Effect Of A Non-linear Elastic Foundation On The Mode Shapes In Stability And Vibration Problems Of Uniform Columns/beams

1993 ◽  
Vol 160 (2) ◽  
pp. 369-371 ◽  
Author(s):  
K.Kanaka Raju ◽  
G.Venkateswara Rao
Keyword(s):  
Elastic Foundation ◽  
Mode Shapes ◽  
Linear Elastic ◽  
Non Linear ◽  
Vibration Problems

NON-LINEAR VIBRATIONS OF A SLIGHTLY CURVED BEAM RESTING ON A NON-LINEAR ELASTIC FOUNDATION

1998 ◽  
Vol 212 (2) ◽  
pp. 295-309 ◽  
Author(s):  
H.R. Öz ◽  
M. Pakdemirli ◽  
E. Özkaya ◽  
M. Yilmaz
Keyword(s):  
Elastic Foundation ◽  
Curved Beam ◽  
Linear Elastic ◽  
Non Linear ◽  
Linear Vibrations

scholarly journals Dynamic Buckling of a Clamped Finite Column Resting on a Non – linear Elastic Foundation

2019 ◽  
pp. 1-47
Author(s):  
E. Julius, Bassey ◽  
M. Anthony, Ette ◽  
U. Joy, Chukwuchekwa ◽  
C. Atulegwu, Osuji
Keyword(s):  
Elastic Foundation ◽  
Governing Equation ◽  
Asymptotic Expansions ◽  
Perturbation Technique ◽  
Dynamic Buckling ◽  
Buckling Load ◽  
Linear Elastic ◽  
Non Linear ◽  
Relevant Variables ◽  
Independent Parameters

The analysis of the dynamic buckling of a clamped finite imperfect viscously damped column lying on a quadratic-cubic elastic foundation using the methods of asymptotic and perturbation technique is presented. The proposed governing equation contains two small independent parameters (δ and ϵ) which are used in asymptotic expansions of the relevant variables. The results of the analysis show that the dynamic buckling load of column decreases with its imperfections as well as with the increase in damping. The results obtained are strictly asymptotic and therefore valid as the parameters δ and ϵ become increasingly small relative to unity.

scholarly journals Natural Frequency and Mode Shapes of Exponential Tapered AFG Beams on Elastic Foundation

2016 ◽  
Vol 9 ◽  
pp. 9-25 ◽  
Author(s):  
Hareram Lohar ◽  
Anirban Mitra ◽  
Sarmila Sahoo
Keyword(s):  
Elastic Foundation ◽  
Static Problem ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Dimensionless Frequency ◽  
Energy Principle ◽  
Various Boundary Conditions ◽  
Non Linear ◽  
Beams On Elastic Foundation ◽  
Minimum Potential Energy Principle

A displacement based semi-analytical method is utilized to study non-linear free vibration and mode shapes of an exponential tapered axially functionally graded (AFG) beam resting on an elastic foundation. In the present study geometric nonlinearity induced through large displacement is taken care of by non-linear strain-displacement relations. The beam is considered to be slender to neglect the rotary inertia and shear deformation effects. In the present paper at first the static problem is solved through an iterative scheme using a relaxation parameter and later on the subsequent dynamic analysis is carried out as a standard eigen value problem. Energy principles are used for the formulation of both the problems. The static problem is solved by using minimum potential energy principle whereas in case of dynamic problem Hamilton’s principle is employed. The free vibrational frequencies are tabulated for exponential taper profile subject to various boundary conditions and foundation stiffness. The dynamic behaviour of the system is presented in the form of backbone curves in dimensionless frequency-amplitude plane and in some particular case the mode shape results are furnished.

scholarly journals Oscillations of a Beam on a Non-Linear Elastic Foundation under Periodic Loads

2006 ◽  
Vol 13 (4-5) ◽  
pp. 273-284 ◽  
Author(s):  
Donald Mark Santee ◽  
Paulo Batista Gonçalves
Keyword(s):  
Mathematical Model ◽  
Elastic Foundation ◽  
Period Doubling ◽  
Algebraic Expression ◽  
Geometrical Parameters ◽  
Flexible Beam ◽  
Algebraic Relation ◽  
Structural Element ◽  
Linear Elastic ◽  
Non Linear

The complexity of the response of a beam resting on a nonlinear elastic foundation makes the design of this structural element rather challenging. Particularly because, apparently, there is no algebraic relation for its load bearing capacity as a function of the problem parameters. Such an algebraic relation would be desirable for design purposes. Our aim is to obtain this relation explicitly. Initially, a mathematical model of a flexible beam resting on a non-linear elastic foundation is presented, and its non-linear vibrations and instabilities are investigated using several numerical methods. At a second stage, a parametric study is carried out, using analytical and semi-analytical perturbation methods. So, the influence of the various physical and geometrical parameters of the mathematical model on the non-linear response of the beam is evaluated, in particular, the relation between the natural frequency and the vibration amplitude and the first period doubling and saddle-node bifurcations. These two instability phenomena are the two basic mechanisms associated with the loss of stability of the beam. Finally Melnikov's method is used to determine an algebraic expression for the boundary that separates a safe from an unsafe region in the force parameters space. It is shown that this can be used as a basis for a reliable engineering design criterion.

Deflection of Non-Uniform Beams Resting on a Non-Linear Elastic Foundation Using (GDQM)

2017 ◽  
pp. 52-56 ◽  
Author(s):  
Ramzy M. Abumandour ◽  
Islam M. Eldesoky ◽  
Mohamed A. Safan ◽  
R. M. Rizk-Allah ◽  
Fathi A. Abdelmgeed
Keyword(s):  
Elastic Foundation ◽  
Linear Elastic ◽  
Non Linear
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