Bevel Gears

Part of Gear Making

Making and using bevel gears to transmit power between shafts meeting at an angle, typically 90 degrees.

Why This Matters

Most early machinery needs to change the direction of power transmission. A waterwheel turns on a horizontal axis, but the millstone it drives turns on a vertical axis. A hand-cranked machine turns in one plane, but the driven element needs to rotate in a perpendicular plane. The bevel gear is the standard solution to this problem, transmitting rotation between shafts at an angle — most often 90 degrees but any angle is possible.

Without bevel gears, a machine designer has limited options for changing shaft direction: belt and pulley arrangements can work but lose efficiency, and some direction changes are geometrically impossible with flat drives. The bevel gear provides a compact, rigid, high-efficiency solution that has been used in mills, agricultural machinery, and mechanical devices for centuries.

Making bevel gears by hand is more demanding than making spur gears because the tooth form must taper correctly from the large end to the small end — the outer diameter is larger than the inner, and the tooth proportions change continuously along the tooth length. This requires more care in layout and cutting but is well within the capability of a skilled craftsman.

How Bevel Gears Work

Bevel gears operate on the principle of two rolling cones. If you imagine two cones with their apices meeting at a point, rolling against each other without slip, the ratio of their surface speeds at any given radius equals the ratio of their pitch circle radii at that point. Bevel gear geometry is derived from this cone-rolling model.

The pitch cone of each bevel gear has its apex at the common apex point (the center of the shaft intersection). The half-angle of each pitch cone is called the pitch cone angle. For a 90-degree shaft angle, the sum of the two pitch cone angles equals 90 degrees. For equal-size gears at 90 degrees, each pitch cone angle is 45 degrees — this is a miter gear pair.

The teeth of a bevel gear are tapered — they are taller and wider at the outer end (back face) and shorter and narrower at the inner end (front face). The tooth profile at any cross-section through the cone is a standard involute or cycloidal profile, but scaled to the diameter at that section.

For hand-cut bevel gears, the standard approach is to cut the teeth using the pitch cone geometry, aiming for correct tooth form at the mid-face (mean cone) cross-section and accepting some error at the extremes. This works adequately for low-to-moderate speed, moderate-load applications.

Laying Out Bevel Gear Blanks

Step 1: Calculate the pitch cone angle. For shaft angle Σ (usually 90°) and gear ratio i = N₂/N₁ (where N is number of teeth):

• Pitch cone angle of gear 1: α₁ = arctan(N₁/N₂)
• Pitch cone angle of gear 2: α₂ = Σ - α₁

For a 90° shaft angle with gear ratio 2:1 (24 teeth and 12 teeth):

• α₁ = arctan(12/24) = arctan(0.5) ≈ 26.6°
• α₂ = 90° - 26.6° = 63.4°

Step 2: Establish the mean cone distance. The mean cone distance Rm is the distance from apex to the mid-face of the tooth. Choose this based on the required face width and tooth strength.

Step 3: Calculate mean pitch diameter. Mean pitch diameter = 2 × Rm × sin(α).

Step 4: Mark the gear blank. The blank is a truncated cone, not a cylinder. The face angle (angle of the tooth-bearing face to the shaft axis) equals the pitch cone angle plus the face advance angle (a small correction, typically 1–3 degrees). Cut the blank to the correct cone angles and face width on the lathe or by careful filing from a cylindrical blank.

Cutting Bevel Gear Teeth by Hand

Method 1: Filing to scribed lines. Mark the tooth spaces on the large face of the blank using dividers, then transfer the spacing to the small face, scaling the spacing proportionally (by the ratio of small face diameter to large face diameter). Connect corresponding marks with scribed lines along the tooth face. File away the waste material between lines, keeping the file angled to follow the taper. Check frequently with the pitch cone template.

Method 2: Using a formed cutter in a jig. If a milling-type setup exists (rotary file, formed disc cutter), use a gear-cutting jig that indexes the blank and controls the cutter path. For bevel gears, the cutter passes through the apex point direction rather than parallel to the shaft axis.

Method 3: Generated by rolling. In professional gear cutting, bevel gear teeth are generated by the rolling action of a rack-form tool. Approximations of this can be made with a scraper tool traversing along the tooth face at the correct angle.

The teeth must taper in height and width from the outer to inner face. The tapering is critical — if the teeth are parallel (spur gear form), they will bear only at the outer ends under load, quickly wearing and breaking.

Assembly and Adjustment

Bevel gear pairs must be adjusted so the pitch cone apices coincide — both cones must point to the same apex point. Misalignment of this kind causes load concentration at the large or small end of the teeth.

Setting backlash. Move one gear along its axis (shimming the bearing position) until the correct backlash is achieved. The contact pattern check (Prussian blue) is especially important for bevel gears — examine the pattern and adjust axial position of each gear until the contact is centered on the tooth face.

Thrust bearings. Bevel gears generate an axial (thrust) force component that tries to push the gear away from or into the mesh. Thrust bearings — flanged bushings, thrust washers, or angular-contact arrangements — must restrain this force at each shaft. Do not rely solely on the gear housing or radial bearings to carry thrust loads; dedicated thrust surfaces are necessary.

Applications in Rebuilding Machinery

Bevel gears at 90 degrees are needed in:

• Grain mills (vertical waterwheel to horizontal millstone shaft)
• Horizontal windmills to vertical power shaft
• Hand drill braces (the classic brace drill uses a pair of small bevel gears)
• Post drill (converting handle rotation to drill rotation)
• Agricultural machinery (potato diggers, threshers, seed drills with angled drives)
• Any machinery where power must turn a corner