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.

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.*

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 A Study on the Complexity of a New Chaotic Financial System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Yi Liao ◽  
Yiran Zhou ◽  
Fei Xu ◽  
Xiao-Bao Shu
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Financial System ◽  
System Of Differential Equations ◽  
Bifurcation Diagrams ◽  
Complex Dynamic ◽  
System Parameters ◽  
Wide Range

The interaction of elements in a financial system can exhibit complex dynamic behaviours. In this article, we use a system of differential equations to model the evolution of a financial system and study its complexity. Numerical simulations show that the system exhibits a variety of rich dynamic behaviours, including chaos. Bifurcation diagrams show that the system behaves chaotically over a wide range of system parameters.

scholarly journals Visualization of Transition's Scenarios from Harmonic to Chaotic Flexible Nonlinear-elastic Nano Beam's Oscillations

2019 ◽  
Author(s):  
Вадим Крысько ◽  
Vadim Krys'ko ◽  
Ирина Папкова ◽  
Irina Papkova ◽  
Екатерина Крылова ◽  
...  
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Initial Conditions ◽  
Couple Stress Theory ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Physical Nonlinearity ◽  
Variable Load ◽  
Wide Range ◽  
The Cauchy Problem

In this study, a mathematical model of the nonlinear vibrations of a nano-beam under the action of a sign-variable load and an additive white noise was constructed and visualized. The beam is heterogeneous, isotropic, elastic. The physical nonlinearity of the nano-beam was taken into account. The dependence of stress intensity on deformations intensity for aluminum was taken into account. Geometric non-linearity according to Theodore von Karman’s theory was applied. The equations of motion, the boundary and initial conditions of the Hamilton-Ostrogradski principle with regard to the modified couple stress theory were obtained. The system of nonlinear partial differential equations to the Cauchy problem by the method of finite differences was reduced. The Cauchy problem by the finite-difference method in the time coordinate was solved. The Birger variable method was used. Data visualization is carried out from the standpoint of the qualitative theory of differential equations and nonlinear dynamics were carried out. Using a wide range of tools visualization allowed to established that the transition from ordered vibrations to chaos is carried out according to the scenario of Ruelle-Takens-Newhouse. With an increase of the size-dependent parameter, the zone of steady and regular vibrations increases. The transition from regular to chaotic vibrations is accompanied by a tough dynamic loss of stability. The proposed method is universal and can be extended to solve a wide class of various problems of mechanics of shells.

Gyroscopic Drifts Associated With Rotational Vehicle Vibrations

1968 ◽  
Vol 35 (3) ◽  
pp. 553-559 ◽  
Author(s):  
R. A. Wenglarz ◽  
T. R. Kane
Keyword(s):  
Digital Computer ◽  
Small Amplitude ◽  
Equations Of Motion ◽  
Degree Of Freedom ◽  
Arbitrary Orientation ◽  
Moving Vehicle ◽  
System Parameters ◽  
Drift Theory ◽  
Two Degree Of Freedom

Exact dynamical and kinematical equations governing motions of a two-degree-of-freedom gyroscope mounted on a moving vehicle are formulated. From these equations, a set of simpler ones is derived by taking advantage of a number of experimentally verifiable facts, and this set is solved for a vehicle performing oscillations of small amplitude about an axis having an arbitrary orientation. The validity of the resulting drift theory is established by comparing its predictions with those of digital computer solutions of the exact equations of motion, and formulas relating drift rates to system parameters are presented.

SOLVING OPERATOR DIFFERENTIAL EQUATIONS IN TERMS OF THE WEYL-ORDERED POLYNOMIALS

2002 ◽  
Vol 17 (30) ◽  
pp. 2009-2017 ◽  
Author(s):  
ZENG-BING CHEN ◽  
HUAI-XIN LU ◽  
JUN LI
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Systematic Approach ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Adiabatic Invariants ◽  
Formal Aspect ◽  
First Order ◽  
Heisenberg Equations ◽  
Heisenberg Equations Of Motion

A systematic approach to integrate the Heisenberg equations of motion is proposed by using the Weyl-ordered polynomials. The solutions of the Heisenberg equations of motion, i.e. P(t) and Q(t), are expanded as a sum over the Weyl-ordered polynomials Tm,n(P(t),Q(t)) at time t = 0. The coefficients of the expansions satisfy two sets of first-order ordinary differential equations resulting from the Heisenberg equations of motion for time-independent systems. This general approach for time-independent systems is also tractable in obtaining the adiabatic invariants of the time-dependent systems. In this paper, interest is mainly focused on the formal aspect of the approach.

scholarly journals Determination of Differential Equations of Motion and Parameters of an Elastic Internal Combustion Engine Crankshaft

2016 ◽  
Vol 40 (2) ◽  
pp. 83-95
Author(s):  
Aleksandar Milašinović ◽  
Zdravko Milovanović ◽  
Darko Knežević ◽  
Indir Mujanić
Keyword(s):  
Differential Equations ◽  
Internal Combustion Engine ◽  
Combustion Engine ◽  
Equations Of Motion ◽  
Internal Combustion ◽  
Engine Crankshaft

scholarly journals Eigenfrequencies of generally restrained beams

2003 ◽  
Vol 2003 (10) ◽  
pp. 503-516 ◽  
Author(s):  
Ricardo Oscar Grossi ◽  
Carlos Marcelo Albarracín
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Natural Boundary ◽  
Wide Range ◽  
Exact Determination ◽  
Natural Boundary Conditions ◽  
Elastic Constraints ◽  
Range Of Values ◽  
Intermediate Constraints

We deal with the exact determination of eigenfrequencies of a beam with intermediate elastic constraints and generally restrained ends. It is the purpose of this paper to use the calculus of variations to obtain the equations of motion and the natural boundary conditions, and particularly those at the intermediate constraints. Numerical values for the first five natural frequencies are presented in a tabular form for a wide range of values of the restraint parameters. Several particular cases are presented and some of these cases have been compared with those available in the literature.

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