For an ideal crank circle of radius r and external slotted-lever pivot at center distance d, real tangent dead-center positions require r<d. With α=acos(r/d), return angle=2α, work angle=360°−2α and constant-speed time ratio=work/return.
Slotted Lever · Ideal Quick-Return Angle Split
Explore the exact tangent-angle split of an ideal crank and pivoted slotted lever, including incompatible geometry states.
Equation Used
- The crank pin follows a circle of radius r about a fixed center.
- The slotted lever pivots a distance d from the crank center and its centerline passes through the pin.
- Input speed is constant, so time ratio equals input-angle ratio.
- Real tangent dead centers require r<d; incompatible inputs are not clamped.
- Ram linkage, loads, friction, clearance, wear, strength and safety are excluded.
The two tangent positions divide a crank revolution into the work and return intervals.
Crank pin constrained by a pivoted slot
The crank pin follows a fixed circle and the ideal slotted lever always passes through the external pivot and pin center. The two tangent lever positions define the stroke reversals and split one crank revolution into unequal input-angle intervals.
What the calculator determines
Select center distance and crank radius. For r<d the calculator reports exact work and return input angles, their constant-speed ratio and signed radial clearance d−r. For r≥d it flags incompatible tangent geometry rather than clamping.
Ideal tangent-angle geometry
α=acos(r/d); θreturn=2α; θwork=360°−2α; TR=θwork/θreturn; clearance=d−r.
Selected-input example
For d=60 mm and r=30 mm, α=60°, return angle is 120°, work angle is 240° and the constant-speed ratio is 2.00:1.
Angle split only
Lever output length, connecting link and ram guide are absent, so ram stroke, velocity and force are not solved. Clearance, friction, impact, loads, inertia, wear, strength and safety are excluded.
Questions
Why are the two motion intervals unequal?
The lever reverses at the two tangents from its pivot to the crank circle. These tangent points divide the circle into unequal arcs; a constant-speed crank takes longer to traverse the larger arc.
What happens when r is at least d?
No real tangent from the external pivot to the crank circle exists, so the quick-return angle split is undefined.
References
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
