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.

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.

Response variations of a cantilever beam–tip mass system with nonlinear and linearized boundary conditions

2018 ◽  
Vol 25 (3) ◽  
pp. 485-496 ◽  
Author(s):  
Vamsi C. Meesala ◽  
Muhammad R. Hajj
Keyword(s):  
Boundary Conditions ◽  
Cantilever Beam ◽  
Nonlinear Boundary ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Boundary Conditions ◽  
Distributed Parameter ◽  
Governing Equations ◽  
Mass System ◽  
Tip Mass

The distributed parameter governing equations of a cantilever beam with a tip mass subjected to principal parametric excitation are developed using a generalized Hamilton's principle. Using a Galerkin's discretization scheme, the discretized equation for the first mode is developed for simpler representation assuming linear and nonlinear boundary conditions. The discretized governing equation considering the nonlinear boundary conditions assumes a simpler form. We solve the distributed parameter and discretized equations separately using the method of multiple scales. Through comparison with the direct approach, we show that accounting for the nonlinear boundary conditions boundary conditions is important for accurate prediction in terms of type of bifurcation and response amplitude.

scholarly journals Bifurcation from infinity for reaction–diffusion equations under nonlinear boundary conditions

2017 ◽  
Vol 147 (3) ◽  
pp. 649-671
Author(s):  
Nsoki Mavinga ◽  
Rosa Pardo
Keyword(s):  
Maximum Principle ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Existence Of Solutions ◽  
Reaction Diffusion ◽  
Nonlinear Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Asymptotically Linear ◽  
Bifurcation From Infinity

We consider reaction–diffusion equations under nonlinear boundary conditions where the nonlinearities are asymptotically linear at infinity and depend on a parameter. We prove that, as the parameter crosses some critical values, a resonance-type phenomenon provides solutions that bifurcate from infinity. We characterize the bifurcated branches when they are sub- or supercritical. We obtain both Landesman–Lazer-type conditions that guarantee the existence of solutions in the resonant case and an anti-maximum principle.

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.

Optimal Shape and Nonhomogeneity of a Timoshenko Beam for Maximum Load

1979 ◽  
Vol 23 (04) ◽  
pp. 229-234
Author(s):  
Sarp Adali ◽  
Ibrahim Sadek
Keyword(s):  
Optimal Design ◽  
Young’S Modulus ◽  
Timoshenko Beam ◽  
Young's Modulus ◽  
Maximum Load ◽  
Optimal Shape ◽  
Graphical Form ◽  
Sectional Area ◽  
Cross Sectional ◽  
Closed Form Solutions

The best possible distributions of Young's modulus or the cross-sectional area or both are determined explicitly for a cantilevered Timoshenko beam which, for a given volume and end deflection, carries the maximum possible load at its free end. Closed-form solutions are given for the optimal height, width and/or Young's modulus functions. Numerical results are presented in graphical form. It is found that the inclusion of shear deformation decreases the efficiency of the optimal design, and that optimization with respect to Young's modulus in addition to shape increases the efficiency considerably in comparison with optimization of the shape only.

scholarly journals ALGORITHM FOR CONTROLLING THE SPECTRUM OF EIGENFREQUENCIES AND VIBRATION MODES BY CHARDING THE GEOMETRIC PARAMETERS OF MAST SESTEMS

2021 ◽  
pp. 6-18
Author(s):  
Vladimir Grinyov ◽  
Vitaliy Vynogradov
Keyword(s):  
Sensitivity Analysis ◽  
Finite Element ◽  
Maximum Principle ◽  
Optimality Conditions ◽  
Natural Frequency ◽  
Natural Frequencies ◽  
Sectional Area ◽  
Cross Sectional ◽  
The Maximum Principle ◽  
Natural Frequency Spectrum

The article considers a model of a mast with six levels of fastening of cables. The main attention in the work is considered to the methods of control of the natural frequency spectrum, due to the use of methods of sensitivity analysis and optimization. The above task is achieved by varying the cross-sectional area of the pipes - racks. Automation of computational processes is provided by programming the built-in module in the Revit program. For more convenient and faster control of the natural frequency spectrum, the algorithm described above was written in a free add-on for Revit - Dynamo. With the help of so-called nodes, an application was created that took data from the depicted 3D model Revit and performed calculations. This allows you to easily use optimality conditions similar to the maximum principle. The sensitivity analysis for the first and second own is carried out in the work. The mechanism of their management within the limits of the investigated model is shown. The relations in the case of the problem of finding the natural frequency extremum with a given number are given, provided that the total amount of varied bands is fixed. The numerical control algorithm is based on the necessary optimality conditions in the form of the maximum principle for rod models. A variant of varying the area of the belts along the height of the mast is proposed. The sensitivity analysis for the first and second natural frequencies is carried out and its use for construction of effective computational process is shown. Based on the results of the work, a working software algorithm was created for fast analysis of mast oscillations on extensions. Graphs of zones of possible change of the first and second frequencies are resulted. The distribution of the cross-sectional area for frequencies is shown. To compare the results of natural frequency calculations on other calculation models, the first and second natural frequencies of bending oscillations were calculated by the finite element method in the SCAD complex. The errors for the points of the curves (constant in the height of the mast area of the belts) do not exceed 10%. It should be noted that the consideration of optimization problems of the above type on the basis of finite element models is quite difficult; for them it is not possible to formulate the necessary conditions of optimality similar to the principle of maximum.

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