Critical Force Analysis of a Cantilever Column Subjected to a Subtangential Follower Force and a Vertical Force

The stability of a cantilever column subjected to a subtangential follower force and a vertical force is discussed for investigating the effects of these factors on the critical force. The governing differential equation of the system and the corresponding boundary conditions are established, and the exact solution is found out by integrating the differential equation. Based on the exact solution, the effects of the parameters relating to the subtangential follower force and the vertical force on the critical force are analyzed, and discussions for these results are performed， the influence curves of the subtangential follower force parameter and the vertical force parameter to the critical force are plotted. The results show that the parameter of a subtangential follower force can be bigger than 1/2, even equal to 1.0, with taking into account of the effect of the vertical force.

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Static Instability of a Restrained Cantilever Column with a Tip Mass Subjected to a Subtangential Follower Force

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.345.341 ◽

2013 ◽

Vol 345 ◽

pp. 341-344

Author(s):

Zhen Chao Su ◽

Yan Xia Xue

Keyword(s):

Differential Equation ◽

Mass Parameter ◽

Follower Force ◽

Force Parameter ◽

Euler Beam ◽

Numerical Computations ◽

Divergence Instability ◽

Static Instability ◽

Tip Mass

Based on the theory of Bernoulli-Euler beam, the differential equation of a restrained cantilever column with a tip mass subjected to a subtangential follower force is constructed, the solution of the differential equation is found, and the existence of regions of divergence instability of the system is discussed. The influence of the follower force parameter η, the tip mass parameter β and an end elastic end support on the divergence instability of the column is investigated. Several numerical computations of some cases have completed.

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Calculation of wooden beams on the stability of a flat bending shape enhancement

MATEC Web of Conferences ◽

10.1051/matecconf/201819601003 ◽

2018 ◽

Vol 196 ◽

pp. 01003 ◽

Cited By ~ 9

Author(s):

Anton Chepurnenko ◽

Vera Ulianskaya ◽

Serdar Yazyev ◽

Ivan Zotov

Keyword(s):

Differential Equation ◽

Cross Section ◽

Stability Problem ◽

Secular Equation ◽

Rectangular Cross Section ◽

Center Of Gravity ◽

Critical Force ◽

Distributed Load ◽

Matlab Package ◽

The Stability

Flat bending stability problem of constant rectangular cross section wooden beam, loaded by a distributed load is considered. Differential equation is provided for the cases when load is located not in the center of gravity. The solution of the equation is performed numerically by the method of finite differences. For the case of applying a load at the center of gravity, the problem reduces to a generalized secular equation. In other cases, the iterative algorithm developed by the authors is implemented, in the Matlab package. A relationship between the value of the critical force and the position of the load application point is obtained. A linear approximating function is selected for this dependence.

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The Instability of a Cantilever on an Elastic Foundation under the Influence of a Follower Force

Journal of Mechanical Engineering Science ◽

10.1243/jmes_jour_1975_017_032_02 ◽

1975 ◽

Vol 17 (4) ◽

pp. 219-222 ◽

Cited By ~ 5

Author(s):

I. F. A. Wahed

Keyword(s):

Differential Equation ◽

Elastic Foundation ◽

Critical Frequency ◽

Critical Force ◽

Follower Force ◽

Damping Coefficient ◽

Viscous Damping ◽

Lateral Vibration ◽

Galerkin’S Method ◽

Instability Boundary

The instability of a uniform cantilever compressed by a follower force at its free tip is investigated. The cantilever is supported on an elastic foundation and subjected to external viscous damping. The differential equation of lateral vibration of the cantilever is solved simply by Galerkin's method and the instability boundary is determined by applying Routh's criterion. It is found that the cantilever becomes unstable by flutter and that the critical force and the critical frequency depend on both damping coefficient and foundation modulus. Only with no damping is the critical force independent of foundation modulus, a phenomenon reported by other investigators.

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Buckling of Locally Loaded Isotropic, Orthotropic, and Composite Cylindrical Shells

Journal of Applied Mechanics ◽

10.1115/1.3173693 ◽

1988 ◽

Vol 55 (2) ◽

pp. 425-429

Author(s):

Wei Xiao ◽

Shun Cheng

Keyword(s):

Differential Equation ◽

Critical Loads ◽

Cylindrical Shells ◽

External Pressure ◽

Local Loading ◽

Eighth Order ◽

Governing Differential Equation ◽

The Stability ◽

First Time ◽

Composite Cylindrical Shells

This paper incorporates an analysis of the stability of orthotropic or isotropic cylindrical shells subjected to external pressure applied over all or part of their surfaces. An eighth-order governing equation for buckling of orthotropic, isotropic, and composite cylindrical shells is deduced. This governing differential equation can facilitate the analysis and enable us to resolve the buckling problem. The formulas and results, deduced for the first time in this paper, may be readily applied in determining critical loads for local loading of orthotropic, isotropic, and composite cylindrical shells.

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Critical Force Analysis of Thin-Walled Symmetrical Open-Section Beams

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.827.283 ◽

2016 ◽

Vol 827 ◽

pp. 283-286

Author(s):

Diana Šimić Penava ◽

Maja Baniček

Keyword(s):

Cross Section ◽

Cross Sections ◽

Local Buckling ◽

Fem Analysis ◽

Critical Force ◽

The Finite Element Method ◽

Force Analysis ◽

Standard Specimens ◽

Thin Walled ◽

The Stability

This paper analyzes critical forces and stability of steel thin-walled C-cross-section beams without lateral restraints. Mechanical properties of the rods material are determined by testing standard specimens in a laboratory. Based on the obtained data, the stability analysis of rods is carried out and critical forces are determined: analytically by using the theory of thin-walled rods, numerically by using the finite element method (FEM), and experimentally by testing the C-cross-section beams. The analysis of critical forces and stability shows that the calculation according to the theory of thin-walled rods does not take the effect of local buckling into account, and that the resulting critical global forces do not correspond to the actual behaviour of the rod. The FEM analysis and experimental test show that the simplifications, which have been introduced into the theory of thin-walled rods with open cross-sections, significantly affect final results of the level of the critical force.

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Hyers-Ulam Stability of Nonhomogeneous Linear Differential Equations of Second Order

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2009/576852 ◽

2009 ◽

Vol 2009 ◽

pp. 1-7 ◽

Cited By ~ 17

Author(s):

Yongjin Li ◽

Yan Shen

Keyword(s):

Differential Equation ◽

Exact Solution ◽

Approximate Solution ◽

Differential Equations ◽

Second Order ◽

Linear Differential Equations ◽

Ulam Stability ◽

Linear Differential ◽

The Stability

The aim of this paper is to prove the stability in the sense of Hyers-Ulam of differential equation of second ordery′′+p(x)y′+q(x)y+r(x)=0. That is, iffis an approximate solution of the equationy′′+p(x)y′+q(x)y+r(x)=0, then there exists an exact solution of the equation near tof.

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Stability of Cracked Fluid-Conveying Pipeline on Elastic Foundation under Distributed Follower Forces

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.1065-1069.2076 ◽

2014 ◽

Vol 1065-1069 ◽

pp. 2076-2079

Author(s):

Ye Zhou Sheng ◽

Chang Qing Guo ◽

Wei Bin Hong

Keyword(s):

Differential Equation ◽

Elastic Foundation ◽

Bending Moment ◽

Follower Force ◽

Crack Location ◽

Characteristic Values ◽

Transfer Method ◽

Distributed Follower Force ◽

The Stability ◽

Fluid Conveying Pipes

The differential equation of fluid-conveying pipes considering distributed follower force and elastic foundation is established. The equation is discreted and solved by Galerkin method and the frequency characteristic values are solved by bending moment transfer method. The effects of crack location and elastic foundation stiffness to the form of instability of the pipes under the distributed follower force are analyzed. Results show that the elastic foundation stiffness can enforce the stability of the pipes effectively, and the effects are more obvious when the crack location is closer to the middle of the pipe.

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AN APPLICATION OF DIFFERENTIAL TRANSFORMATION TO STABILITY ANALYSIS OF HEAVY COLUMNS

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455406001988 ◽

2006 ◽

Vol 06 (03) ◽

pp. 317-332 ◽

Cited By ~ 19

Author(s):

Y. H. CHAI ◽

C. M. WANG

Keyword(s):

Differential Equation ◽

Transformation Method ◽

Differential Transformation Method ◽

Recursive Equation ◽

Transformation Technique ◽

Critical Buckling Load ◽

Differential Transformation ◽

Governing Differential Equation ◽

Support Conditions ◽

The Stability

This paper uses a recently developed technique, known as the differential transformation, to determine the critical buckling load of axially compressed heavy columns of various support conditions. In solving the problem, it is shown that the differential transformation technique converts the governing differential equation into an algebraic recursive equation, which must be solved together with the differential transformation of the boundary conditions. Although a fairly large number of terms are required for convergence of the solution, the differential transformation method is nonetheless efficient and fairly easy to implement. The method is also shown to be very accurate when compared with a known analytical solution. The stability of heavy columns is further examined using approximate formulae currently available in the literature. In this case, the differential transformation method offers a reference for assessing the accuracy of the approximate buckling formulae.

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Difference Schemes for Nonlinear BVPS on the Semiaxis

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2007-0002 ◽

2007 ◽

Vol 7 (1) ◽

pp. 25-47 ◽

Cited By ~ 4

Author(s):

I.P. Gavrilyuk ◽

M. Hermann ◽

M.V. Kutniv ◽

V.L. Makarov

Keyword(s):

Differential Equation ◽

Exact Solution ◽

Boundary Value Problem ◽

Difference Scheme ◽

Adaptive Algorithm ◽

Order Differential Equation ◽

Constructive Method ◽

Numerical Examples ◽

Exact Difference ◽

The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

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