A Dual Lagrange Multiplier Approach for the Dynamics of the Mechanical Systems

[1] Milic, P., Marinkovic, D., Klinge, S., Cojbašic, Ž., Reissner-Mindlin Based Isogeometric Finite Element Formulation for Piezoelectric Active Laminated Shells, Tehnicki Vjesnik, 30(2), 2020, 416-425,

[2] Rama, G., Marinkovic, D., Zehn, M., High performance 3-node shell element for linear and geometrically nonlinear analysis of composite laminates, Composites Part B: Engineering, 151, 2018, 118-126,

[3] Mitrev, R., Janoševic, D., Marinkovic, D., Dynamical modelling of hydraulic excavator considered as a multibody system, Tehnicki Vjesnik, 24, 2017, 327-338.

[4] Yau, H.T., Kuo, P.H., Luan, P.C., Tseng, Y.R., Proximal policy optimization‐based controller for chaotic systems, International Journal of Robust and Nonlinear Control, 34(1), 2024, 586-601.

[5] Kuo, P.H., Chang, C.W., Tseng, Y.R., Yau, H.T., Efficient, automatic, and optimized portable Raman-spectrum-based pesticide detection system, Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy, 308, 2024, 123787.

[6] Rabiee, F., Jafari, A.A., Nonlinear forced vibration of rectangular plates by modified multiple scale method, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 236(9), 2022, 21-30.

[7] Li, X., Wang, D., Saeed, T., Multi-scale numerical approach to the polymer filling process in the weld line region, Facta Universitatis, Series: Mechanical Engineering, 20(2), 2022, 363-380.

[8] Tao, Z.L., Chen, G.H., An effective modification of Ji-Huan He’s variational approach to nonlinear singular oscillator, Journal of Low Frequency Noise, Vibration and Active Control, 38(3-4), 2019, 1023-1027.

[9] Koochi, A., Abadyan, M., Nonlinear Differential Equations in Micro/Nano Mechanics: Application in Micro/Nano Structures and Electromechanical Systems, Elsevier, 2020.

[10] Arif, M.S., Nawaz, Y., Bibi, M., Fayyaz, R., Naz, M., A modification of He’s variational approach using the least square method to nonlinear oscillators, Journal of Low Frequency Noise, Vibration and Active Control, 38(3-4), 2019, 996-1007.

[11] Anjum, N., He, J.H., Ain, Q.T., Tian, D., Li-He’s modified homotopy perturbation method for doubly-clamped electrically actuated microbeams-based microelectromechanical system, Facta Universitatis, Series: Mechanical Engineering, 19(4), 2021, 01-12.

[12] He, C.H., Amer, T.S., Tian, D., Abolila, A.F., Galal, A.A., Controlling the kinematics of a spring-pendulum system using an energy harvesting device, Journal of Low Frequency Noise, Vibration and Active Control, 41(3), 2022, 34-57.

[13] He, C.H., El-Dib, Y.O., A heuristic review on the homotopy perturbation method for non-conservative oscillators, Journal of Low Frequency Noise, Vibration and Active Control, 41(2), 2022, 572-603.

[14] He, C.H., Tian, D., Moatimid, G.M., Salman, H.F., Zekry, M.H., Hybrid Rayleigh–van der pol–duffing oscillator: stability analysis and controller, Journal of Low Frequency Noise, Vibration and Active Control, 41(1), 2022, 44-68.

[15] He, J.H., Verma, L., Pandit, B., Verma, A.K., Agarwal, R.P., A new Taylor series based numerical method: Simple, reliable, and promising, Journal of Applied and Computational Mechanics, 9(4), 2023, 22-34.

[16] Lin, M.X., Deng, C.Y., Chen, C.O.K., Free vibration analysis of non-uniform Bernoulli beam by using Laplace Adomian decomposition method, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 236(13), 2022, 68-78.

[17] Mehdipour, I., Ganji, D.D., Mozaffari, M., Application of the energy balance method to nonlinear vibrating equations, Current Applied Physics, 10(1), 2010, 104-112.

[18] El-Dib, Y.O., Insightful and comprehensive formularization of frequency–amplitude formula for strong or singular nonlinear oscillators, Journal of Low Frequency Noise, Vibration and Active Control, 42(1), 2023, 89-109.

[19] Nofal, T.A., Ismail, G.M., Mady, A.A.M., Abdel-Khalek, S., Analytical and approximate solutions to the fee vibration of strongly nonlinear oscillators, Journal of Electromagnetic Analysis and Applications, 5(10), 2013, 88-92.

[20] Ma, H., Simplified Hamiltonian-based frequency-amplitude formulation for nonlinear vibration systems, Facta Universitatis, Series: Mechanical Engineering, 20(2), 2022, 445-455.

[21] He, J.H., Variational iteration method–a kind of non-linear analytical technique: some examples, International Journal of Non-linear Mechanics, 34(4), 1999, 699-708.

[22] Anjum, N., He, J.H., He, C.H., Ashiq, A., A Brief Review on the Asymptotic Methods for the Periodic‎ Behaviour of Microelectromechanical Systems, Journal of Applied and Computational Mechanics, 8(3), 2022, 20-40.

[23] Farrokhabadi, A., Koochi, A., Kazemi, A., Abadyan, M., Effects of size-dependent elasticity on stability of nanotweezers, Applied Mathematics and Mechanics, 35, 2014, 1573–1590.

[24] Wu, G.C., Baleanu, D., Deng, Z.G., Variational iteration method as a kernel constructive technique, Applied Mathematical Modelling, 39(15), 2015, 78-84.

[25] Wang, Y.H., Wu, G.C., Baleanu, D., Variational iteration method-a promising technique for constructing equivalent integral equations of fractional order, Open Physics, 11(10), 2013, 92-98.

[26] Farrokhabadi, A., Mokhtari, J., Koochi, A. et al., A theoretical model for investigating the effect of vacuum fluctuations on the electromechanical stability of nanotweezers, Indian Journal of Physics, 89, 2015, 599-609.

[27] Ramos, J.I., On the variational iteration method and other iterative techniques for nonlinear differential equations, Applied Mathematics and Computation, 199(1), 2008, 39-69.

[28] Dogan, D.D., Konuralp, A., Fractional variational iteration method for time-fractional non-linear functional partial differential equation having proportional delays, Thermal Science, 22(1), 2018, 33-46.

[29] He, C.H., Liu, C., Variational principle for singular waves, Chaos, Solitons and Fractals, 172, 2023, 113566.

[30] He, C.H., A variational principle for a fractal nano/microelectromechanical (N/MEMS) system, International Journal of Numerical Methods for Heat and Fluid Flow, 33(1), 2022, 351-359.

[31] Khuri, S.A., Sayfy, A., Generalizing the variational iteration method for BVPs: proper setting of the correction functional, Applied Mathematics Letters, 68, 2017, 68-75.

[32] Khuri, S.A., Sayfy, A., Variational iteration method: Green’s functions and fixed point iterations perspective, Applied Mathematics Letters, 32, 2014, 28-34.

[33] Tian, D., Ain, Q.T., Anjum, N., He, C.H., Cheng, B., Fractal N/MEMS: from pull-in instability to pull-in stability, Fractals, 29(2), 2021, 1-8.

[34] Tian, D., He, C.H., A fractal micro-electromechanical system and its pull-in stability, Journal of Low Frequency Noise, Vibration and Active Control, 40(3), 2021, 80-86.

[35] He, J.H., Anjum, N., Skrzypacz, P.S., A Variational Principle for a Nonlinear Oscillator Arising in the‎ Microelectromechanical System, Journal of Applied and Computational Mechanics, 7(1), 2021, 78-83.

[36] Marinca, V., Herişanu, N., Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method, Journal of Sound and Vibration, 329(9), 2010, 50-59.

[37] Ganji, D.D., Azimi, M., Mostofi, M., Energy balance method and amplitude frequency formulation-based simulation of strongly non-linear oscillators, Indian Journal of Pure and Applied Physics, 50, 2012, 70-75.

[38] Davodi, A.G., Ganji, D.D., Azami, R., Babazadeh, H., Application of improved amplitude-frequency formulation to nonlinear differential equation of motion equations, Modern Physics Letters B, 23(28), 2009, 27-36.

[39] Billington, D.P., The tower and the bridge: the new art of structural engineering, Princeton University Press, 1985.

[40] Akbarzade, M., Khan, Y., Dynamic model of large amplitude non-linear oscillations arising in the structural engineering: analytical solutions, Mathematical and Computer Modelling, 55(3-4), 2012, 80-89.

[41] Mathews, P.M., Lakshmanan, M., On a unique nonlinear oscillator, Quarterly of Applied Mathematics, 32(2), 1974, 15-18.