Relative Velocity Analysis-GATE Mechanical Notes

GATE Mechanical Short Notes on Relative Velocity Analysis. With this help of these you will be able to describe the velocity of a rigid body in terms of translation and rotation components and perform a relative-motion velocity analysis of a point on the body. 

As the slider block A moves horizontally to the left with vA, it causes the link CB to rotate counterclockwise. Thus vB is directed tangent to its circular path.

Which link is undergoing general plane motion? How can its angular velocity, ω, be found?

Planetary gear systems are used in many automobile automatic transmissions. By locking or releasing different gears, this system can operate the car at different speeds.

How can we relate the angular velocities of the various gears in the system?

Relative Motion Analysis: Displacement

When a body is subjected to general plane motion, it undergoes a combination of translation and rotation.

Point A is called the base point in this analysis. It generally has a known motion. The x’-y’ frame translates with the body, but does not rotate. The displacement of point B can be written:

Relative Motion Analysis: Velocity

The velocity at B is given as : (drB/dt) = (drA/dt) + (drB/A/dt) or vB = vA + vB/A 

Since the body is taken as rotating about A, vB/A = drB/A/dt = ω x rB/A Here ω will only have a k component since the axis of rotation is perpendicular to the plane of translation.

When using the relative velocity equation, points A and B should generally be points on the body with a known motion. Often these points are pin connections in linkages.

Here both points A and B have circular motion since the disk and link BC move in circular paths. The directions of vA and vB are known since they are always tangent to the circular path of motion.

When a wheel rolls without slipping, point A is often selected to be at the point of contact with the ground. Since there is no slipping, point A has zero velocity

Furthermore, point B at the center of the wheel moves along a horizontal path. Thus, vB has a known direction, e.g., parallel to the surface.

Procedure of analysis

The relative velocity equation can be applied using either a Cartesian vector analysis or by writing scalar x and y component equations directly.

Scalar Analysis:

  1. Establish the fixed x-y coordinate directions and draw a kinematic diagram for the body. Then establish the magnitude and direction of the relative velocity vector vB/A
  2. Write the equation vB = vA + vB/A and by using the kinematic diagram, underneath each term represent the vectors graphically by showing their magnitudes and directions.
  3. Write the scalar equations from the x and y components of these graphical representations of the vectors. Solve for the unknowns.

Vector Analysis:

  1. Establish the fixed x-y coordinate directions and draw the kinematic diagram of the body, showing the vectors vA, vB, rB/A and ω. If the magnitudes are unknown, the sense of direction may be assumed
  2. Express the vectors in Cartesian vector form and substitute into vB = vA + ω x rB/A. Evaluate the cross product and equate respective i and j components to obtain two scalar equations
  3. If the solution yields a negative answer, the sense of direction of the vector is opposite to that assumed.

Example:

Given: Block A is moving down at 2 m/s.

Find: The velocity of B at the instant 

Plan: 

  1. Establish the fixed x-y directions and draw a kinematic diagram.
  2. Express each of the velocity vectors in terms of their i, j, k components and solve vB = vA + ω x rB/A.

Solution:

vB = vA + ωAB x rB/A

vBi = -2 j + (ω k x (0.2 sin 45 i - 0.2 cos 45 j ))

vBi = -2 j + 0.2 ω sin 45 j + 0.2 ω cos 45 i

Equating the i and j components gives:

vB = 0.2 ω cos 45

0 = -2 + 0.2 ω sin 45

Solving: ω = 14.1 rad/s or ωAB = 14.1 rad/s k

vB = 2 m/s or vB = 2 m/s i