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 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 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.

Approximate Response of Beam-Type Resonant Biosensors

2018 ◽  
Author(s):  
Pezhman A. Hassanpour
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Exponential Function ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Impact Force ◽  
Equations Of Motion ◽  
Varying Coefficients ◽  
Time Varying Coefficients ◽  
The Impact

In this paper, the effect of absorption of antigens to the functionalized surface of a biosensor is modeled using a single degree-of-freedom mass-spring-damper system. The change in the mass of the system due to absorption is modeled with an exponential function. The governing equations of motion is derived considering the change in the mass of the system as well as the impact force due to absorption. It has been demonstrated that this equation is a linear second-order ordinary differential equation with time-varying coefficients. The solution of this differential equation is approximated by expanding the exponential function with a Taylor series and applying the method of multiple scales. The advantage of using the method of multiple scales to derive an approximate solution is in the insight it provides on the effect of each parameter on the response of the system. The free vibration response of the biosensor is derived using the approximate solution under different conditions, namely, with and without viscous damping, with and without considering the impact force, and for different binding rates.

scholarly journals Motions of a Constrained Spherical Pendulum in an Arbitrarily Moving Reference Frame: The Automobile Seatbelt Inertial Sensor

1995 ◽  
Vol 2 (3) ◽  
pp. 227-236 ◽  
Author(s):  
A. Peter Allan ◽  
Miles A. Townsend
Keyword(s):  
Numerical Simulation ◽  
Reference Frame ◽  
Inertial Sensor ◽  
Equations Of Motion ◽  
Sensor Design ◽  
Spherical Pendulum ◽  
Crash Data ◽  
Kane’S Method ◽  
Kane's Method ◽  
Automobile Crash

A common automatic seatbelt inertial sensor design, comprised of a constrained spherical pendulum, is modeled to study its motions and possible unintentional release during vehicle emergency maneuvers. The kinematics are derived for the system with the most general inputs: arbitrary pivot motions. The influence of forces due to gravity and constraint torque functions is developed. The equations of motion are then derived using Kane's method. The equations of motion are used in a numerical simulation with both actual and hypothetical automobile crash data.

Nonlinear Oscillation of a Cylinder Containing a Flowing Fluid

1969 ◽  
Vol 91 (4) ◽  
pp. 1147-1155 ◽  
Author(s):  
A. L. Thurman ◽  
C. D. Mote
Keyword(s):  
Approximate Solution ◽  
Fluid Transport ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Practical Interest ◽  
Longitudinal Motion ◽  
Flowing Fluid ◽  
Weakly Nonlinear ◽  
Efficient Calculation

The fundamental and second periods of transverse oscillation of a cylinder containing a flowing fluid are theoretically determined for the approximate solution of two, coupled, nonlinear partial differential equations describing the transverse and longitudinal motion. The calculations indicate that the existence of the fluid transport velocity reduces all cylinder natural periods of oscillation and increases the relative importance of non-linear terms in the equations of motion. Accordingly, in many cases of practical interest the linear analysis is shown to be severely limited in its applicability. Curves are presented that will assist one to estimate both the accuracy of the linear period and the approximate nonlinear period in selected examples. A new approximate solution method is utilized that permits accurate and efficient calculation of the nonlinear period. This method can be applied to the period determination of additional cylindrical models not examined herein; the method appears to be semi-generally applicable to the periodic solution of weakly nonlinear systems.

scholarly journals The Problem of Balancing an Inverted Spherical Pendulum on an Omniwheel Platform

2021 ◽  
Vol 17 (4) ◽  
pp. 507-525
Author(s):  
A. S. Shaura ◽  
◽  
V. A. Tenenev ◽  
E. V. Vetchanin ◽  
◽  
...  
Keyword(s):  
Genetic Algorithm ◽  
Inverted Pendulum ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Hybrid Genetic Algorithm ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Spherical Pendulum ◽  
Pendulum System ◽  
End Point

This paper addresses the problem of balancing an inverted pendulum on an omnidirectional platform in a three-dimensional setting. Equations of motion of the platform – pendulum system in quasi-velocities are constructed. To solve the problem of balancing the pendulum by controlling the motion of the platform, a hybrid genetic algorithm is used. The behavior of the system is investigated under different initial conditions taking into account a necessary stop of the platform or the need for continuation of the motion at the end point of the trajectory. It is shown that the solution of the problem in a two-dimensional setting is a particular case of three-dimensional balancing.

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