Kinematic modeling of flexure hinges and compliant mechanisms

In this paper, the kinematic performance of flexure hinges and compliant mechanisms calculated by conventional modeling techniques are compared. As these exhibit certain drawbacks with regard to control strategies, mainly large number of degrees of freedom or unacceptable errors, a novel modeling approach for flexure hinges is presented. Instead of the entire flexure hinge only its significant regions are modeled by 3-D structural solids. These master patterns are positioned appropriately and connected by rigid constraint conditions to build a compliant mechanism. The resulting model is characterized by considerably fewer degrees of freedom than a full solid model as well as a marginal deviation of the deflection compared to that of pseudo-rigid-body models, 3-D tapered finite beam elements and analytical Timoshenko beam theory. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An improved modeling approach for circular flexure hinges with application to geometry optimization

Within this paper, a modeling approach for flexure hinges based on the Euler-Bernoulli beam theory for beams of variable cross section is investigated in a static analysis. The proposed approach is implemented in a finite beam element routine, for which two different discretizations are discussed. The results are compared to a full scale three dimensional model. It is shown that a circular flexure hinge cannot be modeled accurately with one element. An improved model with three elements across the flexure hinge length is presented which shows excellent accordance with the reference model. A geometry optimization is realized based on the improved, low-DOF model. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Model order reduction methods and their possible application in compliant mechanisms

Model order reduction (MOR) is commonly used to approximate large-scale linear time-invariant dynamical systems. A new feed unit based on a compliant mechanism consisting of flexure hinges can be described by a discrete system of n ordinary differential equations. A projection framework using modal and Krylov subspace techniques is applied to reduce the order of the system to lower computational cost and make the model feasible for control, analysis and optimization. Single flexure hinges are investigated numerical, analytical and experimental and compared to reduced models via modal and tangential Krylov subspace methods regarding the first eigenfrequency. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of a Compliant Mechanism for Positioning in the cm-Range

The kinematics and kinetics of a compliant mechanism are analyzed quasi-statically. The mechanism consists of a parallelkinematic (Biglide type) with the particular feature, that its revolute joints are implemented by flexure hinges. In the considered example the flexure hinges are uniform beams similar to leaf springs. The preliminary considerations with a rigid body system and by linear theory of elasticity result in an initial geometry. In order to predict the large displacement behavior more precisely the theory of Elastica is used. An optimization method is applied to solve the mechanism's state of deformation. It was found that some aspects can be approximated by a pseudo rigid body system, while others, namely the rotation, cannot be rendered. The decreased stiffness in the highly deformed operating state appears as key problem. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Failure analysis of highly predeformed beams used as flexure hinges

The trend to extend the working ranges of flexure hinges implies large deformations during operation. To conduct a failure analysis the total deformation is decomposed into desired deformation and deviations. In particular, a flexure hinge of leaf-spring type is examined. It is modeled by the theory of elastica. The resulting boundary value problem is solved numerically for the static case by Ritz's method. It is discretized into trial functions and their free coefficients are determined from the minimum of potential energy by optimization methods. The crucial point is that the elastic energy stored in the beam is formulated intrinsically, while the potential of external conservative loads is formulated in a space-fixed coordinate system. The well-known special case of buckling of a straight cantilever beam is used for verification. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new Jacobi operational matrix: An application for solving fractional differential equations

In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.     

Laplace transform for the solution of higher order deformation equations arising in the homotopy analysis method

A new analytic approach for solving nonlinear ordinary differential equations with initial conditions is proposed. First, the homotopy analysis method is used to transform a nonlinear differential equation into a system of linear differential equations; then, the Laplace transform method is applied to solve the resulting linear initial value problems; finally, the solutions to the linear initial value problems are employed to form a convergent series solution to the given problem. The main advantage of the new approach is that it provides an effective way to solve the higher order deformation equations arising in the homotopy analysis method.     

Using artificial reaction force to design compliant mechanism with multiple equality displacement constraints

Monolithic compliant mechanisms are elastic workpieces which transmit force and displacement from an input position to an output position. Continuum topology optimization is suitable to generate the optimized topology, shape and size of such compliant mechanisms. The optimization strategy for a single input single output compliant mechanism under volume constraint is known to be best implemented using an optimality criteria or similar mathematical programming method. In this standard form, the method appears unsuitable for the design of compliant mechanisms which are subject to multiple outputs and multiple constraints. Therefore an optimization model that is subject to multiple design constraints is required. With regard to the design problem of compliant mechanisms subject to multiple equality displacement constraints and an area constraint, we here present a unified sensitivity analysis procedure based on artificial reaction forces, in which the key idea is built upon the Lagrange multiplier method. Because the resultant sensitivity expression obtained by this procedure already compromises the effects of all the equality displacement constraints, a simple optimization method, such as the optimality criteria method, can then be used to implement an area constraint. Mesh adaptation and anisotropic filtering method are used to obtain clearly defined monolithic compliant mechanisms without obvious hinges. Numerical examples in 2D and 3D based on linear small deformation analysis are presented to illustrate the success of the method.     

A simple SLP algorithm for solving a class of nonlinear programs

Successive linear programming (SLP) algorithms solve nonlinear optimization problems via a sequence of linear programs. We present an approach for a special class of nonlinear programming problems, which arise in multiperiod coal blending. The class of nonlinear programming problems and the solution approach considered in this paper are quite different from previous work. The algorithm is very simple, easy to apply and can be applied to as large a problem as the linear programming code can handle. The quality of solution, produced by the proposed algorithm, is discussed and the results of some test problems, in the real world environment, are provided.     

A reliable technique for solving the weakly singular second-kind Volterra-type integral equations

In this paper, the Adomian decomposition method and the phenomenon of the self-canceling “noise” terms are used for solving the weakly singular Volterra-type, linear and nonlinear, integral equations. The solution obtained is in the form of a convergent power series with elegantly computable terms. Comparing this scheme with many collocation-type methods that use the nonpolynomial basis shows that the present approach is effective and powerful. Many test modeling problems from mathematical physics, linear and nonlinear, are discussed to illustrate the effectiveness and the performance of the decomposition method.     

Approximate analytic solutions to Radulet equations for the analysis of electromagnetic transients in single-phase power transmission lines

In this work, an analytic development for a transmission line with a corona effect for simulating an electromagnetic transient is presented. The asymptotic solution for the Radulet equations in which a nonlinear term is presented is obtained. The study is carried out for a single-phase transmission line. The electrical parameters for an overhead line are defined and several formulations for their calculation are presented. The frequency dependence of the electrical parameters is considered. In the first part, the linear problem solution is found; the Fourier and Laplace transforms are applied with respect to distance and time respectively. After finding the solution in the Fourier–Laplace domain, which is expressed in terms of a Green’s function integral, an approximate analytical solution is obtained in the distance–time domain by means of asymptotic methods. Finally, the nonlinear solution is found using as a first approach the linear solution. The results obtained show an attenuation in the voltage wave due to the corona effect.     

Accurate analytical solutions to nonlinear oscillators by means of the Hamiltonian approach

The purpose of this paper is to apply the Hamiltonian approach to nonlinear oscillators. The Hamiltonian approach is applied to derive highly accurate analytical expressions for periodic solutions or for approximate formulas of frequency. A conservative oscillator always admits a Hamiltonian invariant, H , which stays unchanged during oscillation. This property is used to obtain approximate frequency–amplitude relationship of a nonlinear oscillator with high accuracy. A trial solution is selected with unknown parameters. Next, the Ritz–He method is used to obtain the unknown parameters. This will yield the approximate analytical solution of the nonlinear ordinary differential equations. In contrast with the traditional methods, the proposed method does not require any small parameter in the equation. Copyright © 2011 John Wiley & Sons, Ltd.     

Wavelet Galerkin schemes for higher order time dependent partial differential equations

Fourth‐order derivatives appearing in different linear and nonlinear transient higher order multidimensional equations modeling several physical problems have always posed computational challenges to widely prevailing numerical approaches such as FDM, FEM, and so forth. In this study, we address the issue effectively using the special features of Daubechies wavelets such as orthogonality, compact support, arbitrary regularity, high‐order vanishing moments, and good localization. An efficient compression strategy is proposed to reduce the computational cost significantly. Implicitly stable backward Euler scheme is used for time marching the evolving solution. A priori error estimates have been derived to prove the convergence of the numerical scheme. The proposed approach is successfully tested on few linear and nonlinear multidimensional PDEs.     

Solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach

The Solid Transportation Problem (STP) arises when bounds are given on three item properties. The Fuzzy Solid Transportation Problem (FSTP) appears when the nature of the data problem is fuzzy. This paper deals with the FSTP in the case in which the fuzziness affects the constraint set, and a fuzzy solution to the problem is required. Moreover, an arbitrary linear or nonlinear objective function is considered. In order to find a fuzzy solution to the problem, a parametric approach is used to obtain an auxiliary Parametric Solid Transportation Problem (PSTP) associated to the original problem. As there are no well-known solution methods proposed in literature to solve effectively the PSTP, in this paper an Evolutionary Algorithm (EA) based solution method is proposed to solve it, which can finally be applied to find a “good” fuzzy solution to the FSTP. Comparisons with another conventional method are presented and the results show the EA based approach to be better as a whole.     

A nonlinear differential delay equation

A procedure reported elsewhere for solution of linear and nonlinear, deterministic or stochastic, delay differential equations developed by the authors as an extension of the first author's methods for nonlinear stochastic differential equations is now applied to a nonlinear delay-differential equation arising in population problems and studied by Kakutani and Markus. Examples involving time-dependent constants and even stochastic coefficients and delays can also be done.     

Nonlinear homogenization of metal-ceramic composites with lamellar microstructure

Using nonlinear homogenization methods for the computation of the material response of metal-ceramic composites with lamellar microstructure is a power approach to do computation less costly in comparison to finite elements modeling. A modified secant homogenization method is utilized in this study for simulation of inelastic behaviors of the composite micro-constituents. A nonlinear homogenization method is based on a linear homogenization scheme for multilayer composites. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Localization analysis of compact invariant sets of multi-dimensional nonlinear systems and symmetrical prolongations

In our paper we study the localization problem of compact invariant sets of nonlinear systems. Methods of a solution of this problem are discussed and a new method is proposed which is based on using symmetrical prolongations and the first-order extremum condition. Our approach is applied to the system modeling the Rayleigh–Bénard convection for which the symmetrical prolongation with the Lorenz system has been constructed.     

Model reduction and clustering techniques for crash simulations

Model reduction in car crash simulations is a fairly new research field. In this paper, a possible workflow is presented: Since nonlinear behavior can occur, parts with linear and nonlinear behavior need to be separated with clustering methods such as k-means or spectral clustering. For the latter, a nonlinear reduction technique such as POD-DEIM needs to be applied. A longitudinal chassis beam of a 2001 Ford Taurus is used to examine the different clustering methods. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linearly constrained global optimization via piecewise-linear approximation

This paper considers the problem of optimizing a continuous nonlinear objective function subject to linear constraints via a piecewise-linear approximation. A systematic approach is proposed, which uses a lattice piecewise-linear model to approximate the nonlinear objective function on a simplicial partition and determines an approximately globally optimal solution by solving a set of standard linear programs. The new approach is applicable to any continuous objective function rather than to separable ones only and could be useful to treat more complex nonlinear problems. A numerical example is given to illustrate the practicability.     

Online identification of Takagi–Sugeno fuzzy models based on self-adaptive hierarchical particle swarm optimization algorithm

This paper presents an approach for online learning of Takagi–Sugeno (T-S) fuzzy models. A novel learning algorithm based on a Hierarchical Particle Swarm Optimization (HPSO) is introduced to automatically extract all fuzzy logic system (FLS)’s parameters of a T–S fuzzy model. During online operation, both the consequent parameters of the T–S fuzzy model and the PSO inertia weight are continually updated when new data becomes available. By applying this concept to the learning algorithm, a new type T–S fuzzy modeling approach is constructed where the proposed HPSO algorithm includes an adaptive procedure and becomes a self-adaptive HPSO (S-AHPSO) algorithm usable in real-time processes. To improve the computational time of the proposed HPSO, particles positions are initialized by using an efficient unsupervised fuzzy clustering algorithm (UFCA). The UFCA combines the K-nearest neighbour and fuzzy C-means methods into a fuzzy modeling method for partitioning of the input–output data and identifying the antecedent parameters of the fuzzy system, enhancing the HPSO’s tuning. The approach is applied to identify the dynamical behavior of the dissolved oxygen concentration in an activated sludge reactor within a wastewater treatment plant. The results show that the proposed approach can identify nonlinear systems satisfactorily, and reveal superior performance of the proposed methods when compared with other state of the art methods. Moreover, the methodologies proposed in this paper can be involved in wider applications in a number of fields such as model predictive control, direct controller design, unsupervised clustering, motion detection, and robotics.