The Transient Dynamics of a Beam Mounted on Spring Supports and Equipped with the Nonlinear Energy Sink


erospace Engineering Department, Faculty of New Technologies and Engineering, Shahid Beheshti University, Tehran, Iran


The transient dynamics of a beam mounted on springer-damper support and equipped with a nonlinear energy sink (NES) is investigated under the effects of shock loads. The equations of motion are derived using the Hamilton’s principle leading to four hybrid ordinary and partial differential equations and descritized using the Galerkin method. An adaptive Newmark method is employed for accurate and efficient numerical simulation and the results are used to assess the efficiency of the NES by conducting various parametric studies. The mechanisms of targeted energy transfer from the beam to the NES are indicated using the wavelet transform and Hilbert–Huang transform of the responses. Numerous modes are recognized to contribute to the response and the modes with smaller particiaption of the rigid-body motions are found to be strongly engaged in the transient resonance capture (TRC) at the initial stage of the motion. The modes with dominant rigid-body motions are, however found to be less engaed in an effective TRC. The enhanced empirical mode decomposition, with different masking signals are used to extract narrow-band intrinsic mode functions (IMFs) and simultaneous 1:1 transient resonances are observed between different IMFs of the responses especially at the initial stage of the motion.


1.     Pourzeynali, S. and Esteki, S., "Optimization of the tmd parameters to suppress the vertical vibrations of suspension bridges subjected to earthquake excitations", Iranian International Journal of Engineering, IJE Transaction B (Application),  Vol. 22, No. 1, (2009), 23-34.
2.     Saberi, L. and Nahvi, H., "Vibration analysis of a nonlinear system with a nonlinear absorber under the primary and super-harmonic resonances", International Journal of Engineering-Transactions C: Aspects,  Vol. 27, No. 3, (2013), 499-507.
3.     Lee, Y., Vakakis, A.F., Bergman, L., McFarland, D., Kerschen, G., Nucera, F., Tsakirtzis, S. and Panagopoulos, P., "Passive non-linear targeted energy transfer and its applications to vibration absorption: A review", Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics,  Vol. 222, No. 2, (2008), 77-134.
4.     Georgiades, F. and Vakakis, A., "Dynamics of a linear beam with an attached local nonlinear energy sink", Communications in Nonlinear Science and Numerical Simulation,  Vol. 12, No. 5, (2007), 643-651.
5.     Tsakirtzis, S., Lee, Y., Vakakis, A., Bergman, L. and McFarland, D., "Modelling of nonlinear modal interactions in the transient dynamics of an elastic rod with an essentially nonlinear attachment", Communications in Nonlinear Science and Numerical Simulation,  Vol. 15, No. 9, (2010), 2617-2633.
6.     Tsakirtzis, S., Vakakis, A.F. and Panagopoulos, P., "Broadband energy exchanges between a dissipative elastic rod and a multi-degree-of-freedom dissipative essentially non-linear attachment", International Journal of Non-Linear Mechanics,  Vol. 42, No. 1, (2007), 36-57.
7.     Besanjideh, M. and Mahani, M.F., "Nonlinear and non-stationary vibration analysis for mechanical fault detection by using emd-fft method", International Journal of Engineering-Transactions C: Aspects,  Vol. 25, No. 4, (2012), 363-372.
8.     Lee, Y.S., Tsakirtzis, S., Vakakis, A.F., Bergman, L.A. and McFarland, D.M., "Physics-based foundation for empirical mode decomposition", AIAA journal,  Vol. 47, No. 12, (2009), 2938-2963.
9.     Kerschen, G., Vakakis, A.F., Lee, Y., McFarland, D. and Bergman, L., "Toward a fundamental understanding of the hilbert-huang transform in nonlinear structural dynamics", Journal of Vibration and Control,  Vol. 14, No. 1-2, (2008), 77-105.
10.   Georgiades, F. and Vakakis, A.F., "Passive targeted energy transfers and strong modal interactions in the dynamics of a thin plate with strongly nonlinear attachments", International Journal of Solids and Structures,  Vol. 46, No. 11, (2009), 2330-2353.
11.   Ahmadabadi, Z.N. and Khadem, S., "Nonlinear vibration control of a cantilever beam by a nonlinear energy sink", Mechanism and Machine Theory,  Vol. 50, (2012), 134-149.
12.   Parseh, M., Dardel, M., Ghasemi, M.H. and Pashaei, M.H., "Steady state dynamics of a non-linear beam coupled to a non-linear energy sink", International Journal of Non-Linear Mechanics,  Vol. 79, (2016), 48-65.
13.   Kani, M., Khadem, S., Pashaei, M. and Dardel, M., "Vibration control of a nonlinear beam with a nonlinear energy sink", Nonlinear Dynamics,  Vol. 83, No. 1-2, (2016), 1-22.
14.   Ebrahimzade, N., Dardel, M. and Shafaghat, R., "Performance comparison of linear and nonlinear vibration absorbers in aeroelastic characteristics of a wing model", Nonlinear Dynamics,  Vol. 86, No. 2, (2016), 1075-1094.
15.   Bab, S., Khadem, S., Shahgholi, M. and Abbasi, A., "Vibration attenuation of a continuous rotor-blisk-journal bearing system employing smooth nonlinear energy sinks", Mechanical Systems and Signal Processing,  Vol. 84, (2017), 128-157.
16.   Baruh, H., "Analytical dynamics, WCB/McGraw-Hill Boston,  (1999).
17.   Heydari, H., Ghazavi, M., Najafi, A. and Rahmanian, S., "Nonlinear dynamics of the rotational slender axially moving string with simply supported conditions", International Journal of Engineering-Transactions C: Aspects,  Vol. 29, No. 6, (2016), 834-841.
18.   Abdollahi, D., Ivaz, K. and Shabani, R., "A numerical improvement in analyzing the dynamic characteristics of an electrostatically actuated micro-beam in fluid loading with free boundary approach", International Journal of Engineering-Transactions A: Basics,  Vol. 29, No. 7, (2016), 1005-114.
19.   Zeng, L.F., Wiberg, N.E., Li, X. and Xie, Y., "A posteriori local error estimation and adaptive time‐stepping for newmark integration in dynamic analysis", Earthquake Engineering & Structural Dynamics,  Vol. 21, No. 7, (1992), 555-571.