A six degree of freedom passive vibration isolator with quasi-zero-stiffness-based supporting

Abstract

A six degree of freedom nonlinear passive vibration isolator is proposed based on Stewart platform configuration with the quasi-zero-stiffness structure as its legs. Due to the high static stiffness and low dynamic stiffness of each leg, the proposed six degree of freedom system can realize very good vibration isolation performance in all six directions while keeping high static load-bearing capacity in a pure passive manner. The mechanic model of the proposed six degree of freedom isolator and the dynamic equation of the isolator are established successively. Theoretical analysis on cross coupling stiffness reveals that the system can demonstrate quasi-zero-stiffness property in all six degree of freedom. Moreover, an analysis on stability shows that the condition of structural parameters for the isolator to realize quasi-zero-stiffness is also the stability boundary of the system. A series of numerical simulations on displacement transmissibilities in coupled degree of freedoms, the coupling effects of transmissibility, and a dynamic response in random excitation are carried out to show the effectiveness of the proposed six degree of freedom isolator, as well as the influence of structural parameters on vibration attenuation performance. Considering its high performance in a simple passive manner, it can be foreseen that the proposed six degree of freedom isolator will be applied in various engineering practices with multi-degree of freedom vibration isolation.

Introduction

Various engineering applications would be impossible without a qualified multi-degree of freedom or six degree of freedom (6-DOF) vibration isolation, such as vehicle- or ship-mounted optical instruments, atomic force microscopes, space telescopes, and gravitational waves interferometer.^1–8 However, the traditional passive isolator is effective only for high-frequency excitation. Typically, the active vibration control of Stewart platforms is employed to isolate 6-DOF vibrations in wide frequency range.^9–13

With respect to active or semiactive vibration control systems, pure passive vibration system commonly has higher reliability, lower development cost, easier maintenance, and without independence of external power. ¹⁴ However, to extend the effective bandwidth of passive linear isolator a big mass and/or a small stiffness is desired, but it also results in the system being cumbersome or of lower loading capacity.^12,15 Ideally, a passive 6-DOF vibration isolator should possess not only a low dynamic stiffness to effectively isolate vibration over a large bandwidth, but also a high static stiffness to firmly support a large weight mass, which implies a typical nonlinear stiffness characteristic. ¹⁶ Nonlinear systems are reported to be beneficial for vibration isolation over wider frequency range and could perform very well at low frequency.^17–21 Generally, most of the exiting quasi-zero-stiffness (QZS) isolators were designed for attenuating the transmission of vertical translational excitations. In recent years, several types of 6-DOF nonlinear isolators have been proposed and demonstrated to have very good vibration isolation performance. Platus ²² proposed a compact 6-DOF QZS isolator with natural frequencies as low as 0.2–0.5 Hz. Zhu et al. ²³ used magnetic levitation combined with a stabilizing control system to develop a 6-DOF QZS isolator. Wu et al. ²⁴ used X-shape structures as the legs to assemble a 6-DOF QZS Stewart platform and analyzed the equivalent stiffness and the displacement transmissibility in the six decoupled DOFs. Zhou et al. ²⁵ used compact QZS strut and studied the cross coupling stiffness and force/moment transmissibility to develop a 6-DOF QZS isolator.

In this study, a simple passive 6-DOF isolator is developed with Stewart platform configuration and QZS leg. The QZS leg is structured by a load-bearing spring and two oblique springs which serve as negative stiffness corrector. The structure is simple, thus a low processing cost and a high accuracy can be guaranteed. By properly designing the structural parameters of the 6-DOF system, it can realize QZS in all six directions with large loading capacity, and greatly enhance the performance of the 6-DOF isolator in a pure passive manner.

Structure of the 6-DOF isolator

Structure of the Stewart platform

Figure 1(a) shows the structure of 6-DOF passive Stewart isolator. The system contains a top platform with radius $r$ and a base platform with radius $R$ which are connected by six symmetrical legs (i.e. $L_1$ , $L_2$ … $L_6$ ) with length $L_0$ after assembling mass $m$ , and the height of Stewart platform is $H$ , as shown in Figure 1(a). In Figure 1(a), the static leg length $L_0=\sqrt{R^2+r^2+H^2-Rr}$ . Each leg $L_{\text{i}}\hspace{0.17em}(i=1,\hspace{0.17em}\hspace{0.17em}2,\hspace{0.17em}\dots 6)$ of Figure 1(a) is structured as a QZS system shown as Figure 1(b).

Figure 1.

(a) The schematic diagram of the 6-DOF Stewart platform and (b) the structure of $i$ th QZS-based leg $L_i$ .

Structure of QZS-based leg

As shown in Figure 1, each QZS-based leg is a single DOF system attached between base platform and top platform, in which a vertical spring serves as the positive stiffness element and two oblique springs as the negative stiffness element. Unlike typical research of QZS isolator, in this study, the QZS isolator is plugged obliquely in parallel to the connecting line of mounting point, as shown in Figure 1(a).

The stiffness of the vertical and oblique springs are $K_v$ and $K_h$ , respectively. Figure 2 shows the assembly process of the legs. As shown in Figure 2(a), in the rest position, the springs are not compressed and the length of legs and oblique springs are $\mu +L_0$ and $L_h$ , respectively. While in the equilibrium position, the vertical spring and oblique springs of $i$ th leg are compressed with $\mu$ and ${\lambda }_i$ , respectively, as shown in Figure 2(b). In the equilibrium position, the oblique springs should be perpendicular to the axis of corresponding leg. From Figures 1 and 2, for load $m$ of the top platform, the static deformation $\mu$ should satisfy

$$\mu =\frac{mg{L}_0}{6K_{v}H}$$

(1)

Figure 2.

Scheme diagram of the QZS leg. (a) Position before assembling mass and (b) equilibrium position.

Static analysis of the 6DOF isolator

Stiffness of the QZS leg

When the QZS leg vibrates around equilibrium position, by analyzing the mechanical relationship in Figure 2(b), the dynamic term ${\tilde{F}}_i$ of $i$ th leg force $F_i$ can be written as

$${\tilde{F}}_i=K_v{L}_i-2K_h{L}_i\frac{L_h-\sqrt{{(L_h-{\lambda }_i)}^2+L_i^2}}{\sqrt{{(L_h-{\lambda }_i)}^2+L_i^2}}\text{(i=1,2,}\mathrm{\dots }\text{6)}$$

(2)

where $L$ is a column vector with six elements, and the $i$ th element $L_i$ denotes the leg stretching about its equilibrium position. $\tilde{F}$ is a column vector with six elements, and the $i$ th element ${\tilde{F}}_i$ denotes the dynamic force of the $i$ th leg. To simplify calculations, the third-order Taylor series expansion of ${\tilde{F}}_i$ is employed as follows

$$\begin{array}{c}{\tilde{F}}_i\left(L_i\right)=K_{u1}{L}_i+K_{u3}{L}_i^3+o(L_i^5)\text{(i=1,2,}\mathrm{\dots }\text{6)}\\ K_{u1}=\frac{d{\tilde{F}}_i\left(L_i\right)}{d_{L_i}1!}=K_v+2K_h-\frac{2K_h{L}_h}{L_h-\lambda },K_{u3}=\frac{d{\tilde{F}}_i{\hspace{0.17em}}^{(3)}\left(L_i\right)}{d_{L_i}3!}=\frac{K_h{L}_h}{{(L_h-\lambda )}^3}\end{array}$$

(3)

In Figure 3, the values of the applied force expanded by the third-order Taylor series are close to the original expression of equation (2). Therefore, in the following discussion, $\tilde{F}$ can be approximately replaced by its third-order Taylor expansion.

Figure 3.

Comparison of (lines) the applied force of (2) and (dots) the third-order Taylor expansion as (3) for different $\text{λ}$ s.

Modeling

Define $P={\left[\begin{array}{cc}\begin{array}{ccc}{P}_{\text{x}}& P_{\text{y}}& P_{\text{z}}\end{array}& \begin{array}{ccc}{P}_{\alpha }& P_{\beta }& P_{\gamma }\end{array}\end{array}\right]}^T$ is the vibration response of top platform, and $Q={\left[\begin{array}{cc}\begin{array}{ccc}{Q}_{\text{x}}& Q_{\text{y}}& Q_{\text{z}}\end{array}& \begin{array}{ccc}{Q}_{\alpha }& Q_{\beta }& Q_{\gamma }\end{array}\end{array}\right]}^T$ is the vibration excitation of base platform. Based on Newton’s second law, the dynamic equation of the system can be given by

$$M\ddot{P}+C{J}^T\dot{L}(U)+F(U)=0$$

(4)

$M$ is the $6\times 6$ inertia matrix of the top platform and it can be given by

$$M=diag(\begin{array}{cccccc}m& m& m& I_{\alpha }& I_{\beta }& I_{\gamma }\end{array})$$

(5)

$C$ is the damping coefficient of each leg. $U=P-Q$ is the relative vibration between base platform and top platform. $F(U)$ is the general dynamic force on top platform, and its relation to dynamic leg force $\tilde{F}$ is

$$F(U)=J^T\tilde{F}(U)$$

(6)

$L(U)$ is the leg dynamic stretching vector, which can be expressed as

$$L(U)={\left[\begin{array}{cccc}{L}_1(U)& L_2(U)& \mathrm{\dots }& L_6(U)\end{array}\right]}^T=JU$$

(7)

$J$ is the Jacobian matrix of the Stewart platform. Following Yang et al., ¹³ $J$ can be expressed as

$$J=\frac{1}{2L_0}\left[\begin{array}{cccccc}\hfill \sqrt{3}r\hfill & \hfill r-2R\hfill & \hfill 2H\hfill & \hfill 2HR\hfill & \hfill 0\hfill & \hfill -\sqrt{3}Rr\hfill \\ \hfill -\sqrt{3}\left(R-r\right)\hfill & \hfill r+R\hfill & \hfill 2H\hfill & \hfill -HR\hfill & \hfill -\sqrt{3}HR\hfill & \hfill \sqrt{3}Rr\hfill \\ \hfill -\sqrt{3}R\hfill & \hfill R-2r\hfill & \hfill 2H\hfill & \hfill -HR\hfill & \hfill -\sqrt{3}HR\hfill & \hfill -\sqrt{3}Rr\hfill \\ \hfill \sqrt{3}R\hfill & \hfill R-2r\hfill & \hfill 2H\hfill & \hfill -HR\hfill & \hfill \sqrt{3}HR\hfill & \hfill \sqrt{3}Rr\hfill \\ \hfill \sqrt{3}\left(R-r\right)\hfill & \hfill r+R\hfill & \hfill 2H\hfill & \hfill -HR\hfill & \hfill \sqrt{3}HR\hfill & \hfill -\sqrt{3}Rr\hfill \\ \hfill -\sqrt{3}r\hfill & \hfill r-2R\hfill & \hfill 2H\hfill & \hfill 2HR\hfill & \hfill 0\hfill & \hfill \sqrt{3}Rr\hfill \end{array}\right]$$

(8)

Assume the precompressions ${\lambda }_i$ of each leg have the same value $\lambda$ . Then substitute equation (3) into equation (6), and the general dynamic applied force $F$ can be reexpressed as follows

$$\begin{array}{c}F(U)\hspace{0.17em}\text{=}\hspace{0.17em}{K}_{u1}{J}^{T}L(U)+K_{u3}{J}^T{\left[\begin{array}{cccc}{L}_1{\hspace{0.17em}}^3(U)& L_2{\hspace{0.17em}}^3(U)& \mathrm{\dots }& L_6{\hspace{0.17em}}^3(U)\end{array}\right]}^T\\ =K_{u1}{J}^{T}L(U)+K_{u3}{J}^T{\left\{\begin{array}{cccc}{\left[{\displaystyle \sum _{j=1}^6{J}_{1j}{U}_x}\right]}^3& {\left[{\displaystyle \sum _{j=1}^6{J}_{2j}{U}_y}\right]}^3& \mathrm{\dots }& {\left[{\displaystyle \sum _{j=1}^6{J}_{6j}{U}_{\gamma }}\right]}^3\end{array}\right\}}^T\end{array}$$

(9)

Clearly, for certain parameters of Stewart structure and QZS legs, $J$ , $K_v$ , $K_h$ , $L_h$ , and $\lambda$ are all constants, and $F(U)$ can be expressed as the sum of first- and third-order function of relative vibration $U$ .

Cross coupling stiffness

From equation (9), the stiffness of the 6-DOF isolator can be given by

$$\begin{array}{c}K(U)=\frac{dF(U)}{dU}\hspace{0.17em}\text{=}\hspace{0.17em}{K}_{\text{1}}\text{+}{K}_{\text{3}}(U)\\ K_{\text{1}}=\frac{d{F}_1(U)}{dU}=J^{T}J{K}_{u1}\end{array}$$

$$\begin{array}{c}{K}_{\text{3}}(U)=\frac{d{F}_3(U)}{dU}\hspace{0.17em}\text{=}\hspace{0.17em}3K_{u3}{J}^T\left(\begin{array}{cccc}\hfill L_1^2(U)\frac{\partial L_1(U)}{\partial U_x}\hfill & \hfill L_1^2(U)\frac{\partial L_1(U)}{\partial U_y}\hfill & \hfill \cdots \hfill & \hfill L_1^2(U)\frac{\partial L_1(U)}{\partial U_{\gamma }}\hfill \\ \hfill L_2^2(U)\frac{\partial L_2(U)}{\partial U_x}\hfill & \hfill L_2^2(U)\frac{\partial L_2(U)}{\partial U_y}\hfill & \hfill \cdots \hfill & \hfill L_2^2(U)\frac{\partial L_2(U)}{\partial U_{\gamma }}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill L_6^2(U)\frac{\partial L_6(U)}{\partial U_x}\hfill & \hfill L_6^2(U)\frac{\partial L_6(U)}{\partial U_y}\hfill & \hfill \cdots \hfill & \hfill L_6^2(U)\frac{\partial L_6}{\partial U_{\gamma }}\hfill \end{array}\right)\\ =\frac{3K_3{J}^T}{8L_0^3}\left[\begin{array}{cccccc}\hfill 3\sqrt{3}r{V}_1\hfill & \hfill 3\sqrt{3}\left(r-R\right)V_1\hfill & \hfill -3\sqrt{3}R{V}_1\hfill & \hfill 3\sqrt{3}R{V}_1\hfill & \hfill 3\sqrt{3}\left(R-r\right)V_1\hfill & \hfill -3\sqrt{3}r{V}_1\hfill \\ \hfill \left(r-2R\right)V_2\hfill & \hfill \left(r+R\right)V_2\hfill & \hfill \left(R-2r\right)V_2\hfill & \hfill \left(R-2r\right)V_2\hfill & \hfill \left(r+R\right)V_2\hfill & \hfill \left(r-2R\right)V_2\hfill \\ \hfill 8H^3{V}_3\hfill & \hfill 8H^3{V}_3\hfill & \hfill 8H^3{V}_3\hfill & \hfill 8H^3{V}_3\hfill & \hfill 8H^3{V}_3\hfill & \hfill 8H^3{V}_3\hfill \\ \hfill 2R^3{H}^3{V}_4\hfill & \hfill -R^3{H}^3{V}_4\hfill & \hfill -R^3{H}^3{V}_4\hfill & \hfill -R^3{H}^3{V}_4\hfill & \hfill -R^3{H}^3{V}_4\hfill & \hfill 2R^3{H}^3{V}_4\hfill \\ \hfill 0\hfill & \hfill -3\sqrt{3}{R}^3{H}^3{V}_5\hfill & \hfill -3\sqrt{3}{R}^3{H}^3{V}_5\hfill & \hfill 3\sqrt{3}{R}^3{H}^3{V}_5\hfill & \hfill 3\sqrt{3}{R}^3{H}^3{V}_5\hfill & \hfill 0\hfill \\ \hfill -3\sqrt{3}{R}^3{r}^3{V}_6\hfill & \hfill 3\sqrt{3}{R}^3{r}^3{V}_6\hfill & \hfill -3\sqrt{3}{R}^3{r}^3{V}_6\hfill & \hfill 3\sqrt{3}{R}^3{r}^3{V}_6\hfill & \hfill -3\sqrt{3}{R}^3{r}^3{V}_6\hfill & \hfill 3\sqrt{3}{R}^3{r}^3{V}_6\hfill \end{array}\right]\\ V_1={\left[R\left(U_y+U_z-U_{\alpha }-U_{\beta }\right)+r\left(-U_x-U_y+U_{\beta }+U_{\gamma }\right)\right]}^2\\ V_2={\left[R\left(-2U_x+U_y+U_z+U_{\alpha }+U_{\beta }-2U_{\gamma }\right)+r\left(U_x+U_y-2U_z-2U_{\alpha }+U_{\beta }+U_{\gamma }\right)\right]}^2\\ V_3={\left[U_x+U_y+U_z+U_{\alpha }+U_{\beta }+U_{\gamma }\right]}^2\\ V_4={\left[-2U_x+U_y+U_z+U_{\alpha }+U_{\beta }-2U_{\gamma }\right]}^2\\ V_5={\left[U_y+U_z-U_{\alpha }-U_{\beta }\right]}^2\\ V_6={\left[U_x-U_y+U_z-U_{\alpha }+U_{\beta }-U_{\gamma }\right]}^2\end{array}$$

(10)

Clearly, for specified structural parameters of the isolator, the first part of system stiffness $K_1$ is a constant and the second part $K_3$ is a quadratic function of vibration $U$ . Thus, for small amplitude of base vibration, the equivalent stiffness $K$ is mainly determined by $K_1$ . And then, the QZS condition (i.e. $K_1(U=0)=0$ ) of the 6-DOF isolator can be achieved by

$${\lambda }_0=\frac{K_v{L}_h}{K_v+2K_h}$$

(11)

With the increase of precompression $\lambda$ from zero to ${\lambda }_0$ , system stiffness $K_1$ gradually decreases from $J^{T}J{K}_v$ to 0. As equation (10) indicated, the second part of equivalent stiffness $K_3$ is a quadratic function of $U$ . Thus, for micro-vibration $K_3$ can also be very small. Particularly, $K_3$ can even be zero at position $U=0$ , which means the 6-DOF isolator can realize QZS performance at equilibrium position $U=0$ . If $\lambda$ increases continuously, the equivalent stiffness can be negative.

From equation (9), the restoring force is fully coupled with displacements in all 6-DOFs. Figures 4 and 5 show the force/torque–displacement relationships of $F_z$ and $F_{\alpha }$ for system realize QZS ( $\lambda ={\lambda }_0$ ) in cross coupling planes, by letting other four displacement variables out of the cross coupling plane be zeros. For the sake of brevity, other four forces/torques are not shown, which are similar to $F_z$ and $F_{\alpha }$ . Considering the symmetry on horizontal and pitching directions, values of DOFs $y$ and $\beta$ are set to zeros in the following discussions.

Figure 4.

Force–displacement relationship of $F_z$ in (a) $\text{x}z$ plane, (b) $\alpha z$ plane, and (c) $\gamma z$ plane.

Figure 5.

Torque–displacement relationship of $F_{\alpha }$ in (a) $\text{x}\alpha$ plane, (b) $z\alpha$ plane, and (c) $\gamma \alpha$ plane.

Stability analysis

From the analysis above, it can be found that the linear part of the stiffness decreases with the precompression increasing. However, if the structure parameters of the 6-DOF isolator are improperly changed, the system can be unstable, which will induce danger in practice. Thus, to maintain the stability of the system at equilibrium position $U=0$ , the precompression $\lambda$ and the other structural parameters are required to stay in a reasonable range. Ignoring the damping and excitation terms in equation (4), the dynamical equation of the system can be written as

$$\ddot{P}+K_{u1}{M}^{-1}{J}^{T}JP+K_{u3}{M}^{-1}{J}^T{\{\begin{array}{cccc}{\left[{\displaystyle \sum _{j=1}^6{J}_{1j}{P}_x}\right]}^3& {\left[{\displaystyle \sum _{j=1}^6{J}_{2j}{P}_y}\right]}^3& \mathrm{\dots }& {\left[{\displaystyle \sum _{j=1}^6{J}_{6j}{P}_{\gamma }}\right]}^3\end{array}\}}^T=0$$

(12)

The stability of a noncritical stable nonlinear system can be determined by its first-order approximation. ²⁶ For $K_1\ne 0$ , the 6-DOF isolation system is noncritically stable and its stability can be determined by

$$\left(\begin{array}{c}\dot{P}\\ \ddot{P}\end{array}\right)=\left(\begin{array}{cc}0& 1\\ -K_{u1}{M}^{-1}{J}^{T}J& 0\end{array}\right)\left(\begin{array}{c}P\\ \dot{P}\end{array}\right)$$

(13)

The stability of the equilibrium depends on the eigenvalues of the system.^27–29 For the Stewart platform proposed in “Structure of the 6-DOF isolator” section

$$J^{T}J\hspace{0.17em}\text{=}\hspace{0.17em}\left[\begin{array}{cccccc}{\delta }_1& 0& 0& 0& {\delta }_7& 0\\ 0& {\delta }_2& 0& -{\delta }_7& 0& 0\\ 0& 0& {\delta }_3& 0& 0& 0\\ 0& -{\delta }_7& 0& {\delta }_4& 0& 0\\ {\delta }_7& 0& 0& 0& {\delta }_5& 0\\ 0& 0& 0& 0& 0& {\delta }_6\end{array}\right]$$

(14)

where ${\delta }_1={\delta }_2,\hspace{0.17em}\hspace{0.17em}{\delta }_3,\hspace{0.17em}\hspace{0.17em}{\delta }_4={\delta }_5,\hspace{0.17em}\hspace{0.17em}{\delta }_6$ are all positive numbers, and ${\delta }_7$ is also a real number

$$\left[\begin{array}{c}\hfill {\delta }_1\hfill \\ \hfill {\delta }_3\hfill \\ \hfill {\delta }_4\hfill \\ \hfill {\delta }_6\hfill \\ \hfill {\delta }_7\hfill \end{array}\right]=\frac{1}{R_b^2+R_t^2+H^2-R_b{R}_t}\left[\begin{array}{c}\hfill 3\left(R_b^2+R_t^2-R_b{R}_t\right)\hfill \\ \hfill 6H^2\hfill \\ \hfill 3R_b^2{H}^2\hfill \\ \hfill \frac{9}{2}{R}_b^2{R}_t^2\hfill \\ \hfill \frac{3}{2}{R}_{b}H\left(2R_b\text{-}{R}_t\right)\hfill \end{array}\right]$$

(15)

Then the eigenvalues $\nu$ of $J^{T}J$ can be expressed as

$$\nu =\left[\begin{array}{c}{\nu }_1\\ {\nu }_2\\ {\nu }_3\\ {\nu }_4\\ {\nu }_5\\ {\nu }_6\end{array}\right]=eig(J^{T}J)=\left[\begin{array}{c}\hfill {\delta }_3\hfill \\ \hfill {\delta }_6\hfill \\ \hfill \frac{1}{2}{\delta }_1+\frac{1}{2}{\delta }_5-\frac{1}{2}\sqrt{{({\delta }_1-{\delta }_5)}^2+4{\delta }_7^2}\hfill \\ \hfill \frac{1}{2}{\delta }_1+\frac{1}{2}{\delta }_5+\frac{1}{2}\sqrt{{({\delta }_1-{\delta }_5)}^2+4{\delta }_7^2}\hfill \\ \hfill \frac{1}{2}{\delta }_2+\frac{1}{2}{\delta }_4-\frac{1}{2}\sqrt{{({\delta }_2-{\delta }_4)}^2+4{\delta }_7^2}\hfill \\ \hfill \frac{1}{2}{\delta }_2+\frac{1}{2}{\delta }_4+\frac{1}{2}\sqrt{{({\delta }_2-{\delta }_4)}^2+4{\delta }_7^2}\hfill \end{array}\right]$$

(16)

Obviously, ${\nu }_1,{\nu }_2,{\nu }_4={\nu }_6$ are all positive numbers. On the other hand

$${\left({\delta }_1+{\delta }_5\right)}^2\text{-}\left[{({\delta }_1-{\delta }_5)}^2+4{\delta }_7^2\right]\hspace{0.17em}\text{=}\hspace{0.17em}\frac{27R_b^2{R}_t^2{H}^2}{{\left(R_b^2+R_t^2+H^2-R_b{R}_t\right)}^2}>0$$

(17)

Therefore, ${\nu }_3={\nu }_5$ are also positive numbers.

Because $\lambda$ is less than ${\lambda }_0$ , $K_1$ will be greater than zero; as a consequence all of the eigenvalues of system will be negative numbers and the system will be stable in equilibrium $U=0$ . Similarly, for precompression $\lambda$ greater than ${\lambda }_0$ , $K_1$ will be less than zero; as a consequence all of the eigenvalues will be positive numbers and the system will be unstable in equilibrium $U=0$ .

For $\lambda ={\lambda }_0$ , $K_1$ will also equal zero, and the system will be critically stable and its stability can be determined by

$$\left(\begin{array}{c}\dot{P}\\ \ddot{P}\end{array}\right)=\left(\begin{array}{cc}0& 1\\ \text{-}{K}_{non}& 0\end{array}\right)\left(\begin{array}{c}P\\ \dot{P}\end{array}\right)$$

(18)

where

$$K_{non}=\frac{d\overline{F}}{dU}\hspace{0.17em}\text{=}\hspace{0.17em}{M}^{-1}{K}_{v}G(U),\hspace{1em}G(U)=K_3(U)/K_v$$

(19)

A Hermitian matrix is positive semidefinite if and only if all of its principal minors are nonnegative. ³⁰ Obviously, $K_{non}$ is a real symmetric matrix, and the expressions of $i$ -order principal minors $D_i$ ( $i=1,2,\dots 6)$ of $G$ are listed in Appendix 2. Because of the structural parameters of Stewart platform $H$ , $R_b$ , $R_t$ are all positive numbers. And the sum of squared vibration displacements $S_1$ , $S_2$ , $S_3$ is also nonnegative numbers. All principal minors of $G$ are all nonnegative, thus the eigenvalues of $G$ are all nonnegative, and then the eigenvalues of system $K_{non}$ are all nonpositive. Considering the above discussion, the equilibrium $U=0$ of the 6-DOF isolator is stable for $\lambda \le {\lambda }_0$ , and it is unstable for $\lambda >{\lambda }_0$ .

Dynamic analysis

Displacement transmissibility

Utilizing the relative vibration as the variables, the dynamic equation (4) can be rewritten as

$$M\ddot{Q}+M\ddot{U}+C{J}^{T}J\dot{U}+F(U)=0$$

(20)

The first approximation of the primary resonance is obtained by the harmonic balance method. Assume the base displacement excitation $Q=Q_{\text{c}}\hspace{0.17em}\cos \hspace{0.17em}\Omega \tau$ and $U=U_{\text{c}}\hspace{0.17em}\cos \hspace{0.17em}\Omega \tau +U_{\text{s}}\hspace{0.17em}\sin \hspace{0.17em}\Omega \tau$ . Where $Q_{\text{c}}={\left[\begin{array}{cc}\begin{array}{ccc}{Q}_{\text{cx}}& Q_{\text{cy}}& Q_{\text{cz}}\end{array}& \begin{array}{ccc}{Q}_{\text{c}\alpha }& Q_{\text{c}\beta }& Q_{\text{c}\gamma }\end{array}\end{array}\right]}^T$ , $U_{\text{c}}={\left[\begin{array}{cc}\begin{array}{ccc}{U}_{\text{cx}}& U_{\text{cy}}& U_{\text{cz}}\end{array}& \begin{array}{ccc}{U}_{\text{c}\alpha }& U_{\text{c}\beta }& U_{\text{c}\gamma }\end{array}\end{array}\right]}^T$ , and $U_{\text{s}}={\left[\begin{array}{cc}\begin{array}{ccc}{U}_{\text{sx}}& U_{\text{sy}}& U_{\text{sz}}\end{array}& \begin{array}{ccc}{U}_{\text{s}\alpha }& U_{\text{s}\beta }& U_{\text{s}\gamma }\end{array}\end{array}\right]}^T$

$$-{\Omega }^{2}M{Q}_0\hspace{0.17em}\cos \hspace{0.17em}\Omega \tau -M{\Omega }^2\left(U_c\hspace{0.17em}\cos \hspace{0.17em}\Omega \tau +U_s\hspace{0.17em}\sin \hspace{0.17em}\Omega \tau \right)+\Omega C{J}^{T}J\left(U_s\hspace{0.17em}\cos \hspace{0.17em}\Omega \tau -U_c\hspace{0.17em}\sin \hspace{0.17em}\Omega \tau \right)+F(U)=0$$

(21)

By ignoring the high-order harmonic terms, $F(U)$ can be approximated as

$$F(U)\hspace{0.17em}\approx \hspace{0.17em}{F}_c\hspace{0.17em}\cos \hspace{0.17em}\Omega \tau +F_s\hspace{0.17em}\hspace{0.17em}\sin \hspace{0.17em}\hspace{0.17em}\Omega \tau$$

(22)

where $F_c$ and $F_s$ are functions of $U_{\text{c}}$ and $U_{\text{s}}$ and independent of time $\tau$ . By substituting equation (26) into equation (25) and equating coefficients of $\hspace{0.17em}\cos \hspace{0.17em}\Omega \tau$ and $\hspace{0.17em}\sin \hspace{0.17em}\Omega \tau$

$$\{\begin{array}{c}-M{\Omega }^2\left(Q_C+U_c\right)+\Omega C{J}^{T}J{U}_s+F_c=0\\ -M{\Omega }^2{U}_s-\Omega C{J}^{T}J{U}_c+F_s=0\end{array}$$

(23)

The displacement amplitudes $U_{\text{c}}$ and $U_{\text{s}}$ can be solved numerically by the above third-order nonlinear algebraic equations. Define the displacement transmissibility $T={\left[\begin{array}{cc}\begin{array}{ccc}{T}_{\text{x}}& T_{\text{y}}& T_{\text{z}}\end{array}& \begin{array}{ccc}{T}_{\alpha }& T_{\beta }& T_{\gamma }\end{array}\end{array}\right]}^T$ as the total amplitude ratio from excitation $Q$ to response $P$ , and it can be given by

$$T=\frac{|P|}{|Q|}=\frac{|Q+U|}{|Q|}=\frac{1}{Q_C}\sqrt{{\left(U_c+Q_C\right)}^2+U_s^2}$$

(24)

Figure 6 shows the comparison of displacement transmissibility of the 6-DOF Stewart isolator with QZS-based legs for different $\lambda$ s in the four coupled directions. To simplify calculation, only the sweep frequencies from left to right are discussed. It can be seen from the curves that while the precompression of legs $\lambda$ increases from $0.18$ to $0.195\text{\hspace{0.17em}m}$ , the resonance peaks move left and the vibration isolation frequency ranges are expanded gradually. For $\lambda ={\lambda }_0=0.2\text{m}$ (i.e. system reach QZS), even the resonance peak disappears and the vibrations of base platform are little transmitted to top platform within wide frequency ranges in all of the four directions. In combination with the stability analysis of “Stability analysis” section, the optimal selection of $\lambda$ is ${\lambda }_0$ . And in the following simulations, the default value of $\lambda$ is selected as ${\lambda }_0$ . The default values of the structural parameters are shown in Appendix 1. And damping coefficient $C=300\text{N}\text{s/m}$ , base excitation $Q_0=\left[\begin{array}{ccc}\begin{array}{cc}\text{2}\text{\hspace{0.17em}mm}& \text{0}\end{array}& \begin{array}{cc}\text{2\hspace{0.17em}}\text{mm}& \text{2}\text{mrad}\end{array}& \begin{array}{cc}\text{0}& \text{2}\text{mrad}\end{array}\end{array}\right]$ .

Figure 6.

Comparison of the displacement transmissibility of the isolator for different precompressions: (black) $\lambda =0.18 \text{m}$ , (blue) $\lambda =0.185 \text{m}$ , (red) $\lambda =0.19 \text{m}$ , (green) $\lambda =0.195 \text{m}$ , (cyan) $\lambda =0.2 \text{m}$ .

Figure 7 shows the comparison of displacement transmissibility of the 6-DOF Stewart isolator with different damping coefficient $C$ . It can be seen that the influence of the damping coefficient of legs on the vibration isolation performance is obvious. For $C$ decreases from $400\hspace{0.17em}$ to $250\hspace{0.17em}\text{N}\text{s/m}$ , the dynamic behavior of the proposed nonlinear system is similar to an overdamped linear isolator that less damping induces better vibration isolation in mid–high frequency range. When damping coefficient $C$ decreases further to $200\text{N}\text{s/m}$ , jump phenomena emerge quickly around resonance peaks, and for frequency higher than resonance peaks, less damping also induces better vibration isolation. Therefore, a proper damping coefficient $C$ is important for specified application of the proposed isolator. And in the simulations, the default value of $C$ is selected as $300\text{N}\text{s/m}$ .

Figure 7.

Comparison of the displacement transmissibility of the isolator for different damping coefficients: (black) $C=200 \text{N}\text{s/m}$ , (blue) $C=250 \text{N}\text{s/m}$ , (red) $C=300 \text{N}\text{s/m}$ , (green) $C=350 \text{N}\text{s/m}$ , (cyan) $C=400 \text{N}\text{s/m}$ .

Figure 8 shows the vibration isolation performance of the 6-DOF Stewart isolator with different base excitation amplitudes. For small $Q_0$ , such as $Q_0\le 3\text{\hspace{0.17em}mm\hspace{0.17em}(mrad)}$ , the vibration isolation performance is little affected by excitation amplitude $Q_0$ and an excellent vibration isolation performance is achieved over full frequency range. When $Q_0=3.5\hspace{0.17em}\text{mm (mrad)}$ , small resonance peaks emerge in DOFs $x$ , $z$ _, and $\alpha$ , and the vibration isolation performance outside of the resonance ranges is also close to that of $Q_0\le 3\hspace{0.17em}\text{mm\hspace{0.17em}(mrad)}$ . When $Q_0$ increases further to $4\hspace{0.17em}\text{mm\hspace{0.17em}(mrad)}$ , jump phenomena emerge quickly around resonance peaks. And for outside of the resonance range, the vibration isolation performance is little affected by the increase of $Q_0$ . Through the above discussion, it can be seen that the proposed system is not suitable for vibration isolation of large excitation. The default value of $Q_0$ is selected as $2\hspace{0.17em}\text{mm\hspace{0.17em}(mrad)}$ .

Figure 8.

Comparison of the displacement transmissibility of the isolator for different excitation amplitudes: (black) $Q_0=2 \text{mm (mrad)}$ , (blue) $Q_0=2.5 \text{mm (mrad)}$ , (red) $Q_0=3 \text{mm (mrad)}$ , (green) $Q_0=3.5 \text{mm (mrad)}$ , (cyan) $Q_0=4 \text{mm (mrad)}$ .

Effects of coupling stiffness

From equation (10), Figures 4 and 5 the stiffness of each direction is not only related to current DOF, but also coupled with all 6-DOFs. From Figure 9, with the increase of excitation DOFs increase from 1 to 3, the coupling effects is more and more significant. And particularly, when the excitation DOFs increase from 1 ( $x$ ), 2 ( $x$ and $z$ ) to 3 ( $x$ , $z$ and $\alpha )$ , the coupled vibration transmissibility $T_{\gamma }$ is also increased from 0.009, 0.012 to 0.05, respectively. But the overall influence of coupling effects on transmissibility is still smaller than 0.07 in all 4-DOFs.

Figure 9.

Comparison of the displacement transmissibility of the amount of excitation DOFs: (black) $Q_0=\left[\begin{array}{ccc}\begin{array}{cc}2 \text{mm}& 0\end{array}& \begin{array}{cc}0& 0\end{array}& \begin{array}{cc}0& 0\end{array}\end{array}\right]$ , (red) $Q_0=\left[\begin{array}{ccc}\begin{array}{cc}2 \text{mm}& 0\end{array}& \begin{array}{cc}2 \text{mm}& 0\end{array}& \begin{array}{cc}0& 0\end{array}\end{array}\right]$ , (blue) $Q_0=\left[\begin{array}{ccc}\begin{array}{cc}2 \text{mm}& 0\end{array}& \begin{array}{cc}2 \text{mm}& 2\text{mrad}\end{array}& \begin{array}{cc}0& 0\end{array}\end{array}\right]$ .

To show the coupling effects of the isolator more clearly, define $6\times 6$ coupling transmissibility matrix $\Gamma$

$$\Gamma =\left[\begin{array}{cccc}\hfill {\Gamma }_{xx}\hfill & \hfill {\Gamma }_{xy}\hfill & \hfill \cdots \hfill & \hfill {\Gamma }_{x\gamma }\hfill \\ \hfill {\Gamma }_{yx}\hfill & \hfill {\Gamma }_{yy}\hfill & \hfill \cdots \hfill & \hfill {\Gamma }_{y\gamma }\hfill \\ \hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {\Gamma }_{\gamma x}\hfill & \hfill {\Gamma }_{\gamma y}\hfill & \hfill \cdots \hfill & \hfill {\Gamma }_{\gamma \gamma }\hfill \end{array}\right]$$

(25)

Each element ${\Gamma }_{ij}$ ( $i=x,y\dots \gamma ,\hspace{0.17em}j=x,y\dots \gamma$ ) of $\Gamma$ represents the coupling transmissibility from DOF $Q_i$ to $P_j$ . And it also can be represented by solutions

$$T_{ij}=\frac{|P_j|}{|Q_i|}=\frac{|Q_j+U_j|}{|Q_i|}=\frac{1}{Q_{ci}}\sqrt{{\left(U_{cj}+Q_{cj}\right)}^2+U_{sj}^2}=\frac{1}{Q_{ci}}\sqrt{U_{cj}^2+U_{sj}^2},\text{(}i=x,y\mathrm{\dots }\gamma ,j=x,y\mathrm{\dots }\gamma )$$

(26)

Figure 10 shows the coupling transmissibility of the isolator. It can be seen from the curves that, for system reaches QZS and micro-vibration, the coupling transmissibility is little except for diagonal element. ${\Gamma }_{x\gamma }$ is about 0.095, ${\Gamma }_{\alpha z}$ is about 0.003, and other elements are very close to 0.

Figure 10.

The coupling transmissibility of the isolator. (black, ${\Gamma }_x\cdot$ ) $Q_c=\left[\begin{array}{ccc}\begin{array}{cc}5 \text{mm}& 0\end{array}& \begin{array}{cc}0& 0\end{array}& \begin{array}{cc}0& 0\end{array}\end{array}\right]$ , (blue, ${\Gamma }_z\cdot$ ) $Q_c=\left[\begin{array}{ccc}\begin{array}{cc}0& 0\end{array}& \begin{array}{cc}5 \text{mm}& 0\end{array}& \begin{array}{cc}0& 0\end{array}\end{array}\right]$ , (red, ${\Gamma }_{\alpha }\cdot$ ) $Q_c=\left[\begin{array}{ccc}\begin{array}{cc}0& 0\end{array}& \begin{array}{cc}0& 5\text{mrad}\end{array}& \begin{array}{cc}0& 0\end{array}\end{array}\right]$ , (green, ${\Gamma }_{\gamma }\cdot$ ) $Q_c=\left[\begin{array}{ccc}\begin{array}{cc}0& 0\end{array}& \begin{array}{cc}0& 0\end{array}& \begin{array}{cc}0& 5\text{mrad}\end{array}\end{array}\right]$ .

Comparisons with the simplified linear case

Figure 11 shows the comparison of the displacement transmissibility of the 6-DOF isolator with different types of legs. The spring stiffness of the linear legs is equal to that of vertical spring in QZS legs. And the linear legs have no oblique spring. Both the two systems present vibration isolation performance in all directions. Figure 11 shows that compared with the 6-DOF isolator with linear legs, the proposed system has no resonance peak and significantly smaller vibration mitigation in all 4-DOF.

Figure 11.

Comparison of the displacement transmissibility of the 6-DOF isolator with (black) QZS leg and with (blue) corresponding linear stiffness leg.

Simulation with random excitation

Figure 12 shows the time series of the random excitation $Q(t)$ . The mean value of the random input signal is zero. The unbiased variance of translational DOFs $Q_x\left(t\right)=Q_y\left(t\right)=Q_z\left(t\right)$ is $1.5\text{\hspace{0.17em}mm}$ , and the unbiased variance of rotational DOFs $Q_{\alpha }\left(t\right)=Q_{\beta }\left(t\right)=Q_{\gamma }\left(t\right)$ is $1.5\text{\hspace{0.17em}mrad}$ . The frequency range of $Q(t)$ is $\left[0,20\right]\text{\hspace{0.17em}Hz}$ .

Figure 12.

Time series of the random excitations of the 6-DOF isolator.

Figure 13 shows the time series of the dynamic vibration response of the 6-DOF isolator $P(t)$ . Due to beneficial nonlinear stiffness characteristics of the 6-DOF isolator, the dynamic vibration responses of the 6-DOF isolator are very small. The root mean square (RMS) values of the dynamic response $P(t)$ in directions $x$ , $y$ , $\hspace{0.17em}z$ , $\hspace{0.17em}\alpha$ , $\hspace{0.17em}\beta$ , and $\hspace{0.17em}\gamma$ are $0.12$ , $0.15$ , $0.16\hspace{0.17em}\text{mm}$ , $0.21$ , $0.17$ , and $0.26\hspace{0.17em}\text{mrad}$ , respectively. By contrast, the time series of the dynamic response of corresponding linear system is shown in Figure 14. The RMS values of the dynamic response of the corresponding linear system in directions $x$ , $y$ , $\hspace{0.17em}z$ , $\hspace{0.17em}\alpha$ , $\hspace{0.17em}\beta$ and $\hspace{0.17em}\gamma$ are $2.5\hspace{0.17em}$ , $2.2$ , $2.4\hspace{0.17em}\text{mm}$ , $3.6$ , $3.7$ , and $2.5\text{\hspace{0.17em}mrad}$ , respectively. Clearly, under the same base excitation, the vibration isolation performance of the 6-DOF passive Stewart with QZS legs is significantly superior to the corresponding linear system.

Figure 13.

Time series of the vibrations of the top platform of 6-DOF isolator with QZS-based legs.

Figure 14.

Time series of the vibrations of the top platform of 6-DOF isolator with linear legs.

Conclusions

In this paper, a 6-DOF passive Stewart vibration isolation platform with QZS-based leg is studied and its nonlinear stiffness advantages in 6-DOF vibration isolation have been presented. The feature of the nonlinear stiffness is employed by simple QZS-based legs, and each QZS leg is composed of two precompressed oblique springs as negative stiffness corrector and a loaded linear stiffness spring. By properly designing the structural parameters of each leg, the 6-DOF isolator can achieve very small stiffness even zero stiffness around the equilibrium position of all six directions. The stability analysis of the equilibrium point of the 6-DOF isolator shows that the condition of structural parameters reach QZS is also the boundary of its stability. Both theoretical analysis and simulation studies vindicate clearly that the 6-DOF Stewart isolator can realize very good vibration isolation in all 6-DOF simultaneously in a passive manner. Future work will focus on building a prototype of the 6-DOF isolator and carrying out experimental investigations to confirm its effectiveness.