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

Liu, Tao; Li, Aiqun; Zhang, Hengyuan

doi:https://doi.org/10.1155/2023/9756226

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Research Article | Open Access

Volume 2023 | Article ID 9756226 | https://doi.org/10.1155/2023/9756226

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

Tao Liu,¹Aiqun Li,¹and Hengyuan Zhang¹

Academic Editor: Łukasz Jankowski

Received04 Nov 2022

Revised09 Apr 2023

Accepted28 Jun 2023

Published18 Jul 2023

Abstract

Compared with the linear isolation system, the quasi-zero-stiffness (QZS) nonlinear isolation system has the characteristics of high static stiffness and low dynamic stiffness, which has better low-frequency vibration isolation performance. However, most of the existing QZS isolators only consider the quasi-zero-stiffness characteristic at the static equilibrium position achieved by the parallel connection of positive and negative stiffness 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. The device can not only achieve quasi-zero-stiffness at the static equilibrium position but also expand the interval of quasi-zero-stiffness through parameter optimization design to optimize the stiffness characteristics of the QZS system, thus effectively improving the vibration isolation performance. The QZS nonlinear systems with the optimal parameters were analyzed dynamically, and the nonlinear motion equations were approximately solved based on the fifth-order polynomials fitted 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.

1. Introduction

A quasi-zero-stiffness isolation system is generally composed of the positive stiffness mechanism for bearing the main load and the negative stiffness mechanism for offsetting the positive stiffness around the static equilibrium position. The high-static-low-dynamic stiffness characteristics make the QZS system have a wider isolation frequency band and better isolation performance than the linear stiffness system [1]. A great deal of research and applications have been carried out on QZS isolators with different structures, such as multispring structures [2–5], special spring structures [6–9], origami structures [10–12], buckling beam structures [13–16], geometric nonlinear structures [17–19], magnetic structures [20–22], bioinspired structures [23–25], and other types of structures [26–30]. All these QZS systems can be effectively used in low-frequency vibration isolation.

Carrella et al. and Kovacic et al. first studied the quasi-zero-stiffness system proposed by Molyneux [31] and simplified the model into a third-order approximate Duffing equation. They further analyzed the static problems [32], force and displacement transmissibility [33, 34], and its application in rotor isolation [35]. The dynamic analysis is mainly based on the first-order approximate solution obtained by the harmonic balance method. It is considered that the isolation performance of the QZS isolator can be reflected only when the downward jump frequency of the QZS system is less than the corresponding resonance frequency of the linear system. Neild and Wagg [36–38] studied the amplitude-frequency response characteristics of the above QZS isolator through a fifth-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-stiffness and expand the interval of quasi-zero-stiffness to optimize the stiffness characteristics of the system and effectively improve the vibration isolation performance has become a key problem to be solved [39–41].

This paper focuses on the optimal parameter design of the proposed device which uses quasi-zero-stiffness around the static equilibrium position, by comparing the values of dynamic stiffness 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 three-spring 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 flexible adjustment and use in practical engineering.(2)The proposed device can improve the vibration isolation performance by expanding the quasi-zero-stiffness range through parameter optimization design. The 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 significantly 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. The displacement transmissibility of the quasi-zero-stiffness vibration isolation system and the equivalent linear vibration isolation system were compared and analyzed by the shaking table test, which verified the good low-frequency vibration isolation performance of the device.

2. Isolation Device Description and Static Analysis

The 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. The three main components are integrated through a central connection block, enabling the device to achieve quasi-zero-stiffness at the static equilibrium position through the parametric adjustment of the spring stiffness characteristics, as shown in Figure 1.

The negative stiffness 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 coefficients of the inclined spring can be easily adjusted, thus changing the stiffness characteristics of the negative stiffness system. Similarly, by replacing different vertical springs, the vertical stiffness system can be controlled according to different needs. Then the two systems can be connected in parallel to achieve the adjustable quasi-zero-stiffness characteristics.

The inclined springs of the device are supported internally by a piston rod to maintain stability in compression, a damping fluid can be added inside the piston rod, and the springs can be replaced by other stiffness elements with better compressibility and elasticity to meet the actual needs in different projects. The 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.

The isolation device introduces the following parameters: linear stiffness coefficient for the vertical spring; linear stiffness coefficient and softening cubic stiffness coefficient for the precompressed inclined spring; precompression coefficient and vertical projection length for the oblique spring system; the displacement parameter of the isolated object from the static equilibrium position; and distance 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 is not zero. The relationship between the vertical applied force and the resulting displacement can be given as follows.

When where is the symbolic function and is the absolute value function, further introducing the dimensionless parameter: , , , , , and , and the force expression of the nondimensional form can be derived as follows:

The expressions of each of these simplified parameters are as follows: and

The nondimensional stiffness can be further derived as follows:

It can be seen from the Figure 2 that when , the upper and bottom compression stroke of the device is the vertical projection length of the oblique spring, and the force analysis of the system when is not considered for the time being.

If is specified at and , then the value of and can be obtained as follows:when , the oblique spring is soft, and when , 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 , further simplification is given in the following equation:

Substituting equation (6) into equation (5) and making and lead to . Therefore, when considering the precompression of the oblique spring system, the value of is best chosen between 0 and 0.5 in order to ensure that the mechanical properties of the QZS isolation system are meaningful.

3. Optimum Parameter Design of the QZS Device

In order to optimize and extend the quasi-zero-stiffness interval of the device, it is necessary to keep the dimensionless stiffness -surface as close to the zero axis as possible at the static equilibrium position (), while ensuring the positive stiffness characteristics of the device over the entire range of compression strokes. When , bringing equation (5) into equation (3) yields

Figure 3 shows the three-dimensional surface diagram of the dimensionless stiffness with parameters and for and , as well as the − diagram and the − diagram for different viewpoints. It can be seen from Figure 3 that the abrupt change in the value of dimensionless stiffness is at and . This is due to the fact that the conditions and need to be satisfied in equation (5); otherwise, the expression is meaningless.

Considering that the stiffness ratio needs to be greater than 0, a three-dimensional surface plot of the stiffness ratio with parameters and can be made for , as well as − diagram and − diagram from different viewpoints, as shown in Figure 4. It can be seen that the sudden change of the stiffness ratio is at . When and , the stiffness ratio of the QZS device is greater than 0. Note that the values of and are taken only in consideration of the parameters that may be taken in practical engineering applications and do not imply that the stiffness ratio is greater than 0 only when and are taken in this interval.

Figures 5 and 6 show the − view and − view when . It can be seen that the stiffness ratio of the QZS device is greater than 0 for , , and for , , . From Figures 4 to 6, it can be seen that when the precompression factor and the softening cubic stiffness factor , the stiffness ratio can meet the needs of the actual project.

According to the selection range of parameters, the three-dimensional surface plots of its dimensionless stiffness with parameters and for different values , ; , ; , ; and , when the softening cubic stiffness coefficient , 0.25, 0.5, and 0.75 are shown in Figure 7. It can be seen that for different values of and , the concavity and convexity of the 3D surface are transformed at a certain value of . When the value of is in the small range, the dimensionless stiffness is a 3D concave surface, and when the value of is in the large range, the dimensionless stiffness is a 3D convex surface.

To see this property more clearly, the 3D surface plots of the dimensionless stiffness with parameters and are made by taking , , and , as well as the − diagram and the − diagram views from different viewpoints, as shown in Figure 8. It can be seen from Figures 7 and 8 that the dimensionless stiffness surface produces a shift in concavity at a critical value of . Therefore, the critical parameter value is found by solving for .

By deriving equation (7), the first-order derivative and the second-order derivative can be further derived as follows:

By equation (9), the equation of can be obtained as equation (11), and the corresponding curves of the critical parameters and when , can be made, as shown in Figure 9. It can be seen that when the parameter is small, the change of the critical parameter is also small, and when the parameter is large, the change of the critical parameter also increases, especially when , the variation of the critical parameter increases sharply. For practical use, considering the limited compression coefficient of the spring, a smaller value of is chosen in the optimal parameter selection.

From the correspondence between the values of the critical parameter and in Figure 10, the relationship between the nonlinear coefficients and the critical parameter and can be further obtained, as shown in Figure 10. It can be seen that for different values of and critical parameters and , the nonlinear coefficients show increasing and then decreasing characteristics, and when and are larger, the nonlinear coefficients are negative, which is not allowed to occur in practical applications.

(a)

(b)

(c)

(d)

When , a further derivative of equation (10) is required in order to perform a Taylor series expansion of the dimensionless restoring force . The third-order derivative and fourth-order derivative and can be further derived, as shown in equations (12)–(14). Further combining equations (11) and (14), the dimensionless restoring force at can be expanded in a fifth-order Taylor series, as shown in equation (15).

In summary, when considering different softening cubic stiffness coefficients , we can obtain four different sets of optimal parameters for the QZS device as follows: , , , ; , , , , ; , , , , and ; and , , , , and .

4. Dynamic Analysis of the QZS Device under Optimal Parameter

4.1. Comparison of Exact and Approximate Expressions

Figure 11 shows the comparison curves between the exact expression of the dimensionless restoring force and the fifth-order Taylor series expansion for four different sets of optimization parameters. It can be seen that the fifth-order Taylor series expansion fits 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 of the QZS system. When , the fifth-order Taylor series expansion fits best, and along with the increase of , the fifth-order expansion fits gradually worse, and when the dimensionless displacement is less than 0.05, it fits well for different values of .

(a)

(b)

(c)

(d)

4.2. 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

By dimensionless processing: , , , , , , , and , the equations can be further derived as

Assuming the so-called T-periodic solutions and , then applying the harmonic balance method to equations (16) and (17), the following equations can be derived as

Note that the dimensionless force transmitted to the isolated structure by the QZS system is . Assuming and and expanding into Fourier series, it can be obtained as . Combined with equation (19), the force transmissibility can be further deduced as

The conventional QZS system consists of a vertical spring connected at point O with two oblique springs, as shown in Figure 12 [33, 42]. The force transmissibility of conventional QZS system was given in equation (21) where was used at . When considering , we can also obtain a set of optimal parameter for the QZS device from equation (2) as follows: , , , and

Similarly combining equation (19), the relative displacement transmissibility can be further derived as

The relative displacement transmissibility of conventional QZS system was given in equation (23) where was used at [42].

Based on equations (18) and (20), the force transmissibility of the QZS system and the equivalent linear system is plotted for the four different 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-stiffness 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 decreases or the nonlinear coefficients 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 find 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 of the QZS system and the equivalent linear system is plotted for two different sets of parameters, as shown in Figure 15(a). One set of parameters is the optimal parameters when the nonlinear coefficient , and the other one is the one for the QZS system when without considering the nonlinear coefficient . It is worth noting from Figure 15(a) that when the nonlinear coefficient , 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 coefficient and the equivalent linear system.

(a)

(b)

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, and . 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.

5. Dynamic Experiments of the QZS Device

Although considering the nonlinear coefficient of the tilted precompression springs can optimize the stiffness characteristics of the QZS system and thus improve the vibration isolation, the nonlinear springs are not readily available on the market, so we first 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 stiffness system and simulate the equivalent linear system with positive stiffness springs.

(a)

(b)

The Frame diagram of the shaking table experiment platform is shown in Figure 17. The accelerometers are arranged on the shaking table and the isolated object, and the picked up acceleration signal is collected by the INV-3602A signal collector and subsequently processed by the Coinv DASP V10 software. To compare the vibration isolation performance of the QZS system and the equivalent linear system, displacement excitations were set in the frequency ranges of 3 Hz, 4 Hz, 5 Hz, 6 Hz, 8 Hz, 10 Hz, 12 Hz, 15 Hz, 17 Hz, 21 Hz, and 25 Hz, all with an excitation amplitude of 0.4 mm and a sampling frequency of 1024 Hz.

The 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). The stiffness of the equivalent linear system in the experiment is N/m, the load mass is 7.5 kg, and the resonant frequency can be obtained by 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 , and the theoretical displacement transmissibility of the equivalent linear system is further obtained, as shown in Figure 18(b).

(a)

(b)

The dimensionless vertical force in the dynamical analysis of the proposed QZS system uses , which is obtained by fitting the restoring force curve under the experimental parameters. The experimental displacement excitation amplitude of 0.4 mm is dimensionlessly treated as , 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. The absolute displacement transmissibility shown in Figure 18 demonstrated the vibration isolation advantage of the proposed QZS system with respect to the equivalent linear system.

6. Conclusions

A new isolation device with spring systems has been proposed to improve vibration isolation performance by achieving quasi-zero-stiffness characteristics at the static equilibrium position and enlarge the QZS range through the optimal parameter design. The main conclusions are as follows:(1)The static characteristics of the isolation device were theoretically studied for the proposed QZS system, and four sets of optimal stiffness curves are obtained by considering the nonlinear softening coefficient of the preloaded springs. These optimal stiffness curves can enlarge the QZS range and have lower dynamic stiffness 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 coefficient and the equivalent linear system.(3)The 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 findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

All authors have contributed to the creation of this manuscript for important intellectual content and read and approved the final manuscript.

Acknowledgments

This research was financially supported by the authors.

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