Optimizing the dynamic parameters of the vibration equation of motion with the jerk effect – part-I

Abstract

In common differential equations of the second order of vibration of the mass and spring system, the internal forces include only the effects of displacement and its first and second derivatives, and if there are higher order derivatives, these effects should be considered. In the type of loading and in the physical structure of mass and spring, the reasons for the existence of this force are presented, and by solving the 3rd order differential equation of motion in the state of free vibration, limit points have been determined. The compliance of the structural response is shown in the 2nd and 3rd order motion equations and the test results With harmonic loading, the response spectrum of structural forces in the 3rd order equation has been compared with the 2nd order equation. When the ratio of the loading frequency to the natural frequency of the structure is equal to 3.67 the jerk force will be equal to the damping force. With the increase of loading frequency in large damping ratios, the jerk force tends to the amplitude of the harmonic loading force and other forces decrease, which indicates the intensity of this force in loading with large frequencies. Therefore, in high-frequency earthquakes, in high-rise structures or structures in non-linear behavior that have a long period of time, with the increase of the damping force, the contribution of the jerk force in the balance of internal forces will increase.