, δ₂₀ = 0.025m, 0.015m, 0.01m, L₁ = 0.2m, L₂ = 0.4m, L₃ = 0.8m), θ₁ = 55°, θ₂ = 55°, θ₃ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m; a, c, e, g represent the amplitude-frequency curves; b, d, f, h represent the transmissibility curves

Fig. 32

The effects of parameters on the resonant frequency and response amplitude of the BQZS-AA with the double-layer BCS. a describes the effects of θ₁ when λ₂ = 14 (k_p = 60e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m), θ₁ ∈ (50°, 65°), θ₂ = 55°, θ₃ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m; b describes the effects of θ₂ when λ₂ = 14 (k_p = 70e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125, λ₅ = 0.0625 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m, L₃ = 0.8m), θ₁ = 55°,θ₂ ∈ (50°, 65°), θ₃ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m; c describes the effects of θ₃ when λ₂ = 14 (k_p = 70e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125, λ₅ = 0.0625 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m, L₃ = 0.8m), θ₁ = 55°,θ₂ = 55°,θ₃ ∈ (50°, 65°), c = 80N⋅s/m, z₀ = 0.5e−3m; d describes the effects of λ₁ when λ₂ = 9 (k_p = 45e4N/m, k_l = 5e4N/m), λ₁ ∈ (2, 5), λ₃ = 0.25, λ₄ = 0.125, λ₅ = 0.0625 (L_s = 0.05m, δ₂₀ ∈ (0.01m, 0.025m), L₁ = 0.2m, L₂ = 0.4m, L₃ = 0.8m), θ₁ = 55°, θ₂ = 55°, θ₃ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m

The increase of θ_i (i = 1, 2, 3) and the decrease of λ₁ result in the progressive right-distortion of resonant peaks for the BQZS-AI with the triple-layer BCS. Conversely, the BQZS-AA with the triple-layer BCS, through the introduction of the triple-amplified damping coefficient, exhibits straightened resonance peak. These parameter variations significantly enhance its performance, in rems of resonance frequency, resonant amplitude, peak transmissibility, and isolation frequency band.

In terms of high-frequency field, the response amplitude and transmissibility of the BQZS-AA are higher than that of the BQZS-AI when θ_i (i = 1, 2, 3) reaches 65°or λ₁ decreases to 2. However, that does not mean a deterioration in the high-frequency performance of BQZS-AA with the triple-layer BCS. On the contrary, the high-frequency performance of the triple-layer BQZS-AA outperforms that of the double-layer system, encompassing both the double-layer and initial systems, exhibiting lower transmissibility.

6 Conclusions

A novel multi-layer bionic cobweb structure (BCS), integrated with QZS isolators, is proposed to improve isolation performance. The layer-by-layer amplification effect of the BCS enhances the attributes of the embedded isolation elements, augmenting negative stiffness and damping coefficient.

The effects of BCS geometrical parameters on amplified negative stiffness and damping are investigated for mutual validation with the dynamic behavior. Subsequently, dynamic equations for multi-layer bionic QZS isolation systems with BCS are formulated and solved. Dynamic analysis considering parameters affecting amplified stiffness and damping characteristics indicates the following conclusions. (I) the increments of the number of BCS layer, θ_i (i = 1, 2, 3), and the decrement of λ₁ can significantly amplify the negative stiffness and damping coefficient at the equilibrium position; (II) The triple-layer configuration can not only amplify the negative stiffness and damping coefficient but can also significantly weaken the right offset of the stiffness curves and mitigate the nonlinearities of the combined restoring force and damping coefficient. (III) The incorporation of negative stiffness and damping elements in the BCS mitigates resonant peaks and nonlinearities in the BQZS-AA, ensuring adequate negative stiffness for heavy-load applications while avoiding the hardening phenomenon observed in the BQZS-AI due to individual negative stiffness amplification. (IV) For the BCS with a fixed layer, augmenting θ_i (i = 1, 2, 3) and diminishing λ₁ can benefits the isolation performance of the BQZS-AA; (V) Under identical parameters, an increased number of structural layers significantly improves isolation performance, characterized by lower resonant frequency, reduced amplitude, and broader isolation bandwidth.

The BCS demonstrates the potential to balance high loading capacity and low-frequency isolation when incorporared into the QZS system at low cost. The amplification effect and BCS can enhance performance and be combined with other attenuation elements, such as inerters. This work pioneers the link between spider hunting and vibration isolation, identifying the cobweb as a biomimetic structure, and inspiring future research.

Appendices

Appendix A: The nondimensionalization of the stiffness and damping

The nondimensional forms of F_kni (i = 0, 1, 2, 3) and $\hat{K}_{ni}$ $\left( {K_{ni} L_s /\left( {k_l \delta_{20} } \right),\begin{array}{*{20}c} {\,} \\ \end{array} i = 0,1,2,3} \right)$ are illustrated in Eq. (A.1).

$$ \begin{gathered} \left\{ {\begin{array}{*{20}l} {\hat{F}_{kn0} = - 2\hat{y} - \left( {1 - \lambda_1 } \right)\hat{y}^3 ,\;\;\;\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\hat{F}_{kn{\text{i}}} = \hat{F}_{kni1} \hat{y} + \hat{F}_{kni2} \hat{y}^2 + \hat{F}_{kni3} \hat{y}^3 ,\;\;\;i = 1,2,3} \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill \\ {\hat{K}_{kn0} { = } - 2 - 3\left( {1 - \lambda_1 } \right)\hat{y}^2 ,\;\;\;\begin{array}{*{20}c} {\begin{array}{*{20}c} {\hat{K}_{ni} = \frac{{\partial \hat{F}_{kni} }}{{\partial \hat{y}}} = 3\hat{F}_{kni3} \left( {\hat{y} + \frac{{\hat{F}_{kni2} }}{{3\hat{F}_{kni3} }}} \right)^2 + \hat{F}_{kni1} - \frac{{\hat{F}_{kni2}^2 }}{{3\hat{F}_{kni3} }},\;\;\;i = 1,2,3} \\ \end{array} } \\ \end{array} } \hfill \\ \end{array} } \right. \hfill \\ \hfill \\ \end{gathered} $$

(A.1)

where $\hat{F}_{kni} = \frac{{F_{kni} }}{{k_l \delta_{20} }},\begin{array}{*{20}c} {\,} \\ \end{array} \hat{F}_{kni1} = \frac{{k_{ni1} L_s }}{{k_l \delta_{20} }},\begin{array}{*{20}c} {\,} \\ \end{array} \hat{F}_{kni2} = \frac{{k_{ni2} L_s^2 }}{{k_l \delta_{20} }},\begin{array}{*{20}c} {\,} \\ \end{array} \hat{F}_{kni3} = \frac{{k_{ni3} L_s^3 }}{{k_l \delta_{20} }},\begin{array}{*{20}c} {\,} \\ \end{array} i = 0,1,2,3$, $\hat{y}{ = }\frac{y}{L_s }$, $\lambda_1 = \frac{{L_s^{\,} }}{{\delta_{20} }}$, $\lambda_2 = \frac{k_p }{{k_l }}$, $\lambda_3 = \frac{{L_s^{\,} }}{L_1 }$, $\lambda_4 = \frac{{L_s^{\,} }}{L_2 }$, $\lambda_5 = \frac{{L_s^{\,} }}{L_3 }$.

F_ki (F_ki = F_ni + k_py, i = 0, 1, 2, 3) denote the combined restoring force of the initial QZS isolation system and the bionic QZS isolation systems with the i-layer BCS (i = 1,2,3). The nondimensional F_ki are denoted as $\hat{F}_{ki}$ $\left( {F_{ki} /k_l \delta_{20} ,\begin{array}{*{20}c} {\,} \\ \end{array} i = 0,1,2,3} \right)$, and the nondimensional combined stiffness of the initial QZS isolation system and the BQZS-BCSs are taken as $\hat{K}_i$($K_i L_s^{\,} /\left( {k_l \delta_{20} } \right),\begin{array}{*{20}c} {\,} \\ \end{array} i = 0,1,2,3$). $\hat{F}_{ki}$ and $\hat{K}_i$ are presented in Eq. (A.2).

$$ \left\{ {\begin{array}{*{20}l} {\hat{F}_{k0} = \lambda_1 \lambda_2 \hat{y} + \hat{F}_{kn0} { = (}\lambda_1 \lambda_2 - 2)\hat{y} - \left( {1 - \lambda_1 } \right)\hat{y}^3 ,\begin{array}{*{20}c} {\,} \\ \end{array} \hat{F}_{ki} = \lambda_1 \lambda_2 \hat{y}{ + }\hat{F}_{kni} = (\lambda_1 \lambda_2 { + }\hat{F}_{kni1} )\hat{y} + \hat{F}_{kni2} \hat{y}^2 + \hat{F}_{kni3} \hat{y}^3 ,\begin{array}{*{20}c} {\,} \\ \end{array} i = 1,2,3} \hfill \\ {\hat{K}_0 { = }\frac{{k_p L_s^{\,} }}{{k_l \delta_{20} }} - 2 - 3\left( {1 - \frac{{L_s^{\,} }}{{\delta_{20} }}} \right)\hat{y}^2 = \lambda_1 \lambda_2 - 2 - 3(1 - \lambda_1 )\hat{y}^2 ,\begin{array}{*{20}c} {\,} \\ \end{array} \begin{array}{*{20}c} {\hat{K}_i = \lambda_1 \lambda_2 { + }\hat{K}_{ni} = 3\hat{F}_{kni3} (\hat{y} + \frac{{\hat{F}_{kni2} }}{{3\hat{F}_{kni3} }})^2 + \lambda_1 \lambda_2 { + }\hat{F}_{kni1} - \frac{{\hat{F}_{kni2}^2 }}{{3\hat{F}_{kni3} }},\begin{array}{*{20}c} {\,} \\ \end{array} i = 1,2,3} \\ \end{array} } \hfill \\ \end{array} } \right.. $$

(A.2)

The nonlinearity of the isolation systems is defined as the ratio of the nonlinear and linear parts of the nondimensional combined restoring force. The nonlinearity of the initial QZS isolation system and the bionic QZS isolation systems with the i-layer BCS (i = 1,2,3) are exhibited as taken as η (i = 0,1,2,3), both of which are expressed in Eq. (A. 3).

$$ \eta { = }\left\{ {\begin{array}{*{20}l} {\left| { - \left( {1 - \lambda_1 } \right)\hat{y}^3 /\left[ {(\lambda_1 \lambda_2 - 2)\hat{y}} \right]} \right|,\;\;\;i = 0} \hfill \\ {\left| {\left( {\hat{F}_{kni2} \hat{y}^2 + \hat{F}_{kni3} \hat{y}^3 } \right)/\left[ {\left( {\lambda_1 \lambda_2 + \hat{F}_{kni1} } \right)\hat{y}} \right]} \right|,\;\;\;i = 1,2,3} \hfill \\ \end{array} } \right.. $$

(A.3)

The nondimensional f is processed as $\hat{f}_{ci}$ ($f_{ci} /c,\begin{array}{*{20}c} {\,} \\ \end{array} i = 0,1,2,3$). $\hat{f}_{c0}$ can be nondimensionalized as 1, and f_ci (i = 1,2,3) can be yielded as Eq. (A.4). Their nonlinearities are defined as the ratio of the respective nonlinear and linear parts, which are derived in Eq. (A.5).

$$ \hat{f}_{ci} = \hat{f}_{ci3} \hat{y}^3 { + }\hat{f}_{ci2} \hat{y}^2 { + }\hat{f}_{ci1} \hat{y} + \hat{f}_{ci0} ,\;\;\;i = 1,2,3, $$

(A.4)

where $\hat{f}_{ci3} = c_{i3} L_s^3 /c$, $\hat{f}_{ci2} = c_{i2} L_s^2 /c$, $\hat{f}_{ci1} = c_{i1} L_s /c$, $\hat{f}_{ci0} = c_{i0} /c$.

$$ \eta { = }\left| {\frac{{\hat{f}_{ci3} \hat{y}^3 + \hat{f}_{ci2} \hat{y}^2 }}{{\hat{f}_{ci1} \hat{y}}}} \right|,\;\;\;i = 1,2,3. $$

(A.5)

Appendix B: The influences of θ ₁, θ ₂, θ ₃, λ ₁ on the double-layer amplified stiffness

See Figs. 33, 34.

Fig. 33

The effects of parameters on the non-dimensional double-layer and single-layer amplified stiffnesses when λ₂ = 7 (k_p = 35e4N/m, k_l = 5e4N/m), λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, L₁ = 0.2m, L₂ = 0.4m), c = 80N⋅s/m, z₀ = 0.5e−3m. a and b the effect of θ₁, setting θ₁ = 50°,55°,60°,65°, θ₂ = 55°; c and d the effect of θ₂, setting θ₂ = 50°, 55°,60°,65°, θ₁ = 55°. e and f the effect of λ₁, setting λ₁ = 0.67, 1, 2, 3.3, 5 (δ₂₀ = 0.075m, 0.05m, 0.025m, 0.015m, 0.01m), θ₁ = 55°, θ₂ = 55°. a, c, and d, the negative stiffness; b, d, and f, the combined stiffness

Fig. 34

The effects of parameters on the nonlinearities of the non-dimensional combined restoring forces with double-layer and single-layer BCSs when λ₂ = 7 (k_p = 35e4N/m, k_l = 5e4N/m), λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, L₁ = 0.2m, L₂ = 0.4m), c = 80N⋅s/m, z₀ = 0.5e−3m. a The effect of θ₁, θ₁ = 50°, 55°, 60°, 65°, θ₂ = 55°; b The effect of θ₂, θ₂ = 50°, 55°, 60°, 65°, θ₁ = 55°. c The effect of λ₁, λ₁ = 0.67, 1, 2, 3.3, 5 (δ₂₀ = 0.075m, 0.05m, 0.025m, 0.015m, 0.01m), θ₁ = 55°, θ₂ = 55°

Appendix C: The influences of θ ₁, θ ₂, θ ₃, λ ₁ on the double-layer amplified damping coefficient

See Figs. 35, 36.

Fig. 35

The effects of parameters on the non-dimensional double-layer and one-layer damping coefficients when λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m), c = 80N⋅s/m, z₀ = 0.5e−3m. a The effects of θ₁, setting θ₁ = 50°,55°,60°,65°, θ₂ = 55°. b The effects of θ₁, setting θ₁ = 55°, θ₂ = 50°, 55°, 60°, 65°

Fig. 36

The effects of parameters on the nonlinearities of the non-dimensional double-layer and one-layer damping coefficients when λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m), c = 80N⋅s/m, z₀ = 0.5e−3m. a The effects of θ₁, setting θ₁ = 50°,55°,60°,65°, θ₂ = 55°. b The effects of θ₁, setting θ₁ = 55°, θ₂ = 50°, 55°, 60°, 65°

Appendix D: The dynamic behavior of the bionic QZS isolation system with the double-layer BCS

According to Sect. 4, "Appendices B, and C", the double-amplified negative stiffness and damping coefficient enhance with increasing θ₁, θ_2,, and decreasing λ₁. For BQZS-AI with identical BCS layers (Fig. 37), θ_i (i = 1,2) and smaller λ₁ result in a more severe resonance peak distortion and substantial degradation of the isolation performance when compared to the one-layer and initial systems. The simultaneous implementation of the amplified negative stiffness and damping coefficient endows the BQZS-AA with more straightened characteristic curves for the same set of parameters. The BQZS-AA exhibited an anti-resonant behavior with a narrower or even disappeared (θ_i = 65°) unstable region, a lower resonant frequency, a smaller resonant amplitude and peak transmissibility, and a narrower isolation frequency band than that of the one-layer and initial systems (Fig. 38). The isolation performance of the BQZS-AA gets improved with the increase of the number of BCS layer. Compared to BQZS-AI, the introduction of the double-amplified damping coefficient for the BQZS-AA has almost no effect on the high-frequency performance, except when the θ_i (i = 1, 2) changes to 65°. Nevertheless, the high-frequency transmissibility of the double-layer BQZS-AA gets reduced with the increase of θ_i (i = 1, 2) and the decrease of λ₁ in comparison with the initial and one-layer system.

Fig. 37

The effects of parameters on the BQZS-AA and BQZS-AI with the double-layer BCS. a and b describe the effects of θ₁ when λ₂ = 7 (k_p = 35e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m), θ₁ = 50°,55°,60°,65°, θ₂ = 55°,c = 80N⋅s/m, z₀ = 0.5e−3m; c and d describe the effects of θ₂ when λ₂ = 7 (k_p = 35e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m), θ₁ = 55°,θ₂ = 50°,55°,60°,65°, c = 80N⋅s/m, z₀ = 0.5e−3m; e and f describe the effect of λ₁ when λ₂ = 7 (k_p = 35e4N/m, k_l = 5e4N/m), λ₁ = 2, 3.3, 5 λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.025m, 0.015m, 0.01m, L₁ = 0.2m, L₂ = 0.4m), θ₁ = 55°, θ₂ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m; a, c, e represent the amplitude-frequency curves; b, d, f represent the transmissibility curves

Fig. 38

The effects of parameters on the resonant frequency and response amplitude of the BQZS-AA with the double-layer BCS. a Describes the effects of θ₁ when λ₂ = 4 (k_p = 20e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m), θ₁ ∈ (50°, 65°), c = 80N⋅s/m, z₀ = 0.5e−3m; b describes the effects of θ₂ when λ₂ = 7 (k_p = 35e4N/m, k_l = 5e4N/m), λ₁ = 3.3, λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ = 0.015m, L₁ = 0.2m, L₂ = 0.4m), θ₁ = 55°, θ₂ ∈ (50°, 65°), c = 80N⋅s/m, z₀ = 0.5e−3m; c describes the effects of λ₁ when λ₂ = 7 (k_p = 35e4N/m, k_l = 5e4N/m), λ₁ ∈ (2, 5), λ₃ = 0.25, λ₄ = 0.125 (L_s = 0.05m, δ₂₀ ∈ (0.01m, 0.025m), L₁ = 0.2m, L₂ = 0.4m), θ₁ = 55°, θ₂ = 55°, c = 80N⋅s/m, z₀ = 0.5e−3m