The pursuit of advanced manufacturing, as exemplified by initiatives like “Made in China 2025,” hinges critically on the development of high-precision, reliable core components for industrial robots. Among these, the rotary vector reducer stands out as a pivotal element. Its exceptional performance in terms of high transmission efficiency, compactness, and large reduction ratio makes it indispensable for precision motion control in robotic joints. Consequently, the vibration characteristics of a rotary vector reducer are not merely secondary attributes but are fundamental indicators of its operational quality, directly influencing positional accuracy, service life, and overall system stability. The vibration signal emanating from a rotary vector reducer during operation is a rich source of information, encoding the dynamic interactions and potential anomalies within its complex assembly of gears and bearings. Therefore, sophisticated analysis of these signals is paramount for enhancing product quality, enabling predictive maintenance, and advancing the state-of-the-art in rotary vector reducer design and manufacturing.
Traditional vibration analysis in the frequency domain often struggles with the complex, non-linear, and non-stationary signals generated by rotary vector reducers. Characteristic frequencies from multiple meshing events and rotational components often overlap and are obscured by noise, making precise fault identification challenging. This work presents a comprehensive investigation into the torsional vibration of rotary vector reducers, employing Variational Mode Decomposition (VMD) as a powerful adaptive signal processing tool to demystify these complex signals. We establish a dedicated wireless torsional vibration test bench, calculate the fundamental theoretical vibration frequencies, and apply VMD to decompose acquired acceleration signals. The extracted Intrinsic Mode Functions (IMFs) are then analyzed to correlate spectral features with specific mechanical components and, ultimately, to diagnose the root cause of excessive vibration in substandard units. This methodology provides a robust framework for quality assurance and performance optimization of rotary vector reducers.
Structural Composition and Operational Principles of the Rotary Vector Reducer
The distinctive performance of the rotary vector reducer arises from its two-stage, compound planetary design. It ingeniously combines a first-stage planetary gear train with a second-stage cycloidal pin-wheel mechanism. Key components include the input gear, planetary gears, crankshafts, cycloid gears, a pin housing (or ring gear), needle pins, main bearings, and the output flange. This sophisticated arrangement is what allows the rotary vector reducer to achieve high reduction ratios in a robust and compact package.
The transmission principle can be described in two phases. In the first stage, the input shaft, driven by a servo motor, rotates the central sun gear. This sun gear meshes with multiple planetary gears (typically two or three), causing them to rotate on their own axes. This constitutes the initial speed reduction. The crankshafts are fixed to these planetary gears and thus rotate with them. In the second stage, the rotation of each crankshaft imparts an eccentric motion to a cycloid gear mounted on it via bearings. This cycloid gear meshes with a set of stationary needle pins housed in the pin shell. Due to the epitrochoidal geometry of the cycloid gear and its interaction with the fixed pins, the eccentric motion is converted into a reverse, slower rotation of the cycloid gear itself. This reverse rotation of the cycloid gear causes the crankshaft to revolve around the central axis of the reducer. Finally, this revolution is transferred to the output flange, completing the second and final stage of speed reduction. The overall high reduction ratio is the product of the ratios from these two stages.
Theoretical Vibration Frequency Analysis for the Rotary Vector Reducer
The complex motion within a rotary vector reducer generates a spectrum of characteristic vibration frequencies. These frequencies are deterministic and can be calculated from the kinematic relationships and the geometry of the components. Identifying these theoretical frequencies is crucial for interpreting the measured vibration spectra. For a given rotary vector reducer model, the following fundamental frequencies can be derived:
The rotational frequency of the input shaft is directly proportional to the motor speed:
$$f_{in} = \frac{n_{in}}{60}$$
where $n_{in}$ is the input shaft speed in revolutions per minute (RPM).
The rotational frequency of the output flange is:
$$f_{out} = \frac{n_{out}}{60} = \frac{n_{in}}{60 R}$$
where $R$ is the total reduction ratio of the rotary vector reducer.
More critical for vibration analysis are the meshing and component frequencies. The auto-rotation frequency of the planetary gears and their attached crankshafts is given by:
$$f_{p\_auto} = \frac{z_1 z_2}{(z_1 + z_2)} \cdot f_{in}$$
where $z_1$ is the number of teeth on the input sun gear and $z_2$ is the number of teeth on each planetary gear.
The revolution frequency of the planetary gear carrier (which is often the output in the first stage, driving the crankshaft revolution) is:
$$f_{p\_rev} = \frac{z_1}{(z_1 + z_2)} \cdot f_{in}$$
The auto-rotation frequency of the cycloid gear relative to the crankshaft is:
$$f_{c\_auto} = \frac{z_4}{(z_3 – z_4)} \cdot f_{p\_rev}$$
where $z_3$ is the number of lobes on the cycloid gear and $z_4$ is the number of needle pins in the housing. The term $(z_3 – z_4)$ is typically 1, defining the fundamental reduction of the cycloid stage.
The revolution frequency of the cycloid gear, which is the primary motion driving the output, is equal to the output frequency $f_{out}$.
The most significant excitation sources are the meshing frequencies. The planetary gear meshing frequency is:
$$f_{mesh\_p} = z_2 \cdot f_{p\_auto} = \frac{z_1 z_2}{(z_1 + z_2)} \cdot f_{in}$$
The cycloid gear meshing frequency, a primary source of vibration and noise, is:
$$f_{mesh\_c} = z_4 \cdot f_{c\_auto} = \frac{z_1 z_4}{(z_1 + z_2)} \cdot f_{in}$$
For the specific 190BX model of rotary vector reducer analyzed in this study, the parameters are as follows:
| Parameter | Symbol | Value |
|---|---|---|
| Reduction Ratio | $R$ | 121 |
| Input Sun Gear Teeth | $z_1$ | 12 |
| Planetary Gear Teeth | $z_2$ | 36 |
| Cycloid Gear Lobes | $z_3$ | 39 |
| Needle Pins | $z_4$ | 40 |
Based on these parameters, the key theoretical vibration frequencies for different output speeds can be calculated, as summarized in the table below for an output speed of 2500 RPM (corresponding input speed $n_{in} = 2500 \times 121$ RPM).
| Frequency Description | Symbol | Theoretical Value (Hz) @ 2500 RPM Output |
|---|---|---|
| Output Shaft Frequency | $f_{out}$ | 41.67 |
| Planetary/Crankshaft Auto-rotation | $f_{p\_auto}$ | 13.77 |
| Cycloid Gear Meshing | $f_{mesh\_c}$ | 537.19 |
| Planetary Gear Meshing | $f_{mesh\_p}$ | 495.87 |
Torsional Vibration Test Platform and Signal Acquisition
To experimentally investigate the vibration characteristics, a dedicated wireless torsional vibration test platform was constructed. The core of the setup is the rotary vector reducer unit under test. A servo motor is directly coupled to the input shaft of the reducer. The housing (pin shell) of the rotary vector reducer is rigidly mounted to a fixed base plate via a flange, ensuring that all measured vibration is due to internal dynamics and not external mounting effects. The motor drives the rotary vector reducer under no-load conditions to isolate the vibration signature of the reducer itself from applied load variations.
The key instrumentation is a high-sensitivity wireless tri-axial accelerometer. To specifically capture torsional vibration, the sensor is adhesively mounted at the edge of the output flange of the rotary vector reducer. Careful alignment is ensured so that one of its measurement axes is precisely tangential to the direction of rotation. This tangential acceleration is directly proportional to the angular acceleration, providing a clear signal of torsional vibration. Data is acquired at a sampling frequency of 4000 Hz, which, according to the Nyquist theorem, allows for the analysis of frequency components up to 2000 Hz, well beyond the expected major meshing frequencies of the rotary vector reducer.
Two distinct quality grades of the 190BX rotary vector reducer were tested: units classified as “superior” based on standard quality control checks and post-assembly testing, and units classified as “inferior” which exhibited subpar performance in initial evaluations. For each unit, data was collected under multiple operating conditions: with the output shaft rotating at 1500 RPM, 2000 RPM, and 2500 RPM, and in both clockwise and counterclockwise directions. Multiple datasets were recorded for each condition to ensure consistency and reliability.
The raw time-domain acceleration signals for both a superior and an inferior rotary vector reducer at 2500 RPM are shown conceptually (descriptions of amplitude variations over time). Initial observation in the time domain reveals that the inferior rotary vector reducer generally exhibits a larger amplitude of vibration compared to the superior one. However, the specific frequency content and the root cause of the excess vibration are not discernible from the time-domain plot alone.
A quantitative initial assessment can be made by calculating the Root Mean Square (RMS) value of the acceleration signal, a standard metric for vibration intensity. For a discrete signal $x_i$ of length $N$, the RMS is defined as:
$$X_{RMS} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} x_i^2 }$$
The calculated RMS values for the tested units under different speeds clearly demonstrate the performance gap.
| Output Speed (RPM) | Product Quality | Sample 1 RMS (g) | Sample 2 RMS (g) | Sample 3 RMS (g) |
|---|---|---|---|---|
| 1500 | Superior | 0.0943 | 0.0921 | 0.0905 |
| Inferior | 0.1636 | 0.1626 | 0.1734 | |
| 2000 | Superior | 0.1899 | 0.1685 | 0.1584 |
| Inferior | 0.2538 | 0.2620 | 0.2537 | |
| 2500 | Superior | 0.1792 | 0.1719 | 0.1881 |
| Inferior | 0.3465 | 0.3595 | 0.3661 |
The table confirms that the inferior rotary vector reducer consistently produces stronger vibration across all tested speeds. While the RMS value is a useful overall indicator, it provides no insight into which component or process within the rotary vector reducer is responsible for the degraded performance. For this, a more sophisticated signal decomposition technique is required.
Principle of Variational Mode Decomposition (VMD)
Variational Mode Decomposition is a fully adaptive, non-recursive signal processing technique designed to effectively handle non-linear and non-stationary signals. Its core strength lies in its ability to concurrently decompose a complex signal into a discrete number of band-limited Intrinsic Mode Functions (IMFs), denoted as $u_k(t)$. Each IMF $u_k(t)$ is compact around a corresponding center pulsation $\omega_k$, which is determined along with the decomposition.
The method is founded on solving a constrained variational problem. The goal is to minimize the sum of the estimated bandwidths of all modes. The problem is formulated as follows:
$$\min_{\{u_k\},\{\omega_k\}} \left\{ \sum_{k} \left\| \partial_t \left[ \left( \delta(t) + \frac{j}{\pi t} \right) * u_k(t) \right] e^{-j\omega_k t} \right\|_2^2 \right\}$$
$$\text{subject to} \quad \sum_{k} u_k = f(t)$$
where $f(t)$ is the original signal, $\delta(t)$ is the Dirac delta distribution, $j$ is the imaginary unit, $*$ denotes convolution, and $\partial_t$ is the partial derivative with respect to time. The term inside the norm calculates the analytic signal of $u_k(t)$ and shifts its frequency spectrum to baseband, and its gradient squared L2-norm effectively estimates the bandwidth of that mode.
To render the problem unconstrained and solvable, two elements are introduced: a quadratic penalty term $\alpha$ to enforce reconstruction fidelity in the presence of noise, and a Lagrangian multiplier $\lambda(t)$ to enforce strict constraint satisfaction. The augmented Lagrangian $\mathcal{L}$ becomes:
$$
\begin{aligned}
\mathcal{L}(\{u_k\},\{\omega_k\},\lambda) &= \alpha \sum_{k} \left\| \partial_t \left[ \left( \delta(t) + \frac{j}{\pi t} \right) * u_k(t) \right] e^{-j\omega_k t} \right\|_2^2 \\
&+ \left\| f(t) – \sum_{k} u_k(t) \right\|_2^2 \\
&+ \left\langle \lambda(t), f(t) – \sum_{k} u_k(t) \right\rangle
\end{aligned}
$$
The problem is then solved using the Alternate Direction Method of Multipliers (ADMM). This involves iterative updates in the frequency domain for all $u_k$, $\omega_k$, and $\lambda$ until convergence. The update equations in the frequency domain are:
$$\hat{u}_k^{n+1}(\omega) = \frac{\hat{f}(\omega) – \sum_{i \neq k} \hat{u}_i(\omega) + \frac{\hat{\lambda}(\omega)}{2}}{1 + 2\alpha (\omega – \omega_k)^2}$$
$$\omega_k^{n+1} = \frac{\int_0^\infty \omega |\hat{u}_k^{n+1}(\omega)|^2 d\omega}{\int_0^\infty |\hat{u}_k^{n+1}(\omega)|^2 d\omega}$$
The Lagrangian multiplier is updated as:
$$\hat{\lambda}^{n+1}(\omega) = \hat{\lambda}^{n}(\omega) + \tau \left( \hat{f}(\omega) – \sum_{k} \hat{u}_k^{n+1}(\omega) \right)$$
where $\tau$ is an update parameter. The iteration stops when a convergence criterion is met:
$$\frac{\sum_k \|\hat{u}_k^{n+1} – \hat{u}_k^{n}\|_2^2}{\|\hat{u}_k^{n}\|_2^2} < \epsilon$$
The result is a set of $K$ IMFs, $u_1(t), u_2(t), …, u_K(t)$, with distinct center frequencies $\omega_1 < \omega_2 < … < \omega_K$. For the vibration signal of the rotary vector reducer, this process effectively disentangles the overlapping frequency components related to different physical phenomena within the gearbox.
Signal Decomposition and Feature Extraction Results
The VMD algorithm was applied to the torsional acceleration signals from both superior and inferior rotary vector reducer units operating at 2500 RPM. A critical parameter in VMD is the number of modes $K$. To avoid over-decomposition (creating spurious modes) or under-decomposition (failing to separate important components), $K$ was set to 20. This choice was guided by the prior theoretical analysis, which indicated significant vibration energy concentrated in frequency bands from near 0 Hz (shaft speeds) up to approximately 600 Hz (meshing frequencies and their sidebands). A larger $K$ allows the algorithm to sufficiently separate these components.
The VMD process successfully decomposed the complex signal into 20 IMFs. The lower-order IMFs (e.g., IMF1-IMF5) typically capture very high-frequency or noise components. The intermediate and higher-order IMFs contain the most mechanically significant information. A visual inspection of the time-domain waveforms of selected IMFs (e.g., IMF17-IMF20) for both product grades shows distinct oscillatory patterns, but the spectral content is the key to interpretation.
Performing a Fast Fourier Transform (FFT) on each relevant IMF reveals its dominant frequency components. The spectral plots for IMFs 15 through 20 for the inferior rotary vector reducer and their counterparts for the superior unit were analyzed. The results show a striking alignment between the dominant peaks in specific IMFs and the theoretical vibration frequencies calculated earlier.
- The spectrum of IMF 17 (superior) and IMF 15 (inferior) exhibited a dominant peak at approximately 539.1 Hz. This aligns almost perfectly with the theoretical cycloid gear meshing frequency $f_{mesh\_c} = 537.2$ Hz for the rotary vector reducer.
- The spectrum of IMF 18 (superior) and IMF 16 (inferior) showed a strong peak near 498.0 Hz, closely matching the theoretical planetary gear meshing frequency $f_{mesh\_p} = 495.9$ Hz.
- The spectrum of the highest-order mode, IMF 20 for both units, contained low-frequency peaks. Most notably, a peak at 41.0 Hz corresponds to the output shaft rotational frequency $f_{out}=41.67$ Hz. Its harmonics at approximately 82.0 Hz and 125.0 Hz (2x and 3x $f_{out}$) are also present, indicating non-linear effects or other periodic excitations at the output stage.
This successful correlation validates the effectiveness of VMD in adaptively filtering and isolating the characteristic vibration signatures of a rotary vector reducer from the raw, composite signal. The normalized frequency errors between the VMD-extracted peaks and the theoretical values are all below 2%, demonstrating high fidelity.
| Theoretical Frequency (Hz) | VMD-Extracted Frequency (Hz) | Normalized Error | Related IMF (Superior/Inferior) |
|---|---|---|---|
| 537.19 ($f_{mesh\_c}$) | 539.06 | 0.35% | IMF17 / IMF15 |
| 495.87 ($f_{mesh\_p}$) | 498.05 | 0.44% | IMF18 / IMF16 |
| 41.67 ($f_{out}$) | 41.01 | 1.58% | IMF20 / IMF20 |
Fault Identification and Diagnosis for the Inferior Rotary Vector Reducer
The true diagnostic power of combining theoretical frequency analysis with VMD-based decomposition becomes apparent when comparing the detailed spectra of corresponding IMFs from the superior and inferior rotary vector reducers. While both units show peaks at the major meshing frequencies, a critical difference emerges in the low-frequency spectrum of the highest IMF (IMF20).
A focused analysis of the IMF20 spectrum in the 0-50 Hz range reveals a prominent peak at 13.67 Hz for both units. This frequency corresponds precisely to the theoretical auto-rotation frequency of the planetary gears and crankshafts $f_{p\_auto} = 13.77$ Hz, which is kinematically identical to the revolution frequency of the cycloid gear carrier in this specific architecture. The presence of this component is expected. However, the amplitude of this peak is drastically different.
In the superior rotary vector reducer, the amplitude of the 13.67 Hz component is very low, measured at approximately 0.0073g. In stark contrast, the same frequency component in the inferior rotary vector reducer has an amplitude of about 0.0467g, which is over six times larger. This significant amplitude disparity was consistently observed across all three inferior units tested and was present at all operating speeds (1500, 2000, and 2500 RPM), although the absolute frequency shifted proportionally with speed.
This finding is of paramount diagnostic importance. The frequency at 13.67 Hz is directly linked to the motion of the planetary gears/crankshafts and the associated cycloid gear wobble (eccentric revolution before reduction). An abnormally high vibration at this fundamental kinematic frequency strongly suggests an issue related to the assembly or component quality of these specific parts within the rotary vector reducer.
The most plausible root causes, deduced from the frequency signature, are:
- Planetary Gear or Crankshaft Imbalance/Runout: Eccentricity or mass imbalance in the planetary gears or their attached crankshafts would generate a once-per-revolution force at frequency $f_{p\_auto}$, directly exciting the system at 13.67 Hz. This could result from manufacturing tolerances in gear blank machining, bearing seat concentricity errors on the crankshaft, or improper assembly.
- Cycloid Gear Eccentricity or Mounting Error: While the cycloid gear is designed for eccentric motion, deviations from the ideal eccentricity profile or misalignment in its mounting on the crankshaft bearing can introduce anomalous forces at its revolution frequency, which is also $f_{p\_auto}$.
- Inconsistent Clearance or Preload in Crankshaft Bearings: Variations in the bearing fit or preload on the multiple crankshafts can cause uneven load distribution and alter the dynamics of the planetary stage, potentially amplifying vibration at the characteristic frequencies of that stage.
The VMD analysis, by cleanly isolating this specific low-frequency component from the overwhelming meshing frequencies, pinpointed the excitation source to the first-stage planetary/crankshaft assembly or its interface with the cycloid gears. This conclusion directs quality inspection efforts towards the machining accuracy of planetary gears and crankshafts, the consistency of bearing fits, and the assembly process of the planetary stage in the rotary vector reducer, providing a clear and actionable insight for manufacturing process improvement.
Discussion and Engineering Implications
The successful application of VMD for the vibration analysis of the rotary vector reducer demonstrates a significant advancement over conventional spectral analysis. Traditional FFT of the raw vibration signal from the rotary vector reducer typically produces a complex spectrum with a forest of peaks, where meshing frequency harmonics, sidebands due to modulation, and shaft frequency harmonics are densely packed and often overlap. Disentangling the specific source of a problem, especially one related to a lower-frequency kinematic error, is exceedingly difficult from such a spectrum.
VMD overcomes this by performing an adaptive, quasi-orthogonal filter bank operation. It automatically tunes the center frequencies and bandwidths of the IMFs to match the dominant oscillatory modes present in the signal. For the rotary vector reducer, this resulted in nearly pure tones corresponding to the planetary meshing, cycloid meshing, and shaft frequencies being separated into distinct IMFs. This separation is not merely mathematical; it has a direct physical interpretation, associating each IMF with a specific mechanical excitation process within the rotary vector reducer.
The parameter selection for VMD, particularly the number of modes $K$ and the penalty factor $\alpha$, requires careful consideration. In this study, $K=20$ was chosen based on the known bandwidth of interest for the rotary vector reducer. A systematic approach, such as observing the center frequency distribution or using criteria like the envelope entropy, could be employed for automated parameter optimization in future industrial applications. The value of $\alpha$ influences the bandwidth of each IMF; a higher $\alpha$ produces modes with stricter bandwidth constraints.
The diagnostic finding—that excessive vibration at the planetary/crankshaft auto-rotation frequency is a key marker for inferior rotary vector reducers—has immediate and profound engineering implications. It shifts quality control from a passive, post-assembly vibration level check (e.g., RMS threshold) to an active, diagnostic procedure. Production lines can implement this VMD-based analysis as a final test step. Units exhibiting abnormally high amplitude in the IMF containing the $f_{p\_auto}$ frequency can be flagged for rework or detailed teardown inspection, focusing on the first-stage components. This targeted approach is more efficient and informative than rejecting units based on overall vibration level alone.
Furthermore, this method provides invaluable feedback to the design and manufacturing departments. Persistent identification of this fault mode indicates a potential weakness or tolerance stack-up issue in the planetary gear train design or its manufacturing process for the rotary vector reducer. Engineers can then review tolerances for gear concentricity, crankshaft journal runout, and bearing clearance specifications to mitigate the issue at its source.
Conclusion
This study has established a robust methodology for the advanced torsional vibration analysis of rotary vector reducers, integrating experimental testing, theoretical frequency modeling, and the adaptive signal processing technique of Variational Mode Decomposition. The construction of a dedicated wireless test platform enabled the accurate capture of torsional vibration signals under controlled conditions.
The core achievement lies in the effective deployment of VMD to decompose the complex, non-linear vibration signal of the rotary vector reducer into a set of physically meaningful Intrinsic Mode Functions. The spectral analysis of these IMFs revealed a high-degree correlation with the theoretically calculated characteristic frequencies of the rotary vector reducer, including the output shaft frequency, the planetary gear meshing frequency, and the dominant cycloid gear meshing frequency. This validated VMD as a superior tool for isolating specific vibration sources within the densely packed spectrum of a rotary vector reducer.
Beyond validation, the comparative analysis of IMF spectra between superior and inferior rotary vector reducer units delivered a precise diagnostic outcome. The significant amplitude enhancement observed specifically at the planetary gear/crankshaft auto-rotation frequency (approximately 13.67 Hz at 2500 RPM output) in inferior units definitively identified the root cause of their excessive vibration. This anomaly is attributed to imbalances, eccentricities, or assembly inconsistencies in the first-stage planetary gear train or its interface with the cycloid gears.
In conclusion, the VMD-based approach transforms vibration analysis for the rotary vector reducer from a general health monitoring tool into a precise diagnostic and quality assurance instrument. It provides a clear, frequency-based signature for a common manufacturing or assembly defect, offering direct guidance for corrective actions in production and design refinement. This methodology significantly contributes to the ongoing effort to enhance the precision, reliability, and overall performance of rotary vector reducers, which are essential for the advancement of high-end robotic and automation systems.