ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Nonlinear Dynamical Modeling and Vibration Responses of An L-Shaped Beam-Mass Structure

Journal of Applied Nonlinear Dynamics 6(1) (2017) 91--104 | DOI:10.5890/JAND.2017.03.007

Jin Wei; Dengqing Cao; Yang Yang; Wenhu Huang

Division of Dynamics and Control, School of Astronautics, Harbin Institute of Technology, Harbin 150001, PR China.

Abstract

The global modal approach is employed to obtain a set of ordinary differential equations of motion describing the nonlinear dynamics of an L-shaped beam structure in this paper. Firstly, the Lagrangian of nonlinear dynamics for the whole system is formulated. The linear partial differential equations of transverse motion are derived for each beam, along with their boundary and matching conditions. Consequently, the characteristic equation is formulated for the whole system. The natural frequencies and global mode shapes of the system are determined, and orthogonality relations of the global mode shapes are established. Then, the Lagrange's procedure is employed to obtain the nonlinear ODEs of motion for the structure with multiple- DoF. A comparison between the natural frequencies obtained by the proposed method and those from finite element method is given to illustrate the validity of our approach. Through the nonlinear ODEs presented in this article, a study on the variation of dynamic responses for the systems with different number of global modes is performed to give a suggestion of how many modes should be taken for vibration analysis of the structure.

Acknowledgments

We gratefully acknowledge the National Natural Science Foundation of China (Grant Nos. 91216106 and 11472089) for the financial support of this work.

References

1.  [1] Erturk A, Renno J M, Inman D J. (2009), 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: 529-544.
2.  [2] Iseki T, Okumura M, Sugawara T. (2012), Deflection properties of a MEMS optical scanner with four torsion beams and L-shaped arms, Sensors and Actuators A: Physical, 178:154-163.
3.  [3] Abou-Rayan A, Nayfeh A, Mook D, (1993), Nonlinear response of a parametrically excited buckled beam, Nonlinear Dynamics, 4(5): 499-525.
4.  [4] Ashworth R, Barr A. (1987), The resonances of structures with quadratic inertial non-linearity under direct and parametric harmonic excitation, Journal of Sound and Vibration, 118(1): 47-68.
5.  [5] Crespo da Silva M, Glynn C. (1978), Nonlinear flexural-flexural-torsional dynamics of inextensional beams. I. Equations of motion, Journal of Structural Mechanics, 6(4): 437-448.
6.  [6] Crespo da Silva M, Glynn C. (1978), Nonlinear flexural-flexural-torsional dynamics of inextensional beams. II. Forced motions, Journal of Structural Mechanics, 6(4): 449-461.
7.  [7] Da Silva M C, Zaretzky C. (1990), Non-linear modal coupling in planar and non-planar responses of inextensional beams, International Journal of Non-Linear Mechanics, 25(2): 227-239.
8.  [8] Da Silva M C, Zaretzky C. (1994), Nonlinear flexural-flexural-torsional interactions in beams including the effect of torsional dynamics. I: Primary resonance, Nonlinear Dynamics, 5(1): 3-23.
9.  [9] Dwivedy S, Kar R. (2001), Non-linear dynamics of a slender beam carrying a lumped mass under principal parametric resonance with three-mode interactions, International Journal of Non-Linear Mechanics, 36(6): 927-945.
10.  [10] Zaretzky C, da Silva M C. (1994), Nonlinear flexural-flexural-torsional interactions in beams including the effect of torsional dynamics. II: Combination resonance, Nonlinear Dynamics, 5(2): 161-180.
11.  [11] Zavodney L, Nayfeh A. (1989), The non-linear response of a slender beam carrying a lumped mass to a principal parametric excitation: theory and experiment, International Journal of Non-Linear Mechanics, 24(2): 105-125.
12.  [12] Balachandran B, Nayfeh A. (1990), Nonlinear motions of beam-mass structure, Nonlinear Dynamics, 1(1): 39-61.
13.  [13] Balachandran B, Nayfeh A. (1991), Observations of modal interactions in resonantly forced beam-mass structures, Nonlinear Dynamics, 2(2): 77-117.
14.  [14] El-Bassiouny A. (2003), Modal interaction of resonantly forced oscillations of 2-DoF structure, Applied mathematics and computation, 134(2): 217-242.
15.  [15] Haddow A, Barr A, Mook D. (1984), Theoretical and experimental study of modal interaction in a 2-DoF structure, Journal of Sound and Vibration, 97(3): 451-473.
16.  [16] Nayfeh A, Balachandran B. (1990), Experimental investigation of resonantly forced oscillations of a 2-DoF structure, International Journal of Non-Linear Mechanics, 25(2): 199-209.
17.  [17] Nayfeh A, Balachandran B, Colbert M, (1989), An experimental investigation of complicated responses of a 2-DoF structure, Journal of Applied Mechanics, 56(4): 960-967.
18.  [18] Nayfeh A, Zavodney L. (1988), Experimental observation of amplitude-and phase-modulated responses of two internally coupled oscillators to a harmonic excitation, Journal of Applied Mechanics, 55(3): 706-710.
19.  [19] Onozato N, Nagai K-i, Maruyama S, (2012), Chaotic vibrations of a post-buckled L-shaped beam with an axial constraint, Nonlinear Dynamics, 67(4): 2363-2379.
20.  [20] Georgiades F, Warminski J, Cartmell M P. (2012), Linear Modal Analysis of L-shaped Beam Structuresâ€“ Parametric Studies, Proceedings of the Journal of Physics: Conference Series, IOP Publishing, 012006.
21.  [21] Georgiades F, Warminski J, Cartmell M P. (2013), Linear modal analysis of L-shaped beam structures, Mechanical Systems and Signal Processing, 38(2): 312-332.
22.  [22] Georgiades F, Warminski J, Cartmell M P. (2013), Towards linear modal analysis for an L-shaped beam: Equations of motion, Mechanics Research Communications, 47: 50-60.
23.  [23] Bux S, Roberts J. (1986), Non-linear vibratory interactions in systems of coupled beams, Journal of Sound and Vibration, 104(3): 497-520.
24.  [24] Cartmell M, Roberts J. (1988), Simultaneous combination resonances in an autoparametrically resonant system, Journal of Sound and Vibration, 123(1): 81-101.
25.  [25] Warminski J, Cartmell M, Bochenski M, (2008), Analytical and experimental investigations of an autoparametric beam structure, Journal of Sound and Vibration, 315(3): 486-508.
26.  [26] Cao D, Song M, Zhu W, (2012), Modeling and analysis of the in-plane vibration of a complex cable-stayed bridge, Journal of Sound and Vibration, 331(26): 5685-5714.
27.  [27] Song M, Cao D, Zhu W. (2011), Dynamic analysis of a micro-resonator driven by electrostatic combs, Communications in Nonlinear Science and Numerical Simulation, 16(8): 3425-3442.
28.  [28] Wei J, Cao D, Yang Y, (2015), Global modal approach for nonlinear dynamical modeling of an L-shaped beam-mass structure, Dynamical Systems Theory and Applications: Mechatronics and Life Sciences, ISBN 978-83-7283-707-3:567
29.  [29] Nyfeh A, Mook D, (1995), Nonlinear Oscillations, NewYork: Wiley-Interscience, ISBN 0-471-12142-8:469