Elastic Spur Gear Mechanism Explained: How It Works, Diagram, Parts, Formula and Uses

A toothed rim coupled to the shaft by springs

Hiscox movement 127 shows an elastic spur gear with the toothed gear loose on its shaft and a cam-like center fixed to the shaft. Compression springs lie between shoulders on the two members. The original source describes the arrangement as intended to prevent backlash.

The animation exposes that spring-coupled load path. For clarity, the central carrier is held stationary while the rim loads and unloads. Both retain their shape; the outer teeth do not distort. Four connected spring symbols shorten as the relative angle increases.

The illustration simplifies the historical center into visible arms and shoulders. The spring symbols follow curved guides to remain legible across the mathematical range. Their dimensions are not an actual spring specification or a reproduction of the historical cam profile.

This is distinct from a split, preloaded scissor gear and from a bonded elastomer coupling. Each construction has its own stiffness, preload and loss characteristics.

Explore a supplied torque–deflection relationship

Enter maximum torque, total torsional stiffness in N·m per degree, and the gear pitch diameter. The stiffness represents the complete spring-coupled assembly; the calculator does not derive it from the four drawn springs.

The replay gradually applies and removes the entered torque. Its live angle and stored energy follow that load. The result cards report values at the entered maximum, so they can be compared consistently while the diagram moves.

Changing pitch diameter changes the physical tangential displacement and pitch-circle force. The drawing automatically uses a common view size; it is not an absolute-scale manufacturing drawing.

Relative angle, energy and tangential force

With torque T and supplied stiffness k in N·m/degree, the relative angle is θ_deg=T/k. Convert to radians using θ_rad=θ_degπ/180.

At pitch radius r=D/2, tangential arc displacement is s=rθ_rad. Use matching length units. The ideal stored elastic energy is U=½Tθ_rad, which requires radians rather than degrees. Tangential pitch-circle force is F_t=T/r, with r in meters for force in newtons.

These relations assume a linear, lossless torsional spring model from zero load. They do not calculate tooth-root stress, contact pressure, radial gear force, damping, preload or a permissible load.

The tangential pitch-circle force describes the load associated with the entered torque. Compare it with the actual gear’s allowable load separately.

Default spring-coupled gear example

For T=40 N·m, k=10 N·m/degree and D=60 mm, the relative angle is 4° or 0.069813 radians. The pitch-circle arc shift is 30×0.069813=2.0944 mm.

The stored energy is ½×40×0.069813=1.3963 J, and the tangential pitch-circle force is 40/0.03=1333.3 N.

Doubling stiffness at the same torque halves relative angle, arc shift and stored energy, while pitch-circle force remains unchanged. Doubling diameter doubles arc shift and halves tangential force, without changing the angle or energy.

Elasticity is not a damping or backlash guarantee

An ideal spring stores and returns energy. Dissipating vibration requires a loss mechanism that is not calculated here. The page therefore makes no numerical noise-reduction or damping claim.

Backlash behavior depends on the arrangement, preload, clearances and direction of loading. The historic mechanism’s intended purpose does not establish zero backlash for an arbitrary assembly.

The sliders allow large calculated angles to expose the mathematical relationship. Real spring travel, coil clearance and nonlinearity may limit usable movement far earlier. The diagram remains visible and flags large relative angles; it does not certify them as mechanically achievable.

The teeth are illustrative involute outlines with no manufacturing root treatment. There is no mating pinion or tooth-contact simulation in this load-deflection view.

Elastic spur gear questions

Why does the center stay still?

Holding the shaft is a way to show relative wind-up under a changing load. The replay does not claim that the shaft stays still in normal rotating service.

Are the four individual spring rates calculated?

No. The input is total assembly stiffness. Individual rates, preload and travel require a separate geometric design.

Why does stored energy use radians?

Torque multiplied by angular travel gives work when the angle is in radians. Using degrees directly would overstate the energy.

Does stiffness reduce tangential tooth force?

At unchanged torque and pitch radius, F_t=T/r is unchanged. Stiffness changes wind-up, not that static force relationship.

Is this the same as a scissor gear?

No. The illustrated arrangement couples a loose rim to a shaft-fixed center through springs. A split preloaded gear uses a different construction.