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.

Development of the flow induced by a piston moving impulsively in a dusty gas

1985 ◽  
Vol 397 (1813) ◽  
pp. 295-309 ◽  
Keyword(s):  
Shock Wave ◽  
Equations Of Motion ◽  
Wave Structure ◽  
Elastic Collision ◽  
Dust Particles ◽  
Dusty Gas ◽  
Impulsive Motion ◽  
Dependent Change ◽  
Piston Speed ◽  
Time Dependent Change

The flow resulting from the impulsive motion of a piston moving at constant speed in a dusty gas is studied analytically and numerically. An idealized equilibrium-gas approximation is used to discuss the effects of piston speed and mass concentration of dust particles on the eventually formed shock wave. The detailed time-dependent change of the flow structure is studied by solving the equations of motion numerically. A partly dispersed shock-wave structure is formed at a high piston speed and a fully dispersed shock at a low piston speed. Two situations are considered, where the particles striking the piston experience an elastic collision, or where they stick to its surface. Significant effects on the flow produced by particles that reflect from the piston surface are discussed and clarified.

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.

The Effects of Flare and Overhangs on the Motions of a Yacht in Head Seas

1993 ◽  
Author(s):  
John C. Kuhn ◽  
Eric C. Schlageter
Keyword(s):  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Order Effects ◽  
Third Order ◽  
Numerical Work ◽  
Strip Theory ◽  
Hull Forms ◽  
The Time Domain ◽  
The Impact

The coupled heave and pitch motions of hull forms with flare and overhangs are examined numerically. The presence of flare and overhangs is numerically modelled with nonlinear hydrostatic and Froude-Krylov forces based on integrals over the instantaneous wetted surface. Forces due to radiation and diffraction are computed with a linear strip-theory. These forces are combined in two coupled nonlinear differential equations of motion that are solved in the time domain with a fourth-order Runge-Kutta integration method. An assessment of the impact of flare and overhangs on motions is obtained by comparing these nonlinear solutions with solutions of the traditional linear equations of motion, which do not contain forces due to flare and overhangs. For an example based on an International America's Cup Class yacht design, it is found that the nonlinear heave and pitch motions are smaller than the linear motions. This is primarily due to reduced first-order response components, which are coupled with nonlinear response components. Comparisons of these results with towing tank data demonstrate that the nonlinear procedure improves prediction quality relative to linear results. In support of this numerical work, the hydrostatic and Froude­Krylov force integrals are expanded in Taylor series with respect to wave elevation. These results indicate how hydrostatic and Froude-Krylov forces change with changing flare and overhang angles, revealing that sectional slope has second and third-order effects on forces while sectional curvature and overhang angles produce third-order effects.

Optimization of Nonlinear Unbalanced Flexible Rotating Shaft Passing Through Critical Speeds

2021 ◽  
Author(s):  
Sadegh Amirzadegan ◽  
Mohammad Rokn-Abadi ◽  
R. D. Firouz-Abadi
Keyword(s):  
Degrees Of Freedom ◽  
Nonlinear Oscillations ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Angular Acceleration ◽  
Beam Theory ◽  
Rotor System ◽  
Rotating Shaft ◽  
Critical Speeds ◽  
Applied Torque

This work studies the nonlinear oscillations of an elastic rotating shaft with acceleration to pass through the critical speeds. A mathematical model incorporating the Von-Karman higher-order deformations in bending is developed to investigate the nonlinear dynamics of rotors. A flexible shaft on flexible bearings with springs and dampers is considered as rotor system for this work. The shaft is modeled as a beam and the Euler–Bernoulli beam theory is applied. The kinetic and strain energies of the rotor system are derived and Lagrange method is then applied to obtain the coupled nonlinear differential equations of motion for 6 degrees of freedom. In order to solve these equations numerically, the finite element method (FEM) is used. Furthermore, for different bearing properties, rotor responses are examined and curves of passing through critical speeds with angular acceleration due to applied torque are plotted. Then the optimal values of bearing stiffness and damping are calculated to achieve the minimum vibration amplitude, which causes to pass easier through critical speeds. It is concluded that the value of damping and stiffness of bearing change the rotor critical speeds and also significantly affect the dynamic behavior of the rotor system. These effects are also presented graphically and discussed.

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 ?

Nonlinear Dynamic Modeling and Simulation of an Insect-Like Flapping Wing

2014 ◽  
Vol 555 ◽  
pp. 3-10 ◽  
Author(s):  
Afshin Banazadeh ◽  
Neda Taymourtash
Keyword(s):  
Modeling And Simulation ◽  
Reference Frames ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Added Mass ◽  
Open Loop ◽  
Flapping Wing ◽  
Loop Dynamics ◽  
Three Degree Of Freedom ◽  
Nonlinear Dynamic Modeling

The main objective of this paper is to present the modeling and simulation of open loop dynamics of a rigid body insect-like flapping wing. The most important aerodynamic mechanisms that explain the nature of the flapping flight, including added mass, rotational lift and delayed stall, are modeled. Wing flapping kinematics is described using appropriate reference frames and three degree of freedom for each wing with respect to the insect body. In order to simulate nonlinear differential equations of motion, 6DOF model of the insect-like flapping wing is developed, followed by an evaluation of the simulation results in hover condition.

Torquewhirl—A Theory to Explain Nonsynchronous Whirling Failures of Rotors With High-Load Torque

1978 ◽  
Vol 100 (2) ◽  
pp. 235-240
Author(s):  
J. M. Vance
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
High Power ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Rotating Machinery ◽  
High Load ◽  
Driving Mechanism ◽  
Load Torque ◽  
Driving Torque

Numerous unexplained failures of rotating machinery by nonsynchronous shaft whirling point to a possible driving mechanism or source of energy not identified by previously existing theory. A majority of these failures have been in machines characterized by overhung disks (or disks located close to one end of a bearing span) and/or high power and load torque. This paper gives exact solutions to the nonlinear differential equations of motion for a rotor having both of these characteristics and shows that high ratios of driving torque to damping can produce nonsynchronous whirling with destructively large amplitudes. Solutions are given for two cases: (1) viscous load torque and damping, and (2) load torque and damping proportional to the second power of velocity (aerodynamic case). Criteria are given for avoiding the torquewhirl condition.

Stability of the High-Speed Journal Bearing Under Steady Load: 1—The Incompressible Film

1962 ◽  
Vol 84 (3) ◽  
pp. 351-357 ◽  
Author(s):  
M. M. Reddi ◽  
P. R. Trumpler
Keyword(s):  
High Speed ◽  
Journal Bearing ◽  
Orbital Motion ◽  
Hydrodynamic Force ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Static Equilibrium ◽  
The Stability ◽  
External Loads ◽  
Steady Load

The phenomenon of oil-film whirl in bearings subjected to steady external loads is analyzed. The journal, assumed to be a particle mass, is subjected to the action of two forces; namely, the external load acting on the bearing and the hydrodynamic force developed in the fluid film. The resulting equations of motion for a full-film bearing and a 180-deg partial-film bearing are developed as pairs of second-order nonlinear differential equations. In evaluating the hydrodynamic force, the contribution of the shear stress on the journal surface is found to be negligible for the full-film bearing, whereas for the partial-film bearing it is found to be significant at small attitude values. The equations of motion are linearized and the coefficients of the resulting characteristic equations are studied for the stability of the static-equilibrium positions. The full-film bearing is found to have no stable static-equilibrium position, whereas the 180-deg partial-film bearing is found to have stable static-equilibrium positions under certain parametric conditions. The equations of motion for the full-film bearing are integrated numerically on a digital computer. The results show that the journal center, depending on the parametric conditions, acquired either an orbital motion or a dynamical path of increasing attitude terminating in bearing failure.

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