Nonlinear Modeling of Bolted Lap Jointed Structure with Large Amplitude Vibration of Timoshenko Beams

Document Type: Research Paper


Department of Aerospace Engineering, Space Research Institute, Malek Ashtar University of Technology, Tehran, Iran


This paper aims at investigating the nonlinear behavior of a system which is consisting of two free-free beams which are connected by a nonlinear joint. The nonlinear system is modelled as an in-extensional beam with Timoshenko beam theory. In addition, large amplitude vibration assumption is taken into account in order to obtain exact results. The nonlinear assumption in the system necessities existence of the curvature-related and inertia-related nonlinearities. The nonlinear partial differential equations of motion for the longitudinal, transverse, and rotation are derived using the Hamilton’s principle. A set of coupled nonlinear ordinary differential equations are further obtained with the aid of Galerkin method. The frequency-response curves are presented in the section of numerical results to demonstrate the effect of the different dimensionless parameters. It is shown that the nonlinear bolted-lap joint structure exhibits a hardening-type behavior. Furthermore, it is found that by adding a nonlinear spring the system exhibits a stronger hardening-type behavior. In addition, it is found that the system shows nonlinear behavior even in the absence of the nonlinear spring due to the nonlocal nonlinearity assumption. Moreover, it is shown that considering different engineering beam theories lead to different results and it is found that the Euler-Bernoulli beam theory over-predict the resonance frequency of the structure by ignoring rotary inertia and shear deformation.


Main Subjects

[1] Lee U., 2001, Dynamic characterization of the joints in a beam structure by using spectral element method, Shock and Vibration 8: 357-366.
[2] Haines R., 1980, Survey: 2-dimensional motion and impact at revolute joints, Mechanism and Machine Theory 15: 361-370.
[3] Flores P., Ambrósio J., 2004, Revolute joints with clearance in multibody systems, Computers & Structures 82: 1359-1369.
[4] Ahmadian H., Jalali H., 2007, Identification of bolted lap joints parameters in assembled structures, Mechanical Systems and Signal Processing 21: 1041-1050.
[5] Ma X., Bergman L., Vakakis A., 2001, Identification of bolted joints through laser vibrometry, Journal of Sound and Vibration 246: 441-460.
[6] Jalali H., Ahmadian H., Mottershead J.E., 2007, Identification of nonlinear bolted lap-joint parameters by force-state mapping, International Journal of Solids and Structures 44: 8087-8105.
[7] Liao X., Zhang J., 2016, Energy balancing method to identify nonlinear damping of bolted-joint interface, Key Engineering Materials 693: 318-323.
[8] Chatterjee A., Vyas N.S., 2003, Non-linear parameter estimation with Volterra series using the method of recursive iteration through harmonic probing, Journal of Sound and Vibration 268: 657-678.
[9] Chatterjee A., Vyas N.S., 2004, Non-linear parameter estimation in multi-degree-of-freedom systems using multi-input Volterra series, Mechanical Systems and Signal Processing 18: 457-489.
[10] Kerschen G., Worden K., Vakakis A.F., Golinval J.-C., 2007, Nonlinear system identification in structural dynamics: current status and future directions, In 25th International Modal Analysis Conference, Orlando.
[11] Thothadri M., Moon F., 2005, Nonlinear system identification of systems with periodic limit-cycle response, Nonlinear Dynamics 39: 63-77.
[12] Thothadri M., Casas R.A., Moon F.C., D'Andrea R., Johnson Jr C.R., 2003, Nonlinear system identification of multi-degree-of-freedom systems, Nonlinear Dynamics 32: 307-322.
[13] Hajj M.R., Fung J., Nayfeh A.H., Fahey S.O'F., 2000, Damping identification using perturbation techniques and higher-order spectra, Nonlinear Dynamics 23: 189-203.
[14] Noël J.P., Kerschen G., 2016, 10 years of advances in nonlinear system identification in structural dynamics: A review, In Proceedings of ISMA 2016-International Conference on Noise and Vibration Engineering.
[15] Di Maio D., 2016, Identification of dynamic nonlinearities of bolted structures using strain analysis, Nonlinear Dynamics 1: 387-414.
[16] Ahmadian H., Azizi H., 2011, Stability analysis of a nonlinear jointed beam under distributed follower force, Journal of Vibration and Control 17: 27-38.
[17] Cooper S., Di Maio D., Ewins D, 2017, Nonlinear vibration analysis of a complex aerospace structure, Nonlinear Dynamics 1: 55-68.
[18] Jahani K., Nobari A., 2008, Identification of dynamic (Young’s and shear) moduli of a structural adhesive using modal based direct model updating method, Experimental Mechanics 48: 599-611.
[19] Li WL., 2002, A new method for structural model updating and joint stiffness identification, Mechanical Systems and Signal Processing 16: 155-167.
[20] Ratcliffe M., Lieven N., 2000, A generic element-based method for joint identification, Mechanical Systems and Signal Processing 14: 3-28.
[21] Wang J., Chuang S., 2004, Reducing errors in the identification of structural joint parameters using error functions, Journal of Sound and Vibration 273: 295-316.
[22] Segalman D.J., Paez T., Smallwood D., Sumali A., Urbina A., 2003, Status and Integrated Road-Map for Joints Modeling Research, Sandia National Laboratories, Albuquerque.
[23] Shokrollahi S., Adel F., 2016, Finite element model updating of bolted lap joints implementing identification of joint affected region parameters, Journal of Theoretical and Applied Vibration and Acoustics 2: 65-78.
[24] Ren Y., Beards C., 1998, Identification of’effective’linear joints using coupling and joint identification techniques, Journal of Vibration and Acoustics 120: 331-338.
[25] Ghorbanpour Arani A., Atabakhshian V., Loghman A., Shajari A.R., Amir S., 2012, Nonlinear vibration of embedded SWBNNTs based on nonlocal Timoshenko beam theory using DQ method, Physica B: Condensed Matter 47: 2549-2555.
[26] Ghorbanpour Arani A., Kolahchi R., 2014, Nonlinear vibration and instability of embedded double-walled carbon nanocones based on nonlocal Timoshenko beam theory, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228: 690-702.
[27] Ghorbanpour Arani A., Dashti P., Amir S., Yousefi M., 2015, Nonlinear vibration of coupled nano- and microstructures conveying fluid based on Timoshenko beam model under two-dimensional magnetic field, Acta Mechanica 226: 2729-2760.
[28] Rao S.S., 2007, Vibration of Continuous Systems, John Wiley & Sons.
[29] Stanton S.C., Erturk A., Mann B.P., Inman D.J., Inman D.J., 2012, Nonlinear nonconservative behavior and modeling of piezoelectric energy harvesters including proof mass effects, Journal of Intelligent Material Systems and Structures 23: 183-199.
[30] Firoozy P., Khadem S.E., Pourkiaee S.M., 2017, Broadband energy harvesting using nonlinear vibrations of a magnetopiezoelastic cantilever beam, International Journal of Engineering Science 111: 113-133.
[31] Ghayesh M.H., Amabili M., Farokhi H., 2013, Three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams, International Journal of Engineering Science 71: 1-14.
[32] Reddy J.N., 2004, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC press.
[33] Firoozy P., Khadem S.E., Pourkiaee S.M., 2017, Power enhancement of broadband piezoelectric energy harvesting using a proof mass and nonlinearities in curvature and inertia, International Journal of Mechanical Sciences 133: 227-239.
[34] Pourkiaee S.M., Khadem S.E., Shahgholi M., 2016, Parametric resonances of an electrically actuated piezoelectric nanobeam resonator considering surface effects and intermolecular interactions, Nonlinear Dynamics 84: 1943-1960.
[35] Han S.M., Benaroya H., Wei T., 1999, Dynamics of transversely vibrating beams using four engineering theories, Journal of Sound and Vibration 225: 935-988.