Mass minimization of an Euler-Bernoulli beam with coupled bending and axial vibrations at prescribed fundamental frequency
Само за регистроване кориснике
2021
Чланак у часопису (Објављена верзија)
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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.
Кључне речи:
Singular optimal control / Pontryagin's maximum principle / Optimization / Mechanical vibrations / Mass minimization / Euler-Bernoulli beam / Axial-bending vibrationИзвор:
Engineering Structures, 2021, 228Издавач:
- Elsevier Sci Ltd, Oxford
Финансирање / пројекти:
- Министарство науке, технолошког развоја и иновација Републике Србије, институционално финансирање - 200105 (Универзитет у Београду, Машински факултет) (RS-MESTD-inst-2020-200105)
- Министарство науке, технолошког развоја и иновација Републике Србије, институционално финансирање - 200108 (Универзитет у Крагујевцу, Машински факултет, Краљево) (RS-MESTD-inst-2020-200108)
Напомена:
- Peer reviewed version of the article: https://machinery.mas.bg.ac.rs/handle/123456789/4331
Повезане информације:
- Друга верзија
https://machinery.mas.bg.ac.rs/handle/123456789/4331
DOI: 10.1016/j.engstruct.2020.111538
ISSN: 0141-0296
WoS: 000607486500005
Scopus: 2-s2.0-85097086598
Колекције
Институција/група
Mašinski fakultetTY - JOUR AU - Obradović, Aleksandar AU - Šalinić, Slaviša AU - Grbović, Aleksandar PY - 2021 UR - https://machinery.mas.bg.ac.rs/handle/123456789/3628 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 . .