DTAS邀您探索单孔销浮动之奥秘，快来围观吧！
点击关注，一起探讨加油！https://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9wbmcvSmFGdlB2dkEySjNPWmFOdVJpYTgzd1RpYW81UFlyUFpiZzl5ZHRyS3hiRnl1SjkxTFM5d3RSZEhoTnVuVkZRZ2ljblNPYm9sTUh3clVFR1JMYXZreGZ2UkEvMD93eF9mbXQ9cG5n单孔销浮动初探（一）https://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9wbmcvSmFGdlB2dkEySjNPWmFOdVJpYTgzd1RpYW81UFlyUFpiZzl5ZHRyS3hiRnl1SjkxTFM5d3RSZEhoTnVuVkZRZ2ljblNPYm9sTUh3clVFR1JMYXZreGZ2UkEvMD93eF9mbXQ9cG5n 概要：本文以一简单的单孔销浮动案例用理论计算结果与模拟仿真计算结果做对比，验证了仿真计算的精度，同时为我们公差仿真计算提供一个理论校核的可借鉴步骤。 https://bcn.135editor.com/uploadword/11609120/202203/6225c743-528c-4bdc-8bd1-529fac10006c.png问题：假设孔销直径公差不考虑，孔销相切浮动时，销在竖直方向的波动量为多少?
我们用DTAS3D 建立孔销虚拟装配和沿着竖直方向的虚拟测量，我们用蒙特卡洛法模拟5000次，动画模拟如上图所示，各种统计参数结果如下图所示，最大值最小值为±5，均值接近0，方差为12.517，柱状图拟合分布曲线形状奇特，不是正态分布。（仿真结果会随着初始随机种子的不同略有不同）。https://bcn.135editor.com/uploadword/11609120/202203/6225c577-7154-439c-a43c-740fac10006c.png仿真动画
仿真结果https://bcn.135editor.com/uploadword/11609120/202203/6225c548-fe68-4c1f-bc5e-37a4ac10006c.png
本文尝试通过严格的数学理论推导来对比模拟结果与理论计算的差异，其中主要涉及概率论与微积分等相关知识。主要知识点如下：
1.概率密度函数（pdf）累计分布函数的cdf, pdf是cdf的求导
2.连续性随机变量X的期望 https://bcn.135editor.com/files/users/1160/11609120/202203/2XyBN73U_QCB8.png
3.随机变量函数的分布及期望方差等，如Y=g(X) 的期望https://bcn.135editor.com/files/users/1160/11609120/202203/tqfP6mVS_zTxf.png
4.微分积分公式(包括三角和反三角函数等) 如arcsin x求导为https://bcn.135editor.com/files/users/1160/11609120/202203/Dnm3KWNI_mzt5.png
5.三角函数倍角公式
https://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9naWYvSmFGdlB2dkEySjNPWmFOdVJpYTgzd1RpYW81UFlyUFpiZ05WSXQ4VjFjZmVOY3dyeUFvZUJGZDUydlc4d2lhMk1Rb0ZJYTUwZ0FqN3V6M0FtNmxXbG1iQmcvMD93eF9mbXQ9Z2lmhttps://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9wbmcvSmFGdlB2dkEySjNPWmFOdVJpYTgzd1RpYW81UFlyUFpiZ0tpYm1pY245S3VNMnZVUHdDQzA0VVBmakJ0THY5NTgwWUdpYlJvcG5lUjRaa05OcVRaM2ljTTFlamcvMD93eF9mbXQ9cG5n一、数学模型https://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9wbmcvSmFGdlB2dkEySjNPWmFOdVJpYTgzd1RpYW81UFlyUFpiZzl5ZHRyS3hiRnl1SjkxTFM5d3RSZEhoTnVuVkZRZ2ljblNPYm9sTUh3clVFR1JMYXZreGZ2UkEvMD93eF9mbXQ9cG5n孔销浮动中，在公差仿真分析中幅值方向我们经常设为相切浮动，角度θ设为0到2Π的均匀分布，那么实际问题转化为如下数学模型：已知随机变量θ的概率密度函数(pdf)为:https://bcn.135editor.com/files/users/1160/11609120/202203/pxUYYSJ4_rvGg.png那么随机变量Y=R*sinθθ∈【0，2Π】，R=5的分布形式如何呢，其方差是多少呢？https://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9naWYvSmFGdlB2dkEySjNPWmFOdVJpYTgzd1RpYW81UFlyUFpiZ05WSXQ4VjFjZmVOY3dyeUFvZUJGZDUydlc4d2lhMk1Rb0ZJYTUwZ0FqN3V6M0FtNmxXbG1iQmcvMD93eF9mbXQ9Z2lmhttps://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9wbmcvSmFGdlB2dkEySjNPWmFOdVJpYTgzd1RpYW81UFlyUFpiZ0tpYm1pY245S3VNMnZVUHdDQzA0VVBmakJ0THY5NTgwWUdpYlJvcG5lUjRaa05OcVRaM2ljTTFlamcvMD93eF9mbXQ9cG5n二、分布形状即概率密度曲线的理论计算
https://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9wbmcvSmFGdlB2dkEySjNPWmFOdVJpYTgzd1RpYW81UFlyUFpiZzl5ZHRyS3hiRnl1SjkxTFM5d3RSZEhoTnVuVkZRZ2ljblNPYm9sTUh3clVFR1JMYXZreGZ2UkEvMD93eF9mbXQ9cG5n 分布形状与概率密度紧密关联。我们要想知道Y=R*sinθ的概率密度函数，需先求出Y的累积概率分布函数F(y)，F(y)=P{Y≤y}=P{RSinθ≤y}=P{Sinθ≤y/R}由于sinθ在【0，2Π】上不是单调函数，故须分类讨论
1)当 0<y/R<1 即 0 <y<Rhttps://bcn.135editor.com/files/users/1160/11609120/202203/4pbInGFW_RUr2.pnghttps://bcn.135editor.com/files/users/1160/11609120/202203/wczXt38b_kn2L.png 2)同理 当-1<y/R<0 即 -R <y<0
https://bcn.135editor.com/files/users/1160/11609120/202203/32tHut4X_QHb2.pnghttps://bcn.135editor.com/files/users/1160/11609120/202203/Wv2eFRNL_52JH.png综上Y=R*sinθ 的累计分布函数(cdf)为：https://bcn.135editor.com/files/users/1160/11609120/202203/Ec7y6LPt_MRcq.png故Y的概率密度为https://bcn.135editor.com/files/users/1160/11609120/202203/CjIjyOZJ_ktGH.png其概率密度曲线如下图所示。将之与仿真计算结果作对比可知，拟合曲线与理论计算结果吻合。
https://bcn.135editor.com/files/users/1160/11609120/202203/dudCRb26_UNQS.pnghttps://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9naWYvSmFGdlB2dkEySjNPWmFOdVJpYTgzd1RpYW81UFlyUFpiZ05WSXQ4VjFjZmVOY3dyeUFvZUJGZDUydlc4d2lhMk1Rb0ZJYTUwZ0FqN3V6M0FtNmxXbG1iQmcvMD93eF9mbXQ9Z2lmhttps://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9wbmcvSmFGdlB2dkEySjNPWmFOdVJpYTgzd1RpYW81UFlyUFpiZ0tpYm1pY245S3VNMnZVUHdDQzA0VVBmakJ0THY5NTgwWUdpYlJvcG5lUjRaa05OcVRaM2ljTTFlamcvMD93eF9mbXQ9cG5n三、方差及标准差的理论计算
https://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9wbmcvSmFGdlB2dkEySjNPWmFOdVJpYTgzd1RpYW81UFlyUFpiZzl5ZHRyS3hiRnl1SjkxTFM5d3RSZEhoTnVuVkZRZ2ljblNPYm9sTUh3clVFR1JMYXZreGZ2UkEvMD93eF9mbXQ9cG5nhttps://bcn.135editor.com/files/users/1160/11609120/202203/te2txxJB_C4gw.png由于R=5，所以方差的理论值为25/2=12.5
因此标准差的理论值为https://bcn.135editor.com/files/users/1160/11609120/202203/9py3hWfJ_9pVz.png 模拟仿真计算结果为3.538，仿真精度满足工程需求。
https://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9naWYvSmFGdlB2dkEySjNPWmFOdVJpYTgzd1RpYW81UFlyUFpiZ05WSXQ4VjFjZmVOY3dyeUFvZUJGZDUydlc4d2lhMk1Rb0ZJYTUwZ0FqN3V6M0FtNmxXbG1iQmcvMD93eF9mbXQ9Z2lmhttps://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9wbmcvSmFGdlB2dkEySjNPWmFOdVJpYTgzd1RpYW81UFlyUFpiZ0tpYm1pY245S3VNMnZVUHdDQzA0VVBmakJ0THY5NTgwWUdpYlJvcG5lUjRaa05OcVRaM2ljTTFlamcvMD93eF9mbXQ9cG5n四、工程应用的思考
https://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9wbmcvSmFGdlB2dkEySjNPWmFOdVJpYTgzd1RpYW81UFlyUFpiZzl5ZHRyS3hiRnl1SjkxTFM5d3RSZEhoTnVuVkZRZ2ljblNPYm9sTUh3clVFR1JMYXZreGZ2UkEvMD93eF9mbXQ9cG5n1.在一维尺寸链公差分析中，我们分析带有孔销浮动X方向的或Y方向的尺寸链，是否可以简单的化为正态分布。位置度的模拟方法其实也类似，也不能简单的用正态分布来代替。2.文中的案例简单，但为我们其它的公差仿真计算提供一个理论校核的借鉴步骤。即建立数学模型，然后运用数学知识求解新的随机变量的累积分布函数、概率密度函数、期望方差等，然后与计算结果作对比。当然了随着模型的复杂，数学模型的建立很困难，这时候就需要借助专业软件。大多数数值模拟仿真有一定的使用条件或假设，具备一定的理论知识对辨别计算结果的合理与否有很大的帮助。3.有兴趣的可以想一下如果案例中R也为一个随机变量，即孔销直径带有公差或孔销不相切，结果又会怎么样呢？后续我们继续介绍这些背后的原理，敬请期待！https://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9naWYvSmFGdlB2dkEySjNPWmFOdVJpYTgzd1RpYW81UFlyUFpiZ05WSXQ4VjFjZmVOY3dyeUFvZUJGZDUydlc4d2lhMk1Rb0ZJYTUwZ0FqN3V6M0FtNmxXbG1iQmcvMD93eF9mbXQ9Z2lm- 扫码关注 -https://bcn.135editor.com/files/users/1160/11609120/202203/cJO2e2WY_U5NV.jpghttps://image2.135editor.com/cache/remote/aHR0cHM6Ly9tbWJpei5xbG9nby5jbi9tbWJpel9qcGcvSmFGdlB2dkEySjNqVkhDUUxVNjlQQnpONm40cHZFUEE3VjU0a1BqVVczRWFyZmFIMm5IZGliWTluVU1yYWhRaWMyampKb09PcjA4dUpKWlBxY0FWVk81US8wP3d4X2ZtdD1qcGVn
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