用動態勁性法於正齒輪動態之研究

Abstract

由於高性能機械之要求，使得齒輪動態分析日趨重要。不同於等效彈簧與等效質量之傳統離散方法，本研究以連體模式，應用動態勁性法，考慮隨著正齒輪運轉囓合點位置之改變，質量矩陣與勁性矩陣之時變特性，分析運轉囓合過程正齒輪之動態。 本文先以動態勁性法，分析旋轉非均勻截面歐拉樑之動態，考慮離心力造成之軸向硬化效應，以級數解推導出樑元素之動態勁性矩陣，並計算旋轉樑之動態變形。不同於有限元素法訴諸於元素之細分割，本研究僅以少數分割元素與少數個模態，即可獲得旋轉非均勻樑動態之準確結果。 本文應用動態勁性法，進一步分析正齒輪運轉中之動態。將輪齒分割為四個非均勻截面的提末辛科樑，每個樑元素以二階多項式來表示其外型。推導出樑元素之動態勁性矩陣之後，組合樑元素之動態勁性矩陣與非線性赫茲應力理論之勁性彈簧，構成齒輪系統之動態勁性矩陣，計算齒輪對囓合過程中輪齒之動態齒頂變形和齒根應變。齒輪實驗中，分別用 CCD 影像處理方法與應變計方法，量測齒頂變形與齒根應變，以驗證理論模式之計算結果。 論文之第五章利用所提出之分析方法，探討齒輪剛體運動產生之離心力對齒輪動態應變之影響。並探討齒輪設計參數，包括移位係數、中心距與背隙，對齒輪動態之影響。最後以多齒對之模式，計算齒輪對之齒根應變。

With an increasing demand for high performance machinery, dynamic analysis for gears has become important. Instead of discrete mass-spring models, this study presents a dynamic stiffness method to analyze spur gear dynamics. The dynamic responses of single gears and gear pairs are investigated. This continuum approach takes into account time-varying stiffness and mass matrices due to moving meshing points. Firstly, using a dynamic stiffness method, dynamic analysis for rotating nonuniform cross-section Euler beams is carried out. Longitudinally stiffening effect is taken into account by including a geometrically nonlinear term. The dynamic stiffness of a beam element is formulated using power series. Tip deformations of the beams are calculated. In contrast to the finite element method that has to refine the element mesh, accurate dynamic response can be obtained with only few elements and few modes in the dynamic stiffness method. The dynamic stiffness method for a Timoshenko beam is further developed and applied to spur gear dynamic analysis. Firstly, quadratic polynomials are used to represent profiles of the nonuniform cross-section beam elements for the gear blank and tooth. Then, the dynamic stiffness matrix of a Timoshenko beam element is derived. Next, assemble the dynamic stiffness matrices of the beam elements and a nonlinear Hertzian contact stiffness to form the dynamic stiffness matrix of the gear system. Finally, tip deformations and fillet strains of gears are calculated by performing modal analysis and the Runge-Kutta method. In experiment, tip deformations and fillet stains are respectively obtained using CCD image processing and a strain gauge. Experimental results serve to verify calculation results by the proposed method. The final part of this study investigates the effect of the centrifugal force arising from the gear rotation on gear dynamics. In addition, this study also examines influences of gear parameters including the correction factor, backlash, and nonstandard center distance on gear dynamics. Fillet strains of a gear pair are calculated using a multi-tooth pair model to account for multi-tooth pairs in contact.