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Equivalent Nonlinear Beam Model for Static and Free Vibration Analysis of the Beamlike Truss

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Abstract

Purpose

In this study, an equivalent nonlinear beam model (ENBM) of the beamlike truss considering the geometric nonlinearity is proposed based on the equivalent modeling approach. The ENBM can promote significantly computational efficiency and has the advantage of analytical solution for nonlinear dynamic analysis; moreover, the equivalent model provides great convenience for controller design of the beamlike truss.

Methods

The ENBM contains nonlinear stretching force that can capture the large displacement effect of the beamlike truss, unlike most researches currently focus on equivalent beam model for linear beamlike truss (LBT). The novel equivalent nonlinear model is developed by introducing the von Karman nonlinear strain–displacement relationship in the equivalent linear beam model (ELBM). In effect, the nonlinear characteristic of the beamlike truss is related to nonlinearity of each member.

Results

To check the validity of the proposed ENBM, two aspects including static deflection and free vibration response are investigated utilizing nonlinear finite-element method to establish the full-scale beamlike truss model. This paper employs the Hamilton principle to work out the governing partial differential equations of motion of the ENBM with two pinned ends. The governing partial differential equations are further discretized with the aid of the Galerkin approach.

Conclusions

Comparisons between results of the ENBM and those obtained from the finite-element simulation of the nonlinear beamlike truss (NBT) show an excellent agreement, which confirms the validity and accuracy of the proposed method for imitating the static deflection and free vibration response of the original beamlike truss. The influences of the nonlinearity on the deflection and frequency of the equivalent beam model and the beamlike truss are discussed. In addition, the result of the calculation time demonstrates that the proposed approach can provide a more efficient nonlinear response analysis with significant less computational cost compared with the finite-element method.

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Data availability

The data that support the findings of this study are available from the corresponding author, upon reasonable request.

References

  1. Jones TC, Bart-Smith H, Mikulas M, Watson J (2007) Finite element modeling and analysis of large pretensioned space structures. J Spacecr Rocket 44(1):183–193. https://doi.org/10.2514/1.23116

    Article  Google Scholar 

  2. Murakami H (2001) Static and dynamic analyses of tensegrity structures. Part 1. Nonlinear equations of motion. Int J Solids Struct 38(20):3599–3613. https://doi.org/10.1016/S0020-7683(00)00232-8

    Article  MATH  Google Scholar 

  3. Natsuki T, Endo M (2005) Structural dependence of nonlinear elastic properties for carbon nanotubes using a continuum analysis. Appl Phys A 80(7):1463–1468. https://doi.org/10.1007/s00339-004-3146-4

    Article  Google Scholar 

  4. Yang P, Huang Z (2019) Effect of truss number on the dynamic response of truss. DEStech Trans Comput Sci Eng. https://doi.org/10.12783/dtcse/ammso2019/30163

    Article  Google Scholar 

  5. Li W, Ma H (2019) A novel model order reduction scheme for fast and accurate material nonlinear analyses of large-scale engineering structures. Eng Struct 193:238–257. https://doi.org/10.1016/j.engstruct.2019.04.036

    Article  Google Scholar 

  6. Rezaiee-Pajand M, Hashemian M, Bohluly A (2017) A novel time integration formulation for nonlinear dynamic analysis. Aerosp Sci Technol 69:625–635. https://doi.org/10.1016/j.ast.2017.07.032

    Article  Google Scholar 

  7. Salehian A, Inman D, Cliff E M (2006) Natural frequencies of an innovative space based radar antenna by continuum modeling. 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 14th AIAA/ASME/AHS Adaptive Structures Conference 7th. https://doi.org/10.2514/6.2006-2101

  8. Piccardo G, Tubino F, Luongo A (2019) Equivalent Timoshenko linear beam model for the static and dynamic analysis of tower buildings. Appl Math Model 71:77–95. https://doi.org/10.1016/j.apm.2019.02.005

    Article  MathSciNet  MATH  Google Scholar 

  9. Santana MV, Gonçalves PB, Silveira RA (2019) Nonlinear oscillations and dynamic stability of an elastoplastic pyramidal truss. Nonlinear Dyn 98(4):2847–2877. https://doi.org/10.1007/s11071-019-05072-9

    Article  MATH  Google Scholar 

  10. Nuhoglu A, Korkmaz KA (2011) A practical approach for nonlinear analysis of tensegrity systems. Eng Comput 27(4):337–345. https://doi.org/10.1007/s00366-010-0203-9

    Article  Google Scholar 

  11. Shi H, Salim H, Shi Y, Wei F (2015) Geometric and material nonlinear static and dynamic analysis of space truss structures. Mech Based Des Struct Mach 43(1):38–56. https://doi.org/10.1080/15397734.2014.925808

    Article  Google Scholar 

  12. Faroughi S, Lee J (2014) Geometrical nonlinear analysis of tensegrity based on a co-rotational method. Adv Struct Eng 17(1):41–51. https://doi.org/10.1260/1369-4332.17.1.41

    Article  Google Scholar 

  13. Driemeier L, Proenca SPB, Alves M (2005) A contribution to the numerical nonlinear analysis of three-dimensional truss systems considering large strains, damage and plasticity. Commun Nonlinear Sci Numer Simul 10(5):515–535. https://doi.org/10.1016/j.cnsns.2003.12.002

    Article  MATH  Google Scholar 

  14. Tran HC, Lee J (2011) Geometric and material nonlinear analysis of tensegrity structures. Acta Mech Sin 27(6):938–949. https://doi.org/10.1007/s10409-011-0520-2

    Article  MathSciNet  MATH  Google Scholar 

  15. Van Do VN, Lee CH (2017) Bending analyses of FG-CNTRC plates using the modified mesh-free radial point interpolation method based on the higher-order shear deformation theory. Compos Struct 168:485–497. https://doi.org/10.1016/j.compstruct.2017.02.055

    Article  Google Scholar 

  16. Witteveen W, Pichler F (2014) Efficient model order reduction for the dynamics of nonlinear multilayer sheet structures with trial vector derivatives. Shock Vib. https://doi.org/10.1155/2014/913136

    Article  Google Scholar 

  17. Guzmán AM, Rosales MB, Filipich CP (2019) Continuous one-dimensional model of a spatial lattice. Deformation, vibration and buckling problems. Eng Struct 182:290–300. https://doi.org/10.1016/j.engstruct.2018.12.074

    Article  Google Scholar 

  18. Zhang D, Li F, Shao F, Fan C (2019) Evaluation of equivalent bending stiffness by simplified theoretical solution for an FRP–aluminum Deck–truss structure. KSCE J Civ Eng 23(1):367–375. https://doi.org/10.1007/s12205-018-1093-4

    Article  Google Scholar 

  19. McCallen DB, Romstad K (1988) A continuum model for the nonlinear analysis of beamlike lattice structures. Comput Struct 29(2):177–197. https://doi.org/10.1016/0045-7949(88)90252-0

    Article  MATH  Google Scholar 

  20. Liu H, Lv J (2017) An equivalent continuum multiscale formulation for 2D geometrical nonlinear analysis of lattice truss structure. Compos Struct 160:335–348. https://doi.org/10.1016/j.compstruct.2016.10.072

    Article  Google Scholar 

  21. Wu L, Tiso P (2016) Nonlinear model order reduction for flexible multibody dynamics: a modal derivatives approach. Multibody Syst Dyn 36(4):405–425. https://doi.org/10.1007/s11044-015-9476-5

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu F, Wang L, Jin D, Wen H (2019) Equivalent continuum modeling of beamlike truss structures with flexible joints. Acta Mech Sin 35(5):1067–1078. https://doi.org/10.1007/s10409-019-00872-z

    Article  MathSciNet  Google Scholar 

  23. Liu M, Cao D, Zhu D (2020) Equivalent dynamic model of the space antenna truss with initial stress. AIAA J 58(4):1851–1863. https://doi.org/10.2514/1.J058647

    Article  Google Scholar 

  24. Noor AK, Anderson MS, Greene WH (1978) Continuum models for beam-and platelike lattice structures. AIAA J 16(12):1219–1228. https://doi.org/10.2514/3.61036

    Article  Google Scholar 

  25. Farokhi H, Ghayesh MH (2017) Nonlinear resonant response of imperfect extensible Timoshenko microbeams. Int J Mech Mater Des 13(1):43–55. https://doi.org/10.1007/s10999-015-9316-z

    Article  Google Scholar 

  26. Payette GS, Reddy JN (2010) Nonlinear quasi-static finite element formulations for viscoelastic Euler-Bernoulli and Timoshenko beams. Int J Numer Methods Biomed Eng 26(12):1736–1755. https://doi.org/10.1002/cnm.1262

    Article  MathSciNet  MATH  Google Scholar 

  27. Ansari R, Mohammadi V, Shojaei MF, Gholami R, Rouhi H (2014) Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory. Eur J Mech-A/Solids 45:143–152. https://doi.org/10.1016/j.euromechsol.2013.11.002

    Article  MathSciNet  MATH  Google Scholar 

  28. Ansari R, Gholami R, Rouhi H (2015) Size-dependent nonlinear forced vibration analysis of magneto-electro-thermo-elastic Timoshenko nanobeams based upon the nonlocal elasticity theory. Compos Struct 126:216–226. https://doi.org/10.1016/j.compstruct.2015.02.068

    Article  Google Scholar 

  29. Ghayesh MH, Amabili M, Farokhi H (2013) Three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams. Int J Eng Sci 71:1–14. https://doi.org/10.1016/j.ijengsci.2013.04.003

    Article  MathSciNet  MATH  Google Scholar 

  30. Ke LL, Wang YS, Yang J, Kitipornchai S (2012) Nonlinear free vibration of size-dependent functionally graded microbeams. Int J Eng Sci 50(1):256–267. https://doi.org/10.1016/j.ijengsci.2010.12.008

    Article  MathSciNet  MATH  Google Scholar 

  31. Şimşek M, Kocatürk T, Akbaş ŞD (2013) Static bending of a functionally graded microscale Timoshenko beam based on the modified couple stress theory. Compos Struct 95:740–747. https://doi.org/10.1016/j.compstruct.2012.08.036

    Article  Google Scholar 

  32. Ghayesh MH (2018) Nonlinear vibrations of axially functionally graded Timoshenko tapered beams. J Comput Nonlinear Dyn 13(4):041002. https://doi.org/10.1115/1.4039191

    Article  Google Scholar 

  33. Yapanmış BE, Togun N, Bağdatlı SM, Akkoca S (2021) Magnetic field effect on nonlinear vibration of nonlocal nanobeam embedded in nonlinear elastic foundation. Struct Eng Mech 79(6):041002. https://doi.org/10.12989/sem.2021.79.6.723

    Article  Google Scholar 

  34. Yapanmış BE, Bağdatlı SM (2022) Investigation of the non-linear vibration behaviour and 3:1 internal resonance of the multi supported nanobeam. Zeitschrift für Naturforschung A 77(4):305–321. https://doi.org/10.1515/zna-2021-0300

    Article  Google Scholar 

  35. Liu M, Cao D, Zhang X, Wei J, Zhu D (2021) Nonlinear dynamic responses of beamlike truss based on the equivalent nonlinear beam model. Int J Mech Sci 194:106197. https://doi.org/10.1016/j.ijmecsci.2020.106197

    Article  Google Scholar 

  36. Asghari M, Kahrobaiyan MH, Ahmadian MT (2010) A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int J Eng Sci 48(12):1749–1761. https://doi.org/10.1016/j.ijengsci.2010.09.025

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhang W, Wu R, Behdinan K (2019) Nonlinear dynamic analysis near resonance of a beam-ring structure for modeling circular truss antenna under time-dependent thermal excitation. Aerosp Sci Technol 86:296–311. https://doi.org/10.1016/j.ast.2019.01.018

    Article  Google Scholar 

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Funding

National Natural Science Foundation of China, 11732005, Dengqing Cao.

Author information

Authors and Affiliations

  1. College of Shipbuilding Engineering, Harbin Engineering University, Harbin, 150001, People’s Republic of China

    Mei Liu

  2. School of Electromechanical and Automotive Engineering, Yantai University, Yantai, 264005, People’s Republic of China

    Jin Wei

  3. School of Astronautics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Xiaoyun Zhang & Dengqing Cao

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  1. Mei Liu
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  2. Jin Wei
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Correspondence to Mei Liu.

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Liu, M., Wei, J., Zhang, X. et al. Equivalent Nonlinear Beam Model for Static and Free Vibration Analysis of the Beamlike Truss. J. Vib. Eng. Technol. 11, 4039–4051 (2023). https://doi.org/10.1007/s42417-022-00800-9

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  • DOI: https://doi.org/10.1007/s42417-022-00800-9

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