Critical Force in the Buckling of Drill Bits

1979 ◽  
Vol 46 (2) ◽  
pp. 461-462 ◽  
Author(s):  
L. E. Bobisud ◽  
C. O. Christenson
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Fourth Order ◽  
Critical Force ◽  
The Other ◽  
Maximum Force ◽  
System Of Differential Equations ◽  
Linearized Equations ◽  
Drill Bits

We consider a fourth-order nonlinear system of differential equations that describe the slope of a steadily rotating flexible rod, one end of which is clamped, and the other end of which is “hinged.” There is a force directed along the length of the rod. We graph against speed of rotation the maximum force that can be applied before buckling occurs, using linearized equations.

Strongly Nonlinear Vibrations of a Hyperelastic Thin-walled Cylindrical Shell Based on the Modified Lindstedt–Poincaré Method

2019 ◽  
Vol 19 (12) ◽  
pp. 1950160 ◽  
Author(s):  
Jing Zhang ◽  
Jie Xu ◽  
Xuegang Yuan ◽  
Wenzheng Zhang ◽  
Datian Niu
Keyword(s):  
Cylindrical Shell ◽  
Nonlinear System ◽  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Harmonic Excitation ◽  
System Of Differential Equations ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Hardening Behavior ◽  
Thin Walled

Some significant behaviors on strongly nonlinear vibrations are examined for a thin-walled cylindrical shell composed of the classical incompressible Mooney–Rivlin material and subjected to a single radial harmonic excitation at the inner surface. First, with the aid of Donnell’s nonlinear shallow-shell theory, Lagrange’s equations and the assumption of small strains, a nonlinear system of differential equations for the large deflection vibration of a thin-walled shell is obtained. Second, based on the condensation method, the nonlinear system of differential equations is reduced to a strongly nonlinear Duffing equation with a large parameter. Finally, by the appropriate parameter transformation and modified Lindstedt–Poincar[Formula: see text] method, the response curves for the amplitude-frequency and phase-frequency relations are presented. Numerical results demonstrate that the geometrically nonlinear characteristic of the shell undergoing large vibrations shows a hardening behavior, while the nonlinearity of the hyperelastic material should weak the hardening behavior to some extent.

scholarly journals A Modified SIRD Model to Study the Evolution of the COVID-19 Pandemic in Spain

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 723
Author(s):  
Vicente Martínez
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
System Of Differential Equations ◽  
Short Term ◽  
Exponential Expansion ◽  
Harmful Effects

In this paper, we use an SIRD model to analyze the evolution of the COVID-19 pandemic in Spain, caused by a new virus called SARS-CoV-2 from the coronavirus family. This model is governed by a nonlinear system of differential equations that allows us to detect trends in the pandemic and make reliable predictions of the evolution of the infection in the short term. This work shows this evolution of the infection in various changing stages throughout the period of maximum alert in Spain. It also shows a quick adaptation of the parameters that define the disease in several stages. In addition, the model confirms the effectiveness of quarantine to avoid the exponential expansion of the pandemic and reduce the number of deaths. The analysis shows good short-term predictions using the SIRD model, which are useful to influence the evolution of the epidemic and thus carry out actions that help reduce its harmful effects.

Snap-Through Phenomenon and Self-Contact of Spatial Elastica Subjected to Mid-Torque

2015 ◽  
Vol 07 (04) ◽  
pp. 1550057
Author(s):  
Boonchai Phungpaingam ◽  
Lawrence N. Virgin ◽  
Somchai Chucheepsakul
Keyword(s):  
Differential Equations ◽  
Shooting Method ◽  
Contact Point ◽  
Point Load ◽  
The Other ◽  
Arc Length ◽  
System Of Differential Equations ◽  
Kinematic Singularity ◽  
Euler Parameters ◽  
Equilibrium Configurations

This paper presents the snap-through phenomenon and effect of self-contact of the spatial elastica subjected to mid-length torque. One end of the elastica is clamped while the other end is placed in a sleeve joint. The total arc-length of the elastica can be varied by sliding the end through the sleeve joint. At a certain value of total arc-length, the sleeve joint is clamped and an external torque is applied at the mid-length of the elastica. The system of governing differential equations is derived from the equilibrium of an elastica segment and geometric relations of the inextensible elastica. The transformation matrix formulated in terms of Euler parameters is utilized to avoid the kinematic singularity. To display the behavior of the elastica, the system of differential equations needs to be integrated numerically from one end to the other end. The integration is performed so that the boundary conditions and some constraint conditions of the problem are satisfied, i.e., a shooting method is used. The effect of self-contact is taken into account by considering the contact force as a point load applying at contact point. From the results, the snap-through phenomenon, effect of self-contact and equilibrium configurations are highlighted herein.

scholarly journals L-STABLE (4,2)-METHOD OF THE FOURTH ORDER FOR SOLVING STIFF PROBLEMS

2017 ◽  
Vol 17 (8) ◽  
pp. 59-68
Author(s):  
E.A. Novikov
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Maximal Order ◽  
Stiff Problems ◽  
System Of Differential Equations ◽  
Maximum Order ◽  
Order Of Accuracy ◽  
Right Hand ◽  
The Right

(M,k)-methods for solving stiff problems, in which on each step two times the right-hand side of the system of differential equations is calculated are investigated. It is shown that the maximum order of accuracy of the L-stable (m,2)-method is equal to four. (4,2)-method of maximal order is built.

THE COMPOSITION OF THE POSITIVE COLUMN OF A HELIUM GLOW DISCHARGE AT INTERMEDIATE PRESSURES

1979 ◽  
Vol 40 (C7) ◽  
pp. C7-127-C7-128
Author(s):  
F. Dothan ◽  
Yu. M. Kagan
Keyword(s):  
Glow Discharge ◽  
Differential Equations ◽  
Positive Column ◽  
Molecular Ions ◽  
Time Change ◽  
The Other ◽  
System Of Differential Equations ◽  
Plasma Parameters ◽  
Electron Collisions

The concentration of atomic and molecular ions and metastables is investigated for the positive column of a helium glow discharge. Recently (1,2) the system of differential equations for the helium afterglow describing the time change of these plasma parameters was written and solved. In the stationary positive column we can neglect some processes which are importent in the afterglow. On the other side we must take into account processes of excitation and ionization by electron collisions which can be neglected in the afterglow.

PERIODIC ORBITS IN THE NEWTON–LEIPNIK SYSTEM

2002 ◽  
Vol 12 (03) ◽  
pp. 511-523 ◽  
Author(s):  
BENJAMIN A. MARLIN
Keyword(s):  
Periodic Solution ◽  
Nonlinear System ◽  
Differential Equations ◽  
Periodic Orbits ◽  
Closed Orbit ◽  
Numerical Results ◽  
Asymptotically Stable ◽  
System Of Differential Equations ◽  
Promising Candidate ◽  
Closed Orbits

This paper considers an autonomous nonlinear system of differential equations derived in [Leipnik, 1979]. A criterion for the existence of closed orbits in similar systems is presented. Numerical results are made rigorous by the use of interval analytic techniques in establishing the existence of a periodic solution which is not asymptotically stable. The limitations of the method of locating orbits are considered when a promising candidate for a closed orbit is shown not to intersect itself.

Sign in / Sign up

Export Citation Format

Share Document