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《理论力学》双语教学 第十章 刚体的一般运动

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双语理论力学 刚体的一般运动 理论力学的双语教学 理论力学双语教学 理论力学双语教学第 理论力学
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Theoretical mechanics 1 2 Chapter 10 General motion of a rigid body 第十章 刚体的一般运动 §10–1 Composition of rotations about two parallel axes §10–1 刚体绕平行轴转动的合成 3 Chapter 10 General motion of a rigid body 第十章 刚体的一般运动 §10-1 Composition of rotations about two parallel axes §10-1 刚体绕平行轴转动的合成 Composition problem of rotations about parallel axes is often encountered in mechanical engineering. For instance, the planet gear wheel in a planet gear mechanism has plane motion. 刚体绕平行轴转动的合成问题在机械中经常遇到。例如,行星 圆柱齿轮机构,行星轮作平面运动。 4 In previous study, we dealt with a plane motion of a planet gear by decomposing it into a translation and a rotation. However, it is more convenient to view the plane motion of a planet gear as a composition of two rotations. 前面所研究的平面运动是把它看成为平动和转动 的合成运动,但是在分析行星轮系的传动问题时,将 行星轮的平面运动看成为转动与转动的合成运动则比 较方便。 5 Take a planet gear mechanism as an example. 例如在行星轮系中 Static system is O1xy 静系:O1xy Moving system is O1x'y' 动系: O1x'y' 6 Relative motion: rotation about axis O2, 相对运动: 绕O2轴转动, r is the relative angular velocity. r为相对角速度。 7 Embroiling motion: rotation about axis O1, 牵连运动: 绕O1轴转动, e is the embroiling angular velocity. e为牵连角速度。 (翻页请看动画) (Look for the animation on the next page) 8 9 10 The figure shows that Differentiating it with respect to t gives: 由图看出 对 t 求导: At moment t, it is in position O2A; At moment t +  t, it is in position O2'A‘. t时刻,O2A 位置; t +  t 时刻, O2'A'位置 11 In conclusion: The absolute angular velocity a of the plane figure (the planet gear) equals to the algebraic sum of the embroiling angular velocity e and the relative one r . 即:平面图形(这里指行星轮)的绝对角速度a等于牵连角速 度e 与相对角速度r的代表和. 12 当e 与r 转向相同时 转向与两者相同. 当e与r 转向相异时 转向与大者的相同. When e and r have same direction, the direction of a is the same as them. When e and r have opposite directions, the direction of a is the same as the bigger one. 13 Next, we determine the position of the instantaneous velocity center for the plane figure S. 下面来确定图形S的瞬心的位置 e and r have the same direction e 与r同转向 e and r have the opposite directions e 与r同 反向 Since ve=vr, and they have opposite directions, vp=0, P is the instantaneous center of velocity. Hence, 由于ve=vr, 且方向相 反, 因此vp=0 , P为速 度瞬心。此时 14 P点为图形的速度瞬心, 通过点P且与轴O1、O2平行 的轴称为瞬时轴, 该轴上各点的速度都等于零。 P is the instantaneous center of velocity of the figure. The axis across point P and perpendicular to axes O1 and O2 is called instantaneous axis, every point of which has zero velocity. e and r have the same direction e 与r同转向 e and r have the opposite directions e 与r同 反向 15 When these rotations turn to the same direction, the instantaneous axis lies between the two axes, . Otherwise, it is in the extension of the line segment between the two axes and lies in the side of the axis with larger angular velocity . 同向转动时,瞬轴在两轴之间, ,转向 与两者相同;反向转动时, 瞬轴 在两轴之外, 在角速度值大的一侧, ,转 向与大者的相同。 16 Conclusion:Two rotations of a rigid body about two parallel axes can be composed into a rotation about a instantaneous axis. The instantaneous axis is parallel to the original axes,and the ratio of its distance to one axis and that to the other is inversely proportional to the ratio of their angular velocities. 即:刚体绕两平行轴的转动可合成为绕瞬轴的转动, 瞬轴与原两轴共面且平行,到两轴的距离与两角速 度大小成反比。 17 [Example 1] Gear  and  have the same radius R, and the radius of gear  is r. They are meshed one by one, and gear  is fixed. Gear  and  is pined to the crank O1O3 respectively, and they can rotate around axes O2 and O3. Crank O1O3 rotate with  0 . [例1] 齿轮、半径均为R, 齿轮半径为 r ,依次互啮合, 轮 固 定不动,轮 和轮 装在曲柄O1O3上,可分别绕O2、O3轴转 动。设曲柄O1O3以 0顺时针转动. 18 Fine the angular velocity 3 r of gear III relative to the crank , its absolute angular velocity 3 , and the velocities of point A and B respectively at the instant shown in the figure. 试求齿轮III相对于曲柄转动的角速度3 r 和齿轮的绝对角速 度3 以及图示瞬时A、 B 两点的速度。 19 Solution:Take O1O3 as the moving coordinate system, and denote 1 r ,2 r and 3r as the angular velocities of gear ,  and  relative to the connecting rod. The transmission ratio can be expressed as 解:取系杆O1O3 为动系, 1 r 、2 r 、 3r 分别是 、  、  轮相对于系杆的角速度, 根据传动比公式, 可得 20 由平行轴转动的合成理论,得 Employing the theory developed above, we have 21 It is clear that gear  is in translation. Noting that all points in a translating rigid body have the same velocity,we obtain 由此可知,齿轮作平动,平动刚体上各点的速度相同,故 22 方向:垂直于O1O3杆,指向朝下. Direction:perpendicular to rod O1O3,and pointing downward . 23 24 25
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