The mesh force is a vector, not a scalar
Spiral bevel gears carry motion between intersecting shafts in axle drives, rotorcraft transmissions, machine-tool spindles and industrial reducers. Their curved teeth engage gradually along a diagonal contact line, which gives high contact ratio and smooth transfer at speed, but it also makes the dynamic behaviour sensitive to the exact geometry of the tooth surface. The design lever for that geometry is the contact pattern: its position, size and orientation are set during generation by the cutter head parameters and by the relative position and motion of the cutter and the workpiece. Shifting a wheel position by a few tenths of a millimetre moves the contact pattern, changes the load distribution along the contact line and, with it, the direction and magnitude of the resultant force transmitted through the mesh.
Conventional optimization treats that force as a scalar amplitude or as a transmission-error curve and iterates on contact pattern, loaded tooth contact analysis and a dynamic simulation of the whole transmission. Two surfaces that pass the same contact-pattern criterion can still load the shaft–bearing structure very differently, because a normal mesh force resolved into six generalized components — three forces and three moments — can be distributed in different proportions. The approach described here makes those six components the design output, uses the elastic compliance of the gear–shaft structure as the transfer path, and selects machine settings that reduce the vibration amplitude in the direction that dominates the response.
Reducing a twelve-degree-of-freedom transmission to an equivalent single-shaft model
A pinion–wheel pair is modelled in two local frames whose axes follow the two gear axes and whose x-axes are kept parallel. Each gear is given six degrees of freedom — three translations and three rotations — so the pair carries twelve. Equivalent mass, damping and stiffness for each degree of freedom follow from the dimensions and materials of the gear body, the shaft segments and the bearings. Along the axis, the assembly is discretized into nodes, the gear body is represented as a tapered shaft segment, bearings enter as spring–damper elements, and the mesh itself is replaced by an externally applied, time-varying six-component force rather than by a coupled mesh element.
That replacement is what makes the loop practical. The six-component mesh force is computed once from the tooth contact and loaded contact analyses of the actual surface, and is then reused as the excitation of a linear harmonic solution on a single-stage output-shaft finite element model. Because the coupled two-shaft dynamic problem no longer has to be solved at every design point, the finite-element stage of an evaluation drops from hours to seconds for a pair of this class, which is the difference between a one-off verification and a settings search.
From cutter settings to the loaded mesh force
The tooth surface is generated from the kinematics of a mechanical face-milling machine. A machine frame is placed with its origin on the cradle centre line, a cutter frame with its origin at the cutter head centre, and a generating plane whose origin coincides with the cutter centre. The unit tangent, unit normal and tip radius of the cutting edge are defined in the cutter frame and then rotated by an angle about the generating axis, so that they describe any chosen section of the generating surface. A general rotation matrix for an arbitrary axis and angle carries these vectors into the machine frame, and the position vector of the instantaneous conjugate contact point follows from the radial vector to the cutter centre, the displacement of the workpiece axis from the machine origin and the distance from the tip along the cutting direction.
Five settings enter this chain: the cradle angle qr, the machine root cone angle Γm, the horizontal wheel position Xr, the horizontal bed position Xbr and the vertical wheel position Em. Varying the last three displaces the generated flank while leaving the generating motion itself unchanged, which is why they are convenient optimization variables. A tooth contact analysis on the generated surfaces yields the contact path and the contact point; a loaded contact analysis distributes the torque along the contact line and gives the normal mesh force Fn as a function of the roll position.
The normal force is then projected onto the six generalized directions using the pressure angle αn, the spiral angle βm and the pitch cone angle δ1. For a gear axis aligned with z, the transverse component takes the familiar form
$$F_x = F_n \cos\alpha_n \cos\beta_m$$
with the remaining force components obtained by resolving the same normal force through the pitch cone angle, and the three moments from the corresponding moment arms ix, iy, iz measured from the rotation centre to the lines of action:
$$M_x = F_x i_x,\qquad M_y = F_y i_y,\qquad M_z = F_z i_z$$
The result is the excitation vector F = [Fx, Fy, Fz, Mx, My, Mz]T, whose components are all functions of the machine settings through Fn and through the orientation of the contact.
A compliance matrix as the transfer from force to vibration
Within the linear elastic range and under the superposition principle, the six-component displacement response δ = [δx, δy, δz, θx, θy, θz]T is related to the excitation by a 6 × 6 matrix whose entries are unit-load flexibilities:
$$C_{ij}=\frac{\delta_i}{F_j},\qquad \boldsymbol{\delta}=\mathbf{C}\,\mathbf{F}$$
Each Cij is the displacement in direction i produced by a unit generalized force in direction j; the matrix is real, symmetric and positive definite. Its eigenpairs
$$\mathbf{C}\mathbf{v}_i=\lambda_i \mathbf{v}_i,\qquad \lambda_1\ge\lambda_2\ge\cdots\ge\lambda_6>0$$
define six principal directions and six principal flexibilities. A large λi means that a load applied along vi produces a comparatively large deformation, that is, the structure is soft in that direction; a small value indicates a stiff direction. Because the eigenvalues carry the dimension of displacement per unit force (or angle per unit moment), they can be compared with each other only after normalization, but their ordering is a direct statement about which deformation mode will dominate the vibration response of a given surface.
Scalarizing six objectives with self-normalizing weights
Reducing all six principal flexibilities at once is a six-objective problem in a design space spanned by the machine settings. A weighted sum collapses it to a single scalar,
$$\min_{\mathbf{x}}\; U(\mathbf{x})=\sum_{i=1}^{6}\omega_i\,\lambda_i(\mathbf{x}),\qquad \omega_i=\frac{\lambda_i}{\sum_{j=1}^{6}\lambda_j},\qquad \sum_{i=1}^{6}\omega_i=1$$
where x collects the adjustable settings and the weights are normalized to remove the dimensional influence. The weighting rule has a specific consequence worth stating explicitly: substituting the definition of ωi gives
$$U(\mathbf{x})=\frac{\sum_{i=1}^{6}\lambda_i^{2}}{\sum_{i=1}^{6}\lambda_i}$$
so the objective is dominated by the largest eigenvalue without any external weighting decision. The dominant soft direction therefore sets the optimization target automatically, and the reported weight ω1 doubles as a scalar indicator of how strongly the response is concentrated in a single mode. In the case below, qr and Γm are held fixed and only Xr, Xbr and Em are varied within their permissible ranges, so any change in the response can be attributed to the repositioning of the generated flank.
Case study: a 28/41-tooth pair at 400 N·m
The procedure was applied to a 90° shaft-angle pair with 28 pinion teeth and 41 wheel teeth, a transverse module of 3.75 mm and an outer cone distance of 93.0914 mm. Both members and the shaft are 45 steel, modelled with a Young’s modulus of 210 GPa, a Poisson ratio of 0.3 and a density of 7.86 g·cm-3. The operating torque for the comparison is 400 N·m.
| Parameter | Pinion | Wheel |
|---|---|---|
| Number of teeth | 28 | 41 |
| Transverse module (mm) | 3.7500 | 3.7500 |
| Pressure angle (°) | 20 | 20 |
| Spiral angle (°) | 18.3000 | 18.3000 |
| Outer cone distance (mm) | 93.0914 | 93.0914 |
| Shaft angle (°) | 90.0000 | |
| Addendum (mm) | 3.9675 | 2.4075 |
| Dedendum (mm) | 3.1125 | 4.6725 |
| Addendum angle (°) | 2.8734 | 1.9150 |
| Dedendum angle (°) | 1.9150 | 2.8734 |
| Pitch diameter (mm) | 105 | 153.75 |
| Pitch cone angle (°) | 34.3302 | 55.6698 |
Two parameter sets were compared. Set 1 is the original flank; set 2 is the flank obtained after the settings were adjusted on the basis of the sensitivity result. The cradle angle and root cone angle were unchanged.
| Setting | Set 1 (original) | Set 2 (adjusted) |
|---|---|---|
| Horizontal wheel position Xr (mm) | 18.30 | 18.04 |
| Horizontal bed position Xbr (mm) | 34.30 | 34.33 |
| Vertical wheel position Em (mm) | 18.30 | 17.00 |
For set 1 the projection of the loaded normal force delivered a dominant first weight ω1 = 0.70, with the principal direction aligned essentially with the x axis of the local frame. The harmonic response of the single-shaft model under the corresponding six-component excitation therefore had its largest amplitude in that direction, and the x amplitude became the quantity to be reduced. After the settings were changed to set 2, the first weight fell to ω1 = 0.45, a reduction of 35.7% in the concentration of the response, and the x amplitude fell from −5.7637 × 10-3 mm to −4.5986 × 10-3 mm, a reduction of 20.21% at the same excitation frequency of 1 Hz.
| Quantity | Set 1 | Set 2 | Change |
|---|---|---|---|
| Dominant weight ω1 | 0.70 | 0.45 | −35.7% |
| Normal mesh force amplitude Fn (N) | 8034 | 7995 | −0.5% |
| x-direction response amplitude (mm) | −5.7637 × 10-3 | −4.5986 × 10-3 | −20.21% |
Interpretation: the load path matters more than the load magnitude
The two flanks differ by only 0.26 mm in wheel position and 1.30 mm in vertical position, and the resulting change in the normal mesh force is under one percent. Yet the vibration amplitude in the dominant direction falls by a fifth. The mechanism is visible in the intermediate quantities: the force amplitude is almost unchanged, so the improvement cannot come from a lighter load, and the drop in ω1 from 0.70 to 0.45 shows that the six components are distributed differently over the principal directions of the structure. The adjusted flank steers the resultant force away from the softest direction, so the same nominal load produces a smaller projection onto the mode that dominates the response.
This has a practical consequence for tooth-surface design targets. A contact-pattern specification that controls only the position and size of the contact ellipse, and a mesh-force specification that controls only the peak normal force, can both be satisfied while the response in the soft direction remains large. Adding the direction of the resultant mesh force — or, equivalently, the weighting vector ωi — to the acceptance criteria of a flank design couples the surface geometry directly to the structural dynamics of the housing, shaft and bearings that will carry it.
Scope of the formulation
The chain rests on linear elasticity and superposition, with the compliance matrix evaluated at the operating point defined by a single torque level; outside the small-deformation range, or when the contact pattern migrates substantially across the face under load, the eigenvalues no longer describe the response and the matrix must be rebuilt. The projection vector is defined at the design reference point, which fixes the load line and suppresses the effect of contact-point migration along the tooth. The comparison is made at a single excitation frequency, so the method ranks surfaces by their response at one point in the spectrum rather than by a speed sweep or by resonance proximity.
The weighted-sum scalarization returns one compromise design and does not expose the trade-offs among the six principal flexibilities; a Pareto formulation would be needed if, for example, a gain in one direction had to be paid for in another. Only three position settings were varied here, with the cradle angle and root cone angle held constant, so the attainable improvement is bounded by that subspace; the generating motion parameters, cutter tilt and swivel angle offer additional and largely unexplored degrees of freedom for the same objective. Extending the procedure to several torque levels, to a frequency sweep, and to measured contact patterns and noise on a running pair would establish how far the numerical reduction of the dominant amplitude carries over into the transmission as built.