scholarly journals Optimal shape of a column with clamped-elastically supported ends positioned on elastic foundation

2015 ◽  
Vol 42 (3) ◽  
pp. 191-200 ◽  
Author(s):  
Branislava Novakovic
Keyword(s):  
Optimality Conditions ◽  
Elastic Foundation ◽  
Compressive Force ◽  
First Integral ◽  
Optimal Shape ◽  
Cross Sectional ◽  
Bimodal Optimization ◽  
Elastic Column ◽  
Elastically Supported ◽  
Linear Boundary Value Problem

We determine optimal shape of an elastic column positioned on elastic foundation of Winkler type. The Euler-Bernoulli model of beam is considered. The column is loaded by a compressive force and has one clamped end and the other elastically supported end. In deriving the optimality conditions, the Pontryagin?s principle was used. The optimality conditions for the case of bimodal optimization are derived. Optimal cross-sectional area is obtained from the solution of a non-linear boundary value problem. A first integral (Hamiltonian) is used to monitor accuracy of integration. This system is solved by using standard Math CAD procedure. New numerical results are obtained.

scholarly journals Bimodal optimization with constraints: Critical value of the constraint and post-critical configurations

2011 ◽  
Vol 38 (2) ◽  
pp. 107-124
Author(s):  
Teodor Atanackovic ◽  
Alexander Seyraniany
Keyword(s):  
Total Energy ◽  
Critical Value ◽  
Optimal Shape ◽  
Cross Sectional Area ◽  
Point Of View ◽  
Sectional Area ◽  
Cross Sectional ◽  
Critical Configurations ◽  
Bimodal Optimization ◽  
Elastic Column

By using a method based on Pontryagin?s principle, formulated in [13], and [14] we study optimal shape of an elastic column with constraints on the minimal value of the cross-sectional area. We determine the critical value of the minimal cross-sectional area separating bi from unimodal optimization. Also we study the post-critical shape of optimally shaped rod and find the preferred configuration of the bifurcating solutions from the point of view of minimal total energy.

scholarly journals Optimal Shape and First Integrals for Inverted Compressed Column

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 334
Author(s):  
Enes Kacapor ◽  
Teodor M. Atanackovic ◽  
Cemal Dolicanin
Keyword(s):  
Variational Principle ◽  
First Integrals ◽  
Arbitrary Cross Section ◽  
Optimal Shape ◽  
Equilibrium Equations ◽  
Concentrated Force ◽  
Cross Sectional ◽  
Nonlinear Equilibrium ◽  
Elastic Column ◽  
Special Case

We study optimal shape of an inverted elastic column with concentrated force at the end and in the gravitational field. We generalize earlier results on this problem in two directions. First we prove a theorem on the bifurcation of nonlinear equilibrium equations for arbitrary cross-section column. Secondly we determine the cross-sectional area for the compressed column in the optimal way. Variational principle is constructed for the equations determining the optimal shape and two new first integrals are constructed that are used to check numerical integration. Next, we apply the Noether’s theorem and determine transformation groups that leave variational principle Gauge invariant. The classical Lagrange problem follows as a special case. Several numerical examples are presented.

Optimal Shape of a Rotating Rod With Unsymmetrical Boundary Conditions

2007 ◽  
Vol 74 (6) ◽  
pp. 1234-1238 ◽  
Author(s):  
Teodor M. Atanackovic
Keyword(s):  
Maximum Principle ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Optimal Shape ◽  
Elastic Rods ◽  
Sectional Area ◽  
Governing Equations ◽  
Cross Sectional ◽  
Bimodal Optimization ◽  
Multiplicity Of An Eigenvalue

Governing equations of a compressed rotating rod with clamped–elastically clamped (hinged with a torsional spring) boundary conditions is derived. It is shown that the multiplicity of an eigenvalue of this system can be at most equal to two. The optimality conditions, via Pontryagin’s maximum principle, are derived in the case of bimodal optimization. When these conditions are used the problem of determining the optimal cross-sectional area function is reduced to the solution of a nonlinear boundary value problem. The problem treated here generalizes our earlier results presented in Atanackovic, 1997, Stability Theory of Elastic Rods, World Scientific, River Edge, NJ. The optimal shape of a rod is determined by numerical integration for several values of parameters.

On the Optimal Shape of a Rotating Rod

2001 ◽  
Vol 68 (6) ◽  
pp. 860-864 ◽  
Author(s):  
T. M. Atanackovic
Keyword(s):  
Maximum Principle ◽  
Boundary Value Problem ◽  
Numerical Integration ◽  
Nonlinear Boundary ◽  
Variational Principles ◽  
Boundary Value ◽  
First Integral ◽  
Optimal Shape ◽  
Sectional Area ◽  
Cross Sectional

By using Pontryagin’s maximum principle we determine the shape of the lightest rotating rod, stable against buckling. It is shown that the cross-sectional area function is determined from the solution of a nonlinear boundary value problem. Three variational principles for this boundary value problem are formulated and a first integral is constructed. The optimal shape of a rod is determined by numerical integration.

APPLICATION OF PONTRYAGIN'S PRINCIPLE TO BIMODAL OPTIMIZATION OF NANO RODS

2012 ◽  
Vol 12 (03) ◽  
pp. 1250012 ◽  
Author(s):  
TEODOR M. ATANACKOVIC ◽  
BRANISLAVA N. NOVAKOVIC ◽  
ZORA VRCELJ
Keyword(s):  
Maximum Principle ◽  
Characteristic Length ◽  
Optimization Procedure ◽  
Optimal Shape ◽  
Cross Sectional Area ◽  
Sectional Area ◽  
Elastic Rod ◽  
Cross Sectional ◽  
Bimodal Optimization ◽  
Nano Rods

By using the Pontryagin's maximum principle, we determine optimal shape of a nonlocal elastic rod clamped at both ends. In the optimization procedure, we imposed restriction on the minimal value of the cross-sectional area. We showed that the optimization may be both unimodal and bimodal depending on the value of the restrictions and the value of characteristic length. Several concrete examples are treated in detail and the increase in buckling capacity is determined.

FLUTTER SUPPRESSION OF AN ELASTICALLY SUPPORTED PLATE WITH ELECTRO-RHEOLOGICAL FLUID CORE UNDER YAWED SUPERSONIC FLOWS

2013 ◽  
Vol 13 (01) ◽  
pp. 1250073 ◽  
Author(s):  
SEYYED M. HASHEMINEJAD ◽  
M. NEZAMI ◽  
M. E. ARYAEE PANAH
Keyword(s):  
Elastic Foundation ◽  
Sandwich Plate ◽  
Sliding Mode ◽  
Plate Theory ◽  
System Response ◽  
First Order ◽  
Fluid Core ◽  
Pasternak Elastic Foundation ◽  
Elastically Supported ◽  
Er Fluid

This paper investigates the active control of the supersonic flutter motion of an elastically supported rectangular sandwich plate, which has a tunable electrorheological (ER) fluid core and rests on a Winkler–Pasternak elastic foundation, subjected to an arbitrary flow of various yaw angles. The classical thin plate theory is adopted. The ER fluid core is modeled as a first order Kelvin–Voigt material, and the quasi-steady first order supersonic piston theory is employed for the aerodynamic loading. The generalized Fourier expansions in conjunction with Galerkin method are employed to formulate the governing equations in the state-space domain. The critical dynamic pressures at which unstable panel oscillations occur are obtained for a square sandwich plate, with or without an interacting soft/stiff elastic foundation, for selected applied electric field strengths and flow yaw angles. The Runge–Kutta method is then used to calculate the open-loop aeroelastic response of the system in various basic loading configurations. Subsequently, a sliding mode control (SMC) synthesis is set up to actively suppress the closed loop system response in yawed supersonic flight conditions with imposed excitations. The results demonstrate the performance, effectiveness, and insensitivity with respect to the spillover of the proposed SMC-based control system.

scholarly journals Free oscillations of thin-walled bimetallic pipelines of large diameter on an elastic foundation

2018 ◽  
Vol 193 ◽  
pp. 02027
Author(s):  
Vladimir Sokolov ◽  
Igor Razov ◽  
Evgeniy Koynov
Keyword(s):  
Elastic Foundation ◽  
Nuclear Power ◽  
Large Diameter ◽  
Compressive Force ◽  
Free Oscillations ◽  
Working Pressure ◽  
Momentum Theory ◽  
Internal Working ◽  
Nonlinear Version ◽  
Thin Walled

In the article, solutions are obtained for a thin-walled bimetallic pipeline. Solutions are obtained, and the frequencies of free oscillations are investigated taking into account the internal working pressure, the longitudinal compressive force, and the elastic foundation. The solutions were obtained on the basis of a geometrically nonlinear version of the semi-momentum theory of cylindrical shells of the middle bend. The proposed calculations can find application in the nuclear power industry, aviation, and the petrochemical industry.

Impact of a Mass on a Damped Elastically Supported Beam

1948 ◽  
Vol 15 (2) ◽  
pp. 125-136
Author(s):  
W. H. Hoppmann
Keyword(s):  
Differential Equation ◽  
Elastic Foundation ◽  
Forced Vibration ◽  
Numerical Solutions ◽  
Coefficient Of Restitution ◽  
Tabular Form ◽  
Single Degree Of Freedom ◽  
Simply Supported ◽  
Single Degree ◽  
Elastically Supported

Abstract In this paper a study is made of the problem of the central impact of a mass on a simply supported beam on an elastic foundation with considerations of internal and external damping. The differential equation for the forced vibration of the beam is developed. It is solved for the case in which the force is a function of time and is concentrated at the center of the beam. Formulas are obtained for the deflections. An expression is developed for the coefficient of restitution which is essential in determining the deflections and the strains. Criteria are devised for determining the cases in which the beam may be considered as a single-degree-of-freedom system when damping and an elastic foundation are considered. The importance of these criteria is discussed. A numerical example illustrating the theory developed in the paper is worked out in detail. Results of computations for several numerical solutions are given in tabular form.

Passive Control of Vibration of Elastically Supported Beams Subjected to Moving Loads

2005 ◽  
Author(s):  
D. Younesian ◽  
E. Esmailzadeh ◽  
M. H. Kargarnovin
Keyword(s):  
High Speed ◽  
Passive Control ◽  
Vibration Suppression ◽  
Damping Ratio ◽  
Equations Of Motion ◽  
Compressive Force ◽  
Moving Loads ◽  
Axial Compressive Force ◽  
Before And After ◽  
Elastically Supported

Vibration suppression of elastically supported beams subjected to moving loads is investigated in this work. For a Timoshenko beam with an arbitrary number of elastic supports, subjected to a constant axial compressive force, and having a tuned mass damper (TMD) installed at the mid-span, the equations of motion are derived and using the Galerkin approach the solution is sought. The optimum values of the frequency and damping ratio are determined both analytically and numerically and presented as some design curves directly applicable in the TMD design for bridge structures. To show the efficiency of the designed TMD, computer simulation for two real bridges, subjected to a S.K.S Japanese high-speed train, is carried out and the results obtained are compared for before and after the installation of the TMD system.

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