Effective Frequency Range and Jump Behavior of Horizontal Quasi-Zero Stiffness Isolator

Abstract

The quasi-zero stiffness (QZS) isolator shows excellent characteristics of low-frequency vibration isolation. However, the jump behavior caused by the strong nonlinearity is a primary reason for the failure of QZS isolators. In order to grasp the effective frequency range and failure mechanism of a horizontal QZS isolator comprehensively, the dynamics of the isolator were studied in the following two cases. In the first case, the isolator is subject to a base displacement excitation; in the second case, the isolator is installed on a linear structure that is subject to a harmonic force. The nonlinear algebraic equations describing the steady-state response of the two systems were derived via the complexification-averaging method, and the results obtained using the derived expressions were verified by comparing the results of the complexification-averaging method and the Runge–Kutta method. The effective frequency ranges of the isolator were then obtained, and the jump phenomena in the response amplitude induced by the strong nonlinearity of the isolator were analyzed. The results show that when the excitation amplitude is small, the vibration isolation system does not exhibit jumping behavior and the effective frequency range is relatively wide. With increases in the excitation amplitude, the system can exhibit jumping behavior when an additional impact load is considered, and this phenomenon leads to a narrowing of the effective frequency range. The characteristics of the jump phenomena produced in the two cases were analyzed, and the differences in the jump behaviors were elucidated. Furthermore, the effect of the isolator parameters on the effective frequency range was investigated.

1. Introduction

Vibrations are harmful in most engineering applications. Restricting the transmission of vibrations via the installation of vibration isolators is an effective method to reduce the harm induced by vibrations. Linear isolators generally have an extremely small stiffness for reducing the lowest effective isolation frequency and have relatively large damping for suppressing the resonance peak. Therefore, the loading capacity of the linear isolators is generally low and the displacement transmissibility in the high-frequency band is high [1]. The application of nonlinearity permits a wide range of novel design concepts in engineering; nonlinear design underpins the concept of quasi-zero stiffness (QZS) within the field of vibration isolation, which expands the effective isolation frequency range and ensures the isolator has a sufficient load capacity [2].

Although the QZS isolator was first proposed over decades ago, there is still considerable room for improvements in its design, and its potential for widespread use thus attracts considerable research attention. Smooth and discontinuous oscillators are widely used in the design of QZS elements and have become a basic component in the field [3]. Xu et al. [4] introduced a permanent magnet spring in conjunction with a vertical linear spring and then formed an isolator. Qiang et al. [5] proposed a compact QZS isolator using repulsive magnet rings and a space-saving wave spring. Zhou et al. [6] designed a QZS isolator utilizing cam–roller–spring mechanisms. Liu et al. [7] used buckled, parallel Euler beams and linear springs to construct a QZS vibration isolator. Yu et al. [8] proposed a QZS isolator using a torsion bar spring and negative stiffness structure. Sun et al. [9] proposed a QZS isolator composed of a cantilever beam and parallel magnets. Ding et al. [10] designed a variable constant force configuration that can be used to provide the QZS characteristic in an isolator. Hao et al. [11] studied an isolator system comprising an isolator with high-static low-dynamic stiffness (HSLDS) characteristics and a flexible plate with an arbitrary boundary. Carrella et al. [12] analyzed the force and displacement transmissibility of an HSLDS isolator. Yang et al. [13] designed a nonlinear mechanical oscillator using the high-order QZS mechanism; the proposed oscillator has potential applications in vibration isolation as well as energy harvesting. Ding et al. [14] studied the vibration isolation performance of parallel QZS isolators attached to fluid-conveying pipes. Palomares et al. [15] designed a negative stiffness system to isolate vibrations and then improve the comfort of vehicle seats. In addition, electromagnetic mechanisms have been widely used in the design of active vibration isolation devices [16].

The X-shaped structure is a novel component in the design of vibration isolators. Xiong et al. [17] designed a QZS isolator by connecting two X-shaped structures to a linear vibration isolation platform. Sun et al. [18] studied the vibration isolation of a platform using an n-layer, scissor-like truss structure. Mao et al. [19] proposed a novel X-shape QZS isolator, in which oblique springs and a cam–roller–spring mechanism were used to enhance its isolation performance. Wang et al. [20] proposed an n-layer, vertically asymmetric, X-shaped structure for passive vibration isolation. Bioinspired designs provide an effective route to the realization of QZS mechanisms, and X-shaped structures belong to this category of structures [21]. Metamaterials such as origami and auxetic structures provide a large degree of tailorability and can be used to create structures or materials with extraordinary mechanical properties; such designs can be introduced to the field of vibration isolation [22,23,24,25]. Cellular structures with negative stiffness components and origami-based structures with bi- and multi-stable states can be used to create QZS isolators [26,27,28]. Furthermore, tensegrity structure [29] and symmetrical structure [30] are also potential base units for isolators because of their designability. The QZS characteristic can also be realized by the other various methods and its applications are not limited to constructing an isolator. Ding et al. [31] designed a constant force structure which is also a QZS structure by using the folding beam and bi-stable beam, and the novel structure can be used for polishing and deburring.

However, the strong nonlinearity of the isolator leads to a jump phenomenon in systems under a certain external energy input. When this phenomenon occurs, the amplitude of the system increases by a large factor in a short time; this means that the isolator will lose efficacy and even amplify the input amplitude. Motivated by the fact that horizontal QZS isolators are widely used to protect large-scale electronic equipment and valuable relics, and thus the failure of this component is likely to lead to considerable economic losses, in this study the variation of the effective frequency range and the failure induced by jump phenomena of the horizontal QZS isolator was studied; the cases where the isolator is excited by a base displacement excitation and installed on a linear structure subject to harmonic forces were particularly focused on. In this work, the jump processes were simulated by exerting an additional impact load on the harmonic excited vibration isolation systems. The effective frequency ranges that are dependent on the isolator characteristics were analyzed, and the influences of the jump phenomenon on the effective frequency ranges were established.

2. Isolator Subjected to Base Excitation

2.1. Dynamic Model

The model of the horizontal QZS isolator subject to a base displacement excitation is shown in Figure 1. The system is composed of a mass, a nonlinear spring, and a damping element. The QZS property of the system is achieved by two opposing linear springs, as shown in Figure 2. The mass element comprises the isolator platform and the isolated structure, which can translate along the guide rail. The dynamic equation describing the vibration isolation system can be expressed as,

$$M\ddot{x}+\mu \left(\dot{x}-P\mathsf{\Omega }\cos \mathsf{\Omega }t\right)+k_1\left(x-P\sin \mathsf{\Omega }t\right)+k_2{\left(x-P\sin \mathsf{\Omega }t\right)}^3=0,$$

(1)

where $k_1=2K-\frac{2KL}{l}$ and $k_2=\frac{KL}{l^3}$ [32,33]. $k_{\mathrm{qzs}}$ represents the stiffness of the isolator, $M$ is the total mass of the isolator platform and the isolated structure, $x$ is the displacement of the mass element, $K$ is the stiffness of the linear springs, $L$ is the original length of the springs, $l$ is the distance between the fixed hinge support and the mass element, $\mu$ is the damping coefficient, $P$ is the input displacement amplitude, and $\mathsf{\Omega }$ is the excitation frequency.

In this study, a semi-analytical method and a numerical method were applied. The semi-analytical method was mainly used to study the effective frequency range and the numerical method was used to study the jump behaviors. The results obtained from the two methods were compared for mutual verification.

For obtaining the first-order equation that describes the system, the complexification-averaging method [34,35] was adopted. The transformation $\dot{x}+i\mathsf{\Omega }x=\alpha e^{i\mathsf{\Omega }t}$ is considered, which yields:

$$\begin{gathered} x=\frac{\alpha e^{i\mathsf{\Omega }t}-\stackrel{-}{\alpha }{e}^{-i\mathsf{\Omega }t}}{2i\mathsf{\Omega }}, \dot{x}=\frac{\alpha e^{i\mathsf{\Omega }t}+\stackrel{-}{\alpha }{e}^{-i\mathsf{\Omega }t}}{2}, \\ \ddot{x}=\frac{1}{2}\left(\dot{\alpha }{e}^{i\mathsf{\Omega }t}+i\mathsf{\Omega }\alpha e^{i\mathsf{\Omega }t}+\dot{\stackrel{-}{\alpha }}{e}^{-i\mathsf{\Omega }t}-i\mathsf{\Omega }\stackrel{-}{\alpha }{e}^{-i\mathsf{\Omega }t}\right), \end{gathered}$$

where $\stackrel{-}{\alpha }$ is the conjugate of $\alpha$, and i represents the symbol of imaginary number ($i=\sqrt{-1}$). Substituting the above expressions into Equation (1) and retaining the slowly varying part, then obtaining:

$$\dot{\alpha }+i\mathsf{\Omega }\alpha +\frac{\mu }{M}\left(\alpha -P\mathsf{\Omega }\right)+\frac{k_1}{M}\left(\frac{\alpha }{i\mathsf{\Omega }}-\frac{P}{i}\right)+\frac{3k_2{(\alpha -P\mathsf{\Omega })}^2\left(\stackrel{-}{\alpha }-P\mathsf{\Omega }\right)}{4M{\mathsf{\Omega }}^{3}i}=0.$$

(2)

Introducing $\alpha =a+ib$ into Equation (2) yields:

$$\begin{gathered} \dot{a}+i\dot{b}+i\mathsf{\Omega }\left(a+ib\right)+\frac{\mu }{M}\left(a+ib-P\mathsf{\Omega }\right)+\frac{k_1\left(a+ib-P\mathsf{\Omega }\right)}{iM\mathsf{\Omega }} \\ +\frac{3k_2\left[{\left(a-P\mathsf{\Omega }\right)}^2+b^2\right]\left(a-P\mathsf{\Omega }+ib\right)}{4iM{\mathsf{\Omega }}^3}=0. \end{gathered}$$

(3)

The real and imaginary parts of Equation (3) can then be separated giving:

$$\dot{a}=b\mathsf{\Omega }-\frac{\mu }{M}\left(a-P\mathsf{\Omega }\right)-\frac{k_{1}b}{M\mathsf{\Omega }}-\frac{3k_2[\left(a-P\mathsf{\Omega }{)}^2+b^2\right]b}{4M{\mathsf{\Omega }}^3},$$

$$\dot{b}=-a\mathsf{\Omega }-\frac{\mu }{M}b+\frac{k_1\left(a-P\mathsf{\Omega }\right)}{M\mathsf{\Omega }}+\frac{3k_2[\left(a-P\mathsf{\Omega }{)}^2+b^2\right]\left[a-P\mathsf{\Omega }\right]}{4M{\mathsf{\Omega }}^3}.$$

(4)

In order to obtain the steady-state response of the system, $\dot{a}=0$ and $\dot{b}=0$ are set. The amplitude of the isolator can then be obtained by solving Equation(4):

$$A=\frac{\sqrt{a^2+b^2}}{\mathsf{\Omega }}.$$

2.2. Jump Behavior

In order to verify the accuracy of this derivation, the semi-analytical solutions obtained via the complexification-averaging method and the numerical solutions obtained using the Runge–Kutta method were compared, the results are shown in Figure 3. The parameters that describe the system being modeled here are:

$$M=10 \mathrm{kg}, \mu =10 \mathrm{N}\cdot \mathrm{s}/\mathrm{m}, K=5000 \mathrm{N}/\mathrm{m}, L=0.2 \mathrm{m}, \mathrm{and} l=0.201 \mathrm{m}.$$

Figure 3a,b shows the amplitude–frequency response curves of the system obtained for $P=0.006 \mathrm{m}$ and $P=0.009 \mathrm{m}$, respectively. The semi-analytical solutions are obtained by solving the steady-state response equations derived from Equation (4) using the least squares method, and the numerical solutions are obtained by solving Equation (1) directly via the Runge–Kutta method. The solution processes of the least squares method and Runge–Kutta method were both carried out in the Matlab software package. The results indicate that the solutions obtained via the complexification-averaging method and the Runge–Kutta method are in good agreement under most excitation frequencies and the deviation between the solutions of the peak point is acceptable, which verifies the correctness of the theoretical derivation. In addition, the response of the vibration isolation system in Figure 3a is close to that of a linear system, and all solutions are stable. However, in Figure 3b, the frequency–response curves are clearly tilted to the right, and there exist three solutions within the frequency range considered; two of them are stable, and the third solution is unstable.

The existence of multiple solutions is a necessary precondition for the jump phenomenon to occur. The response of the system will jump from a lower value to a higher value when the jump condition is met. In this work, the jump-up phenomenon is mainly studied (jump-down is not involved) because a nonlinear vibration isolator can lose its efficacy when the jump-up occurs. Typically, the different solutions that exist at the given frequency can be obtained by assigning different initial conditions to the system in the numerical simulations. In this paper, when the system is in a given harmonic excitation, an impact load is applied to induce the jump behavior in the system; that is, the system vibrates according to the solution in the lower branch before the jump occurs and vibrates according to the solution in the higher branch after the impact has occurred. This simulation method is more consistent with the failure process in the actual world.

Figure 4 shows the displacement transmissibility of the vibration isolation system for two different input displacement amplitudes. The isolator is effective when the transmissibility is less than 1, and the isolator loses efficacy when the transmissibility is greater than 1. When the displacement transmissibility is large, the isolator amplifies the vibrations present in the system, which is not conducive to the protection of the isolated structure.

Figure 5a shows the response of the isolator for $P=0.009 \mathrm{m}, \mathrm{and} \mathsf{\Omega }=4.5 \mathrm{rad}/\mathrm{s}$. The isolator generates periodic vibrations under harmonic excitation. The amplitude of the isolator is significantly increased after the impact load is applied to the isolator. The expression of the impact load is as follows:

$$F=\{\begin{array}{cc}f\sin \left(\frac{2\pi t}{T}\right),& 0\le t\le \frac{T}{2}\\ 0,& t>\frac{T}{2}\end{array}$$

where $T=\frac{0.4}{\pi }$ and $f=5 \mathrm{N}$.

Figure 5b shows the local magnification of the low-amplitude steady-state response of the isolator prior to impact with the half-wave load; the figure also shows that the isolator is effective. Figure 5c shows the local amplification of the high-amplitude steady-state response of the isolator after the half-wave impact is applied. After the occurrence of the jumping phenomenon, the amplitude of the isolator is seen to increase by a factor of approximately 10. The jumping phenomenon induces failure in the isolator.

The jumping phenomenon of the isolator cannot be excited even for an impact of 20 N when the excitation frequency is set to 7 rad/s and for an input displacement amplitude of 0.009 m, as shown in Figure 6. This is because the system does not have multiple solutions when subject to this excitation; in this state, the preconditions for the occurrence of the jump phenomenon are not met. Therefore, the jumping phenomenon cannot be excited irrespective of how large the impact force is. The amplitude of the isolator will increase rapidly under the action of the impact load, but the system will return to its original periodic vibration with a small amplitude after a small duration of time.

The excitation frequency is maintained at 7 rad/s and the input displacement amplitude is increased to 0.01 m in Figure 7. The vibration isolation system generates a jump when the impact amplitude is 10 N. It is shown that the input energy of harmonic load is a prerequisite for the occurrence of the jump behavior, and the impact load induces the jump. With the increase in the input displacement amplitude, the lowest effective frequency of the isolator increases. Under this excitation, the amplitude after the jump is more than 30 times greater than that observed prior to the jump.

2.3. Parametric Study of the Effective Frequency Range

Figure 8 shows the variation in the effective frequency range of the isolator with the variation of the input amplitude. The x-axis is the input displacement amplitude, and the y-axis represents the excitation frequency. This plot was obtained considering 10 values of the input amplitude from 1 to 10 mm and obtaining the amplitude–frequency curve for each value of the input amplitude. Furthermore, the boundary curve for the effective frequency can be obtained based on the amplitude–frequency curves. The effective frequency range is above the boundary curve, i.e., within the white region shown in Figure 8 (the frequency range in which the transmissibility is always less than 1 is referred to as the effective frequency range). It is shown that the effective frequency range becomes narrower with the increase in excitation energies. The lowest effective frequency increases significantly for input amplitudes greater than 8 mm, which is due to the occurrence of the jumping behavior.

Figure 9a shows the effect of damping on the effective frequency range of the isolator for $P=0.006 \mathrm{m}$; this figure demonstrates that the effective frequency range is enlarged for increasing levels of damping. The effective frequency range changes significantly when the damping is in the range of $6–8 \mathrm{N}\cdot \mathrm{s}/\mathrm{m}$. This is because the system exhibits jumping behavior when the damping is small, so the lowest effective frequency of the isolator increases significantly. When the damping is larger than $8 \mathrm{N}\cdot \mathrm{s}/\mathrm{m}$, the effective frequency range changes slightly because the system does not exhibit jumping behavior, i.e., the width of the effective frequency range cannot be stably increased by increasing the damping.

Figure 9b shows the influence of the elongation of the linear springs on the effective frequency range of the isolator. The linear stiffness of the isolator varies with the change in the elongation of the two linear springs. Therefore, the linear natural frequency of the isolator increases with the elongating of the two springs. Consequently, the resonance peak is increased, which leads to the effective frequency range becoming narrower. This finding indicates that the elongation of the springs should not be too large. It is shown from the above analysis that the effective frequency range of the horizontal QZS isolator changes significantly with changes in damping and spring elongation. When the damping is large and the spring elongation is small, the effective frequency range of the isolator is wider. More importantly, the occurrence of the jumping behavior significantly influences the effective frequency range of the isolator.

3. Isolator Installed on Linear Structure

3.1. Dynamic Model

Here, a QZS isolator is installed on a harmonically excited linear structure to further study the dynamic behavior of the isolator; this system is shown in Figure 10. The parameters $m$, $k_0$, and $\lambda$ represent the mass, stiffness, and damping of the primary structure, respectively; $p$ is the harmonic excitation forcing. The other parameters take the same definition as given in the previous section.

The dynamic equation of the two-degree-of-freedom vibration isolation system can be expressed by the following equations,

$$m{\ddot{x}}_1+\lambda {\dot{x}}_1+k_0{x}_1+\mu \left({\dot{x}}_1-{\dot{x}}_2\right)+k_1\left(x_1-x_2\right)+k_2{\left(x_1-x_2\right)}^3=p\sin \mathsf{\Omega }t,$$

$$M{\ddot{x}}_2+\mu \left({\dot{x}}_2-{\dot{x}}_1\right)+k_1\left(x_2-x_1\right)+k_2{\left(x_2-x_1\right)}^3=0,$$

(5)

where $x_1$ and $x_2$ are the displacements of the primary structure and the isolator, respectively. Introducing ${\dot{x}}_1+i\mathsf{\Omega }{x}_1={\alpha }_1{e}^{i\mathsf{\Omega }t}$ and ${\dot{x}}_2+i\mathsf{\Omega }{x}_2={\alpha }_2{e}^{i\mathsf{\Omega }t}$, then obtaining,

$$\begin{gathered} x_1=\frac{{\alpha }_1{e}^{i\mathsf{\Omega }t}-{\stackrel{-}{\alpha }}_1{e}^{-i\mathsf{\Omega }t}}{2i\mathsf{\Omega }},{\dot{x}}_1=\frac{{\alpha }_1{e}^{i\mathsf{\Omega }t}+{\stackrel{-}{\alpha }}_1{e}^{-i\mathsf{\Omega }t}}{2}, \\ {\ddot{x}}_1=\frac{1}{2}\left({\dot{\alpha }}_1{e}^{i\mathsf{\Omega }t}+i\mathsf{\Omega }{\alpha }_1{e}^{i\mathsf{\Omega }t}+{\dot{\stackrel{-}{\alpha }}}_1{e}^{-i\mathsf{\Omega }t}-i\mathsf{\Omega }{\stackrel{-}{\alpha }}_1{e}^{-i\mathsf{\Omega }t}\right), \\ {x}_2=\frac{{\alpha }_2{e}^{i\mathsf{\Omega }t}-{\stackrel{-}{\alpha }}_2{e}^{-i\mathsf{\Omega }t}}{2i\mathsf{\Omega }}, {\dot{x}}_2=\frac{{\alpha }_2{e}^{i\mathsf{\Omega }t}+{\stackrel{-}{\alpha }}_2{e}^{-i\mathsf{\Omega }t}}{2}, \\ {\ddot{x}}_2=\frac{1}{2}\left({\dot{\alpha }}_2{e}^{i\mathsf{\Omega }t}+i\mathsf{\Omega }{\alpha }_2{e}^{i\mathsf{\Omega }t}+{\dot{\stackrel{-}{\alpha }}}_2{e}^{-i\mathsf{\Omega }t}-i\mathsf{\Omega }{\stackrel{-}{\alpha }}_2{e}^{-i\mathsf{\Omega }t}\right). \end{gathered}$$

Substituting the above expressions into Equation (5) and retaining the slowly varying parts, yields,

$${\dot{\alpha }}_1+i\mathsf{\Omega }{\alpha }_1+\frac{\lambda }{m}{\alpha }_1+\frac{k_0}{m}\frac{{\alpha }_1}{i\mathsf{\Omega }}+\frac{\mu }{m}\left({\alpha }_1-{\alpha }_2\right)+\frac{k_1}{m}\frac{{\alpha }_1-{\alpha }_2}{i\mathsf{\Omega }}+\frac{3k_2{\left({\alpha }_1-{\alpha }_2\right)}^2\left({\stackrel{-}{\alpha }}_1-{\stackrel{-}{\alpha }}_2\right)}{4m{\mathsf{\Omega }}^{3}i}=\frac{p}{i},$$

$$\epsilon \left({\dot{\alpha }}_2+i\mathsf{\Omega }{\alpha }_2\right)+\frac{\mu }{m}\left({\alpha }_2-{\alpha }_1\right)+\frac{k_1}{m}\frac{{\alpha }_2-{\alpha }_1}{i\mathsf{\Omega }}-\frac{3k_2{\left({\alpha }_1-{\alpha }_2\right)}^2\left({\stackrel{-}{\alpha }}_1-{\stackrel{-}{\alpha }}_2\right)}{4m{\mathsf{\Omega }}^{3}i}=0,$$

(6)

where $\epsilon =\frac{M}{m}$. Substituting ${\alpha }_1=a_1+i{b}_1$ and ${\alpha }_2=a_2+i{b}_2$ into Equation (6), yields,

$$\begin{gathered} {\dot{a}}_1+i{\dot{b}}_1+i\mathsf{\Omega }\left(a_1+i{b}_1\right)+\frac{\lambda }{m}\left(a_1+i{b}_1\right)+\frac{k_0}{m}\frac{a_1+i{b}_1}{i\mathsf{\Omega }}+\frac{\mu }{m}\left(a_1+i{b}_1-a_2-i{b}_2\right) \\ +\frac{k_1}{m}\frac{\left(a_1+i{b}_1-a_2-i{b}_2\right)}{i\mathsf{\Omega }}+\frac{3k_2\left[{\left(a_1-a_2\right)}^2+{\left(b_1-b_2\right)}^2\right]\left[a_1-a_2+i\left(b_1-b_2\right)\right]}{4m{\mathsf{\Omega }}^{3}i}=\frac{p}{i}, \end{gathered}$$

$$\begin{gathered} \epsilon \left[{\dot{a}}_2+i{\dot{b}}_2+i\mathsf{\Omega }\left(a_2+i{b}_2\right)\right]+\frac{\mu }{m}\left(a_2+i{b}_2-a_1-i{b}_1\right)+\frac{k_1}{m}\frac{\left(a_2+i{b}_2-a_1-i{b}_1\right)}{i\mathsf{\Omega }} \\ -\frac{3k_2\left[{\left(a_1-a_2\right)}^2+{\left(b_1-b_2\right)}^2\right]\left[a_1-a_2+i\left(b_1-b_2\right)\right]}{4m{\mathsf{\Omega }}^{3}i}=0. \end{gathered}$$

(7)

Separating the real and imaginary parts of Equation (7), yields,

$${\dot{a}}_1=\mathsf{\Omega }{b}_1-\frac{\lambda }{m}{a}_1-\frac{k_0{b}_1}{m\mathsf{\Omega }}-\frac{\mu }{m}\left(a_1-a_2\right)-\frac{k_1\left(b_1-b_2\right)}{m\mathsf{\Omega }}-\frac{3k_2\left[{\left(a_1-a_2\right)}^2+{\left(b_1-b_2\right)}^2\right]\left(b_1-b_2\right)}{4m{\mathsf{\Omega }}^3}$$

$${\dot{b}}_1=-\mathsf{\Omega }{a}_1-\frac{\lambda }{m}{b}_1+\frac{k_0{a}_1}{m\mathsf{\Omega }}-\frac{\mu }{m}\left(b_1-b_2\right)+\frac{k_1\left(a_1-a_2\right)}{m\mathsf{\Omega }}+\frac{3k_2\left[{\left(a_1-a_2\right)}^2+{\left(b_1-b_2\right)}^2\right]\left(a_1-a_2\right)}{4m{\mathsf{\Omega }}^3}-p,$$

$${\dot{a}}_2=\mathsf{\Omega }{b}_2-\frac{\mu }{m\epsilon }\left(a_2-a_1\right)-\frac{k_1\left(b_2-b_1\right)}{m\mathsf{\Omega }\epsilon }+\frac{3k_2\left[{\left(a_1-a_2\right)}^2+{\left(b_1-b_2\right)}^2\right]\left(b_1-b_2\right)}{4m{\mathsf{\Omega }}^3\epsilon },$$

$${\dot{b}}_2=-\mathsf{\Omega }{a}_2-\frac{\mu }{m\epsilon }\left(b_2-b_1\right)+\frac{k_1\left(a_2-a_1\right)}{m\mathsf{\Omega }\epsilon }-\frac{3k_2\left[{\left(a_1-a_2\right)}^2+{\left(b_1-b_2\right)}^2\right]\left(a_1-a_2\right)}{4m{\mathsf{\Omega }}^3\epsilon }.$$

(8)

The amplitudes of the primary structure and the isolator can then be obtained; it is found that,

$$A_1=\frac{\sqrt{a_1^2+b_1^2}}{\mathsf{\Omega }} \mathrm{and} A_2=\frac{\sqrt{a_2^2+b_2^2}}{\mathsf{\Omega }}.$$

3.2. Jump Behavior

The accuracy of the derivation shown above was also verified here by comparing the semi-analytical and numerical solutions, and the results are shown in Figure 11a,b. The parameters of the isolator are the same as those adopted in the previous section; the mass, stiffness, and damping of the primary structure are $m=100 \mathrm{kg},$ $k_0=40000 \mathrm{N}/m$, and $\lambda =10 \mathrm{N}\cdot \mathrm{s}/\mathrm{m}$. The paper aims to demonstrate the common problems of the quasi-zero stiffness isolators, and thus the parameters of the primary structure are not taken from actual engineering. In essence, the behaviors which are the qualitatively same as those shown in the paper can be obtained as long as the excitation is appropriate even if the parameters of the primary structure are changed.

Figure 11 shows the amplitude–frequency response of the system for different excitation amplitudes. It is found that the primary structure only has one resonance band, and the isolator has two resonance bands for $p=0.5,\hspace{0.17em}1, \mathrm{and} \hspace{0.17em}2 \mathrm{N}$. The system produces unstable responses around the resonance frequency after the excitation amplitude is increased to a certain threshold. In addition, the response of the system at $p=3 \mathrm{N}$ is significantly different from the system excited by a relatively small force. It is shown that the multiple solutions occur in a frequency band that is lower than the resonance frequency of the primary structure. The characteristics of the multiple solutions in the two-degree-of-freedom vibration isolation system are similar to those observed in systems with nonlinear energy sinks [36,37,38]. Moreover, it is noted that the secondary oscillator does not introduce a new resonance band in the primary oscillator, even for excitation amplitudes that are extremely large, which is a beneficial characteristic for nonlinear energy sinks. The realization of a purely nonlinear stiffness (that is, the linear stiffness is exactly equal to zero) is difficult to achieve in practice; in practical situations, a small positive or negative linear stiffness inevitably arises. The small positive linear stiffness only slightly affects the response of the primary structure, which demonstrates that the configurations of the QZS vibration isolator can be modified to the QZS vibration absorber according to the theory of nonlinear energy sink conveniently by amending parameters.

In order to describe the effective frequency range of the isolator directly, the amplitude ratio of the isolator and primary structure was obtained. The isolator is effective in the frequency range in which the amplitude ratio is smaller than one. The isolator loses efficacy in the frequency range slightly higher than zero-frequency as the excitation amplitude is small, as shown in Figure 12a. However, the isolator can produce large amplitude vibration in the resonance band of the primary structure. With the increase in the excitation amplitude, the amplitude of the isolator exceeds that of the primary structure in the resonance band of the primary structure, the isolator acts as a vibration absorber in these cases, as shown in Figure 12b,c. Moreover, the system produces chaotic responses in the resonance bands of the primary structure; the solutions are unstable in these regimes. A typical chaotic response is depicted in Figure 13.

In Figure 12d, there exist multiple solutions in three frequency ranges. In the lowest frequency range, the curve is similar to the counterpart considered in Figure 4b; the jump phenomenon is also similar to that observed in the previous section. In the second frequency range (9.96–18.84 rad/s), two stable solutions exist, which is a requirement for the jumping behavior. When the system has an additional impact load imposed on it, the jump can occur, as shown in Figure 14 and Figure 15. In Figure 14, the results are shown for an impact load that is exerted on the primary structure, and in Figure 15, the results are shown for a system in which the impact load is exerted on the isolator. It is seen that the amplitude of the primary structure almost doubled, and the amplitude of the isolator has been increased by a factor of greater than thirty after the impact load imposed. The isolator is likely to be damaged in the process of the jump. In the third frequency range (21.21–23.37 rad/s), a stable branch and an unstable branch coexist. In this case, the system can only produce stable responses; unstable responses cannot be generated in this case.

3.3. Parametric Study for Effective Frequency Range

The variation of the effective frequency range of the isolator as a function of the excitation amplitude is shown in Figure 16. The effective frequency range is shown as the white region of this figure. It is shown that the lowest boundary curve for effective frequency varies slightly with the excitation amplitude. However, the amplitude of the isolator exceeds that of the primary structure after the excitation amplitude reaches 1.5 N in the resonance band of the primary structure, and the frequency range in which the isolator loses efficacy becomes wider with further increases in the force. The higher branches of response [30,31] occur after the excitation amplitude reaches 2.5 N, and a new frequency range in which the isolator loses efficacy arises between the two resonance frequencies of the system.

The influence of damping on the effective frequency range for $p=2 \mathrm{N}$ is shown in Figure 17a. When the damping is relatively weak, the system produces higher branches of response, which leads to the narrowing of the effective frequency range. Interestingly, the higher branches can be eliminated by increasing the damping; for this reason, the effective frequency range increases significantly for damping greater than $8 \mathrm{N}\cdot \mathrm{s}/\mathrm{m}$. However, the frequency ranges around zero and the resonance frequency of the primary structure remains unaffected and changes only slightly with variations in the damping. The influence of the elongation of the linear springs is shown in Figure 17b. The linear natural frequency of the isolator increases with increasing elongation, which leads to an increase in the lowest boundary of the effective frequency. The boundaries around the resonance frequency of the primary structure remain almost constant despite variations in the damping and the elongation of the linear springs.

4. Conclusions

The jump behavior caused by the strong nonlinearity of the horizontal QZS isolator is a primary reason for its failure. The dynamics of the isolator were studied to grasp the effective frequency range and failure mechanism using the complexification-averaging method and the Runge–Kutta method. The accuracy of the semi-analytical results was verified by comparing the results obtained from the two methods. The amplitude–frequency response curves of the system were presented.

For small excitation amplitudes, the isolator exhibits good vibration isolation performance. With an increase in the excitation amplitude, the systems exhibit jump behavior around the natural frequency of the isolator and then increase the lowest effective frequency of the isolator. In the two-degree-of-freedom system, the isolator loses efficacy in the resonance band of the primary structure when the excitation exceeds a certain threshold. Furthermore, the two-degree-of-freedom system generates higher branches of response after the excitation reaches a second threshold, which leads to a narrower effective frequency range between the two natural frequencies of the system. In addition, the influences of the isolator parameters on the effective frequency range were studied. The results show that increases in the damping of the system can effectively eliminate the jump phenomena and expand the effective frequency range. It is also found that decreases in the elongation of the two linear springs reduce the lowest effective frequency, but such changes do not significantly affect the jump behavior.

Low-frequency and broadband vibration isolation are important issues in engineering practices. The jump phenomenon is the biggest obstacle to achieving these two goals. This study can deepen the understanding of the jump behaviors induced by the horizontal QZS isolator and provide design guidance for this kind of isolator to avoid jumps. However, the factors that affect the dynamic characteristics of the isolator in actual engineering are not considered. In the future, the jump behaviors that lead to the narrowing of the effective frequency range should be investigated under the full consideration of the engineering environments.