Optimal Design and Dynamic Analysis of a New Quasi-Zero-Stiffness Isolation Device

. Compared with the linear isolation system, the quasi-zero-stifness (QZS) nonlinear isolation system has the characteristics of high static stifness and low dynamic stifness, which has better low-frequency vibration isolation performance. However, most of the existing QZS isolators only consider the quasi-zero-stifness characteristic at the static equilibrium position achieved by the parallel connection of positive and negative stifness structures. To optimize the isolation performance of the QZS system, a new isolation device based on the parallel connection of oblique springs and vertical springs was proposed. Te device can not only achieve quasi-zero-stifness at the static equilibrium position but also expand the interval of quasi-zero-stifness through parameter optimization design to optimize the stifness characteristics of the QZS system, thus efectively improving the vibration isolation performance. Te QZS nonlinear systems with the optimal parameters were analyzed dynamically, and the nonlinear motion equations were approximately solved based on the ffth-order polynomials ftted by the restoring force curves. A prototype was further designed and fabricated to compare and analyze the vibration isolation performance of the QZS system and the equivalent linear system through a shaking table test.

Carrella et al. and Kovacic et al. frst studied the quasizero-stifness system proposed by Molyneux [31] and simplifed the model into a third-order approximate Dufng equation. Tey further analyzed the static problems [32], force and displacement transmissibility [33,34], and its application in rotor isolation [35]. Te dynamic analysis is mainly based on the frst-order approximate solution obtained by the harmonic balance method. It is considered that the isolation performance of the QZS isolator can be refected only when the downward jump frequency of the QZS system is less than the corresponding resonance frequency of the linear system. Neild and Wagg [36][37][38] studied the amplitude-frequency response characteristics of the above QZS isolator through a ffth-order polynomial function. QZS systems are often limited in practical engineering applications, one of the important reasons being that they only form near the static equilibrium position region, which means that the optimal performance of the QZS system is limited to a small excitation amplitude. Considering that some vibration will produce a larger displacement response, how to achieve quasi-zero-stifness and expand the interval of quasi-zero-stifness to optimize the stifness characteristics of the system and efectively improve the vibration isolation performance has become a key problem to be solved [39][40][41].
Tis paper focuses on the optimal parameter design of the proposed device which uses quasi-zero-stifness around the static equilibrium position, by comparing the values of dynamic stifness in the entire compression stroke range of the device, and the innovations are described as follows: (1) In contrast to the conventional QZS system, a threespring QZS system was proposed by Molyneux, and the physical and geometric parameters of the proposed device can be adjusted and controlled through a slide-link connection system and mechanical assembly, allowing for more fexible adjustment and use in practical engineering. (2) Te proposed device can improve the vibration isolation performance by expanding the quasi-zerostifness range through parameter optimization design. Te nonlinear dynamics analysis shows that the proposed QZS system has a smaller vibration isolation starting frequency and a larger vibration isolation frequency range than the conventional QZS system and the equivalent linear system for vibration isolation, and the performance improves more signifcantly as the excitation amplitude decreases. (3) In this paper, a new QZS isolation device based on the parallel connection of oblique springs and vertical springs has been developed, fabricated, and tested by the authors. Te displacement transmissibility of the quasi-zero-stifness vibration isolation system and the equivalent linear vibration isolation system were compared and analyzed by the shaking table test, which verifed the good lowfrequency vibration isolation performance of the device.

Isolation Device Description and Static Analysis
Te device consists of three parts: the bearing plate and connecting rod; the vertical and oblique spring systems; and upper and bottom limit plates, vertical rails, and sliders. Te three main components are integrated through a central connection block, enabling the device to achieve quasi-zerostifness at the static equilibrium position through the parametric adjustment of the spring stifness characteristics, as shown in Figure 1. Te negative stifness system consists of eight symmetrically arranged precompressed inclined springs. By adjusting the distance between the upper and bottom limit plates or the distance between the four vertical rails, the precompression coefcients of the inclined spring can be easily adjusted, thus changing the stifness characteristics of the negative stifness system. Similarly, by replacing diferent vertical springs, the vertical stifness system can be controlled according to diferent needs. Ten the two systems can be connected in parallel to achieve the adjustable quasizero-stifness characteristics.
Te inclined springs of the device are supported internally by a piston rod to maintain stability in compression, a damping fuid can be added inside the piston rod, and the springs can be replaced by other stifness elements with better compressibility and elasticity to meet the actual needs in diferent projects. Te geometric parameters of the QZS device and the physical parameters of the spring system are shown in Figure 2, with the device in a static equilibrium position, which is the force state of the device under the design mass load.
Te isolation device introduces the following parameters: linear stifness coefcient k 2 for the vertical spring; linear stifness coefcient k 1 and softening cubic stifness coefcient k 3 for the precompressed inclined spring; precompression coefcient δ and vertical projection length h for the oblique spring system; the displacement parameter x of the isolated object from the static equilibrium position; and distance e of the upper and lower threaded caps from the central connection block.
Note that does not represent the original length of the spring, as the precompression δ is not zero. Te relationship between the vertical applied force f and the resulting displacement x can be given as follows. When where sgn(x) is the symbolic function and abs(x) is the absolute value function, further introducing the dimensionless parameter: α � k 1 /k 2 , and β � k 3 (a 2 + b 2 + c 2 )/k 2 , and the force expression of the nondimensional form can be derived as follows:

Structural Control and Health Monitoring
Te expressions of each of these simplifed parameters are as follows: ���������� � x 2 + 2hx + 1, ∆ 1 � (1 + δ)/P 1 and ∆ 2 � (1 + δ)/P 2 Te nondimensional stifness K can be further derived as follows: It can be seen from the Figure 2 that when e � 0, the upper and bottom compression stroke of the device is the vertical projection length h of the oblique spring, and the force analysis of the system when e ≠ 0 is not considered for the time being.
If K � 0 is specifed at x � 0 and e � 0, then the value of α and β can be obtained as follows: when β > 0, the oblique spring is soft, and when β < 0, the oblique spring is hard. In order to ensure that the oblique springs on both sides are not in a state of tension, the individual parameters of the springs also need to satisfy the following: Considering the small value of h 2 , further simplifcation is given in the following equation: Substituting equation (6) into equation (5) and making α > 0 and β > 0 lead to 0 < h < 0.577. Terefore, when considering the precompression of the oblique spring system, the value of h is best chosen between 0 and 0.5 in order to ensure that the mechanical properties of the QZS isolation system are meaningful.

Optimum Parameter Design of the QZS Device
In order to optimize and extend the quasi-zero-stifness interval of the device, it is necessary to keep the dimensionless stifness K-surface as close to the zero axis as possible at the static equilibrium position (x � 0), while ensuring the positive stifness characteristics of the device over the entire range of compression strokes. When e � 0, bringing equation (5) into equation (3) yields (7) Figure 3 shows the three-dimensional surface diagram of the dimensionless stifness K with parameters x and δ for h � 0.2 and α � 1, as well as the K − x diagram and the K − δ diagram for diferent viewpoints. It can be seen from Figure 3 that the abrupt change in the value of dimensionless stifness K is at δ � 3h 2 /(1 − h 2 ) and δ � 0. Tis is due to the fact that the conditions δ ≠ 3h 2 /(1 − h 2 ) and δ ≠ 0 need to be satisfed in equation (5); otherwise, the expression is meaningless.
Considering that the stifness ratio α needs to be greater than 0, a three-dimensional surface plot of the stifness ratio α with parameters β and δ can be made for h � 0.1, as well as α − β diagram and α − δ diagram from diferent viewpoints, as shown in Figure 4. It can be seen that the sudden change of the stifness ratio α is   taken in practical engineering applications and do not imply that the stifness ratio α is greater than 0 only when β and δ are taken in this interval. Figures 5 and 6 show the α − β view and α − δ view when h � 0.2 ∼ 0.5. It can be seen that the stifness ratio α of the QZS device is greater than 0 for h � 0.
From Figures 4 to 6, it can be seen that when the precompression factor δ > h 2 /(1 − h 2 ) and the softening cubic stifness factor 0 < β < 4, the stifness ratio α > 0 can meet the needs of the actual project.
According to the selection range of parameters, the three-dimensional surface plots of its dimensionless stifness K with parameters x and δ for diferent values h � 0.1, 14,5]; and h � 0.4, δ∈ [0. 22,5] when the softening cubic stifness coefcient β � 0.05, 0.25, 0.5, and 0.75 are shown in Figure 7. It can be seen that for diferent values of β and h, the concavity and convexity of the 3D surface are transformed at a certain value of δ. When the value of δ is in the small range, the dimensionless stifness K is a 3D concave surface, and when the value of δ is in the large range, the dimensionless stifness K is a 3D convex surface.
To see this property more clearly, the 3D surface plots of the dimensionless stifness K with parameters x and δ are made by taking β � 0.5, h � 0.1, and δ∈ [0.04, 5], as well as the K − x diagram and the K − δ diagram views from different viewpoints, as shown in Figure 8. It can be seen from Figures 7 and 8 that the dimensionless stifness K surface produces a shift in concavity at a critical value of δ. Terefore, the critical parameter δ value is found by solving By deriving equation (7), the frst-order derivative K ′ and the second-order derivative K ″ can be further derived as follows: By equation (9), the equation of K ″ (0) � 0 can be obtained as equation (11), and the corresponding curves of the critical parameters δ and h when K ″ (0) � 0, β � 0.05 can be made, as shown in Figure 9. It can be seen that when the parameter h is small, the change of the critical parameter δ is also small, and when the parameter h is large, the change of the critical parameter δ also increases, especially when h > 0.3, the variation of the critical parameter δ increases sharply. For practical use, considering the limited compression coefcient δ of the spring, a smaller value of h is chosen in the optimal parameter selection.
From the correspondence between the values of the critical parameter δ and h in Figure 10, the relationship between the nonlinear coefcients χ and the critical parameter δ and h can be further obtained, as shown in Figure 10. It can be seen that for diferent values of β and critical parameters δ and h, the nonlinear coefcients χ show increasing and then decreasing characteristics, and when δ and h are larger, the nonlinear coefcients χ are negative, which is not allowed to occur in practical applications.
When K ″ (0) � 0, a further derivative of equation (10) is required in order to perform a Taylor series expansion of the dimensionless restoring force f. Te third-order derivative K ‴ and fourth-order derivative K ⁗ and K ⁗ (0) can be further derived, as shown in equations (12)- (14). Further combining equations (11) and (14), the dimensionless restoring force f at x � 0 can be expanded in a ffth-order Taylor series, as shown in equation (15). In summary, when considering diferent softening cubic stifness coefcients β, we can obtain four diferent sets of optimal parameters for the QZS device as follows: β � 0.05, α � 0.542, h � 0.108, δ � 2.66, χ � 0.1640; β � 0. 25 Figure 11 shows the comparison curves between the exact expression of the dimensionless restoring force f and the ffthorder Taylor series expansion for four diferent sets of optimization parameters. It can be seen that the ffth-order Taylor series expansion fts poorly at the end of the compression stroke of the device, and the error between the approximate expression and the exact expression increases with the increase of the displacement x of the QZS system. When β � 0.05, the ffth-order Taylor series expansion fts best, and along with the increase of β, the ffth-order expansion fts gradually worse, and when the dimensionless displacement x is less than 0.05, it fts well for diferent values of β.

Force and Displacement Transmissibility.
Assuming that the device has viscous damping in the vertical direction, when a simple harmonic force excitation is applied to the isolated object or a displacement excitation is applied to the foundation, the equation of motion can be derived as
Based on equations (18) and (20), the force transmissibility T F of the QZS system and the equivalent linear system is plotted for the four diferent sets of optimal parameters, as shown in Figure 13. As can be seen from Figure 13, compared to the equivalent linear system, the  quasi-zero-stifness system with optimal parameters has a wider isolation frequency range and a lower amplitude of vibration, which can isolate lower-frequency vibrations. For the QZS system, as the excitation amplitude F decreases or the nonlinear coefcients χ decreases, the maximum value of the force transmissibility and the corresponding resonant frequency decrease, resulting in better isolation performance compared to the equivalent linear system.
As shown in Figure 14, the force transmissibility of the proposed QZS system and the corresponding conventional QZS system are plotted for two amplitudes of force excitation based on equations (20) and (21). It is easy to fnd that the proposed QZS system has a lower amplitude, a smaller isolation start frequency, and a larger isolation frequency range compared with the conventional QZS system and equivalent linear system, and the isolation performance becomes better as the amplitude of force excitation decreases.
Based on equation (22), the relative transmissibility T Z of the QZS system and the equivalent linear system is plotted for two diferent sets of parameters, as shown in Figure 15(a). One set of parameters is the optimal parameters when the nonlinear coefcient χ � 0.1640, and the other one is the one for the QZS system when h � 0.1 without considering the nonlinear coefcient β. It is worth noting from Figure 15(a) that when the nonlinear coefcient χ � 0.1640, the maximum value of relative transmissibility under harmonic excitation conditions disappears, which gives the QZS system with optimal parameters better low-frequency vibration isolation performance than the QZS system without considering the nonlinear coefcient β and the equivalent linear system. Figure 15(b) shows a comparison of the relative displacement transferability between the proposed QZS system and the corresponding conventional QZS system under the same damping and excitation, Z e � 0.0449 and β � 0. It can be seen that the proposed QZS system has a better vibration isolation performance than the conventional QZS system due to the construction of the device with two sets of symmetrically arranged precompressed tilting springs, which further optimizes the mechanical properties of the QZS system.

Dynamic Experiments of the QZS Device
Although considering the nonlinear coefcient of the tilted precompression springs can optimize the stifness characteristics of the QZS system and thus improve the vibration isolation, the nonlinear springs are not readily available on the market, so we frst manufactured the proposed device using linear springs to test and evaluate the vibration isolation performance of the QZS system, as shown in Figure 16(a). Figure 16(b) shows the device to remove the negative stifness system and simulate the equivalent linear system with positive stifness springs.     Te root-mean-square value of the obtained time-domain displacement signal is used as the corresponding excitation amplitude and response amplitude, and the absolute displacement transmittance expressed in decibels (dB) is further obtained by the ratio of the two, as shown in Figure 18(a). Te stifness of the equivalent linear system in the experiment is k 2 � 18572 N/m, the load mass is 7.5 kg, and the resonant frequency can be obtained by ω 0 � (k 2 /m) 0.5 /2π � 7.92 Hz. Based on the maximum value of the experimental displacement transmissibility of 19.95 dB, the damping ratio of the linear system can be obtained as ξ � 0.045, and the theoretical  displacement transmissibility of the equivalent linear system is further obtained, as shown in Figure 18(b).
Te dimensionless vertical force in the dynamical analysis of the proposed QZS system uses f � 2.051x 3 , which is obtained by ftting the restoring force curve under the experimental parameters. Te experimental displacement excitation amplitude of 0.4 mm is dimensionlessly treated as Z e � 0.00148, and the theoretical displacement transmissibility of the QZS system can be further obtained, as shown in Figure 18(b).
When the harmonic excitation amplitude is 0.4 mm, the absolute displacement transmissibility of the QZS system has no peak value compared to the equivalent linear system and has a smaller vibration isolation starting frequency and a larger vibration isolation frequency range. Te absolute displacement transmissibility shown in Figure 18 demonstrated the vibration isolation advantage of the proposed QZS system with respect to the equivalent linear system.

Conclusions
A new isolation device with spring systems has been proposed to improve vibration isolation performance by achieving quasi-zero-stifness characteristics at the static equilibrium position and enlarge the QZS range through the optimal parameter design. Te main conclusions are as follows: (1) Te static characteristics of the isolation device were theoretically studied for the proposed QZS system, and four sets of optimal stifness curves are obtained by considering the nonlinear softening coefcient of the preloaded springs. Tese optimal stifness curves can enlarge the QZS range and have lower dynamic stifness in the entire compression stroke range. (2) By analyzing the force and displacement transmissibility, the results show that (a) the QZS system with optimal parameters has a wider isolation frequency range and lower amplitude of vibration compared to the equivalent linear system and (b) the maximum value of relative transmissibility under harmonic excitation conditions disappears, which gives the QZS system with optimal parameters better low-frequency vibration isolation performance than the QZS system without considering the nonlinear coefcient and the equivalent linear system. (3) Te shaking table test demonstrated the vibration isolation advantage of the proposed QZS system with respect to the equivalent linear system by comparing and analyzing the experimental and theoretical absolute displacement transmissibility.

Data Availability
All data, models, or codes that support the fndings of this study are available from the corresponding author upon reasonable request.