In modern mechanical engineering, worm gears play a critical role in various applications, including CNC machine tools, automotive systems, and precision instruments, due to their ability to achieve high transmission ratios, smooth operation, low noise, and self-locking characteristics. However, the performance and accuracy of worm gears are heavily dependent on the precision of their tooth pitch, which directly influences transmission stability and reliability. Traditional methods for measuring worm gear tooth pitch often involve contact-based techniques, such as coordinate measuring machines (CMMs) or non-contact optical systems, which can be inefficient, expensive, and limited in real-world operational scenarios. These limitations have driven the need for a more efficient and accurate detection system that can operate continuously in practical environments. In this research, we address this challenge by developing a novel detection system based on the synchronous displacement sampling principle, which enables high-speed, continuous measurement of worm gear tooth pitch with improved accuracy and efficiency. Our approach leverages advanced signal processing, FPGA-based data acquisition, and USB communication to create a compact and reliable system. By integrating key components such as photoelectric encoders and a customized algorithm for pulse subdivision counting, we aim to overcome the drawbacks of existing methods and provide a solution that is both practical and scalable for industrial use.
The core of our detection system relies on the synchronous displacement sampling principle, which involves measuring the angular displacement of worm gears in relation to a reference component, typically a standard worm. This principle allows us to convert rotational displacements into tooth pitch data through mathematical relationships. Specifically, we employ a method where the worm’s rotation serves as a benchmark, and the worm gear’s displacement is sampled at synchronized intervals. This eliminates the need for complex mechanical setups and reduces measurement errors associated with traditional techniques. To implement this, we use photoelectric encoders attached to both the worm and worm gear shafts. These encoders generate electrical signals proportional to the angular positions, which are then processed to extract tooth pitch information. The system’s hardware includes signal conditioning circuits to convert encoder outputs into TTL-compatible signals, an FPGA module for real-time data processing, and a USB interface for data transmission to a PC-based software platform. The software, developed in C#, provides a user-friendly interface for data visualization, storage, and analysis, enabling dynamic display of tooth pitch measurements and facilitating further evaluation of worm gear performance.
One of the key innovations in our system is the pulse subdivision counting algorithm, which enhances measurement resolution by interpolating fractional parts of encoder pulses using a high-frequency clock signal. This algorithm is implemented in Verilog HDL on an FPGA, allowing for precise counting of both integer and fractional pulse components. The mathematical foundation of this approach is derived from the transmission error theory, where the actual tooth pitch is calculated based on the difference between theoretical and measured angular displacements. For instance, the theoretical tooth pitch \( P \) for a worm gear can be expressed as \( P = \pi \cdot m \), where \( m \) is the module. However, due to manufacturing imperfections, the actual tooth pitch \( P_i’ \) for the \( i \)-th tooth is determined by the angular displacement relationship. Using the synchronous sampling principle, we define the sampling period based on the worm’s encoder signals, and within each period, we count the number of pulses from the worm gear’s encoder. The total displacement \( \Delta \phi_i \) in the \( i \)-th sampling period is given by:
$$ \Delta \phi_i = n_i + \frac{T_{c(i)} – T_{c(i-1)}}{N_{H(i)}} $$
where \( n_i \) is the integer number of pulses, \( T_{c(i)} \) and \( T_{c(i-1)} \) are the counts of high-frequency clock pulses for the fractional parts in consecutive periods, and \( N_{H(i)} \) is the number of clock pulses per worm gear encoder pulse. The actual tooth pitch \( P_i’ \) is then calculated as:
$$ P_i’ = \frac{\Delta \phi_i}{P_0} \cdot \pi \cdot m $$
where \( P_0 \) is the nominal pulse count per sampling period under ideal conditions. This formula ensures that even minor deviations in tooth pitch are captured with high precision, as the fractional part is resolved to sub-micron levels through clock interpolation. To illustrate the system parameters and their relationships, we summarize key variables in Table 1.
| Parameter | Symbol | Description | Typical Value |
|---|---|---|---|
| Worm Gear Module | \( m \) | Module of the worm gear | 3.8 mm |
| Number of Teeth | \( Z \) | Total teeth on worm gear | 144 |
| Encoder Resolution | \( \lambda \) | Lines per revolution of encoder | 2048 (worm), 32768 (gear) |
| Sampling Frequency | \( f_s \) | Frequency of synchronous sampling | 4 kHz |
| Clock Frequency | \( f_t \) | High-frequency interpolation clock | 40 MHz |
| Theoretical Tooth Pitch | \( P \) | Ideal tooth pitch based on module | \( \pi \cdot m \) mm |
In the experimental phase, we constructed a test platform using a CNC hobbing machine equipped with photoelectric encoders on both the worm and worm gear shafts. The encoders used were a Heidenhain ERN180 for the worm shaft and a RCN8581 for the worm gear shaft, providing high-resolution angular measurements. The signal conditioning circuit, designed with operational amplifiers and comparators, converted the encoder outputs into clean TTL signals for FPGA processing. The FPGA, a Cyclone IV series device, was programmed to execute the pulse subdivision algorithm, which involved modules for frequency multiplication, fractional extraction, and data framing. For instance, the frequency multiplication module increased the effective resolution of the encoders by a factor of four, allowing us to detect finer angular changes. Data from the FPGA was buffered using an internal FIFO and transmitted via a USB 2.0 interface using a CY7C68013A chip configured in slave FIFO mode. This setup ensured real-time data transfer to the PC, where the custom software displayed the tooth pitch measurements as a dynamic graph and saved the data for further analysis.
To validate our system, we conducted comparative tests with a semi-automatic pitch measuring instrument, which served as a reference. The tests involved measuring the tooth pitch of a worm gear with known parameters under identical conditions. Our system achieved measurement results that closely matched those of the reference instrument, with a maximum deviation of less than 3.0 μm, which is within the acceptable error range of 5.0 μm for such applications. Moreover, the efficiency of our system was significantly higher, reducing measurement time from approximately 5 minutes per cycle to just 2 minutes, thanks to the continuous sampling capability. The results, summarized in Table 2, demonstrate the accuracy and speed of our approach. For example, the left flank measurements showed a maximum tooth pitch of 11.9346 mm with the reference instrument and 11.9333 mm with our system, both at tooth position 31, indicating consistent performance across different methods.
| Tooth Flank | Reference Instrument (Max/Min, mm) | Proposed System (Max/Min, mm) | Error (μm) |
|---|---|---|---|
| Left Flank | 11.9346 / 11.9296 | 11.9333 / 11.9319 | 3.0 |
| Right Flank | 11.9352 / 11.9293 | 11.9350 / 11.9300 | 2.8 |
The mathematical models underlying our detection system also account for potential errors, such as those introduced by encoder inaccuracies. The error contribution \( \Delta P \) from the encoder to a single tooth pitch measurement can be estimated using the formula:
$$ \Delta P = \frac{\Delta \phi \cdot r}{1296} $$
where \( \Delta \phi \) is the peak-to-peak encoder error in arcseconds, and \( r \) is the pitch radius of the worm gear. For our setup, with \( \Delta \phi = 1.54 \) arcseconds and \( r = 273.6 \) mm, the error per tooth is approximately 0.0142 μm, which is negligible for practical purposes. This highlights the robustness of our system against common sources of inaccuracy. Additionally, the use of FPGA-based logic allows for flexibility in adjusting parameters such as clock frequencies and sampling rates, making the system adaptable to various worm gear configurations. In terms of software functionality, the C# application not only displays real-time data but also exports results in text format for offline analysis, enhancing its utility in industrial settings.
In conclusion, our research presents an efficient and accurate detection system for worm gear tooth pitch based on synchronous displacement sampling. By integrating advanced hardware and software components, we have developed a solution that addresses the limitations of existing methods, offering higher speed, better precision, and greater practicality. The system’s ability to perform continuous measurements in real-time makes it suitable for quality control in manufacturing processes, particularly for high-precision worm gears used in critical applications. Future work could focus on further miniaturizing the hardware, enhancing noise immunity, and extending the system to other types of gear inspections. Overall, this study contributes to the ongoing advancement of measurement technologies for worm gears, supporting the broader goal of improving mechanical transmission systems in various industries.