Why the Velocity Analysis is required:
In the past few posts we have talked about the geometry of some of the mechanisms. When we go for designing a mechanism, which will be a part of a machine, the mere knowledge of geometry is not enough. We all know that machines are required for doing a specific work, which is not possible without the motion of the mechanisms, that are a part of the machine.
Consequently, the analysis and calculation of motion, of the mechanisms, is a prerequisite to designing.
The simplest method of analysis is to pick a point in the mechanism and write down the equation of displacement of the point as a function of time. There are difficulties in formulating simple equations for all the mechanisms, except the Crank & Slider and some of its variants. One has to take the help of some kind of Vector Analysis. Some of the commonly used methods will be discussed.
Finding linear velocities by Composition and resolution:
The mechanism, shown above, is a Four-Bar mechanism in which the Driver Link-2 is transmitting motion to Link-4. The pin connected, intermediate link-3 is capable of transmitting motion or force either under tension or compression, therefore the line of transmission must lie on Link-3.
Suppose, the driver has an angular velocity of ω2 in a anticlockwise direction, then point 'A', will have velocity (with respect to the fixed link) equal to the Vector AM2, which is equal, in magnitude, to O2Aω2. The component of this velocity, which actually drives the mechanism, AS2 lies along the Link-3. Since the Link-3 is rigid, point 'B' will have the same component of velocity - (BS4 = AS2). Considering that the Link-4 is constrained to rotate about point O4 and the velocity of point 'B' must be perpendicular to link-4 and the magnitude, that it has, a component BS4 along Link-3. The required velocity of point 'B' as given by the Vector BM4.
Further we have the angular velocity expressed as;
Where ω = angular velocity
V = absolute Velocity of a point on the body
R = radius from the centre of rotation
Therefore, we have;
In the past few posts we have talked about the geometry of some of the mechanisms. When we go for designing a mechanism, which will be a part of a machine, the mere knowledge of geometry is not enough. We all know that machines are required for doing a specific work, which is not possible without the motion of the mechanisms, that are a part of the machine.
Consequently, the analysis and calculation of motion, of the mechanisms, is a prerequisite to designing.
The simplest method of analysis is to pick a point in the mechanism and write down the equation of displacement of the point as a function of time. There are difficulties in formulating simple equations for all the mechanisms, except the Crank & Slider and some of its variants. One has to take the help of some kind of Vector Analysis. Some of the commonly used methods will be discussed.
Finding linear velocities by Composition and resolution:
The mechanism, shown above, is a Four-Bar mechanism in which the Driver Link-2 is transmitting motion to Link-4. The pin connected, intermediate link-3 is capable of transmitting motion or force either under tension or compression, therefore the line of transmission must lie on Link-3.
Suppose, the driver has an angular velocity of ω2 in a anticlockwise direction, then point 'A', will have velocity (with respect to the fixed link) equal to the Vector AM2, which is equal, in magnitude, to O2Aω2. The component of this velocity, which actually drives the mechanism, AS2 lies along the Link-3. Since the Link-3 is rigid, point 'B' will have the same component of velocity - (BS4 = AS2). Considering that the Link-4 is constrained to rotate about point O4 and the velocity of point 'B' must be perpendicular to link-4 and the magnitude, that it has, a component BS4 along Link-3. The required velocity of point 'B' as given by the Vector BM4.
Further we have the angular velocity expressed as;
ω = V/R
Where ω = angular velocity
V = absolute Velocity of a point on the body
R = radius from the centre of rotation
Therefore, we have;
ω2 = AM2/O2A and also ω4 = BM4/O4B
The ratio ω2/ω4 = AM4/O2A x O4B/BM4
The line O2F2 and O4F4 pass though the centres of ration O2 and O4 respectively. These are also perpendicular to the line of transmission.
Further, the triangles O2AF2 and AM2S2 are similar, since the corresponding sides are perpendicular. The same applies to treiangles O4BF4 and BM4S4, also the triangles LO2F2 and LO4F4 are similar.
Now, by proper substitution, the ratio equation can be written as :
ω2/ω4 = O4F4/O2F2
and also
ω2/ω4 = O4L/O2L
The relationship expressed by the above equations is called the Angular Velocity Theorem, which is stated as:
The angular velocities of the driver and the follower are inversely proportional to the length of the perpendiculars from the centres of rotation to the line of transmission OR inversely as the segments into which the line of centres is cut by the centre line of transmission.
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