In this paper, a semi-analytical approach is proposed to solve natural frequencies and natural modes for circular plates with multiple circular holes by using the indirect formulation in conjunction with degenerate kernels and
In this paper, natural frequencies and natural modes of a circular plate with multiple circular holes are theoretically derived and numerically determined by using the indirect boundary integral formulation, the addition theorem, and the complex Fourier series. Owing to the addition theorem, all kernel functions are expanded into degenerate forms and further expressed in the same polar coordinates centered at one circle where the boundary conditions are specified. Not only the computation of the principal value is avoided but also the calculation of higher-order derivatives can be easily determined. By matching boundary conditions, a coupled infinite system of linear algebraic equations is derived as an analytical model for the free vibration of a circular plate with multiple circular holes. The direct-searching approach is utilized in the truncated finite system to determine the natural frequency through singular value decomposition. After determining the unknown Fourier coefficients, the corresponding mode shapes are obtained by using the indirect boundary integral formulations. Some numerical eigensolutions are presented and then utilized to explain some physical phenomenon such as the beating and the dynamic stress concentration. Good accuracy and fast rate of convergence are the main features of the present method, thanks to the analytical approach.
A control system to improve the efficiency of machining a workpiece with varying thickness in the wire electrical discharge machining (WEDM) process is proposed. The abnormal ratio R ab defined by the proportion of abnormal sparks in a sampling period is taken as the controlled variable. It is allowed to reduce temporarily as the cutting thickness is changing. A gain self-tuning fuzzy control algorithm is used so that the transient situation as the cutting thickness is suddenly increasing can be suppressed immediately, and a stable performance can be achieved. In addition, the grey predictor is adopted to compensate the time-delayed R ab caused by the low-pass filter data processing. Experiments reveal that there is a slight variance in the optimal reference of R ab when the cutting thickness is larger than 20 mm, and its value is set to 55% in these cases. Three cases were tested: the constant machining parameters, the constant R ab and the proposed adaptive R ab . The results show that the cutting speed can be obviously improved by the proposed control strategy.
Wire rupture in the wire electrical discharge machining (WEDM) process is one of the most troublesome problems in practical applications. In this paper, the abnormal ratio R ab , defined as the proportion of abnormal sparks in a sampling period, is taken to represent the gap state in machining. The grey predictor is adopted to compensate the time-delayed R ab caused by the low pass filter data processing. A gain self-tuning fuzzy control system has been developed to cope with the conditions that often occur with wire rupture in the WEDM process, such as an improper setting of machining parameters, machining the workpiece with varying thickness, etc. Experimental results of several cases show that the proposed controller results in a satisfactory performance. Not only can it immediately suppress transient situation once there is a sudden change of workpiece thickness, but a stable performance can also be achieved during machining a workpiece of constant thickness. As a result, wire rupture problems in most WEDM processes can be successively solved by the proposed control strategy.
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