A free-running wheel arrangement lets each wheel rotate on its own bearings around a fixed axle. On a curve, equal-diameter wheels following concentric paths need different rotational speeds. The calculator finds those speeds from curve radius, wheel-path spacing and a reference wheel RPM at the middle of the axle.
Free Running Axle Interactive Calculator
Vary curve radius, wheel spacing, and centerline RPM to see the independent inner and outer wheel speeds on a mine-car free running axle.
Equation Used
The calculation assumes equal rolling diameters and pure rolling along concentric paths of radius R + G/2 and R - G/2. Independent wheels can rotate at the resulting different speeds. A conventional rigid railway wheelset can also accommodate a curve through different effective rolling radii on profiled treads, which this equal-diameter model does not represent.
- Curve radius R is measured to the axle centerline between the two wheels.
- Wheel rolling diameters are equal and there is no tread slip.
- The axle is fixed to the frame and each wheel rotates freely on its own bearings.
- Wheel spacing G is treated as the distance between the left and right wheel rolling paths.
Independent wheels around a fixed shaft
The axle supports the wheels through their hub bearings but does not force them to turn together. When the axle points radially across an ideal circular curve, the outer wheel has farther to travel during the same turn. Equal rolling diameters therefore require higher outer-wheel RPM.
The shaft, wheel hubs, bearings, retainers and frame attachments carry the vehicle loads. Their dimensions, lubrication and adjustment depend on the actual assembly; wheel-speed arithmetic does not specify bearing fits or service intervals.
Independent rotation and steering are different questions. A rigid railway wheelset with profiled treads can use different effective rolling radii while both wheels have the same angular speed. Independently rotating wheels change that self-steering behavior and still need appropriate guidance. Independent bearings do not guarantee zero flange contact, wear or slip.
The upper view follows the axle over concentric circular rolling paths. Their curvature uses the entered ratio R/G. The axle stays fixed to its frame while the wheels turn on separate bearings. The two enlarged wheel faces below show their different angular speeds at 8% of the entered rate. This illustrates pure rolling geometry, not wheel-contact forces or vehicle guidance.
Where independent rotation matters
Independent wheel rotation is useful to study in rail vehicles, carts and wheeled carriers whose left and right rolling paths differ. A fixed shaft with bearing-mounted wheels is one way to provide it. Driven road vehicles often use a differential instead, allowing unequal wheel speeds while transmitting torque.
For a rail vehicle, compare the complete wheel profiles, suspension and guidance system. The presence of a curve alone does not establish which wheel arrangement is appropriate. This calculator can quantify an ideal speed difference, but cannot select a mine-car axle or assess a particular vehicle's running behavior.
Wheel speeds from concentric paths
Let R be the radius to the midpoint between the rolling paths and G their lateral separation. G is a rolling-path dimension, not necessarily the nominal track gauge measured between rail faces. With the axle radial to the curve, the inner and outer path radii are R − G/2 and R + G/2.
Center RPM n_c is a reference wheel speed at radius R for the assumed rolling diameter. It is not the vehicle's yaw rate and does not require a real wheel at the axle midpoint. If centerline speed v and common wheel diameter D are known, n_c = 60v/(Ï€D), using consistent length units.
The outputs are n_inner = n_c(R − G/2)/R and n_outer = n_c(R + G/2)/R. Speed Difference is n_outer − n_inner = n_c G/R. Outer Increase is 100(n_outer/n_inner − 1), expressed relative to the inner wheel, not relative to n_c.
The model assumes pure rolling, equal wheel diameters and a common angular rate about the curve center. R must exceed G/2; every setting offered by these controls satisfies that requirement. Speed difference is not a prediction of actual slip, contact force, bearing load or wear.
Example: a 4.95 m curve
At the defaults R = 4.95 m, G = 0.9 m and n_c = 120 RPM, the inner path radius is 4.5 m and the outer radius is 5.4 m. The wheel speeds are 109.09 and 130.91 RPM.
The difference is 21.82 RPM. Relative to the inner wheel, the outer wheel is 20% faster. These values describe the assumed rolling geometry; they do not establish a bearing rating or whether a vehicle can negotiate the curve.
At R = 2 m, G = 1.2 m and n_c = 200 RPM, the outputs become 140 and 260 RPM, a difference of 120 RPM and an outer increase of 85.71%. At R = 12 m, G = 0.45 m and n_c = 80 RPM, they become 78.5 and 81.5 RPM, a difference of 3 RPM and an increase of 3.82%.
Increasing reference RPM scales both speeds and their difference equally. It does not change the percentage increase, which depends only on G/R.
Independent wheels and rigid wheelsets
Independent wheels provide separate angular speeds and require separate rotating bearing interfaces. A rigid wheelset ties the two angular speeds together, while wheel profiles and contact positions can create different effective rolling radii. A geared differential permits different speeds within a driven axle.
These arrangements have different guidance and load paths. Their bearing life, cost, permissible curve radius and maintenance needs cannot be ranked from the three inputs here. Vehicle-specific drawings and running-performance data are needed for those decisions.
Frequently asked questions
What does Center RPM mean?
It is the reference RPM for the assumed wheel diameter moving along the midpoint path. It is not the rate at which the vehicle turns about the curve center.
Is Wheel Spacing the same as track gauge?
Here it is the distance between the wheel rolling paths. Nominal railway gauge is measured between rail faces and may differ from that distance.
Why is the outer wheel faster?
For equal rolling diameters, the longer outer path requires more wheel revolutions during the same time.
Does the percentage compare the outer wheel with Center RPM?
No. Outer Increase compares outer RPM with inner RPM. Speed Difference is the arithmetic difference between those two wheel speeds.
Must a rigid railway wheelset skid on every curve?
No. Profiled wheels can develop different effective rolling radii even though their angular speeds are equal. The calculator assumes equal diameters and omits that behavior.
Does free rotation eliminate flange wear?
No. Alignment, guidance, wheel profiles and contact forces still matter. They are not modeled here.
Can these results size bearings or set preload?
No. The outputs contain no load, life, fit or temperature calculation. Use the assembly and bearing specifications for those tasks.
Are the animated rail shapes the calculated paths?
Yes. Both drawn paths share a curve center, with radii R − G/2 and R + G/2 in a common scale. The view follows the axle, and wheel rotation is slowed to 8% for inspection.
Wheelset references
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