Concept of Instant Centers:
The method of Instant centers is one of the easiest means of determination of the velocity of a point in a mechanism. It is based upon the assumption that at any given instant the motion of one body, with respect to another body, is such that there is a 'point', in the plane of motion, where the two bodies have no relative velocity. This point may or may not lie on the actual links, it may even be at infinity.
Note: The picture will become clear after you go through the whole post and come back to revisit the above statement.
Further, you may try to understand that since there is no relative velocity between the points, therefore, at this point both the bodies must have the same absolute velocity.
You may appreciate that the other points (other than the 'point' discussed above) on the bodies do not have zero relative velocity(ies). Also, since the links are rigid bodies, the relative velocities are because of the relative rotation about a common point.
The 'point' in fact represents the instantaneous axis of rotation of one link with respect to the other. This point is known as the instant center or velocity pole or centro.
The best way to understand any statement in engineering is by using a sketch or a diagram.
We take up Fig. 1 first. From the Fig. 1, you may notice that the link-2 can only rotate with respect to the fixed link-1, around point O21. Please note that the notation O21 indicates the instant center of rotation of link-2 relative to link-1.
Here, O21 doesn't have any motion, hence is called the Fixed Center. Now, for any point on the link-2 the direction of rotation is perpendicular to it. The magnitude of the velocity is is proportional to the radius from the center O21.
Next, we take up Fig. 2. In this case the fixed centers are O21 and O41. The instant center of rotation of link-3 relative to link-2 will be at the point O32, the point at which the two are mechanically connected. Another point to be noted is that O32 could also be considered as the instant center of link-2 relative to link-3. Similarly the other point is O34 is also an instant center. You may take a note of the fact that because the center O32 has come into existence due to the mechanical connection between link-3 and link-2, as such its position with respect to either Link-2 or link-3 remains unchanged, it may be called the permanent center.
You may appreciate that it is always possible, and is also easy, to identify the permanent centers by merely looking at the diagram. The method proves to be useful because of the fact that the links having complicated motion, such as link-3, have instant centers, relative to the members they are mechanically connected to and also have instant centers, relative to all the other members in the mechanism. Moreover, it can be seen that the instant center O31 is of great importance because (considering the velocities) it is the center of rotation of link-3 with respect to link-1.
In the preceding post you have already seen that the velocity of any point on a link is proportional to the distance from the point to the center of the link's rotation relative to the ground. Therefore, if the velocity of any point on link-3 and the location of the instant center O31 are known, the velocities of all the points on link-3 can be determined by drawing similar triangles which gives the desired proportional relationships.
Now, our real problem at this point is to locate the center O31. This brings us to Fig. 3 posted below:
The procedure of finding O31 is shown in the Fig. 3. The centers O32 and O34 have velocities perpendicular to the link-2 and Link-4 respectively. As such if O32 and O34 are considered as pints on link-3, the instant center of rotation of link-3, relative to the ground will lie on the lines which are perpendicular to the velocities of O32 and O34; thus it is an extensions of the lines O21-O34 and O41-O34 which intersect at O31.
P.S. Your views and comments are solicited, please don't forget to share if you find the post interesting and worth sharing!!!
The method of Instant centers is one of the easiest means of determination of the velocity of a point in a mechanism. It is based upon the assumption that at any given instant the motion of one body, with respect to another body, is such that there is a 'point', in the plane of motion, where the two bodies have no relative velocity. This point may or may not lie on the actual links, it may even be at infinity.
Note: The picture will become clear after you go through the whole post and come back to revisit the above statement.
Further, you may try to understand that since there is no relative velocity between the points, therefore, at this point both the bodies must have the same absolute velocity.
You may appreciate that the other points (other than the 'point' discussed above) on the bodies do not have zero relative velocity(ies). Also, since the links are rigid bodies, the relative velocities are because of the relative rotation about a common point.
The 'point' in fact represents the instantaneous axis of rotation of one link with respect to the other. This point is known as the instant center or velocity pole or centro.
The best way to understand any statement in engineering is by using a sketch or a diagram.
We take up Fig. 1 first. From the Fig. 1, you may notice that the link-2 can only rotate with respect to the fixed link-1, around point O21. Please note that the notation O21 indicates the instant center of rotation of link-2 relative to link-1.
Here, O21 doesn't have any motion, hence is called the Fixed Center. Now, for any point on the link-2 the direction of rotation is perpendicular to it. The magnitude of the velocity is is proportional to the radius from the center O21.
Next, we take up Fig. 2. In this case the fixed centers are O21 and O41. The instant center of rotation of link-3 relative to link-2 will be at the point O32, the point at which the two are mechanically connected. Another point to be noted is that O32 could also be considered as the instant center of link-2 relative to link-3. Similarly the other point is O34 is also an instant center. You may take a note of the fact that because the center O32 has come into existence due to the mechanical connection between link-3 and link-2, as such its position with respect to either Link-2 or link-3 remains unchanged, it may be called the permanent center.
You may appreciate that it is always possible, and is also easy, to identify the permanent centers by merely looking at the diagram. The method proves to be useful because of the fact that the links having complicated motion, such as link-3, have instant centers, relative to the members they are mechanically connected to and also have instant centers, relative to all the other members in the mechanism. Moreover, it can be seen that the instant center O31 is of great importance because (considering the velocities) it is the center of rotation of link-3 with respect to link-1.
In the preceding post you have already seen that the velocity of any point on a link is proportional to the distance from the point to the center of the link's rotation relative to the ground. Therefore, if the velocity of any point on link-3 and the location of the instant center O31 are known, the velocities of all the points on link-3 can be determined by drawing similar triangles which gives the desired proportional relationships.
Now, our real problem at this point is to locate the center O31. This brings us to Fig. 3 posted below:
The procedure of finding O31 is shown in the Fig. 3. The centers O32 and O34 have velocities perpendicular to the link-2 and Link-4 respectively. As such if O32 and O34 are considered as pints on link-3, the instant center of rotation of link-3, relative to the ground will lie on the lines which are perpendicular to the velocities of O32 and O34; thus it is an extensions of the lines O21-O34 and O41-O34 which intersect at O31.
P.S. Your views and comments are solicited, please don't forget to share if you find the post interesting and worth sharing!!!
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