臺灣博碩士論文加值系統

Schatz機構之完整力分析
學生：莊宗憲 指導教授：李聰慶
國立高雄應用科技大學模具工程系碩士班
摘 要
Schatz機構是由六根連桿及六個迴轉對接頭所組成的空間機構，因其特殊機構幾何尺寸的限制，為具拘束運動的過度拘束機構。在實際工業應用中，此機構被使用於液體及粉末的混煉用途上。爲了避免機構在驅動運轉時，連桿及各迴轉對接頭承受過大的作用力及作用力矩，造成機構破損發生。所以促使本論文以Schatz機構作為研究對象，而研究目的為針對此機構進行完整力的分析，且利用數學符號運算軟體建立完整力的分析程式，預測各桿件所承受的作用力及作用力矩，並提供分析及降低機構在運轉時產生的搖撼力及搖撼力矩之參考。
本文首先對於Schatz機構的幾何構形作簡單描述，再針對此機構進行運動特性及完整力的分析。而運動特性分析是將該機構以 坐標轉換矩陣列出其各桿間坐標關係，並利用坐標變換矩陣相乘原理及其代數運算，得到各接頭變數之閉合解且以單一輸入變數表示之。接著，將接頭位置對時間微分，推導出各接頭的速度、加速度及急跳度方程式，以提供Schatz機構運動分析的基本方程式。關於耦桿點的運動分析，則由三個構造參數來說明，並藉由坐標轉換矩陣和表示耦桿點位置的行矩陣相乘，求得耦桿點位置的參數方程式，進而得到耦桿點的速度及加速度方程式，以利後續之分析探討。接著，在機構完整力的分析方面，包括靜力、搖撼力及搖撼力矩與動態力等的分析。在靜力分析中，由運動分析所得結果，利用靜力學的觀念，將各桿件取分離體圖，列出力及力矩平衡方程式，以矩陣形式表示，再解其矩陣方程式來決定各桿之間的作用力及作用力矩的大小分佈。當改變桿件為均質圓桿或非均質圓桿時，由靜力分析的結果可知機構桿件為非均質圓桿時，受力有減少且在某些特定的角度會有奇異解現象發生。
再者，各桿件在運動時會有慣性力存在，進而產生搖撼力及搖撼力矩，造成機構不正常的震動和搖晃。因此，對此機構有必要進一步探討其搖撼力及搖撼力矩的大小變化。文中嘗試以改變輸入速度的方式，觀察其變化情形。且假設桿件為非均質圓桿時，即由配重或改變桿件的形狀，來改變運動桿件的質心及機構整體的質心位置，藉此比較搖撼力與搖撼力矩的大小分佈變化。除與各桿件為均質圓桿的情況互相比較外，並利用空間機構等力矩橢圓球理論，描述其均方根值搖撼力矩分佈特性，比較平衡前後個別的最小均方根值搖撼力矩。最後，考慮桿件加速度、角速度及角加速度的影響，對此機構進行動態力分析。由各桿件的分離體圖，根據牛頓-尤拉運動方程式，獲得各桿件力與力矩平衡方程式且以矩陣形式表示之，分析各桿在動態情形下，所承受作用力及作用力矩的大小分佈情況。進而比較桿件為均質圓桿或非均質圓桿時，其動態力變化情形。

The Complete-Force Analysis of Schatz Mechanism
Student：Tsung-Hsien Chuang Advisor：Prof.Chung-Ching LEE
Institute of Tool and Die Making Engineering
National Kaohsiung University of Applied Sciences
ABSTRACT
Schatz mechanism is composed of six binary links and six revolute pairs. Due to its specific geometrical constraints, it is a spatial overconstrained mechanism with the constrained motion. In the practical industrial application, this mechanism is used for a spatial mixing machine. To identify joint reaction forces & moments and reduce the shaking forces & shaking moments for increasing mechanism life, we arouse and encourage the topic of this thesis. The purpose of this work is focused on the complete force analysis of Schatz mechanism with the help of the mathematic symbolic program.
In this thesis, we first describe dimensional constraints and geometrical properties of the Schatz mechanism and then proceed with the motion and complete force analysis of mechanism. The D-H matrix method is used to analyze the kinematic problem. Through the multiplication of the relative coordinate transformation matrices and after the algebraic manipulations, we can obtain kinematic closed-form solutions, such as the displacement, velocity, the acceleration and the Jerk. These will provide the basic motion equations for the analysis of the of Schatz mechanism. In the analysis of coupler-point motion, based on the coordinate-transformation matrix and in terms of three defined parameters, we derive the parametric displacement equation of the coupler point, which functions as the input angle only. In the complete force analysis of mechanism, based on the motion equations provided by coordinate-transformation matrix method, we get position vector with respect to a global coordinate system. By taking free-body diagram from the statics, we determine forces and moments of each link by using matrix-equation systems. The results of static-force analysis show that the non-uniform link will offer a better reduction on the force of reaction and the singular phenomena exist at some specific angles. In the singular conditions, we must directly solve the equilibrium equations to determine forces & moments of each link.
When the inertia force of moving link of mechanism exists, it produces the shaking force & moment of mechanism and leads the mechanism to vibrate and shake irregularly. Therefore, it is necessary that we must analyze the shaking force & moment. In this work, we tried to change input-velocity pattern and discuss the influence on the shaking force & moment. Assume that non-homogeneous link can be used to change mass distribution or geometry of the link; we discuss the variation of shaking force & moment. In both cases, we also apply the root mean square shaking moment and theory of isomomental ellipses to predict the minimum shaking moment after applying a balancing technique. Finally, we consider the dynamic force analysis under the influence of the linear acceleration, angular velocity and angular acceleration. From the free-body diagrams, we apply Newton-Euler equations to determine the equilibrium equations of each link in matrix form. Furthermore, we analyze the distribution of dynamic forces & moments and make a comparison between the homogenous and non-homogenous links.

目 錄
中文摘要 ---------------------------------------- i
英文摘要 ---------------------------------------- iii
誌謝 -------------------------------------------- v
目錄 -------------------------------------------- vi
圖目錄 ------------------------------------------ vii
一、 前言---------------------------------------- 1
二、 運動分析方程式------------------------------ 5
2.1 機構描述------------------------------------- 5
2.2 角位移分析----------------------------------- 6
2.3 角速度及角加速度----------------------------- 9
2.4 急跳度分析----------------------------------- 12
2.5 各桿角速度及角加速度------------------------- 13
三、 耦桿點運動分析------------------------------ 22
3.1 參數方程式----------------------------------- 22
3.2 各迴轉對接頭位置的位移、速度及加速度--------- 29
3.3 各桿任意點之位移、速度及加速度--------------- 31
3.4 各桿任意點及質心的速度及加速度--------------- 34
四、 靜力分析------------------------------------ 42
4.1 靜平衡方程式--------------------------------- 42
4.2 均質桿之迴轉對接頭受力----------------------- 55
4.3 非均質桿之迴轉對接頭受力--------------------- 68
五、 搖撼力及搖撼力矩---------------------------- 76
5.1 搖撼力及搖撼力矩----------------------------- 76
5.2 均方根值搖撼力矩的分佈探討------------------- 82
5.3 減少均方根值搖撼力矩------------------------- 89
5.4 改變輸入速度探討搖撼力及搖撼力矩的變化------- 94
六、 動力分析------------------------------------ 96
6.1 動平衡方程式--------------------------------- 96
6.2 均質圓桿之迴轉對接頭受力分佈----------------- 106
6.3 非均質桿之迴轉對接頭受力分佈----------------- 120
七、 結論與建議---------------------------------- 129
參考文獻 ---------------------------------------- 131

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