Abstract
Objectives. The longitudinal oscillations of a vertical rod of a continually discrete system with kinematic seismic disturbances in the form of a stationary random process are considered.Method. A method for determining the variance of the output process of displacements, using the representation of the input random process as a sum of harmonic deterministic perturbations, is proposed and implemented.Result. The dependence function of the dispersion of displacements on the longitudinal coordinate is determined. Longitudinal vibrations of vertical rods near the epicenter of earthquakes are dangerous for their strength and stability. The methods of finite differences and coordinate descent allow you to create universal algorithms and computer programs that easily solve complex spectral problems.Conclusion. To date, research on random vibrations of buildings and structures, as well as regulatory documents, has been devoted to horizontal seismic effects and transverse bending vibrations caused by them. Examples indicate the need to expand the scope of research with the inclusion of other types of vibrations: combinations of longitudinal with transverse, angular, torsional, parametric, etc. This design can be easily adapted to vibrations of rods of variable cross section, to vibrations of continually discrete rods.
Highlights
Longitudinal vibrations of vertical rods near the epicenter of earthquakes are dangerous for their strength and stability
Examples indicate the need to expand the scope of research with the inclusion of other types of vibrations: combinations of longitudinal with transverse, angular, torsional, parametric, etc. This design can be adapted to vibrations of rods of variable cross section, to vibrations of continually discrete rods
Free vibration of a functionally graded rotating Timoshenko beam using FEM // International Journal of Advanced Structural Engineering
Summary
Для изучения свободных колебаний изобразим элементарную частицу стержня и приложенные к ней внутренние силы в сечениях N, N+dN, инерционную силу dI и силу демпфирования dR на рис. Что элементарные выкладки приведут к дифференциальному уравнению в частных производных гиперболического типа для свободных продольных колебаний mü + ηmu - b u'' = 0, b =ES, m = ρS x ε (0, l), t > - ∞. Из (13) следует, что коэффициент затухания и частота свободных колебаний должны определяться из системы двух нелинейных алгебраических уравнений f1 , 0 , f2 , 0.
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