Why the headstock gear pair sets the vibration limit
A conditioning grinder removes surface defects, scale and cracks from steel billets and slabs with an abrasive wheel carried on a grinding head. The headstock is the structural and kinematic core of that machine: it houses the wheel spindle, its bearings and the right-angle reduction gearing inside a shell that must survive two distinct load paths. The first is external — impact and friction from the billet surface, transmitted through the wheel and spindle into the housing. The second is internal — the periodic mesh force of the gear pair and the bearing forces it generates. Both paths feed the same structure, and both are broadband enough to be dangerous if any component of them lands on a natural frequency of the head assembly.
Gleason spiral bevel gears are the usual choice for this right-angle drive because the curved, gradually engaging tooth flank gives a high contact ratio, smooth transfer of load from tooth to tooth, high torque capacity and comparatively low noise. The same overlap that smooths the transmission also makes the mesh a distributed, load-dependent stiffness element: several tooth pairs share the load simultaneously, and the contact patch grows and migrates across the flank as the gears roll. A modal model that ignores this contact state treats the pair as a single elastic body with traction-free flanks, which is not the structure that operates in the machine.
The engineering question is therefore narrow and quantitative: where do the lowest elastic modes of the gear pair sit, and how far are they from the excitation orders of the drive? Modal analysis answers the first part; the second part follows from the shaft speeds and the tooth counts. If the lowest elastic frequency clears the rotation orders and their low mesh harmonics with margin, the design is dynamically sound and the alternating stress at the tooth root is governed by static load rather than by resonant amplification.
Formulation: eigenvalue problem and stress stiffening
For an n-degree-of-freedom system the structural dynamics equation is
$$\mathbf{M}\ddot{\mathbf{x}} + \mathbf{C}\dot{\mathbf{x}} + \mathbf{K}\mathbf{x} = \mathbf{F}(t)$$
where M, C and K are the mass, damping and stiffness matrices, F is the force vector, and the vectors of displacement, velocity and acceleration are x, its first and second time derivatives. Setting damping and external force to zero leaves the free-vibration problem, whose harmonic solution x = φ sin(ωt + α) leads to the generalized eigenvalue problem
$$\left(\mathbf{K} – \omega^{2}\mathbf{M}\right)\boldsymbol{\varphi} = \mathbf{0}$$
Nontrivial solutions require the coefficient determinant to vanish,
$$\left|\mathbf{K} – \omega^{2}\mathbf{M}\right| = 0$$
which is an n-th order polynomial in ω². Its positive roots ω0i are the natural frequencies of the undamped system, and the corresponding vectors φi are the mode shapes, assembled into the modal matrix. For a real, undamped structure these shapes are real vectors and can be normalized to unit modal mass.
Contact modal analysis adds the operating boundary conditions before the eigenvalue extraction. A linear static analysis is solved first under the working torque, bearing constraints and frictional contact, giving a stress field σ0 and from it a stress stiffness matrix S. The modal problem is then linearized about that stressed state:
$$\left(\mathbf{K} + \mathbf{S} – \omega_{0i}^{2}\mathbf{M}\right)\boldsymbol{\varphi}_{i} = \mathbf{0}$$
The stress stiffness term is the formal reason contact modes sit above free modes. A tooth flank under compressive contact and a root fillet under bending carry a stress state whose geometric contribution stiffens the structure, so every natural frequency computed on the loaded configuration is higher than the corresponding traction-free value. The magnitude of the shift depends on how much of the elastic energy of a given mode is stored in the loaded region, which is why the lowest, most compliant mode generally shifts most.
Extraction used the Lanczos algorithm, which recovers a bounded number of low-order modes from a large model far more cheaply than subspace iteration and is the default eigensolver in the Ansys environment in which the contact formulation is implemented. Only the first six contact modes were retained: low-order modes dominate the dynamic response of a stiff, lightly damped assembly, while modes far above the excitation band absorb negligible energy.
Gear pair, loads and finite element model
The case is a Gleason spiral bevel pair designed for the headstock of a conditioning grinder at a steel works. The large wheel is the driving member on the input shaft, overhung at one end of that shaft; the pinion sits on the grinding spindle, supported between bearings in the housing. The shaft angle is 90°. The design brief and the resulting model parameters are collected below.
| Parameter | Value |
|---|---|
| Input torque on the large wheel | 2 100 N·m |
| Resisting torque on the spindle | 890 N·m |
| Input shaft speed (large wheel, driving) | 1 500 r/min |
| Spindle speed (pinion) | 2 000–3 500 r/min |
| Shaft angle Σ | 90° |
| Normal pressure angle αn | 20° |
| Addendum coefficient ha* | 0.85 |
| Clearance coefficient c* | 0.188 |
| Mean spiral angle βm | 35° |
| Total contact ratio | 2.3 |
| Friction coefficient at the flank | 0.06 |
| Material | 20CrMnTi carburizing steel |
| Elastic modulus / Poisson ratio / density | 2.07 × 105 MPa / 0.254 / 7 830 kg·m-3 |
| Average element edge length | 2 mm, automatic tetrahedral mesh |
The solid model was simplified before meshing, following the Saint-Venant principle: keys, fixing bolts and their holes, end-face threads, chamfers and transition fillets were removed because they carry no part of the load path in service, and the removed mass was redistributed over the neighbouring geometry. What remains is the two toothed bodies with their hubs, which keeps the element count within a practical budget without altering the global stiffness that governs the modes of interest.
Contact ratio drives the contact model. With the total contact ratio given by
$$\varepsilon_{r} = \sqrt{\varepsilon_{\alpha}^{2} + \varepsilon_{\beta}^{2}}$$
equal to 2.3, up to three tooth pairs can share the load at any instant, so three face-to-face contact pairs were defined rather than one. The large wheel flank acts as the contact surface and the pinion flank as the target, with a friction coefficient of 0.06. This is a deliberate choice: a single-pair model would underestimate the mesh stiffness and overestimate the flank stress, whereas the three-pair configuration reproduces the load sharing that the contact ratio implies.
Constraints follow the bearing arrangement. Cylindrical supports at the bearing locations allow rotation about the shaft axis only, blocking radial and axial translation. The operating load case applies 2 100 N·m to the input shaft and 890 N·m of resisting torque to the spindle. The free-free case, by contrast, applies no constraint and no load at all, and serves only as the reference against which the effect of the operating state is measured.
Free-free modes
The first ten eigenvalues of the unconstrained pair are listed below with the character of each shape.
| Order | Frequency / Hz | Mode character |
|---|---|---|
| 1 | 0 | Rigid-body translation |
| 2 | 5.26 × 10-4 | Rigid-body translation |
| 3 | 9.73 × 10-4 | Rigid-body translation |
| 4 | 33.72 | Quasi-rigid; one end of each gear moves axially |
| 5 | 42.10 | Quasi-rigid; similar to order 4 |
| 6 | 59.70 | Quasi-rigid; diagonal axial motion of the wheel, one-end motion of the pinion |
| 7 | 711.47 | First elastic mode; circumferential rolling of the wheel, lateral sway of the pinion |
| 8 | 3 115.2 | Diagonal axial motion of the wheel, one-end axial motion of the pinion |
| 9 | 4 373.8 | Circumferential rolling of the wheel, diagonal axial motion of the pinion |
| 10 | 4 708.9 | Four-end axial motion of the wheel, circumferential folding of the pinion |
The three near-zero orders are the rigid-body freedoms of a body with no support, and they carry no engineering information about the installed machine. Orders 4 to 6 are still rigid-body in character — the two bodies move relative to each other without elastic deformation of the teeth — but their frequencies are finite because the contact between the flanks and the hubs provides a compliant coupling. Only order 7 at 711.47 Hz represents genuine elastic deformation of the gear bodies, with the wheel rolling circumferentially and the pinion swaying sideways. Reporting “the first mode” of such a model without this distinction is misleading: the number that matters for resonance is 711.47 Hz, not the quasi-rigid orders two orders of magnitude below it.
Static contact response under operating torque
The static pass that precedes the contact modal extraction also yields the working stress state. Peak equivalent stress reaches 311.37 MPa at the pinion tooth root, at the instant the pinion tooth enters mesh — the position where the moment arm from the contact patch to the root is shortest and the tooth behaves most nearly as a cantilever. Away from that instant the load spreads over the flank and over the compressive and tensile sides of the root, and the peak falls. The peak equivalent displacement, 1.396 × 10-3 m, occurs at the driven flank contact rather than at the root, which is consistent with a contact-dominated deformation field.
With friction included, the peak contact stress is 278.208 MPa on the large wheel and 311.37 MPa on the pinion. Both lie below the allowable contact fatigue stress of the carburized 20CrMnTi, so the pair is not flank-limited at the rated torque. The three-pair load sharing implied by the contact ratio of 2.3 is what keeps these values moderate: the same torque carried by a single pair would raise the flank pressure substantially. Because the contact stress is much larger than the root bending stress, the root field is barely visible on the same colour scale as the flank — a plotting artifact rather than evidence that the root is unloaded, and one that matters when the results are read for fatigue assessment.
Contact modes: stiffening without a change of shape
The prestressed, constrained extraction gives the first six contact modes. Frequencies, the corresponding free-free values and the reported modal deformations are compared below.
| Order | Free-free / Hz | Contact / Hz | Change | Reported deformation / mm |
|---|---|---|---|---|
| 1 | 711.47 | 1 389.2 | +95.3% | 8.947 |
| 2 | 3 115.2 | 4 368.3 | +40.2% | 14.398 |
| 3 | 4 373.8 | 4 707.0 | +7.6% | 33.245 |
| 4 | 4 708.9 | 5 040.9 | +7.0% | 33.245 |
| 5 | 5 056.1 | 5 051.1 | −0.1% | 26.415 |
| 6 | 5 215.0 | 5 496.8 | +5.4% | 28.935 |
The frequency shift is strongly order-dependent. The lowest elastic mode nearly doubles, from 711.47 Hz to 1 389.2 Hz, because its deformation is concentrated in the tooth contact and hub region where the operating stress state adds the most stiffness. The second order gains 40%, and orders three to six gain between nothing and 8%, their elastic energy being stored increasingly in rim and web material far from the loaded flanks. The fifth order is essentially unchanged, within the numerical noise of the extraction.
What does not change is the shape. Each contact mode reproduces the deformation pattern of its free-free counterpart: circumferential rolling of both bodies with radial and tensile motion of the driven gear for order 1; vertical motion of one end of the driven gear for order 2; side-to-side motion for order 3; relative folding for order 4; four-direction torsion with circumferential folding for order 5; and six-end telescoping of the driving gear with circumferential folding of the driven gear for order 6. The operating load therefore rescales the frequency spectrum but does not reorder or reinterpret the mode shapes, which means a free-free analysis is adequate for identifying which parts of the structure deform, while a contact analysis is required for the frequency values themselves.
One caution applies to the deformation column. Values of 8.9 to 33.2 mm are large compared with the tooth height and reflect the mass-normalized scaling that the eigensolver applies to mode shapes, not physical tooth deflection under load; they indicate where motion is concentrated, not how far the gear actually moves.
Resonance clearance and design implications
Shaft rotation orders follow directly from the speeds: 1 500/60 = 25 Hz on the input shaft, and 2 000–3 500/60 = 33.3–58.3 Hz on the spindle. Mesh orders are the rotation frequency multiplied by the tooth count,
$$f_{r} = \frac{n}{60}, \qquad f_{h} = \frac{nZ}{60}$$
so the mesh spectrum extends upward in integer multiples of the shaft orders. Against this, the lowest elastic contact mode sits at 1 389.2 Hz. The margin is large: the first elastic frequency is more than twenty times the highest shaft order and above the mesh harmonics of a spiral bevel pair of this size at these speeds. On that basis the internal mesh excitation does not drive the pair into severe vibration, and the design is dynamically acceptable for the rated operating point.
The spectrum does contain closely spaced pairs, and these define the bands to keep excitation away from. In the free-free set, orders 5 and 6 at 5 056.1 Hz and 5 215.0 Hz are only 3.1% apart; in the contact set, orders 2 and 3 at 4 368.3 Hz and 4 707.0 Hz are 7.8% apart. A lightly damped structure excited between two nearly coincident modes responds in both at once, and the resulting beat can be more damaging than either mode alone because the response no longer decays between pulses. In practice no mesh harmonic or bearing defect order should be allowed to settle near 1.39 kHz or in the 4.3–5.5 kHz band.
The trend across loads is the other design lever. Because frequency rises with stiffness and the tooth contact is the most compliant element in the load path, the first elastic frequency is the most load-sensitive quantity in the model — a 95% shift between the traction-free and the rated-torque state. A design that clears resonance at rated torque may not clear it at part load, where the mesh is less stiff and the first mode sits closer to the free-free value. Conversely, increasing contact ratio, tooth thickness or the stiffness of the hub and bearing supports raises the lowest mode and widens the margin.
Scope and limitations
The results are numerical and linearized about a single static state: one input torque of 2 100 N·m, one friction coefficient of 0.06, no damping, no lubricant film and no thermal or centrifugal prestress. The lubricated contact in service will differ from the dry frictional model, and the damping that the model omits is what actually bounds the resonant amplitude, so the frequency comparison supports a resonance-avoidance argument rather than a vibration-level prediction. The geometry is simplified, with keys, bolts and fillets removed, which slightly alters local stiffness near the hubs. Most importantly, the frequencies are not backed by a modal test; an impact test on the assembled headstock would be needed to confirm the predicted separation between the first elastic mode and the excitation orders, and to detect the support compliance that a bearing modelled as a cylindrical constraint cannot capture. Within those limits, the comparison between free-free and contact states establishes the quantitative point that matters for the design: contact and prestress raise the natural frequencies of a spiral bevel pair without altering its mode shapes, and the lowest elastic mode of this headstock pair sits far above the orders the drive can generate.