A grinding head concentrates the dynamic problems of a machine tool into a small volume. Its shell, bearing seats and gear pair must absorb two distinct excitation paths: the periodic mesh force generated inside the gearbox, and the broadband impact transmitted from the workpiece surface through the spindle. When either path approaches a natural frequency of the head assembly, the consequences range from audible noise and accelerated tooth wear to a shift of the contact pattern and, in the limit, fatigue failure of the teeth. Modal data are therefore the entry point for any dynamic assessment of the head, but the modal data must be obtained under boundary conditions that resemble the real ones.
The drive examined here uses a Gleason spiral bevel gear pair. Spiral bevel gears are preferred in aerospace, machine tool and heavy machinery transmissions because the curved tooth trace gives a gradual overlap, high load capacity and comparatively low noise. Free–free modal analysis, the usual first step, describes a gear pair that is supported by nothing and loaded by nothing. A pair clamped in bearing seats and transmitting a rated torque is a different structure: the teeth are pressed together over one or more contact patches, the contact stiffness adds to the elastic stiffness, and the resulting natural frequencies can move. To what extent they move, and whether the tooth contact changes the mode shapes enough to alter the resonance margin, is the question addressed by the analysis below.
Design point and tooth geometry
The pair was specified for the head of a grinder built for a steel producer, with the large gear on the input shaft acting as the driver and the small gear mounted on the grinding spindle. The rated input torque is 2 100 N·m at 1 500 r/min, which corresponds to roughly 330 kW; the spindle carries a resisting torque of 890 N·m at its upper speed of 3 500 r/min, giving about 326 kW at that point, so the two figures describe the same rated operating condition. Lower spindle speeds, down to 2 000 r/min, correspond to proportionally lower transmitted power.
The Gleason blank data are a normal pressure angle of 20°, an addendum coefficient of 0.85, a clearance coefficient of 0.188 and a mean spiral angle of 35°, with a shaft angle of 90° between the two axes. The large gear is overhung at one end of the input shaft, while the small gear is straddle-mounted between two bearings in the housing — an asymmetry worth noting, because the overhung member contributes a compliant, low-frequency path to the assembly. The tooth data were used to generate a two-dimensional profile and a three-dimensional solid model of the pair, which is the geometry imported into the finite element environment.
Contact ratio and the choice of a three-tooth model
The total contact ratio of a spiral bevel pair combines the transverse and the longitudinal components:
$$\varepsilon_{r}=\sqrt{\varepsilon_{\alpha}^{2}+\varepsilon_{\beta}^{2}}$$
where $\varepsilon_{\alpha}$ is the transverse contact ratio and $\varepsilon_{\beta}$ the overlap ratio. For this design $\varepsilon_{r}=2.3$, so as many as three tooth pairs share the load instantaneously. Analysing a single pair would therefore misrepresent both the load split and the stiffness of the mesh. The finite element model consequently contains three tooth pairs in simultaneous contact, defined as surface-to-surface contacts with the large gear as the contact surface, the small gear as the target surface and a friction coefficient of 0.06.
Finite element model and boundary conditions
The gears are made of 20CrMnTi case-hardening steel, entered with a Young’s modulus of $2.07\times10^{5}$ MPa, a Poisson ratio of 0.254 and a density of 7 830 kg/m³. An average element edge length of 2 mm with automatic meshing was judged sufficient to resolve the contact patch without exceeding the available computing budget.
Local features that do not participate in load transfer were removed according to Saint-Venant’s principle: keyways, fixing bolts and their holes, threads and chamfers on the end faces, and the small transition fillets, with their mass redistributed equivalently. This is a standard trade in gear contact modelling — the fillet radii removed here are, however, exactly the locations where root stress concentrates, so root stresses should be read as engineering estimates rather than as local peak values.
Two analyses were run on the same mesh. The free modal analysis used no constraint and no load. The contact modal analysis applied the working boundary conditions: a torque of 2 100 N·m on the input shaft, a resisting torque of 890 N·m on the spindle, and cylindrical supports at the bearing seats that permit rotation about the axis only. A linear static solution under this load precedes the modal extraction, and the Lanczos algorithm available in Ansys is used to extract the requested number of modes from the resulting large model.
Free modes and preloaded contact modes: the governing equations
For an $n$-degree-of-freedom system the structural dynamic equation is
$$\mathbf{M}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}=\mathbf{F}(t)$$
with the mass matrix $\mathbf{M}$, damping matrix $\mathbf{C}$, stiffness matrix $\mathbf{K}$, force vector $\mathbf{F}$ and the displacement, velocity and acceleration vectors $\mathbf{x}$, $\dot{\mathbf{x}}$, $\ddot{\mathbf{x}}$. Setting $\mathbf{C}=0$ and $\mathbf{F}=0$ leaves the free-vibration problem
$$\mathbf{M}\ddot{\mathbf{x}}+\mathbf{K}\mathbf{x}=0$$
which, with the harmonic solution $\mathbf{x}=\boldsymbol{\varphi}\sin(\omega t+\alpha)$, becomes the generalised eigenvalue problem
$$(\mathbf{K}-\omega^{2}\mathbf{M})\boldsymbol{\varphi}=0,\qquad \det(\mathbf{K}-\omega^{2}\mathbf{M})=0$$
The roots $\omega_{0i}$ ($i=1,2,\dots,n$) are the undamped natural frequencies and the corresponding vectors $\boldsymbol{\varphi}_{i}$ the mode shapes; assembled as columns they form the modal matrix of the system.
A contact mode is the natural frequency and mode shape of the same structure while the teeth are in contact under load. The static solution provides a stress state $\sigma_{0}$ and hence a stress stiffness matrix $\mathbf{S}$, and the modes follow from
$$(\mathbf{K}+\mathbf{S}+\omega_{0i}^{2}\mathbf{M})\boldsymbol{\varphi}_{i}=0$$
The additional term is the physical reason for the frequency shift reported below: a tensile stress state softens a structure, a compressive one stiffens it, and a loaded tooth contact is predominantly compressive in the region where the mode shapes localise.
Free-modal baseline
The first ten free modes are listed in Table 1. The first three are rigid-body translations in the three coordinate directions, with frequencies indistinguishable from zero, and carry no engineering information.
| Order | Natural frequency (Hz) | Character of the mode |
|---|---|---|
| 1 | 0 | Rigid-body translation |
| 2 | 5.261 × 10−4 | Rigid-body translation |
| 3 | 9.731 × 10−4 | Rigid-body translation |
| 4 | 33.723 | One end of each gear vibrating axially |
| 5 | 42.095 | Axial motion at one end of each gear, similar to mode 4 |
| 6 | 59.700 | Diagonal axial motion of the large gear, one-end axial motion of the small gear |
| 7 | 711.47 | First elastic mode: circumferential rolling of the large gear, lateral sway of the small gear |
| 8 | 3 115.2 | Diagonal axial motion of the large gear, one-end axial motion of the small gear |
| 9 | 4 373.8 | Circumferential rolling of the large gear, diagonal axial motion of the small gear |
| 10 | 4 708.9 | Four-end axial motion of the large gear, circumferential folding of the small gear |
Modes 4 to 6 are whole-body motions of the two gears relative to each other rather than elastic deformation of the teeth, and their low frequencies reflect the absence of any support in the free–free configuration. The first genuinely elastic mode is mode 7 at 711.47 Hz, and it is separated from the next elastic mode by a factor of more than four — a gap that is favourable for a structure intended to operate far below its first elastic resonance.
Contact stress under rated load
The static solution under 2 100 N·m and 890 N·m places the stress maxima in the contact zone, with secondary concentrations on the compression and tension flanks of the tooth root. The peak von Mises stress, 311.37 MPa, occurs at the root of the small gear at the instant its tooth enters mesh, that is, at the beginning of the meshing cycle where the load is transferred to a single fresh contact. Considering friction at the interface, the peak contact stress is 278.208 MPa on the large gear and 311.37 MPa on the small gear. Both values remain below the allowable contact fatigue stress adopted for the case-hardened 20CrMnTi, so the design is structurally acceptable at its rated point.
Because the contact stress is substantially higher than the root stress, the root concentration appears only weakly in a contour plot drawn on a common scale; the root stress nevertheless follows the travel of the contact imprint along the tooth, and the small gear is the more severely loaded member throughout. This is consistent with the geometry: the small gear completes more load cycles per unit time and its tooth counts fewer contacting pairs over the same arc of roll.
Contact modes and the frequency shift
Six contact modes were extracted, matching the six elastic modes of the free pair, and are compared in Table 2. The peak modal displacement is reported as obtained from the mass-normalised eigenvectors; because mode shapes are defined only up to an arbitrary scale, these magnitudes indicate relative deformation within a mode, not a physical amplitude.
| Order | Free natural frequency (Hz) | Contact natural frequency (Hz) | Change (%) | Peak modal displacement (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 | +7.1 | 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 shift is strongly order-dependent. The first elastic mode rises from 711.47 Hz to 1 389.2 Hz, an increase of 95%, because in that mode the two gears move relative to one another and the loaded tooth contact acts as a stiff spring that the free model lacks. The higher modes, whose shapes are dominated by local bending and folding of the rims and teeth rather than by relative motion of the two bodies, change by 5–8%, and the fifth mode is essentially unchanged at −0.1%, which is within the resolution of the extraction. In no case does the load alter the character of a mode shape; the same deformation pattern reappears at a higher frequency, and the free-modal analysis therefore remains a valid qualitative guide to how the pair deforms.
In the contact condition the first mode combines circumferential rolling of the driving gear with radial and tensile motion of the driven gear. The second shows the driven gear vibrating up and down at one end; the third swings it from one side to the other; the fourth folds it towards the opposite flank; the fifth twists it along four directions with circumferential folding; and the sixth combines extension–contraction at the six ends of the driving gear with circumferential folding of the driven gear. The driven gear, being the lighter and more compliant member and the one mounted on the spindle, dominates the modal response in five of the six modes.
Resonance margin at the operating speeds
The two excitation frequencies that matter are the shaft rotation frequency and the tooth-mesh frequency:
$$f_{r}=\frac{n}{60},\qquad f_{h}=\frac{nZ}{60}$$
with $n$ in r/min and $Z$ the number of teeth of the member concerned. At 1 500 r/min the input shaft rotates at 25 Hz; the spindle at 2 000–3 500 r/min covers 33.3–58.3 Hz. The first contact natural frequency of 1 389.2 Hz lies more than twenty times above the highest rotation frequency, so the rotational orders cannot excite the pair into resonance, and the design is dynamically well conditioned with respect to shaft-speed excitation.
The mesh frequency, however, scales with tooth count and must be checked case by case, since a tooth-mesh harmonic of the order of 1.4 kHz is reachable at the top spindle speed for a moderate tooth count. Two further features of the modal set deserve attention in that check. First, within the contact set, the fourth and fifth modes at 5 040.9 Hz and 5 051.1 Hz are separated by only 0.2%, a near-degenerate pair in which a single excitation can drive both modes and produce a strongly coupled response. Second, the free and contact analyses place the first elastic mode in quite different positions, so a resonance assessment based on free-modal data alone would be conservative by nearly a factor of two in frequency but would also, and more importantly, misplace the band that must be avoided. Preloaded contact modal data are the appropriate basis for the excitation spectrum that the grinding head will actually see, combining mesh harmonics with the impact spectrum transmitted from the workpiece.
Scope and limitations of the result
The analysis is a linear perturbation about a single static load step. Centrifugal stiffening and gyroscopic effects at the spindle speeds of up to 3 500 r/min are not included, nor is the stiffness of the lubricating film, which in practice lowers the effective mesh stiffness relative to dry contact and adds damping. The three-tooth contact model represents the tooth pairs sharing load at one instant rather than a rotating contact pattern, and the friction coefficient is a fixed value rather than a function of sliding velocity and lubrication state. The comparison between free and contact conditions should therefore be read as the difference between an unconstrained and a rated-load state of the same structure, not as a prediction of in-service frequencies to better than the modelling assumptions allow. Within that scope, the conclusion is that the tooth contact raises the natural frequencies of the pair without changing its mode shapes, that the resulting first elastic mode at 1 389.2 Hz leaves a wide margin over the rotational excitation, and that the mesh-frequency harmonics and the near-degenerate mode pair around 5 045 Hz define the frequency bands to be avoided in the final design of the grinding head.