The Practice of Engineering Dynamics. Ronald J. Anderson

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      which, after considerable effort and again noting that images, expands to,

      We now simply add the relative acceleration vectors to arrive at the absolute acceleration of point images. That is,

      (1.36)equation

      Relative angular velocity vectors can be added together in the same way that relative velocity vectors were in Section 1.5. That is, having established the angular velocity of one body in a chain of bodies with respect to a stationary body (i.e. the absolute angular velocity of the body), we simply go through the chain adding the relative angular velocity of neighboring bodies as we pass through the joints connecting them.

      For example, the absolute angular velocity of body images in Figure 1.5 can be determined as follows,

      where the joint at images constrains images to rotate about a vertical axis relative to the ground, so that,

      is the absolute angular velocity of images. The joint at images constrains images to rotate about the axis of images with an angular velocity that is relative to images giving,

      (1.40)equation

      The absolute angular acceleration of images is, by definition, the time rate of change of the absolute angular velocity vector of images. In this example we note that the angular velocity vector is expressed in a rotating coordinate system so that there will be both a rate of change of magnitude and a rate of change of direction. The coordinate system has angular velocity images. Using the symbol images for angular acceleration we can write,

      (1.41)equation

      which becomes, upon differentiation,

      (1.43)equation

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