Kinematic Analysis of Mechanisms(1)

General Plane Motion Representation

General plane motion include rotational motion and transitional motion. Purelly translational motion happens when every particle of body has the same instantaneous velocity as every other particle then path traced out by any particle is exactly parallel to the path traced out by every other particle in the body. Under the translational motion the change in position of a rigid body is specified completely by three coordinates such as x, y and z giving displacement of any point such that the center of mass is fixed to the rigid body.

Purely rotational motion happens when every particle in body moves in a circle about a single line. This line is called axis of rotation. Then the radius vectors from axis to all particles undergo the same angular displacement in same time. The axis of rotation need not to go through the body. In general any rotation can be specified completely by three angular displacements with respect to rectangular-coordinate axes x, y, and z. Any change in the position of the rigid body is then completely described by three translational with three rotational coordinates.

Fig: Motion of a link

Consider the rigid link AB which moves from the position AB to A₁B₁ as shown in fig. The link has both rotational and transitional motion. The fig (a) has first transitional AB to A₁B’ and then rotational about A₁ from B’ to B_1. Similarly in the fig (b) the motion is rotational and then transitional. Such a motion of link AB to A₁B₁ is an example of combined motion of rotation and translation. Then it being immaterial whether the motion of rotation takes first or translation.

Instantaneous centers of velocity (ICV)

Any point on a rigid body or on its extension that has zero velocity is called Instantaneous Center of Velocity of the body. Assuming one knows the ICV of a body one can estimate the velocity of any point A on the body using the equation and recognizing that be definition. The instant centre of rotation also called instantaneous velocity center or also instantaneous centre or instant centre which is the point fixed to a body undergoing the planar movement that has zero velocity at a particular instant of time. At this instant the velocity vectors of trajectories of other points in the body generate a circular field around this point which is identical to that is generated by a pure rotation.

Fig: Instantenous center of velocity

In actual practice the motion of link AB is so gradual that it is difficult to see two separate motions. But we see the two separate motions though the point B moves faster than point A. So this combined motion of rotation and translation of link AB may be assumed to be a motion of pure rotation about some of the centre I which is known as instantaneous centre of rotation or velocity (also called centro or virtual centre. Since the points A and B of the link has moved to A₁ and B₁ respectively under the motion of rotation as assumed above. Therefore the position of centre of rotation must lie on the intersection of the right bisectors of chords AA₁ and BB₁. Let these bisectors intersect at I as shown in fig. which is the instantaneous centre of rotation or virtual centre of link AB.

From above we can see that the position of the link AB goes on changing, therefore the centre about which motion is assumed to take place i.e. the instantaneous centre of rotation also goes on changing. So the instantaneous centre of a moving body may be defined as that centre which goes on changing from one instant to another one. The locus of all such instantaneous centres is known as centrode. A line drawn through an instantaneous centre and perpendicular to the plane centre of rotation of motion is called instantaneous axis. The locus of this axis is known as axode.

Methods for Determining the Velocity of a Point on a Link

There are many methods for determining that velocity of any point on a link in a mechanism whose direction of motion (i.e. path) and velocity of some other point on same link is known in the magnitude and direction and there are following two methods.

1. Instantaneous centre method
2. Relative velocity method.

Velocity of a Point on a Link by Instantaneous Centre Method

The displacement of a body (or a rigid link) having motion in one plane, can be considered as a pure rotational motion of a rigid link as a whole about some centre which is known as instantaneous centre or virtual centre of rotation.

Fig: Velocity of a point on a link

Consider the two points A and B on rigid link. Let v_A and v_B be velocities of points A and B whose directions are given by angles α and β which is shown in fig. If v_A is known in the magnitude and direction and v_B in direction only then the magnitude of v_B can be determined by instantaneous centre method.

Draw AI and BI perpendiculars to the directions v_A and v_B respectively. Suppose these lines intersect at I which is known as instantaneous centre or virtual centre of link. The complete rigid link which is to rotate or turn about centre I. Since A and B are the points on a rigid link therefore there cannot be any type of relative motion between them along the line AB.

Now resolving the velocities along AB,

\[\begin{array}{l}
{V_A}\cos \alpha = {V_B}\cos \beta \\
\;\;\;or\;\;\;\frac{{{V_A}}}{{{V_B}}} = \frac{{\cos \beta }}{{\cos \alpha }} = \frac{{\sin (90 - \beta )}}{{\sin (90 - \alpha )}}..............(i)
\end{array}\]

Applying Lami’s theorem to triangle ABI,

\[\frac{{AI}}{{\sin (90 - \beta )}} = \frac{{BI}}{{\sin (90 - \alpha )}}\;\;\;\;\;or\;\;\;\;\frac{{AI}}{{BI}} = \frac{{\sin (90 - \beta )}}{{\sin (90 - \alpha )}}...........(ii)\]

From equation (i) and (ii),

\[\frac{{{V_A}}}{{{V_B}}} = \frac{{AI}}{{BI}}\;\;\;\;or\;\;\;\;\frac{{{V_A}}}{{AI}} = \frac{{{V_B}}}{{BI}} = \omega ................(iii)\]

Where ω = Angular velocity of the rigid link.

If C is any other point on the link, then from the above equation then,

\[\frac{{{V_A}}}{{AI}} = \frac{{{V_B}}}{{BI}} = \frac{{{V_C}}}{{CI}}................(iv)\]

Fron above equation,

• If v_A is known in magnitude and direction and v_B in direction only then velocity of point B or any other point C lying on same link may be determined in magnitude and direction.
• The magnitude of velocities of the points on a rigid link is inversely proportional to the distances from the points to the instantaneous centre and is perpendicular to the line joining the point to the instantaneous centre.

References:
1. H.H. Mabie and C. F. Reinholtz, “Mechanism and Dynamics of Machinery”, Wiley.
2. J.S. Rao & R.V. Dukkipati Mechanisms and Machine Theory, New Age International (P) Limited..
3. J.E. Shigley and J.J. Uicker, Jr., “ Theory of Machines and Mechanisms”, McGraw Hill.
4. B. Paul, “Kinematics and Dynamics of Planar Machinery”, Prentice Hall.
5. C. E. Wilson, J.P. Sadler and W.J. Michels, “Kinematics and Dynamics of Machinery”, Harper Row.