Design and experimental study of a quasi-zero-stiffness vibration isolator incorporating transverse groove springs

Abstract

The concept of quasi-zero-stiffness (QZS) vibration isolator was proposed in recent decades to improve the low-frequency isolation performance without increasing the static displacement. This work is devoted to the concrete realization of a QZS isolator by utilizing transverse groove springs. Firstly, the QZS isolator is theoretically analyzed and some dynamical indices are analytically calculated. Then, the transverse groove springs are designed and the isolator prototype is assembled; the QZS feature of the prototype is basically fulfilled. Finally, the experiments are conducted by means of an electrodynamic shaker which generates sinusoidal base excitation for the isolator prototype; the experimental results clearly show the good isolation performance of the QZS isolator and meanwhile reflect some practical factors that should be noticed in actual applications.

Introduction

Vibration is a common phenomenon in the natural world. Even for the micro-/nano-sized structure, vibration still exists and plays an important role in influencing its overall characteristics: Demir et al. [1] studied the torsional and longitudinal vibrations of microtubules; Numanoglu et al. [2] studied the longitudinal free vibration behaviors of one-dimensional nanostructures with various boundary conditions; and Akgöz et al. [3] studied the longitudinal vibration of microbars based on strain gradient elasticity theory. However, in the vast majority of engineering practices, vibration is harmful and usually causes undesirable consequences such as noise generation, passenger comfort reduction and wear of machine components. In order to reduce the unfavorable consequences caused by vibration, the vibration isolation is the most direct and convenient class of methods [4], which means inserting a device (usually called vibration isolator) between the vibration source and the vibration receiver to attenuate the vibration transmission. Vibration isolation methods are widely used in astronautics field: Chen et al. [5] designed a dual micro-vibration isolation system for optical satellite flywheel; Deng et al. [6] adopted vibration isolation method to reduce the negative effects of flywheel disturbance on space camera image quality; and Oh et al. [7] proposed an on-orbit vibration isolation system for the spaceborne cryocooler. Vibration isolation methods have also been widely used in other engineering fields like civil engineering, mechanical engineering and vehicle engineering: Sitharam et al. [8] utilized open trenches to achieve vibration isolation of the buildings housed with sensitive equipment; Li et al. [9] experimentally studied the vibration isolation characteristics of hydraulic engine mounts; Bai et al. [10] employed the integrated semi-active control for both longitudinal and vertical vibration isolation of seat suspension. The earliest passive vibration isolation model is purely linear, which, however, has an inherent weakness that it involves a trade-off between the frequency range of isolation and the static deflection, so its isolation performance at low frequencies is relatively poor. This weakness can be addressed by introducing active (semi-active) control if necessary, although the active (semi-active) control has some shortcomings on the aspects of reliability and complexity. In fact, an ideal purely passive vibration isolator is supposed to possess both low dynamic stiffness to widen the frequency range of isolation and high static stiffness to avoid large static deflection, which needs to introduce nonlinearity. Many kinds of nonlinear passive isolators are found in a review article by Ibrahim [11]. The concept of high-static-low-dynamic-stiffness (HSLDS) vibration isolator, as a typical nonlinear isolator, was proposed in recent decades to overcome the disadvantages of linear isolator in a passive manner, which is implemented by connecting a negative stiffness module in parallel with a linear spring element [12]. Most of the HSLDS isolators have symmetric stiffness characteristics with respect to the static equilibrium point, and the minimum dynamic stiffness of the HSLDS isolator can be designed to zero to achieve the lowest natural frequency, which is done by proper selection of the system parameters to ensure that the dynamic stiffness of the negative stiffness module at the static equilibrium point exactly counterparts the stiffness of the linear spring element, and then the HSLDS isolator becomes a so-called quasi-zero-stiffness (QZS) vibration isolator [13].

A number of QZS/HSLDS vibration isolators can be found in the literature, the most archetypal one of which is the three-spring isolator; two oblique springs act as negative stiffness module, and a vertical spring provides positive stiffness and withstands the isolated mass; this vibration isolator is often taken as a classical model for analyzing nonlinear isolator and has been widely studied by many researchers. Carrella et al. [14] might be the first group of people who have did preliminary theoretical works on this kind of isolator, including static analysis, static optimization [15], force transmissibility analysis [16], comparison between force and displacement transmissibility [17] and application to rotating shaft [18]. Kovacic et al. and Ramirez et al. have also studied the three-spring isolator, which considered physical nonlinearity [19] and focused on shock isolation [20], respectively. Liu et al. [21] established a more accurate dynamic model for the three-spring isolator and then conducted a more accurate analysis. Inspired by the three-spring QZS isolator model, researchers have designed and proposed more isolators with various forms and structures. Zhou et al. [22] designed a QZS isolator employing cam-roller-spring mechanism (CRSM) which has piecewise linear–nonlinear stiffness and thus exhibits special dynamic characteristics. Sun et al. [23] proposed a scissor-like isolator with HSLDS characteristic, which has many structural parameters with strong designability; the biggest difference between the scissor-like isolator and other HSLDS isolators is that it may have asymmetric stiffness and therefore the vibration center deviates from the static equilibrium point. Liu et al. [24] integrated the scissor-like structure into lever-type isolation system with additional inertial element. Meng et al. [25] proposed a QZS isolator using disk spring. The utilization of magnets or magnetic springs to develop nonlinear isolators has also aroused great interest in the past decade, because the negative stiffness produced by magnetic system does not require direct contact and thus the magnetic isolators are relatively convenient to manufacture and assemble. Carrella et al. [26] conceptually proposed a simple magnetic HSLDS isolator in which the magnets were treated as magnetic points. Actual magnets are, of course, a little more complicated than the conceptual magnetic points. Ring-shaped magnets [27] are commonly adopted to construct magnetic QZS/HSLDS isolators; Dong et al. [28] designed a magnetic HSLDS isolator using Ring-shape magnets integrated with spiral flexure spring; Zheng et al. [29] also designed a QZS isolator by employing ring-shaped magnets but for torsional vibration isolation. Cube-shaped magnets can also be configured to design QZS/HSLDS isolators, which was studied by Robertson et al. [30]. Li et al. [31] proposed a negative stiffness isolator using magnetic spring combined with rubber membrane. Generally speaking, an essential reason for the different characteristics of these magnetic nonlinear isolators is the arrangement and shape of the magnets. Another category of material that can be exploited to build QZS/HSLDS isolators is the beams with large deformation, which is based on a well-known fact that a beam will possess nonlinear characteristics after it experiences large deformation; for example, Liu et al. [32] employed Euler buckled beams as negative stiffness corrector to build QZS isolator; Huang et al. [33] studied the characteristics of an ultra-low-frequency nonlinear isolator using sliding beam as negative stiffness module.

Considering that the QZS/HSLDS isolator is a nonlinear system which has more complex dynamic behaviors than linear system, some researchers carried out more in-depth studies for a better understanding of it. Kovacic et al., Wang et al. and Abolfathi et al. studied the effects of system imperfections on the QZS isolator, which are caused by existence of a static force [34], parameter errors [35] and mistuning of the system [36], respectively. Hao et al. [37] conducted a very comprehensive nonlinear dynamic analysis on the classical three-spring QZS isolator, finding that it may exhibit various complex nonlinear behaviors, such as chaotic seas, transient chaos and chaotic saddles; meanwhile, they proposed a two-sided damping constraint control strategy [38] for the QZS isolator, which can effectively isolate shock and swiftly stabilize the transient states into the desired steady-state responses. Dong et al. [39] improved the isolation performance of the HSLDS isolator by introducing damping nonlinearity. Liu et al. [40] studied the superharmonic resonance of a QZS isolator and its effect on the isolation performance. Valeev [41] studied the dynamic behaviors of a group of QZS isolators with slightly different parameters. Kovacic et al. [42] studied the behaviors of a forced damped purely nonlinear (QZS) oscillator at different excitation frequencies.

In addition to single-stage QZS/HSLDS isolators such as those mentioned above, the two-stage QZS vibration isolation systems were also studied to further improve the isolation performance, especially when a fast roll-off of the high-frequency transmissibility is required [43]; Wang et al. [44] comprehensively compared the dynamic performances of the single-degree-of-freedom (SDOF) and 2DOF QZS isolation systems; Liu et al. [45] proposed a two-stage HSLDS isolator with auxiliary system which can suppress the jump phenomenon. The QZS/HSLDS isolation strategy can also be extended to multi-DOF or 6DOF vibration isolation to achieve simultaneous multi-DOF or 6DOF low-frequency isolation; for example, Sun et al. [46] formulated a multi-direction QZS vibration isolation model by employing geometrical nonlinearity; Liu et al. [47] proposed an eight-legged isolation platform with HSLDS characteristic using dual-pyramid-shape struts; Zhou et al. [48] proposed a 6DOF QZS isolation platform using CRSM struts as legs; Zhu et al. [49] designed a 6DOF nonlinear vibration isolation model using QZS magnetic levitation.

Most of the existing studies of QZS/HSLDS isolators focus on the theoretical analyses, while there are also some researchers who have conducted experiments on QZS/HSLDS isolators. Lan et al. [50] utilized planar springs to materialize the three-spring QZS isolator model and did the experiments. Le et al. [51] presented an experimental prototype using coil springs and bars for vibration isolation of vehicle seat. Huang et al. [52] carried out an experimental study on the HSLDS isolator using buckled beams as negative stiffness module; Fulcher et al. [53] also did the experiments on buckled-beam-type QZS isolator but focused on the shock isolation. Wu et al. [54] and Xu et al. [55] have conducted the experiments on magnetic QZS isolators. Lu et al. [56] experimentally investigated a two-stage HSLDS isolation system using bi-stable composite plates.

This study is concerned with a QZS vibration isolator composed of five transverse groove springs, four of which are configured laterally to act as negative stiffness module, and another is vertical to provide positive stiffness (Fig. 1). In fact, the five-spring QZS conceptual model is essentially similar to the three-spring QZS conceptual model; the reason for using four lateral springs instead of two is to improve the stability in the third direction. This paper presents a systematic process of theoretical analysis, prototype design and experiments. This research process is also suitable for studying the micro-/nano-vibration of some NEMS/MEMS systems. In the theoretical part, the stiffness characteristics, dynamic response and isolation performance are analyzed, and the analytical methods for obtaining the peak response, stability boundary, jump frequencies, peak transmissibility and isolation frequency are proposed. The experimental prototype is constructed by utilizing transverse groove springs which have many advantages over conventional coil springs such as the compact structure and strong designability. The experiments are conducted by means of an electrodynamic shaker which generates sinusoidal base excitation with controllable excitation amplitude and frequency, and the experiments on the equivalent linear isolator are also performed for comparison. This work can provide some reference for the prototype design of QZS isolator and an experimental demonstration of its good isolation performance. Most importantly, this study can reveal some practical factors that should be noticed in actual situations and thus is of significance for promoting the application of QZS isolator.

Fig. 1
figure1

Simplified mechanical analysis model of the five-spring QZS vibration isolator (this position is just the static equilibrium position)

This paper is organized as follows: Sect. 2 gives the analysis of static characteristics. Section 3 focuses on the dynamic characteristics and the analytical calculations of some dynamical indices. Section 4 presents the design process of the experimental prototype. Section 5 shows the experimental results and discussions. The conclusions are drawn in Sect. 6.

Static analysis

The simplified mechanical model of the QZS vibration isolator being considered is depicted in Fig. 1. It is comprised of five springs, one of which is vertical to withstand the isolated mass and provide positive stiffness, and the other four lateral springs have the same length and same stiffness. The overall system is symmetrical with respect to the axis of the vertical spring, which is used to isolate the vibration in the vertical direction. The position shown in Fig. 1 is just the static equilibrium position where the four lateral springs become horizontal. At this position, the lateral springs are in the compressed state, thereby producing negative stiffness in the vertical direction, and then, the combination of the four lateral springs is the negative stiffness module (NSM).

When the isolated mass experiences a vertical displacement \( X \) from the static equilibrium position, the force exerted by each of the lateral springs is

$$ F_{l} = K_{\text{l}} \left( {L_{0} - \sqrt {L_{1}^{2} + X^{2} } } \right) $$
(1)

where \( K_{\text{l}} \) is the stiffness of the lateral spring, \( L_{0} \) is the original length of the lateral spring without deformation, and \( L_{1} \) is the projection length of the lateral spring on the horizontal plane. The horizontal components of the forces produced by the four lateral springs cancel each other out, and the sum of their vertical components is the restoring force of the NSM:

$$ F_{\text{NSM}} = - 4F_{\text{l}} \sin \alpha $$
(2)

where \( \alpha \) is the angle between the axis of the lateral spring and the horizontal plane, and \( \sin \alpha \) is given by

$$ \sin \alpha = \frac{X}{{\sqrt {L_{1}^{2} + X^{2} } }} $$
(3)

Substituting Eqs. (1) and (3) into Eq. (2) gives the restoring force–displacement relationship of the NSM:

$$ F_{\text{NSM}} \left( X \right) = - 4K_{\text{l}} \left( {\frac{{L_{0} }}{{\sqrt {L_{1}^{2} + X^{2} } }} - 1} \right)X $$
(4)

The QZS isolator is a parallel combination of the NSM and the vertical spring, whose force–displacement relationship is calculated by

$$ F_{\text{QZS}} \left( X \right) = F_{\text{NSM}} \left( X \right) + K_{\text{v}} X $$
(5)

where \( K_{\text{v}} \) is the stiffness of the vertical spring. Note that the constant term, which is equal to the weight of the isolated mass \( mg \) (\( m \) is the mass and \( g \) is the gravitational acceleration), is omitted in this expression. Substituting for \( F_{\text{NSM}} \left( X \right) \) from Eq. (4) into Eq. (5) yields

$$ F_{\text{QZS}} \left( X \right) = K_{\text{v}} X - 4K_{l} \left( {\frac{{L_{0} }}{{\sqrt {L_{1}^{2} + X^{2} } }} - 1} \right)X $$
(6)

By introducing \( f_{\text{QZS}} = {{F_{\text{QZS}} } \mathord{\left/ {\vphantom {{F_{\text{QZS}} } {\left( {K_{\text{v}} L_{0} } \right)}}} \right. \kern-0pt} {\left( {K_{\text{v}} L_{0} } \right)}} \), \( \lambda = {{K_{\text{l}} } \mathord{\left/ {\vphantom {{K_{\text{l}} } {K_{\text{v}} }}} \right. \kern-0pt} {K_{\text{v}} }} \), \( \mu = {{L_{1} } \mathord{\left/ {\vphantom {{L_{1} } {L_{0} }}} \right. \kern-0pt} {L_{0} }} \) and \( x = {X \mathord{\left/ {\vphantom {X {L_{0} }}} \right. \kern-0pt} {L_{0} }} \), the force–displacement relationship of the QZS isolator can be written in the non-dimensional form:

$$ f_{\text{QZS}} \left( x \right) = x - 4\lambda \left( {\frac{1}{{\sqrt {\mu^{2} + x^{2} } }} - 1} \right)x $$
(7)

The non-dimensional stiffness–displacement relationship is obtained by differentiating Eq. (7) with respect to the non-dimensional displacement \( x \) to give

$$ k_{\text{QZS}} \left( x \right) = 1 + 4\lambda - \frac{{4\lambda \mu^{2} }}{{\left( {\mu^{2} + x^{2} } \right)^{{\frac{3}{2}}} }} $$
(8)

It can be seen that the stiffness is a minimum at the static equilibrium point, which is \( k_{\text{eq}} = 1 + 4\lambda \left( {1 - {1 \mathord{\left/ {\vphantom {1 \mu }} \right. \kern-0pt} \mu }} \right) \) (non-dimensional form). In order to realize the quasi-zero stiffness, we need to set \( k_{\text{eq}} \) to zero, and therefore, the relationship between \( \lambda \) and \( \mu \) that yields QZS characteristic is

$$ \lambda_{\text{QZS}} = \frac{\mu }{{4\left( {1 - \mu } \right)}} $$
(9)

If \( \lambda \) is larger than \( \lambda_{\text{QZS}} \), \( k_{\text{eq}} \) will be negative, which implies instability of the static equilibrium point; if \( \lambda \) is smaller than \( \lambda_{\text{QZS}} \), \( k_{\text{eq}} \) will be positive and the QZS feature is lost. In the subsequent analysis, Eq. (9) is always held to ensure a stable quasi-zero stiffness, and for conciseness, the value of \( \lambda \), which is dependent on the value of \( \mu \), is not mentioned hereafter.

Figure 2 shows the (non-dimensional) force–displacement and stiffness–displacement relationships for various values of \( \mu \). It can be seen that within a certain displacement range (related to \( \mu \)), the stiffness of the QZS isolator is lower than that of the equivalent linear isolator; however, outside this range, the low stiffness does not hold. The displacement range in which the stiffness of the QZS isolator is lower than the equivalent linear isolator (hereafter referred to as low stiffness–displacement range) is an important indicator of the stiffness characteristics, which is derived by setting \( k_{\text{QZS}} \left( x \right) < 1 \) to give

$$ \left| x \right| < x_{d} = \mu^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} \sqrt {1 - \mu^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} } $$
(10)

The length of the low stiffness–displacement range is \( L_{\text{lsdr}} = 2\mu^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} \sqrt {1 - \mu^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} } \). Figure 3 shows the length of low stiffness–displacement range as a function of geometrical parameter \( \mu \), it can be seen that \( L_{\text{lsdr}} \) has a maximum value, which can be obtained by differentiating \( L_{\text{lsdr}} \) with respect to \( \mu \) and setting it to zero; the maximum \( L_{\text{lsdr}} \) is calculated to be \( \hbox{max} \left( {L_{\text{lsdr}} } \right) = {{4\sqrt 3 } \mathord{\left/ {\vphantom {{4\sqrt 3 } 9}} \right. \kern-0pt} 9} \approx 0. 7 6 9 8 \), at which \( \tilde{\mu } = \left( {{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}} \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} \approx 0. 5 4 4 3 \).

Fig. 2
figure2

Static characteristics of the QZS isolator for various values of geometrical parameter \( \mu \): a force–displacement relationship and b stiffness–displacement relationship. Note that the stiffness ratio \( \lambda \) is dependent on \( \mu \) according to Eq. (9) to ensure a stable quasi-zero stiffness

Fig. 3
figure3

(Non-dimensional) length of low stiffness–displacement range as a function of geometrical parameter \( \mu \). Low stiffness–displacement range means the displacement range in which the stiffness of the QZS isolator is lower than the equivalent linear isolator

The subsequent analysis would be significantly simplified if the force–displacement relationship could be expressed in a polynomial form, which can be done by employing the Taylor series expansion, and it is natural to set the center of expansion at the static equilibrium point:

$$ f_{\text{QZS}} \left( x \right) = \frac{{x^{3} }}{{2\left( {1 - \mu } \right)\mu^{2} }} - \frac{{3x^{5} }}{{8\left( {1 - \mu } \right)\mu^{4} }} + \frac{{5x^{7} }}{{16\left( {1 - \mu } \right)\mu^{6} }} - \frac{{35x^{9} }}{{128\left( {1 - \mu } \right)\mu^{8} }} + \cdots $$
(11)

The approximate force–displacement relationships truncated to different orders are plotted in Fig. 4a, b for two values of \( \mu \) (0.7 and 0.25); it can be seen that for a relatively large \( \mu \), higher-order approximation gives rise to higher accuracy, whereas for a small \( \mu \), the accuracy of polynomial form approximation does not necessarily increase with the increase in truncation order; this can also be clearly seen from Fig. 4c, which shows the (relative) approximation error evaluated at the maximum displacement with low stiffness (\( x_{d} = \mu^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} \sqrt {1 - \mu^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} } \)) as \( \mu \) changes. Therefore, a reasonable order of truncation should be selected. In this study, we use fifth-order approximate expression:

$$ f_{\text{QZS}} \left( x \right) \approx \beta x^{3} + \rho x^{5} $$
(12)

where

$$ \beta = \frac{1}{{2\left( {1 - \mu } \right)\mu^{2} }};\quad \rho = - \frac{3}{{8\left( {1 - \mu } \right)\mu^{4} }} $$
(13)
Fig. 4
figure4

Comparison between the exact and approximate (non-dimensional) force–displacement relationship of the QZS isolator when a\( \mu = 0.7 \) and b\( \mu = 0.25 \). c Relative approximation error evaluated at \( x_{d} = \mu^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} \sqrt {1 - \mu^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} } \) as \( \mu \) is varied

Dynamic analysis

Dynamic response

There are two categories of vibration isolation problems [17]: (a) the force excitation is applied on the isolated mass, and the goal is to minimize the disturbance force transmitted to the base on which the vibration isolation system is mounted; (b) the source of vibration is the base, in which case the purpose of vibration isolation is to attenuate the vibration of the isolated mass. The former case is referred to as the force transmission problem, while the latter as the motion transmission problem. This work is concerned with the motion transmission problem, which is consistent with the type of excitation in the experiments.

The equation of motion under base excitation is

$$ m\ddot{X} + c\dot{X} + F_{\text{QZS}} \left( X \right) = - m\ddot{Z} $$
(14)

where \( X \) is the relative displacement between the base and the isolated mass, and the acceleration of the base excitation is assumed to be \( \ddot{Z} = A_{0} \cos \left( {\omega t} \right) \). To non-dimensionalize the equation of motion, the following non-dimensional constants and variables are introduced:

$$ \omega_{n} = \sqrt {\frac{{K_{v} }}{m}} ;\quad \zeta = \frac{c}{{2\sqrt {mK_{v} } }};\quad \, a_{0} = \frac{{A_{0} }}{{\omega_{n}^{2} L_{0} }};\quad \, \varOmega = \frac{\omega }{{\omega_{n} }};\quad \, \tau = \omega_{n} t;\quad \quad \, x = \frac{X}{{L_{0} }} $$
(15)

where \( \omega_{n} \) is the natural frequency of the equivalent linear isolator, \( \zeta \) is the damping ratio, \( a_{0} \) is the non-dimensional excitation acceleration amplitude, \( \varOmega \) is the non-dimensional excitation frequency and \( \tau \) is the non-dimensional time, \( x \) is the non-dimensional relative displacement. Then, the non-dimensional equation of motion with respect to the non-dimensional relative displacement is found to be

$$ x^{\prime\prime} + 2\zeta x^{\prime} + \beta x^{3} + \rho x^{5} = - a_{0} \cos \left( {\varOmega \tau } \right) $$
(16)

where \( \left( \bullet \right)^{\prime \prime } \) and \( \left( \bullet \right)^{\prime } \) denote differentiation with respect to the non-dimensional time \( \tau \). For succinct statement, the word “non-dimensional” is omitted in subsequent sections.

The averaging method is employed to conduct the dynamic analysis, which can not only find the response but also be helpful for analyzing the stability of the steady-state response. This method has also been adopted in Ref. [57] for analyzing nonlinear vibration isolator. The displacement response and velocity response are assumed to be

$$ x\left( \tau \right) = H\cos \left( {\varOmega \tau + \varphi } \right) $$
(17a)
$$ x^{\prime}\left( \tau \right) = - \varOmega H\sin \left( {\varOmega \tau + \varphi } \right) $$
(17b)

where the amplitude \( H \) and phase \( \varphi \) can vary with \( \tau \) slowly. Differentiating Eq. (17) with respect to \( \tau \) gives

$$ x^{\prime}\left( \tau \right) = H^{\prime}\cos \left( {\varOmega \tau + \varphi } \right) - \varphi^{\prime}H\sin \left( {\varOmega \tau + \varphi } \right) - \varOmega H\sin \left( {\varOmega \tau + \varphi } \right) $$
(18a)
$$ x^{\prime\prime}\left( \tau \right) = - \varOmega H^{\prime}\sin \left( {\varOmega \tau + \varphi } \right) - \varOmega \varphi^{\prime}H\cos \left( {\varOmega \tau + \varphi } \right) - \varOmega^{2} H\cos \left( {\varOmega \tau + \varphi } \right) $$
(18b)

Comparison of Eqs. (17b) and (18a) gives

$$ H^{\prime}\cos \left( {\varOmega \tau + \varphi } \right) - \varphi^{\prime}H\sin \left( {\varOmega \tau + \varphi } \right) = 0 $$
(19)

Substituting Eqs. (17) and (18b) into the dynamic equation Eq. (16) results in

$$ \varOmega H^{\prime}\sin \left( {\varOmega \tau + \varphi } \right) + \varOmega \varphi^{\prime}H\cos \left( {\varOmega \tau + \varphi } \right) = \varLambda \left( {H,\varphi ,\varPhi \left( {\varphi ,\tau } \right)} \right) $$
(20)

where

$$ \begin{aligned} \varLambda \left( {H,\varphi ,\varPhi } \right) & = - \varOmega^{2} H\cos \varPhi - 2\zeta \varOmega H\sin \varPhi + \beta H^{3} \cos^{3} \varPhi \\ & \quad + \rho H^{5} \cos^{5} \varPhi + a_{0} \cos \left( {\varPhi - \varphi } \right) \\ \end{aligned} $$
(21a)
$$ \varPhi \left( {\varphi ,\tau } \right) = \varOmega \tau + \varphi $$
(21b)

Combining Eqs. (19) and (20), one can obtain

$$ \left\{ {\begin{array}{*{20}l} {H^{\prime} = \frac{{\varLambda \left( {H,\varphi ,\varPhi \left( {\varphi ,\tau } \right)} \right)\sin \varPhi \left( {\varphi ,\tau } \right)}}{\varOmega }} \hfill \\ {\varphi^{\prime} = \frac{{\varLambda \left( {H,\varphi ,\varPhi \left( {\varphi ,\tau } \right)} \right)\cos \varPhi \left( {\varphi ,\tau } \right)}}{\varOmega H}} \hfill \\ \end{array} } \right. $$
(22)

Considering that \( H \) and \( \varphi \) vary slowly, it is feasible to use the average values of \( H^{\prime} \) and \( \varphi^{\prime} \) over a cycle of oscillation, which are calculated by

$$ \begin{aligned} H^{\prime} & = \frac{\varOmega }{2\pi }\int_{\tau }^{{\tau + \frac{2\pi }{\varOmega }}} {\frac{{\varLambda \left( {H,\varphi ,\varPhi \left( {\varphi ,\tau } \right)} \right)\sin \varPhi \left( {\varphi ,\tau } \right)}}{\varOmega }{\text{d}}\tau } = \frac{1}{2\pi \varOmega }\int_{0}^{2\pi } {\varLambda \left( {H,\varphi ,\varPhi } \right)\sin \varPhi {\text{d}}\varPhi } \\ & = - \zeta H + \frac{{a_{0} }}{2\varOmega }\sin \varphi \\ \end{aligned} $$
(23a)
$$ \begin{aligned} \varphi^{\prime} & = \frac{\varOmega }{2\pi }\int_{\tau }^{{\tau + \frac{2\pi }{\varOmega }}} {\frac{{\varLambda \left( {H,\varphi ,\varPhi \left( {\varphi ,\tau } \right)} \right)\cos \varPhi \left( {\varphi ,\tau } \right)}}{\varOmega H}} {\text{d}}\tau = \frac{1}{2\pi \varOmega H}\int_{0}^{2\pi } {\varLambda \left( {H,\varphi ,\varPhi } \right)\cos \varPhi } {\text{d}}\varPhi \\ & = - \frac{\varOmega }{2} + \frac{3\beta }{8\varOmega }H^{2} + \frac{5\rho }{16\varOmega }H^{4} + \frac{{a_{0} }}{2\varOmega H}\cos \varphi \\ \end{aligned} $$
(23b)

The steady-state response can be found by setting \( H^{\prime} = 0 \) and \( \varphi^{\prime} = 0 \) to give

$$ 2\zeta \varOmega H = a_{0} \sin \varphi $$
(24a)
$$ - \varOmega^{2} H + \frac{3}{4}\beta H^{3} + \frac{5}{8}\rho H^{5} = - a_{0} \cos \varphi $$
(24b)

Squaring Eqs. (24a) and (24b) and adding them can eliminate \( \varphi \) and result in an equation in \( H^{2} \):

$$ \frac{25}{64}\rho^{2} H^{10} + \frac{15}{16}\rho \beta H^{8} + \left( {\frac{9}{16}\beta^{2} - \frac{5}{4}\rho \varOmega^{2} } \right)H^{6} - \frac{3}{2}\beta \varOmega^{2} H^{4} + \left[ {\varOmega^{4} + \left( {2\zeta \varOmega } \right)^{2} } \right]H^{2} - a_{0}^{2} = 0 $$
(25)

Solving this equation gives the steady-state response amplitude, and substituting it into either Eqs. (24a) or (24b) gives the steady-state response phase. Figure 5 shows the comparison between the analytical and numerical responses. It can be seen that the analytical solutions are in good agreement with the numerical solutions (there is only a little gap at very low frequencies), which indicates the satisfactory accuracy of the averaging method employed in this paper. It can also be seen that both the amplitude and the phase of the high-frequency response gradually approach zero as the excitation frequency increases.

Fig. 5
figure5

Comparison between the analytical and numerical responses: a steady-state response amplitude and b steady-state response phase (\( \mu = {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3} \), \( \zeta = 0.12 \),\( a_{0} = 0.02 \))

The influences of excitation amplitude and damping on the frequency response are shown in Fig. 6. The frequency response curve has a potential trend of bending to the right, thereby probably resulting in a frequency range where three analytical response amplitudes coexist, which is attributed to the hardening stiffness; this bending causes a jump in the amplitude when the excitation frequency is swept from left to right or right to left, which is known as the jump phenomenon. Increasing excitation amplitude will cause both the peak response amplitude and the peak response frequency to increase and exacerbate the jump phenomenon. Increasing damping can reduce the peak response amplitude and peak response frequency and suppress the jump phenomenon, while it has very small effect on the high-frequency response amplitude.

Fig. 6
figure6

Frequency response curves: a under different excitation amplitudes (when \( \mu = {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3} \), \( \zeta = 0.1 \)) and b under different damping ratios (when \( \mu = {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3} \), \( a_{0} = 0.02 \))

Analysis for the peak response

The response amplitude equation Eq. (25) can also be expressed as a quadratic equation in \( \varOmega^{2} \), which leads to:

$$ \begin{aligned} \varOmega_{1,2} & = \left[ { \pm \sqrt {\left( {\frac{5}{8}\rho H^{4} + \frac{3}{4}\beta H^{2} - 2\zeta^{2} } \right)^{2} - \left( {\frac{25}{64}\rho^{2} H^{8} + \frac{15}{16}\rho \beta H^{6} + \frac{9}{16}\beta^{2} H^{4} - \frac{{a_{0}^{2} }}{{H^{2} }}} \right)} } \right. \\ & \quad + \left. {\left( {\frac{5}{8}\rho H^{4} + \frac{3}{4}\beta H^{2} - 2\zeta^{2} } \right)} \right]^{{\frac{1}{2}}} \\ \end{aligned} $$
(26)

\( \varOmega_{1} \) and \( \varOmega_{2} \) given in Eq. (26) can be seen as the abscissae of the two intersections of the amplitude-frequency curve and a horizontal line in \( \left( {\varOmega ,H} \right) \) plane. At the peak response point, the two intersections coincide with each other, in which case there must be \( \varOmega_{1} = \varOmega_{2} \), and thus the inner radicand in Eq. (26) becomes zero, which leads to an equation for the peak response amplitude (denoted as \( H_{p} \)):

$$ \frac{5}{2}\rho H_{p}^{6} + 3\beta H_{p}^{4} - 4\zeta^{2} H_{p}^{2} - \frac{{a_{0}^{2} }}{{\zeta^{2} }} = 0 $$
(27)

The frequency at which the peak response occurs (denoted as \( \varOmega_{p} \)) can also be obtained by setting \( \varOmega_{1} = \varOmega_{2} \) to give

$$ \varOmega_{p} = \sqrt {\frac{5}{8}\rho H_{p}^{4} + \frac{3}{4}\beta H_{p}^{2} - 2\zeta^{2} } $$
(28)

Figure 7 shows the peak response amplitude and peak response frequency with varying the damping ratio and excitation amplitude. The amplitude and frequency of the peak response are both increased by decreasing \( \zeta \) or increasing \( a_{0} \).

Fig. 7
figure7

Peak response versus excitation amplitude \( a_{0} \) and damping ratio \( \zeta \) (when \( \mu = {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3} \)): a peak response amplitude and b frequency at the peak response

The above procedure for finding the peak response is based on an assumption that the frequency response curve does have a peak that occurs at \( \varOmega > 0 \). However, if \( \zeta \) is too large or \( a_{0} \) is too small, it is very likely that the frequency response curve becomes a monotonic decreasing function, in which case there is no peak response but only maximum response at \( \varOmega = 0 \). In fact, when under the critical condition that makes the frequency response curve monotonically decrease, the “peak response frequency” calculated by Eq. (28) is equal to zero. Solving \( \varOmega_{p} = 0 \) gives

$$ H_{p}^{2} = \frac{1}{\rho }\left( {\sqrt {\frac{9}{25}\beta^{2} + \frac{16}{5}\rho \zeta^{2} } - \frac{3}{5}\beta } \right) $$
(29)

Substituting Eq. (29) into Eq. (27) gives the critical parameter condition for the disappearance of peak response:

$$ a_{0} = 2\zeta^{2} \sqrt {\left( {\frac{1}{\rho }\sqrt {\frac{9}{25}\beta^{2} + \frac{16}{5}\rho \zeta^{2} } - \frac{3\beta }{5\rho }} \right)} $$
(30)

If \( a_{0} \) is larger than the expression given in Eq. (30), the peak response exists, and the amplitude and frequency of the peak response are given in Eqs. (27) and (28); if \( a_{0} \) is smaller than the expression given in Eq. (30), the frequency response curve decreases monotonically, and thus, the maximum response occurs at \( \varOmega = 0 \) (i.e., there is no peak response in the conventional sense).

Stability boundary and jump frequencies

The stability of the analytical steady-state response can be determined by inspection of the eigenvalues of the Jacobian matrix. Calculating the Jacobian matrix of Eq. (23) and substituting \( \cos \varphi \) and \( \sin \varphi \) as functions of \( H \) into it gives

$$ {\mathbf{J}} = \left[ {\begin{array}{*{20}c} { - \zeta } & {\frac{\varOmega H}{2} - \frac{3\beta }{8\varOmega }H^{3} - \frac{5\rho }{16\varOmega }H^{5} } \\ {\frac{9\beta }{8\varOmega }H + \frac{25\rho }{16\varOmega }H^{3} - \frac{\varOmega }{2H}} & { - \zeta } \\ \end{array} } \right] $$
(31)

The eigenvalues (denoted as \( \sigma \)) are solved from \( \sigma^{2} - {\text{tr}}\left( {\mathbf{J}} \right)\sigma + \det \left( {\mathbf{J}} \right) = 0 \). The trace of the Jacobian matrix \( {\text{tr}}\left( {\mathbf{J}} \right) = - 2\zeta \) is definitely negative, so whether or not all the eigenvalues have negative real parts depends on the sign of \( \det \left( {\mathbf{J}} \right) \). If \( \det \left( {\mathbf{J}} \right) > 0 \), all the eigenvalues have negative real parts, and thus, the steady-state response is stable; If \( \det \left( {\mathbf{J}} \right) < 0 \), one of the eigenvalues has positive real part, in which case the steady-state \( \left( {a,\varphi } \right) \) is a unstable saddle point. Therefore, the boundary between the stable and unstable regions is given by \( \det \left( {\mathbf{J}} \right) = 0 \), i.e.,

$$ \frac{125}{64}\rho^{2} H^{8} + \frac{15}{4}\rho \beta H^{6} + \left( {\frac{27}{16}\beta^{2} - \frac{15}{4}\rho \varOmega^{2} } \right)H^{4} - 3\beta \varOmega^{2} H^{2} + \varOmega^{2} \left( {\varOmega^{2} + 4\zeta^{2} } \right) = 0 $$
(32)

Figure 8a, b shows the unstable regions when under the same parameter conditions as Fig. 6a, b, respectively. It can be seen that the unstable region is reduced by increasing the damping ratio, but independent of the excitation amplitude (this can also be seen from the stability boundary equation Eq. (32) in which \( a_{0} \) does not appear). The most important observation is that, when the bending of the frequency response curve is serious enough to induce jump phenomenon, the intermediate branch of the frequency response curve is always unstable, whereas the upper and lower branches are stable. This observation can be exploited to analytically calculate the jump frequencies.

Fig. 8
figure8

Unstable regions in the \( \left( {H,\varOmega } \right) \) plane: a when \( \mu = {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3} \),\( \zeta = 0.1 \) and b under different values of \( \zeta \) with \( \mu = {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3} \). The value of \( a_{0} \) does not need to be mentioned because the unstable region is independent of the excitation amplitude

The intermediate branch (if exists) of the frequency response curve, which is always unstable, lies between the jump-up and jump-down frequencies, so the jump phenomena must occur at the two intersections of the frequency response curve and the stability boundary. Therefore, the jump frequencies can be obtained by solving simultaneous equations of Eqs. (25) and (32). The jump-down and jump-up frequencies (\( \varOmega_{d} \) and \( \varOmega_{u} \)) under different excitation amplitudes and damping ratios are shown in Fig. 10, in which the circles denote the critical conditions that trigger jump phenomenon, the solid lines on the right of the circles denote jump-down frequency, and the dashed lines denote jump-up frequency. It can be seen that both the jump-down and jump-up frequencies are increased when increasing the excitation amplitude, and smaller damping also gives rise to larger jump frequencies, but the effect of damping on the jump-up frequency is not very significant. The minimum excitation amplitude that can trigger jump phenomenon increases when increasing the damping ratio.

Vibration isolation performance

The vibration isolation performance in motion transmission problem is quantified by the motion transmissibility, which is defined as the ratio of the absolute motion amplitude of the isolated mass to the base motion amplitude. Since the isolated mass and the base vibrate at the same frequency, we can directly use acceleration amplitude to give the definition of motion transmissibility, which is calculated as

$$ \begin{aligned} T & = \frac{{\left| {a_{0} \cos \left( {\varOmega \tau } \right) + x^{\prime\prime}} \right|}}{{a_{0} }} = \frac{{\sqrt {\varOmega^{4} H^{2} - 2\varOmega^{2} Ha_{0} \cos \varphi + a_{0}^{2} } }}{{a_{0} }} \\ & = \sqrt {\left( {\frac{\varOmega H}{{a_{0} }}} \right)^{2} \left( {\frac{5}{4}\rho H^{4} + \frac{3}{2}\beta H^{2} - \varOmega^{2} } \right) + 1} \\ \end{aligned} $$
(33)

For a given QZS isolator under a given excitation amplitude, the transmissibility varies with the excitation frequency, and therefore, the overall isolation performance is actually characterized by the transmissibility-frequency curve. Figure 9a shows the comparison between the transmissibility curves of the QZS isolator and the equivalent linear isolator; Fig. 9b, c shows the transmissibility curves of the QZS isolator under different parameter conditions. Note that the transmissibility shown in Fig. 9 takes the decibel form (\( T\left( {\text{dB}} \right) = 20\lg T \)). For the convenience of discussion, the frequency and transmissibility at the peak of the transmissibility-frequency curve are called the peak frequency and peak transmissibility, respectively. It can be seen that: (a) the QZS isolator has a much better isolation performance than the equivalent linear isolator considering its much wider frequency range of isolation and lower peak transmissibility; (b) increasing excitation amplitude results in a reduction in the frequency range of isolation and an increase in the peak transmissibility, which degrades the isolation performance, while excitation amplitude has little effect on the high-frequency transmissibility; (c) increasing damping ratio can effectively suppress the peak transmissibility and peak frequency, but it will cause the high-frequency transmissibility to increase. Therefore, smaller excitation amplitude and appropriate damping are beneficial for the overall isolation performance.

Fig. 9
figure9

Transmissibility curves (decibel form) of the QZS isolator: a compared with the equivalent linear isolator (\( \mu = {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3} \), \( \zeta = 0.12 \),\( a_{0} = 0.02 \)), b under different excitation amplitudes with \( \mu = {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3} \), \( \zeta = 0.1 \) and c under different damping ratios with \( \mu = {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3} \), \( a_{0} = 0.02 \)

Peak transmissibility is an important measure of the isolation performance. Comparison of Figs. 6 and 9b, c suggests that the transmissibility reaches its peak value approximately when the peak response occurs. Therefore, the peak transmissibility (denoted as \( T_{p} \)) can be calculated by substituting for \( H_{p} \) and \( \varOmega_{p} \) from Eqs. (27) and (28) into the transmissibility expression Eq. (33) to give

$$ T_{p} = \sqrt {\frac{{H_{p}^{2} }}{{a_{0}^{2} }}\left[ {\left( {\frac{5}{8}\rho H_{p}^{2} + \frac{3}{4}\beta } \right)^{2} H_{p}^{4} - 4\zeta^{4} } \right] + 1} $$
(34)

where \( H_{p} \) is solved from Eq. (27). Note that this expression for peak transmissibility is not very accurate when under a very large damping or a very small excitation amplitude.

The frequency above which isolation can occur is another very important measure of the isolation performance, which characterizes the effective frequency range with vibration isolation effect; this key frequency is called the isolation frequency hereafter. It is obvious that, if there is no jump phenomenon, the isolation frequency is equal to the crossing frequency, namely the frequency at which the transmissibility curve crosses the 0 dB line. However, if jump phenomenon occurs, there are three possible values of transmissibility at frequencies between the crossing frequency and the jump-down frequency, and the upper and intermediate values are larger than unity (decibel form larger than zero), in which case the isolation frequency should be equal to the jump-down frequency. The method of calculating the jump-down frequency has be given in Sect. 3.3; in order to analytically find the isolation frequency, it is necessary to calculate the crossing frequency first. The relationship between the crossing frequency (denoted as \( \varOmega_{c} \)) and crossing amplitude (the response amplitude at the crossing frequency, denoted as \( H_{c} \)) is obtained by setting \( T = 1 \) to give

$$ \frac{5}{4}\rho H_{c}^{4} + \frac{3}{2}\beta H_{c}^{2} - \varOmega_{c}^{2} = 0 $$
(35)

The response amplitude equation Eq. (25) can be rewritten in the following form:

$$ q_{5} H^{6} Q\left( {H,\varOmega } \right) + q_{4} H^{4} Q\left( {H,\varOmega } \right) + q_{3} H^{2} Q\left( {H,\varOmega } \right) + q_{2} Q\left( {H,\varOmega } \right) + r_{1} H^{2} + r_{0} = 0 $$
(36)

where

$$ Q\left( {H,\varOmega } \right) = \frac{5}{4}\rho H^{4} + \frac{3}{2}\beta H^{2} - \varOmega^{2} $$
(37)
$$ \left[ {\begin{array}{*{20}c} {q_{5} \left( \varOmega \right)} \\ {q_{4} \left( \varOmega \right)} \\ {q_{3} \left( \varOmega \right)} \\ {q_{2} \left( \varOmega \right)} \\ {r_{1} \left( \varOmega \right)} \\ {r_{0} \left( \varOmega \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\frac{5}{4}\rho } & 0 & 0 & 0 & 0 & 0 \\ {\frac{3}{2}\beta } & {\frac{5}{4}\rho } & 0 & 0 & 0 & 0 \\ { - \varOmega^{2} } & {\frac{3}{2}\beta } & {\frac{5}{4}\rho } & 0 & 0 & 0 \\ 0 & { - \varOmega^{2} } & {\frac{3}{2}\beta } & {\frac{5}{4}\rho } & 0 & 0 \\ 0 & 0 & { - \varOmega^{2} } & {\frac{3}{2}\beta } & 1 & 0 \\ 0 & 0 & 0 & { - \varOmega^{2} } & 0 & 1 \\ \end{array} } \right]^{ - 1} \left[ {\begin{array}{*{20}c} {\frac{25}{64}\rho^{2} } \\ {\frac{15}{16}\rho \beta } \\ {\frac{9}{16}\beta^{2} - \frac{5}{4}\rho \varOmega^{2} } \\ { - \frac{3}{2}\beta \varOmega^{2} } \\ {\varOmega^{2} \left( {\varOmega^{2} + 4\zeta^{2} } \right)} \\ { - a_{0}^{2} } \\ \end{array} } \right] $$
(38)

Substituting the relationship between \( \varOmega_{c} \) and \( H_{c} \) from Eq. (35), i.e., \( Q\left( {H_{c} ,\varOmega_{c} } \right) = 0 \), into Eq. (36) leads to

$$ H_{c}^{2} = - \frac{{r_{0} \left( {\varOmega_{c} } \right)}}{{r_{1} \left( {\varOmega_{c} } \right)}} = \frac{{4{\mkern 1mu} a_{0}^{2} }}{{16{\mkern 1mu} \zeta^{2} \varOmega_{c}^{2} {\mkern 1mu} + \varOmega_{c}^{4} }} $$
(39)

Substituting Eq. (39) into Eq. (35) results in an equation in the crossing frequency:

$$ \varOmega_{c}^{10} + 32\zeta^{2} \varOmega_{c}^{8} + 256\zeta^{4} \varOmega_{c}^{6} - 6\beta a_{0}^{2} \varOmega_{c}^{4} - 96\beta \zeta^{2} a_{0}^{2} \varOmega_{c}^{2} - 20\rho a_{0}^{4} = 0 $$
(40)

Figure 10 shows the isolation frequency under different excitation amplitudes and damping ratios, in which the circles denote the critical conditions that trigger the jump phenomenon; on the left of the circle, the crossing frequency is taken as the isolation frequency, while on the right of the circle, the jump-down frequency is taken as the isolation frequency. It is seen that the isolation frequency is increased when increasing the excitation amplitude or decreasing the damping ratio.

Fig. 10
figure10

The isolation frequency and jump frequencies versus excitation amplitude when under different damping ratios (\( \mu = {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3} \)). The circles denote the critical conditions that trigger jump phenomenon. Solid lines denote isolation frequency; solid lines on the right of the circles also denote jump-down frequency; solid lines on the left of the circles also denote crossing frequency; dashed lines denote jump-up frequency

Design of the experimental prototype

Stiffness formulation for the transverse groove spring

The most important components of the experimental prototype are the vertical spring and the four lateral springs. The transverse groove springs are utilized to construct the experimental prototype in this study. As its name implies, the transverse groove spring is fabricated by cutting some pieces away from a hollow cylinder, as depicted in Fig. 11a. Compared with conventional springs, the transverse groove spring mainly has the following advantages: (1) it has a compact structure and thus can save space; (2) it has strong design ability with many key structural parameters which can be well designed to obtain the required stiffness; (3) the assembly structure can be easily integrated with the spring structure when designing it, thereby reducing the number of mechanical components and the assembly complexity.

Fig. 11
figure11

a General structure of the transverse groove spring; b simplified mechanical analysis model of a quarter of a layer of the transverse groove spring

The stiffness of the transverse groove spring is dominantly determined by the beam structure of each layer. The key design parameters on which the stiffness of the transverse groove spring depends include the outer diameter \( D \), the inner diameter \( d \), the thickness of the beam of each layer \( h \), the number of layers \( n \) and material properties. It is instructive to analytically find the relationship between the spring stiffness and these key parameters. Since all the layers are identical to each other and each individual layer is symmetrical with respect to the axis of the spring, we first analyze a quarter of a layer, which is modeled as a 1/4 circular beam (Fig. 11b) with rectangular cross-section. The radius of the 1/4 circular beam and the width of the rectangular cross-section are \( R = {{\left( {D + d} \right)} \mathord{\left/ {\vphantom {{\left( {D + d} \right)} 4}} \right. \kern-0pt} 4} \) and \( e = {{\left( {D - d} \right)} \mathord{\left/ {\vphantom {{\left( {D - d} \right)} 2}} \right. \kern-0pt} 2} \), respectively. In our design of the transverse groove spring, we set the width \( e \) larger than the thickness \( h \), which is beneficial for the stability of the transverse groove spring when compressed. Due to the structural symmetry of each layer and the fact that each end of the 1/4 circular beam is fixed to a small wedge between adjacent layers, it is reasonable to think that the two ends of the beam do not experience rotation. Generally speaking, the constraint at one end of a curved beam consists of the forces in three directions and the moments in three directions. However, for the 1/4 circular beam being considered currently, only the force in the axial direction of the spring and the moment around the radial direction of the spring are predominant, as shown in Fig. 11b where the constraint at one end of the beam is abstracted as a force \( F_{0} \) and a moment \( M_{0} \). The angle \( \phi \) is used to indicate and denote a specific location on the 1/4 circular beam, for the convenience of description. The bending moment and the torque at location \( \phi \) are, respectively:

$$ M\left( \phi \right) = F_{0} R\sin \phi - M_{0} \cos \phi $$
(41)
$$ T\left( \phi \right) = F_{0} R\left( {1 - \cos \phi } \right) - M_{0} \sin \phi $$
(42)

The arc length from the force bearing point to the location \( \phi \) is \( s = R\phi \), so we have \( {\text{d}}s = R{\text{d}}\phi \). When the infinitesimal element \( {\text{d}}s \) is subjected to the bending moment \( M\left( \phi \right) \), the bending angle of this infinitesimal element is

$$ {\text{d}}\theta { = }\frac{M\left( \phi \right)}{EI}{\text{d}}s = \frac{M\left( \phi \right)}{EI}R{\text{d}}\phi $$
(43)

where \( E \) is the Young’s modulus and \( I \) is the cross-section area moment of inertia. When the infinitesimal element \( {\text{d}}s \) is subjected to the torque \( T\left( \phi \right) \), the angle of twist of this infinitesimal element is

$$ {\text{d}}\psi { = }\frac{T\left( \phi \right)}{{\gamma Geh^{3} }}{\text{d}}s = \frac{T\left( \phi \right)}{{\gamma Geh^{3} }}R{\text{d}}\phi $$
(44)

where \( G \) is the shear modulus and \( \gamma \) is a factor depending only on the ratio of the long side length to the short side length of the cross-section (\( {e \mathord{\left/ {\vphantom {e h}} \right. \kern-0pt} h} \)). According to Mechanics of Materials, some values of \( \gamma \) with respect to the ratio of the long side length to the short side length are shown in Table 1.

Table 1 \( \gamma \) value with respect to \( {e \mathord{\left/ {\vphantom {e h}} \right. \kern-0pt} h} \)

The strain energy of an infinitesimal element includes the bending strain energy, torsion strain energy and shear strain energy; for a slender beam, the shear strain energy is very small compared to the bending and torsion strain energy and can be ignored. Therefore, the strain energy of the infinitesimal element \( {\text{d}}s \) is

$$ \begin{aligned} {\text{d}}W & = \frac{1}{2}M\left( \phi \right){\text{d}}\theta + \frac{1}{2}T\left( \phi \right){\text{d}}\psi = \left\{ {\frac{{\left[ {M\left( \phi \right)} \right]^{2} }}{2EI} + \frac{{\left[ {T\left( \phi \right)} \right]^{2} }}{{2\gamma Geh^{3} }}} \right\}R{\text{d}}\phi \\ & = \frac{R}{{Eeh^{3} }}\left\{ {6\left[ {M\left( \phi \right)} \right]^{2} + \frac{1 + \nu }{\gamma }\left[ {T\left( \phi \right)} \right]^{2} } \right\}{\text{d}}\phi \\ \end{aligned} $$
(45)

where \( I = {{eh^{3} } \mathord{\left/ {\vphantom {{eh^{3} } {12}}} \right. \kern-0pt} {12}} \) and \( G = {E \mathord{\left/ {\vphantom {E {\left[ {2\left( {1 + \nu } \right)} \right]}}} \right. \kern-0pt} {\left[ {2\left( {1 + \nu } \right)} \right]}} \) (\( \nu \) is Poisson’s ratio) are used. The total strain energy of the whole 1/4 circular beam is thus

$$ W \approx \frac{R}{{Eeh^{3} }}\int_{0}^{{\frac{\pi }{2}}} {\left\{ {6\left[ {M\left( \phi \right)} \right]^{2} + \frac{1 + \nu }{\gamma }\left[ {T\left( \phi \right)} \right]^{2} } \right\}} {\text{d}}\phi $$
(46)

According to Castigliano’s second theorem, the angular displacement corresponding to the moment \( M_{0} \) is calculated by

$$ \frac{\partial W}{{\partial M_{0} }} = \frac{2R}{{Eeh^{3} }}\int_{0}^{{\frac{\pi }{2}}} {\left[ {6M\left( \phi \right)\frac{\partial M\left( \phi \right)}{{\partial M_{0} }} + \frac{1 + \nu }{\gamma }T\left( \phi \right)\frac{\partial T\left( \phi \right)}{{\partial M_{0} }}} \right]} {\text{d}}\phi $$
(47)

It has been mentioned earlier in this subsection that the two ends of the 1/4 circular beam do not experience rotation, so the angular displacement corresponding to the moment \( M_{0} \) must be zero. By substituting for \( M\left( \phi \right) \) and \( T\left( \phi \right) \) from Eqs. (41) and (42) into Eq. (47) and solving \( {{\partial W} \mathord{\left/ {\vphantom {{\partial W} {\partial M_{0} }}} \right. \kern-0pt} {\partial M_{0} }} = 0 \), we can obtain the relationship between \( M_{0} \) and \( F_{0} \):

$$ M_{0} = \frac{2}{\pi }RF_{0} $$
(48)

According to Castigliano’s second theorem again, the displacement corresponding to the force \( F_{0} \) is

$$ \begin{aligned} \delta_{0} & = \frac{\partial W}{{\partial F_{0} }} = \frac{2R}{{Eeh^{3} }}\int_{0}^{{\frac{\pi }{2}}} {\left[ {6M\left( \phi \right)\frac{\partial M\left( \phi \right)}{{\partial F_{0} }} + \frac{1 + \nu }{\gamma }T\left( \phi \right)\frac{\partial T\left( \phi \right)}{{\partial F_{0} }}} \right]} {\text{d}}\phi \\ & = \left[ {\frac{3}{2}\pi - \frac{12}{\pi } + \frac{1 + \nu }{\gamma }\left( {\frac{3}{2}\pi - \frac{2}{\pi } - 4} \right)} \right]\frac{{R^{3} F_{0} }}{{Eeh^{3} }} \\ \end{aligned} $$
(49)

The parameter \( \gamma \) is only related to the ratio of width to thickness of the beam cross-section (\( {e \mathord{\left/ {\vphantom {e h}} \right. \kern-0pt} h} \)), as shown in Table 1. It would be beneficial for subsequent design if \( \gamma \) is expressed as an explicit function of \( {e \mathord{\left/ {\vphantom {e h}} \right. \kern-0pt} h} \). To achieve this, we can fit the relationship between \( \gamma \) and \( {e \mathord{\left/ {\vphantom {e h}} \right. \kern-0pt} h} \), and the fitted \( \gamma \) − \( {e \mathord{\left/ {\vphantom {e h}} \right. \kern-0pt} h} \) relationship is

$$ \gamma = \frac{{p_{1} }}{{{e \mathord{\left/ {\vphantom {e h}} \right. \kern-0pt} h} + p_{2} }} + \frac{1}{3} = \frac{13.8322}{{1 - 67.1141\left( {{e \mathord{\left/ {\vphantom {e h}} \right. \kern-0pt} h}} \right)}} + \frac{1}{3} $$
(50)

The above analysis is for the 1/4 circular beam; now we consider the whole spring. Assume that a force \( F_{s} \) is applied on the transverse groove spring in the axial direction, and the deformation of the spring is denoted as \( \delta_{s} \); the relationship between \( F_{0} \) and \( F_{s} \), and the relationship between \( \delta_{0} \) and \( \delta_{s} \) are \( F_{0} = {{F_{s} } \mathord{\left/ {\vphantom {{F_{s} } 4}} \right. \kern-0pt} 4} \) and \( \delta_{0} = {{\delta_{s} } \mathord{\left/ {\vphantom {{\delta_{s} } n}} \right. \kern-0pt} n} \) where \( n \) is the number of layers. The final formula for the stiffness of the transverse groove spring is derived by calculating the ratio of \( F_{s} \) to \( \delta_{s} \), which gives

$$ \begin{aligned} K_{tgs} & = \frac{{F_{s} }}{{\delta_{s} }} = \frac{{4F_{0} }}{{n\delta_{0} }} = {{\frac{{4Eeh^{3} }}{{nR^{3} }}} \mathord{\left/ {\vphantom {{\frac{{4Eeh^{3} }}{{nR^{3} }}} {\left[ {\frac{3}{2}\pi - \frac{12}{\pi } + \frac{1 + \nu }{\gamma }\left( {\frac{3}{2}\pi - \frac{2}{\pi } - 4} \right)} \right]}}} \right. \kern-0pt} {\left[ {\frac{3}{2}\pi - \frac{12}{\pi } + \frac{1 + \nu }{\gamma }\left( {\frac{3}{2}\pi - \frac{2}{\pi } - 4} \right)} \right]}} = \frac{52.7919}{{11.7814 + {{\left( {1 + \nu } \right)} \mathord{\left/ {\vphantom {{\left( {1 + \nu } \right)} \gamma }} \right. \kern-0pt} \gamma }}} \cdot \frac{{Eeh^{3} }}{{nR^{3} }} \\ & = \frac{{Eh^{3} \left( {D - d} \right)\left[ {23930.4134h - 18896.4312\left( {D - d} \right)} \right]}}{{n\left( {D + d} \right)^{3} \left[ {\left( {167.8898 + \nu } \right)h - \left( {165.34 + 33.5571\nu } \right)\left( {D - d} \right)} \right]}} \\ \end{aligned} $$
(51)

Design of the springs

The stiffness ratio \( \lambda \) (ratio of the lateral spring stiffness to the vertical spring stiffness) is closely related to the geometrical parameter \( \mu \) as given by Eq. (9) so that the quasi-zero stiffness can be realized. Note that too small value of \( \mu \) will lead to excessive deformation of the lateral spring; in order to avoid this consequence which might cause damage to the lateral spring, we design \( \mu \) value within a reasonable range of 0.85–0.9, and thus, the stiffness ratio \( \lambda \) is calculated to be within the range of 1.417–2.25.

Since the masses of springs have not been taken into consideration in the theoretical analysis, the material selected for fabricating the transverse groove springs should possess a relatively low density. As is well known, the aluminum alloy is a kind of alloy material with low density and has good processability and excellent corrosion resistance. Due to its favorable material properties, the aluminum alloy is used to fabricate the transverse groove springs for the experimental prototype in this study, whose Young’s modulus is 71GPa and Poisson’s ratio is 0.33. We preliminarily choose appropriate values of the key structural parameters of the transverse groove springs according to Eq. (51) to obtain the desired stiffness ratio (1.417–2.25), which are shown in Table 2.

Table 2 Key design parameters of the springs

The analytical stiffnesses of the lateral and vertical springs calculated using Eq. (51) are 13.7855 N/mm and 6.9899 N/mm, respectively, and thus, the stiffness ratio is 1.972.

The actual structures of the lateral and vertical springs are, of course, a little more complex than shown in Fig. 11a, for the reason that both ends of the lateral spring should be able to rotate around two parallel axes and both ends of the vertical spring should be able to be assembled with other parts. Figure 12a, b shows the actual structures of the lateral and vertical springs. The distance between the axes of the revolution joints at both ends of the lateral spring is \( L_{0} = 100.3\;{\text{mm}} \); the length of the vertical spring is 118 mm.

Fig. 12
figure12

Finite element mesh diagrams of the springs in ABAQUS: a lateral spring and b vertical spring. The simulated deformations under different static loading forces and the fitted force–deformation relationship: c lateral spring and d vertical spring

In order to obtain more accurate stiffnesses of the springs designed in this study and validate the analytical stiffness formula of the transverse groove spring, simulations are conducted by using the finite element analysis software ABAQUS. The process of obtaining the simulated stiffness is as follows: we apply various values of static axial load on one end of the spring with the other end fixed and measure the static displacement of the loaded end, which results in a set of force–deformation data; then, we perform a linear fitting using this set of data, and the slope of the fitted equation is taken as the simulated stiffness. The simulated deformations under various values of force and the fitted force–deformation relationship are shown in Fig. 12c, d. The simulated stiffnesses of the lateral spring and vertical spring are 12.8249 N/mm and 6.6142 N/mm, respectively. The relative error between the analytical stiffness and the simulated stiffness is shown in Table 3. It can be seen that the analytical stiffness formula for the transverse groove spring has satisfactory accuracy, which is thus able to offer preliminary guidelines on the design of transverse groove spring for researchers.

Table 3 Comparison between the analytical stiffness and simulated stiffness

Prototype design

After the lateral spring and vertical spring have been designed and the actual stiffness ratio is \( \lambda = {{K_{l} } \mathord{\left/ {\vphantom {{K_{l} } {K_{v} }}} \right. \kern-0pt} {K_{v} }} = 1.939 \), the geometrical parameter \( \mu \) that ensures quasi-zero stiffness should be calculated as \( \mu = {{4\lambda } \mathord{\left/ {\vphantom {{4\lambda } {\left( {1 + 4\lambda } \right)}}} \right. \kern-0pt} {\left( {1 + 4\lambda } \right)}} = 0.886 \). However, if the isolator prototype is really designed to possess quasi-zero stiffness, a small manufacturing or assembling tolerance might give rise to negative stiffness, which is undesirable because of the instability it implies. Consequently, it is better to make the isolator prototype possess a very small positive stiffness, which can be done by increasing the value of \( \mu \) slightly. The actual horizontal distance between the axes of the revolution joints at both ends of the lateral spring is designed to be \( L_{1} = 89\;{\text{mm}} \), and thus, we have \( \mu = {{L_{1} } \mathord{\left/ {\vphantom {{L_{1} } {L_{0} }}} \right. \kern-0pt} {L_{0} }} = 0.8873 \), which is a litter larger than 0.886. Figure 13 shows the exploded view of the designed QZS isolator prototype on which the isolated mass block has been mounted (the holding frame and the pins are not shown). The prototype is composed of four lateral springs, four lateral spring seats, a vertical spring, a vertical spring seat, eight pins, a loading platform, a holding frame and four adjustable threaded rods. The mass block is mounted on the loading platform. The adjustable threaded rods are used to adjust the height of the bottom of the vertical spring so that the loading platform is at the same height as the lateral spring seats when the mass block is mounted, which ensures that the lateral springs are horizontal at the static equilibrium position. All the parts are made of aluminum alloy except that the mass block is made of steel.

Fig. 13
figure13

Exploded view of the designed QZS isolator prototype and mass block (the holding frame and the pins are not shown)

The actual parameters of the isolator prototype and the isolated mass are shown in Table 4. Note that the isolated mass \( m \) is the sum of the masses of the mass block and the loading platform.

Table 4 Actual parameters of the isolator prototype

Based on the actual parameters of the isolator prototype, it can be thought that the isolator prototype approximately possesses quasi-zero stiffness. In order to further confirm this, we have conducted quasi-static experiments with different loading velocities to test the actual force–displacement relationship; the displacement range for the quasi-static tests is from − 15 to 15 mm. Although different loading velocities inevitably cause differences in the quasi-static tested results, the tested force–displacement curves (Fig. 14) still indicate that the zero dynamic stiffness can be roughly achieved at the equilibrium point.

Fig. 14
figure14

Tested force–displacement curves of the isolator prototype with different loading velocities

Experiments

Experimental principle

The experiments are conducted by means of an electrodynamic shaker which can generate sinusoidal base excitation for the isolator prototype. The isolated mass block is mounted on the loading platform of the isolator prototype, and the bottom of the isolator prototype is attached to the shaker table. The block diagram of the experimental principle is depicted in Fig. 15a, which consists of a computer, a multifunctional data acquisition instrument, a power amplifier, a shaker, the isolator prototype with the mass block and two accelerometers. The multifunctional data acquisition instrument utilized in the experiments has the functions of data acquisition, data analysis and control signal generation and thus can be treated as not only a data acquisition unit but also data analyzer and controller. The excitation signal is inputted via the computer and then transmitted to the multifunctional data acquisition instrument; this control signal is amplified by the power amplifier to drive the shaker table. One of the accelerometers is placed at the bottom of the isolator prototype, and another is placed at the top of the mass block. The former accelerometer measures the acceleration signal of the shaker table; this signal is fed back to the multifunctional data acquisition instrument for negative feedback control to improve the motion accuracy of the shaker table. The latter accelerometer is used to measure the acceleration signal of the mass block; this signal is transmitted to the multifunctional data acquisition instrument for data processing. Figure 15b shows the photograph of the experimental setup; Fig. 15c, d shows the photographs of the lateral spring and vertical spring, respectively.

Fig. 15
figure15

a Experimental principle; b experimental setup; c lateral spring; d vertical spring

Experimental results and discussions

The springs are made of aluminum alloy, and the mass block is made of steel. The material properties and the geometrical parameters are given in Tables 2 and 4. Three sets of experiments on the QZS isolator prototype have been conducted; each set comprises a series of sinusoidal excitation tests with the same excitation amplitude (i.e., the acceleration amplitude of the shaker table) but different excitation frequencies. The excitation amplitudes of the three sets of experiments are 1.2 m/s2, 2.5 m/s2 and 3.8 m/s2, respectively. The excitation frequency ranges from 2 to 15 Hz; the frequency interval is 0.25 Hz for 2–5 Hz and is 2 Hz for 5–15 Hz. Experiments on the equivalent linear isolator have also been conducted for comparison; the linear isolator prototype is obtained by simply removing the four lateral springs from the QZS isolator prototype; the excitation amplitude for the linear isolator is 2.5 m/s2; and the excitation frequency ranges from 2 to 15 Hz with an interval of 1 Hz.

The experimental transmissibility curves (dB) of the QZS isolator prototype under different excitation amplitudes are plotted in Fig. 16, and the experimental transmissibility curve (dB) of the equivalent linear isolator prototype is also presented in the same figure for direct comparison. At ultra-low frequencies (for example 2 Hz), the transmissibility of the QZS isolator is a little higher than that of the linear isolator; however, after the excitation frequency is increased to 3 Hz, the transmissibility of the QZS isolator starts to decrease rapidly, whereas the transmissibility of the linear isolator keeps increasing until 6 Hz and then decreases, which results in a direct observation that the experimental transmissibility of the QZS isolator is much lower than the equivalent linear isolator in a very wide frequency range. The maximum gap between the transmissibility of the QZS isolator and the linear isolator occurs at the resonant frequency of the linear isolator; at higher frequencies, this gap gradually decreases, and it can be forecasted that the gap may disappear at very high frequencies, which implies that the advantages of the QZS isolator over the equivalent linear isolator are mainly reflected in low frequencies (note that the word “low” is a relative term and means “not very high”). There are two important measures of the overall vibration isolation performance: isolation frequency and peak transmissibility; the former means the frequency above which isolation can occur (for the experimental results presented in Fig. 16, the isolation frequency can be simply seen as the frequency at which the transmissibility curve crosses 0 dB line), which characterizes the effective frequency range with vibration isolation effect; the latter means the transmissibility at the peak of the transmissibility-frequency curve. In general, the smaller the two indices are, the better the overall isolation performance is. Table 5 shows the experimental isolation frequency and experimental peak transmissibility (dB) of the QZS isolator and the equivalent linear isolator; it can be seen that the isolation frequency and peak transmissibility of the QZS isolator are only 34.52–39.53% and 23.76–45.05% of those of the equivalent linear isolator, respectively, which is another clear manifestation of the good vibration isolation performance of the QZS isolator.

Fig. 16
figure16

Experimental transmissibility curves (dB) under different excitation amplitudes

Table 5 Experimental isolation frequency and experimental peak transmissibility (dB) of the QZS isolator and the equivalent linear isolator

The following focus is placed on the consequences of different excitation amplitudes to seek more insightful findings. Obviously, as a nonlinear dynamical system, the dynamic behavior of the QZS isolator is related to the excitation amplitude, and thus, different excitation amplitudes lead to slightly different isolation performances as indicated in Fig. 16. It is seen from Table 5 that both the experimental isolation frequency and experimental peak transmissibility of the QZS isolator are increased when increasing the excitation amplitude, which is agreement with the theoretical findings in Sect. 3.4. However, at higher frequencies (note that the word “higher” is a relative term and means the frequencies higher than the isolation frequency), a discrepancy between the experimental and theoretical results arises: The theoretical analysis indicates that the high-frequency transmissibility of the QZS isolator is expected to be almost unaffected by the excitation amplitude (Fig. 9b), whereas in the experiments, smaller excitation amplitude leads to higher transmissibility when at high frequencies (at frequencies higher than about 3.5 Hz). In order to ascertain the reason for this discrepancy, the experimental results and the analytically predicted results obtained by following the procedure in Sect. 3 are directly compared, as shown in Fig. 17; the non-dimensional structural parameter \( \mu \) and stiffness ratio \( \lambda \) for theoretical calculation are given in Sect. 4.3; the non-dimensional excitation amplitude \( a_{0} \) is calculated to be 0.0093, 0.0194 and 0.0295, respectively; the damping ratios of the QZS isolator and the linear isolator are set to 0.142 and 0.103, respectively, to match the peak value of the experimental transmissibility curve. Some errors are seen between the experimental results and the theoretical results for the QZS isolator prototype, while the experimental results of the linear isolator prototype are basically in agreement with the theoretically predicted results. Obviously, the approximation introduced by the averaging method is a reason for the errors between the experiment results and the theoretical results, but this reason is only enough to explain the errors at low frequencies, so we need to seek for other reasons. A very interesting phenomenon is that the profile and trend of the experimental transmissibility curve of the QZS isolator prototype basically agree with the theoretical transmissibility curve, and the experimental curve looks like a “right translation” of the corresponding theoretical curve. This phenomenon is called the “translation phenomenon.” Due to this phenomenon, the experimental transmissibility of the QZS isolator is a little larger than the theoretical transmissibility in a wide frequency range, which implies a slightly worse isolation performance than in theory. The most meaningful consequence of this phenomenon is that the experimental peak frequency is moved to a larger value compared to the theoretical value, which could, in a sense, reflect some crucial attributes of the QZS isolator prototype. As is well known, the peak frequency is closely related to the mass and stiffness properties and can be qualitatively determined by the ratio of stiffness to mass. The actual total mass of the moving parts is a little larger than the theoretical mass 5.146 kg, because the lateral springs are considered massless in the theoretical analysis. Larger mass is supposed to cause smaller peak frequency, but the experimental peak frequency, however, is larger than the theoretical value. Therefore, it can be deduced that the reason for the translation phenomenon must be attributed to that the actual stiffness of the QZS isolator prototype is increased by some factors, and the degree of the increase in stiffness is much larger than the increase in mass. In the quasi-static tests on the force–displacement relationship of the QZS isolator prototype, the quasi-zero-stiffness characteristic has been roughly confirmed (see Sect. 4.3), so how can the stiffness be increased? The only explanation is that the stiffness becomes larger only when the QZS isolator prototype vibrates. Specifically, the lateral spring possesses not only axial stiffness but also bending stiffness; the axial stiffness can produce negative stiffness in the vertical direction when under compressed state, but the bending stiffness will cause an increase in the overall vertical stiffness of the isolator prototype. As a contrast, the experimental transmissibility curve of the linear isolator (which does not incorporate lateral springs) basically agrees with the theoretical curve, which further confirms that the reason for the translation phenomenon lies in the lateral springs. In quasi-static tests on the QZS isolator prototype, the isolated mass moves slowly, in which case the lateral spring is not likely to experience bending deformation and thus does not exhibit bending stiffness. On the contrary, when the isolated mass vibrates under the excitation of shaker, the bending stiffness is manifested due to some practical factors for example, the friction in the revolution joints and the inertia of the lateral springs; the revolution joint friction causes a little hindrance to the free revolution of the lateral spring and thus causes bending deformation; the lateral spring inertia itself will slightly lower the nature frequency but it can probably exacerbate the bending deformation due to local modes. Comparison of Fig. 17a–c indicates that larger excitation amplitude gives rise to smaller degree of translation phenomenon. This is attributed to that larger excitation amplitude results in larger response amplitude and thus larger angle between the axis of the lateral spring and the horizontal plane; obviously, as the inclination angle of the lateral spring increases, the vertical component of the force produced by bending deformation is decreased. The translation phenomenon and its cause might remind us that a smaller bending stiffness is better for vibration isolation purposes; however, an adequate bending stiffness is needed to avoid buckling instability of the lateral spring since it is always in the compressed state. Therefore, one of the legitimate approaches for attenuating the effect of the bending stiffness of lateral springs is to reduce the friction in revolution joints as much as possible.

Fig. 17
figure17

Comparison between the experimental and analytical transmissibility curves: ac QZS isolator prototype under 1.2 m/s2, 2.5 m/s2 and 3.8 m/s2 excitation amplitudes, respectively, and d linear isolator prototype under 2.5 m/s2 excitation amplitude

In summary, the experiments do demonstrate the good isolation performance of the QZS isolator especially for its wide frequency range of isolation and low peak transmissibility, which is much superior to the equivalent linear isolator especially at low frequencies. However, there are also some factors that cause the lateral spring to exhibit bending stiffness when vibrating, which increases the overall stiffness in the vertical direction and thus slightly degrades the vibration isolation performance.

Conclusions

In this work, the prototype of a QZS vibration isolator composed of five transverse groove springs is constructed and studied. The stiffness characteristics, dynamic response, vibration isolation performance and parameters’ effects are theoretically analyzed, and the peak response, stability boundary, jump frequencies, peak transmissibility and isolation frequency are analytically calculated. The transverse groove spring utilized in this work has many advantages such as the compact structure and the strong designability. The force–displacement curve of the designed prototype is tested and the QZS characteristic is basically realized. The vibration experiments are conducted by means of an electrodynamic shaker which can generate sinusoidal base excitation with controllable excitation amplitude and frequency.

The peak response amplitude, peak response frequency, unstable region, jump frequencies, peak transmissibility and isolation frequency can, in a sense, characterize the dynamic properties or the vibration isolation performance; analytical calculations show that all these indices are reduced when decreasing the excitation amplitude or increasing the damping ratio, except that the unstable region is independent of the excitation amplitude. The analytically derived stiffness formula of the transverse groove spring is validated by simulations, which can offer preliminary guidelines on the design of the transverse groove springs and the isolator prototype. Experimental results clearly demonstrate the good isolation performance of the QZS isolator for its wide frequency range of isolation and low peak transmissibility, which is much superior to the equivalent linear isolator especially at low frequencies. The experimental results also imply that there are some factors, for example the revolution joint friction and the inertia of lateral spring, that cause the lateral spring to exhibit bending stiffness when vibrating; the bending stiffness will increase the overall vertical stiffness of the isolator prototype and thus slightly degrades the actual vibration isolation performance. Therefore, it reminds us that reducing the revolution joint friction is beneficial for exploiting the advantages of the QZS isolator in practical applications.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11372084), which is gratefully acknowledged.

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Correspondence to Kaiping Yu.

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Liu, C., Yu, K. Design and experimental study of a quasi-zero-stiffness vibration isolator incorporating transverse groove springs. Archiv.Civ.Mech.Eng 20, 67 (2020). https://doi.org/10.1007/s43452-020-00069-3

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Keywords

  • Quasi-zero stiffness
  • Vibration isolation
  • Hardening nonlinearity
  • Transverse groove spring
  • Prototype design