scholarly journals Compliance of a microfibril subjected to shear and normal loads

2008 ◽  
Vol 5 (26) ◽  
pp. 1087-1097 ◽  
Author(s):  
Jingzhou Liu ◽  
Chung-Yuen Hui ◽  
Lulin Shen ◽  
Anand Jagota
Keyword(s):  
Nonlinear Response ◽  
Closed Form Solution ◽  
Elliptic Functions ◽  
Surface Region ◽  
Form Solution ◽  
Compliant Structure ◽  
Strongly Nonlinear ◽  
Indentation Experiment ◽  
Backing Layer ◽  
Nonlinear Deformations

Many synthetic bio-inspired adhesives consist of an array of microfibrils attached to an elastic backing layer, resulting in a tough and compliant structure. The surface region is usually subjected to large and nonlinear deformations during contact with an indenter, leading to a strongly nonlinear response. In order to understand the compliance of the fibrillar regions, we examine the nonlinear deformation of a single fibril subjected to a combination of shear and normal loads. An exact closed-form solution is obtained using elliptic functions. The prediction of our model compares well with the results of an indentation experiment.

Nonlinear Isolator Optimization Using RMS Cost Function

2004 ◽  
Author(s):  
A. Narimani ◽  
M. F. Golnaraghi
Keyword(s):  
Closed Form Solution ◽  
Optimization Method ◽  
Relative Displacement ◽  
Form Solution ◽  
Jump Phenomenon ◽  
Nonlinear Parameters ◽  
Strongly Nonlinear ◽  
Damping Characteristics ◽  
Stiffness And Damping ◽  
Boundary Limits

In this paper using a modified averaging method the frequency response of a general nonlinear isolator is obtained. Stiffness and damping characteristics are considered cubic functions of displacement and velocity through the isolator. Analytical results are compared with those obtained by numerical integration in order to validate the closed form solution for strongly nonlinear isolator. While increasing the nonlinearity in the system improves the response of the isolator, stability and jump avoidance conditions set boundary limits for the parameters. The effects of nonlinear parameters to avoid jump phenomenon are discussed in detail. The set of parameters where the system behaves regularly are found and the nonlinear isolator is optimized based on RMS optimization method. Using this method the RMS function of absolute acceleration of the sprung mass is minimized versus the RMS function of relative displacement.

scholarly journals Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs

2007 ◽  
Vol 2007 ◽  
pp. 1-25
Author(s):  
M. P. Markakis
Keyword(s):  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Form Solution ◽  
Nonlinear Odes ◽  
Analytical Technique ◽  
Closed Form Solutions ◽  
Strongly Nonlinear ◽  
Parameter Values

We establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the original equation is expressed. This treatment is applied to another nonlinear equation, subjected to the same general class as that of Gambier, by constructing step by step an appropriate analytical technique. The developed procedure yields a general exact closed form solution of this equation, valid for specific values of the parameters involved and containing two arbitrary (free) parameters evaluated by the relevant initial conditions. We finally verify this technique by applying it to two specific sets of parameter values of the equation under consideration.

scholarly journals A PAIR OF PERFECTLY CONDUCTING DISKS IN AN EXTERNAL FIELD

2015 ◽  
Vol 20 (2) ◽  
pp. 273-288 ◽  
Author(s):  
Natalia Rylko
Keyword(s):  
Closed Form Solution ◽  
Elliptic Functions ◽  
Convergent Series ◽  
Fiber Composites ◽  
Form Solution ◽  
Local Fields ◽  
Trigonometric Functions ◽  
Method Of Images ◽  
Asymptotic Investigation ◽  
Asymptotic Formulae

A pair of non-overlapping perfectly conducting equal disks embedded in a two-dimensional background was investigated by the classic method of images, by Poincar´e series, by use of the bipolar coordinates and by the elliptic functions in the previous works. In particular, successive application of the inversions with respect to circles were applied to obtain the field in the form of a series. For closely placed disks, the previous methods yield slowly convergent series. In this paper, we study the local fields around closely placed disks by the elliptic functions. The problem of small gap is completely investigated since the obtained closed form solution admits a precise asymptotic investigation in terms of the trigonometric functions when the gap between the disks tends to zero. The exact and asymptotic formulae are extended to the case when a prescribed singularity is located in the gap. This extends applications of structural approximations to estimations of the local fields in densely packed fiber composites in various external fields.

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.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

Plane Wave Resonance in the Tire Air Cavity as a Vehicle Interior Noise Source

1995 ◽  
Vol 23 (1) ◽  
pp. 2-10 ◽  
Author(s):  
J. K. Thompson
Keyword(s):  
Closed Form Solution ◽  
Form Solution ◽  
Interior Noise ◽  
Cavity Resonance ◽  
Test Results ◽  
Vehicle Interior ◽  
Air Cavity ◽  
Vehicle Interior Noise ◽  
Wave Resonance ◽  
Resonance Frequencies

Abstract Vehicle interior noise is the result of numerous sources of excitation. One source involving tire pavement interaction is the tire air cavity resonance and the forcing it provides to the vehicle spindle: This paper applies fundamental principles combined with experimental verification to describe the tire cavity resonance. A closed form solution is developed to predict the resonance frequencies from geometric data. Tire test results are used to examine the accuracy of predictions of undeflected and deflected tire resonances. Errors in predicted and actual frequencies are shown to be less than 2%. The nature of the forcing this resonance as it applies to the vehicle spindle is also examined.

Capacity Analysis for Correlated Multi-Hop MIMO Channels under Colored Noise

2015 ◽  
Vol 1 ◽  
pp. 41
Author(s):  
Nguyen N. Tran ◽  
Ha X. Nguyen
Keyword(s):  
Computational Complexity ◽  
Mutual Information ◽  
Closed Form Solution ◽  
Optimal Solution ◽  
Form Solution ◽  
Maximization Problem ◽  
Capacity Analysis ◽  
Mimo Channels ◽  
Average Mutual Information ◽  
Channel Output

A capacity analysis for generally correlated wireless multi-hop multi-input multi-output (MIMO) channels is presented in this paper. The channel at each hop is spatially correlated, the source symbols are mutually correlated, and the additive Gaussian noises are colored. First, by invoking Karush-Kuhn-Tucker condition for the optimality of convex programming, we derive the optimal source symbol covariance for the maximum mutual information between the channel input and the channel output when having the full knowledge of channel at the transmitter. Secondly, we formulate the average mutual information maximization problem when having only the channel statistics at the transmitter. Since this problem is almost impossible to be solved analytically, the numerical interior-point-method is employed to obtain the optimal solution. Furthermore, to reduce the computational complexity, an asymptotic closed-form solution is derived by maximizing an upper bound of the objective function. Simulation results show that the average mutual information obtained by the asymptotic design is very closed to that obtained by the optimal design, while saving a huge computational complexity.

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