Contributions to the Determination of the Equations of Motion for Multidegree of Freedom Systems

1971 ◽  
Vol 93 (1) ◽  
pp. 191-195 ◽  
Author(s):  
Desideriu Maros ◽  
Nicolae Orlandea
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
System Of Differential Equations ◽  
Original Contribution ◽  
Deductive Method ◽  
Method Of Solution ◽  
Further Development ◽  
Three Degrees Of Freedom

This paper is a further development of the kinematic problem presented in our 1967 paper [1] in which we have obtained the transmission functions for different orders of plane systems with many degrees of freedom. This paper establishes the corresponding system of differential equations of motion beginning with these functions. The purpose of this paper is to facilitate computer programming. Our study is based on the work of R. Beyer [2, 3] and is the first original addition to his papers. A second original contribution to Beyer’s theories is the deductive method of solution, from general to particular, which we have, incorporated in our work. Beyer concluded that the cases having two or three degrees of freedom can be considered as particular solutions to the results obtained.

scholarly journals Analytical Model of the Two-Mass Above Resonance System of the Eccentric-Pendulum Type Vibration Table

2020 ◽  
Vol 25 (4) ◽  
pp. 116-129
Author(s):  
O.S. Lanets ◽  
V.T. Dmytriv ◽  
V.M. Borovets ◽  
I.A. Derevenko ◽  
I.M. Horodetskyy
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Experimental Studies ◽  
Oscillation Mode ◽  
Oscillatory System ◽  
Forced Oscillations ◽  
System Of Differential Equations ◽  
Lagrange Equations

AbstractThe article deals with atwo-mass above resonant oscillatory system of an eccentric-pendulum type vibrating table. Based on the model of a vibrating oscillatory system with three masses, the system of differential equations of motion of oscillating masses with five degrees of freedom is compiled using generalized Lagrange equations of the second kind. For given values of mechanical parameters of the oscillatory system and initial conditions, the autonomous system of differential equations of motion of oscillating masses is solved by the numerical Rosenbrock method. The results of analytical modelling are verified by experimental studies. The two-mass vibration system with eccentric-pendulum drive in resonant oscillation mode is characterized by an instantaneous start and stop of the drive without prolonged transient modes. Parasitic oscillations of the working body, as a body with distributed mass, are minimal at the frequency of forced oscillations.

Investigation of Train Dynamics in Passing Through Curves Using a Full Model

Joint Rail ◽  
2004 ◽  
Author(s):  
Mohammad Durali ◽  
Mohammad Mehdi Jalili Bahabadi
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Full Model ◽  
Bogie Frame ◽  
Nonlinear Characteristics ◽  
Train Derailment ◽  
Train Dynamics

In this article a train model is developed for studying train derailment in passing through bends. The model is three dimensional, nonlinear, and considers 43 degrees of freedom for each wagon. All nonlinear characteristics of suspension elements as well as flexibilities of wagon body and bogie frame, and the effect of coupler forces are included in the model. The equations of motion for the train are solved numerically for different train conditions. A neural network was constructed as an element in solution loop for determination of wheel-rail contact geometry. Derailment factor was calculated for each case. The results are presented and show the major role of coupler forces on possible train derailment.

A Hybrid Highly Parallelizable Algorithm for the Dynamics of Rigid Body Chain Systems

1999 ◽  
Author(s):  
Shanzhong Duan ◽  
Kurt S. Anderson
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Constraint Forces ◽  
Dynamics Of Rigid Body ◽  
A Chain ◽  
Explicit Determination ◽  
Order Algorithm

Abstract The paper presents a new hybrid parallelizable low order algorithm for modeling the dynamic behavior of multi-rigid-body chain systems. The method is based on cutting certain system interbody joints so that largely independent multibody subchain systems are formed. These subchains interact with one another through associated unknown constraint forces f¯c at the cut joints. The increased parallelism is obtainable through cutting the joints and the explicit determination of associated constraint loads combined with a sequential O(n) procedure. In other words, sequential O(n) procedures are performed to form and solve equations of motion within subchains and parallel strategies are used to form and solve constraint equations between subchains in parallel. The algorithm can easily accommodate the available number of processors while maintaining high efficiency. An O[(n+m)Np+m(1+γ)Np+mγlog2Np](0<γ<1) performance will be achieved with Np processors for a chain system with n degrees of freedom and m constraints due to cutting of interbody joints.

Model Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes

1999 ◽  
Author(s):  
E. Pesheck ◽  
C. Pierre ◽  
S. W. Shaw
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Single Equation ◽  
Rotating Beam ◽  
Model Size ◽  
Nonlinear Normal

Abstract Equations of motion are developed for a rotating beam which is constrained to deform in the transverse (flapping) and axial directions. This process results in two coupled nonlinear partial differential equations which govern the attendant dynamics. These equations may be discretized through utilization of the classical normal modes of the nonrotating system in both the transverse and extensional directions. The resultant system may then be diagonalized to linear order and truncated to N nonlinear ordinary differential equations. Several methods are used to determine the model size necessary to ensure accuracy. Once the model size (N degrees of freedom) has been determined, nonlinear normal mode (NNM) theory is applied to reduce the system to a single equation, or a small set of equations, which accurately represent the dynamics of a mode, or set of modes, of interest. Results are presented which detail the convergence of the discretized model and compare its dynamics with those of the NNM-reduced model, as well as other reduced models. The results indicate a considerable improvement over other common reduction techniques, enabling the capture of many salient response features with the simulation of very few degrees of freedom.

scholarly journals Numerical simulation of oscillations of a nonlinear mechanical system with a spring-loaded viscous friction damper

2019 ◽  
Vol 265 ◽  
pp. 05030
Author(s):  
Viktor Nekhaev ◽  
Viktor Nikolaev ◽  
Marina Safronova
Keyword(s):  
Differential Equations ◽  
Mechanical System ◽  
Degrees Of Freedom ◽  
External Disturbance ◽  
Viscous Friction ◽  
Force Characteristic ◽  
Friction Damper ◽  
System Of Differential Equations ◽  
Nonlinear Force ◽  
Linear Spring

The dynamics of a mechanical system consisting of a parallel-connected main elastic element, an external disturbance compensator having a nonlinear force characteristic, and a viscous friction damper sprung by a linear spring are studied. The resulting system of differential equations describing the behavior of the system has one and a half degrees of freedom and has specific properties depending on the ratio of stiffness of the main spring and the spring suspension of a viscous friction damper. It is established that a single nonlinear system with one and a half degrees of freedom has either one or two harmonics. In the general solution of the system of differential equations, there are always two harmonics in the above-resonance zone, one of which is always equal to the disturbance frequency, and the second one is sufficiently close to the frequency k0. In the linear conservative case and the absence of suspension of the viscous friction damper, the natural frequency of the displacement of the system k0 =14.046 s-1.

XIV.—On General Dynamics:—I. Hamilton's Partial Differential Equations and the Determination of their Complete Integrals

1913 ◽  
Vol 32 ◽  
pp. 164-174
Author(s):  
A. Gray
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Equations ◽  
Equations Of Motion ◽  
Partial Differential ◽  
General Dynamics ◽  
Elementary Manner ◽  
Derivatives Of

The present paper contains the first part of a series of notes on general dynamics which, if it is found worth while, may be continued. In § 1 I have shown how the first Hamiltonian differential equation is led up to in a natural and elementary manner from the canonical equations of motion for the most general case, that in which the time t appears explicitly in the function usually denoted by H. The condition of constancy of energy is therefore not assumed. In § 2 it is proved that the partial derivatives of the complete integral of Hamilton's equation with respect to the constants which enter into the specification of that integral do not vary with the time, so that these derivatives equated to constants are the integral equations of motion of the system.*

The Vibrations of the Engine with Neo-Hookean Suspension

2013 ◽  
Vol 430 ◽  
pp. 53-59 ◽  
Author(s):  
Nicolae Doru Stanescu ◽  
Dinel Popa
Keyword(s):  
Degrees Of Freedom ◽  
Stable Equilibrium ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Harmonic Force ◽  
Small Oscillations ◽  
Excited System ◽  
Beat Phenomenon ◽  
Three Degrees Of Freedom

Our paper realizes a study of the vibrations of an engine excited by a harmonic force and sustained by four identical neo-Hookean springs of negligible masses. The considered model is one with three degrees of freedom (one translation and two rotations) and we obtain for it the equations of motion. Using these equations, we determine for the unexcited system the equilibrium positions and their stability. We also study the small oscillations about the stable equilibrium positions and we find the fundamental eigenpulsations of the system. For the case of the excited system we perform a numerical study considering the situation when the pulsation of the excitation is far away from the eigenpulsations and the situation when the pulsation of the excitation is closed to one eigenpulsation, highlighting the beat phenomenon.

Determination of Gyroscopic Drifts Associated With Angular Support Motions

1970 ◽  
Vol 37 (2) ◽  
pp. 279-286
Author(s):  
R. A. Wenglarz
Keyword(s):  
Differential Equations ◽  
Digital Computer ◽  
Systematic Approach ◽  
Equations Of Motion ◽  
System Parameters ◽  
Constant Rate ◽  
Wide Range ◽  
Fixed Line

A previously proposed systematic approach for the analysis of gyroscopic drifts associated with angular support motions is further developed. For a wide range of support motions, the problem of determination of drifts is reduced to the evaluation of four integrals. The validity of the theory is tested by applying it to a gyroscope experiencing a constant rate about a fixed line and comparing the resulting predictions with those of digital computer solutions of the exact differential equations of motion, and formulas relating steady drifts to system parameters are presented.

scholarly journals Decomposition of the Equations of Motion in the Analysis of Dynamics of a 3-DOF Nonideal System

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Roman Starosta ◽  
Grażyna Sypniewska-Kamińska
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Dynamical Behaviour ◽  
Engineering Systems ◽  
System Operation ◽  
Unbalanced Rotor ◽  
Nonideal System ◽  
Analysis Of Dynamics ◽  
Three Degrees Of Freedom

The dynamic response of a nonlinear system with three degrees of freedom, which is excited by nonideal excitation, is investigated. In the considered system the role of a nonideal source is played by a direct current motor, where the central axis of the rotor is not coincident with the axis of rotation. This translation generates a torque whose magnitude depends on the angular velocity. During the system operation a general coordinate assigned to the nonideal source grows rapidly as a result of rotation. We propose the decomposition of the equations of motion in such a way to extract the solution which is directly related to the rotation of an unbalanced rotor. The remaining part of the solution describes pure oscillation depending on the dynamical behaviour of the whole system. The decomposed equations are solved numerically. The influence of selected system parameters on the rotor vibration is examined. The presented approach can be applied to separate vibration and rotation of motions in many other engineering systems.

Sign in / Sign up

Export Citation Format

Share Document