Keywords

1 Introduction

Gearboxes are common transmission components in mechanical systems and are widely used in quay cranes. However, since quay cranes are often in normal operation, many ports now adopt regular inspection strategies, making it difficult to obtain various fault signals. Therefore, it is necessary to model and simulate the quay crane gearbox to obtain simulated signals.

Currently, common methods for obtaining simulation signals can be mainly divided into several categories: using finite element simulation to generate data, constructing vibration models based on phenomenological theory, setting up laboratory test rigs to obtain experimental data, and solving data based on dynamic models, etc. Jiawei Xiang [1] took mechanical transmission systems such as bearings, gear transmissions, and rotor systems as examples, constructed a complete structural finite element model, carried out model corrections, and obtained a simulation model with a certain degree of accuracy. Ming Zhang [2] and others proposed a digital twin framework, which provides a fault signal generation mechanism for rolling bearings based on phenomenological theory. Liang Guo [3] set up a laboratory test rig and used experimental data to implement unsupervised transfer learning. Yiming Xiao [4] and others proposed a bearing dynamic model and successfully applied the simulation signals to local transfer learning. Jing Liu [5] and others considered the lateral, longitudinal, and torsional vibrations of the shaft, and established a rigid shaft gear transmission dynamic system model for multi-stage transmission systems. Zhaoyang Tian [6] and others considered time-varying meshing stiffness, backlash, gyroscopic effects, and transmission error excitation, and established a finite element nodal dynamic model for the gear-bearing-Squeeze Film Damper system. Yuhang Hu [7] and others established a six-degree-of-freedom interval dynamic model for the gear system, considering the internal excitation changes caused by shaft misalignment, which can better predict the actual dynamic characteristics of the gear system compared to traditional models. Josef Koutsoupakis [8] and others used multibody dynamics software to build a dynamic model of a two-stage gearbox and used simulation signals to validate the proposed algorithm. Jianbo Yu [9] and others built a dynamic model of a planetary gear system through multibody dynamics software, and proposed a signal fusion model based on digital twins, which can accurately identify gearbox faults at the signal level.

When the aforementioned scholars model transmission systems, they often set the load as a constant torque. However, in the operation of quay cranes handling containers, the output shaft of the gearbox is connected to the wire rope, wheel slide group, and the container, resulting in a coupling phenomenon between the gearbox, wire rope, and container. If there are accelerations, decelerations, or if the container sways in the air, or even changes in stiffness during the meshing process of the gears, it will cause changes in the internal excitation of the gearbox, leading to dynamic loads. Furthermore, changes in the weight of the container itself can also cause variations in dynamic loads.

In response to this situation, this paper proposes a multibody dynamics model of the gearbox based on dynamic load simulation, used to generate simulation signals. First, the gearbox is modeled, then meshing is performed on key components, and finally, the multibody dynamics simulation software is used to build the dynamic model. Two dynamic model control schemes, dynamic load and static load, were designed, and an actual scaled-down quay crane test rig was set up. Through the dynamic modeling of normal and faulty gears and the analysis of simulation signals, it was proven that the dynamic model considering dynamic loads can better predict the actual dynamic characteristics of the gearbox system than the dynamic model with constant loads.

2 Common Fault Characteristics of Gears

During the operation of a gearbox, it transfers a very large load. When gears mesh and rotate, the number of teeth involved in the meshing alternates, resulting in significant alternating changes in the meshing stiffness of the gears. When a pair of gears (with the number of teeth for Gear 1 and Gear 2 being z1 and z2, and their rotation frequencies being f1 and f1, respectively) engage in transmission, several frequencies will appear in the vibration signal spectrum collected from Shaft 1. One is the rotation frequency f1 of Gear 1 and Shaft 1. The second is the meshing frequency of each pair of teeth, fm = f1 × z1 = f2 × z2. When Gear 1 has a localized fault (such as pitting, broken teeth, etc.), it will produce periodic impacts when meshing with Gear 2. The spectral line of the rotation frequency f1 in the collected vibration signal spectrum will increase. At the same time, the gear fault frequency will also induce a gear meshing modulation effect in the vibration signal [10]. As shown in Fig. 1b, the meshing frequency fm and the sidebands around its harmonics are the fault frequency f1. If one uses the spectrum to observe whether the bearing has a fault, it is time-consuming and labor-intensive. Envelope spectrum analysis of the vibration signal can demodulate the vibration signal. As shown in Fig. 1c, compared to the spectrum, the envelope spectrum of the faulty bearing can more accurately locate the fault frequency characteristic f1 and its harmonics.

Fig. 1
3 schematic diagrams. A. A graph with a simulation scale that ranges from 0 to 1.5. B. A graph with a spectrum that ranges from 0 to 800 hertz. C. A graph with an envelope spectrum that ranges from 0 to 800 hertz.

Time domain waveform, frequency domain waveform, and envelope spectrum of gear fault simulation signal. a Simulated vibration acceleration signal of the faulty gear b Spectrum of the signal c Envelope spectrum of the signal

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3 Proposed Dynamic Modeling Method

3.1 Construction of Dynamic Equations

Without considering the elastic deformation of the shaft, bearings, casing, etc., in the gearbox transmission process, the cylindrical gear system can be simplified as a pure torsional vibration system. Its vibration can be simplified as a single degree of freedom vibration equation. Using the lumped mass method to establish the dynamic model of the auxiliary gear system's torsional vibration, as shown in Fig. 2, the dynamic model of Gear 1 (driving gear) in the meshing process with Gear 2 (driven gear) is as follows:

$$ m\ddot{x} + c\dot{x} + k(t)[x - e(t)] = (M_{2} - M_{1} )/r_{1} $$
(1)
Fig. 2
A schematic diagram of gear meshing. It illustrates gears 1 and 2 rotating with torques m 1 and m 2, shafts 1 and 2, both of which are connected through wire rope. A mass m c and a spring k c.

Schematic diagram of gear meshing. In the diagram, the blue part corresponds to the constant torque in Eq. 1, while the red part corresponds to the wire rope + container section considered in Eq. 2

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where x is the relative displacement of the gears on the meshing line; m is the equivalent mass, \(m = \frac{{m_{1} m_{2} }}{{m_{1} + m_{2} }}\); c is the meshing damping; k(t) is the time-varying meshing stiffness; e(t) is the transmission error; M1 and M2 are the torques acting on the gears; r1 is the base circle radius of Gear 1.

During the operation of quay cranes, containers are transported using a wire rope + pulley system. The weight of the transported container is unknown and might be much larger than the total weight of the transmission system. The wire rope also has associated stiffness and mass properties. In the actual operation of the quay crane, there is a coupling phenomenon between the wire rope + container and the gearbox transmission system. In Eq. 1, the load on Gear 2 is a constant torque M2, which does not consider important influences such as the inertia of the load in the transmission system and cannot directly reflect the real response state of the gearbox.

If the influence of the rope and the weight of the container is considered, the dynamic model is as follows.

$$ m\ddot{x} + c\dot{x} + k(t)[x - e(t)] = ((m_{c} g + k_{c} x_{c} ) \times r_{2} - M_{1} )/r_{1} $$
(2)

where mc represents the total mass of the wire rope and the container; g is the gravitational acceleration; rc is the drum radius where the wire rope and container are connected to the gear; xc is the elongation of the wire rope.

3.2 Construction of the Multibody Dynamics Model

Considering the limitations of the gear dynamic equation Formula 1 in Sect. 3.1 and the difficulty in solving Eq. 2, this paper proposes a digital modeling method for the multibody dynamics of the quay crane gearbox transmission system. The modeling process is illustrated in Fig. 3.

Fig. 3
A flowchart of the modeling method. It illustrates the process, starting with part modeling and going through assembly, meshing, modeling, importing the model, defining joints and forces, loading the bearing module, loading the cable module, and simulation.

Flowchart of the modeling method

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Step 1: Based on the actual quay crane, use 3D modeling software to model the gearbox components. Perform interference checks on the model to ensure that there is no interference between components during system operation.

Step 2: Use finite element software to mesh the key components of the gearbox and produce flexible body files that can be used by multibody dynamics simulation software.

Step 3: Import the models and flexible body files created in Steps 1 and 2 into the multibody dynamics simulation software. Create constraints for the imported models, solve the calculations, and obtain the required vibration signals.

3.2.1 Component Modeling

Based on the actual object, this paper has drawn a 1:4 scaled model of the quay crane gearbox hoisting mechanism. The hoisting mechanism adopts a dual-input and dual-output structure, with a total of four shafts and three pairs of meshing helical gears. The gearbox input comes from two motors, and the output leads to two double drums, with a wire rope winding ratio of 2. Detailed parameters of the gearbox gears are shown in Table 1.

Table 1 Gear parameters of the quay crane hoisting mechanism gearbox
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3.2.2 Meshing

To obtain vibration simulation signals from the dynamic model that are similar to the actual ones, it's necessary to perform meshing on the gearbox established in Sect. 3.2.1. After meshing the key components of the quay crane gearbox and generating the flexible body file, it's essential to set key points that contact other objects. In this paper, 23 key points were set for the flexible body file of the gearbox, including 6 fixed points with the ground, 8 fixed points of the casing with the transmission shaft, and 9 sensor fixed points. The mesh model of the gearbox and its key points are shown in Fig. 4.

Fig. 4
2 schematics of 3-D diagrams. A. A mesh model of the gearbox. B. A mesh diagram presents the key points of the gearbox flexible body file.

a Meshing diagram of the gearbox b Key points of the gearbox flexible body file; yellow points represent fixed points with the ground, red points represent connection points of the casing with the transmission shaft, and green points represent sensor fixed points

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3.2.3 Construction of the Dynamic Model

When modeling the quay crane gearbox transmission system using multibody dynamics software, the gearbox components established in Sect. 3.2.1 and the flexible body files created in Sect. 3.2.2 are imported. The specific construction process is as follows.

Step 1: Modify the material properties of the imported model. Step 2: Create constraints. Establish fixed pairs for the 6 key points on the gearbox with the ground. Create two motor dummies at both ends of the input shaft and set rotational pairs to simulate the two input motors. Constrain the shaft and the gear on the shaft with fixed pairs. Also, constrain the output shaft and the drum with fixed pairs. Step 3: Define motion. Apply rotational motion to the two rotational pairs of the motor dummies created in Step 2. Step 4: Define forces. Add contact forces between each pair of driving and driven gears. Add a rotational torque to the two drums on the output shaft. Apply a sleeve force between the input shaft and the motor dummy that only restricts the rotation direction of the motor. Step 5: Define the bearing module. Use the virtual bearing module provided by the multibody dynamics software to create virtual bearings connecting the gearbox shaft and the casing. Step 6: Define the rope module. To address the issue of dynamic load simulation difficulty raised in Sect. 3.1, use the cable module provided by the multibody dynamics software to simulate the rope. Create a dummy at each end of the rope as an anchor point, with one dummy fixed and the other having a translational pair and force applied. Define the mutual conversion relationship between the rope module and the output shaft and drum, as shown in Fig. 5.

Fig. 5
A schematic diagram of the rope module. It presents an Ancher, pulley, and output shaft connected by a cable.

Schematic diagram of the rope module and the mutual conversion relationship between force and displacement

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By following the above steps, the dynamic modeling of the quay crane hoisting mechanism gearbox is completed. The dynamic model is shown in Fig. 6.

Fig. 6
A schematic diagram of the quay crane's dynamic model. It depicts the dynamic modeling of the quay crane hoisting mechanism gearbox completed.

Schematic diagram of the quay crane's dynamic model

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4 Simulation Experiment

4.1 Introduction to the Simulation Conditions of the Multibody Dynamics Model

To verify the effectiveness of the modeling method proposed in this paper, the constructed dynamic model was simulated under different state categories, different loading methods, and different operating conditions. Vibration acceleration signals from key sensor points on the gearbox casing were obtained across various dimensions.

4.1.1 Types of Gear Faults

This paper depicts different states of the driven gear 1 on the intermediate shaft 1, as shown in Fig. 7, mainly divided into normal state and missing tooth state.

Fig. 7
2 3-D schematic diagrams. A. 3-D diagram of the gear in its normal state. B. 3-D diagram of the gear in the missing tooth state.

Model of the driven gear 1 on intermediate shaft 1. a Normal state b Missing tooth state

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4.1.2 Different Load Weights

To verify the impact between different loads as proposed in Sect. 3.1, in addition to creating a dynamic load dynamic model based on the rope, this paper also produced a dynamic model with an equivalent constant torque as a static load, as shown in Fig. 8.

Fig. 8
2 schematics of the 3-D diagrams. A. An illustration of a static load gearbox dynamic model. B. An illustration of a dynamic load model.

a Static load gearbox dynamic model b Dynamic load dynamic model

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4.1.3 Different Operating Conditions

During the actual operation of the quay crane, the transported container is first lifted to a certain height and then lowered onto the truck transporting the container. In the quay crane hoisting mechanism gearbox, there are two different operating conditions: forward rotation and reverse rotation. To simulate the relevant operating process of the gearbox, this paper set the weight of the transported container to 1000 kg. The motor and input shaft speeds are as shown in the Fig. 9. The gearbox first increases the input shaft speed from 0 to 900Rpm between 0.2 s and 0.3 s (with a maximum angular velocity of -5400 degrees/sec in the figure). Between 2.6 s and 2.9 s, the speed drops from 900Rpm to -900Rpm (with a maximum angular velocity of 5400 degrees/sec in the figure). The total simulation time is 5 s, with 50,000 simulation steps.

Fig. 9
A line graph of angular velocity versus time presents gear 0.1 C M angular velocity Y, Shuruzhou driver body 0.1 C M angular velocity Y, and Shuruzhou driver body 0 2 CM angular velocity Y. All three lines illustrate a fluctuating trend.

Motor and input shaft speeds

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4.2 Verification with the Built Test Rig

To verify the effectiveness of the method proposed in this paper, a scaled test rig of an actual quay crane hoisting mechanism was constructed. Sensors were installed at corresponding positions, and faulty components were fabricated, as shown in Figs. 10 and 11. The sensor sampling frequency fs = 10 kHz.

Fig. 10
3 photographs of a scaled test rig of an actual quay crane hoisting mechanism, along with installed sensors at corresponding positions and faulty fabricated components.

Scaled test rig of the quay crane gearbox and sensor installation positions (numbers in the picture indicate sensor labels)

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Fig. 11
A photograph of the scale test rig of the quay crane gearbox and the faulty gear component.

Scaled test rig of the quay crane gearbox and the faulty gear component

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4.3 Analysis of Simulation Results

Since the driven gear 1 of the Intermediate shaft 1 was selected as the faulty gear, the vibration acceleration signal collected from sensor channel 3 was chosen as the research subject. Under the conditions proposed in this paper, when this gear is faulty, the gear rotation frequency fr = 5.455 Hz and the meshing frequency fm = 300Hz. The simulation signal from 0.5 s to 2.5 s is taken as the container lifting phase signal, and the simulation signal from 3 to 5 s is taken as the container lowering phase signal.

By observing and comparing the simulation signals of the dynamic load dynamic model during the rising and falling phases, as shown in Figs. 12 and 13, it can be found that compared to the container lowering phase, the container lifting phase can more accurately and clearly capture the modulation phenomenon of the rotation frequency and meshing frequency in the envelope spectrum. The rotation frequency and its harmonics can be more precisely captured in the envelope spectrum.

Fig. 12
6 line graphs of waveform, spectrum, and envelope spectrum present simulation signals of the dynamic model in the container lifting phase and container lowering phase. Dark and light lines represent the fault and normal signals, respectively.

Simulation signals of the dynamic load dynamic model a Container lifting phase b Container lowering phase

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Fig. 13
6 line graphs of waveform, spectrum, and envelope spectrum present a simulation signal of the static load dynamic model during the rising phase and from the scaled test rig. Dark and light lines represent faults and normal signals, respectively.

a Simulation signal of the static load dynamic model during the rising phase b Signal from the scaled test rig

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5 Conclusion

This paper proposes a gearbox multibody dynamic model based on dynamic load simulation and has experimentally proven the effectiveness of the proposed method. It also reveals that the simulation signals with dynamic loads are more in line with actual conditions than those with static loads. Additionally, during the experimental process, it was found that the fault characteristics of the gearbox during the rising phase are more pronounced than those during the lowering phase. The above results can provide a basis for fault diagnosis of quay crane hoisting gearboxes under dynamic loads and can offer simulation data support for intelligent diagnostic models that lack fault samples.