Determination of the Velocity of a Point of a Body Using the Instantaneous Centre of Zero Velocity. Centrodes.

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Ex.1. Look at Appendix  1 and read the following mathematical symbols.

     , , , ,

     Another simple and visual method of determining the velocity of any point of a body performing plane motion is based on the concept of instantaneous centre of zero velocity. The instantaneous centre of zero velocity is a point belonging to the section S of a body or its extension, which at the given instant is momentarily at rest. It will be readily noticed that if a body is in non-translational motion, such one and only one point always exists at any instant t. Let points A and B in section S of a body (Fig. 22) have, at time t, non-parallel velocities and . Then point P of intersection of perpendiculars Aa to vector  and Bb to vector  will be the instantaneous centre of zero velocity, as . For, if we assumed that , then, by the theorem of the projections of the velocities of the points of a body, vector  would have to be simultaneously perpendicular to AP (as ) and to BP (as ), which is impossible. It also follows from the theorem that, at the given instant, no other point of section S can have zero velocity (e.g., for point a, the projection of  on Ba is not zero and consequently ).

Fig. 22

If, now, we take a point P as the pole at time t, the velocity of point A will, by Eq. (50), be

,

as . The same result can be obtained for any other point of the body. Thus, the velocity of any point of a body lying in section S is equal to the velocity of its rotation about the instantaneous centre of zero velocity P. From Eqs. (51) we have

                         , etc.                  (53)

It also follows from Eqs. (53) that

                                           ,                                      (54)

i.e., that the velocity of any point of a body is proportional to its distance from the instantaneous centre of zero velocity.

These results lead to the following conclusions:

 (1) To determine the instantaneous centre of zero velocity, it is sufficient to know the directions of the velocities  and  of any two points A and B of a section of a body (or their paths); the instantaneous centre of zero velocity lies at the intersection of the perpendiculars erected from points A and B to their respective velocities, or to the tangents to their paths.

(2) To determine the velocity of any point of a body, it is necessary to know the magnitude and direction of the velocity of any point A of that body and the direction of the velocity of another point B of the same body. Then, by erecting from points A and B perpendiculars to  and , we obtain the instantaneous centre of zero velocity P and, from the direction of , the sense of rotation of the body. Next, knowing , we can find from Eq. (54) the velocity  of any point M of the body. Vector  is perpendicular to  in the direction of the rotation.

(3) The angular velocity of a body, as can be seen from Eqs. (53), is at any given instant equal to the ratio of the velocity of any point belonging to the section S’ to its distance from the instantaneous centre of zero velocity P:

                                                                                     (55)

Let us evolve another expression for ω.  It follows from Eqs. (50) and (51) that  and  whence

                        .                                  (56)

When = 0 (point A is the instantaneous centre of zero velocity), Eq. (56) transforms into Eq. (55).

Fig. 23

Eqs. (55) and (56) give the same quantity, it follows that the rotation of the section S about either point A or point P takes place with the same angular velocity.

It is easy to verify that both equations give the same answer.

Let us consider some special cases of the instantaneous centre of zero velocity.

(a) If plane motion is performed by a cylinder rolling without slipping along a fixed cylindrical surface, the point of contact P (for the section shown in Fig. 23) is momentarily at rest and, consequently, is the instantaneous centre of zero velocity (  because if there is no slipping, the contacting points of both bodies must have the same velocity, and the second body is motionless). An example of such motion is that of a wheel running on a rail.

(b) If the velocities of points A and B of the body are parallel to each other, and AB is not perpendicular to  (Fig. 24 a) the instantaneous centre of zero velocity lies in infinity, and the velocities of all points are parallel to . From the theorem of the projections of velocities it follows that , i.e., ; the result is the same for all other points of the body. Consequently, in this case the velocities of all points of the body are equal in magnitude and direction at every instant, i.e., the instantaneous distribution of the velocities of the body is that of translation (this state of motion is also called instantaneous translation). It will be found from Eq. (56) that the angular velocity ω of the body at the given instant is zero.

Fig. 24

(c) If the velocities of points A and B are parallel and AB is perpendicular to , the instantaneous centre of zero velocity P can be located by the construction shown in Fig. 24 b. The validity of this construction follows from the proportion (54). In this case, unlike the previous ones, we have to know the magnitudes of velocities  and  to locate the instantaneous centre of zero velocity P.

(d) If the velocity vector  of a point in section S and the angular velocity ω are known, the position of the instantaneous centre of zero velocity P, lying on the perpendicular to   can be immediately found from Eq. (55), which yields .

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