三维振动里兹法在变幅杆谐振特性分析中的应用研究

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The recent situation of one dimensional vibration solution theory for a horn and gear ultrasonic machining was introduced. Ritz method based on energy-variational principle and three dimensional elastic dynamic equations was presented here. Three vibrations including longitudinal, bending and torsional vibrations, their natural frequencies and modal shapes for conical, exponential and catenary kinds of horns could be solved. Convergence to four-digit was demonstrated and natural frequencies for the three vibration types of horns with different length-diameter ratios and diameter ratios were tabulated. Three circular cross-section exponential ultrasonic horns including big, medium and small sizes were designed and machined. Their modal tests were conducted. Analytical solutions based on the three dimensional elastic vibration theory had more accurate answers than ones based on the classical Euler-Bernoulli theory. The results with 1 D classical Euler-Bernoulli beam theory, three-dimensional elastic vibration Ritz method, finite elements method, and modal tests were compared each other. It was shown that the results with three-dimensional elastic vibration Ritz method are more accurate, this method provides a new approach for vibration characteristics analysis of acoustic horns with non uniform cross-section.针对变幅杆振动分析的一维振动求解理论和齿轮超声加工研究现状，提出基于三维弹性动力学方程，利用能量变分原理，采用里兹法求解特征值的方法，统一了圆锥、圆截面指数形和悬链形变幅杆的扭转、纵向、弯曲振动的固有频率和振型求解方法；计算了不同直径比、长径比变幅杆的四位有效数字无量纲固有圆频率。设计加工了大、中、小截面指数形变幅杆，并利用锤击激励法做了模态实验。对其一维欧拉一伯努利、三维振动里兹数值法、有限单元法、实验模态法进行对比分析，结果表明：三维振动里兹数值法求解结果比一维振动求解理论准确，可以作为其他数值求解方法的验证标准；为其他非均匀截面变幅杆的振动特性分析提供了一种新的求解方法。