Maximum Allowable Step Research Articles

Comparative Monte Carlo efficiency by Monte Carlo analysis

We propose a modified power method for computing the subdominant eigenvalue λ{2} of a matrix or continuous operator. While useful both deterministically and stochastically, we focus on defining simple Monte Carlo methods for its application. The methods presented use random walkers of mixed signs to represent the subdominant eigenfunction. Accordingly, the methods must cancel these signs properly in order to sample this eigenfunction faithfully. We present a simple procedure to solve this sign problem and then test our Monte Carlo methods by computing λ{2} of various Markov chain transition matrices. As |λ{2}| of this matrix controls the rate at which Monte Carlo sampling relaxes to a stationary condition, its computation also enabled us to compare efficiencies of several Monte Carlo algorithms as applied to two quite different types of problems. We first computed λ{2} for several one- and two-dimensional Ising models, which have a discrete phase space, and compared the relative efficiencies of the Metropolis and heat-bath algorithms as functions of temperature and applied magnetic field. Next, we computed λ{2} for a model of an interacting gas trapped by a harmonic potential, which has a mutidimensional continuous phase space, and studied the efficiency of the Metropolis algorithm as a function of temperature and the maximum allowable step size Δ. Based on the λ{2} criterion, we found for the Ising models that small lattices appear to give an adequate picture of comparative efficiency and that the heat-bath algorithm is more efficient than the Metropolis algorithm only at low temperatures where both algorithms are inefficient. For the harmonic trap problem, we found that the traditional rule of thumb of adjusting Δ so that the Metropolis acceptance rate is around 50% is often suboptimal. In general, as a function of temperature or Δ , λ{2} for this model displayed trends defining optimal efficiency that the acceptance ratio does not. The cases studied also suggested that Monte Carlo simulations for a continuum model are likely more efficient than those for a discretized version of the model.

A new decision algorithm of maximum allowable step length for five‐axis table‐tilting type machining

Traditional CAM algorithms evaluate the cutting error only based on the geometry of the sculptured surfaces. The kinematic parameters, such as rotational angles, machine structure, and tool geometry, are not considered until the post‐processing stage. When using the tool paths generated by traditional algorithms, the corresponding cutting errors may be larger than expected due to rotation. Therefore, the estimation algorithm presented in this paper considers not only surface geometry but also kinematic parameters. Compared to traditional CAM algorithms, simulation results show that the step lengths are varied adaptively, and cutting errors are reduced.

Cutting Error Analysis for Table-Tilting Type Four-Axis NC Machines

Current computer-aided manufacturing (CAM) systems generate cutter contact points (CC points), cutter location points (CL points) and cutter orientation angles (CO angles) based only on the sculptured surface geometry. However, under certain conditions, the cutting errors of multiple-axis NC machines are larger than expected when using these CC, CL and CO data. This is because the cutting errors of multiple-axis NC machines depend on the machines structure data as well as on the geometry of the machined part. An algorithm is developed in this paper to estimate the cutting error and calculate the maximum allowable step length for table-tilting type four-axis NC machines. It is shown that the cutting error is proportional to the distance between the machined arc and the centre of the rotational axis. Therefore, the step length should be chosen to be smaller when the machined arc is displaced from the rotational axis.

Estimation of Maximum Allowable Step Length for Five-Axis Cylindrical Machining

Five-axis cylindrical machining is the only way to machine some complicated sculptured surfaces, such as the wheel surfaces of turbo compressors. Current computer-aided manufacturing (CAM) algorithms for five-axis cylindrical machining only consider the surfaces’ geometric information when generating cutter contact points (CC points). Hence, the step length, which is the distance between two CC points, is determined only by the surface parameters for these CAM algorithms. Because of tool-axis rotation, the actual cut trajectory within each step is no longer a straight line passing through two consecutive CC points for five-axis machining. Thus the cut error and the maximum allowable step length should depend on the structural parameters of NC machines as well as the surface's geometry. This paper develops a new algorithm to estimate the maximum allowable step length based on the cut error of five-axis cylindrical machining, and shows that this algorithm provides better estimation than traditional algorithms.

Cutting errors analysis for spindle-tilting type 5-axis NC machines

Current computer-aided manufacturing (CAM) systems generate cutter contact points (CC points), cutter location points (CL points) and cutter orientation angles (CO angles) based only on the sculptured surface geometry. However, under certain situations, the cutting errors of 5-axis NC machines are larger than expected using these CC, CL and CO data. It is shown in this paper that the cutting error depends on the structures of 5-axis NC machines as well as the surface geometry. A new algorithm has been developed to estimate the cutting error and calculate the maximum allowable step length for spindle-tilting type 5-axis NC machines. Simulation results show that the new algorithm provides better estimation on the cutting errors for spindle-tilting type 5-axis NC machines.