In my investigation of a screw gears transmission unit used in a cold-environment mechanism, I observed a clear loss of reversing response when the ambient temperature fell to -40 °C. The unit was originally built around a worm and worm wheel pair, which I treat here as a form of screw gears because the meshing action depends on sliding contact along helical screw surfaces. The slow reversal was not caused by the motor alone. I measured the shaft torque at normal temperature and at low temperature, and I found that the increase in internal resistance torque was the dominant cause. This led me to examine the supporting structure of the screw gears, the thermal behavior of the housing, the bearing arrangement, and the eccentric adjustment of the wheel shaft. I then developed an optimized support concept that replaces the original two-end fixed arrangement with a fixed-floating arrangement and replaces separate eccentric end caps with a single eccentric seat. The result is a screw gears drive with lower parasitic torque, better cold-start behavior, and improved safety.
1. Observed Performance of Screw Gears at Low Temperature
I began by comparing the transmission behavior at 18 °C and at -40 °C. The mechanism reversed very slowly at -40 °C. When I isolated the motor and measured the torque required to turn the input shaft, the normal-temperature torque was higher than the low-temperature motor torque in some phases, but the total resistance torque at the screw gears shaft was significantly higher in the cold state. This indicated that the screw gears support structure was adding friction instead of allowing free rotation. I recorded the following representative measurements.
| Condition | Ambient temperature (°C) | Input shaft torque at start (N·m) | Input shaft torque after warm-up (N·m) | Reversing time (s) | Observed behavior |
|---|---|---|---|---|---|
| Normal | 18 | 1.82 | 1.74 | 0.9 | Prompt reversal, stable screw gears motion |
| Cold soak | -40 | 4.65 | 2.10 | 3.4 | Slow reversal, high initial resistance |
| Cold soak with improved support | -40 | 2.05 | 1.78 | 1.1 | Reversal close to normal behavior |
The difference between the first and third rows shows that the support structure itself was responsible for most of the cold-state resistance. The screw gears were not failing because of tooth breakage or gross misalignment. Instead, the bearings were being preloaded by thermal contraction of the housing, and the resulting friction torque was large enough to dominate the drive.
2. Housing Deformation and Thermal Mismatch in Screw Gears
The external housing of the screw gears transmission was made of an aluminum alloy. The bearing rings and the screw gears shaft were made of steel. Because aluminum has a higher coefficient of linear expansion than steel, the housing shrinks more than the steel components when the temperature drops. This differential contraction changes the fit between the housing bore and the bearing outer ring. It also changes the axial position of the screw gears shaft. The material properties I used are summarized below.
| Component | Material | Coefficient of linear expansion (10-5/°C) | Elastic modulus (GPa) | Poisson ratio |
|---|---|---|---|---|
| Housing | Aluminum alloy | 2.30 | 70 | 0.33 |
| Shaft and bearing rings | Steel | 1.12 | 210 | 0.30 |
| Fixed seat | Steel | 1.12 | 210 | 0.30 |
| Eccentric seat | Steel or ductile iron | 1.12 to 1.20 | 170 to 210 | 0.30 |
The free thermal contraction of any dimension can be written as:
$$ \Delta L = L_0 \alpha \Delta T $$
where $$\Delta L$$ is the change in length, $$L_0$$ is the original length, $$\alpha$$ is the coefficient of linear expansion, and $$\Delta T$$ is the temperature change. For a cold soak from 18 °C to -40 °C, the temperature change is:
$$ \Delta T = -40 – 18 = -58 \text{ °C} $$
The relative contraction difference between the aluminum housing and the steel screw gears shaft is therefore:
$$ \Delta_{rel} = L_0 \left(\alpha_{Al} – \alpha_{steel}\right) \Delta T $$
For a nominal length of 270 mm, I obtained:
$$ \Delta_{rel} = 270 \times \left(2.30 \times 10^{-5} – 1.12 \times 10^{-5}\right) \times (-58) $$
$$ \Delta_{rel} = 270 \times 1.18 \times 10^{-5} \times (-58) \approx -0.185 \text{ mm} $$
This means the housing pulls inward about 0.185 mm more than the steel screw gears components over a 270 mm span. That extra contraction can close the bearing clearance, remove the axial float, and create a preload in the screw gears support structure. I also examined the measured housing dimensions at the two temperatures.
| Temperature | L1 (mm) | L2 (mm) | L3 (mm) | L4 (mm) | X1 (mm) | X2 (mm) | X3 (mm) |
|---|---|---|---|---|---|---|---|
| 18 °C | 160.01 | 270.03 | 18.95 | 312.85 | 41.97 | 39.97 | 62.97 |
| -40 °C | 159.60 | 269.58 | 18.96 | 312.59 | 41.95 | 26.95 | 62.95 |
| Difference | -0.41 | -0.45 | +0.01 | -0.26 | -0.02 | -13.02 | -0.02 |
The reduction in L1, L2, and L4 shows that the housing becomes smaller in the axial direction. The large change in X2 shows that one local support dimension is especially sensitive to temperature. When these dimensional changes act on the bearing seats, they can force the screw gears shaft toward one side, reduce the designed clearance, and increase friction. In the original design, the screw gears support structure used a two-end fixed arrangement. Each end was intended to locate the shaft, but the thermal contraction of the housing was not accommodated. The result was an increase in axial force and a corresponding increase in friction torque.
3. Force and Torque Analysis of Screw Gears
To quantify the problem, I analyzed the forces acting on the screw gears. In a worm and worm wheel pair, the worm drives the wheel through sliding contact. The tangential force on the worm depends on the input torque and the worm pitch diameter:
$$ F_{t1} = \frac{2 T_1}{d_1} $$
where $$T_1$$ is the input torque and $$d_1$$ is the worm pitch diameter. The axial force on the worm is related to the tangential force by the lead angle:
$$ F_{a1} = F_{t1} \tan \gamma $$
The radial force is:
$$ F_{r1} = \frac{F_{t1} \tan \alpha_n}{\cos \gamma} $$
where $$\gamma$$ is the lead angle and $$\alpha_n$$ is the normal pressure angle. The normal force on the screw gears tooth surface can be approximated as:
$$ F_n = \frac{F_t}{\cos \alpha_n \cos \gamma} $$
These forces are transmitted to the bearings and then to the housing. If the housing contracts and removes the bearing clearance, the bearing internal clearance becomes zero or negative, and the bearing raceways and rolling elements are pressed together. The additional friction torque from an axial preload can be written as:
$$ T_f = \mu F_a r_m $$
where $$\mu$$ is an equivalent friction coefficient, $$F_a$$ is the axial preload, and $$r_m$$ is the effective bearing radius. The total resistance torque at the screw gears shaft becomes:
$$ T_{total} = T_{load} + T_{mesh} + T_{bearing} + T_{seal} + T_{preload} $$
In the cold state, the term $$T_{preload}$$ became dominant. I estimated that a preload of only a few hundred newtons could increase the friction torque by more than 1 N·m, which is enough to make the screw gears reverse slowly. This explains why the mechanism performed poorly even though the tooth surfaces were not damaged.
4. Weaknesses of the Original Two-End Fixed Support
The original support structure located the screw gears shaft at both ends. One end used a bearing fitted into the housing bore with a clearance fit. An end cap was placed outside the bearing, and a gap was left between the cap and the bearing outer ring. Adjustment shims were used to set the axial position. I identified several weaknesses in this arrangement.
| Issue | Cause | Effect on screw gears |
|---|---|---|
| Axial clearance loss | Aluminum housing shrinks more than steel shaft | Bearing preload, increased friction torque |
| Radial clearance loss | Housing bore contracts onto bearing outer ring | Higher rolling resistance, possible brinelling |
| Two-end fixed location | Both ends resist thermal contraction | Large internal force, poor cold reversal |
| Shim adjustment | Manual shim stack changes with temperature | Unstable clearance and assembly variation |
| Separate eccentric caps | Upper and lower caps adjusted independently | Axes not perpendicular, uneven screw gears contact |
| Contact area effects | Full housing bore contact with bearing ring | High force from thermal contraction |
I also noted that the two-end fixed arrangement was sensitive to the difference between the housing material and the bearing material. When the temperature dropped, the housing pressed on the bearing outer ring from all sides. The radial pressure produced an elastic contraction of the outer ring. This contraction reduced the internal clearance of the bearing. The axial contraction simultaneously reduced the end clearance. The two effects combined to create a stiff, high-friction support. In the case of screw gears, even a small change in bearing drag can produce a noticeable change in reversing time because the screw gears pair already has sliding friction at the tooth mesh.
5. Fixed-Floating Support Optimization
Based on the failure mode, I changed the screw gears support structure from a two-end fixed arrangement to a fixed-floating arrangement. The fixed end carries the axial load and locates the screw gears shaft. The floating end allows relative axial movement between the shaft and the housing. This prevents the housing contraction from generating a large axial preload. The optimized arrangement is summarized in the following table.
| End | Bearing type | Primary function | Thermal behavior |
|---|---|---|---|
| Fixed end | Deep groove ball bearing plus thrust ball bearing on a steel fixed seat | Carries radial load and bidirectional axial load; sets axial position | Steel seat has low thermal expansion and keeps preload stable |
| Floating end | Cylindrical roller bearing without outer ring flange | Carries radial load; permits axial sliding | Outer ring can move relative to housing during contraction |
| Wheel shaft | Eccentric seat with tapered roller bearings | Controls center distance and backlash | Single eccentric seat avoids independent cap misalignment |
The fixed end uses a deep groove ball bearing for radial support and a thrust ball bearing for axial support. The thrust bearing is mounted in a steel fixed seat. Because the steel seat expands and contracts less than the aluminum housing, the axial clearance is less sensitive to temperature. The thrust bearing also allows the axial clearance to be adjusted in a controlled way. The floating end uses a cylindrical roller bearing whose outer ring has no flange. This allows the outer ring to slide axially in the housing bore when the housing contracts. The screw gears shaft is therefore free to expand or contract without being over-constrained.
The required floating allowance can be estimated from the thermal mismatch. For a span L, the free contraction difference is:
$$ \delta_{thermal} = L \left(\alpha_{housing} – \alpha_{shaft}\right) \left|\Delta T\right| $$
For L = 160 mm and $$\Delta T = -58 \text{ °C}$$, I obtained:
$$ \delta_{thermal} = 160 \times \left(2.30 \times 10^{-5} – 1.12 \times 10^{-5}\right) \times 58 $$
$$ \delta_{thermal} = 160 \times 1.18 \times 10^{-5} \times 58 \approx 0.110 \text{ mm} $$
For L = 270 mm, the same calculation gives:
$$ \delta_{thermal} = 270 \times 1.18 \times 10^{-5} \times 58 \approx 0.185 \text{ mm} $$
Therefore, the floating support must allow at least 0.19 mm of relative movement for the longer span and at least 0.11 mm for the shorter span. I designed the screw gears shaft with a nominal axial float of 0.20 mm at the L1 position, and I added a margin for manufacturing tolerances and bearing seat deformation. The fixed end controls the remaining axial position. This division of function prevents the housing from preloading the screw gears bearings.
6. Contact Pressure and Deformation Control
Even with a floating end, the fixed end and the bearing seats must be checked for contact pressure. When the housing contracts onto a bearing outer ring, the contact pressure depends on the radial interference, the housing wall thickness, and the diameter. For a thin-walled housing, the hoop stress can be approximated as:
$$ \sigma_\theta = \frac{p D}{2 t} $$
where $$p$$ is the contact pressure, $$D$$ is the outer diameter, and $$t$$ is the wall thickness. Rearranging gives:
$$ p = \frac{2 t \sigma_\theta}{D} $$
The contact force is the product of pressure and contact area:
$$ F_c = p A_c $$
If the contact pressure is held nearly constant by the thermal fit, then reducing the contact area reduces the total force acting on the bearing outer ring. I applied this idea by reducing the effective contact area of the bearing seat. Instead of a full cylindrical contact over the entire bore length, I used a controlled seat band. When the contact area was reduced by about 50%, the total radial force from thermal contraction was also reduced substantially. The resulting deformation of the bearing outer ring became smaller than the designed internal clearance. This ensured that the screw gears could still rotate freely at -40 °C.
I also checked the radial deformation of the bearing outer ring. For a bearing outer ring under external pressure, the diametral contraction is approximately:
$$ \Delta D \approx \frac{p D^2}{E t} $$
where $$E$$ is the elastic modulus. By keeping the contact pressure low and the seat band narrow, I kept $$\Delta D$$ below the bearing clearance. This is important for screw gears because any loss of bearing clearance directly increases the sliding resistance in the mesh. The screw gears pair already has a lower mechanical efficiency than many other gear types, so the support structure must not add unnecessary drag.
7. Eccentric Seat Optimization for the Screw Gears Wheel Shaft
The original wheel shaft assembly used a worm wheel, a wheel shaft, bearings, and two eccentric end caps. The eccentric caps were adjusted independently to set the screw gears center distance and backlash. In practice, the upper and lower caps were not synchronized. One cap could be rotated more than the other, which tilted the wheel shaft and made the screw gears axes non-perpendicular. This produced uneven contact across the tooth width and a fluctuating torque. To solve this, I replaced the two separate eccentric caps with a single eccentric seat.
| Feature | Original separate caps | Optimized eccentric seat |
|---|---|---|
| Adjustment points | Two independent caps | One seat with a single eccentric motion |
| Axis perpendicularity | Poor, depends on operator | Maintained by the seat bore |
| Backlash control | Unstable and uneven | Continuous and repeatable |
| Thermal response | Clearance changes unpredictably | Center distance changes in a controlled direction |
| Assembly time | High | Reduced |
The eccentric seat has an outer circle that fits into the housing bore and an inner bore that holds the wheel shaft bearings. The center of the inner bore is offset from the center of the outer circle by an eccentricity $$e$$. When the seat is rotated by an angle $$\theta$$, the center distance between the screw gears changes approximately as:
$$ \Delta a \approx e \cos \theta $$
More exactly, if the initial center distance is $$a_0$$, then:
$$ a(\theta) = a_0 + e \cos \theta $$
This allows fine adjustment of the screw gears backlash. The normal backlash is related to the center distance change by:
$$ j_n = 2 \Delta a \tan \alpha_n $$
where $$\alpha_n$$ is the normal pressure angle. For a small eccentricity, the adjustment is smooth and can be locked in place after setting. Because there is only one eccentric seat, the wheel shaft remains parallel to its intended axis. The screw gears contact pattern remains centered, and the torque variation is greatly reduced.
I also considered the perpendicularity error between the worm shaft and the wheel shaft. If the two bearing bores are offset by $$\Delta y$$ over a span $$L$$, the angular error is:
$$ \gamma = \arctan \left(\frac{\Delta y}{L}\right) $$
For screw gears, even a small angular error can concentrate the contact at one end of the tooth. The single eccentric seat reduces this error because the seat is machined in one piece and the bearing bore is aligned during manufacturing. The adjustment rotation only changes the center distance; it does not tilt the wheel shaft.
8. Bearing Selection, Life, and Load Capacity
After changing the support concept, I selected bearings that matched the new load path. The fixed end must carry both radial and axial loads. The floating end carries mainly radial load. The wheel shaft bearings must carry radial and axial loads from the screw gears mesh. I used the following bearing selection logic.
| Location | Bearing | Radial capacity | Axial capacity | Reason for selection |
|---|---|---|---|---|
| Worm fixed end | Deep groove ball bearing | Good | Moderate | Simple radial support and axial location |
| Worm fixed end | Thrust ball bearing | None | High | Carries bidirectional axial force from screw gears |
| Worm floating end | Cylindrical roller bearing | High | None | Allows axial sliding during thermal contraction |
| Wheel shaft | Tapered roller bearing | High | High | Handles combined load and provides rigidity |
The dynamic load rating of a rolling bearing is used to estimate its life. The basic rating life in millions of revolutions is:
$$ L_{10} = \left(\frac{C}{P}\right)^p $$
where $$C$$ is the dynamic load rating, $$P$$ is the equivalent dynamic load, and $$p$$ is 3 for ball bearings and 10/3 for roller bearings. In hours, the life is:
$$ L_{10h} = \frac{10^6}{60 n} \left(\frac{C}{P}\right)^p $$
where $$n$$ is the rotational speed in rpm. The equivalent dynamic load for combined radial and axial load is:
$$ P = X F_r + Y F_a $$
where $$X$$ and $$Y$$ are bearing factors. For the thrust ball bearing, the equivalent load is primarily axial:
$$ P_a = F_a $$
I checked the fixed-end thrust bearing against the maximum axial force from the screw gears. The axial force from the screw gears mesh is:
$$ F_a = F_t \tan \gamma $$
For the expected torque and lead angle, this axial force is the main load on the thrust bearing. The floating-end cylindrical roller bearing was selected for radial load only. Because it has no axial locating function, it does not resist the thermal contraction of the housing. This is the key reason the optimized screw gears support structure has lower cold-state friction.
9. Thermal Clearance Model and Design Calculations
I built a simple thermal clearance model to predict whether the screw gears support would remain free at low temperature. The axial clearance at temperature $$T$$ can be written as:
$$ c(T) = c_0 + \Delta L_s(T) – \Delta L_h(T) – \delta_b(T) $$
where $$c_0$$ is the initial assembly clearance, $$\Delta L_s$$ is the change in shaft length, $$\Delta L_h$$ is the change in housing length, and $$\delta_b$$ is the change in bearing width or seat deformation. Since both $$\Delta L_s$$ and $$\Delta L_h$$ are negative when the temperature decreases, the clearance becomes smaller when the housing contracts more than the shaft. The condition for free operation is:
$$ c(T) > 0 $$
If the clearance becomes negative, the support is preloaded. The preload force can be estimated from the axial stiffness $$k_a$$:
$$ F_a = k_a \left[-\left(c_0 + \Delta L_s – \Delta L_h\right)\right]_{+} $$
where the subscript plus means that only the negative clearance produces force. The friction torque from this preload is:
$$ T_f = \mu F_a r_m $$
I calculated the clearance at several temperatures for the original and optimized designs.
| Temperature (°C) | Original axial clearance (mm) | Original preload (N) | Original friction torque (N·m) | Optimized axial clearance (mm) | Optimized preload (N) | Optimized friction torque (N·m) |
|---|---|---|---|---|---|---|
| 18 | 0.18 | 0 | 0.02 | 0.20 | 0 | 0.02 |
| 0 | 0.12 | 0 | 0.03 | 0.18 | 0 | 0.02 |
| -20 | 0.02 | 40 | 0.14 | 0.14 | 0 | 0.03 |
| -40 | -0.08 | 260 | 0.92 | 0.11 | 0 | 0.04 |
The original design loses all clearance before -40 °C. The optimized design keeps a positive clearance because the floating end absorbs the differential contraction. The friction torque is reduced by more than 90% in the cold state. This explains the improved reversing time.
I also calculated the required floating allowance with a safety factor. The minimum required float is:
$$ \delta_{req} = c_0 + \Delta L_h – \Delta L_s + \delta_{margin} $$
For the longest span, $$\Delta L_h – \Delta L_s \approx 0.185 \text{ mm}$$. With an initial clearance of 0.05 mm and a margin of 0.05 mm, the required float is:
$$ \delta_{req} = 0.05 + 0.185 + 0.05 = 0.285 \text{ mm} $$
I therefore provided a total floating allowance greater than 0.30 mm in the bearing seat, distributed between the cylindrical roller bearing outer ring and the housing bore. The actual working clearance at the screw gears shaft remains positive because the fixed end takes up part of the tolerance stack.
10. Validation Results
After implementing the optimized support structure, I compared the screw gears behavior before and after the change. The comparison included cold-start torque, steady torque, reversing time, axial clearance, and safety factor.
| Performance metric | Original two-end fixed support | Optimized fixed-floating support | Improvement |
|---|---|---|---|
| Cold-start torque at -40 °C (N·m) | 4.65 | 2.05 | 56% lower |
| Steady torque at -40 °C (N·m) | 2.10 | 1.78 | 15% lower |
| Reversing time at -40 °C (s) | 3.4 | 1.1 | 68% shorter |
| Axial clearance at -40 °C (mm) | -0.08 | +0.11 | Preload removed |
| Friction torque increase from 18 °C to -40 °C | +152% | +8% | Thermal sensitivity reduced |
| Bearing safety factor | 1.2 | 1.8 | 50% higher |
| Backlash variation | High | Low | More stable screw gears contact |
The optimized structure also improved the tooth contact of the screw gears. Because the eccentric seat keeps the wheel shaft aligned, the contact pattern remains centered. The measured torque fluctuation was reduced. The screw gears did not experience the same level of localized loading that occurred with the separate eccentric caps.
I also checked the thermal stress in the housing. The maximum housing stress under the optimized contact condition was below the yield strength of the aluminum alloy. The fixed seat carried the axial load without significant deformation. The thrust ball bearing maintained its clearance over the full temperature range. The floating bearing allowed the required axial movement without binding.
11. Assembly, Lubrication, and Maintenance Rules
The optimized screw gears support structure requires careful assembly and maintenance. I established the following rules to ensure that the design intent is preserved in production and service.
| Stage | Requirement | Reason |
|---|---|---|
| Housing bore preparation | Control bore tolerance and surface finish | Ensures predictable contact pressure and bearing fit |
| Fixed seat installation | Use steel seat with controlled axial preload | Keeps axial clearance stable during thermal cycling |
| Floating bearing installation | Verify axial sliding freedom in the housing bore | Allows thermal contraction without preload |
| Eccentric seat adjustment | Set backlash with a single rotation and lock it | Maintains screw gears center distance and perpendicularity |
| Lubrication | Use low-temperature grease or oil compatible with steel and aluminum | Reduces friction at cold start and protects screw gears teeth |
| Initial run-in | Cycle the screw gears at low load before full load | Seats the bearing and confirms free reversal |
| Maintenance | Inspect axial clearance and backlash periodically | Detects wear before it affects screw gears performance |
I also recommend that the screw gears assembly be checked at both normal and low temperature before final acceptance. A simple cold-chamber test can reveal whether the axial clearance remains positive. If the reversing torque is still high, the fixed-end thrust bearing clearance should be rechecked first, followed by the floating-end sliding fit. The eccentric seat should be adjusted only after the bearing support is confirmed to be free.
12. Conclusion
My analysis showed that the slow reversing of the screw gears transmission at -40 °C was caused primarily by the original two-end fixed support structure. The aluminum housing contracted more than the steel screw gears shaft, which reduced the bearing clearance and created an axial preload. The preload increased the friction torque and made the screw gears difficult to reverse. By changing the support structure to a fixed-floating arrangement, I allowed the housing and shaft to contract independently. The fixed end uses a deep groove ball bearing and a thrust ball bearing on a steel seat to carry axial load and set position. The floating end uses a cylindrical roller bearing to permit axial movement. This prevents thermal contraction from generating a large preload.
I also replaced the separate eccentric end caps with a single eccentric seat. This improved the perpendicularity between the screw gears axes, stabilized the backlash, and reduced torque fluctuation. The optimized design keeps a positive axial clearance at -40 °C, reduces cold-start torque by more than half, shortens reversing time by about two-thirds, and increases the bearing safety factor. The screw gears pair operates with lower parasitic friction and more uniform tooth contact. The supporting structure is therefore no longer the limiting factor in cold-environment operation. The same principles can be applied to other screw gears systems where thermal mismatch, bearing preload, and backlash control are important for reliability and efficiency.
For future work, I would extend the model to include transient temperature gradients, since the housing and shaft may not cool at the same rate. A transient analysis would help refine the floating allowance and the break-in procedure. I would also measure the actual contact pattern of the screw gears under load to confirm that the eccentric seat adjustment remains stable over time. With these additional steps, the optimized support structure can be tuned even more precisely for specific screw gears applications.