关于用来确定机构速度瞬心的“三心定理”的直接证明及若干注记

Direct Proof and Some Notes on "Three-Centers Theorem" Used for Determining Instantaneous Centers of Velocity in Mechanism

本文根据三构件中两构件相对于第三构件的三种运动情况,分别讨论了机构中三构件间速度瞬心的位置关系。当构件1和2相对构件3分别以不同的角速度绕瞬心P_(13)和P_(23)转动时,则构件1和2的瞬心P_(12)位于瞬心P_(13)和P_(23)的连线上。对这一结论的讨论方法能够代替通常用于证明“三心定理”的反证法。当构件1和2相对构件3以相同的角速度分别绕瞬心P_(13)和P_(23)转动时,则瞬心P_(12)应位于瞬心P_(13)和P_(23)连线的平行线上无穷远处。当构件1和2相对构件3分别是绕瞬心P_(13)的转动和以速度V_(23)作平动时,则瞬心P_(12)位于过瞬心P_(13)且垂直于速度V_(23)的直线上。当构件1和2相对构件3都作平动时,则构件1和2之间以速度V_(12)作相对平动,其瞬心P_(12)位于V_(12)的垂线上无穷远处。文中给出了两个应用实例。

In this paper, a discussion is made on the relationship of the locations of three instantaneous centers of velocity for three links in a mechanism according to three kinds of two links' motions relative to the third link in three links. When links 1 and 2 rotate relative to link 3 about instantaneous centers P13 and P23 respectively in unequal angular velocity, the instantaneous center P12 of links 1 and 2 relative to each other is on a straight line passing through P13 and P23 . The method of discussing this conclusion may be substituted for the reduction-to-absurdity traditionally used for proving the 'Three-Centers Theorem'. When links 1 and 2 rotate relative to link 3 respectively about instantaneous centers P13 and P23 in equal angular velocity, the instantaneous center P12 lies at infinity on lines parallel to the straight line joining P13 and P23. When links 1 and 2 are each in rotation about the instantaneous center P13 and in translation with velocity V23 relative to link 3, the instantaneous center P12 is on a straight line through P13 and perpendicular to velocity V23. When the links 1 and 2 are all in translation relative to link 3, they are in translation with velocity V12 relative to each other, and the instantaneous center P12 is at infinity on the lines perpendicular to the velocity VJ2. Two examples are shown in the article.