scholarly journals Closed-form solution for column buckling optimization

2021 ◽  
Author(s):  
Vladimir Kobelev
Keyword(s):  
Closed Form ◽  
Dimensional Analysis ◽  
Optimization Problem ◽  
Closed Form Solution ◽  
Elliptic Functions ◽  
Isoperimetric Inequalities ◽  
Form Solution ◽  
Algebraic Equations ◽  
Closed Form Solutions ◽  
Axial Forces

Abstract An optimization problem for a column, loaded by axial forces, whose direction and value remain constant, is studied in this article. The dimensional analysis introduces the dimensionless mass and rigidity factors, which simplicities the mathematical technique for the optimization problem. With the method of dimensional analysis, the solution of the nonlinear algebraic equations for the Lagrange multiplier is superfluous. The closed-form solutions for Sturm-Liouville and mixed types boundary conditions are derived. The solutions are expressed in terms of the higher transcendental function. The principal results are the closed form solution in terms of the hypergeometric and elliptic functions, the analysis of single- and bimodal regimes, and the exact bounds for the masses of the optimal columns. The proof of isoperimetric inequalities exploits the variational method and the Hölder inequality. The isoperimetric inequalities for Euler’s column are rigorously verified.

scholarly journals Existence Conditions and General Solutions of Closed-form Inverse Kinematics for Revolute Serial Robots

2019 ◽  
Vol 9 (20) ◽  
pp. 4365 ◽  
Author(s):  
Wang Shanda ◽  
Luo Xiao ◽  
Luo Qingsheng ◽  
Han Baoling
Keyword(s):  
Closed Form ◽  
Inverse Kinematics ◽  
Degrees Of Freedom ◽  
Closed Form Solution ◽  
Form Solution ◽  
Algebraic Equations ◽  
Closed Form Solutions ◽  
Serial Robots ◽  
Existence Conditions ◽  
Low Degrees

This study proposes a method for judging the existence of closed-form inverse kinematics solutions based on the Denavit–Hartenberg (DH) model. In this method, serial robots with closed-form solutions are described using three types of sub-problems from the viewpoint of solving algebraic equations. If a serial robot can be described using these three types of sub-problems, i.e., if the inverse kinematics problems can be solved by several basic problems, then there is a closed-form solution. Based on the above method, we design a set of universal closed-form inverse kinematics solving algorithms. Since there is a definite formula solution for the three types of sub-problems, the joint angles can be rapidly determined. In addition, because the DH parameters can directly reflect the linkage of the robot, the judgment of the sub-problems is also quick and accurate. More importantly, the algorithm can be applied to serial robots with low degrees of freedom. This enables the algorithm to not only quickly and accurately solve inverse kinematics problems but also to exhibit high universality. This proposed theory improves the existence conditions for closed-form reverse solutions and further promotes the development of motion control techniques for serial robots.

scholarly journals Closed-Form Solutions to Differential Equations via Differential Evolution

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
L. Mex ◽  
Carlos A. Cruz-Villar ◽  
F. Peñuñuri
Keyword(s):  
Differential Equations ◽  
Differential Evolution ◽  
Closed Form ◽  
Optimization Problem ◽  
Closed Form Solution ◽  
Form Solution ◽  
Closed Form Solutions ◽  
Independent Variables ◽  
Effectiveness And Efficiency ◽  
Derivatives Of

We focus on solving ordinary differential equations using the evolutionary algorithm known as differential evolution (DE). The main purpose is to obtain a closed-form solution to differential equations. To solve the problem at hand, three steps are proposed. First, the problem is stated as an optimization problem where the independent variables areelementaryfunctions. Second, as the domain of DE is real numbers, we propose a grammar that assigns numbers to functions. Third, to avoid truncation and subtractive cancellation errors, to increase the efficiency of the calculation of derivatives, the dual numbers are used to obtain derivatives of functions. Some examples validating the effectiveness and efficiency of our method are presented.

Validation of Nitinol SMA Characteristics Using Finite Element Analysis and Closed Form Solutions

2013 ◽  
Vol 856 ◽  
pp. 147-152
Author(s):  
S.H. Adarsh ◽  
U.S. Mallikarjun
Keyword(s):  
Experimental Data ◽  
Finite Element Analysis ◽  
Finite Element ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Fe Analysis ◽  
Element Analysis ◽  
Complex Behaviour ◽  
Closed Form Solutions

Shape Memory Alloys (SMA) are promising materials for actuation in space applications, because of the relatively large deformations and forces that they offer. However, their complex behaviour and interaction of several physical domains (electrical, thermal and mechanical), the study of SMA behaviour is a challenging field. Present work aims at correlating the Finite Element (FE) analysis of SMA with closed form solutions and experimental data. Though sufficient literature is available on closed form solution of SMA, not much detail is available on the Finite element Analysis. In the present work an attempt is made for characterization of SMA through solving the governing equations by established closed form solution, and finally correlating FE results with these data. Extensive experiments were conducted on 0.3mm diameter NiTinol SMA wire at various temperatures and stress conditions and these results were compared with FE analysis conducted using MSC.Marc. A comparison of results from finite element analysis with the experimental data exhibits fairly good agreement.

scholarly journals Reconstruction of three-dimensional maps based on closed-form solutions of the variational problem of multisensor data registration

2019 ◽  
Vol 484 (6) ◽  
pp. 672-677
Author(s):  
A. V. Vokhmintcev ◽  
A. V. Melnikov ◽  
K. V. Mironov ◽  
V. V. Burlutskiy
Keyword(s):  
Closed Form ◽  
Large Scale ◽  
Three Dimensional ◽  
Dynamic Environment ◽  
Closed Form Solution ◽  
Point Clouds ◽  
Accurate Method ◽  
Form Solution ◽  
Closed Form Solutions ◽  
Reconstruction Methods

A closed-form solution is proposed for the problem of minimizing a functional consisting of two terms measuring mean-square distances for visually associated characteristic points on an image and meansquare distances for point clouds in terms of a point-to-plane metric. An accurate method for reconstructing three-dimensional dynamic environment is presented, and the properties of closed-form solutions are described. The proposed approach improves the accuracy and convergence of reconstruction methods for complex and large-scale scenes.

scholarly journals A New Closed-Form Solution of the Side Abutment Pressure Distribution of Roadway

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Liang Cheng ◽  
Yidong Zhang
Keyword(s):  
Plastic Zone ◽  
Pressure Distribution ◽  
Closed Form ◽  
Abutment Pressure ◽  
Closed Form Solution ◽  
Evaluation Process ◽  
Friction Angle ◽  
Form Solution ◽  
Closed Form Solutions ◽  
Roadway Stability

Instability of coal wall is one of the hot-button and difficult issues in the study of coal mine ground control. The shallow side coal of roadway in the coal measures is usually weak and consequently easy to bring about failure. Hence, the side abutment pressure redistributes and dramatically influences the roadway stability. Since the previous closed-form solutions of the side abutment pressure do not take into account all the necessary parameters which include the properties of the coal and the interface between coal and roof/floor, the roadway height, and the support strength, a mechanical model is established based on the equilibrium of the plastic zone, and a new closed-form solution is derived in this paper. Moreover, a numerical investigation is conducted to validate the accuracy of the closed-form solution. The numerical results of the side abutment pressure distribution are in good agreement with the closed-form solution. Afterwards, a parametric analysis of the width of the plastic zone is carried out, and the results show that the width of the plastic zone is nearly negatively linearly correlated with the friction angle and the cohesion of the coal, the interfacial cohesion, and the support strength. By contrast, it is positively linearly correlated with the roadway height and negatively exponentially correlated with the interfacial friction angle. The results obtained in the present study could be useful for the evaluation process of roadway stability.

Use of Resultants and the Bernshtein Formula in the Dimensional Synthesis of Linkage Components

1994 ◽  
Vol 116 (4) ◽  
pp. 1171-1172 ◽  
Author(s):  
Chuen-Sen Lin ◽  
Bao-Ping Jia
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Polynomial Equations ◽  
Dimensional Synthesis ◽  
Fourth Degree ◽  
Finite Precision ◽  
Closed Form Solutions ◽  
System Of Polynomial Equations ◽  
Resultant Theory

The applications of resultants and the Bernshtein formula for the dimensional synthesis of linkage components for finite precision positions are discussed. The closed-form solutions, which are derived from systems of polynomials in multiple unknowns by applying resultant theory, are in forms of polynomial equations of a single unknown. For the case of two compatibility equations, the closed form solution is a sixth degree solution polynomial. For the case of three compatibility equations, the solution is a fifty-fourth degree solution polynomial. For each case, the Bernshtein formula is applied to calculate the number of solutions of the system of polynomial equations. The calculated numbers of solutions match the degrees of the solution polynomials for both cases.

scholarly journals Comparison of closed-form solutions to experimental magnetic force between two cylindrical magnets

2021 ◽  
pp. 141-146
Author(s):  
Sampart Cheedket ◽  
Chitnarong Sirisathitkul
Keyword(s):  
Magnetic Field ◽  
Closed Form ◽  
Magnetic Force ◽  
Permanent Magnets ◽  
Magnetic Levitation ◽  
Closed Form Solution ◽  
Vertical Tube ◽  
Form Solution ◽  
Closed Form Solutions ◽  
Set Up

The force between permanent magnets implemented in many engineering devices remains an intriguing problem in basic physics. The variation of magnetic force with the distance x between a pair of magnets cannot usually be approximated as x-4 because of the dipole nature and geometry of magnets. In this work, the force between two identical cylindrical magnets is accurately described by a closed-form solution. The analytical model assumes that the magnets are uniformly magnetized along their length. The calculation, based on the magnetic field exerted by one magnet on the other along the direction of their orientation, shows a reduction in the magnetic force with the distance x and a dependence on the size parameters of magnets. To verify the equation, the experiment was set up by placing two cylindrical neodymium iron boron type magnets in a vertical tube. The repulsive force between the identical upper and lower magnets of 2.5 cm in diameter and 7.5 cm in length was measured from the weight on the top of the upper magnet. The resulting separation between the magnets was recorded as x. The forces measured at x=0.004-0.037 m differ from the values calculated using the analytic solution by -0.55 % to -13.60 %. The calculation also gives rise to a practical remnant magnetic field of 1.206 T. When x is much large than the equation of force is approximated as a simple form proportional to 1/x-4. The finding can be directly used in magnetic levitation as well as applied in calculating magnetic fields and forces in other systems incorporating permanent magnets.

scholarly journals Axisymmetric Large Deflection Elastic Analysis of Hollow Annular Membranes under Transverse Uniform Loading

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1770
Author(s):  
Jun-Yi Sun ◽  
Qi Zhang ◽  
Xue Li ◽  
Xiao-Ting He
Keyword(s):  
Closed Form ◽  
Circular Plate ◽  
Large Deflection ◽  
Closed Form Solution ◽  
Circular Ring ◽  
Form Solution ◽  
Elastic Response ◽  
Geometrically Nonlinear ◽  
Closed Form Solutions ◽  
Annular Membrane

The anticipated use of a hollow linearly elastic annular membrane for designing elastic shells has provided an impetus for this paper to investigate the large deflection geometrically nonlinear phenomena of such a hollow linearly elastic annular membrane under transverse uniform loads. The so-called hollow annular membranes differ from the traditional annular membranes available in the literature only in that the former has the inner edge attached to a movable but weightless rigid concentric circular ring while the latter has the inner edge attached to a movable but weightless rigid concentric circular plate. The hollow annular membranes remove the transverse uniform loads distributed on “circular plate” due to the use of “circular ring” and result in a reduction in elastic response. In this paper, the large deflection geometrically nonlinear problem of an initially flat, peripherally fixed, linearly elastic, transversely uniformly loaded hollow annular membrane is formulated, the problem formulated is solved by using power series method, and its closed-form solution is presented for the first time. The convergence and effectiveness of the closed-form solution presented are investigated numerically. A comparison between closed-form solutions for hollow and traditional annular membranes under the same conditions is conducted, to reveal the difference in elastic response, as well as the influence of different closed-form solutions on the anticipated use for designing elastic shells.

Quasi-Steady Axisymmetric Bingham-Plastic Model of Magnetorheological Flow Damper Behavior

Aerospace ◽  
2005 ◽  
Author(s):  
Jin-Hyeong Yoo ◽  
Norman M. Wereley
Keyword(s):  
Closed Form ◽  
Yield Stress ◽  
Dynamic Range ◽  
Closed Form Solution ◽  
Form Solution ◽  
Damping Capacity ◽  
Stress Profile ◽  
Mr Fluid ◽  
Bingham Plastic ◽  
Closed Form Solutions

A typical magnetorheological (MR) flow mode damper consists of a piston attached to a shaft that travels in a tightly fitting hydraulic cylinder. The piston motion makes fluid flow through an annular valve in the MR damper. An electro-magnet applies magnetic field to the MR fluid as it flows through the MR valve, and changes its yield stress. An MR fluid, modeled as a Bingham-plastic material, is characterized by a field dependent yield stress, and a (nearly constant) postyield plastic viscosity. Although the analysis of such an annular MR valve is well understood, a closed form solution for the damping capacity of a damper using such an MR valve has proven to be elusive. Closed form solutions for the velocity and shear stress profile across the annular gap are well known. However, the location of the plug must be computed numerically. As a result, closed form solutions for the dynamic range (ratio of field on to field off damper force) cannot be derived. Instead of this conventional theoretic procedure, an approximated closed form solution for a dampers dynamic range, damping capacity and other key performance metrics is derived. And the approximated solution is used to validate a rectangular duct simplified analysis of MR valves for small gap condition. These approximated equations are derived, and the approximation error is also provided.

Closed Form Solutions for the Motion of Electrically Excited Micro-Cantilever Beams

2006 ◽  
Author(s):  
Seyed Babak Ghaemi Oskouei ◽  
Mohammad Taghi Ahmadian
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Mode Shapes ◽  
Dynamical Response ◽  
Accurate Analysis ◽  
Nonlinear Part ◽  
Closed Form Solutions ◽  
Micro Cantilever ◽  
The Galerkin Method

The differential equation governing the motion of an electrically excited capacitive microcantilever beam is a nonlinear PDE [1]. Accurate analysis about its motion is of great importance in MEMS' dynamical response. In this paper first the nonlinear 4th order 2 point boundary value problem (ODE) governing the static deflection of the system is solved using three methods. 1. The nonlinear part is linearized and its exact solution is obtained. 2. For low applied DC voltages (not near pull-in) the solutin is found using the direct straight forward perturbation analysis. 3. Numerical computer solutions which are used for the previous solution's verifications. The next parts are devoted to the dynamic solution. The nonlinear time variant 4th order PDE governing the dynamic deflection of an electrically excited microbeam is scrutinized. First using the Galerkin Method the mode shapes and the first three mode temporal equations of the linearized equation are found. Considering no damping, using the perturbations method the temporal equations are solved in three states: far from resonance, near 1:1 resonance and near 1:2 resonance. Finally the damped equation is solved using the aforementioned method. In the literature no closed form solution for this problem is presented.

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