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Design and characteristic analysis of an X-shaped negative stiffness structure

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Abstract

The quasi-zero stiffness (QZS) mechanism is typically implemented by paralleling negative and positive stiffness structures to enhance the vibration isolation effect. As the core component, a novel bistable X-shaped negative stiffness structure (XNSS) is designed. Based on the geometric relationship between angular and displacement coordinates, the single-layer and multilayer mechanical model is derived from the principle of virtual work. After the non-dimensional process, a set of system parameters is summarized. Ensuring that these system parameters satisfy constraints, the comprehensive effects on XNSS are studied in detail. The parallel connection of XNSS and linear stiffness mechanism can constitute a QZS vibration isolator, and the influence of system parameters on loading capacity, stability of equilibriums, and dynamic stiffness are discussed in detail. The amplitude-frequency response and displacement transmissibility of nonlinear QZS vibration isolation model show excellent vibration isolation performance, compared with the corresponding linear system. The results show that the static analysis of the XNSS system parameters can be a good guide to the design of the QZS vibration isolator in order to obtain better dynamic performance.

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Abbreviations

c S :

Viscous damping coefficient in N/m/s

D :

Dissipation function

F NS :

Negative force in N

F NS max, F NS min :

Maximum and minimum value of negative stiffness force in N

F S :

Restoring force in N

h :

Vertical distance between the spring fixing point and the center of rotation of the bar l2 in meter

J :

Jacobian Matrix

k NS :

Spring stiffness for negative stiffness structure in N/m

K NS :

Stiffness of negative stiffness force in N/m

K NS min :

Minimum of negative stiffness in N/m

K NS 1, K NS 2 :

Maximum positive stiffness values in compression and tension stages in N/m

K S :

Dynamical stiffness in N/m

k P S :

Stiffness of linear spring in N/m

l max, l min :

Maximum and minimum value of \(\sqrt s\) in meter

l 1 :

Length of the long bar in meter

l 2 :

Length of the short bar in meter

l NS 0 :

Free length of springs for negative stiffness structure in meter

l PS 0 :

Free length of linear spring in meter

L :

Lagrange function

m :

Mass in kg

n :

Layer number of negative stiffness structure

OB :

Length of spring for negative stiffness structure in meter

r :

Horizontal distance between the spring fixing point and the center of rotation of the bar l2 in meter

R NS :

Relative range of maximum positive stiffness values in compression and tension stages

s :

Square of the distance between highest point of first layer in negative stiffness structure and center of rotation of bar l2 in m2

Skew(F NS):

Skewness of negative stiffness force

t :

Time in s

T :

Kinetic energy

T AD :

Absolute displacement transmissibility in dB

V :

Potential energy

V NS :

Potential energy of negative stiffness structure

x B , y B :

Horizontal and vertical displacement of the free end of the bar l2 in meter

y A :

Vertical displacement of the highest point of negative stiffness structure in meter

y A 0 :

Center position of negative stiffness structure in meter

y AT :

Travel of negative stiffness structure in meter

z :

Displacement of the mass relative to the base in meter

z b :

Vertical excitation from the base in meter

z m :

Absolute displacement response of mass in meter

Z amp :

Amplitude of steady-state response in meter

Z b :

Vertical excitation amplitude from the base in meter

α, β, γ :

Square of root of cubic resolvent equation

α 0, α 1, α 2, α 3 :

Configurative parameters of negative stiffness structure

α 4 :

Static equilibrium position with only linear spring for load bearing

α 5 :

Stiffness ratio

α b :

Dimensionless vertical excitation amplitude from the base

θ :

Angle between l2 and horizontal direction

θ 0, θ 1, θ 2 :

Angle corresponding to the three equilibrium positions in negative stiffness structure

θ max, θ min :

Maximum and minimum value of θ

ξ :

Damping ratio

ω :

Vertical excitation frequency from the base in rad/s

ω 0 :

Cube root of 1

ω n :

Un-damped natural frequency in rad/s

Ω:

Frequency ratio

^:

Denotes dimensionless quantity

•:

Denotes time derivative

′:

Denotes displacement derivative

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Acknowledgements

This work was supported by the Institute of Vibration and Noise Control at Beijing Institute of Technology. The authors would like to thank the reviewers for their precious reviews and comments.

CRediT authorship contribution statement

Bingyi Liu contributed to conceptualization, investigation, methodology, writing—original draft; writing—review and editing. Liang Gu contributed to funding acquisition and supervision. Mingming Dong supervised the study.

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  1. School of Mechanical Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Bingyi Liu, Liang Gu & Mingming Dong

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  1. Bingyi Liu
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Appendices

Appendix A

The expressions of c, d, and e in Eq. (26) are as follows:

$$\begin{aligned} c & = n^{4} \left[ { - \frac{1}{2}w^{2} + 4\left( {n - 1} \right)^{2} r^{2} w} \right. + 4\left( {n - 1} \right)^{4} r^{4} + 2n^{2} \left( {n - 1} \right)^{2} \left( {l_{\max }^{2} l_{\min }^{2} - 2r^{4} - r^{2} \left( {l_{\max } - l_{\min } } \right)^{2} } \right) \\ &\left. { - 4n\left( {n - 1} \right)\left( {3n - 2} \right)r^{2} l_{\max } l_{\min } } \vphantom{ { - \frac{1}{2}w^{2} + 4\left( {n - 1} \right)^{2} r^{2} w}} \right], \\ \end{aligned}$$
(A.1)
$$\begin{aligned}d = 4n^{6} \left( {n - 1} \right)^{2} r^{2} \left[ {w - n^{2} \left( {l_{\max }^{2} + l_{\min }^{2} } \right)} \right]\left[ {w + 2n\left( {n - 1} \right)l_{\max } l_{\min } - \left( {2n - 1} \right)r^{2} } \right], \end{aligned}$$
(A.2)
$$\begin{aligned} e & = \frac{1}{16}n^{8} w^{4} + n^{8} \left( {n - 1} \right)^{2} r^{2} w^{3} - \frac{1}{2}n^{8} \left( {n - 1} \right)\left[ {n^{2} \left( {n - 1} \right)l_{\max }^{2} l_{\min }^{2} } \right. + 2\left( {2n - 1} \right)\left( {n - 1} \right)r^{4} + 2n\left( {n - 2} \right)r^{2} l_{\max } l_{\min } \\ & \left. + {3n^{2} \left( {n - 1} \right)r^{4} \left( {l_{\max } - l_{\min } } \right)^{2} } \right]w^{2} - 2n^{10} \left( {n - 1} \right)^{2} r^{2} \left[ {2n\left( {n - 1} \right)l_{\max } l_{\min } \left( {l_{\max }^{2} + l_{\min }^{2} } \right)} \right. - \left( {2n - 1} \right)r^{2} \left( {l_{\max }^{2} + l_{\min }^{2} } \right) \\ &\left. { - 2n^{2} l_{\max }^{2} l_{\min }^{2} } \right]w + n^{12} \left( {n - 1} \right)^{4} \left[ {\left( {r^{2} l_{\max }^{2} + r^{2} l_{\min }^{2} - l_{\max }^{2} l_{\min }^{2} } \right)^{2} } \right.\left. { + 4r^{4} l_{\max }^{2} l_{\min }^{2} } \right] + 4n^{13} \left( {n - 1} \right)^{3} r^{2} l_{\max } l_{\min } \left( {r^{2} l_{\max }^{2} } \right. \\ &\left. { + r^{2} l_{\min }^{2} + l_{\max }^{2} l_{\min }^{2} } \right). \\ \end{aligned}$$
(A.3)

Appendix B

The expressions of q and δ in Eq. (31) are as follows:

$$p = - \frac{1}{48}c^{2} - \frac{1}{4}e,$$
(B.1)
$$q = - \frac{1}{864}c^{3} + \frac{1}{24}ec - \frac{1}{64}d^{2},$$
(B.2)
$$\delta = \left( \frac{p}{3} \right)^{3} + \left( \frac{q}{2} \right)^{2}.$$
(B.3)

Appendix C

The coefficient expressions in Eqs. (35)–(44) are as follows:

$$\begin{aligned} a_{1} &= \alpha_{0} \alpha_{2} ,\;a_{2} = \alpha_{0} \alpha_{1} \alpha_{2} ,\;\;a_{3} = 2\alpha_{0} \alpha_{1} \alpha_{2}^{3} ,\;a_{4} = 2\alpha_{1} \alpha_{2}^{2} ,\;a_{5} = \alpha_{1}^{2} \alpha_{2} + \alpha_{2}^{3} + \alpha_{1}^{2} \alpha_{2}^{3} ,\;a_{6} = 2\alpha_{1}^{2} \alpha_{2}^{3} ,\\ a_{7}& = \alpha_{0} \alpha_{2}^{2} \left( {1 + \alpha_{1}^{2} } \right),\;a_{8} = 2\alpha_{0} \alpha_{1}^{2} \alpha_{2}^{2} ,\;a_{9} = \left( 1 \right. + \left. {\alpha_{1}^{2} } \right)\alpha_{2} ,\;a_{10} = 2\alpha_{1}^{2} \alpha_{2} ,\,a_{11} = \alpha_{0}^{2} \alpha_{3}^{2} + \frac{{\alpha_{3}^{2} \left( {1 - \alpha_{1} } \right)^{2} }}{{4\alpha_{2}^{2} }},\\ a_{12}& = \frac{{\alpha_{0} \alpha_{3}^{2} }}{{\alpha_{2} }},\,a_{13} = \frac{{\alpha_{3}^{2} }}{{\alpha_{2} }},\;a_{14} = \alpha_{1} \alpha_{3}^{2} ,a_{15} = \frac{{2\left( {n - 1} \right)^{2} }}{{3n^{2} }}\alpha_{2}^{2} , \end{aligned}$$
$$a_{16} = \frac{{2\left( {n - 1} \right)^{4} }}{{3n^{4} }}\alpha_{2}^{4} + \frac{{\left( {n - 1} \right)^{2} }}{{3n^{2} }}\left[ {\alpha_{1}^{2} } \right. - 2\alpha_{2}^{4} - \left. {\alpha_{2}^{2} \left( {1 - \alpha_{1} } \right)^{2} } \right] - \frac{{2\left( {n - 1} \right)\left( {3n - 2} \right)}}{{3n^{3} }}\alpha_{1} \alpha_{2}^{2} ,a_{17} = \frac{{\left( {n - 1} \right)^{2} }}{{144n^{2} }}\alpha_{2}^{2} ,$$
$$a_{18} = \frac{{\left( {n - 1} \right)^{2} \left( {4n^{2} - 10n + 5} \right)}}{{144n^{4} }}\alpha_{2}^{4} - \frac{{\left( {n - 1} \right)^{2} }}{{72n^{2} }}\alpha_{2}^{2} \left( {1 - \alpha_{1} } \right)^{2} + \frac{{\left( {n - 1} \right)}}{{72n^{3} }}\alpha_{1} \alpha_{2}^{2} - \frac{{\left( {n - 1} \right)^{2} }}{{144n^{2} }}\alpha_{1}^{2} ,$$
$$\begin{aligned} a_{19} & = \frac{{\left( {n - 1} \right)^{4} \left( {2n^{2} - 10n + 5} \right)}}{{54n^{6} }}\alpha_{2}^{6} - \frac{{\left( {n - 1} \right)^{2} \left( {10n^{2} - 26n + 13} \right)}}{{144n^{4} }}\alpha_{2}^{4} \left( {1 - \alpha_{1} } \right)^{2} + \frac{{\left( {n - 1} \right)^{2} \left( {11n - 8} \right)}}{{72n^{5} }}\alpha_{1} \alpha_{2}^{4} \\ & \quad- \frac{{\left( {n - 1} \right)^{3} }}{{24n^{3} }}\alpha_{1} \alpha_{2}^{2} \left( {1 - \alpha_{1} } \right)^{2} - \frac{{\left( {n - 1} \right)^{2} \left( {4n^{2} - 8n + 1} \right)}}{{72n^{4} }}\alpha_{1}^{2} \alpha_{2}^{2} , \\ \end{aligned}$$
$$\begin{aligned} a_{20} & = \frac{{\left( {n - 1} \right)^{4} \left( {2n - 1} \right)\left( {4n^{2} - 10n + 5} \right)}}{{36n^{8} }}\alpha_{2}^{8} + \frac{{\left( {n - 1} \right)^{4} \left( {4n^{2} - 30n + 15} \right)}}{{72n^{6} }}\alpha_{2}^{6} \left( {1 - \alpha_{1} } \right)^{2} - \frac{{\left( {n - 1} \right)^{3} \left( {20n^{2} - 33n + 12} \right)}}{{36n^{7} }}\alpha_{1} \alpha_{2}^{6} \\ & \quad - \frac{{5\left( {n - 1} \right)^{4} }}{{72n^{4} }}\alpha_{2}^{4} \left( {1 - \alpha_{1} } \right)^{4} + \frac{{\left( {n - 1} \right)^{3} \left( {3n^{2} - 12n + 11} \right)}}{{36n^{5} }}\alpha_{1} \alpha_{2}^{4} \left( {1 - \alpha_{1} } \right)^{2} + \frac{{\left( {n - 1} \right)^{2} \left( {32n^{4} - 88n^{3} + 62n^{2} - 11} \right)}}{{72n^{6} }}\alpha_{1}^{2} \alpha_{2}^{4} \\ & \quad- \frac{{\left( {n - 1} \right)^{4} }}{{36n^{4} }}\alpha_{1}^{2} \alpha_{2}^{2} \left( {1 - \alpha_{1} } \right)^{2} + \frac{{\left( {n - 1} \right)^{3} }}{{36n^{5} }}\alpha_{1}^{3} \alpha_{2}^{2} - \frac{{\left( {n - 1} \right)^{4} }}{{36n^{4} }}\alpha_{1}^{4} , \\ \end{aligned}$$
$$\begin{aligned} a_{21} & = \frac{{\left( {n - 1} \right)^{6} \left( {2n - 1} \right)^{2} }}{{9n^{10} }}\alpha_{2}^{10} + \frac{{\left( {n - 1} \right)^{4} \left( {2n - 1} \right)\left( {4n^{2} - 14n + 7} \right)}}{{36n^{8} }}\alpha_{2}^{8} \left( {1 - \alpha_{1} } \right)^{2} + \frac{{\left( {n - 1} \right)^{4} \left( {2n - 1} \right)\left( {6n^{2} - 17n + 8} \right)}}{{18n^{9} }}\alpha_{1} \alpha_{2}^{8} \\ & \quad - \frac{{\left( {n - 1} \right)^{4} \left( {n^{2} + 4n - 2} \right)}}{{18n^{6} }}\alpha_{2}^{6} \left( {1 - \alpha_{1} } \right)^{4} - \frac{{\left( {n - 1} \right)^{3} \left( {12n^{4} - 30n^{3} + 37n^{2} - 32n + 10} \right)}}{{18n^{7} }}\alpha_{1} \alpha_{2}^{6} \left( {1 - \alpha_{1} } \right)^{2} \\ & \quad - \frac{{\left( {n - 1} \right)^{3} \left( {12n^{5} - 40n^{4} + 32n^{3} + 3n^{2} - 15n + 5} \right)}}{{9n^{8} }}\alpha_{1}^{2} \alpha_{2}^{6} + \frac{{\left( {n - 1} \right)^{5} }}{{3n^{5} }}\alpha_{1} \alpha_{2}^{4} \left( {1 - \alpha_{1} } \right)^{4} + \frac{{\left( {n - 1} \right)^{5} }}{{6n^{5} }}\alpha_{1}^{3} \alpha_{2}^{2} \left( {1 - \alpha_{1} } \right)^{2} \\ & \quad + \frac{{\left( {n - 1} \right)^{4} \left( {34n^{2} - 38n - 5} \right)}}{{36n^{6} }}\alpha_{1}^{2} \alpha_{2}^{4} \left( {1 - \alpha_{1} } \right)^{2} + \frac{{\left( {n - 1} \right)^{3} \left( {13n^{2} - 11n + 4} \right)}}{{18n^{7} }}\alpha_{1}^{3} \alpha_{2}^{4} + \frac{{\left( {n - 1} \right)^{4} \left( {2n^{2} - 4n - 1} \right)}}{{18n^{6} }}\alpha_{1}^{4} \alpha_{2}^{2} , \\ \end{aligned}$$
$$\begin{aligned} a_{22} & = \frac{{\left( {n - 1} \right)^{6} \left( {2n - 1} \right)^{3} }}{{27n^{12} }}\alpha_{2}^{12} + \frac{{\left( {n - 1} \right)^{6} \left( {2n - 1} \right)^{2} }}{{18n^{10} }}\alpha_{2}^{10} \left( {1 - \alpha_{1} } \right)^{2} + \frac{{\left( {n - 1} \right)^{5} \left( {2n - 1} \right)^{2} \left( {3n - 2} \right)}}{{9n^{11} }}\alpha_{1} \alpha_{2}^{10} \\ & \quad - \frac{{\left( {n - 1} \right)^{4} \left( {2n - 1} \right)\left( {4n^{2} + 10n - 5} \right)}}{{72n^{8} }}\alpha_{2}^{8} \left( {1 - \alpha_{1} } \right)^{4} - \frac{{\left( {n - 1} \right)^{4} \left( {2n - 1} \right)\left( {12n^{3} - 6n^{2} + 7n - 4} \right)}}{{18n^{9} }}\alpha_{1} \alpha_{2}^{8} \left( {1 - \alpha_{1} } \right)^{2} \\ & \quad - \frac{{\left( {n - 1} \right)^{4} \left( {2n - 1} \right)\left( {24n^{4} - 16n^{3} - 20n^{2} + 28n - 9} \right)}}{{18n^{10} }}\alpha_{1}^{2} \alpha_{2}^{8} - \frac{{\left( {n - 1} \right)^{6} }}{{27n^{6} }}\alpha_{2}^{6} \left( {1 - \alpha_{1} } \right)^{6} + \frac{{\left( {n - 1} \right)^{5} }}{{9n^{7} }}\alpha_{1}^{5} \alpha_{2}^{2} + \frac{{\left( {n - 1} \right)^{6} }}{{27n^{6} }}\alpha_{1}^{6} \\ & \quad- \frac{{\left( {n - 1} \right)^{5} \left( {6n^{2} - 18n + 7} \right)}}{{18n^{7} }}\alpha_{1} \alpha_{2}^{6} \left( {1 - \alpha_{1} } \right)^{4} - \frac{{\left( {n - 1} \right)^{4} \left( {12n^{4} - 58n^{3} + 73n^{2} - 20n - 2} \right)}}{{18n^{8} }}\alpha_{1}^{2} \alpha_{2}^{6} \left( {1 - \alpha_{1} } \right)^{2} \\ & \quad- \frac{{\left( {n - 1} \right)^{3} \left( {72n^{4} - 183n^{3} + 174n^{2} - 78n + 14} \right)}}{{27n^{9} }}\alpha_{1}^{3} \alpha_{2}^{6} - \frac{{7\left( {n - 1} \right)^{6} }}{{18n^{6} }}\alpha_{1}^{2} \alpha_{2}^{4} \left( {1 - \alpha_{1} } \right)^{4} \\ & \quad - \frac{{\left( {n - 1} \right)^{4} \left( {24n^{4} - 52n^{3} + 52n^{2} - 20n + 3} \right)}}{{18n^{8} }}\alpha_{1}^{4} \alpha_{2}^{4} - \frac{{\left( {n - 1} \right)^{6} }}{{9n^{6} }}\alpha_{1}^{4} \alpha_{2}^{2} \left( {1 - \alpha_{1} } \right)^{2} - \frac{{\left( {n - 1} \right)^{5} \left( {15n^{2} - 18n + 2} \right)}}{{9n^{7} }}\alpha_{1}^{3} \alpha_{2}^{4} \left( {1 - \alpha_{1} } \right)^{2} , \\ \end{aligned}$$
$$\begin{aligned} a_{23} &= \frac{{\left( {2n - 1} \right)}}{{n^{2} }}\alpha_{2}^{2} - \frac{{2\left( {n - 1} \right)}}{n}\alpha_{1} ,\;a_{24} = \frac{{2\left( {n - 1} \right)}}{{n^{2} }}\alpha_{1} \alpha_{2}^{2} - \frac{{2\left( {n - 1} \right)}}{n}\alpha_{1}^{2} - \frac{{\left( {n - 1} \right)}}{n}\alpha_{2}^{2} \left( {1 + \alpha_{1} } \right)^{2} ,\;a_{25} = \frac{{\left( {n - 1} \right)^{4} }}{{1728n^{4} }}\alpha_{2}^{4} , \\ a_{26} &= \frac{{\left( {n - 1} \right)^{6} }}{{216n^{6} }}\alpha_{2}^{6} - \frac{{\left( {n - 1} \right)^{5} }}{{864n^{5} }}\alpha_{2}^{4} \left( {1 + \alpha_{1} } \right)^{2} - \frac{{\left( {n - 1} \right)^{4} }}{{1728n^{6} }}\alpha_{2}^{4} \left( {1 + \alpha_{1}^{2} } \right),\\ a_{27} & = \frac{{\left( {n - 1} \right)^{6} }}{{1728n^{6} }}\alpha_{2}^{4} \left( {1 + \alpha_{1}^{2} } \right)^{2} + \frac{{\left( {n - 1} \right)^{5} \left( {2n^{2} - 2n + 1} \right)}}{{432n^{7} }}\alpha_{1} \alpha_{2}^{4} \left( {1 + \alpha_{1}^{2} } \right) + \frac{{\left( {n - 1} \right)^{4} \left( {2n^{2} - 2n + 1} \right)^{2} }}{{1728n^{8} }}\alpha_{1}^{2} \alpha_{2}^{4} \\ & \quad- \frac{{\left( {n - 1} \right)^{6} \left( {16n^{2} - 6n + 3} \right)}}{{864n^{8} }}\alpha_{2}^{6} \left( {1 + \alpha_{1}^{2} } \right) + \frac{{\left( {n - 1} \right)^{5} \left( {4n^{2} + 2n - 1} \right)}}{{432n^{7} }}\alpha_{1} \alpha_{2}^{6} + \frac{{\left( {n - 1} \right)^{6} \left( {2n^{2} - 6n + 3} \right)}}{{216n^{8} }}\alpha_{2}^{8} , \\ \end{aligned}$$
$$\begin{aligned} a_{28} & = \frac{{\left( {n - 1} \right)^{7} }}{{432n^{7} }}\alpha_{1} \alpha_{2}^{4} \left( {1 + \alpha_{1}^{2} } \right)^{2} + \frac{{\left( {n - 1} \right)^{6} \left( {2n^{2} - 2n + 1} \right)}}{{432n^{8} }}\alpha_{1}^{2} \alpha_{2}^{4} \left( {1 + \alpha_{1}^{2} } \right) + \frac{{\left( {n - 1} \right)^{5} \left( {2n - 1} \right)^{2} }}{{432n^{9} }}\alpha_{1}^{3} \alpha_{2}^{4} + \frac{{\left( {n - 1} \right)^{8} \left( {2n - 1} \right)}}{{54n^{10} }}\alpha_{2}^{10} \\ & \quad- \frac{{\left( {n - 1} \right)^{6} \left( {20n^{2} - 22n + 11} \right)}}{{864n^{8} }}\alpha_{2}^{6} \left( {1 + \alpha_{1}^{2} } \right)^{2} + \frac{{\left( {n - 1} \right)^{5} \left( {4n^{2} - 2n + 1} \right)^{2} }}{{864n^{9} }}\alpha_{1} \alpha_{2}^{6} \left( {1 + \alpha_{1}^{2} } \right) + \frac{{\left( {n - 1} \right)^{7} \left( {2n - 1} \right)}}{{54n^{9} }}\alpha_{1} \alpha_{2}^{8} \\ & \quad+ \frac{{\left( {n - 1} \right)^{6} \left( {8n^{4} - 40n^{3} + 32n^{2} - 12n + 3} \right)}}{{432n^{10} }}\alpha_{2}^{8} \left( {1 + \alpha_{1}^{2} } \right) + \frac{{\left( {n - 1} \right)^{6} \left( {2n - 1} \right)\left( {2n^{2} - 2n + 1} \right)}}{{432n^{10} }}\alpha_{1}^{2} \alpha_{2}^{6} , \\ \end{aligned}$$
$$\begin{aligned} a_{29} & = \frac{{\left( {n - 1} \right)^{8} }}{{432n^{8} }}\alpha_{1}^{2} \alpha_{2}^{4} \left( {1 + \alpha_{1}^{2} } \right)^{2} + \frac{{\left( {n - 1} \right)^{6} \left( {2n - 1} \right)^{2} }}{{432n^{10} }}\alpha_{1}^{4} \alpha_{2}^{4} - \frac{{\left( {n - 1} \right)^{8} }}{{108n^{8} }}\alpha_{2}^{6} \left( {1 + \alpha_{1}^{2} } \right)^{3} + \frac{{\left( {n - 1} \right)^{7} \left( {2n - 1} \right)^{2} }}{{36n^{11} }}\alpha_{1} \alpha_{2}^{10} \\ & \quad+ \frac{{\left( {n - 1} \right)^{6} \left( {2n - 1} \right)\left( {2n^{2} - 10n + 5} \right)}}{{432n^{10} }}\alpha_{1}^{2} \alpha_{2}^{6} \left( {1 + \alpha_{1}^{2} } \right) + \frac{{\left( {n - 1} \right)^{5} \left( {2n - 1} \right)^{2} \left( {3n^{2} - 2n + 1} \right)}}{{216n^{11} }}\alpha_{1}^{3} \alpha_{2}^{6} + \frac{{\left( {n - 1} \right)^{8} \left( {2n - 1} \right)^{2} }}{{108n^{12} }}\alpha_{2}^{12} \\ & \quad+ \frac{{\left( {n - 1} \right)^{6} \left( {4n^{2} - 2n + 1} \right)\left( {4n^{2} - 26n + 13} \right)}}{{1728n^{10} }}\alpha_{2}^{8} \left( {1 + \alpha_{1}^{2} } \right)^{2} + \frac{{\left( {n - 1} \right)^{7} \left( {2n - 1} \right)\left( {8n^{2} - 10n + 5} \right)}}{{432n^{11} }}\alpha_{1} \alpha_{2}^{8} \left( {1 + \alpha_{1}^{2} } \right) \\ & \quad+ \frac{{\left( {n - 1} \right)^{6} \left( {2n - 1} \right)^{2} \left( {13n^{2} - 2n + 1} \right)}}{{432n^{12} }}\alpha_{1}^{2} \alpha_{2}^{8} + \frac{{\left( {n - 1} \right)^{8} \left( {2n - 1} \right)\left( {4n^{2} - 2n + 1} \right)}}{{216n^{12} }}\alpha_{2}^{10} \left( {1 + \alpha_{1}^{2} } \right) \\ & \quad+ \frac{{\left( {n - 1} \right)^{7} \left( {4n^{2} - 2n + 1} \right)}}{{432n^{9} }}\alpha_{1} \alpha_{2}^{6} \left( {1 + \alpha_{1}^{2} } \right)^{2} , \\ \end{aligned}$$
$$a_{30} = \frac{1}{{n^{2} }}\alpha_{2}^{2} + \frac{{2\left( {n - 1} \right)}}{n}\left( {\frac{1}{n}\alpha_{2}^{2} - \alpha_{1} } \right),$$
(C.1)

The coefficient expressions in Eq. (53) are as follows:

$$\begin{aligned}a_{31} & = \frac{1}{1728}\left( {\frac{{4\alpha_{2}^{2} }}{{\left( {1 - \alpha_{1} } \right)^{2} }}} \right. - 1\left. { + \frac{{\alpha_{1} }}{{n\left( {n - 1} \right)\left( {1 - \alpha_{1} } \right)^{2} }}} \right)^{3} + \frac{1}{128}\frac{{\left( {2n - 1} \right)^{2} \alpha_{2}^{2} \left( {1 - \alpha_{1}^{2} } \right)^{2} }}{{n^{2} \left( {n - 1} \right)^{2} \left( {1 - \alpha_{1} } \right)^{6} }},\;a_{32} = \frac{1}{384}\frac{{\left( {2n - 1} \right)\alpha_{2} \left( {1 - \alpha_{1}^{2} } \right)}}{{n\left( {n - 1} \right)\left( {1 - \alpha_{1} } \right)^{3} }}, \end{aligned}$$
$$a_{33} = \frac{4}{3}\left( {\frac{{4\alpha_{2}^{2} }}{{\left( {1 - \alpha_{1} } \right)^{2} }}} \right. - 1\left. { + \frac{{\alpha_{1} }}{{n\left( {n - 1} \right)\left( {1 - \alpha_{1} } \right)^{2} }}} \right)^{3} + 9\frac{{\left( {2n - 1} \right)^{2} \alpha_{2}^{2} \left( {1 - \alpha_{1}^{2} } \right)^{2} }}{{n^{2} \left( {n - 1} \right)^{2} \left( {1 - \alpha_{1} } \right)^{6} }},\;a_{34} = \sqrt[3]{{a_{31} + a_{32} \sqrt {a_{33} } }} + \sqrt[3]{{a_{31} - a_{32} \sqrt {a_{33} } }},$$
$$a_{35} = 1 + \frac{{2\alpha_{2}^{2} }}{{\left( {1 - \alpha_{1} } \right)^{2} }} - \frac{{\alpha_{1} }}{{n\left( {n - 1} \right)\left( {1 - \alpha_{1} } \right)^{2} }},\;a_{36} = \frac{{\alpha_{2} }}{{1 - \alpha_{1} }}\left( {2 + \frac{{1 + \alpha_{1}^{2} }}{{n\left( {n - 1} \right)\left( {1 - \alpha_{1} } \right)^{2} }}} \right),\;\;a_{37} = \frac{{\alpha_{2} }}{{1 - \alpha_{1} }}.$$
(C.2)

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Liu, B., Gu, L. & Dong, M. Design and characteristic analysis of an X-shaped negative stiffness structure. Acta Mech 233, 4549–4587 (2022). https://doi.org/10.1007/s00707-022-03343-y

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