Coupled bending and axial vibrations of axially functionally graded Euler-Bernoulli beams
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2023
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,
Springer
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Purpose This study aims to obtain and present the closed-form solution of coupled
axial-bending vibration problem for the general case of axially functionally graded
(AFG) beams, to create the frequency equation and to propose numerical method for
computing natural frequencies.
Methods The general model of a beam is introduced using two cylindrical springs, one
rotational spring, and a rigid body at each beam end. Mass centers of bodies are
placed eccentrically in axial and transverse direction with respect to end points of the
beam. The general boundary conditions are modelled by linearized Newton-Euler
differential equations and the general case of the in-plane axial-bending vibrations of
AFG beams is covered. It is assumed that the beam is made of a functionally graded
material whose material and cross-sectional characteristics change along its
longitudinal axis without any restrictions. The Euler-Bernoulli constitutive theory is
applied for modelling. Partial differenti...al equations of motion are transformed into a
system of ordinary differential equations with variable coefficients, the form suitable for
the implementation of the symbolic-numeric methods of initial parameters. Natural
frequencies of the beam are computed as numerical solutions of the exact frequency
equation.
Results and conclusions The closed-form solution of coupled axial-bending vibrations
is derived for general case of AFG beams. Orthogonality conditions of mode shapes
are derived, and constants in time function. Also, the frequency equation is derived and
solved numerically in order to obtain natural frequencies. Obtained results of natural
frequencies and mode shapes are compared to those available in open literature.
Ključne reči:
mechanical vibrations / axially functionally graded beam / axial-bending vibrations / Euler-Bernoulli beam / mode shapes / orthogonality conditionsIzvor:
Journal of Vibration Engineering & Technologies, 2023Izdavač:
- Springer
Finansiranje / projekti:
- Ministarstvo nauke, tehnološkog razvoja i inovacija Republike Srbije, institucionalno finansiranje - 200105 (Univerzitet u Beogradu, Mašinski fakultet) (RS-MESTD-inst-2020-200105)
- Ministarstvo nauke, tehnološkog razvoja i inovacija Republike Srbije, institucionalno finansiranje - 200108 (Univerzitet u Kragujevcu, Mašinski fakultet, Kraljevo) (RS-MESTD-inst-2020-200108)
Kolekcije
Institucija/grupa
Mašinski fakultetTY - JOUR AU - Tomović, Aleksandar AU - Šalinić, Slaviša AU - Obradović, Aleksandar AU - Zorić, Nemanja AU - Mitrović, Zoran PY - 2023 UR - https://machinery.mas.bg.ac.rs/handle/123456789/6872 AB - Purpose This study aims to obtain and present the closed-form solution of coupled axial-bending vibration problem for the general case of axially functionally graded (AFG) beams, to create the frequency equation and to propose numerical method for computing natural frequencies. Methods The general model of a beam is introduced using two cylindrical springs, one rotational spring, and a rigid body at each beam end. Mass centers of bodies are placed eccentrically in axial and transverse direction with respect to end points of the beam. The general boundary conditions are modelled by linearized Newton-Euler differential equations and the general case of the in-plane axial-bending vibrations of AFG beams is covered. It is assumed that the beam is made of a functionally graded material whose material and cross-sectional characteristics change along its longitudinal axis without any restrictions. The Euler-Bernoulli constitutive theory is applied for modelling. Partial differential equations of motion are transformed into a system of ordinary differential equations with variable coefficients, the form suitable for the implementation of the symbolic-numeric methods of initial parameters. Natural frequencies of the beam are computed as numerical solutions of the exact frequency equation. Results and conclusions The closed-form solution of coupled axial-bending vibrations is derived for general case of AFG beams. Orthogonality conditions of mode shapes are derived, and constants in time function. Also, the frequency equation is derived and solved numerically in order to obtain natural frequencies. Obtained results of natural frequencies and mode shapes are compared to those available in open literature. PB - Springer T2 - Journal of Vibration Engineering & Technologies T1 - Coupled bending and axial vibrations of axially functionally graded Euler-Bernoulli beams UR - https://hdl.handle.net/21.15107/rcub_machinery_6872 ER -
@article{ author = "Tomović, Aleksandar and Šalinić, Slaviša and Obradović, Aleksandar and Zorić, Nemanja and Mitrović, Zoran", year = "2023", abstract = "Purpose This study aims to obtain and present the closed-form solution of coupled axial-bending vibration problem for the general case of axially functionally graded (AFG) beams, to create the frequency equation and to propose numerical method for computing natural frequencies. Methods The general model of a beam is introduced using two cylindrical springs, one rotational spring, and a rigid body at each beam end. Mass centers of bodies are placed eccentrically in axial and transverse direction with respect to end points of the beam. The general boundary conditions are modelled by linearized Newton-Euler differential equations and the general case of the in-plane axial-bending vibrations of AFG beams is covered. It is assumed that the beam is made of a functionally graded material whose material and cross-sectional characteristics change along its longitudinal axis without any restrictions. The Euler-Bernoulli constitutive theory is applied for modelling. Partial differential equations of motion are transformed into a system of ordinary differential equations with variable coefficients, the form suitable for the implementation of the symbolic-numeric methods of initial parameters. Natural frequencies of the beam are computed as numerical solutions of the exact frequency equation. Results and conclusions The closed-form solution of coupled axial-bending vibrations is derived for general case of AFG beams. Orthogonality conditions of mode shapes are derived, and constants in time function. Also, the frequency equation is derived and solved numerically in order to obtain natural frequencies. Obtained results of natural frequencies and mode shapes are compared to those available in open literature.", publisher = "Springer", journal = "Journal of Vibration Engineering & Technologies", title = "Coupled bending and axial vibrations of axially functionally graded Euler-Bernoulli beams", url = "https://hdl.handle.net/21.15107/rcub_machinery_6872" }
Tomović, A., Šalinić, S., Obradović, A., Zorić, N.,& Mitrović, Z.. (2023). Coupled bending and axial vibrations of axially functionally graded Euler-Bernoulli beams. in Journal of Vibration Engineering & Technologies Springer.. https://hdl.handle.net/21.15107/rcub_machinery_6872
Tomović A, Šalinić S, Obradović A, Zorić N, Mitrović Z. Coupled bending and axial vibrations of axially functionally graded Euler-Bernoulli beams. in Journal of Vibration Engineering & Technologies. 2023;. https://hdl.handle.net/21.15107/rcub_machinery_6872 .
Tomović, Aleksandar, Šalinić, Slaviša, Obradović, Aleksandar, Zorić, Nemanja, Mitrović, Zoran, "Coupled bending and axial vibrations of axially functionally graded Euler-Bernoulli beams" in Journal of Vibration Engineering & Technologies (2023), https://hdl.handle.net/21.15107/rcub_machinery_6872 .