Mathematical modelling of chip thickness in micro-end- milling: A Fourier modelling

This study developed a model of undeformed chip thickness in micro-end-milling for the use in estimating cutting constants based on measured cutting forces. The proposed estimation method is based upon the invertibility of the average milling force model. In this paper, chip thickness in micro-end-milling was estimated by summing the thicknesses of the conventional chip component and the additional chip component. Thickness was then expressed in terms of Fourier series. The analyses showed that the fast convergence of Fourier series gives the Fourier chip thickness model sufficient accuracy when using only five terms of the truncated Fourier series for common micro-end-milling processes. The Fourier coefficients can be expressed in terms of the ratio of feed per tooth to cutter radius for different numbers of cutter teeth. The accuracy and conciseness of the chip thickness model enables the modelling of average cutting force in a closed form, which can be applied to identify the cutting constants. Cutting force experiments verify that the model prediction agrees very well with the experimental results.     

Oscillations of Carbon Brushes in Hand‐held Electrical Tools

A series‐universal motor is usually used in hand‐held electrical tools, such as cutting or grinding machines. The corresponding two carbon brushes are preloaded by springs to maintain a proper contact necessary to reduce their wear and erosion. The oscillations of carbon brushes are coupled with the rotor motion. A model including carbon brushes and the main parts of tool rotor is described. By using the method of multibody systems the equations of motion are derived. The one‐sided contact forces and the electrical forces between commutator and brushes are mainly responsible for the nonlinear oscillations of brushes. The contour of contact surface is one of the major parameters of contact force. There exist periods where the brushes are separated from the commutator, i.e. a free flight motion of carbon brush occurs. The measured and simulated results show that a large spring preload and a small brush mass improve the contact and result in reduced free flight periods. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

张力腿平台有限振幅运动的方程和数值解

论证了张力腿平台（TLP）在波浪作用下发生有限振幅运动时,所受惯性力、粘性力、浮力等载荷不仅与波浪场有关,还与瞬时响应有关,是响应的非线性函数;张力腿拉力也是各自由度位移的非线性函数．所以分析TLP受力时必须考虑平台的瞬时加速度、速度和位移,在瞬时位置建立运动方程．据此推导出TLP发生有限振幅运动时的外力计算公式,建立了TLP 6自由度有限振幅运动非线性控制方程．其中考虑了由6自由度有限位移引起的多种非线性因素,如各自由度之间的耦合、瞬时湿表面、瞬时位置等;还包括自由表面效应、粘性力等因素引起的非线性．用数值方法求解所得到的非线性运动方程．对典型平台ISSC TLP进行了数值分析,求得该平台在规则波作用下的6自由度运动响应．用退化到线性范围的解与已有解进行了对比,吻合良好．数值结果表明,综合考虑非线性因素后响应有明显改变．     

柔长结构气固耦合的线性与非线性气动力理论

针对柔性结构与风在三方向相互作用的特点,在合理的结构节段力学模型的基础上,建立了新的气动力模型,即三分力系数C_i=C_i(β(t),θ),(#em/em#=D,L,M)不仅是瞬时攻角的函数,而且也是转速的函数,并依据“片条理论”与改进的“准静态理论”,提出了推导结构节段模型与风相互作用的线性与非线性气动力项的方法,从而将土木工程中柔性结构与风的相互作用的线性与非线性理论集中到一个模型中.对于线性气动力部分,给出了与经典气动力公式中相对应的颤振导数的半解析表达式.对于非线性气动力部分,给出了扭转气动耦合的非线性气动力表达式,并给出了Tacoma大桥扭转非线性运动的控制方程,其形式与结果与V.F.B-m的相吻合.     

A control scheme based on ER-materials for vibration attenuation of dynamical systems

Based on a bang-bang control scheme acting on so called “electrorheological” fluids (ER-fluids), a vibration suppression method is proposed for a class of n-dimensional systems subjected to unknown perturbations. The proposed controller relates to robustness vis-a-vis unknown but bounded disturbances. Two approaches for designing the control scheme are presented and compared. On the one hand we employ Lyapunov stability theory; on the other hand there is an obvious reason for minimizing rate of energy change due to the spring/damper elements by varying the ER-fluid properties appropriately. The system under investigation is an n-degree of freedom one consisting of masses and spring/damper elements. The spring/damper elements contain an ER-fluid; their stiffness and damping properties are changed by varying an imposed electrical field. The changes in spring and damping properties can be effected in microseconds since the control does not involve the separate dynamics (inertia) of usual actuators. Detailed derivations are presented for a two-dimensional case and simulations are carried out for examples of smooth periodic and discontinuous periodic excitation forces.     

An integrated approach to the one-dimensional cutting stock problem in coronary stent manufacturing

This paper presents a two-stage approach for pattern generation and cutting plan determination of the one-dimensional cutting stock problem. Calculation of the total number of patterns that will be cut and generation of the cutting patterns are performed in the first stage. On the other hand, the second stage determines the cutting plan. The proposed approach makes use of two separate integer linear programming models. One of these models is employed by the first stage to generate the cutting patterns through a heuristic procedure with the objective of minimizing trim loss. The cutting patterns obtained from Stage 1 are then fed into the second stage. In this stage, another integer linear programming model is solved to form a cutting plan. The objective of this model is to minimize a generalized total cost function consisting of material inputs, number of setups, labor hours and overdue time; subject to demand requirements, material availability, regular and overtime availability, and due date constraints. The study also demonstrates an implementation of the proposed approach in a coronary stent manufacturer. The case study focuses on the cutting phase of the manufacturing process followed by manual cleaning and quality control activities. The experiments show that the proposed approach is suitable to the conditions and requirements of the company.     

Modeling of cutting force prediction in machining high-volume SiCp/Al composites

Variations in cutting forces significantly influence the tool wear and part quality in machining high-volume SiC-particle-reinforced aluminum matrix (SiCp/Al) composites. Properties of the reinforcement SiC particles, such as size and volume fraction, contribute to the change in the cutting forces. This paper presents a cutting force model based on the geometrical and mechanical nature of the tool and workpiece, considering the effect of the SiC reinforcement particles. The cutting force is predicted as three components (F_z, F_x, and F_y) and the resultant cutting force F_τs. The cutting force was considered to generate three deformed zones: (a) shear deformed zone, (b) friction deformed zone on the chip–tool interface, (c) plow deformed zone. The effect of SiC reinforcement particles on friction deformed zone is analyzed emphatically. The friction force from friction deformed zone was obtained by calculating the sliding friction force and rolling friction force. To verify the feasibility and validity of the predicted model of cutting force, cutting experiments were performed with different combinations of cutting speed, feed rate, depth of cut, and tool nose radius. The predicted cutting force values demonstrate good agreement with the measured experimental cutting force values in most cutting conditions. The average percentages of the prediction error were 1.93%, 6.20%, and 10.48% for the F_z, F_x, and F_y components, respectively, thus proving the validity and accuracy of the predicted model of cutting forces.     

A new Taylor collocation method for nonlinear Fredholm‐Volterra integro‐differential equations

The aim of this article is to present an efficient numerical procedure for solving nonlinear integro‐differential equations. Our method depends mainly on a Taylor expansion approach. This method transforms the integro‐differential equation and the given conditions into the matrix equation which corresponds to a system of nonlinear algebraic equations with unkown Taylor coefficients. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer program written in Maple10. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A new nonlinear dynamic analysis method of rotor system supported by oil-film journal bearings

The strong nonlinear behavior usually exists in rotor systems supported by oil-film journal bearings. In this paper, the partial derivative method is extended to the second-order approximate extent to predict the nonlinear dynamic stiffness and damping coefficients of finite-long journal bearings. And the nonlinear oil-film forces approximately represented by dynamic coefficients are used to analyze nonlinear dynamic performance of a symmetrical flexible rotor-bearing system via the journal orbit, phase portrait and Poincaré map. The effects of mass eccentricity on dynamic behaviors of rotor system are mainly investigated. Moreover, the computational method of nonlinear dynamic coefficients of infinite-short bearing is presented. The nonlinear oil-film forces model of finite-long bearing is validated by comparing the numerical results with those obtained by an infinite-short bearing-rotor system model. The results show that the representation method of nonlinear oil-film forces by dynamic coefficients has universal applicability and allows one easily to conduct the nonlinear dynamic analysis of rotor systems.     

Multiple Statistical models for simulation and prediction of nonlinear processes

The paper describes a parameterisation procedure for multiple nonlinear equations with intercepts. These equations may consist of numerous members with few practical limitations on their structure. Examples are given on the application of the procedure to simulation in the areas of chemical technology and hydrology. Advantages of the proposed method are simplicity of the fitting procedure, high accuracy of simulations and more reliable simulations beyond the calibration range     

On nonlinear cutting response and tool chatter in turning operation

Tool chatter in turning process is addressed with a new perspective. Turning dynamics is investigated using a 3D model that allows for simultaneous workpiece-tool deflections in response to the exertion of nonlinear regenerative force. The workpiece is modeled as a system of three rotors, namely, unmachined, being machined and machined, connected by a flexible shaft. Such a configuration enables the workpiece motion relative to the tool and tool motion relative to the machining surface to be three-dimensionally established as functions of spindle speed, instantaneous depth-of-cut, material removal rate and whirling. The equations of motion for the model are coupled through the nonlinear cutting force. The model is explored along with its 1D counterpart, which considers only tool motions and disregards workpiece vibrations. Different stages of stability for the workpiece and the tool subject to the same cutting conditions are studied. Numerical simulations reveal diverse, oftentimes inconsistent, tool behaviors described by the two models. Most notably, observations made with regard to the inconsistency in describing machining stability limits raise the concern for using 1D models to obtain stability charts.     

Stress recovery procedure for solving boundary value problems in the mechanics of a deformable solid by the finite element method

A stress recovery procedure, based on the determination of the forces at the mesh points using a stiffness matrix obtained by the finite element method for the variational Lagrange equation, is described. The vectors of the forces reduced to the mesh points are constructed for the known stiffness matrices of the elements using the displacements at the mesh points found from the solution of the problem. On the other hand, these mesh point forces are determined in terms of the unknown forces distributed over the surface of an element and given shape functions. As a result, a system of Fredholm integral equations of the first kind is obtained, the solution of which gives these distributed forces. The stresses at the mesh points are determined for the values of these forces found on the surfaces of the finite element mesh (including at the mesh points) using the Cauchy relations, which relate the forces, stresses and the normal to the surface. The special features of the use of the stress recovery procedure are demonstrated for a plane problem in the linear theory of elasticity.     

一类非线性两级整数规划问题的全局优化方法

本文研究了整数规划连续化的途径,对一类非线性两级整数规划问题的上级规划连续化以后采用模拟退火算法;其对应的下级规划问题采用离散搜索法求解,从而给出了求解一类非线性两级整数规划问题的一种全局优化算法,并通过算例验证了该算法是有效的.     

Adaptive control of uncertain nonlinear systems using mixed backstepping and Lyapunov redesign techniques

In this paper, a robust adaptive control law for a class of uncertain nonlinear systems is proposed. The proposed controller guarantees asymptotic output tracking of systems in the strict-feedback form with unknown static parameters, and matched and unmatched dynamic uncertainties. This controller takes advantages of a robust stability property of the Lyapunov redesign method and a systematic design procedure of the backstepping technique. In fact, the backstepping technique is employed to enrich the Lyapunov redesign method to compensate for not only matched - but also unmatched-uncertainties. On the other hand, using the Lyapunov redesign method in each step of the conventional backstepping technique makes backstepping robust. The suggested controller is designed through repeatedly utilizing the Lyapunov redesign method in each step of the backstepping technique. Simulation results reveal the efficiency of the Lyapunov redesign-based backstepping controller.     

Strongly nonlinear internal soliton load on a small vertical circular cylinder in two-layer fluids

The solutions of MCC theory are used to investigate larger-amplitude strongly nonlinear internal soliton load on a small surface-piercing circular cylinder in two-layer fluids. By comparing the wave profiles and instantaneous horizontal velocities calculated by MCC theory with those of KdV theory and experimental data, we verify the validity of MCC theory for larger-amplitude strongly nonlinear internal soliton. The accelerations are computed, and then force and torque on a small cylinder are estimated based on Morison’s formula for both MCC and KdV theories. Computed results show that the internal soliton force and torque become more and more large and wide with the increase of amplitude for MCC theory. The location of torque crest calculated by MCC theory departs from origin (moving to the right) as the amplitude grows and whenever the inertial term is included or not, the wave forces computed based on the two theories both have small discrepancies for the same amplitude, but when the inertial term is included, the torque obtained by MCC theory will be much larger and the torque obtained by KdV still have a small discrepancy. The reasons are presented in detail. The internal wave force will be underestimated if the traditional KdV theory is used. Therefore, ocean engineers should consider the large-amplitude strongly nonlinear internal soliton load on marine construct carefully.     

The two-grid interpolating element free Galerkin (TG-IEFG) method for solving Rosenau-regularized long wave (RRLW) equation with error analysis

The two-grid method is a technique to solve the linear system of algebraic equations for reducing the computational cost. In this study, the two-grid procedure has been combined with the EFG method for solving nonlinear partial differential equations. The two-grid FEM has been introduced in various forms. The well-known two-grid FEM is a three-step method that has been proposed by Bajpai and Nataraj (Comput. Math. Appl. 2014;68:2277–2291) that the new proposed scheme is an ecient procedure for solving important nonlinear partial differential equations such as Navier–Stokes equation. By applying shape functions of IMLS approximation in the EFG method, a new technique that is called interpolating EFG (IEFG) can be obtained. In the current investigation, we combine the two-grid algorithm with the IEFG method for solving the nonlinear Rosenau-regularized long-wave (RRLW) equation. In other hand, we demonstrate that solutions of steps 1, 2, and 3 exist and are unique and also we achieve an error estimate for them. Moreover, three test problems in one- and two-dimensional cases are given which support accuracy and efficiency of the proposed scheme.     

A new method for control of nonlinear state-dependent switched systems

In this paper, a new method for the control of input-affine nonlinear switched systems is introduced. The system switching conditions are assumed to be state-dependent, rather than the simpler input-dependent case. The main contribution of this research is that the effects of switched dynamics are interpreted as a model uncertainty bounded within a polynomial of states norms, with unknown coefficients. In order to prevent extra conservativeness, coefficients are tuned adaptively, so that a minimal state-varying bound could be achieved. This is unlike the conventional sliding mode control (SMC) scheme, where the existence of a constant and usually large upper bound must be presumed. To address the challenge of coping with such a new concept of uncertainty, an extended form of the original adaptive fuzzy sliding mode control scheme is proposed. Adaptation laws are used to tune a fuzzy controller and also real-time estimation of the instantaneous bound of uncertainties. Closed-loop stability is guaranteed by proposing a group of multiple Lyapunov functions (MLF) with tunable parameters. Except for the mild condition that the largest difference between the magnitudes of the sub-manifolds of the switched system is bounded by a polynomial of states with uncertain coefficients, the proposed method has the distinct advantage that no information about the dynamic equations or switching conditions is required in the control design stage. The proposed method is applied to the two challenging case studies, depicting the outstanding effectiveness of the method.     

An alternative linearization approach applicable to hysteretic systems

In this paper a method is proposed for equivalent linearization of nonlinear restoring forces being governed by differential equations in weakly nonlinear systems. These types of restoring forces cannot be linearized by employing conventional approximate approaches. Two analytical examples are used to show the accuracy of the proposed method. The application of the method to hysteretic systems is examined by constructing equivalent linear representation for Bouc–Wen model in its general formulation. Numerical investigations reveal that the proposed method is efficient in dynamic behavior analysis of weakly nonlinear hysteretic systems.     

On an exact numerical simulation of solitary-wave solutions of the Burgers–Huxley equation through Cardano’s method

In this work, we develop an exact finite-difference methodology to approximate the solution of a diffusive partial differential equation with Burgers advection and Huxley reaction law. The model under investigation possesses solitary-wave solutions which are positive, bounded, and both spatially and temporally monotone. On the other hand, our computational model is a nonlinear technique for which the new approximations are provided as the roots of an uncoupled system of cubic polynomials, in which the constant coefficients are functions of the model parameters and the numerical step-sizes. In this system, each cubic equation is solved using Cardano’s formulas. The method proposed in this work preserves the positivity, the boundedness and the monotonicity of approximations, as well as the constant solutions of the continuous model. The simulations provided in this work show a good agreement with respect to the analytical solutions employed.