Henry T. Brown's movement 154 converts continuous disk rotation into alternating rectilinear motion. Successive studs on the disk act on one arm of a bell crank; a weighted cord passing over a fixed pulley is attached to the other arm and returns the lever after each stud clears. Calculate event rate, time between events, ideal cord tension and counterweight moment.
Studded Disk to Bell-crank Traverse Interactive Calculator
Vary disk speed, equally spaced stud count, weight-side bell-crank arm, counterweight mass and the entered neutral deviation between the cord force and its maximum-torque orientation. The animation follows Brown's studded disk, rigid bell crank, fixed pulley and hanging weight. Results give event timing and ideal quasi-static counterweight values, not impact force, output stroke or a mechanism rating.
Equation Used
- N studs are equally spaced and each produces one event per disk revolution.
- The cord is massless and inextensible, and the pulley is frictionless for the ideal tension calculation.
- Standard gravity is fixed at 9.80665 m/s^2.
- r is the entered distance from bell-crank pivot to cord attachment; alpha is the entered neutral deviation from the maximum-moment orientation.
- The disk turns at constant speed. After a stud releases, the drawing shows a smooth return driven by the hanging weight. Return speed and acceleration are illustrative rather than calculated.
- The source illustration does not specify the bell-crank swing, stud impact law, output stroke, speed limit, strength, wear or manufacturing dimensions.
How Brown's Studded-disk Bell Crank Works
Henry T. Brown describes movement 154 as circular motion converted into alternating rectilinear motion. Studs fixed around the rotary disk act in succession on the short arm of a pivoted bell crank. A cord attached to the other rigid arm passes over a fixed pulley and supports a hanging weight.
During the illustrated drive interval, one stud acts on the short arm and rotates the bell crank. After that stud clears, the hanging weight returns the crank through the cord. The next equally spaced stud repeats the event. The output therefore alternates and returns; it does not accumulate one-way traverse.
Reading the animation
- The disk rotates continuously and carries exactly the entered number of equally spaced studs.
- Only one stud is identified as acting on the short arm during a drive interval.
- The two bell-crank arms remain rigid and perpendicular about one fixed pivot.
- The cord remains tangent to the fixed pulley and keeps a constant total drawn length while the weight moves vertically.
- The weight returns the crank during the clearance interval.
What This Model Is Useful For
Use the model to follow the repeating drive-and-return motion and to calculate event timing, ideal cord tension and the counterweight moment at the entered neutral angle.
It is not a component-selection or manufacturing model. A working mechanism requires verified stud and follower profiles, swing limits, contact speed, impact and rebound behavior, cord and pulley losses, clearances, materials, stresses, fatigue, guarding and application-specific dimensions.
Event Timing and Ideal Counterweight Relations
For disk speed n in revolutions per minute and N equally spaced studs, one revolution produces N events:
The interval between ideal equally spaced events is:
For counterweight mass m and standard gravity g = 9.80665 m/s squared, ideal cord tension is T = mg. If r is the weight-side arm and alpha is the entered neutral deviation from the maximum-moment orientation, the ideal neutral return torque is:
These are timing and quasi-static weight relations, not contact-force or motor-torque predictions.
Worked Example: Three Studs and a Five-kilogram Weight
At 30 rpm with three equally spaced studs, the event rate is 30 times 3 = 90 events per minute and the event interval is 60000/90 = 666.7 ms.
For m = 5 kg, ideal cord tension is 5 times 9.80665 = 49.03 N. With r = 100 mm = 0.100 m and alpha = 15 degrees, ideal neutral return torque is 49.03 times 0.100 times cos(15 degrees) = 4.736 N m.
The example does not infer swing angle, output stroke, impact load or allowable operating speed.
What the model includes
- Historical arrangement: a rotating studded disk, one arm of a bell crank acted on by the studs, and a weighted cord over a pulley attached to the other arm, producing alternating rectilinear motion.
- In the animation: equally spaced studs, one indicated drive contact, two rigid bell-crank arms, a fixed pivot, tangent cord, fixed pulley, vertically moving weight and a clearance/return interval.
- Calculated quantities: nN, 60000/(nN), mg and mg r cos(alpha).
- Outside this model: swing amplitude, output stroke, drive-contact force, motor torque, impact, rebound, friction, wear, safe speed, strength, life or manufacturing geometry.
The disk turns continuously and the weight returns the crank after release. The drawn return is smooth and illustrative; it does not calculate impact, acceleration, rebound or a working design's required clearances.
Frequently asked questions
Does the output advance continuously in one direction?
No. The output alternates: a stud drives the bell crank and the weighted cord returns it after the stud clears.
Why does the default drawing have three studs?
The default follows the three studs shown in Brown's movement 154. Changing the stud count changes the number of events per disk revolution.
Does the event rate equal disk rpm?
Only for one stud. With N equally spaced studs, each revolution produces N events, so the ideal event rate is nN.
Is mg the force exerted by a stud?
No. mg is the ideal hanging-weight cord tension, assuming a massless cord and frictionless pulley. Dynamic stud-contact force is not calculated.
What does neutral return torque mean?
It is the ideal quasi-static moment of the counterweight about the bell-crank pivot at the entered neutral cord deviation. It is not motor torque or an impact load.
Can the calculator determine output stroke?
No. The historical drawing does not give the dimensions and contact profile needed to calculate bell-crank swing and output stroke.
References
- Henry T. Brown, Five Hundred and Seven Mechanical Movements, movement 154: studs on a rotary disk act on one bell-crank arm; a weighted cord over a pulley is attached to the other, converting circular motion to alternating rectilinear motion.
- NIST Reference on Constants, Units, and Uncertainty: standard acceleration of gravity, 9.80665 m/s squared.
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