Mass minimization of an Euler-Bernoulli beam with coupled bending and axial vibrations at prescribed fundamental frequency
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2021
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The problem of determining the optimum shape of a homogeneous Euler-Bernoulli beam of a circular cross-section, in which the coupled axial and bending vibrations arose due to complex boundary conditions, is considered. The beam mass is minimized at prescribed fundamental frequency. The problem is solved applying Pontryagin's maximum principle, with the beam cross-sectional diameter derivative with respect to longitudinal coordinate taken for control variable. This problem involves first-order singular optimal control, the calculations of which allowed the application of the Poisson bracket formalism and the fulfillment of the Kelley necessary condition on singular segments. Numerical solution of the two-point boundary value problem is obtained by the shooting method. An inequality constraint is imposed to the beam diameter derivative. Depending on the size of the diameter derivative boundaries, the obtained solutions are singular along the entire beam or consist of singular and non-sin...gular segments, where the diameter derivative is at one of its boundaries. It is shown that such system is self-adjoint, so that only one differential equation of the costate equations system was integrated and the rest costate variables were expressed via the state variables. Also, the paper shows the fulfillment of necessary conditions for the optimality of junctions between singular and non-singular segments, as well as the percent saving of the beam mass compared to the beams of constant diameter at identical value of the fundamental frequency.
Ključne reči:
Singular optimal control / Pontryagin's maximum principle / Optimization / Mechanical vibrations / Mass minimization / Euler-Bernoulli beam / Axial-bending vibrationIzvor:
Engineering Structures, 2021, 228Izdavač:
- Elsevier Sci Ltd, Oxford
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)
Napomena:
- This is the peer reviewed version of the article: Obradović, A.; Šalinić, S.; Grbović, A. Mass Minimization of an Euler-Bernoulli Beam with Coupled Bending and Axial Vibrations at Prescribed Fundamental Frequency. Engineering Structures 2021, 228. https://doi.org/10.1016/j.engstruct.2020.111538
Povezane informacije:
- Druga verzija
https://doi.org/10.1016/j.engstruct.2020.111538 - Druga verzija
https://machinery.mas.bg.ac.rs/handle/123456789/3628
DOI: 10.1016/j.engstruct.2020.111538
ISSN: 0141-0296
WoS: 000607486500005
Scopus: 2-s2.0-85097086598
Kolekcije
Institucija/grupa
Mašinski fakultetTY - JOUR AU - Obradović, Aleksandar AU - Šalinić, Slaviša AU - Grbović, Aleksandar PY - 2021 UR - https://machinery.mas.bg.ac.rs/handle/123456789/4331 AB - The problem of determining the optimum shape of a homogeneous Euler-Bernoulli beam of a circular cross-section, in which the coupled axial and bending vibrations arose due to complex boundary conditions, is considered. The beam mass is minimized at prescribed fundamental frequency. The problem is solved applying Pontryagin's maximum principle, with the beam cross-sectional diameter derivative with respect to longitudinal coordinate taken for control variable. This problem involves first-order singular optimal control, the calculations of which allowed the application of the Poisson bracket formalism and the fulfillment of the Kelley necessary condition on singular segments. Numerical solution of the two-point boundary value problem is obtained by the shooting method. An inequality constraint is imposed to the beam diameter derivative. Depending on the size of the diameter derivative boundaries, the obtained solutions are singular along the entire beam or consist of singular and non-singular segments, where the diameter derivative is at one of its boundaries. It is shown that such system is self-adjoint, so that only one differential equation of the costate equations system was integrated and the rest costate variables were expressed via the state variables. Also, the paper shows the fulfillment of necessary conditions for the optimality of junctions between singular and non-singular segments, as well as the percent saving of the beam mass compared to the beams of constant diameter at identical value of the fundamental frequency. PB - Elsevier Sci Ltd, Oxford T2 - Engineering Structures T1 - Mass minimization of an Euler-Bernoulli beam with coupled bending and axial vibrations at prescribed fundamental frequency VL - 228 DO - 10.1016/j.engstruct.2020.111538 ER -
@article{ author = "Obradović, Aleksandar and Šalinić, Slaviša and Grbović, Aleksandar", year = "2021", abstract = "The problem of determining the optimum shape of a homogeneous Euler-Bernoulli beam of a circular cross-section, in which the coupled axial and bending vibrations arose due to complex boundary conditions, is considered. The beam mass is minimized at prescribed fundamental frequency. The problem is solved applying Pontryagin's maximum principle, with the beam cross-sectional diameter derivative with respect to longitudinal coordinate taken for control variable. This problem involves first-order singular optimal control, the calculations of which allowed the application of the Poisson bracket formalism and the fulfillment of the Kelley necessary condition on singular segments. Numerical solution of the two-point boundary value problem is obtained by the shooting method. An inequality constraint is imposed to the beam diameter derivative. Depending on the size of the diameter derivative boundaries, the obtained solutions are singular along the entire beam or consist of singular and non-singular segments, where the diameter derivative is at one of its boundaries. It is shown that such system is self-adjoint, so that only one differential equation of the costate equations system was integrated and the rest costate variables were expressed via the state variables. Also, the paper shows the fulfillment of necessary conditions for the optimality of junctions between singular and non-singular segments, as well as the percent saving of the beam mass compared to the beams of constant diameter at identical value of the fundamental frequency.", publisher = "Elsevier Sci Ltd, Oxford", journal = "Engineering Structures", title = "Mass minimization of an Euler-Bernoulli beam with coupled bending and axial vibrations at prescribed fundamental frequency", volume = "228", doi = "10.1016/j.engstruct.2020.111538" }
Obradović, A., Šalinić, S.,& Grbović, A.. (2021). Mass minimization of an Euler-Bernoulli beam with coupled bending and axial vibrations at prescribed fundamental frequency. in Engineering Structures Elsevier Sci Ltd, Oxford., 228. https://doi.org/10.1016/j.engstruct.2020.111538
Obradović A, Šalinić S, Grbović A. Mass minimization of an Euler-Bernoulli beam with coupled bending and axial vibrations at prescribed fundamental frequency. in Engineering Structures. 2021;228. doi:10.1016/j.engstruct.2020.111538 .
Obradović, Aleksandar, Šalinić, Slaviša, Grbović, Aleksandar, "Mass minimization of an Euler-Bernoulli beam with coupled bending and axial vibrations at prescribed fundamental frequency" in Engineering Structures, 228 (2021), https://doi.org/10.1016/j.engstruct.2020.111538 . .