coriolis component of acceleration

CORIOLIS COMPONENT OF ACCELERATION

1. What is it?

• We have seen that the total acceleration of a point with respect to another point on a rigid link is the vector sum of its radial and tangential acceleration components i.e., a_AB = a^r_BA + a^t_AB. This is true only when the distance between the two points is fixed.

• On the contrary, when the distance between the two points is not fixed (i.e., the distance varies), then the total acceleration will have one more additional component known as Coriolis component of acceleration. It is represented by a^c_AB. In such cases, the total acceleration of that link is the vector sum of its radial, tangential and Coriolis acceleration components i.e., a_AB = a^r_AB + a^t_AB + a^c_AB.

• The Coriolis component of acceleration happens only when a point known as coincident point, on one link is sliding along another rotating link. In other words, whenever a coincident point exists in a mechanism, we have to consider Coriolis component of acceleration.

• Examples: The mechanisms such as crank and slotted lever mechanism, Whitworth quick return mechanism, oscillating cylinder mechanism and swivelling joint mechanism require the calculation of Coriolis component of acceleration.

2. Magnitude of Coriolis Component of Acceleration

Consider a link OA which has a slider B which is free to slide, as shown in Fig.2.23. With O as centre, let the link OA move, with a uniform angular velocity w, to its new position OA' such that it is displaced dθ in time dt. The slider B moves outwards with sliding velocity v^s on link OA and occupies the position B' in the same interval of time.

The point C is the coincident point with slider on link OA.

The motion of slider can be explained in the following three stages:

(i) Motion from C to C' due to rotation of link OA. It is caused by tangential component of acceleration a'.

(ii) Motion from C' to B₁ due to outward motion along the link OA. It is caused by radial component ar (or sliding component, a^s) of acceleration.

(iii) Motion from B₁ to B' is caused by Coriolis component of acceleration a^c.

From the geometry of Fig.2.23,

For small value of dθ,

Arc B₁B' = B₁B'

From equations (ii) and (iii), we get

where

v^s = Velocity of sliding, and

ω = Angular velocity of link OA.

3. Direction of Coriolis Component of Acceleration

• The direction of Coriolis component of acceleration a is to rotate the sliding velocity vector vs in the same sense as the angular velocity of OA.

• The direction of Coriolis component of acceleration is obtained by rotating the velocity of sliding vector v3 through 90° in the direction of rotation of angular velocity, ω.

• Since the direction of a^c depends on the direction of v^s and ω, therefore there are four possible cases of direction of a^c, as shown in Fig.2.24.

• It may be noted that in equation (2.15), the outward direction of velocity of sliding v^s is assumed as positive and the counter clockwise direction of ω is assumed as positive.

Example 2.11

The driving crank AB of the quick-return mechanism, as shown in Fig.2.25, revolves at a uniform speed of 200 rpm. Find the velocity and acceleration of the tool-box R, in the position shown, when the crank makes an angle of 60° with the vertical line of centres PA. What is the acceleration of sliding of the block at B along the slotted lever PQ?

Given data:

N_BA = 200 rpm

Solution: Relative velocity method.

Procedure:

Step 1: Configuration diagram: First of all, draw the configuration diagram, to some suitable scale (say, 1 cm 50 mm), as shown in Fig.2.26(a).

Step 2: Velocity of input link:

Step 3: Velocity diagram: Now draw the velocity diagram, to some suitable scale (say, 1 cm = 0.75 m/s), as shown in Fig.2.26(b).

Step 4: Velocity of various links:

By measurement from the velocity diagram, we get

Step 5: Acceleration diagram: The values of radial, tangential and Coriolis components of acceleration of various links are calculated as shown in Table 2.9.

Now using the known values of magnitude and direction of acceleration components, the acceleration diagram can be constructed, to some suitable scale (say, 1 cm = 5.5 m/s²), as shown in Fig.2.26(d).

Step 6: Acceleration of various links:

By measurement from the acceleration diagram, we get