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Vibration response and isolation of X-shaped two-stage vibration isolators: Analysis of multiple parameters

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Abstract

To exploit its advantages in passive vibration isolation, an X-shaped structure is integrated into a two-stage vibration isolation system so as to realize the broadband vibration isolation capability. The Lagrange equation is employed to establish the nonlinear equation of motion of the proposed X-shaped two-stage vibration isolator, which takes into account both rotational and vertical displacements. The incremental harmonic balance method is used to examine the effects of the key parameters of the system on the frequency response characteristics and vibration isolation frequency bands. The correctness and feasibility of the present analytical approach are validated by comparing with the numerical simulation and the experimental results. The investigation results indicate that the vibration isolation frequency of the designed system can be as low as 2–5 Hz, and the performance of low-frequency vibration isolation can be improved considerably by softening nonlinearity. Additionally, systematic revelations behind the occurrence of softening-hardening, softening, and hardening nonlinear behaviors are made. To assess the effectiveness of the system in vibration isolation, random vibration tests are also conducted. In conclusion, the designed X-shaped two-stage mechanism offers a novel point of view on low-frequency vibration isolation.

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References

  1. Ling, P., Miao, L.L., Zhang, W.M., Wu, C.Y., Yan, B.: Cockroach-inspired structure for low-frequency vibration isolation. Mech. Syst. Sig. Process. 171, 108955 (2022)

    Google Scholar 

  2. Jin, Y.B., Zeng, S.X., Wen, Z.H., He, L.S., Li, Y., Li, Y.: Deep-subwavelength lightweight metastructures for low-frequency vibration isolation. Mater. Des. 215, 110499 (2022)

    Google Scholar 

  3. Ibrahim, R.A.: Recent advances in nonlinear passive vibration isolators. J. Sound Vib. 314(3–5), 371–452 (2008)

    Google Scholar 

  4. Liu, C.C., Jing, X.J., Daley, S., Li, F.M.: Recent advances in micro-vibration isolation. Mech. Syst. Sig. Process. 56–57, 55–80 (2015)

    Google Scholar 

  5. Yang, T., Cao, Q.J.: Modeling and analysis of a novel multi-directional micro-vibration isolator with spring suspension struts. Arch. Appl. Mech. 92, 801–819 (2022)

    Google Scholar 

  6. Zhang, F., Xu, M.L., Shao, S.B., Xie, S.L.: A new high-static-low-dynamic stiffness vibration isolator based on magnetic negative stiffness mechanism employing variable reluctance stress. J. Sound Vib. 476, 115322 (2020)

    Google Scholar 

  7. Rezaei, M., Talebitooti, R.: Investigating the performance of tri-stable magneto-piezoelastic absorber in simultaneous energy harvesting and vibration isolation. Appl. Math. Model. 102, 661–693 (2022)

    Google Scholar 

  8. Liu, Y.Q., Ji, W., Xu, L.L., Gu, H.S., Song, C.F.: Dynamic characteristics of quasi-zero stiffness vibration isolation system for coupled dynamic vibration absorber. Arch. Appl. Mech. 91, 3799–3818 (2021)

    Google Scholar 

  9. Kamaruzaman, N.A., Robertson, W.S.P., Ghayesh, M.H., Cazzolato, B.S., Zander, A.C.: Six degree of freedom quasi-zero stiffness magnetic spring with active control: theoretical analysis of passive versus active stability for vibration isolation. J. Sound Vib. 502, 116086 (2021)

    Google Scholar 

  10. Zhang, Y.Q., Yang, T., Du, H.F., Zhou, S.X.: Wideband vibration isolation and energy harvesting based on a coupled piezoelectric-electromagnetic structure. Mech. Syst. Sig. Process. 184, 109689 (2023)

    Google Scholar 

  11. Lu, Z.Q., Chen, L.Q., Brennan, M.J., Yang, T.J., Ding, H., Liu, Z.G.: Stochastic resonance in a nonlinear mechanical vibration isolation system. J. Sound Vib. 370, 221–229 (2016)

    Google Scholar 

  12. Liu, S.W., Peng, G.L., Jin, K.: Design and characteristics of a novel QZS vibration isolation system with origami-inspired corrector. Nonlinear Dyn. 106, 255–277 (2021)

    Google Scholar 

  13. Zou, W., Cheng, C., Ma, R., Hu, Y., Wang, W.P.: Performance analysis of a quasi-zero stiffness vibration isolation system with scissor-like structures. Arch. Appl. Mech. 91, 117–133 (2021)

    Google Scholar 

  14. Zhu, Y.P., Lang, Z.Q.: Beneficial effects of antisymmetric nonlinear damping with application to energy harvesting and vibration isolation under general inputs. Nonlinear Dyn. 108, 2917–2933 (2022)

    Google Scholar 

  15. Wen, G.L., He, J.F., Liu, J., Lin, Y.: Design, analysis and semi-active control of a quasi-zero stiffness vibration isolation system with six oblique springs. Nonlinear Dyn. 106, 309–321 (2021)

    Google Scholar 

  16. Vo, N.Y.P., Nguyen, M.K., Le, T.D.: Analytical study of a pneumatic vibration isolation platform featuring adjustable stiffness. Commun. Nonlinear Sci. Numer. Simulat. 98, 105775 (2021)

    MathSciNet  MATH  Google Scholar 

  17. Ye, K., Ji, J.C., Brown, T.: A novel integrated quasi-zero stiffness vibration isolator for coupled translational and rotational vibrations. Mech. Syst. Sig. Process. 149, 107340 (2021)

    Google Scholar 

  18. Gatti, G.: Statics and dynamics of a nonlinear oscillator with quasi-zero stiffness behaviour for large deflections. Commun. Nonlinear Sci. Numer. Simulat. 83, 105143 (2020)

    MathSciNet  MATH  Google Scholar 

  19. Xiong, Y.H., Li, F.M., Wang, Y.: A nonlinear quasi-zero-stiffness vibration isolation system with additional X-shaped structure: theory and experiment. Mech. Syst. Sig. Process. 177, 109208 (2022)

    Google Scholar 

  20. Liu, C.R., Yu, K.P., Liao, B.P., Hu, R.P.: Enhanced vibration isolation performance of quasi-zero-stiffness isolator by introducing tunable nonlinear inerter. Commun. Nonlinear Sci. Numer. Simulat. 95, 105654 (2021)

    MathSciNet  MATH  Google Scholar 

  21. Song, X.Y., Chai, Z.Y., Zhang, Y.W., Zang, J., Xu, K.F.: Nonlinear vibration isolation via an innovative active bionic variable stiffness adapter (ABVSA). Nonlinear Dyn. 109, 353–370 (2022)

    Google Scholar 

  22. Niu, M.Q., Chen, L.Q.: Nonlinear vibration isolation via a compliant mechanism and wire ropes. Nonlinear Dyn. 107, 1687–1702 (2022)

    Google Scholar 

  23. Chai, Y.Y., Jing, X.J., Guo, Y.Q.: A compact X-shaped mechanism based 3-DOF anti-vibration unit with enhanced tunable QZS property. Mech. Syst. Sig. Process. 168, 108651 (2022)

    Google Scholar 

  24. Robertson, W.S., Kidner, M.R.F., Cazzolato, B.S., Zander, A.C.: Theoretical design parameters for a quasi-zero stiffness magnetic spring for vibration isolation. J. Sound Vib. 326(1), 88–103 (2009)

    Google Scholar 

  25. Yun, H., Liu, L., Li, Q., Yang, H.J.: Investigation on two-stage vibration suppression and precision pointing for space optical payloads. Aerosp. Sci. Technol. 96, 105543 (2020)

    Google Scholar 

  26. Wu, Q.C., Huang, G.T., Liu, C., Xie, S.L., Xu, M.L.: Low-frequency multi-mode vibration suppression of a metastructure beam with two-stage high-static-low-dynamic stiffness oscillators. Acta. Mech. 230, 4341–4356 (2019)

    MathSciNet  MATH  Google Scholar 

  27. Xie, X., Ren, M., Zheng, H., Zhang, Z.: Investigation on a two-stage platform of large stroke for broadband vertical vibration isolation. J. Vib. Control 25(6), 1233–1245 (2019)

    Google Scholar 

  28. Lu, Z.Q., Brennan, M.J., Yang, T.J., Li, X.H., Liu, Z.G.: An investigation of a two-stage nonlinear vibration isolation system. J. Sound Vib. 332(6), 1456–1464 (2013)

    Google Scholar 

  29. Mohanty, S., Dwivedy, S.K.: Traditional and non-traditional active nonlinear vibration absorber with time delay combination feedback for hard excitation. Commun. Nonlinear Sci. Numer. Simulat. 117, 106919 (2023)

    MathSciNet  MATH  Google Scholar 

  30. Yang, K., Harne, R.L., Wang, K.W., Huang, H.: Investigation of a bistable dual-stage vibration isolator under harmonic excitation. Smart Mater. Struct. 23(4), 494–501 (2014)

    Google Scholar 

  31. Eskandary-Malayery, F., Ilanko, S., Mace, B., Mochida, Y., Pellicano, F.: Experimental and numerical investigation of a vertical vibration isolator for seismic applications. Nonlinear Dyn. 109, 303–322 (2022)

    Google Scholar 

  32. Lee, C.M., Goverdovskiy, V.N., Temnikov, A.I.: Design of springs with “negative” stiffness to improve vehicle driver vibration isolation. J. Sound Vib. 302(4–5), 865–874 (2007)

    Google Scholar 

  33. Lee, C.M., Goverdovskiy, V.N.: A multi-stage high-speed railroad vibration isolation system with “negative” stiffness. J. Sound Vib. 331(4), 914–921 (2012)

    Google Scholar 

  34. Xu, Z.D., Huang, X.H., Xu, F.H., Yuan, J.: Parameters optimization of vibration isolation and mitigation system for precision platforms using non-dominated sorting genetic algorithm. Mech. Syst. Sig. Process. 128, 191–201 (2019)

    Google Scholar 

  35. Sun, B., Xu, Z.D.: A computational method for simulating mesoscale competitive fracture process of heterogeneous Quasi-brittle building materials. Iran. J. Sci. Technol. Trans. Mech. Eng. 46, 557–572 (2022)

    Google Scholar 

  36. Choi, J.S., Kim, K.Y., Kim, H.Y., Lee, S.W.: Effect of inertia variations for active vibration isolation systems. Precis. Eng. 66, 507–518 (2020)

    Google Scholar 

  37. Xu, Z.D., Xu, F.H., Chen, X.: Intelligent vibration isolation and mitigation of a platform by using MR and VE devices. J. Aerosp. Eng. 29(4), 04016010 (2016)

    MathSciNet  Google Scholar 

  38. Xu, Z.D., Xu, F.H., Chen, X.: Vibration suppression on a platform by using vibration isolation and mitigation devices. Nonlinear Dyn. 83(3), 1341–1353 (2016)

    Google Scholar 

  39. Xu, Z.D., Guo, Y.F., Wang, S.A., Huang, X.H.: Optimization analysis on parameters of multi-dimensional earthquake isolation and mitigation device based on genetic algorithm. Nonlinear Dyn. 72(4), 757–765 (2013)

    Google Scholar 

  40. Ulgen, D., Ertugrul, O.L., Ozkan, M.Y.: Measurement of ground borne vibrations for foundation design and vibration isolation of a high-precision instrument. Measurement 93, 385–396 (2016)

    Google Scholar 

  41. Toygar, O., Ulgen, D., Fidan, N.B.: Assessing vibration isolation performance of single and coupled wave barriers through field experiments. Constr. Build. Mater. 354, 129156 (2022)

    Google Scholar 

  42. Toygar, O., Ulgen, D.: A full-scale field study on mitigation of environmental ground vibrations by using open trenches. Build. Environ. 4, 108070 (2021)

    Google Scholar 

  43. Matichard, F., Lantz, B., Mason, K., et al.: Advanced LIGO two-stage twelve-axis vibration isolation and positioning platform. Part 1: design and production overview. Precis. Eng. 40, 273–286 (2015)

    Google Scholar 

  44. Matichard, F., Lantz, B., Mason, K., et al.: Advanced LIGO two-stage twelve-axis vibration isolation and positioning platform. Part 2: Experimental investigation and tests results. Precis. Eng. 40, 287–297 (2015)

    Google Scholar 

  45. Ma, X.L., Jin, G.Y., Liu, Z.G.: Active structural acoustic control of an elastic cylindrical shell coupled to a two-stage vibration isolation system. Int. J. Mech. Sci. 79, 182–194 (2014)

    Google Scholar 

  46. Yu, Y.H., Yao, G., Wu, Z.H.: Nonlinear primary responses of a bilateral supported X-shape vibration reduction structure. Mech. Syst. Sig. Process. 140, 106679 (2020)

    Google Scholar 

  47. Luo, H.T., Fan, C.H., Li, Y.X., Liu, G.M., Yu, C.S.: Design and experiment of micro-vibration isolation system for optical satellite. Eur. J. Mech. A-Solids 97, 104833 (2023)

    MathSciNet  MATH  Google Scholar 

  48. Lu, J.J., Yan, G., Qi, W.H., Yan, H., Ma, J., Shi, J.W., Wu, Z.Y., Zhang, W.M.: Sliding-boundary-constrained cantilever structure for vibration isolation via nonlinear stiffness modulation. Int. J. Mech. Sci. 235, 107733 (2022)

    Google Scholar 

  49. Zhao, H.J., Feng, Y., Li, W., Xue, C.: Numerical study and topology optimization of vibration isolation support structures. Int. J. Mech. Sci. 228, 107507 (2022)

    Google Scholar 

  50. Chen, R.Z., Li, X.P., Tian, J., Yang, Z.M., Xu, J.C.: On the displacement transferability of variable stiffness multi-directional low frequency vibration isolation joint. Appl. Math. Model. 112, 690–707 (2022)

    MathSciNet  MATH  Google Scholar 

  51. Liu, K., Han, L., Hu, W.X., Ji, L.T., Zhu, S.X., Wan, Z.S., Yang, X.D., Wei, Y.L., Dai, Z.J., Zhao, Z.A., Li, Z., Wang, P.F., Tao, R.: 4D printed zero Poisson’s ratio metamaterial with switching function of mechanical and vibration isolation performance. Mater. Des. 196, 109153 (2020)

    Google Scholar 

  52. Yao, Z.C., Zhao, R.X., Zega, V., Corigliano, A.: A metaplate for complete 3D vibration isolation. Eur. J. Mech. A-Solids 84, 104016 (2020)

    MATH  Google Scholar 

  53. Wang, M., Hu, Y.Y., Sun, Y., Ding, J.H., Pu, H.Y., Yuan, S.J., Zhao, J.L., Peng, Y., Xie, S.R., Luo, J.: An adjustable low-frequency vibration isolation stewart platform based on electromagnetic negative stiffness. Int. J. Mech. Sci. 181, 105714 (2020)

    Google Scholar 

  54. Chen, S.B., Xuan, M., Xin, J., Liu, Y., Gu, S., Li, J., Zhang, L.: Design and experiment of dual micro-vibration isolation system for optical satellite flywheel. Int. J. Mech. Sci. 179, 105592 (2020)

    Google Scholar 

  55. Ujjawal, K.N., Venkateswarlu, H., Hegde, A.: Vibration isolation using 3D cellular confinement system: A numerical investigation. Soil. Dyn. Earthq. Eng. 119, 220–234 (2019)

    Google Scholar 

  56. Billon, K., Montcoudiol, N., Aubry, A., Pascual, R., Mosca, F., Jean, F., Pezerat, C., Bricault, C., Chesné, S.: Vibration isolation and damping using a piezoelectric flextensional suspension with a negative capacitance shunt. Mech. Syst. Sig. Process. 140, 106696 (2020)

    Google Scholar 

  57. Burdzik, R.: A comprehensive diagnostic system for vehicle suspensions based on a neural classifier and wavelet resonance estimators. Measurement 200, 111602 (2022)

    Google Scholar 

  58. Zhou, S.H., Liu, Y.L., Jiang, Z.Y., Ren, Z.H.: Nonlinear dynamic behavior of a bio-inspired embedded X-shaped vibration isolation system. Nonlinear Dyn. 110, 153–175 (2022)

    Google Scholar 

  59. Cheung, Y.K., Chen, S.H., Lau, S.L.: Application of the incremental harmonic balance method to cubic non-linearity systems. J. Sound Vib. 140, 273–286 (1990)

    Google Scholar 

  60. Leung, A.Y.T., Chui, S.K.: Non-linear vibration of coupled duffing oscillators by an improved incremental harmonic balance method. J. Sound Vib. 181(4), 619–633 (1995)

    MathSciNet  MATH  Google Scholar 

  61. Friedmann, P., Hammond, C.E., Woo, T.H.: Efficient numerical treatment of periodic systems with application to stability problems. Int. J. Numer. Meth. Eng. 11, 17–36 (1977)

    MathSciNet  MATH  Google Scholar 

  62. Wu, Z.H., Zhang, Y.M., Yao, G., Yang, Z.: Nonlinear primary and super-harmonic resonances of functionally graded carbon nanotube reinforced composite beams. Int. J. Mech. Sci. 153–154, 321–340 (2019)

    Google Scholar 

  63. Jing, X.J., Chai, Y.Y., Chao, X., Bian, J.: In-situ adjustable nonlinear passive stiffness using X-shaped mechanisms. Mech. Syst. Sig. Process. 170, 108267 (2022)

    Google Scholar 

  64. Xu, J.W., Yang, X.F., Li, W., Zheng, J.Y., Wang, Y.Q., Fan, M.B., Zhou, W.T., Lu, Y.J.: Design of quasi-zero stiffness joint actuator and research on vibration isolation performance. J. Sound Vib. 479, 115367 (2020)

    Google Scholar 

  65. Li, M., Li, Y.Q., Liu, X.H., Dai, F.H.: A quasi-zero-stiffness vibration isolator using bi-stable hybrid symmetric laminate. Compos. Struct. 299, 116047 (2022)

    Google Scholar 

  66. Yadava, A., Amabili, M., Panda, S.K., Dey, T., Kumar, R.: Forced nonlinear vibrations of circular cylindrical sandwich shells with cellular core using higher-order shear and thickness deformation theory. J. Sound Vib. 510, 116283 (2021)

    Google Scholar 

  67. Wang, Y., Li, H.X., Cheng, C., Ding, H., Chen, L.Q.: A nonlinear stiffness and nonlinear inertial vibration isolator. J. Vib. Control 27(11–12), 1336–1352 (2021)

    MathSciNet  Google Scholar 

  68. Loghman, E., Bakhtiari-Nejad, A.K.F., Abbaszadeh, M., Amabili, M.: On the combined shooting-Pseudo-Arclength method for finding frequency response of nonlinear fractional-order differential equations. J. Sound Vib. 516, 116521 (2022)

    Google Scholar 

  69. Khaniki, H.B., Ghayesh, M.H., Chin, R., Amabili, M.: Large amplitude vibrations of imperfect porous-hyperelastic beams via a modified strain energy. J. Sound Vib. 513, 116416 (2021)

    Google Scholar 

  70. Ravindra, B., Mallik, A.: Performance of non-linear vibration isolators under harmonic excitation. J. Sound Vib. 170(3), 325–337 (1994)

    MATH  Google Scholar 

  71. Londoño, J.M., Neild, S.A., Cooper, J.: Identification of backbone curves of nonlinear systems from resonance decay responses. J. Sound Vib. 348, 224–238 (2015)

    Google Scholar 

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Acknowledgements

This research is supported by the National Natural Science Foundation of China under Grant No. 12072083, and the Heilongjiang Provincial Natural Science Foundation of China under Grant No. TD2020A001.

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  1. College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin, 150001, China

    Yongheng Yu & Fengming Li

  2. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, China

    Guo Yao

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Appendix

Appendix

In Eq. (7), the matrices M, C and H(X) are expressed as

$$ {\mathbf{M}} = \left[ {\begin{array}{*{20}c} {m_{1} } & 0 & 0 & 0 \\ 0 & {m_{2} } & 0 & 0 \\ 0 & 0 & {J_{1} } & 0 \\ 0 & 0 & 0 & {J_{2} } \\ \end{array} } \right] $$
$$ {\mathbf{C}} = \left[ {\begin{array}{*{20}c} {c_{1} + c_{2} } & { - (c_{1} + c_{2} )} & 0 & 0 \\ { - \left( {c_{1} + c_{2} } \right)} & {c_{1} + c_{2} } & 0 & 0 \\ 0 & 0 & {a(c_{1} + c_{2} )} & { - a(c_{1} + c_{2} )} \\ 0 & 0 & { - a(c_{1} + c_{2} )} & {a(c_{1} + c_{2} )} \\ \end{array} } \right] $$
$$ {\mathbf{H}}({\mathbf{X}}) = [\begin{array}{*{20}c} {H_{1} ({\mathbf{X}})} & {H_{2} ({\mathbf{X}})} & {H_{3} ({\mathbf{X}})} & {H_{4} ({\mathbf{X}})} \\ \end{array} ]^{{\text{T}}} $$
$$ \begin{aligned} H_{1} \left( {\mathbf{X}} \right) & = \frac{{2k_{1} \left[ {\sqrt {l^{2} - \left( {l\sin \alpha - y_{1} - a\theta_{1} + y_{2} + a\theta_{2} } \right)^{2} } } \right.\left. { - l\cos \alpha } \right]\left( {2l\sin \alpha - 2y_{1} - 2a\theta_{1} + 2y_{2} + 2a\theta_{2} } \right)}}{{H_{1} \left( {\mathbf{X}} \right)}} \\ & \quad + \frac{{2k_{2} [\sqrt {l^{2} - (l\sin \alpha - y_{1} + a\theta_{1} + y_{2} - a\theta_{2} )^{2} } - l\cos \alpha ](2l\sin \alpha - 2y_{1} + 2a\theta_{1} + 2y_{2} - 2a\theta_{2} )}}{{\sqrt {l^{2} - (l\sin \alpha - y_{1} + a\theta_{1} + y_{2} - a\theta_{2} )^{2} } }} \\ \end{aligned} $$
$$ \begin{aligned} H_{2} ({\mathbf{X}}) & = \frac{{2k_{1} [\sqrt {l^{2} - (l\sin \alpha - y_{1} - a\theta_{1} + y_{2} + a\theta_{2} )^{2} } - l\cos \alpha ]( - 2l\sin \alpha + 2y_{1} + 2a\theta_{1} - 2y_{2} - 2a\theta_{2} )}}{{\sqrt {l^{2} - (l\sin \alpha - y_{1} - a\theta_{1} + y_{2} + a\theta_{2} )^{2} } }} \\ & \quad + \frac{{2k_{2} [\sqrt {l^{2} - (l\sin \alpha - y_{1} + a\theta_{1} + y_{2} - a\theta_{2} )^{2} } - l\cos \alpha ]( - 2l\sin \alpha + 2y_{1} + 2a\theta_{1} - 2y_{2} + 2a\theta_{2} )}}{{\sqrt {l^{2} - (l\sin \alpha - y_{1} + a\theta_{1} + y_{2} - a\theta_{2} )^{2} } }} \\ & \quad + k_{3} (y_{2} + 2a\theta_{2} ) + k_{4} (y_{2} - a\theta_{2} ) \\ \end{aligned} $$
$$ \begin{aligned} H_{3} ({\mathbf{X}}) & = \frac{{4k_{1} [\sqrt {l^{2} - (l\sin \alpha - y_{1} - a\theta_{1} + y_{2} + a\theta_{2} )^{2} } - l\cos \alpha ](l\sin \alpha - y_{1} - a\theta_{1} + y_{2} + a\theta_{2} )a}}{{\sqrt {l^{2} - (l\sin \alpha - y_{1} - a\theta_{1} + y_{2} + a\theta_{2} )^{2} } }} \\ & \quad - \frac{{4k_{2} [\sqrt {l^{2} - (l\sin \alpha - y_{1} + a\theta_{1} + y_{2} - a\theta_{2} )^{2} } - l\cos \alpha ](l\sin \alpha - y_{1} + a\theta_{1} + y_{2} - a\theta_{2} )a}}{{\sqrt {l^{2} - (l\sin \alpha - y_{1} + a\theta_{1} + y_{2} - a\theta_{2} )^{2} } }} \\ \end{aligned} $$
$$ \begin{aligned} H_{4} ({\mathbf{X}}) & = \frac{{4k_{2} [\sqrt {l^{2} - (l\sin \alpha - y_{1} + a\theta_{1} + y_{2} - a\theta_{2} )^{2} } - l\cos \alpha ](l\sin \alpha - y_{1} + a\theta_{1} + 2y_{2} - a\theta_{2} )a}}{{\sqrt {l^{2} - (l\sin \alpha - y_{1} + a\theta_{1} + y_{2} - a\theta_{2} )^{2} } }} \\ & \quad - \frac{{4k_{1} [\sqrt {l^{2} - (l\sin \alpha - y_{1} - a\theta_{1} + y_{2} + a\theta_{2} )^{2} } - l\cos \alpha ](l\sin \alpha - y_{1} - a\theta_{1} + y_{2} + a\theta_{2} )a}}{{\sqrt {l^{2} - (l\sin \alpha - y_{1} - a\theta_{1} + y_{2} + a\theta_{2} )^{2} } }} \\ & \quad + k_{3} (y_{2} + a\theta_{2} )a - k_{4} (y_{2} - a\theta_{2} )a \\ \end{aligned} $$

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Yu, Y., Li, F. & Yao, G. Vibration response and isolation of X-shaped two-stage vibration isolators: Analysis of multiple parameters. Nonlinear Dyn 111, 15891–15910 (2023). https://doi.org/10.1007/s11071-023-08704-3

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