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Nonlinear dynamic modeling of planar moving Timoshenko beam considering non-rigid non-elastic axial effects

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Due to the importance of vibration effects on the functional accuracy of mechanical systems, this research aims to develop a precise model of a nonlinearly vibrating single-link mobile flexible manipulator. The manipulator consists of an elastic arm, a rotary motor, and a rigid carrier, and undergoes general in-plane rigid body motion along with elastic transverse deformation. To accurately model the elastic behavior, Timoshenko’s beam theory is used to describe the flexible arm, which accounts for rotary inertia and shear deformation effects. By applying Newton’s second law, the nonlinear governing equations of motion for the manipulator are derived as a coupled system of ordinary differential equations (ODEs) and partial differential equations (PDEs). Then, the assumed mode method (AMM) is used to solve this nonlinear system of governing equations with appropriate shape functions. The assumed modes can be obtained after solving the characteristic equation of a Timoshenko beam with clamped boundary conditions at one end and an attached mass/inertia at the other. In addition, the effect of the transverse vibration of the inextensible arm on its axial behavior is investigated. Despite the axial rigidity, the effect makes the rigid body dynamics invalid for the axial behavior of the arm. Finally, numerical simulations are conducted to evaluate the performance of the developed model, and the results are compared with those obtained by the finite element approach. The comparison confirms the validity of the proposed dynamic model for the system. According to the mentioned features, this model can be reliable for investigating the system’s vibrational behavior and implementing vibration control algorithms.

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  1. MISHRA, N. and SINGH, S. P. Determination of modes of vibration for accurate modelling of the flexibility effects on dynamics of a two link flexible manipulator. International Journal of Non-Linear Mechanics, 141, 103943 (2022)

    Article  ADS  Google Scholar 

  2. WU, D., ENDO, T., and MATSUNO, F. Exponential stability of two Timoshenko arms for grasping and manipulating an object. International Journal of Control, Automation and Systems, 19, 1328–1339 (2021)

    Article  Google Scholar 

  3. LI, C. Nonlocal thermo-electro-mechanical coupling vibrations of axially moving piezoelectric nanobeams. Mechanics Based Design of Structures and Machines, 45(4), 463–478 (2017)

    Article  Google Scholar 

  4. WEI, J., CAO, D., WANG, L., HUANG, H., and HUANG, W. Dynamic modeling and simulation for flexible spacecraft with flexible jointed solar panels. International Journal of Mechanical Sciences, 130, 558–570 (2017)

    Article  Google Scholar 

  5. MENG, D., SHE, Y., XU, W., LU, W., and LIANG, B. Dynamic modeling and vibration characteristics analysis of flexible-link and flexible-joint space manipulator. Multibody System Dynamics, 43, 321–347 (2018)

    Article  MathSciNet  Google Scholar 

  6. HOMAEINEZHAD, M. R. and ABBASI GAVARI, M. Feedback control of actuation-constrained moving structure carrying Timoshenko beam. International Journal of Robust and Nonlinear Control, 33(3), 1785–1806 (2023)

    Article  MathSciNet  Google Scholar 

  7. MOSLEMI, A. and HOMAEINEZHAD, M. R. Effects of viscoelasticity on the stability and bifurcations of nonlinear energy sinks. Applied Mathematics and Mechanics (English Edition), 44(1), 141–158 (2023)

    Article  MathSciNet  Google Scholar 

  8. WANG, Z., WU, W., GÖRGES, D., and LOU, X. Sliding mode vibration control of an Euler-Bernoulli beam with unknown external disturbances. Nonlinear Dynamics, 110, 1393–1404 (2022)

    Article  Google Scholar 

  9. ZHU, K. and CHUNG, J. Vibration and stability analysis of a simply-supported Rayleigh beam with spinning and axial motions. Applied Mathematical Modelling, 66, 362–382 (2019)

    Article  MathSciNet  Google Scholar 

  10. SOLDATOS, K. P. and SOPHOCLEOUS, C. On shear deformable beam theories: the frequency and normal mode equations of the homogeneous orthotropic Bickford beam. Journal of Sound and Vibration, 242, 215–245 (2001)

    Article  ADS  Google Scholar 

  11. SHAN, J., ZHUANG, C., and LOONG, C. N. Parametric identification of Timoshenko-beam model for shear-wall structures using monitoring data. Mechanical Systems and Signal Processing, 189, 110100 (2023)

    Article  Google Scholar 

  12. LEE, T. S. and ALANDOLI, E. A. A critical review of modelling methods for flexible and rigid link manipulators. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 42, 508 (2020)

    Article  Google Scholar 

  13. HUANG, T. C. The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. Journal of Applied Mechanics, 28, 579–584 (1961)

    Article  ADS  MathSciNet  Google Scholar 

  14. JIN, H., SUI, S., ZHU, C., and LI, C. Axial free vibration of rotating FG piezoelectric nanorods accounting for nonlocal and strain gradient effects. Journal of Vibration Engineering and Technologies, 11(2), 537–549 (2023)

    Article  Google Scholar 

  15. FAN, W. An efficient recursive rotational-coordinate-based formulation of a planar Euler-Bernoulli beam. Multibody System Dynamics, 52, 211–227 (2021)

    Article  MathSciNet  Google Scholar 

  16. HE, W., OUYANG, Y., and HONG, J. Vibration control of a flexible robotic manipulator in the presence of input deadzone. IEEE Transactions on Industrial Informatics, 13(1), 48–59 (2017)

    Article  Google Scholar 

  17. ZHAO, Z. and LIU, Z. Finite-time convergence disturbance rejection control for a flexible Timoshenko manipulator. IEEE/CAA Journal of Automatica Sinica, 8, 157 (2021)

    Article  MathSciNet  Google Scholar 

  18. SINHA, S. K. Nonlinear dynamic response of a rotating radial Timoshenko beam with periodic pulse loading at the free-end. International Journal of Non-Linear Mechanics, 40(1), 113–149 (2005)

    Article  ADS  Google Scholar 

  19. VAKIL, M., SHARBATI, E., VAKIL, A., HEIDARI, F., and FOTOUHI, R. Vibration analysis of a Timoshenko beam on a moving base. Journal of Vibration and Control, 21(6), 1068–1085 (2015)

    Article  MathSciNet  Google Scholar 

  20. XING, X. and LIU, J. PDE model-based state-feedback control of constrained moving vehicle-mounted flexible manipulator with prescribed performance. Journal of Sound and Vibration, 441, 126–151 (2019)

    Article  ADS  Google Scholar 

  21. LI, L. and LIU, J. Consensus tracking control and vibration suppression for nonlinear mobile flexible manipulator multi-agent systems based on PDE model. Nonlinear Dynamics, 111, 3345–3359 (2023)

    Article  Google Scholar 

  22. XUE, H. and HUANG, J. Dynamic modeling and vibration control of underwater soft-link manipulators undergoing planar motions. Mechanical Systems and Signal Processing, 181, 109540 (2022)

    Article  Google Scholar 

  23. KORAYEM, M. H. and DEHKORDI, S. F. Derivation of motion equation for mobile manipulator with viscoelastic links and revolute-prismatic flexible joints via recursive Gibbs-Appell formulations. Robotics and Autonomous Systems, 103, 175–198 (2018)

    Article  Google Scholar 

  24. KORAYEM, M. H. and DEHKORDI, S. F. Dynamic modeling of flexible cooperative mobile manipulator with revolute-prismatic joints for the purpose of moving common object with closed kinematic chain using the recursive Gibbs-Appell formulation. Mechanism and Machine Theory, 137, 254–279 (2019)

    Article  Google Scholar 

  25. COWPER, G. R. The shear coefficient in Timoshenko’s beam theory. Journal of Applied Mechanics, 33(2), 335–340 (1966)

    Article  ADS  Google Scholar 

  26. TIMOSHENKO, P. S. P. LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 41(245), 744–746 (1921)

    Article  Google Scholar 

  27. HAN, S. M., BENAROYA, H., and WEI, T. Dynamics of transversely vibrating beams using four engineering theories. Journal of Sound and Vibration, 225(5), 935–988 (1999)

    Article  ADS  Google Scholar 

  28. MEIROVITCH, L. Analytical Methods in Vibrations, Macmillan Series in Applied Mechanics, 1st ed., Macmillan, New York (1967)

    Google Scholar 

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Correspondence to M. R. Homaeinezhad.

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Abbasi Gavari, M., Homaeinezhad, M.R. Nonlinear dynamic modeling of planar moving Timoshenko beam considering non-rigid non-elastic axial effects. Appl. Math. Mech.-Engl. Ed. 45, 479–496 (2024).

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2010 Mathematics Subject Classification