najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2023-14-3-321-327najo-304Research ArticleMATHEMATICSМАТЕМАТИКАMethod of reference problems for obtaining approximate analytical solution of multiparametric Sturm–Liouville problemsМетод эталонных задач приближенного аналитического решения многопараметрических задач Штурма-Лиувлляhttps://orcid.org/0009-0009-8184-3264ЛукьяновВ. Д.LukyanovV. D.

Валерий Дмитриевич Лукьянов

Valeriy D. Lukyanov

Kondrat’evsky, 72, St. Petersburg, 195271

lukyanovvd@rambler.ruhttps://orcid.org/0009-0009-1207-9266НосоваЛ. В.NosovaL. V.

Людмила Васильевна Носова

Lyudmila V. Nosova

Zhdanovskaya, 13, St. Petersburg, 197198

vka@mil.ruJoint-Stock Company “Avangard”Russian FederationMozhaisky Military Space AcademyRussian Federation202330062025143321327Copyright © Lukyanov V.D., Nosova L.V., 20252025Лукьянов В.Д., Носова Л.В.Lukyanov V.D., Nosova L.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/304

Approximate analytical formulas are obtained for the eigenfrequencies of longitudinal oscillations of an elastic rod with different mechanical fixings of the ends. The eigenfrequencies are found by solving Sturm–Liouville problems with the third kind boundary conditions as roots of transcendental equations. Homogeneous boundary conditions contain one or more parameters whose values are calculated through the indices of mechanical system. Approximate expression for analytical dependencies of the eigenfrequencies on the single parameter are obtained for one-parametric problems, which are called reference ones. We propose a method for obtaining approximate analytical expression for dependencies of the eigenfrequencies on several parameters in boundary conditions by sequentially solving the reference problems. The two-parametric Sturm–Liouville problem is solved by the proposed method.

Получены приближенные аналитические формулы для собственных частот продольных колебаний упругого стержня с различными механическими креплениями концов. Собственные частоты находятся из решения задач Штурма-Лиувилля с граничными условиями третьего рода как корни трансцендентных уравнений. Однородные граничные условия содержат один или более параметров, значения которых вычисляются через показатели механической системы. Для однопараметрических задач, которые называются эталонными, получены приближенные аналитические зависимости собственных частот от единственного параметра задачи. Предложен метод последовательного использования решений эталонных задач для получения приближенных аналитических зависимостей собственных частот от нескольких параметров в граничных условиях. Предложенным методом решена двухпараметрическая задача Штурма-Лиувилля.

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The authors declare that there are no conflicts of interest present.