The generating problem behind the machine architecture
Spiral bevel gears transmit motion between intersecting shafts with a high load capacity, a large contact ratio, smooth running and low noise, and they tolerate a certain amount of mounting error. These properties explain their use in aerospace drives, automotive axles and precision machine tools. The same properties make them difficult to manufacture: the tooth flank is not a simple surface of revolution but the envelope of a cutter head sweeping along a circular path while the workpiece rolls in a prescribed relationship to that path.
Mechanical gear-cutting machines generate this envelope with a long kinematic chain — change gears, a cradle with a worm-and-wheel drive, eccentric adjustments and a swivel head. The chain determines the accuracy and limits the speed, and every additional element adds wear and lost motion. Raising cutting speed and tightening flank tolerances therefore pushes the mechanical architecture beyond its practical envelope, and fully numerically controlled machines have taken over the market.
The direction of travel in gear-cutting equipment is well established: complete numerical control, higher spindle and table speeds, higher flank accuracy, process integration and reduced or eliminated cutting fluid. Production machines in this class run cutter spindles at 5,500 r/min, with some reaching 12,000 r/min, and worktable speeds up to 800 r/min. Integrated platforms such as the WERA Profilator 320 or the Gleason 130SVC carry out loading, deburring, milling and shaving in a single clamping. Domestic Chinese builders have developed their own five-axis machines and control systems, but the gap to the leading gear-cutting platforms remains most visible in spiral bevel gear equipment specifically — which is the segment addressed by the YKH200 machine described here.
Machine configuration and axis layout
The YKH200 is a five-axis horizontal milling machine for spiral bevel gears with a maximum workpiece diameter of φ200 mm and a maximum module of 4 mm. It comprises a bed, a saddle, a column, a cutter box and a workpiece box. Three linear axes are driven by servo motors through ball screws; the two rotary axes are direct driven, which removes the worm drives and the cradle of the mechanical generation train and improves cutting efficiency, surface quality and running smoothness.
| Parameter | Value |
|---|---|
| Maximum workpiece diameter | φ200 mm |
| Maximum module | 4 mm |
| Maximum gear ratio | 10:1 |
| Maximum full tooth depth | 8 mm |
| Maximum face width | 30 mm |
| Tooth-number range | 4–200 |
| Cutter diameter | 2–6 in |
| Maximum pitch diameter with 6 in cutter | 178 mm at 2:1 ratio; 200 mm at 10:1 ratio |
| Workpiece spindle taper | Morse No. 5 |
| Cutter spindle speed | 100–350 r/min |
| Distance, workpiece spindle face to machine centre | 63 mm |
| Axis strokes | X (saddle) 120–160 mm; Y (vertical) 100–120 mm; Z (horizontal) 50–108 mm |
| Drive rating | X, Y, Z: 11 N·m each; B (cutter) 3.7 kW; C (workpiece) 16 N·m |
| Overall dimensions (L×W×H) | 2,000 × 1,500 × 2,000 mm |
| Machine mass | 3,500 kg |
The bed is an HT300 casting and carries the X-axis saddle guideways and ball screw, the column feed guideways, and chip-disposal channels sized for wet or dry cutting. The cutter box assembly — cutter spindle, box, column and direct-drive motor — is built from rigid castings with good vibration absorption. Its Y and Z motions are servo-driven ball-screw infeeds that replace the eccentric adjustment of the cradle worm wheel and the raceway structure of the mechanical machine. Eliminating the reverse motion of the cradle in each cutting cycle redistributes the generating time, and the linear-guide/ball-screw arrangement shortens the overall cycle. The cutter spindle runs in high-precision bearings, and the direct-drive motor raises the available cutting torque at low noise.
The workpiece box is a casting that both supports the workpiece assembly and sets the root cone angle. Because the horizontal wheel-position adjustment plate is removed, one layer of moving parts disappears and the assembly is stiffer; the horizontal zero position is instead set numerically within the part program, which is more accurate and easier to control. Clamping is hydraulic: a rotary cylinder acts through a drawbar, and the clamping cylinder itself is a light aluminium unit.
Spindle and feed supports follow conventional high-speed practice. The cutter spindle front bearing set consists of three angular contact ball bearings in tandem taking axial load in one direction, with a deep-groove ball bearing at the rear for radial load and an encoder for condition feedback; the workpiece spindle uses the same front/rear arrangement. The X and Y ball screws are fixed at one end and supported at the other, the Z screw is fixed–free, and all three nuts are preloaded in double-nut fashion to remove backlash.
Control, hydraulics and auxiliary systems
Control is provided by the Mltor H218i five-axis CNC system. The X and Y axes reproduce the generating roll of the mechanical cradle: eccentric angle, cradle angle and roll ratio are set as program parameters and can be edited directly. The Z axis controls depth of cut and implements helical infeed compensation. Each linear axis carries a three-position limit switch set for forward and reverse overtravel plus the machine home position. Cutter spindle speed is infinitely variable, controlled by program in the automatic cycle and by an override switch in manual mode, with forward, reverse and stop keys on the panel.
The hydraulic station is separate from the machine base and serves two circuits: clamping, switched by a solenoid valve, and the Y-axis counterbalance cylinder, which balances the weight of the direct-drive head during vertical motion so that the motor load stays symmetric. The cooling unit is likewise separate, combining pump and chip tray, and can be run either from the part program or from a soft key on the panel.
Finite-element model of the complete machine
Because the stiffness of bearings and ball screws is distributed rather than local, the assembled model was simplified before solution: bearings and screws were removed and their stiffness lumped into spring contact elements at the guide/slider interfaces, using COMB14-type elements whose stiffness was calculated from the component data and whose damping was set to 0.1. Bolted joints were tied by gluing the mating faces. The bed was fully constrained at the levelling feet. All major castings are HT300, with the properties listed in Table 2, and the mesh used the solver’s default element size.
| Property | Value |
|---|---|
| Elastic modulus E | 130 GPa |
| Density ρ | 7,300 kg·m−3 |
| Poisson’s ratio μ | 0.25 |
Static stiffness under cutting load
The cutter head is mounted at the end of the cutter box, so the cutting force was applied there and resolved into components. With a main cutting force FC = 1,050 N, the components were FZ = 945 N, FX = 630 N and FY = 525 N, applied together with the self-weight of the structure. The maximum static deformation of the complete machine was 11.18 μm. For a machine intended to hold flank tolerances on gears of module 4 mm, that figure represents the elastic share of the total error budget and is the quantity that must be reduced if higher accuracy or higher cutting speed is required.
Modal behaviour of the complete machine
Natural frequencies were extracted from the same model and the same constraint set. For an n-degree-of-freedom system the equation of motion is
$$[M]\{\ddot{x}\}+[C]\{\dot{x}\}+[K]\{x\}=\{F\}$$
where [M], [C] and [K] are the mass, damping and stiffness matrices, {x} is the displacement vector and {F} the load vector. Damping has only a minor influence on natural frequency and mode shape, so the free-vibration problem is solved instead:
$$[M]\{\ddot{x}\}+[K]\{x\}=0$$
Assuming a harmonic solution {x} = {φ} cos ωt and substituting gives the eigenvalue problem
$$\left([K]-\omega^{2}[M]\right)\{\phi\}=0$$
which has non-trivial solutions only when the determinant vanishes:
$$\det\left([K]-\omega^{2}[M]\right)=0$$
The resulting discrete roots ωi2 (i = 1, 2, …, n) give the natural frequencies ωi and the corresponding mode shapes {φi}.
| Order | Frequency (Hz) | Mode shape |
|---|---|---|
| 1 | 94.221 | Column sways left–right; other assemblies essentially stationary |
| 2 | 127.28 | Column sways fore–aft; other assemblies essentially stationary |
| 3 | 204.51 | Workpiece box upper and lower ends swing about its own horizontal Z axis; rest nearly stationary |
| 4 | 231.89 | Column pitches vertically; all other assemblies move with it slightly |
| 5 | 259.88 | Workpiece box swings fore–aft; other assemblies micro-vibrate about horizontal axes |
| 6 | 283.36 | Workpiece box translates laterally; column vibrates vertically with small amplitude; bed bends slightly about its short axis |
Two conclusions follow directly. First, the lowest two modes are column modes, so the column sets the dynamic floor of the whole structure. Second, the workpiece box dominates modes 3, 5 and 6, which means that any excitation in the 200–290 Hz band couples mainly into the workpiece side of the loop.
Matching excitation frequencies to the modal spectrum
The dominant excitation in gear milling is the tooth-passing frequency of the cutter head, f = z n/60, where z is the number of cutter teeth and n the cutter speed in r/min. For the process parameters normally used on this machine, the excitation stays below the first natural frequency, as Table 4 shows.
| Cutter speed (r/min) | Number of teeth | Excitation frequency (Hz) | Relation to first mode (94.221 Hz) |
|---|---|---|---|
| 400 | 12 | 80 | Below |
| 500 | 10 | 83 | Below |
Both operating points sit roughly 12–15 % below the first natural frequency — a positive margin, but a relatively narrow one for a machine tool, and one that shrinks quickly if the cutter speed or the tooth count is increased. That is exactly what happens in high-speed operation. At 800 r/min with a 16-tooth cutter, the tooth-passing frequency becomes 213 Hz, which falls between the third (204.51 Hz) and fourth (231.89 Hz) natural frequencies. The parameter table lists a nominal cutter-spindle range of 100–350 r/min, so this case represents the higher-speed regime the design is intended to open up rather than a standard operating point — and in that regime the machine is driven into a band where the workpiece box and the column both respond.
Resonance here would appear as increased vibration of the workpiece box, degraded flank finish and reduced tool life. The remedies available at the design stage are to raise the stiffness of the structural loop, to add or integrate a damping device, or to avoid the critical speed band through process planning.
Column analysis and topology-based redesign
The whole-machine modes identify the column as the weakest link. The column is 805 mm high, with a 40 mm base plate, 30 mm top and side plates and 20 mm internal ribs, giving a mass of 476.32 kg in HT300. For the static case, the two faces in contact with the guide sliders were fixed, and loads of FX = 2,000 N, FY = 12,025 N and FZ = 1,500 N were applied to the two guideway faces.
The maximum equivalent stress was 1.7963 MPa, located at the front of the column base. Even allowing for stress concentration, this is two orders of magnitude below the 300 MPa strength limit of HT300 under the fourth strength theory. The maximum displacement, 18.025 μm, occurred at the top of the column with the cutter box at its uppermost limit — the least favourable position. Both results point the same way: the original column is stiff and strong enough, with substantial margin, and therefore over-dimensioned. That margin is the resource for a mass-reduction exercise, provided the dynamic behaviour is not degraded.
| Order | Frequency (Hz) | Mode shape |
|---|---|---|
| 1 | 100.25 | Side and top plates sway laterally |
| 2 | 297.16 | Side and top plates sway fore–aft |
| 3 | 377.67 | Torsion about the vertical centre axis |
| 4 | 383.45 | Side plates expand outward |
| 5 | 515.00 | Side plates twist about their own vertical axes |
| 6 | 533.17 | Side plates twist about their individual vertical axes |
The first three shapes — lateral sway, fore–aft sway and torsion about the vertical axis — act directly on the cutting point and therefore on accuracy and cutting performance; the last three matter much less. Raising the low-order frequencies, and with them the dynamic stiffness of the column, is the design objective, and it must be achieved without adding mass.
A topology optimisation was run with the same constraints and loads and a target volume reduction of 45 %. The result marked the base plate as removable material, while the modal shapes indicated that the first mode, a vertical sway of the two side plates, is fed by insufficient stiffness of the upper and lower horizontal plates. The redesign therefore combined two moves: lightening the base plate structure and adding 20 mm arc-section ribs at the lower corners of the column to restore local stiffness. The redesigned column has a mass of 441.44 kg.
| Quantity | Original | Redesigned | Change |
|---|---|---|---|
| Maximum deformation (mm) | 0.018025 | 0.017396 | −3.49 % |
| Maximum equivalent stress (MPa) | 1.7963 | 1.6738 | −6.82 % |
| Mass (kg) | 476.32 | 441.44 | −7.32 % |
| First natural frequency (Hz) | 100.25 | 102.23 | +1.98 Hz (+1.98 %) |
| Second natural frequency (Hz) | 297.16 | 297.89 | +0.73 Hz (+0.25 %) |
The redesign improves static stiffness and both low-order frequencies while removing 7.32 % of the mass, so it is a net gain rather than a trade. The size of the gain should be read honestly, however: a first-frequency shift of about 2 % moves the column mode from 100.25 Hz to 102.23 Hz, which is not enough on its own to resolve the 213 Hz coincidence identified above. The critical band involves the third whole-machine mode (workpiece box) and the fourth (column pitch), and the second column mode barely moved. Selective stiffening of the workpiece box and of the column in pitch, or the addition of damping, is the more direct route to higher cutting speeds.
What the design demonstrates and where it stops
The YKH200 shows that a five-axis horizontal architecture with direct-driven B and C axes, servo ball-screw linear axes and a purpose-built five-axis CNC system can replace the mechanical generating train for spiral bevel gears up to φ200 mm and module 4 mm. Finite-element modelling of the assembled structure, with bearings and screws represented as equivalent joint stiffness, gives a maximum static deformation of 11.18 μm under a 1,050 N cutting force, and a modal spectrum whose first two modes at 94.221 Hz and 127.28 Hz are column sway. The column itself carries a large strength reserve — 1.7963 MPa against a 300 MPa limit — and a topology-guided redesign with corner ribs converts part of that reserve into a 7.32 % mass reduction together with small gains in static stiffness and in the first two natural frequencies.
Operation at conventional cutting parameters, where the tooth-passing frequency stays at 80–83 Hz and below the first mode, has proved stable and reliable in production. The limitation is equally clear: at 800 r/min with a 16-tooth cutter the excitation reaches 213 Hz and lands between the third and fourth natural frequencies, so resonance remains possible in high-speed cutting. Closing that gap requires more than the column optimisation reported here — higher stiffness of the complete structural loop, targeted stiffening of the workpiece box and of the column in pitch, integrated damping, or process planning that keeps the tooth-passing frequency out of the 200–290 Hz band.