針對傳統V溝研磨法，使用球運動學，首次推導出一個一般封閉型解析解之球表面研磨軌跡方程式，並且發展成適用於其它研磨法之研磨軌跡方程式。在同心研磨時，球與磨盤之三接觸點在球表面所造成的研磨軌跡為固定的圓圈，並且長度分別與 、 和 成線性比例。為了符合實際上的應用，設球再度進入磨盤時的方向是隨機的，則經過多次的循環後，研磨軌跡呈現密集的分布，並且每次循環的研磨長度越長，則以較少的循環即可得到最大的研磨面積比。在V溝幾何的設計方面，V溝半角應大於 ，但是為了防止研磨劑的飛濺應小於 。
在偏心研磨時，由於球自轉角速度和自轉角隨著時間呈連續變化，球表面的研磨軌跡不是固定的圓圈。實際的研磨情況，A和B接觸點的研磨面積為互補，全部的研磨面積比隨著偏心量的增加而增加至一飽和值。當滑動比小於0.5時，全部的研磨面積比大於87%。因此在研磨的過程中可藉由改變滑動比來研磨全球表面。較大的V溝半角只需要較小的偏心量即可得到最佳的研磨面積比。
針對磁性流體研磨法，推導力與力矩的平衡式來建立球研磨機的動態數學模型，經由數值分析的方法來瞭解球在研磨過程中的運動行為，進而瞭解球的研磨機制。設一批球的尺寸相同時，在動態分析中只考慮單球。解析結果顯示，當球與主軸及浮盤分離時，自轉角快速上升至接近90°，球因此改變姿勢，並且球表面產生新的研磨軌跡。經過多次的循環後，研磨軌跡可得到大範圍的分布，此為球表面創成機制。
球表面波紋會造成研磨負荷的振動。當 時，球與浮盤有機會分離，並且分離時球所承受的負荷為0。無論球是否與浮盤分離，自轉角皆在穩態值附近作小範圍的變動，研磨軌跡只有極小的範圍，因此僅考慮球表面的波紋效應並無法瞭解球表面的創成機制。
事實上，在研磨的過程中是研磨許多不同直徑的球。為了瞭解球表面創成之球研磨機制，必須同時考慮一批球的研磨情況。針對一批不同直徑的球，每顆球所承受的負荷皆不同。通常球直徑大者球與主軸的摩擦力大，球公轉速度大，因此在研磨時有機會追撞直徑較小的球。為了瞭解一批球在研磨過程中的相互作用情形，因此建立多球的動態分析模型。在球與球相互作用的情況，可改變球的自轉角，而研磨軌跡因此得到較大範圍的分布。一批球在研磨的過程中，球有機會與主軸分離，這可使球改變其研磨姿勢，讓球研磨更均勻。

A general closed-form analytical solution is derived for the lapping tracks with its kinematics for the concentric V-groove lapping system. The lapping tracks on the ball surface for the three contact points are fixed circles, and their lengths of the lapping tracks are linearly proportional to , , and , respectively. In practice, if the orientation is randomized as the ball enters the lap again, then the distribution of the lapping tracks are dense after many cycles, and the larger the lapping length in each cycle, the smaller is the number of cycles required achieving the maximum lapped area ratio. In the geometry design of ball lapping, the V-groove half-angle should be larger than 45°, but to prevent the splash of abrasives, it should be less than 75°.
Since the spin angular speed with its angle continuously varies with time for the eccentric lapping system, lapping tracks are not fixed circles. In practice, the lapped areas are complementary at the contact points of A and B. The total lapped area ratio is higher than 87% for a slip ratio less than 0.5. Hence, it is possible to lap all the surface of a ball by changing the slip ratio during the lapping process. Moreover, the larger the V-groove half-angle, the less is the eccentricity to achieve the optimum lapped area ratio.
In order to understand the ball motion and ball lapping mechanism in the magnetic fluid lapping system, the forces and moments equilibrium equations are derived and numerical methods are analyzed. As the balls traveling in a train are assumed to be the same size, only one ball is considered in the dynamic analysis. Results show that as the ball separates from the shaft and the float, the spin angle increases quickly and approaches to 90°. Hence, the ball changes its attitude and thereby generates a new lapping tracks on the ball surface. Consequently, after repeating many cycles, lapping tracks would be scoping out more space and this is one of the spherical surface generation mechanisms.
Surface waviness of ball causes a variation in the lapping load. When , it is possible to cause the ball separated from float and the lapping load is zero during the separation period. No matter how the ball separates from float, the spin angle always varies in a small range. Hence, only a very small region can be grounded due to the effect of the surface waviness. Therefore, it is not the main lapping mechanism of the spherical surface generation.
In fact, during the lapping process, many balls with different diameters are lapped. To understand the ball’s lapping mechanism of the spherical surface generation, it is necessary to consider a batch of balls. For a batch of balls with different diameters, the applied load on each should be different from each other. Generally, the larger the diameter of a ball, the larger is the friction force between the ball and shaft and the ball circulation speed. Therefore, it is possible to cause the collision between the larger and the smaller balls. To understand the interaction between balls traveling in a train, the dynamic analysis of multiple balls is developed. As the ball interacts with each other, it is possible to change the spin angle, and thereby to achieve the larger variation range of the lapping tracks. During the lapping process of a batch of balls, it is also possible to cause the separation between the shaft and the ball, and it causes the ball to change its attitude and to achieve more uniform lapping tracks.

總目錄 i
圖目錄 iv
表目錄 xvi
符號說明 xvii
摘要(中文) xxii
摘要(英文) xxiv
第一章 緒論
1-1 研究動機與背景 1
1-2 文獻回顧 3
1-2-1 傳統V溝研磨法 4
1-2-2 非傳統研磨法 6
1-3 本論文之研究目的 10
1-4 本論文架構 12
第二章 球研磨系統之運動分析
2-1 傳統V溝研磨法之球運動分析 22
2-1-1 偏心式V溝研磨系統 22
2-1-2 同心式V溝研磨系統 27
2-2 非傳統研磨法之球運動分析 29
2-2-1 上盤偏心式研磨法 29
2-2-2 磁性流體研磨法 35
第三章 球研磨系統之動態分析－單球
3-1 前言 67
3-2 球與浮盤之穩態平衡 68
3-2-1 研磨系統之穩態控制方程式 68
3-2-2 各作用力的計算 69
3-2-3 數值解析方法 71
3-2-4 解析結果與討論 72
3-3 球與浮盤之動態平衡 73
3-3-1 研磨系統之動態控制方程式 74
3-3-2 動態的數值解析方法 76
3-3-3 解析結果與討論 77
3-4 球表面波紋效應 79
3-4-1 球表面波紋影響浮盤的振動 79
3-4-2 解析方法 82
3-4-3 解析結果與討論 82
第四章 球研磨系統之動態分析－多球
4-1 前言 109
4-2 球直徑大小不同影響研磨負荷分布不均 109
4-3 球與浮盤之穩態平衡－球與球不接觸 112
4-4 球與浮盤之穩態平衡－球與球接觸 114
4-5 球與浮盤之動態平衡－球與球不接觸 116
4-6 球與浮盤之動態平衡－球與球接觸 118
第五章 球表面之研磨軌跡
5-1 傳統V溝研磨法之研磨軌跡－同心式 143
5-1-1 球表面研磨軌跡方程式 143
5-1-2 每當球再度進入磨盤時旋轉一角度的影響 146
5-1-3 V溝幾何對研磨軌跡長度的影響 150
5-2 傳統V溝研磨法之研磨軌跡－偏心式 151
5-2-1 球表面研磨軌跡方程式 151
5-2-2 自轉角呈線性變化之研磨軌跡 153
5-2-3 自轉角呈週期函數之研磨軌跡 156
5-2-4 偏心實例之研磨軌跡 158
5-3 磁性流體研磨法之研磨軌跡 161
5-3-1 球表面研磨軌跡方程式 161
5-3-2 Wf呈週期函數之研磨軌跡－單球 162
5-3-3 球表面波紋效應之研磨軌跡 164
5-3-4 Wf呈步階函數之研磨軌跡－球與球不接觸 165
5-3-5 Wf呈步階函數之研磨軌跡－球與球接觸 166
第六章 總論與展望
6-1 總論 191
6-2 未來展望 194
參考文獻 196
附錄A 202
附錄B 204
作者簡介 205
研究著作目錄 206

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