Machine kinematics · Slotted rocker

Crank-Shaper Quick Return

A crank pin slides in a real rocker slot and drives a horizontal ram through a connecting rod. Tune the crank radius to change the time ratio, then test which dimensions change stroke without changing timing.

Ram displacement from rod-length datum $0$

Signed output velocity $0$

Output acceleration $0$

One shaft turn $0$

Working / return time $0$

Ram stroke $0$

Return time $0$

Contact / linkage residual $0$

Turn & compare

Give the return less time.

Shaft phase

\theta

240.00\,{}^\circ

Prescribed shaft rate

\omega

60.00\,{}^\circ/\mathrm s

Crank radius / fixed-pivot spacing

r/d

0.45\,

Rocker pivot to rod joint

L

2.50\,\mathrm m

Connecting rod length

c

1.60\,\mathrm m

Right and up are positive in the motion plane; depth points toward you. Shaft phase increases counterclockwise from the rightward crank reference. Controls use degrees; internal calculations and CSV use radians, metres and seconds.

The fixed-pivot spacing is 1.2 metres. The lower rocker pivot, rotating crank pin, rocker tip and connecting rod determine the right-hand ram branch. The rail height is the lever length minus fixed-pivot spacing. Negative ram velocity is the slower working stroke; positive velocity is the faster return.

Changing the connecting rod changes the speed profile but, within this nonsingular parameter box, not the reversal phases or stroke. A longer rocker changes stroke without changing the time ratio. Output displacement subtracts the connecting rod length from the ram coordinate.

Playback runs one phase interval and stops. All curves and exports are exact ideal kinematics sampled over one turn, not acquired measurements. Positions, velocities and accelerations share the same model. Mass, friction, backlash, elastic deformation, real contact loads and drive safety are outside scope.

The real slot and shoe are shown with schematic transverse dimensions. This is the oscillating crank-and-slotted-lever topology, rather than the fully rotating slotted-arm inversion. No material is actually cut.

Ideal rigid planar links, open slot and horizontal prismatic guide. Constant prescribed input rotation. The working stroke is a kinematic label; no cutting load, material removal, inertia, friction or motor dynamics.

Start with the essentials

Focus question: Which dimension changes the return time without changing motor speed?
One-sentence intuition: The rocker reaches its extremes where its slot is tangent to the crank circle. Those points divide the turn into unequal angles.

Core mathematical model

Crank pin and rocker direction

\mathbf A=(0,-d),\quad\mathbf P=(r\cos\theta,r\sin\theta),\quad\mathbf B=\mathbf A+L\frac{\mathbf P-\mathbf A}{|\mathbf P-\mathbf A|}

The crank pin slides along the rocker slot; the rocker tip stays at fixed distance from its pivot.

Right-hand ram branch

H=L-d,\quad Q_x=B_x+\sqrt{c^2-(H-B_y)^2},\quad Q_y=H

The positive square root selects the displayed connecting-rod assembly. Output displacement is the horizontal coordinate minus the rod length.

Quick-return time ratio

\alpha=\arcsin(r/d),\quad\frac{T_{\rm work}}{T_{\rm return}}=\frac{\pi+2\alpha}{\pi-2\alpha}

For constant shaft speed the two input angle intervals set the two durations.

Stroke and time scaling

S=2L\frac rd,\quad v=\omega\frac{dQ_x}{d\theta},\quad a=\omega^2\frac{d^2Q_x}{d\theta^2}

These stroke extrema hold throughout the permitted rod-length box, where ram position is monotone in rocker angle.

Common difficulties

A kinematic model is not a force model

Typical misconceptionSmooth motion proves the real drive can run at the selected rate.

Better mental modelContact loads, inertia, friction, elastic deformation and drive design need separate models. Parts have schematic thicknesses.

Constant input is not constant output

Typical misconceptionThe working stroke has constant cutting speed.

Better mental modelBoth output speeds vary continuously. Quick return compares full stroke durations, not a constant speed ratio at every instant.

Run the experiment

01
Run one turn
Choose the motion-plane view and run or drag the coral crank.
What to observe: The same ram stroke takes different times in opposite directions.
02
Change the radius ratio
Compare mild and strong quick-return presets.
What to observe: A larger radius ratio creates a stronger timing asymmetry and a longer stroke.
03
Separate stroke from timing
Change rocker length and connecting rod length independently.
What to observe: Rocker length changes stroke; rod length changes curve shape. Neither changes the ideal time ratio.

Geometry & apparatus: CMU · Cams · NITK Virtual Labs · Crank and Slotted Mechanism