The Whitworth quick return mechanism is shown in the figure with link lengths as follows: OP = 300 mm, OA = 150 mm, AR = 160 mm, RS = 450 mm.

The quick return ratio for the mechanism is _________ (round off to one decimal place).

The correct answer is 2.

Step-by-Step Explanation:

We are given the following link lengths from the mechanism diagram:
- Distance between the fixed centers (pivots O and A): OA = 150 mm
- Length of the driving crank: OP = 300 mm
- Other link lengths are AR = 160 mm and RS = 450 mm (these relate to the connecting rod and slider but are not needed for calculating the quick return ratio).

In a Whitworth quick return mechanism, the extreme (limiting) positions of the slotted lever occur when the driving crank OP is perpendicular to the slotted member/lever AP. This creates a right-angled triangle OAP, where the angle at the crank pin P is 90°.

Based on the geometry shown in the diagram, let α be the angle described by the crank during the return stroke, and β be the angle described during the cutting (forward) stroke.

From the right-angled triangle OAP, the half-angle of the return stroke (α/2) satisfies the cosine relation:

$\cos \left(\frac{\alpha }{2}\right)=\frac{\mathrm{OA}}{\mathrm{OP}}=\frac{150}{300}=0.5$

Solving for the half-angle:

$\frac{\alpha }{2}=60°$

Multiplying by 2 to find the total return stroke angle (α):

$\alpha =120°$

The cutting stroke angle (β) is the remaining angle in a full crank rotation:

$\beta =360°-\alpha =360°-120°=240°$

The Quick Return Ratio (QRR) is the ratio of the time taken for the cutting stroke to the time taken for the return stroke, which is proportional to their respective angles:

$\mathrm{QRR}=\frac{\beta }{\alpha }=\frac{360°-120°}{120°}=\frac{240°}{120°}=2$

Thus, the quick return ratio for the mechanism is 2.