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Quantitative validation of the analytical mode shapes of a beam-like structure with a Z-shaped configuration

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Abstract

Beam-like structures are widely used in engineering. The accuracy of the analytical mode shapes of these structures is important for studying their dynamic characteristics. A method is presented in this study to theoretically obtain and quantitatively validate the analytical mode shapes of a beam-like structure with a Z-shaped configuration. The governing equations and boundary conditions of the structure’s planar motion are derived using Hamilton’s principle. Frequencies and analytical mode shapes of the structure are theoretically obtained and compared with the numerical results from finite element method. The comparison of frequencies and modal assurance criterion is used to quantitatively validate the analytical mode shapes. Examples are presented to show the analytical mode shapes of Z-shaped beams with different fold angles. The proposed method is useful for improving the accuracy of analytical mode shapes’ which is beneficial to the design and control of beam-like structures in engineering fields.

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References

  1. A. Erturk, J. M. Renno and D. J. Inman, Modeling of piezoelectric energy harvesting from an l-shaped beam-mass structure with an application to uavs, Journal of Intelligent Material Systems and Structures, 20 (5) (2009) 529–544.

    Article  Google Scholar 

  2. R. T. Faal, M. B. Amiri, A. A. Pirmohammadi and A. Milani, Vibration analysis of undamped, suspended multi-beam absorber systems, Meccanica, 47 (5) (2012) 1059–1078.

    Article  MathSciNet  MATH  Google Scholar 

  3. X. M. Gao, D. P. Jin and H. Y. Hu, Internal resonances and their bifurcations of a rigid-flexible space antenna, International Journal of Non-Linear Mechanics, 94 (2017) 160–173.

    Article  Google Scholar 

  4. B. Balachandran and A. H. Nayfeh, Observations of modal interactions in resonantly forced beam-mass structures, Nonlinear Dynamics, 2 (2) (1991) 77–117.

    Article  Google Scholar 

  5. M. Sabag and A. Rosen, A general nonlinear structural model of a multirod (multibeam) system–ii. Results, Computers & Structures, 61 (4) (1996) 633–644.

    Article  MATH  Google Scholar 

  6. M. C. Da Silva, A reduced-order analytical model for the nonlinear dynamics of a class of flexible multi-beam structures, International Journal of Solids and Structures, 35 (25) (1998) 3299–3315.

    Article  MATH  Google Scholar 

  7. N. A. Chrysochoidis and D. A. Saravanos, Generalized layerwise mechanics for the static and modal response of delaminated composite beams with active piezoelectric sensors, International Journal of Solids and Structures, 44 (25–26) (2007) 8751–8768.

    Article  MATH  Google Scholar 

  8. M. Cao, M. Radzienski, W. Xu and W. Ostachowicz, Identification of multiple damage in beams based on robust curvature mode shapes, Mechanical Systems and Signal Processing, 46 (2) (2014) 468–480.

    Article  Google Scholar 

  9. M. R. M. Crespo da Silva and C. C. Glynn, Nonlinear flexural-flexural-torsional dynamics of inextensional beams. I. Equations of motion, Journal of Structural Mechanics, 6 (4) (1978) 437–448.

    Article  Google Scholar 

  10. W. Zhang, Chaotic motion and its control for nonlinear nonplanar oscillations of a parametrically excited cantilever beam, Chaos, Solitons & Fractals, 26 (3) (2005) 731–745.

    Article  MATH  Google Scholar 

  11. A. Mirzabeigy, V. Dabbagh and R. Madoliat, Explicit formulation for natural frequencies of double-beam system with arbitrary boundary conditions, Journal of Mechanical Science and Technology, 31 (2) (2017) 515–521.

  12. S. Park, H. H. Yoo and J. Chung, Eulerian and lagrangian descriptions for the vibration analysis of a deploying beam, Journal of Mechanical Science and Technology, 27 (9) (2013) 2637–2643.

    Article  Google Scholar 

  13. B. Balachandran and A. H. Nayfeh, Nonlinear motions of beam-mass structure, Nonlinear Dynamics, 1 (1) (1990) 39–61.

    Article  Google Scholar 

  14. D. X. Cao, S. Leadenham and A. Erturk, Internal resonance for nonlinear vibration energy harvesting, The European Physical Journal Special Topics, 224 (14–15) (2015) 2867–2880.

    Article  Google Scholar 

  15. Y. Takahashi, N. Shimizu and K. Suzuki, Study on the frame structure modeling of the beam element formulated by absolute nodal coordinate approach, Journal of Mechanical Science and Technology, 19 (1) (2005) 283–291.

    Article  Google Scholar 

  16. L. Q. Chen, W. A. Jiang, M. Panyam and M. F. Daqaq, A broadband internally resonant vibratory energy harvester, Journal of Vibration and Acoustics, 138 (6) (2016).

    Google Scholar 

  17. F. Georgiades, J. Warminski and M. P. Cartmell, Linear modal analysis of l-shaped beam structures, Mechanical Systems and Signal Processing, 38 (2) (2013) 312–332.

    Article  Google Scholar 

  18. F. Georgiades, Nonlinear equations of motion of l-shaped beam structures, European Journal of Mechanics-A/Solids, 65 (2017) 91–122.

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Güler and H. Karagülle, Finite element analysis of structures with extruded aluminum profiles having complex cross sections, Latin American Journal of Solids and Structures, 13 (8) (2016) 1499–1514.

    Article  Google Scholar 

  20. T. A. Nayfeh, A. H. Nayfeh and D. T. Mook, A theoretical and experimental investigation of a three-degree-of-freedom structure, Nonlinear Dynamics, 6 (3) (1994) 353–374.

    Article  Google Scholar 

  21. F. X. Wang and A. K. Bajaj, Nonlinear dynamics of a three-beam structure with attached mass and three-mode interactions, Nonlinear Dynamics, 62 (1–2) (2010) 461–484.

    Article  MathSciNet  MATH  Google Scholar 

  22. C. Wu, H. Xiang and X. P. Du, Improved three-variable element-free galerkin method for vibration analysis of beam-column models, Journal of Mechanical Science and Technology, 30 (9) (2016) 4121–4131.

    Article  Google Scholar 

  23. P. Cartraud and T. Messager, Computational homogenization of periodic beam-like structures, International Journal of Solids and Structures, 43 (3–4) (2006) 686–696.

    Article  MathSciNet  MATH  Google Scholar 

  24. Z. F. Wang, Q. W. Shi, Q. Li, X. Wang, J. G. Hou, H. Zheng, Y. Yao and J. Chen, Z-shaped graphene nanoribbon quantum dot device, Applied Physics Letters, 91 (5) (2007) 053109.

    Article  Google Scholar 

  25. Y. Zhu, S. R. Moheimani and M. R. Yuce, Bidirectional electrothermal actuator with z-shaped beams, IEEE Sensors Journal, 12 (7) (2012) 2508–2509.

    Article  Google Scholar 

  26. P. Mardanpour and D. H. Hodges, Passive morphing of flying wing aircraft: Z-shaped configuration, Journal of Fluids and Structures, 44 (2014) 17–30.

    Article  Google Scholar 

  27. J. M. Kim and H. H. Yoo, Modal analysis of rotating beam structures having complex configurations employing multi-reference frames, Journal of Mechanical Science and Technology, 20 (1) (2006) 66–75.

    Article  MathSciNet  Google Scholar 

  28. A. Tripathi and A. K. Bajaj, Computational synthesis for nonlinear dynamics based design of planar resonant structures, Journal of Vibration and Acoustics, 135 (5) (2013).

    Google Scholar 

  29. N. Vlajic, T. Fitzgerald, V. Nguyen and B. Balachandran, Geometrically exact planar beams with initial pre-stress and large curvature: Static configurations, natural frequencies, and mode shapes, International Journal of Solids and Structures, 51 (19–20) (2014) 3361–3371.

    Article  Google Scholar 

  30. W. Zhang, W. H. Hu, D. X. Cao and M. H. Yao, Vibration frequencies and modes of a z-shaped beam with variable folding angles, Journal of Vibration and Acoustics, 138 (4) (2016) 041004.

    Article  Google Scholar 

  31. B. Peeters and C. E. Ventura, Comparative study of modal analysis techniques for bridge dynamic characteristics, Mechanical Systems and Signal Processing, 17 (5) (2003) 965–988.

    Article  Google Scholar 

  32. S. S. Alkhfaji and S. D. Garvey, Modal correlation approaches for general second-order systems: Matching mode pairs and an application to campbell diagrams, Journal of Sound and Vibration, 330 (23) (2011) 5615–5627.

    Article  Google Scholar 

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Authors and Affiliations

  1. Tianjin Key Laboratory of the Design and Intelligent Control of the Advanced Mechatronical System, School of Mechanical Engineering, Tianjin University of Technology, Tianjin, 300384, China

    Wenhua Hu, Fengxiang Wang, Jianen Chen & Jingjing Feng

  2. National Demonstration Center for Experimental Mechanical and Electrical Engineering Education, Tianjin University of Technology, Tianjin, 300384, China

    Wenhua Hu, Fengxiang Wang, Jianen Chen & Jingjing Feng

  3. Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, China

    Dongxing Cao

Authors
  1. Wenhua Hu
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  2. Fengxiang Wang
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  3. Dongxing Cao
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  4. Jianen Chen
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  5. Jingjing Feng
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Correspondence to Wenhua Hu.

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Recommended by Associate Editor Gyuhae Park

Wenhua Hu is a lecturer in the School of Mechanical Engineering of Tianjin University of Technology, China. He received his Ph.D. in Engineering Mechanics from Beijing University of Technology in 2015. His research interests include structural dynamics, vibration control, and multi-body dynamics.

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Hu, W., Wang, F., Cao, D. et al. Quantitative validation of the analytical mode shapes of a beam-like structure with a Z-shaped configuration. J Mech Sci Technol 33, 2059–2065 (2019). https://doi.org/10.1007/s12206-019-0409-8

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  • DOI: https://doi.org/10.1007/s12206-019-0409-8

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