Skip to main content
SpringerLink
Log in

Stability of an axially moving laminated composite beam reinforced with graphene nanoplatelets

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

This study investigates the dynamic stability of an axially moving laminated composite beam reinforced by graphene nanoplatelets (GPLs) under constant and harmonically varying velocities. It is assumed that GPLs are distributed into the beam in different patterns through the thickness symmetrically, and the GPL distribution in each layer is uniformly and randomly oriented. The effective Young’s modulus of the nanocomposite is predicated on Halpin–Tsai’s model. The partial differential equations of motion for the axially moving laminated composite beam are derived using Hamilton’s principle, and the ordinary differential governing equations are obtained by Galerkin method. According to the eigenvalues of the coefficient matrix of the ordinary differential equations, the linear stability of the axially moving beam reinforced with GPLs at constant velocity is studied. Finally, the instability region of the axially moving beam in the perspective of resonance under harmonically varying velocity is discussed in detail by utilizing the multiscale method.

Article highlights

  1. 1.

    Governing equations of motion for the axially moving beam are established.

  2. 2.

    Instability regions are obtained considering the damping.

  3. 3.

    Reinforcement of GPLs on the stability of axially moving beam is analyzed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

Data and Code Availability

The data, material and Code used or analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Marynowski K, Kapitaniak T (2014) Dynamics of axially moving continua. Int J Mech Sci 81:26–41

    Article  Google Scholar 

  2. Pham PT, Hong KS (2020) Dynamic models of axially moving systems: a review. Nonlinear Dyn 100:315–349

    Article  Google Scholar 

  3. Chen LQ, Yang XD (2005) Stability in parametric resonance of axially moving viscoelastic beams with time-dependent speed. J Sound Vib 284:879–891

    Article  Google Scholar 

  4. Chen LQ, Yang XD (2005) Steady-state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models. Int J Solids Struct 42:37–50

    Article  MATH  Google Scholar 

  5. Ghayesh MH, Balar S (2008) Non-linear parametric vibration and stability of axially moving visco-elastic Rayleigh beams. Int J Solids Struct 45:6451–6467

    Article  MATH  Google Scholar 

  6. Ding H, Chen LQ (2008) Stability of axially accelerating viscoelastic beams: multi-scale analysis with numerical confirmations. Eur J Mech A/Solids 27:1108–1120

    Article  MathSciNet  MATH  Google Scholar 

  7. Ghayesh MH, Amabili M (2013) Parametric stability and bifurcations of axially moving viscoelastic beams with time-dependent axial speed. Mech Based Des Struct Mach 41:359–381

    Article  Google Scholar 

  8. Chen LH, Zhang W, Yang FH (2010) Nonlinear dynamics of higher-dimensional system for an axially accelerating viscoelastic beam with in-plane and out-of-plane vibrations. J Sound Vib 329:5321–5345

    Article  Google Scholar 

  9. Ghayesh MH (2012) Coupled longitudinal–transverse dynamics of an axially accelerating beam. J Sound Vib 331:5107–5124

    Article  Google Scholar 

  10. Yang XD, Zhang W (2014) Nonlinear dynamics of axially moving beam with coupled longitudinal–transversal vibrations. Nonlinear Dyn 78:2547–2556

    Article  MathSciNet  Google Scholar 

  11. Korspeter GS, Kirillov ON, Hagedorn P (2008) Modeling and stability analysis of an axially moving beam with frictional contact. J Appl Mech 75:031001

    Article  Google Scholar 

  12. Kazemirad S, Ghayesh MH, Amabili M (2013) Thermal effects on nonlinear vibrations of an axially moving beam with an intermediate spring-mass support. Shock Vib 20:385–399

    Article  MATH  Google Scholar 

  13. Zhang W, Wang DM, Yao MH (2014) Using Fourier differential quadrature method to analyze transverse nonlinear vibrations of an axially accelerating viscoelastic beam. Nonlinear Dyn 78:839–856

    Article  MathSciNet  MATH  Google Scholar 

  14. Sarigül M (2019) Parametric vibrations of axially moving beams with multiple edge cracks. Int J Acoust Vib 24(2):241–252

    Article  Google Scholar 

  15. Yang XD, Yang JH, Qian YJ, Zhang W, Melnik RVN (2018) Dynamics of a beam with both axial moving and spinning motion: an example of bi-gyroscopic continua. Eur J Mech/A Solids 69:231–237

    Article  MathSciNet  MATH  Google Scholar 

  16. Li YH, Wang L, Yang EC (2018) Nonlinear dynamic responses of an axially moving laminated beam subjected to both blast and thermal loads. Int J Non Linear Mech 101:56–67

    Article  Google Scholar 

  17. Wang B (2018) Effect of rotary inertia on stability of axially accelerating viscoelastic Rayleigh beams. Appl Math Mech Engl Ed 39(5):717–732

    Article  MathSciNet  MATH  Google Scholar 

  18. Hao Y, Gao ML (2019) Traverse vibration of axially moving laminated SMA beam considering random perturbation. Shock Vib 2019:6341289

    Google Scholar 

  19. Yan T, Yang TZ, Chen LQ (2020) Direct multiscale analysis of stability of an axially moving functionally graded beam with time-dependent velocity. Acta Mech Solida Sin 33(2):150–163

    Article  Google Scholar 

  20. Lee U, Kim J, Oh H (2004) Spectral analysis for the transverse vibration of an axially moving Timoshenko beam. J Sound Vib 271:685–703

    Article  MATH  Google Scholar 

  21. An C, Su J (2014) Dynamic response of axially moving Timoshenko beams: integral transform solution. Appl Math Mech 35(11):1421–1436

    Article  MathSciNet  MATH  Google Scholar 

  22. Yan QY, Ding H, Chen LQ (2014) Periodic responses and chaotic behaviors of an axially accelerating viscoelastic Timoshenko beam. Nonlinear Dyn 78:1577–1591

    Article  MathSciNet  Google Scholar 

  23. Sui SH, Chen L, Li C, Liu XP (2015) Transverse vibration of axially moving functionally graded materials based on Timoshenko beam theory. Math Probl Eng 2015:391452

    Article  MathSciNet  MATH  Google Scholar 

  24. Ding H, Tan X, Zhang GC, Chen LQ (2016) Equilibrium bifurcation of high-speed axially moving Timoshenko beams. Acta Mech 227:3001–3014

    Article  MathSciNet  MATH  Google Scholar 

  25. Ding H, Tan X, Dowell EH (2017) Natural frequencies of a super-critical transporting Timoshenko beam. Eur J Mech A/Solids 66:79–93

    Article  MathSciNet  MATH  Google Scholar 

  26. Lin CC (1997) Stability and vibration characteristics of axially moving plates. Intern J Solids Struct 34(24):3179–3190

    Article  MATH  Google Scholar 

  27. Chen LQ, Zhang W, Liu YQ (2007) Modeling of nonlinear oscillations for viscoelastic moving belt using generalized Hamilton’s principle. J Vib Acoust 129:128–132

    Article  Google Scholar 

  28. Yang XD, Chen LQ, Zu JW (2011) Vibrations and stability of an axially moving rectangular composite plate. J Appl Mech 78:011018

    Article  Google Scholar 

  29. Ghayesh MH, Amabili M, Païdoussis MP (2013) Nonlinear dynamics of axially moving plates. J Sound Vib 332:391–406

    Article  Google Scholar 

  30. Ghayesh MH, Amabili M (2013) Non-linear global dynamics of an axially moving plate. Int J Non Linear Mech 57:16–30

    Article  MATH  Google Scholar 

  31. Yao G, Zhang YM (2016) Dynamics and stability of an axially moving plate interacting with surrounding airflow. Meccanica 51:2111–2119

    Article  MathSciNet  MATH  Google Scholar 

  32. Yao G, Zhang YM, Li CY, Yang Z (2016) Stability analysis and vibration characteristics of an axially moving plate in aero-thermal environment. Acta Mech 227:3517–3527

    Article  MathSciNet  MATH  Google Scholar 

  33. Tang YQ, Zhang DB, Rui MH, Wang X, Zhu DC (2016) Dynamic stability of axially accelerating viscoelastic plates with longitudinally varying tensions. Appl Math Mech Engl Ed 37(12):1647–1668

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang YQ, Huang XB, Li J (2016) Hydroelastic dynamic analysis of axially moving plates in continuous hot-dip galvanizing process. Int J Mech Sci 110:201–216

    Article  Google Scholar 

  35. Wang LH, Hu ZD, Zhong Z (2010) Dynamic analysis of an axially translating plate with time-variant length. Acta Mech 215:9–23

    Article  MATH  Google Scholar 

  36. Zhang W, Sun L, Yang XD, Jia P (2013) Nonlinear dynamic behaviors of a deploying-and-retreating wing with varying velocity. J Sound Vib 332:6785–6797

    Article  Google Scholar 

  37. Zhang W, Lu SF, Yang XD (2014) Analysis on nonlinear dynamics of a deploying laminated composited cantilever plate. Nonlinear Dyn 76:69–93

    Article  MATH  Google Scholar 

  38. Lu SF, Zhang W, Song XJ (2018) Time-varying nonlinear dynamics of a deploying piezoelectric laminated composite plate under aerodynamic force. Acta Mech Sin 34(2):303–314

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhao SY, Zhao Z, Yang ZC, Ke LL, Kitipornchai S, Yang J (2020) Functionally graded graphene reinforced composite structures: a review. Eng Struct 210:110339

    Article  Google Scholar 

  40. Yang J, Wu HL, Kitipornchai S (2017) Buckling and postbuckling of functionally graded multilayer graphene platelet-reinforced composite beams. Compos Struct 161:111–118

    Article  Google Scholar 

  41. Feng C, Kitipornchai S, Yang J (2017) Nonlinear bending of polymer nanocomposite beams reinforced with non-uniformly distributed graphene platelets (GPLs). Compos B 110:132–140

    Article  Google Scholar 

  42. Feng C, Kitipornchai S, Yang J (2017) Nonlinear free vibration of functionally graded polymer composite beams reinforced with graphene nanoplatelets (GPLs). Eng Struct 140:110–119

    Article  Google Scholar 

  43. Kitipornchai S, Chen D, Yang J (2017) Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets. Mater Des 116:656–665

    Article  Google Scholar 

  44. Chen D, Yang J, Kitipornchai S (2017) Nonlinear vibration and postbuckling of functionally graded graphene reinforced porous nanocomposite beams. Compos Sci Technol 142:235–245

    Article  Google Scholar 

  45. Song MT, Kitipornchai S, Yang J (2017) Free and forced vibrations of functionally graded polymer composite plates reinforced with graphene nanoplatelets. Compos Struct 159:579–588

    Article  Google Scholar 

  46. Gholami R, Ansari R (2018) Nonlinear harmonically excited vibration of third-order shear deformable functionally graded graphene platelet-reinforced composite rectangular plates. Eng Struct 156:197–209

    Article  Google Scholar 

  47. Mao JJ, Zhang W (2018) Linear and nonlinear free and forced vibrations of graphene reinforced piezoelectric composite plate under external voltage excitation. Compos Struct 203:551–565

    Article  Google Scholar 

  48. Mao JJ, Zhang W (2019) Buckling and post-buckling analyses of functionally graded graphene reinforced piezoelectric plate subjected to electric potential and axial forces. Compos Struct 216:392–405

    Article  Google Scholar 

  49. Guo XY, Jiang P, Zhang W, Yang J, Kitipornchai S, Sun L (2018) Nonlinear dynamic analysis of composite piezoelectric plates with graphene skin. Compos Struct 206:839–852

    Article  Google Scholar 

  50. Lu SF, Xue N, Zhang W, Song XJ, Ma WS (2021) Dynamic stability of axially moving graphene reinforced laminated composite plate under constant and varied velocities. Thin Walled Struct 167:108176

    Article  Google Scholar 

  51. Wu HL, Yang J, Kitipornchai K (2017) Dynamic instability of functionally graded multilayer graphene nanocomposite beams in thermal environment. Compos Struct 162:244–254

    Article  Google Scholar 

  52. Shen HS, Xiang Y, Lin F (2017) Nonlinear vibration of functionally graded graphene-reinforced composite laminated plates in thermal environments. Comput Methods Appl Mech Eng 319:175–193

    Article  MathSciNet  MATH  Google Scholar 

  53. Yang B, Yang J, Kitipornchai S (2017) Thermoelastic analysis of functionally graded graphene reinforced rectangular plates based on 3D elasticity. Meccanica 52:2275–2292

    Article  MathSciNet  MATH  Google Scholar 

  54. Yang B, Mei J, Chen D, Yu F, Yang J (2018) 3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates. Compos Struct 184:1040–1048

    Article  Google Scholar 

  55. Fan Y, Xiang Y, Shen HS, Hui D (2018) Nonlinear low-velocity impact response of FG-GRC laminated plates resting on visco-elastic foundations. Compos B 144:184–194

    Article  Google Scholar 

  56. Fan Y, Xiang Y, Shen HS (2019) Nonlinear forced vibration of FG-GRC laminated plates resting on visco-Pasternak foundations. Compos Struct 209:443–452

    Article  Google Scholar 

  57. Mao JJ, Zhang W, Lu HM (2020) Static and dynamic analyses of graphene-reinforced aluminium-based composite plate in thermal environment. Aerosp Sci Technol 107:106354

    Article  Google Scholar 

Download references

Funding

The authors sincerely acknowledge the financial support of the National Natural Science Foundation of China (Nos. 11862020, 11962020, 12172182, 12102207), Inner Mongolia Natural Science Foundation (No. 2019MS05065) and 2020 talent introduction project of Inner Mongolia Autonomous Region (DC2100001428).

Author information

Authors and Affiliations

  1. Department of Mechanics, Inner Mongolia University of Technology, Hohhot, 010051, People’s Republic of China

    Shufeng Lu, Ning Xue & Wensai Ma

  2. College of Mechanical Engineering, Inner Mongolia University of Technology, Hohhot, 010051, People’s Republic of China

    Xiaojuan Song

Authors
  1. Shufeng Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ning Xue
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaojuan Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wensai Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Shufeng Lu: Conceptualization, Methodology, Formal analysis, Software, Investigation, Validation, Data curation, Writing - original draft. Ning Xue: Conceptualization, Methodology, Formal analysis, Software, Investigation, Writing - original draft. Xiaojuan Song: Conceptualization, Methodology, Validation, Writing - review & editing, Resources, Supervision, Funding acquisition, Project administration. Wensai Ma: Conceptualization, Methodology, Software, Investigation, Data curation, Investigation.

Corresponding author

Correspondence to Xiaojuan Song.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Appendices

Appendix 1

The coefficients of matrices (24a) and (24b) are given as follows

$$ \begin{aligned} G_{11} & = \frac{16}{3}V_{0} ,\;\;\;G_{12} = - 2\pi^{2} V_{0} \frac{{I_{1} h^{2} }}{{I_{0} l^{2} }}, \\ K_{11} & = - \pi^{2} \frac{{A_{11} }}{{I_{0} }} + \pi^{2} V_{0}^{2} ,\;\;\;K_{12} = \frac{32}{3}\pi^{2} V_{0}^{2} \frac{{I_{1} h^{2} }}{{I_{0} l^{2} }}, \\ G_{21} & = - \frac{16}{3}V_{0} ,\;\;\;G_{22} = - 8\pi^{2} V_{0} \frac{{I_{1} h^{2} }}{{I_{0} l^{2} }}, \\ K_{21} & = - 4\pi^{2} \frac{{A_{11} }}{{I_{0} }} + 4\pi^{2} V_{0}^{2} ,\;\;\;K_{22} = - \frac{8}{3}\pi^{2} V_{0}^{2} \frac{{I_{1} h^{2} }}{{I_{0} l^{2} }}, \\ \eta_{1} & = I_{0} + \pi^{2} I_{2} \frac{{h^{2} }}{{l^{2} }},\;\;\;\eta_{2} = I_{0} + 4\pi^{2} I_{2} \frac{{h^{2} }}{{l^{2} }}, \\ G_{31} & = 2\pi^{2} V_{0} \frac{{I_{1} }}{{\eta_{1} }},\;\;\;\;G_{32} = \frac{16}{3}V_{0} \frac{{I_{0} }}{{\eta_{1} }} + \frac{64}{3}\pi^{2} V_{0} \frac{{I_{2} h^{2} }}{{\eta_{1} l^{2} }}, \\ K_{31} & = - \frac{32}{3}\pi^{2} V_{0}^{2} \frac{{I_{1} }}{{\eta_{1} }},\;\;\;K_{32} = - \pi^{4} D_{11} \frac{{h^{2} }}{{l^{2} \eta_{1} }} + \pi^{2} V_{0}^{2} \frac{{I_{0} }}{{\eta_{1} }} + \pi^{4} V_{0}^{2} \frac{{I_{2} h^{2} }}{{\eta_{1} l^{2} }}, \\ G_{41} & = 8\pi^{2} V_{0} \frac{{I_{1} }}{{\eta_{2} }},\;\;\;G_{42} = - \frac{16}{3}V_{0} \frac{{I_{0} }}{{\eta_{2} }} - \frac{16}{3}\pi^{2} V_{0} \frac{{I_{2} h^{2} }}{{\eta_{2} l^{2} }}, \\ K_{41} & = \frac{8}{3}\pi^{2} V_{0}^{2} \frac{{I_{1} }}{{\eta_{2} }},\;\;\;K_{42} = - 16\pi^{4} D_{11} \frac{{h^{2} }}{{l^{2} \eta_{2} }} + 4\pi^{2} V_{0}^{2} \frac{{I_{0} }}{{\eta_{2} }} + 16\pi^{4} V_{0}^{2} \frac{{I_{2} h^{2} }}{{\eta_{2} l^{2} }}. \\ \end{aligned} $$

Appendix 2

The coefficients of Eq. (25) are given as follows

$$ \begin{aligned} \xi & = \frac{\mu }{{I_{0} }},\;\;\;\zeta_{1} = - \frac{16}{3}V_{0} ,\;\;\;\zeta_{2} = - \frac{16}{3}V_{1} , \\ k_{11} & = \pi^{4} D_{11} \frac{{h^{2} }}{{I_{0} l^{2} }} - \pi^{2} V_{0}^{2} ,\;\;\;k_{12} = - 2\pi^{2} V_{0} V_{1} ,\;\;\;k_{13} = - \pi^{2} V_{1}^{2} , \\ k_{21} & = - \frac{8}{3}\xi V_{0} ,\;\;\;k_{22} = - \frac{8}{3}V_{1} ,\;\;\;k_{23} = - \frac{8}{3}\xi V_{1} , \\ k_{31} & = 16\pi^{4} D_{11} \frac{{h^{2} }}{{I_{0} l^{2} }} - 4\pi^{2} V_{0}^{2} ,\;\;\;k_{12} = - 8\pi^{2} V_{0} V_{1} ,\;\;\;k_{13} = - 4\pi^{2} V_{1}^{2} . \\ \end{aligned} $$

Appendix 3

The coefficients of Eqs. (35), (38), (41), (50), and (59) are given as follows

$$ \begin{aligned} \gamma_{11} & = \frac{{i\xi \omega_{1} - i\xi \omega_{1} p_{1}^{2} + 2k_{21} p_{1} }}{{2(i\omega_{1} - i\omega_{1} p_{1}^{2} + \zeta_{1} p_{1} )}}, \\ \gamma_{12} & = \frac{{2\zeta_{2} \omega_{1} p_{1} - ik_{12} + ik_{32} p_{1}^{2} + 2 k_{22} p_{1} }}{{4(i\omega_{1} - i\omega_{1} p_{1}^{2} + \zeta_{1} p_{1} )}}, \\ \gamma_{13} & = - \frac{{2\zeta_{2} \omega_{1} p_{1} - ik_{12} + ik_{32} p_{1}^{2} - 2 k_{22} p_{1} }}{{4(i\omega_{1} - i\omega_{1} p_{1}^{2} + \zeta_{1} p_{1} )}}, \\ \gamma_{14} & = - \frac{{\zeta_{2} \omega_{1} \overline{p}_{1} + \zeta_{2} \omega_{1} p_{1} + ik_{12} - ik_{32} p_{1} \overline{p}_{1} - 2\omega_{1} k_{22} \overline{p}_{1} - 2\omega_{1} k_{22} p_{1} }}{{4(i\omega_{1} - i\omega_{1} p_{1}^{2} + \zeta_{1} p_{1} )}}, \\ \gamma_{15} & = - \frac{{\zeta_{2} \omega_{2} \overline{p}_{2} + \zeta_{2} \omega_{2} p_{1} + ik_{12} - ik_{32} p_{1} \overline{p}_{2} - k_{22} (p_{1} + \overline{p}_{2} )(\omega_{1} + \omega_{2} )}}{{4(i\omega_{1} - i\omega_{1} p_{1}^{2} + \zeta_{1} p_{1} )}}, \\ \gamma_{16} & = - \frac{{\zeta_{2} \omega_{2} p_{2} + \zeta_{2} \omega_{2} p_{1} - ik_{12} + ik_{32} p_{1} p_{2} - k_{22} (p_{1} + p_{2} )(\omega_{2} - \omega_{1} )}}{{4(i\omega_{1} - i\omega_{1} p_{1}^{2} + \zeta_{1} p_{1} )}}, \\ \gamma_{21} & = \frac{{i\xi \omega_{2} - i\xi \omega_{2} p_{2}^{2} + 2k_{21} p_{2} }}{{2(i\omega_{2} - i\omega_{2} p_{2}^{2} + \zeta_{1} p_{2} )}}, \\ \gamma_{22} & = \frac{{2\zeta_{2} \omega_{2} p_{2} - ik_{12} + ik_{32} p_{2}^{2} + 2 k_{22} p_{2} }}{{4(i\omega_{2} - i\omega_{2} p_{2}^{2} + \zeta_{1} p_{2} )}}, \\ \gamma_{23} & = - \frac{{2\zeta_{2} \omega_{2} p_{2} - ik_{12} + ik_{32} p_{2}^{2} - 2 k_{22} p_{2} }}{{4(i\omega_{2} - i\omega_{2} p_{2}^{2} + \zeta_{1} p_{2} )}}, \\ \gamma_{24} & = - \frac{{\zeta_{2} \omega_{2} \overline{p}_{2} + \zeta_{2} \omega_{2} p_{2} + ik_{12} - ik_{32} p_{2} \overline{p}_{2} - 2\omega_{2} k_{22} \overline{p}_{2} - 2\omega_{2} k_{22} p_{2} }}{{4(i\omega_{2} - i\omega_{2} p_{2}^{2} + \zeta_{1} p_{2} )}}, \\ \gamma_{25} & = - \frac{{\zeta_{2} \omega_{1} \overline{p}_{1} + \zeta_{2} \omega_{1} p_{2} + ik_{12} - ik_{32} \overline{p}_{1} p_{2} - k_{22} (\overline{p}_{1} + p_{2} )(\omega_{1} + \omega_{2} )}}{{4(i\omega_{2} - i\omega_{2} p_{2}^{2} + \zeta_{1} p_{2} )}}, \\ \gamma_{26} & = \frac{{\zeta_{2} \omega_{1} p_{1} + \zeta_{2} \omega_{1} p_{2} - ik_{12} + ik_{32} p_{1} p_{2} + k_{22} (p_{1} + p_{2} )(\omega_{2} - \omega_{1} )}}{{4(i\omega_{2} - i\omega_{2} p_{2}^{2} + \zeta_{1} p_{2} )}}. \\ \end{aligned} $$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lu, S., Xue, N., Song, X. et al. Stability of an axially moving laminated composite beam reinforced with graphene nanoplatelets. Int. J. Dynam. Control 10, 1727–1744 (2022). https://doi.org/10.1007/s40435-022-00950-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-022-00950-4

Keywords

Search

Navigation