i
摘要
磨削加工因為適用於各種高硬度鋼材及硬脆陶瓷等難切削材料
的加工特性，長久以來一直是工程上重要的加工方法之一，並廣泛為工業
界使用。然而因為磨削加工的複雜性，導致一般技術人員往往僅憑經驗操
作，這將限制磨削加工的使用範圍。本文即以影響磨削加工精度的工件溫
升及熱變形研究為主，配合不同的加工條件，探討磨削加工時工件各位置
的溫度變化及熱變形情形，希望藉此瞭解磨削加工的特性，並提出有效的
改善方案。
首先，在本文的研究，可以由磨削區的矩形移動熱源來求解工件各位
置的暫態溫升。本文的分析從點熱源開始，並運用分離變數法來求解三維
的熱傳導問題，因為工件的移動速度相當的小，與工件速度有關的對流項
首先從熱傳導方程式中暫時移除，但工件速度的影響隨後將透過修改點熱
源滑動方向的座標變換而納入本文模式之中。因此，三維的穩態溫升通解
可用未知起始條件的函數和溫升的特解乘積的積分式來表示。初始的溫度
上升可由摩擦所造成的點熱源與Dirac’s Delta 函數的乘積所置換，利用
Dirac’s Delta 函數的定義，可以得到點熱源的溫升值。這種求解方法被更
進一步延伸到求解磨削區的均勻移動熱源。本文模式和Jaeger’s 模式[1]
的理論結果預測及實驗結果比較，顯示本文的模式相當精確，並且通常優
於Jaeger’s 模式。
其次，能量分率對於工件溫升及熱變形有相當的影響，然由文獻回顧
各種模式的能量分率，其分率值範圍相當廣泛。對於氧化鋁磨輪而言，工
件的能量分率值範圍依實際的加工條件約在10 至90 %之間，這對工件溫
升及熱變形的預測造成相當的困擾，所以有必要建立新的能量分率理論模
式，其原理是利用磨削區工件及磨輪接觸面的溫度相等，推導得到的能量
分率與磨輪及工件兩者之間的熱性質、相對速度、真實接觸面積均有關
聯，由理論與實驗結果比較，新的理論模式可提供工件溫升及變形較佳的
預測。
另外，對於精密磨削問題，因熱變形而造成尺寸誤差，也是吾人特別
關注的問題，尤其是低熱導係數的鋼工件材料在磨削期間因為熱累積而造
成的熱變形。本文藉由有限元素法理論來預測工件熱變形，並利用前述所
建立的溫升理論結合有限元素法，以不同的加工條件，實驗量測磨削加工
後工件的變形情形，由理論與實驗量測的磨削表面驗證比較，利用本文模
式可精確有效預測工件的三維熱變形。

iii
Abstract
Grinding processing is especially suitable for the difficult cutting
materials , such as various kinds of high hardness steel and hard
brittleness ceramics,etc. It has been one of the important processing
methods for a long time, and use for the industrial field extensivly. But
because of the grinding complexity, causing the technical staff to
operate only according to experience. This will be exercised restraint in
grinding application. In view of this, the present study is concern about
the temperature rise and thermal deformation of the workpiece. It is
hope to understand the characteristic of the grinding process, and
present the effective improvement scheme.
First,in the present study, the general solutions for a transient state
as well as for the temperature rise formed everywhere in the workpiece
due to a rectangular-shaped moving plane heat source arising at the
grinding zone are derived. The present analysis starts from a point heat
source solution by applying the method of separation of variables to a
three-dimensional heat conduction problem. Because the workpiece
moving velocity is quite small, the convective term related to the
workpiece velocity is first excluded from the heat conduction equation.
This workpiece velocity effect will be included in the model by slightly
modifying the coordinate variable in the sliding direction shown in the
iv
solution of the point heat source. Therefore, the general
three-dimensional solution of the stationary temperature rise can be
expressed in an integral form as a function of the product value of the
unknown initial condition and the particular solution of temperature rise.
The initial condition of temperature rise can be replaced by the
point heat source due to frictional that multiplying the product of the
Dirac delta functions defined for three directions. Using the definition of
the Dirac delta function, the temperature rise solution for a point heat
source can thus be obtained. This solution is further extended to obtain
the moving and uniform heat sources arising in a rectangular grinding
zone. Comparisons among the experimental result and the theoretical
results predicted by the present model and Jaeger’s model [1] show
that the present model is quite accurate and is generally superior to
Jaeger’s model.
Secondly, of particular interest is the energy partition of the total
grinding energy that enters the workpiece, which causes the workpiece
temperature rise and deformation. A review of the energy partition
models indicate a wildly ranges. For the aluminum oxide wheels, the
energy partition to the workpiece typically ranges from 10 to 90 %
depending on the actual grinding situation. In this paper, a new
theoretical energy partition model has been built based on the
temperature of the workpiece surface equal to that of the grinding wheel
v
along the grinding zone. The energy partition between two sliding
bodies depends strongly on the relative magnitude of the thermal
characteristics, velocity, real contact area of each body. The energy
partition obtained by theoretical results are compared with those
obtained experimentally. It was found that the new model provide the
basis for improved prediction of workpiece temperatures and
deformations in grinding.
Thirdly, the geometrical inaccuracy of the workpiece due to thermal
deformation is of particular concern in precision grinding. Especially,
steel of low thermal conductivity will deform to become convex during
grinding due to the accumulated heat in the workpiece, which induced a
concave surface profile of the workpiece after grinding, called
over-grinding. Thermal deformation of a workpiece is theoretically
investigated by means of a finite element method(FEM). The present
model utilize the temperature rise theory and the finite element theory
with different processing conditions. Geometrical accuracy of the
ground surface are investigated both theoretically and experimentally.
The theoretical results are compared with the measured profiles of
ground surface. The present model can be applied to predict the
three-dimensional thermal deformations of the workpiece fairly well.

109
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