A bell crank is a rigid two-arm lever rotating about a fixed pivot. This calculator models only ideal tangential endpoint forces and circular endpoint travel; connected pushrods generally introduce position-dependent force and displacement geometry.
Transmission by Bell Crank Interactive Calculator
Enter two pivot-to-pin arm lengths, their included angle and a one-sided rotation from neutral. Results apply to ideal tangential endpoint quantities only.
Equation Used
- The bell crank is rigid and rotates about one fixed frictionless pivot.
- Input and output pins are at the entered radii from the pivot.
- Forces are tangential to the respective pin circles, so each moment arm equals its entered radius.
- Entered oscillation is the magnitude of one-sided angular rotation from neutral; both arms rotate through that same angle.
- Pushrod direction, anchor geometry, bearing friction, clearance, compliance, inertia, strength and safe design are omitted.
Two Rigid Arms About One Fixed Pivot
Bell-crank mechanisms in patents and technical manuals use two arms joined at a common pivot, with an actuator or push-pull rod connected to one arm and a driven rod or member connected to the other. Rotating the crank redirects motion between the two connections.
In this model, endpoint forces are tangential to the pin circles. Static torque balance then depends only on the two radii. Real straight pushrods are generally not tangent throughout a stroke.
What This Calculator Does
The tool reports ideal tangential output/input force ratio, output/input endpoint arc-travel ratio, and each pin's one-sided circular arc travel for the entered rotation.
The included angle changes the bell-crank shape and direction redirection in the diagram, but not those ideal tangential ratios.
Ideal Tangential Lever Relations
Fout/Fin = Lin/Lout
sin = Lintheta
sout = Louttheta
sout/sin = Lout/Lin
Angle theta is converted from degrees to radians for the arc-length equations.
Worked Ideal Example
For Lin=2 relative units, Lout=1 relative unit and a one-sided rotation of 15 degrees (0.2618 rad), the ideal tangential force ratio is 2.00, the endpoint arc-travel ratio is 0.50, input-pin arc travel is about 0.52 relative unit and output-pin arc travel is about 0.26 relative unit.
Model Boundary
- Mechanism: a two-arm bell crank rotates about a common fixed pivot and links an input member to an output member.
- Calculated: rigid circular endpoint paths and ideal tangential static torque balance.
- Not calculated: fixed pushrod anchors, changing pressure angles, finite rod lengths, exact slider travel, dead-centre conditions, side loads, bearing friction, backlash, clearance, compliance, inertia, acceleration, vibration, stress, buckling, fatigue, pin bearing, materials, fasteners, stops, guarding or safe design.
Frequently Asked Questions
Why does included angle not change the displayed ratios?
Under the stated tangential-force and circular-arc assumptions, torque and arc length depend on radius. Included angle only rotates one arm relative to the other.
Do both arms rotate through the same angle?
Yes. They are parts of one rigid crank.
Does the force ratio apply to straight pushrods?
Only instantaneously when each pushrod force is perpendicular to its arm. Otherwise effective moment arms must be calculated from full geometry.
Is arc travel the same as axial pushrod travel?
No. It is distance along the pin's circular path.
Does the model include pivot friction?
No.
Can the results size a bell crank or linkage?
No. Complete loads, geometry, strength, fatigue, joints and safety requirements are outside the model.
References
- U.S. Patent 6,076,423, Bellcrank mechanism: bell crank supported by a pivot, with distinct input and output arms and actuator-driven rotation.
- U.S. Patent 3,805,635, Remote control linkage for automotive transmission: two-armed bell-crank lever fulcrumed at the junction of its arms and connected into a push-pull linkage.
- U.S. Army Air Corps Technical Manual 1-408: push-pull rods connected to bell-crank arms and control levers by rod ends.
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
