Abstract
A novel twostage quasizero stiffness (QZS) vibration isolator was proposed for the purpose of lowfrequency vibration isolation. Firstly, the dynamic model of the vibration isolation system was established; furthermore, the force transmissibility of the system under harmonic force excitation was derived by the averaging method; finally, the effects on the vibration isolation performance caused by excitation amplitude, mass ratio and damping ratio were discussed. Results show that, compared with the corresponding twostage linear system, twostage QZS system not only has better isolation performance, but also possesses a wider range of isolation frequency provided that the excitation amplitude, mass ratio and damping ratio is appropriate.
1. Introduction
Quasizero stiffness (QZS) isolator can obtain zero stiffness at the static equilibrium position by connecting a positive stiffness element in parallel with a negative stiffness element [1]. By reasonably selecting the geometry and stiffness parameters of the negative stiffness institutions, the vibration isolation system can support a large load statically while have lowfrequency vibration isolation performance. The combination of positive and negative stiffness device possesses the characteristics of highstaticlowdynamic stiffness and can ensure that the system natural frequency is very low at small deformation [2]. Therefore, scholars have conducted indepth research on isolator principle [3], structure design [4], properties analysis [5] and engineering application [6] of QZS, which is widely used in precision instrument isolation, bridge and house antiearthquake, highspeed vehicle vibration reduction, marine machinery noise cancellation and other fields.
With the development of science and technology, the request of stability for the work environment of precision equipment is more and more strictly. Conventional singlestage linear or QZS vibration isolation system cannot meet requirements of industry [7, 8]. When the excitation frequency is greater than $\sqrt{2}$ times the natural frequency, vibration decay ratio is proportional to ${\omega}^{2}$ in the singlestage system, while that is proportional to ${\omega}^{4}$ in the twostage system. So rigid twostage isolation system can replace soft singlestage isolation system, which not only has good isolation effect, but also takes into account the loadbearing capacity and stability. Although the twostage isolator is better isolation performance than the singlestage isolator, but the former has one more resonance peak and longer resonance time. To play the single QZS advantages, also take large vibration decay ratio of twostage isolation system into consideration. Twostage QZS vibration isolator was proposed, whose effects on the vibration isolation performance caused by excitation amplitude, mass ratio and damping ratio were studied.
2. Mechanical model of QZS vibration isolation system
Fig. 1 shows a typical QZS system. The system consists of a suspended mass $m$ with a vertical spring and two identical oblique springs.
The forcedeflection in the vertical direction is given by:
where ${l}_{0}$ is the free length of horizontal springs and $l$ is length in the horizontal position. $x$ is the displacement deviation from equilibrium position. The stiffness of oblique spring is ${k}_{h}$ and the stiffness of horizontal spring is ${k}_{h}$.
Eq. (1) can be written in nondimensional form as:
where $\widehat{x}=x/{x}_{0}$, $\widehat{l}=l/{l}_{0}$ and $\widehat{k}={k}_{h}/{k}_{v}$. Using the Maclaurinseries expansion to the third order for small $\widehat{x}$, then Eq. (2) approximates to:
where ${k}_{a}=12\widehat{k}(1\widehat{l})/\widehat{l}$ and ${k}_{b}=\widehat{k}(1{\widehat{l}}^{2})/{\widehat{l}}^{3}$.
Fig. 1Schematic diagram of the QZS vibration isolation system
Fig. 2Schematic of a twostage QZS system
3. Twostage QZS vibration isolation system
The twostage QZS system is shown in Fig. 2.
It is subject to external harmonic force excitation $F\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Omega}T$. The upper QZS isolator stiffness coefficient is ${K}_{1}$, ${K}_{2}$ and damping coefficient is ${c}_{1}$. The lower QZS isolator stiffness coefficient is ${K}_{3}$, ${K}_{4}$ and damping coefficient is ${c}_{2}$. ${z}_{1}$ and ${z}_{2}$ are the displacement of the vibration object and the middle inertia block when the respective spring in a natural state. ${m}_{1}$ and ${m}_{2}$ are the mass of vibration object and middle inertia block respectively.
Provided that only consider the vertical direction of movement, dynamic equation of twostage QZS system is established based on Newton’s second law:
For clarity of analysis, the parameters ${\mathrm{\Omega}}_{0}=\sqrt{{K}_{1}/{m}_{1}}$, ${z}_{1}={X}_{1}\sqrt{{K}_{1}/{K}_{2}}$, ${z}_{2}={X}_{2}\sqrt{{K}_{1}/{K}_{2}}$, $T={\mathrm{\Omega}}_{0}t$ are introduced. Eq. (4) can be written in nondimensional form as:
where ${\xi}_{1}={c}_{1}/\sqrt{{K}_{1}{m}_{1}}$, $f=F/{K}_{1}\sqrt{{K}_{2}/{K}_{1}}$, $G={m}_{1}g/{K}_{1}\sqrt{{K}_{2}/{K}_{1}}$, ${k}_{1}={K}_{3}/{K}_{1}$, ${k}_{2}={K}_{4}/{K}_{2}$, ${k}_{3}={K}_{2}/{K}_{1}$, ${\xi}_{2}={c}_{2}/\sqrt{{K}_{1}{m}_{1}}$, $w={m}_{2}/{m}_{1}$, $\omega ={\mathrm{\Omega}}_{0}\mathrm{\Omega}$.
In order to analyze conveniently, the first and second equation of Eq. (5) are added, which can make stiffness coupling change into inertia coupling. The gravity term in Eq. (5) can be eliminated by using coordinate transform. Introducing ${X}_{1}={Z}_{1}{h}_{1}$, ${X}_{2}={Z}_{2}{h}_{2}$, $H={h}_{1}{h}_{2}$, ${x}_{1}={Z}_{2}{Z}_{1}$, ${x}_{2}={Z}_{2}$, Eq. (5) can be transferred as:
where $H+{H}^{3}=G$, ${k}_{1}{h}_{2}+{k}_{2}{{h}_{2}}^{3}=(w+1)G$.
Eq. (6) in matrix form is:
where:
$\mathbf{F}=\left[\begin{array}{c}f\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)+(1+3{H}^{2}){x}_{1}+3H{{x}_{1}}^{2}+{{x}_{1}}^{3}\\ f\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)({k}_{1}+3{k}_{2}{{h}_{2}}^{2}){x}_{2}+3{k}_{2}{h}_{2}{{x}_{2}}^{2}{k}_{2}{{x}_{2}}^{3}\end{array}\right].$
By the averaging method, suppose steadystate response solution of the above system as:
where $\mathbf{U}={\left[{U}_{1},{U}_{2}\right]}^{T}$ and $\mathbf{V}={\left[{V}_{1},{V}_{2}\right]}^{T}$ change slowly with time. Differentiate the first equation of Eq. (8) with respect to$t$and eliminate the second equation of Eq. (8):
Differentiate the second equation of Eq. (8) with respect to$t$and take it into Eq. (7):
Based on Eqs. (9) and (10), it can be inferred as:
Note that, $\mathbf{U}$ and $\mathbf{V}$ are the slowly changing function of time. The right side of Eq. (11) can be approximately represented with the average value of $\left(\omega t\right)$ in a period. Provided that $\mathbf{U}$ and $\mathbf{V}$ remain unchangeable in a period, average equation can be obtained as:
According to the orthogonality of trigonometric function, Eq. (12) can be simplified as:
${Q}_{3}=\left(1+3{H}^{2}+\frac{3{V}_{1}^{2}}{4}\right){U}_{1}+\frac{3{U}_{1}^{3}}{4},\mathrm{}{Q}_{4}=\left({k}_{1}3{k}_{2}{{h}_{2}}^{2}\frac{3{k}_{2}{V}_{2}^{2}}{4}\right){U}_{2}\frac{3{U}_{2}^{3}}{4}.$
Then, the response of system is the solution of following equation:
The force transmitted to the base includes elastic restoring force and damping force of the lower vibration isolator, which can be expressed as:
Substitute Eq. (6) in Eq. (14) and neglect higher harmonics:
where:
${f}_{c}=\left({k}_{1}+3{k}_{2}{{h}_{2}}^{2}\right){V}_{2}\omega {\xi}_{2}{U}_{2}+\frac{3{k}_{2}{V}_{2}}{4}\left({{V}_{2}}^{3}+{{U}_{2}}^{2}\right),\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{f}_{d}=\frac{3{k}_{2}{h}_{2}\left({{U}_{2}}^{2}+{{V}_{2}}^{2}\right)}{2}.$
Force transmissibility of twostage QZS system is calculated as:
As a comparison, the two oblique springs are removed.
By successively applying the same procedure as for the model of equivalent twostage linear system, the response of the system is the solution of Eq. (15), where
Similarly, the force transmissibility of twostage linear system can be founded:
4. Influence of the system parameters on force transmissibility
As discussed above, force transmissibility is closely related to $f$, $w$ and $\xi $. Next, the effects of different system parameters on force transmissibility of twostage QZS system are investigated by controlling variables method. The Force transmissibility of twostage linear system at the same condition is also plotted together to compare the isolation performance of two systems. All the force transmissibility results are plotted in dB, i.e. as $20\mathrm{l}\mathrm{o}{\mathrm{g}}_{10}T$.
Fig. 3 shows the force transmissibility of system under different $f.$ Twostage linear system are not influenced by the $f$. However, for the twostage QZS system, two order resonant frequencies and transmissibility peaks increase with the rising of $f$. The less the $f$ is, the better the isolation performance of twostage QZS system compared with the twostage linear system. Fig. 4 shows the force transmissibility of system under different $w$, for the twostage linear system, two order resonant frequencies increase with the rising of $w$, the transmissibility of the first order resonant frequency decreases, while the transmissibility of the second order resonant frequency increases. For the twostage QZS system, the first order resonant frequency transmissibility decreases with the rising of $w$, while the second resonant order frequency and transmissibility increases with the rising of $w$. Which indicates that the isolation bandwidth broadens and isolation performance near the second order resonant frequency weakens with the rising of $w$.
The force transmissibility curves of system under different $\xi $ are illustrated in Fig. 5 and Fig. 6 respectively. Two order resonant frequencies of twostage linear system are not influenced by the ${\xi}_{1}$, while the peaks of corresponding force transmissibility reduce. Force transmissibility of system under different ${\xi}_{1}$ is shown in Fig. 5. The resonance branch of the twostage QZS system shortens and the peaks of corresponding force transmissibility reduce with the rising of ${\xi}_{1}$. But the isolation performance will decrease when the excessive damping ratio. Force transmissibility of system under different ${\xi}_{1}$ is shown in Fig. 6. Two order resonant frequencies increase with the rising of ${\xi}_{2}$. The transmissibility of the first order resonant frequency decreases, while the transmissibility of the second order resonant frequency increases, which indicates that the isolation performance near the second order resonant frequency weakens with the rising of ${\xi}_{2}$.
Fig. 3Force transmissibility under different f
Fig. 4Force transmissibility under different w
Fig. 5Force transmissibility under different ξ1
Fig. 6Force transmissibility under different ξ2
5. Conclusions
In this study, a novel twostage quasizero stiffness vibration isolator was presented. The conclusions were summarized as follows:
1) The dynamic models of twostage QZS and linear vibration isolation system were established. The force transmissibility under harmonic force excitation was derived by using the averaging method.
2) Decrease the excitation amplitude and increase the mass ratio as well as damping ratio properly, which can broaden the isolation bandwidth, increase the vibration decay ratio and enhance the isolation performance of twostage QZS system.
3) Compared to the corresponding twostage linear system, the twostage QZS system not only has smaller initial isolation frequency, wider isolation band and better isolation performance, but also possesses excellent loadbearing capacity and stability.
References

Zhu Tao, Cazzolato Benjamin, Robertson William S. P., et al. Vibration isolation using six degreeoffreedom quasizerostiffness magnetic levitation. Journal of Sound and Vibration, Vol. 358, 2015, p. 4873.

Zou Keguan, Nagarajaiah Satish Study of a piecewise linear dynamic system with negative and positive stiffness. Communications in Nonlinear Science and Numerical Simulation, Vol. 22, 2015, p. 10841101.

Ma Yanhui, He Minghua, Shen Wenhou, et al. A planar shock isolation system with highstaticlowdynamicstiffness characteristic based on cables. Journal of Sound and Vibration, Vol. 358, 2015, p. 267284.

Gatti Gianluca, Kovacic Ivana, M. J. Brennan. On the response of a harmonically excited two degreeoffreedom system consisting of a linear and a nonlinear quasizero stiffness oscillator. Journal of Sound and Vibration, Vol. 329, 2010, p. 18231835.

Lan ChaoChieh, Yang ShengAn, Wu YiSyuan Design and experiment of a compact quasizerostiffness isolator capable of a wide range of loads. Journal of Sound and Vibration, Vol. 333, 2014, p. 48434858.

Liu Xingtian, Huang Xiuchang, Hua Hongxing On the characteristics of a quasizero stiffness isolator using Euler buckled beam as negative stiffness corrector. Journal of Sound and Vibration, Vol. 332, 2013, p. 33593376.

Zhou Jiaxi, Wang Xinlong, Xu Daolin, et al. Nonlinear dynamic characteristics of quasizero stiffness vibration isolator with camrollerspring mechanisms. Journal of Sound and Vibration, Vol. 346, 2015, p. 5369.

Sun Xiuting, Jing Xingjian Multidirection vibration isolation with quasizero stiffness by employing geometrical nonlinearity. Mechanical System and Signal Processing, Vol. 62, Issue 63, 2015, p. 149163.
About this article
The authors gratefully acknowledge the support for this work by the National Natural Science Foundation of China (NSFC) under Grant No. 51579242 and No. 51509253.