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

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.

Dynamical Analysis of a Cantilever Column with a Tip Mass Subjected to Subtangential Follower Force

2013 ◽  
Vol 427-429 ◽  
pp. 346-349
Author(s):  
Yan Xia Xue ◽  
Zhen Chao Su
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Frequency Equation ◽  
Follower Force ◽  
Dynamical Analysis ◽  
Euler Beam ◽  
Tip Mass

Based on the theory of Bernoulli-Euler beam and d Alembert principle, the differential equation of a cantilever column with a tip mass subjected to a subtangential follower force is built, the solution of the differential equation under the specific boundary conditions is found, frequency equation is formed for computing the system frequencies, several cases of this system is investigated.

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

2011 ◽  
Vol 138-139 ◽  
pp. 3-8 ◽  
Author(s):  
Z.C. Su ◽  
Yan Xia Xue ◽  
Cheng Bin Du
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Critical Force ◽  
Follower Force ◽  
Vertical Force ◽  
Force Parameter ◽  
Force Analysis ◽  
Influence Curves ◽  
Governing Differential Equation ◽  
The Stability

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.

Nonlinear Dynamic Modeling of Elastic Beam Fixed on a Moving Cart and Carrying Lumped Tip Mass Subjected to External Periodic Force

2009 ◽  
Author(s):  
Fadi A. Ghaith
Keyword(s):  
Linear Model ◽  
Equations Of Motion ◽  
Elastic Beam ◽  
Open Loop ◽  
Beam Deflection ◽  
Mode Shapes ◽  
Euler Beam ◽  
Periodic Force ◽  
Assumed Mode Method ◽  
Tip Mass

In the present work, a Bernoulli – Euler beam fixed on a moving cart and carrying lumped tip mass subjected to external periodic force is considered. Such a model could describe the motion of structures like forklift vehicles or ladder cars that carry heavy loads and military airplane wings with storage loads on their span. The nonlinear equations of motion which describe the global motion as well as the vibration motion were derived using Lagrangian approach under the inextensibility condition. In order to investigate the influence of the axial movement of the cart on the response of the system, unconstrained modal analysis has been carried out, and accurate mode shapes of the beam deflection were obtained. The assumed mode method was utilized for approximating the beam elastic deformation based on the single unconstrained mode shapes. Numerical simulation has been carried out to estimate the open-loop response of the nonlinear beam-mass-cart model as well as for the simplified linear model under the influence of the periodic excitation force. Also a comparison study between the responses of the linear and nonlinear models was established. It was shown that the maximum values of the beam tip deflection estimated from the nonlinear model are lower than the corresponding values obtained via the linear model, which reveals the importance of considering nonlinear hardening term in formulating the equations of motion for such system in order to come with more accurate and reliable model.

A two parameter eigenvalue problem

1974 ◽  
Vol 10 (1) ◽  
pp. 39-50
Author(s):  
J.A. Rickard
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Order Differential Equation ◽  
Inviscid Fluid ◽  
Free Oscillations ◽  
Numerical Computations ◽  
Second Order Differential Equation ◽  
Independent Variable ◽  
Two Parameters ◽  
Two Parameter

An ordinary second order differential equation is considered in which the coefficients are dependent on two parameters ω and F as well as the independent variable μ. The equation arises in the study of free oscillations of incompressible inviscid fluid in global shells. An asymptotic technique is presented which estimates the eigenvaiues (that is the values of ω for which the solution is bounded for all |μ| ≤ 1) as functions of F, as F → ∞. The agreement of the results with numerical computations is also discussed.

Static instability of an elastically restrained cantilever under a partial follower force

AIAA Journal ◽  
1985 ◽  
Vol 23 (10) ◽  
pp. 1637-1639 ◽  
Author(s):  
Lech Tomski ◽  
Jacek Przybylski
Keyword(s):  
Follower Force ◽  
Static Instability ◽  
Elastically Restrained

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

1975 ◽  
Vol 17 (4) ◽  
pp. 219-222 ◽  
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.

On the eigencharacteristics of a centrifugally stiffened, visco-elastic beam

2009 ◽  
Vol 223 (8) ◽  
pp. 1767-1775 ◽  
Author(s):  
M Gürgöze ◽  
S Zeren
Keyword(s):  
Elastic Beam ◽  
Computer Time ◽  
Fe Modelling ◽  
Euler Beam ◽  
Fe Method ◽  
Cooling Fan ◽  
Out Of Plane ◽  
Method Of Solution ◽  
Set Up ◽  
Tip Mass

The present study is concerned with the out-of-plane vibrations of a rotating, internally damped (Kelvin—Voigt model) Bernoulli—Euler beam carrying a tip mass, which can be thought of as a simplified model of a helicopter rotor blade or a blade of an auto-cooling fan. The differential eigenvalue problem set up is solved by using the Frobenius method of solution in a power series. The developed characteristic equation is then solved numerically. The simulation results are tabulated for a variety of non-dimensional rotational speeds, tip mass, and internal damping parameters. These are compared with the results of conventional finite element (FE) modelling as well and excellent agreement is obtained. Furthermore, it is seen that the numerical calculations according to the proposed solution method need much less computer time as compared to the conventional FE method.

scholarly journals Transverse Motions of Bernoulli-euler Beam Resting on Elastic Foundation and under Two Concentrated Moving Loads

2020 ◽  
pp. 1-21
Author(s):  
A. Adedowole
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Elastic Foundation ◽  
Structural Members ◽  
Moving Loads ◽  
Structural Element ◽  
Euler Beam ◽  
Partial Differential ◽  
Order Four

Aims/Objectives: The aim is to obtain a closed form solutions of single-dimensional structural element of continuously supported by an elastic foundation. Thereafter, we classify the effects of the space d connecting the loads on the relevant partial differential equations governing the motion of the structural members. The study also analysis circumstances under which resonance occur in the dynamical systems involving structural members. Study Design: The single-dimensional structural element is a partial differential equation of order fourth order place on elastic Winkler foundation. The Bernoulli-Euler beam traversed by two moving loads. Place and Duration of Study: Department of Mathematical Sciences, Adekunle Ajasin University P.M.B. 01, Akungba-Akoko, Nigeria, between May 2019 and September 2019. Methodology: The principal equation of the single -dimensional beam model is governing by partial differential equation of the order four. For the single -dimensional beam problem, the solution techniques are based on the Fourier sine transformation. The governing partial differential equation of the order four was reduced to sequence of second order ordinary differential equations.

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