Longitudinal forced harmonic vibrations of vertical rods with concentrated mass.

Abstract

Objective. Longitudinal forced harmonic vibrations of vertical rods with a concentrated mass at the end are considered.Method. The mathematical model of forced kinematically excited longitudinal vibrations is described by a differential equation and boundary conditions arising from the condition of fixing the ends of the rod. The problem will be solved using the finite difference method.Result. With near-resonance oscillation there is a significant difference between the amplitudes and standard deviations. The closeness of β to the natural frequency and the smallness of the value of α make almost all values of the frequencies of seismic action almost resonant. The reliability of the results of problems in deterministic and stochastic formulations has been confirmed. It has been established that the forms and parameters of forced deterministic oscillations significantly depend on the combination of the frequencies of the forcing disturbances with the natural frequencies of the oscillatory system. It has been established that the forms and parameters of forced random oscillations significantly depend on the combination of the spectral density of the random process of disturbances with the discrete spectrum of eigenvalues.Conclusion. It is necessary to expand the scope of research to include other types of vibrations: combinations of longitudinal with transverse, angular, torsional, parametric, etc.

Keywords

• vertical rod
• concentrated mass
• seismic impact
• longitudinal forced harmonic vibrations
• vibration amplitude
• differential equation
• boundary conditions
• finite difference method
• spectral density
• standard deviations