Nonlinear free vibrations of the beam taking into account the longitudinal inertia of the mass element causing the tensile load

• Autorzy: Uzny Sebastian

Identyfikatory

DOI

10.17512/jamcm.2025.4.07

• Języki publikacji: EN

Abstrakty

EN

This paper presents theoretical and numerical studies of the nonlinear natural vibrations of a beam subjected to a tensile load induced by a mass element. The vibration problem was formulated based on Hamilton’s principle, taking into account the Bernoulli - Euler beam theory. Due to the nonlinearity resulting from axial strain defined according to Von Karman’s theory, the boundary value problem was derived using the small parameter method. By taking into account the equations related to the small parameter to the appropri ate power, the linear component of the natural frequencies, the nonlinear component of the internal force in the beam under tension, and the nonlinear component of the natural fre quencies (dependent on the amplitude) were determined. The results of numerical calcula tions of the first three natural frequencies are graphically presented as a function of the tensile load. The theoretical and numerical studies conducted in this paper are introductory to the research on the dynamic properties of the system. In particular, the dependence of the dynamic properties on the longitudinal inertia of the mass element loading the beam is investigated. The omission of damping in this work was aimed at formulating a boundary value problem with the lowest possible degree of complexity, which would enable a prelim inary analysis of the assumed research task.

Słowa kluczowe

EN

tension load; nonlinear free vibration; tension beam; Bernoulli-Euler's theory; small parameter method; longitudinal inertia

PL

obciążenie rozciągające; nieliniowe drgania swobodne; belka rozciągana; teoria Bernoulliego-Eulera; metoda małych parametrów; bezwładność wzdłużna

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2025
• Tom: Vol. 24, nr 4
• Strony: 100--118
• Opis fizyczny: Bibliogr. 15 poz., rys.

Twórcy

autor

Uzny Sebastian

• Department of Mechanics and Machine Design Foundations Czestochowa University of Technology Częstochowa, Poland

Bibliografia

• [1] Przybylski, J., & Gąsiorski, G. (2018). Nonlinear vibrations of elastic beam with piezoelectric actuators. Journal of Sound and Vibration, 437, 150-165.
• [2] Ding, H., Lu, Z.Q., & Chen, L.Q. (2019). Nonlinear isolation of transverse vibration of pre pressure beams. Journal of Sound and Vibration, 442, 738-751.
• [3] Wang, G.X., Ding, H., & Chen, L.Q. (2020). Dynamic effect of internal resonance caused by gravity on the nonlinear vibration of vertical cantilever beams. Journal of Sound and Vibration, 474, 115265.
• [4] Rosenberg, S., & Shoshani, O. (2025). Shape optimization of a curved mechanical beam for transverse vibrations amplification via nonlinear interaction with longitudinal vibrations. Journal of Sound and Vibration, 599, 118908.
• [5] Lenci, S., & Kłoda, Ł. (2025). Nonlinear vibrations of kinematically exact curved beams. Journal of Sound and Vibration, 602, 118951.
• [6] Li, H., & Yao, G. (2025). Nonlinear forced vibration and stability analysis of a rotating three - dimensional cantilever beam with variable cross-section. Thin-Walled Structures, 211, 113104.
• [7] Javadi, M., & Rahmanian, M. (2021). Nonlinear vibration of fractional Kelvin-Voigt viscoelastic beam on nonlinear elastic foundation. Communications in Nonlinear Science and Numerical Simulation, 98, 105784.
• [8] Kłoda, Ł., Lenci, S., Warmiński, J., & Szmit, Z. (2022). Flexural-flexural internal resonances 3:1 in initially straight, extensible Timoshenko beams with an axial spring. Journal of Sound and Vibration, 527, 116809.
• [9] Uzny, S., Kutrowski, Ł., & Osadnik, M. (2021). The non-linear vibrations of simply supported column loaded by the mass element. Applied Mathematical Modelling, 89, 700-709.
• [10] Uzny, S., Kutrowski, Ł., & Skrzypczak, T. (2021). Non-linear free vibrations of the column loaded with a mass element and a local heat source. Journal of Sound and Vibration, 507, 116130.
• [11] Uzny, S., Kutrowski, Ł., & Osadnik, M. (2022). Influence of longitudinal inertia of sliding mass on nonlinear transverse vibrations of hydraulic cylinder piston rod. International Journal of Structural Stability and Dynamics, 22(14). DOI: 10.1142/S0219455422501565.
• [12] Wang, G.-X., Ding H., & Chen L.-Q. (2020). Dynamic effect of internal resonance caused by gravity on the nonlinear vibration of vertical cantilever beams. Journal of Sound and Vibration, 474, 115265.
• [13] Ghayesh, M.H., Kazemirad, S., & Amabili, M. (2012). Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance. Mechanism and Machine Theory, 52, 18-34.
• [14] Ghayesh, M.H. (2011). Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance. International Journal of Mechanical Sciences, 53, 1022-1037.
• [15] Mahmoudkhani, S. (2017). Dynamics of a mass-spring-beam with 0:1:1 internal resonance using the analytical and continuation method. International Journal of Non-Linear Mechanics, 97, 48-67.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2026).