Eigenvalue analysis of rectangular mindlin plates by chebyshev pseudospectral method

A study of free vibration of rectangular Mindlin plates is presented. The analysis is based on the Chebyshev pseudospectral method, which uses test functions that satisfy the boundary conditions as basis functions. The result shows that rapid convergence and accuracy as well as the conceptual simplicity are achieved when the pseudospectral method is applied to the solution of eigenvalue problems. Numerical examples of rectangular Mindlin plates with clamped and simply supported boundary conditions are provided for various aspect ratios and thickness-to-length ratios.     

Temperature rise due to sliding in rolling/sliding elastohydrodynamic lubrication line contacts: an efficient numerical analysis for contact zone temperatures

An efficient numerical method based on Lobatto quadrature analysis is adopted for a rigorous analysis of temperature in elastohydrodynamic lubrication (EHL) line contacts. Temperature distributions are calculated for maximum Hertzian pressures and rolling speeds varying between 0.5 to 2.0 GPa and 1 to 30 m/s, respectively. Significant mid-film temperature and surface temperature increases have been observed at higher rolling speeds with an increase in loads and slip ratios. Results have been compared with the results of Manton, S. M., O'Donoghue, J. P. and Cameron, A., Temperatures at lubricated rolling/sliding contacts. Proceedings of the Institution of Mechanical Engineers, 1967–68, 182(417), 813–824. An empirical equation is presented for the prediction of non-dimensional maximum mid-film temperature in the contact zone in terms of the dimensionless thermal loading parameter Q, dimensionless load W and slip S, as:

     

Stochastic Solution for Uncertainty Propagation in Nonlinear Shallow-Water Equations

This paper presents a stochastic approach to describe input uncertainties and their propagation through the nonlinear shallow-water equations. The formulation builds on a finite-volume model with a Godunov-type scheme for its shock capturing capabilities. Orthogonal polynomials from the Askey scheme provide expansion of the variables in terms of a finite number of modes from which the mean and higher-order moments of the distribution can be derived. The orthogonal property of the polynomials allows the use of a Galerkin projection to derive separate equations for the individual modes. Implementation of the polynomial chaos expansion and its nonintrusive counterpart determines the modal contributions from the resulting system of equations. Examples of long-wave transformation over a submerged hump illustrate the stochastic approach with uncertainties represented by Gaussian distribution. Additional results demonstrate the applicability of the approach with other distributions as well. The stochastic solution agrees well with the results from the Monte Carlo method, but at a small fraction of its computing cost.     

An error-counting network for pattern classification

This paper presents a novel quadratic error-counting network for pattern classification. Two computational issues namely, the network learning issue and the classification error-counting issue have been addressed. Essentially, a linear series functional approximation to network structure and a smooth quadratic error-counting cost function were proposed to resolve these two computational issues within a single framework. Our analysis shows that the quadratic error-counting objective can be related to the least-squares-error objective by adjusting the class-specific normalization factors. The binary classification network is subsequently extended to cater for multicategory problems. An extensive empirical evaluation validates the usefulness of proposed method.     

On the dynamics of ball end milling: modeling of cutting forces and stability analysis

This paper presents a dynamic force model and a stability analysis for ball end milling. The concept of the equivalent orthogonal cutting conditions, applied to modeling of the mechanics of ball end milling, is extended to include the dynamics of cutting forces. The tool is divided into very thin slices and the cutting force applied to each slice is calculated and summed for all the teeth engaged. To calculate the instantaneous chip thickness of each tooth slice, the method of regenerative chip load calculation which accounts for the effects of both the surface undulations and the instantaneous deflection is used. To include the effect of the interference of the flank face of the tool with the finished surface of the work, the plowing force is also considered in the developed model. Experimental cutting forces are obtained using a five-axis milling machine with a rotary dynamometer. The developed dynamic model is capable of generating force and torque patterns with very good agreement with the experimental data. Stability of the ball end milling in the semi-finishing operation of die cavities is also studied in this paper. The tangential and radial forces predicted by the method of equivalent orthogonal condition are fitted by the equations F_t = K_t(Z)bh_av and F_r = K_r(Z)F_t, where b is the depth of cut and h_av is the average chip thickness along the cutting edge and Z is the tool axis coordinate. The polynomial functions K_t(Z) and K_r(Z) are the cutting force constants. The interdependency of the axial and radial depths of cut in ball end milling results in an iterative solution of the characteristic equation for the critical width of cut and spindle speed. In addition, due to different cutting characteristics of the cutting edge at different heights of the ball nose, stability lobes are represented by surfaces. Comparison of the time domain simulation for the shoulder removal process in die cavity machining with the analytical predictions shows that the proposed method is capable of accurate prediction of the stability lobes.     

Irreducibility, reduction and transfer equivalence of nonlinear input–output equations on homogeneous time scales

The purpose of this paper is to present a necessary and sufficient condition for irreducibility of nonlinear input–output delta differential equations. The condition is presented in terms of the common left divisor of two differential polynomials describing the behaviour of the system defined on a homogenous time scale. The concept of reduction is explained. Subsequently, the definition of transfer equivalence based upon the notion of an irreducible differential form of the system is introduced, inspired by the analogous definition for continuous-time systems.     

Algorithms for Modular Counting of Roots of Multivariate Polynomials

Given a multivariate polynomial P(X ₁,…,X _n ) over a finite field , let N(P) denote the number of roots over . The modular root counting problem is given a modulus r, to determine N _r (P)=N(P)mod r. We study the complexity of computing N _r (P), when the polynomial is given as a sum of monomials. We give an efficient algorithm to compute N _r (P) when the modulus r is a power of the characteristic of the field. We show that for all other moduli, the problem of computing N _r (P) is -hard. We present some hardness results which imply that our algorithm is essentially optimal for prime fields. We show an equivalence between maximum-likelihood decoding for Reed-Solomon codes and a root-finding problem for symmetric polynomials. P. Gopalan’s and R.J Lipton’s research was supported by NSF grant CCR-3606B64. V. Guruswami’s research was supported in part by NSF grant CCF-0343672 and a Sloan Research Fellowship.     

An Algorithm for the Exact Numerical Integration of a Special Class of Functions

The classical Gaussian quadrature rules yield very inaccurate results when they are used for the integration of functions of the form f(x)=P(x,\cos(cx+d))\exp(-a^{2}x^{2}+bx) over the infinite interval (-\infty, \infty) , where a\gt 0, b, c, d are real numbers and P is a polynomial. In this paper we propose a new algorithm that computes this type of integrals without error in the machine precision. The algorithm can also be used for the multiple integrals. We give some numerical examples to compare the results obtained by the new algorithm and those obtained by some other methods including the classical Gauss-Hermite integration rules.