In the textile industry, the increase in loom speed has led to a noticeable problem: excessive warp yarn feeding (overfeeding). This phenomenon is closely related to the transmission essence of the worm gear pair in the let-off mechanism. In typical worm gear systems, whether for motion or power transmission, the worm drives the worm wheel. However, in the let-off mechanism of a silk loom, the power source is not on the worm shaft but on the worm wheel shaft. Without the self-locking property of the worm gear, the worm wheel would drive the worm to rotate continuously until the warp tension is completely relaxed. Understanding this principle is crucial for optimizing loom performance.
During weaving, the warp tension is typically around a certain value (e.g., tens of kilograms), and the torque acting on the worm wheel shaft (which is connected to the weaver’s beam) can be substantial. For a medium-weight fabric, with a warp tension of approximately 30 kg and a weaver’s beam diameter of 300 mm, the torque on the worm wheel shaft is about 4500 kg·cm. This significant driving source does not cause continuous rotation of the worm because the worm gear is self-locking. The driving torque is insufficient to overcome the frictional torque between the worm’s helical teeth and the worm wheel’s teeth, as well as the friction in the worm shaft bearings. Therefore, without an additional driving torque on the worm shaft, the worm will not rotate spontaneously.
When feeding occurs, a ratchet wheel applies a driving torque to the worm shaft. In the conventional case where the worm is the driving component and transmits motion to the worm wheel, the torque required to rotate the worm shaft is given by:
$$
T_{\text{worm}} = F_n \cdot \tan(\lambda + \rho) \cdot r_w
$$
where \(F_n\) is the normal force acting on the worm tooth, \(\lambda\) is the lead angle (helix angle) of the worm, \(\rho\) is the friction angle between the worm and worm wheel surfaces (for cast iron materials, \(\rho = \arctan(\mu)\) with \(\mu \approx 0.08-0.12\)), and \(r_w\) is the pitch radius of the worm (in mm).
In the self-locking worm gear pair of the let-off mechanism, to rotate the worm shaft, one must overcome the frictional torque between the helical teeth and the wheel teeth, which can be expressed as:
$$
T_{\text{friction, teeth}} = F_n \cdot \tan(\rho – \lambda) \cdot r_w \quad (\text{for self-locking condition } \lambda < \rho)
$$
Let \(T_{\text{bearing}}\) be the frictional torque between the worm shaft and its bearings. Then the total resisting torque to rotate the worm shaft is:
$$
T_{\text{res}} = T_{\text{friction, teeth}} + T_{\text{bearing}}
$$
During the actual feeding process, the ratchet wheel drives the worm shaft to rotate at a certain angular velocity. The ratchet-worm system possesses kinetic energy. When the ratchet reaches its limit position, the feeding action should normally stop. However, the kinetic energy of the ratchet and worm must be dissipated by the resisting torque \(T_{\text{res}}\). The worm shaft continues to rotate until all kinetic energy is consumed. This is the so-called “parking sliding process.” During this process, the worm shaft rotates further, causing additional warp yarn feeding—the overfeeding defect.
Obviously, under a given amount of kinetic energy, a smaller resisting torque \(T_{\text{res}}\) leads to a longer parking sliding time and thus more excessive warp feed. In general, the value of \(T_{\text{bearing}}\) is relatively small. Therefore, a brake disk is typically installed on the worm shaft to increase the frictional resisting torque, so that the sliding time is minimized and the overfeeding defect becomes imperceptible. With the brake disk, the total resisting torque becomes:
$$
T_{\text{total}} = T_{\text{friction, teeth}} + T_{\text{bearing}} + T_{\text{brake}}
$$
where \(T_{\text{brake}}\) is the frictional torque provided by the brake disk.
As loom speed increases, the angular velocity of the worm shaft during feeding also rises, resulting in larger kinetic energy. Under the same \(T_{\text{total}}\), the parking sliding time extends, and the overfeeding problem reappears. Therefore, to eliminate this defect at higher speeds, the total resisting torque must be appropriately increased.
Quantitative Analysis of the Problem
The kinetic energy \(E_k\) of the rotating system (ratchet plus worm shaft) is:
$$
E_k = \frac{1}{2} J \omega^2
$$
where \(J\) is the moment of inertia and \(\omega\) is the angular velocity. The parking sliding angle \(\theta_{\text{slide}}\) is related to the kinetic energy and resisting torque by:
$$
\theta_{\text{slide}} = \frac{E_k}{T_{\text{total}}}
$$
The amount of warp yarn fed during sliding is proportional to \(\theta_{\text{slide}}\) multiplied by the transmission ratio and beam radius. Thus, reducing \(E_k\) or increasing \(T_{\text{total}}\) reduces overfeeding.
Measures to Counteract Overfeeding
From the analysis, the following strategies can be adopted to mitigate the overfeeding problem at higher loom speeds:
| Measure | Description | Effect on \(T_{\text{total}}\) or \(E_k\) | Practical Considerations |
|---|---|---|---|
| Reduce worm lead angle \(\lambda\) | Decrease \(\lambda\) to increase the self-locking effect and raise \(T_{\text{friction, teeth}}\) | Increases \(T_{\text{total}}\) (since \(T_{\text{friction, teeth}} = F_n \tan(\rho-\lambda)r_w\) becomes larger as \(\lambda\) decreases) | Requires replacing the worm gear pair; may change bearing positions; costly for existing machines; suitable for new designs |
| Increase brake disk pressure | Add more spring force or adjust friction material to raise \(T_{\text{brake}}\) | Directly increases \(T_{\text{total}}\) | Simple and adjustable; may cause excessive wear or heat |
| Add a second brake disk | Install two brake disks on the worm shaft to double \(T_{\text{brake}}\) | Approximately doubles \(T_{\text{total}}\) increment from brakes | Example: some Japanese looms (e.g., Tsudakoma) use two brake disks; effective |
| Reduce the angular velocity of the worm shaft | Change the driving mechanism so that the ratchet is driven from the main shaft at a lower rotational speed, e.g., by using a center shaft transmission | Reduces \(E_k\) (since \(\omega\) decreases, \(E_k \propto \omega^2\)) | Can halve the angular velocity, reducing kinetic energy to one-fourth; may require modifying the drive train |
| Skip feeding cycles | Instead of feeding every pick, feed every other pick (or arranged intervals) by reducing ratchet rotation angle per pick | Effectively reduces the energy per feeding event and allows more time for braking | Suitable for fabrics less sensitive to precise weft density |
Detailed Torque and Energy Relationships
For a self-locking worm gear with lead angle \(\lambda\) and friction angle \(\rho\), the condition \(\lambda < \rho\) ensures that the worm wheel cannot drive the worm. The frictional torque component from the teeth is:
$$
T_{\text{teeth}} = F_n \cdot r_w \cdot \tan(\rho – \lambda)
$$
The normal force \(F_n\) is related to the axial load on the worm shaft from the warp tension. If the warp tension creates a torque \(T_{\text{beam}}\) on the worm wheel shaft, and the transmission ratio is \(i = Z_{\text{wheel}}/Z_{\text{worm}}\) (where \(Z\) are tooth numbers), then the axial thrust on the worm is approximately:
$$
F_{\text{axial}} = \frac{T_{\text{beam}}}{r_w \cdot i} \cdot \frac{1}{\cos(\lambda)}
$$
And the normal force \(F_n\) can be derived from the axial thrust and the pressure angle. For simplicity, assume a 20° pressure angle standard, the relationship involves trigonometric functions. In practice, the exact calculation uses the worm gear geometry.
The resisting torque from bearings (assuming sliding bearings) is:
$$
T_{\text{bearing}} = \mu_b \cdot F_{\text{radial}} \cdot r_{\text{bearing}}
$$
where \(\mu_b\) is the bearing friction coefficient (0.1-0.2 for lubricated bronze bearings), \(F_{\text{radial}}\) is the radial load, and \(r_{\text{bearing}}\) is the bearing journal radius.
The brake torque \(T_{\text{brake}}\) is given by:
$$
T_{\text{brake}} = \mu_d \cdot N_d \cdot R_d
$$
where \(\mu_d\) is the friction coefficient of the brake lining, \(N_d\) is the normal clamping force, and \(R_d\) is the effective radius of the brake disk.
Comparative Table: Effects of Parameter Changes on Parking Sliding
| Parameter Change | Effect on \(J\) | Effect on \(\omega\) | Effect on \(T_{\text{total}}\) | Net Effect on \(\theta_{\text{slide}}\) |
|---|---|---|---|---|
| Increase loom speed (base case) | Unchanged | Increases proportionally | Unchanged | Increases (worse) |
| Reduce \(\lambda\) from 5° to 3° (assuming \(\rho=6°\)) | Unchanged | Unchanged | Increases (since \(\tan(\rho-\lambda)\) becomes larger) | Decreases (improves) |
| Double brake force \(N_d\) | Unchanged | Unchanged | Increases (double \(T_{\text{brake}}\)) | Decreases (improves) |
| Halve ratchet speed by using a 2:1 reduction | May change slightly (ratchet inertia reduces if smaller) | Reduces to half | Unchanged | Reduces to 1/4 (major improvement) |
Practical Example: Loom Speed Increase from 200 rpm to 300 rpm
Assume original angular velocity of worm shaft is \(\omega_0\) at 200 rpm. At 300 rpm, the worm shaft angular velocity becomes \(1.5\omega_0\). The kinetic energy becomes \(2.25\) times the original. If the resisting torque remains constant, the sliding angle increases by 125%, causing noticeable overfeeding. To compensate, the resisting torque must be increased by at least 125%, or the kinetic energy must be reduced by a corresponding factor.
In one practical solution applied in some modern looms (e.g., the Japanese Tsudakoma machine), two brake disks are employed, effectively doubling the brake torque. Combined with a slight decrease in the lead angle of the worm, the overfeeding is suppressed even at higher speeds. Another approach involves redesigning the drive train so that the ratchet is driven from the main shaft via a reduction mechanism, lowering the effective rotational speed of the worm shaft. This reduces kinetic energy by the square of the speed reduction ratio.
Conclusion
The transmission essence of the worm gear in the let-off mechanism is deeply intertwined with the self-locking condition, frictional characteristics, and dynamic energy dissipation. The overfeeding of warp yarns at higher loom speeds is a direct consequence of inadequate braking torque relative to the increased kinetic energy of the rotating parts. By carefully selecting parameters such as the worm lead angle, brake disk configuration, and drive reduction ratio, the parking sliding time can be controlled to an acceptable level. The use of multiple brake disks or decreasing the angular velocity of the worm shaft proves to be the most practical solutions for existing machines, while new designs can also incorporate a smaller lead angle. Understanding these fundamentals allows textile engineers to fine-tune the let-off mechanism for high-speed weaving without compromising warp tension control.