On the time integration of parabolic differential equations

1982 ◽  
pp. 157-168 ◽  
Author(s):  
P. J. van der Houwen
Keyword(s):  
Differential Equations ◽  
Time Integration ◽  
Parabolic Differential Equations

scholarly journals THE PARAMETRIC OSCILLATIONS OF ROTATING ELASTIC RODS UNDER THE ACTION OF THE PERIODIC AXIAL FORCES

2020 ◽  
pp. 56-64
Author(s):  
Petro Lizunov ◽  
Valentyn Nedin
Keyword(s):  
Differential Equations ◽  
Computer Program ◽  
Block Diagram ◽  
Time Integration ◽  
Numerical Differentiation ◽  
Oscillatory Motion ◽  
Elastic Rods ◽  
Axial Forces ◽  
General Block Diagram ◽  
Rotating Rods

The paper presents the results of numerical investigation of the periodic axial forces’ influence on the transverse oscillations of long rotating rods. The gyroscopic inertia forces are taken to account and space oscillating process of rotating rods is considered with account of geometric nonlinearity. The study has been done with computer program with a graphical interface that is developed by authors. The process of numerical solution of the differential equations of oscillations of rotating rods using the method of numerical differentiation of rod’s bend forms by polynomial spline-functions and the Houbolt time integration method is described. A general block diagram of the algorithm is shown. This algorithm describes the process of repeated (cyclical) solving of the system of differential equations of oscillations for every point of mechanical system in order to find the new coordinates of the positions of these points in each next point of time t+∆t. The computer program in which the shown algorithm is realized allows to monitor for the behavior of moving computer model, which demonstrates the process of oscillatory motion in rotation. Moreover, the program draws the graphics of oscillations and changes of angular speeds and accelerations in different coordinate systems. Using this program, the dynamics of a range of objects which are modeled by long elastic rods have been studied. For investigated objects is shown that on various rotational speeds and beat frequencies the oscillatory motion of the rods occurs with different character of behavior. On certain speeds with different frequencies of axial load the oscillations have definite periodicity and occur with beats of amplitude which are the result of the periodic axial force action.

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