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 The gyroscopic forces influence on the oscillations of the rotating shafts

2020 ◽  
pp. 223-231
Author(s):  
Petro Lizunov ◽  
Valentyn Nedin
Keyword(s):  
Differential Equations ◽  
Computer Program ◽  
Block Diagram ◽  
Time Integration ◽  
Numerical Differentiation ◽  
Time Interval ◽  
Gyroscopic Forces ◽  
General Block Diagram ◽  
Inertia Forces ◽  
Transverse Oscillations

The results of numerical investigation of shafts transverse oscillations with account of gyroscopic inertia forces are presented. It is shown what the action and how the gyroscopic forces influence on the transverse oscillations of the shafts during rotation. 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 the system of differential equations of oscillations for every point of mechanical system in order to find the new coordinates of positions of these points in each next point of time t+Dt. 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. Defines the dynamic stability fields and draw the diagrams of found fields. Using this program, the dynamics of a range of objects which are modeled by long elastic rods have been studied. For some objects is shown that on special rotational speeds of shafts with different lengths, in the rotating with shaft coordinate system, the trajectories of center of the section have an ordered character in the form of n-pointed star in time interval from excitation to the start of established circular oscillation with amplitude that harmoniously changes in time. It is noted that such trajectories are fact of the action of gyroscopic inertia forces that arise in rotation.

AUTOMATIC PARALLELIZATION OF ELECTRO-MECHANICAL SYSTEMS GOVERNED BY ODEs

2002 ◽  
Vol 26 (3) ◽  
pp. 347-365
Author(s):  
C.A. Rabbath ◽  
A. Ait El Cadi ◽  
M. Abdonne ◽  
N. Lechevin ◽  
S. Lapierre ◽  
...  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Block Diagram ◽  
Time Integration ◽  
Mechanical Systems ◽  
Automatic Parallelization ◽  
Iteration Step ◽  
Control Laws ◽  
Loop Control ◽  
Novel Method

The paper proposes an effective approach for the automatic parallelization of models of electro-mechanical systems governed by ordinary differential equations. The novel method takes a nominal mathematical model, expressed in block diagram language, and portions in parallel the code to be executed on a set of standard microprocessors. The integrity of the simulations is preserved, the computing resources available are efficiently used, and the simulations are compliant with real-time constraints; that is, the time integration of the ordinary differential equations is performed within restricted time limits at each iteration step. The proposed method is applied to a two-degree-of-freedom revolute joint robotic system that includes an induction motor and two inner-outer loop control laws. Numerical simulations validate the proposed approach.

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.

Dynamic Analysis of Marine Risers With Vortex Excitation

1982 ◽  
Vol 104 (1) ◽  
pp. 14-19 ◽  
Author(s):  
R. P. Nordgren
Keyword(s):  
Finite Element Method ◽  
Time Integration ◽  
Elastic Rods ◽  
Hybrid Finite Element Method ◽  
Marine Risers ◽  
Hybrid Finite Element ◽  
Wake Oscillator ◽  
Transverse Vibrations ◽  
Basic Equations ◽  
Oscillator Equations

The basic equations for nonplanar transverse vibrations of marine risers are derived from the theory of elastic rods. A numerical method is developed for solution of the equations by time integration. Spatial discretization is accomplished by a hybrid finite element method. Vortex excitation is modeled by the coupled wake oscillator proposed by Iwan and Blevins. The vortex oscillator equations are integrated numerically in time along with the riser equations. By way of example, several typical riser problems are analyzed including forced vibration and vortex-induced vibration.

Newmark-Beta Method in Discrete Elastic Rods Algorithm to Avoid Energy Dissipation

2019 ◽  
Vol 86 (8) ◽  
Author(s):  
Weicheng Huang ◽  
Mohammad Khalid Jawed
Keyword(s):  
Energy Dissipation ◽  
Time Integration ◽  
Integration Scheme ◽  
Integration Step ◽  
Elastic Rods ◽  
Computationally Efficient ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Time Integration Scheme

Discrete elastic rods (DER) algorithm presents a computationally efficient means of simulating the geometrically nonlinear dynamics of elastic rods. However, it can suffer from artificial energy loss during the time integration step. Our approach extends the existing DER technique by using a different time integration scheme—we consider a second-order, implicit Newmark-beta method to avoid energy dissipation. This treatment shows better convergence with time step size, specially when the damping forces are negligible and the structure undergoes vibratory motion. Two demonstrations—a cantilever beam and a helical rod hanging under gravity—are used to show the effectiveness of the modified discrete elastic rods simulator.

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