scholarly journals NUMERICAL DIFFERENTIATION OF BEND FORMS OF THE LONG ELASTIC RODS

2021 ◽  
pp. 70-75
Author(s):  
Petro Lizunov ◽  
Valentyn Nedin
Keyword(s):  
Time Integration ◽  
Exact Result ◽  
Numerical Differentiation ◽  
Spline Functions ◽  
Approximation Technique ◽  
Graphic User Interface ◽  
Elastic Rods ◽  
Elastic Line ◽  
Time Integration Method ◽  
Considerable Length

The technique of numerical differentiation of the bend forms of long elastic rods is presented. This technique is based on search for new bend forms of the rod by solving the equations of oscillations with using the time integration method and the polynomial spline-functions that are being described the current bend form. In it, the spline-functions are found by current bend form approximation where each of the found functions is responsible to certain point of rod elastic line and describes the position of nearby points. Using the described approximation technique with subsequent numerical differentiation, the dependences of the derivatives on an arbitrary bend form of the rod with a length that is equal to 100 m are shown. To confirm the reliability, the results of numerical differentiation of the bend forms of the elastic rods described by given functions are presented and the numerical results obtained using the proposed method are compared with the results of analytical differentiation of the original functions. The graphs of values derivatives dependence to rod length are drawn and tables with numerical values of differentiation results are shown. It is concluded that the considered technique of numerical differentiation of rods bend forms allows to do the research of dynamics of rod systems. It gives the exact result of differentiation, provides the continuity and smoothness of all four derivatives functions of spline that are being described the bend form with considerable length. Described technique was realized in a computer program with graphic user interface. Program allows to monitor for dynamics of the oscillatory motion of the modeled system in real-time by calculating and drawing the current band forms of the rotating rod during the oscillation.

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.

scholarly journals An Improved Time Integration Method Based on Galerkin Weak Form with Controllable Numerical Dissipation

2021 ◽  
Vol 11 (4) ◽  
pp. 1932
Author(s):  
Weixuan Wang ◽  
Qinyan Xing ◽  
Qinghao Yang
Keyword(s):  
High Frequency ◽  
Integration Method ◽  
Time Integration ◽  
Low Frequency ◽  
Weak Form ◽  
High Accuracy ◽  
Numerical Dissipation ◽  
Frequency Range ◽  
Time Integration Method ◽  
Numerical Damping

Based on the newly proposed generalized Galerkin weak form (GGW) method, a two-step time integration method with controllable numerical dissipation is presented. In the first sub-step, the GGW method is used, and in the second sub-step, a new parameter is introduced by using the idea of a trapezoidal integral. According to the numerical analysis, it can be concluded that this method is unconditionally stable and its numerical damping is controllable with the change in introduced parameters. Compared with the GGW method, this two-step scheme avoids the fast numerical dissipation in a low-frequency range. To highlight the performance of the proposed method, some numerical problems are presented and illustrated which show that this method possesses superior accuracy, stability and efficiency compared with conventional trapezoidal rule, the Wilson method, and the Bathe method. High accuracy in a low-frequency range and controllable numerical dissipation in a high-frequency range are both the merits of the method.

scholarly journals Impact of difference between explicit and implicit second-order time integration schemes on isotropic/anisotropic steady incompressible turbulence field

2021 ◽  
Vol 2090 (1) ◽  
pp. 012145
Author(s):  
Ryuma Honda ◽  
Hiroki Suzuki ◽  
Shinsuke Mochizuki
Keyword(s):  
Kinetic Energy ◽  
Energy Conservation ◽  
Integration Method ◽  
Time Integration ◽  
Second Order ◽  
Implicit Time ◽  
Explicit Time Integration ◽  
Time Integration Method ◽  
Implicit Time Integration ◽  
Time Integration Methods

Abstract This study presents the impact of the difference between the implicit and explicit time integration methods on a steady turbulent flow field. In contrast to the explicit time integration method, the implicit time integration method may produce significant kinetic energy conservation error because the widely used spatial difference method for discretizing the governing equations is explicit with respect to time. In this study, the second-order Crank-Nicolson method is used as the implicit time integration method, and the fourth-order Runge-Kutta, second-order Runge-Kutta and second-order Adams-Bashforth methods are used as explicit time integration methods. In the present study, both isotropic and anisotropic steady turbulent fields are analyzed with two values of the Reynolds number. The turbulent kinetic energy in the steady turbulent field is hardly affected by the kinetic energy conservation error. The rms values of static pressure fluctuation are significantly sensitive to the kinetic energy conservation error. These results are examined by varying the time increment value. These results are also discussed by visualizing the large scale turbulent vortex structure.

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