In large-scale ship lift systems, the rack and pinion gear mechanism plays a critical role in ensuring stable and reliable vertical movement. This study focuses on analyzing the operating loads and fatigue life of rack and pinion systems used in ship lifts, drawing from extensive operational data. The rack and pinion drive is subjected to varying loads influenced by factors such as water depth in the ship chamber and operational states, including acceleration, constant speed, and deceleration. Understanding these load patterns is essential for assessing the longevity and reliability of the rack and pinion components, which are costly to manufacture and replace. Through statistical analysis and modeling, we develop a load calculation model, compile load spectra for different operational scenarios, and evaluate fatigue life using standardized methods. The results demonstrate that the rack and pinion system exhibits minimal fatigue damage over its design life, supported by full-scale simulation tests.
The rack and pinion mechanism in ship lifts operates under low-speed, heavy-load conditions, where load variations are primarily deterministic, governed by water depth and operational states, combined with random deviations. Data collected from multiple operational cycles reveal that the total load on the rack and pinion gear can be expressed as a function of these factors. For instance, the load model incorporates water depth as a key variable, with the load increasing linearly with water depth deviations from a standard value. Additionally, operational states like direction (up or down) and acceleration phases introduce distinct load components. A significant finding is the presence of load deviations among individual drive points, which follow a normal distribution, indicating random variations in the rack and pinion engagement. This insight allows for a more accurate representation of real-world operating conditions.
To quantify the loads, we define the total rack and pinion load \( F_t \) based on the water depth \( h \) (in meters), operational direction coefficient \( k_1 \) (where \( k_1 = 1 \) for upward movement and \( k_1 = -1 \) for downward movement), and acceleration coefficient \( k_2 \) (where \( k_2 = 0 \) for constant speed, \( k_2 = 1 \) for acceleration, and \( k_2 = -1 \) for deceleration). The deterministic part of the load is derived from the imbalance due to water depth, friction forces, and inertial effects. After analyzing operational data, we fit the parameters to establish the load model. The load on a single rack and pinion drive point \( F_{ti} \) includes a random deviation \( \Delta F_{ti} \), which is normally distributed. Thus, the load model is given by:
$$ F_{ti} = 5905.5(h – 3.5) + 220.5 + 128k_1 + 293.5k_1k_2 + \Delta F_{ti} $$
where \( \Delta F_{ti} \sim N(0, 155^2) \), meaning it follows a normal distribution with a mean of 0 and a standard deviation of 155 kN. This equation highlights how the rack and pinion gear experiences loads that combine predictable and random elements, essential for fatigue analysis.
The distribution of water depth in the ship chamber varies between upward and downward movements, affecting the load on the rack and pinion. We use non-parametric statistical methods, such as Gaussian kernel fitting, to model the water depth distribution. This approach captures the bimodal nature of the data, allowing for accurate extrapolation over the system’s design life. The probability distribution of water depth \( h_j \) for different operational directions is summarized in the table below, which shows the joint probabilities \( P(h_j, k_1) \). This table is crucial for compiling the load spectrum of the rack and pinion.
| Water Depth \( h_j \) (m) | Probability for Upward Movement \( P(h_j, k_1=1) \) (%) | Probability for Downward Movement \( P(h_j, k_1=-1) \) (%) |
|---|---|---|
| 3.38 | 0.26 | 0.09 |
| 3.39 | 1.12 | 1.28 |
| 3.40 | 6.96 | 2.32 |
| 3.41 | 10.70 | 6.72 |
| 3.42 | 9.97 | 6.81 |
| 3.43 | 7.65 | 3.73 |
| 3.44 | 3.97 | 2.63 |
| 3.45 | 2.98 | 2.46 |
| 3.46 | 3.44 | 2.76 |
| 3.47 | 3.24 | 3.50 |
| 3.48 | 3.43 | 4.38 |
| 3.49 | 3.77 | 4.98 |
| 3.50 | 3.58 | 5.50 |
| 3.51 | 3.64 | 6.76 |
| 3.52 | 5.35 | 11.32 |
| 3.53 | 8.63 | 14.38 |
| 3.54 | 10.01 | 9.39 |
| 3.55 | 6.86 | 5.70 |
| 3.56 | 2.95 | 3.24 |
| 3.57 | 1.05 | 1.41 |
| 3.58 | 0.38 | 0.44 |
| 3.59 | 0.02 | 0.18 |
| 3.60 | 0.00 | 0.01 |
Using this distribution, we compile the load spectrum for the rack and pinion gear by considering different operational regions: top, middle, and bottom of the rack. Each region experiences distinct load patterns due to the ship lift’s operational sequence. For example, the top rack primarily bears loads during upward deceleration and downward acceleration, the middle rack during constant speed phases, and the bottom rack during upward acceleration and downward deceleration. The load spectrum is developed by calculating the probability of each load level \( F_{ti} \) occurring, based on the combined probabilities of water depth, operational direction, and acceleration state. The load levels are divided into 32 intervals from -1650 kN to 1650 kN, with a step of 100 kN. The probability for each load level \( P(F_{ti} \leq F_n \leq F_{n+1}) \) is computed using the integral of the normal distribution probability density function over the load interval, combined with the joint probabilities of \( h_j \) and \( k_1 \).
The probability density function \( f(x) \) for the rack and pinion load is defined as:
$$ f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x – \mu)^2}{2\sigma^2}\right) $$
where \( \mu = 5905.5(h – 3.5) + 220.5 + 128k_1 + 293.5k_1k_2 \) and \( \sigma = 155 \) kN. The load spectrum for each region is then derived by integrating this function over the load intervals, weighted by the probabilities of \( h_j \) and \( k_1 \). For instance, the load spectrum for the top rack focuses on cases where \( k_2 = -1 \) for upward movement and \( k_2 = 1 \) for downward movement, while the middle rack uses \( k_2 = 0 \), and the bottom rack uses \( k_2 = 1 \) for upward and \( k_2 = -1 \) for downward movement. This regional approach ensures accurate fatigue life assessment for each part of the rack and pinion system.
The fatigue life of the rack and pinion gear is evaluated using the Miner linear cumulative damage theory, based on the compiled load spectra. The rack and pinion components are designed for a life of \( n = 4.22 \times 10^5 \) operational cycles, with each cycle involving multiple tooth engagements. For the pinion, with 16 teeth, the number of load cycles per tooth is \( N_g = 518n/16 = 1.37 \times 10^7 \), while for the rack, each tooth engages once per cycle, so \( N_r = n = 4.22 \times 10^5 \). The cumulative damage \( U \) is calculated as:
$$ U = \sum \frac{N_i}{N_{fi}} $$
where \( N_i \) is the actual number of cycles at load level \( i \), and \( N_{fi} \) is the number of cycles to failure at that load level, obtained from the S-N curves. The S-N curves for contact fatigue and bending fatigue are derived from ISO 6336 standards, considering material properties and lubrication conditions. The rack and pinion materials are high-strength steel, with the pinion made from 18CrNiMo7-6 (case-hardened to HRC 56-61) and the rack from G35CrNiMo6-6 (induction-hardened to HV 590-630). The nominal contact stress \( \sigma_H \) and bending stress \( \sigma_F \) are calculated as:
$$ \sigma_H = 0.6654 F_{ti} $$
$$ \sigma_{F1} = 9.15 \times 10^{-5} F_{ti} $$
$$ \sigma_{F2} = 8.30 \times 10^{-5} F_{ti} $$
where \( \sigma_{F1} \) and \( \sigma_{F2} \) represent bending stresses for the pinion and rack, respectively. The S-N curves indicate the fatigue limits: contact fatigue limit \( \sigma_{HG} = 1262 \) MPa and bending fatigue limit \( \sigma_{FG} = 781 \) MPa. After applying correction factors for size, load distribution, and surface condition, the effective S-N curves are used to compute \( N_{fi} \) for each load level.
The cumulative damage results for critical regions—pinion lower flank, top rack lower flank, middle rack upper flank, and bottom rack upper flank—show that bending fatigue damage is negligible, and contact fatigue damage is extremely low. For example, the bottom rack upper flank, which experiences the highest loads, has a cumulative damage degree of only \( 1.36 \times 10^{-5} \), corresponding to a fatigue life of \( 3.11 \times 10^{11} \) cycles, far exceeding the design life. The equivalent loads and safety factors for the rack are summarized in the table below, demonstrating high safety margins.
| Region | Equivalent Load \( F_{eq} \) (kN) for Contact | Equivalent Stress \( \sigma_H \) (MPa) | Safety Factor \( S_H \) | Equivalent Load \( F_{eq} \) (kN) for Bending | Equivalent Stress \( \sigma_F \) (MPa) | Safety Factor \( S_F \) |
|---|---|---|---|---|---|---|
| Top Rack | 1040 | 678.5 | 1.86 | 1344 | 111.6 | 6.97 |
| Middle Rack | 779 | 587.2 | 2.15 | 1059 | 87.9 | 8.89 |
| Bottom Rack | 1206 | 730.7 | 1.73 | 1449 | 120.3 | 6.49 |
These results indicate that the rack and pinion gear system has sufficient strength reserves, with contact safety factors above 1.73 and bending safety factors above 6.49, meeting the required thresholds for high-reliability applications (e.g., \( S_{H \min} = 1.50-1.60 \) and \( S_{F \min} = 2.0 \)). Furthermore, the presence of numerous low-load cycles below the fatigue limit may induce strengthening effects, further enhancing the rack and pinion’s durability.
Full-scale simulation tests conducted on a rack and pinion test rig validate these findings. The tests involved \( 4.22 \times 10^5 \) cycles under representative loads, with the rack and pinion operating at 0.2 m/s and a load of 905 kN. Post-test inspections, including surface examination and non-destructive testing, revealed no signs of pitting, cracking, or other damage. The measured root bending stress was approximately 59.2 MPa, lower than the calculated value of 75.1 MPa, confirming the conservative nature of the ISO 6336 standards for large-module rack and pinion gears. Although the test load was slightly lower than the equivalent loads for the top and bottom racks, the high safety factors ensure that the rack and pinion system remains reliable throughout its design life.
In conclusion, the rack and pinion gear in ship lifts exhibits robust performance under operational loads. The load model, incorporating deterministic and random components, accurately captures the load variations, enabling precise fatigue life predictions. The compiled load spectra reveal that loads are concentrated on specific flanks depending on the rack region, with the bottom rack upper flank being the most critical. However, the cumulative fatigue damage is minimal, and the safety factors are well above requirements. This study underscores the reliability of rack and pinion systems in heavy-duty applications and provides a framework for future assessments of similar mechanisms.