The Approximate Solution of the Equations of Motion of an Airplane Moving in a Vertical Plane

1960 ◽  
Vol 27 (5) ◽  
pp. 394-396
Author(s):  
E. F. Trombley
Keyword(s):  
Approximate Solution ◽  
Equations Of Motion ◽  
Vertical Plane

scholarly journals INVESTIGATION OF THE DYNAMICS OF AN OVERHEAD CRANE LIFTING PROCESS IN A VERTICAL PLANE

Transport ◽  
2005 ◽  
Vol 20 (5) ◽  
pp. 176-180 ◽  
Author(s):  
Marijonas Bogdevičius ◽  
Aleksandr Vika
Keyword(s):  
Dynamic Behaviour ◽  
Equations Of Motion ◽  
Asynchronous Motor ◽  
Linear Equations ◽  
Vertical Plane ◽  
Overhead Crane ◽  
Dynamic Forces ◽  
Planet Gear ◽  
Linear Equations Of Motion ◽  
Non Linear Equations

The paper analyses the dynamic behaviour of supporting structure of an overhead crane during the operation of a hoisting mechanism. The crane is expected to operate with a hook and to carry 50 kN of weight. The electric hoist consists of an asynchronous motor with a magnetic brake, a two‐level planet gear, a load drum and an upper block. Non‐linear equations of motion of a crane hoisting mechanism are derived. Real dynamic forces and their influence on the hoisting crane behaviour are obtained. Numerical results of the crane are derived considering two hoisting regimes during the operation of the hoisting.

A Simple Description of the Motion of a Spherical Pendulum

1969 ◽  
Vol 36 (3) ◽  
pp. 408-411 ◽  
Author(s):  
K. F. Johansen ◽  
T. R. Kane
Keyword(s):  
Approximate Solution ◽  
Equations Of Motion ◽  
Spherical Pendulum ◽  
Simple Description

Hamilton-Jacobi theory is used to obtain an approximate solution of the equations of motion of a spherical pendulum. By reference to this solution, the motion is then described in simple geometric terms.

scholarly journals The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
T. S. Amer
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Relative Equilibrium ◽  
Dynamical Behavior ◽  
Vertical Plane ◽  
The Body ◽  
Physical Parameters ◽  
Pendulum Model

In this paper, we will focus on the dynamical behavior of a rigid body suspended on an elastic spring as a pendulum model with three degrees of freedom. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity. The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the fourth-order Runge-Kutta algorithms through Matlab packages. These solutions are represented graphically in order to describe and discuss the behavior of the body at any instant for different values of the physical parameters of the body. The obtained results have been discussed and compared with some previous published works. Some concluding remarks have been presented at the end of this work. The importance of this work is due to its numerous applications in life such as the vibrations that occur in buildings and structures.

scholarly journals Approximately permanent electronic orbits and the origin of spectral series

1915 ◽  
Vol 91 (626) ◽  
pp. 156-170
Keyword(s):  
Approximate Solution ◽  
Exact Solutions ◽  
Experimental Work ◽  
Close Agreement ◽  
Equations Of Motion ◽  
Electronic Systems ◽  
Spectral Series ◽  
Simplifying Assumptions

The electrodynamical theory which we owe to Lorentz and Larmor provides theoretically a logical and consistent scheme whereby the equations of motion of electronic systems may be formulated. But, unfortunately, even most simple cases lead to equations of such complexity that the attempt to deduce exact solutions must at present be abandoned since there is no mathematical machinery available for the purpose. We have accordingly to make some simplifying assumptions, not strictly true, in order to obtain an approximate solution. In many cases, results are thus obtained which give a very close agreement with observation, and this is so far gratifying. But modern experimental work in radiation makes it clear that the phenomena have not yet been co-ordinated with the electrodynamical theory of electrons. It is reasonable to enquire if this is due to the failure of mathematicians to provide an explanation, whether because the approximations used are not accurate enough, or because the conception of the electronic systems considered is not sufficiently general ?

scholarly journals Mathematical Model of Spatial Motion of the Controlled Parachute-Tether System of the Wind Kite Type

2021 ◽  
Vol 24 (4) ◽  
pp. 17-24
Author(s):  
V.M. Churkin ◽  
T.Yu. Churkina ◽  
A.M. Girin
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Solid Body ◽  
Equations Of Motion ◽  
Vertical Plane ◽  
Spatial Motion ◽  
Mathematical Task ◽  
Spatial Movement ◽  
Elastic Thread ◽  
The Mathematical Model

Mathematical modeling is created for the mathematical task of spatial motion of the controlled parachute-tether system of the “wind kite” type. The mathematical model parachute-tether system consists of a model of the main parachute and a model of the braking parachute. The parachutes are connected by the tether. The model of the main parachute is supposed to be the solid body. This solid body has two planes of symmetry. The braking parachute is the solid body with axial symmetry. The tether model is an absolutely flexible elastic thread. The tether is connected by ideal hinges with the main parachute and braking parachute. The control of the main parachute is carried out by changing the length of the control slings. Changing the length causes deformation of the dome. This is the reason for the change in its aerodynamics. Maneuvering of the main parachute occurs in the vertical plane, when the length of the control slings changes simultaneously. Maneuvering of the main parachute in space is carried out when the length of the control slings changes, when the slings are given a travel difference. The system of dynamic and kinematic equations is designed for calculating the controlled spatial movement of the main parachute, braking parachute and tether. The option exists when the mass of the tether and the forces applied to the tether cannot be neglected. The motion of the tether is represented by the equations of motion of an absolutely flexible elastic thread in projections on the axis of a natural trihedron. The mathematical model is represented by a system of ordinary differential equations and partial differential equations. The problem is solved using various numerical methods. The solution is possible with the help of an integrated numerical and analytical approach as well.

scholarly journals Kinematic support modeling in Sage

2020 ◽  
Vol 28 (2) ◽  
pp. 141-153
Author(s):  
Oleg K. Kroytor ◽  
Mikhail D. Malykh ◽  
Sergei P. Karnilovich
Keyword(s):  
Rigid Body ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Seismic Isolation ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Vertical Plane ◽  
Base Plate ◽  
Euler Method ◽  
Point Of View

The article discusses the kinematic support, which allows reducing the horizontal dynamic effects on the building during earthquakes. The model of a seismic isolation support is considered from the point of view of classical mechanics, that is, we assume that the support is absolutely solid, oscillating in a vertical plane above a fixed horizontal solid plate. This approach allows a more adequate description of the interaction of the support with the soil and the base plate of the building. The paper describes the procedure for reducing the complete system of equations of motion of a massive rigid body on a fixed horizontal perfectly smooth plane to a form suitable for applying the finite difference method and its implementation in the Sage computer algebra system. The numerical calculations by the Euler method for grids with different number of elements are carried out and a mathematical model of the support as a perfectly rigid body in the Sage computer algebra system is implemented. The article presents the intermediate results of numerical experiments performed in Sage and gives a brief analysis (description) of the results.

scholarly journals Approximation Solution for the Zener Impact Theory

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2222
Author(s):  
Ping-Kun Tsai ◽  
Cheng-Han Li ◽  
Chia-Chun Lai ◽  
Ko-Jung Huang ◽  
Ching-Wei Cheng
Keyword(s):  
Approximate Solution ◽  
Inelastic Collision ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Elastic Collision ◽  
Theory Model ◽  
Collision Theory ◽  
Approximation Solution ◽  
Model Equations ◽  
Collision Force

Collisions can be classified as completely elastic or inelastic. Collision mechanics theory has gradually developed from elastic to inelastic collision theories. Based on the Hertz elastic collision contact theory and Zener inelastic collision theory model, we derive and explain the Hertz and Zener collision theory model equations in detail in this study and establish the Zener inelastic collision theory, which is a simple and fast calculation of the approximate solution to the nonlinear differential equations of motion. We propose an approximate formula to obtain the Zener nonlinear differential equation of motion in a simple manner. The approximate solution determines the relevant values of the collision force, material displacement, velocity, and contact time.

CFD Approach to Modelling Hydrodynamic Characteristics of Underwater Glider

2019 ◽  
Vol 2019 (4) ◽  
pp. 32-45
Author(s):  
Kamila Stryczniewicz ◽  
Przemysław Drężek
Keyword(s):  
Flow Behavior ◽  
Equations Of Motion ◽  
Vertical Plane ◽  
Dynamic Modelling ◽  
The Body ◽  
Hydrodynamic Characteristics ◽  
Motion Simulation ◽  
Underwater Glider ◽  
Lift And Drag ◽  
Equations Of Motions

Abstract Autonomous underwater gliders are buoyancy propelled vehicles. Their way of propulsion relies upon changing their buoyancy with internal pumping systems enabling them up and down motions, and their forward gliding motions are generated by hydrodynamic lift forces exerted on a pair of wings attached to a glider hull. In this study lift and drag characteristics of a glider were performed using Computational Fluid Dynamics (CFD) approach and results were compared with the literature. Flow behavior, lift and drag forces distribution at different angles of attack were studied for Reynolds numbers varying around 105 for NACA0012 wing configurations. The variable of the glider was the angle of attack, the velocity was constant. Flow velocity was 0.5 m/s and angle of the body varying from −8° to 8° in steps of 2°. Results from the CFD constituted the basis for the calculation the equations of motions of glider in the vertical plane. Therefore, vehicle motion simulation was achieved through numeric integration of the equations of motion. The equations of motions will be solved in the MatLab software. This work will contribute to dynamic modelling and three-dimensional motion simulation of a torpedo shaped underwater glider.

scholarly journals 5. Note on a Singular Problem in Kinetics

1882 ◽  
Vol 11 ◽  
pp. 173-175
Author(s):  
Tait
Keyword(s):  
Kinetic Energy ◽  
Equations Of Motion ◽  
Vertical Plane ◽  
Point Of View ◽  
Singular Problem ◽  
Mixed Potential ◽  
Physical Point ◽  
Peculiar Form ◽  
Subsequent Motion

The following problem presented itself to me nearly thirty years ago. I cannot find any notice of it in books, though it must have occurred to every one who has studied the oscillations of a balance:—Two equal masses are attached to the ends of a cord passing over a smooth pulley (as in Attwood's machine). One of them is slightly disturbed, in a vertical plane, from its position of equilibrium. Find the nature of the subsequent motion of the system.The interest of this case of small motions is twofold. From the peculiar form of the equations of motion, it is of exceptional mathematical difficulty. This is probably the reason for its not having been given as an example in Kinetics. And from the physical point of view it presents a very beautiful example of excessively slow, but continued, transformation of mixed potential and kinetic energy into kinetic energy alone.

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