Simulative and Experimental Investigation of Vibration Transfer Path at Gearboxes

Abstract

Condition monitoring systems are widely used in gearboxes. Gears are one of the most crucial components for power transmission. Hence, the optimal sensor positions for condition monitoring of gears should be investigated to maximize reliability and to minimize costs. This work aims to analyze measured signals from rotating sensors at gears and compare them to signals from housing sensors to find the suitable positions for condition monitoring of the gears. Additionally, the rotational speed and external torque influences on the signal quality have been investigated. These are compared with a simulation model, which considers the vibration excitation from the gear mesh and bearings. The results show that the rotational speed affects the amplitude of the excitation. On this basis, we also investigate the amplitudes of the excitation frequencies of interest. The ratio of the amplitudes of these frequencies related to the mean values of the measurement signals is called the peak-to-mean ratio (PMR), and this PMR corresponds to the speed which is of interest for automatic fault detection in the gearboxes. Additionally, the simulation results show that the intensity of the vibration with the gear mesh frequency hardly reduces during transmission through the tapered roller bearings.

1. Introduction

Gearboxes are widely used in vehicles, ships, wind turbines, etc. To achieve high safety standards and avoid additional costs resulting from any damages, condition monitoring is one way to achieve maximum reliability and availability. By measuring the vibrations under operation and analyzing the amplitude of the measured signal at specific frequencies, it is possible to detect damages and relate them to specific components.

In gearboxes, each rotating component can generate vibrations with different frequencies, so it is important to analyze the vibration transmission through the different components in gearboxes, such as the gears, shafts, bearings, and housing. Ren [1] analyzes the vibration transmission of an entire gearbox using the vibration power flow. He indicates that the bearings and housing are critical for the vibration transmission within the gearbox system. Xiao [2] uses lumped parameter modeling to investigate the vibration transmission of a gearbox and indicates that the maximum attenuation occurs at the transmission from the inner race to the outer race, while the minimum is between the outer race and the housing. Parker [3] analyzes the vibration of a gearbox by using both lumped parameter modeling and finite element analysis and shows that the lumped parameter model is suitable for parametric studies on the dynamic response. Richards [4] defines a periodic structural drive shaft that can reduce gear mesh vibration transmission. A periodic structural shaft means that a few sections of the drive shaft are modeled periodically after each other. Engel [5] investigates vibration transmission to minimize vibration levels by a modification of the gear. He indicates that discontinuity in the gear body is helpful for structure-borne sound insulation.

As the bearing is critical for vibration transmission, several sources [6,7,8] investigate vibration transfer through bearings. Lim [6] uses the lumped parameter model to simulate the dynamic behavior of a generic single-shaft-bearing plate system and investigate the vibration transmission through the bearing. Richter [7] investigates the transfer function of the bearing based on theoretical calculations and experiments. The transfer function describes the ratio of the speed of the vibration between the bearing outer ring and the bearing inner ring. For the investigation of vibration transmission through the housing, the finite element analysis has typically been used because of the complex geometry of the housing.

Several sources [9,10,11,12] investigate the differences in vibration levels at different positions on the gearbox housing in order to find suitable positions for condition monitoring. Knoll [9] evaluates the vibration level in the time domain at different positions of a gearbox based on an experiment and simulation. Yu [10] and Han [11] also investigate the transfer path of a gearbox system by analyzing the vibration with the gear mesh frequency. The gear mesh frequency (GMF) represents the frequency of mating teeth during gear operation. The dynamic model of the gearbox without the housing is used to calculate the dynamic bearing forces caused by the gear mesh. These bearing forces are considered excitation forces for the housing, and the dynamic behavior of the housing is simulated using the finite element method (FEM). Tu [12] evaluates the vibration level caused by a bearing defect at different housing positions.

Thus, there are multiple sources that investigate vibration transfer, the so-called transfer path analysis (TPA). An early source from Verheij [13] investigates the transfer path from a ship engine with the connected machinery and the hull. El Mahmoudi [14] also investigates a powertrain—in this case, a diesel combustion engine in a car. Van der Seijs [15] divides the possibilities of the transfer path analysis into three variants: classical TPA, component-based TPA, and transmissibility TPA.

Other sources investigate complex connection with joints [16,17]. Here, complex contact is modeled using dynamic substructuring. Different methods can be used, such as inverse substructuring, primal decoupling, or dual decoupling [17]. Allen [18] models a multi-point connection which can be compared with the contact between the rollers and the bearing rings at a roller bearing.

Consequently, this work aims to analyze the measured vibrations in the frequency domain at different positions of a gear test rig to find an optimal position for condition monitoring of the gears. Additionally, the data from the simulation of the test rig are compared to the measurements, which allows a better explanation of the cause of the vibration. With this setup, it is possible to investigate the differences in vibration levels at different places on the test rig and to analyze the vibration transmission through different components of the gearbox, whereby the critical component for the vibration transmission can be found. Therefore, the novelty of this work is as follows:

1. In this paper, the vibration execution from the gear mesh and bearings in the simulation are investigated at the same time, based on a particular self-modeled software based on published sources.

2. In this paper, the vibration executions in the frequency domain from the experiment and simulation are discussed and compared.

3. Helical gears and tapered roller bearings are used, which lead to a complex dynamic vibration transfer because of static preloads and dynamic axial loads at the bearings.

Therefore, Section 2 outlines the FZG test rig and the test scenarios, e.g., the different torque and rotational speeds. Section 3 explains the simulation model. Section 4 describes the measurement setups and Section 5 discusses the data from the measurement and the simulation.

2. Test Case

The FZG back-to-back test rig is shown in Figure 1. The data on the modification of the used test rig have already been explained in a previous work [19]. The test rig mainly consists of a drive gearbox, a test gearbox with a center distance of 112.5 mm each, and a motor. The gear parameters are displayed in Table 1.

With the back-to-back setup, the test rig can be operated with one motor. Based on simulations, it can be stated that the long shafts between the gearboxes decouple the vibration excitation of the gearboxes [23]. Therefore, only the vibration excitation of the test gear gearbox can be measured. Torque is applied to the system by using an external load with a lever arm. The test rig offers variable speeds up to 3000 rpm and torques up to 1000 Nm at this setup. In the test gearbox, a helical gear pair and tapered roller bearings of type FAG 32310-A [24] with an axial preload of 10 µm are used. To avoid disturbances, the tests are performed after a sufficient run-in period of the gear set.

In this work, the influence of different torques and rotational speeds on the peak-to-mean ratio of the gear mesh frequency is discussed. The peak-to-mean ratio of the gear mesh frequency in this work can be calculated with Equation (1). In a previous work [9], the influence of the rotational speed on the peak-to-mean ratio for the vibration amplitudes in the frequency domain has already been investigated. The result shows that the peak-to-mean ratio of the measurement at the rotating sensor increases compared to the measurement at the housing sensor when the rotational speed decreases. In this work, the influence of the torque will also be discussed.

(1)$\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}-\mathrm{t}\mathrm{o}-\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}={\displaystyle \frac{\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{g}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{y}}{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{m}}}$

3. Simulation Model

The simulation model was created to evaluate and prove the practical measurements and to expand the results. The validity of the simulation model is verified by comparing it with results from another validated software, the program DZP [25]. Some assumptions have been made to simplify the simulation model. First, only the test gearbox is modeled. That means the influence of the drive gearbox on the vibration of the second gearbox is ignored. Former internal works of FZG show that this assumption can be made caused by the mechanical setup, as already presented in Section 2 [23]. Second, the effects of gravitation and centrifugal force are ignored. Third, for the time-varying gear stiffness used, the interaction with the housing displacement is neglected. For the calculation of the bearing stiffness, only the static displacement is used because the simulation results show that the dynamic displacement of the system is much smaller than the static displacement. The time-varying bearing stiffness is caused by the time-varying distribution of the rolling elements.

Figure 2 shows the simulation model of the test gearbox. The numbers 1–16 in the figure indicate the node numbers on the shafts and housing. The shafts, bearings, gears, and housing are considered in the simulation model. The load with the lever arm can be modeled as external torques applied to the right sides of these two shafts.

3.1. Shafts

Each shaft of the test rig is divided into five cylindrical segments, and each segment is modeled as a Timoshenko beam. The discretization of the shafts is due to the different diameters of each shaft section and the connections with gears and bearings at the shafts. Nodes 1–12 are the discretization points of the shafts. This means that the simulation results only refer to the dynamic behaviors of these nodes. With the Timoshenko beam theory, the stiffness matrix of one segment related to the nodes can be calculated. One segment’s mass matrix can be calculated using consistent mass distribution [26].

3.2. Gear Pair

In this work, the helical gear pair is modeled as one spring element connected with nodes 3 and 9 of the shafts. Figure 3 shows several coordinate systems for the calculation of the gear mesh stiffness matrix in a global coordinate system with index K. The ZE coordinate system is the system of the tooth mesh, which means that the direction of the spring element is identical with _ZEX. To achieve the II coordinate system, the ZE coordinate system should first rotate with the helix angle ß of the gears around the z-axis in the positive direction. To achieve the I coordinate system, the II coordinate system should then be rotated with the operating pressure angle α_wt around the y-axis in the positive direction. Finally, a rotation of 90 degrees around the z-axis in the negative direction achieves the global coordinate system. Based on these rotations, the transfer matrix from ZE to the global coordinate system can be calculated [27].

The gear mesh stiffness matrix in the ZE coordinate system [27] is as follows:

(2)$\left(\begin{array}{c}{f}_{x1}\\ f_{y1}\\ f_{z1}\\ f_{x2}\\ f_{y2}\\ f_{z2}\end{array}\right)=\left(\begin{array}{cccccc}c& 0& 0& -c& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ -c& 0& 0& c& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right)\cdot \left(\begin{array}{c}{u}_{x1}\\ u_{y1}\\ u_{z1}\\ u_{x2}\\ u_{y2}\\ u_{z2}\end{array}\right),$

where $f$ and $u$ are the force and displacement of points A and B on the base circle of the gears and $c$ is the stiffness of the spring element, which is equal to the gear mesh stiffness and has been calculated using the FVA-program DZP [25]. The stiffness matrix in the ZE coordinate system has been transferred to the global coordinate system by multiplying the transfer matrices that were described before.

The stiffness matrix after the coordinate transformation still presents the force and displacement of points A and B. Hence, it is necessary to convert it by multiplying an additional matrix so that the stiffness matrix corresponds to the center point of the gears [26]. In the simulation model, the center points are nodes 3 and 9 in Figure 2.

3.3. Bearings

The four bearings here are modeled as springs. They connect the nodes of the shafts and the housing. The bearing stiffness matrix can be calculated using the methods from Lim [28] and Liew [29]. This simulation considers the time-varying bearing stiffness caused by the time-varying distribution of the rolling elements. In this case, the frequency of the time-varying bearing stiffness is the outer ring defect frequency (BPFO), as the outer ring does not rotate. The calculation of the general bearing defect frequencies is presented in Table 2.

Where $z$ is the number of rolling elements, $f_i$ and $f_o$ are the rotational speeds of the bearing inner ring and outer ring, $\alpha$ is the contact angle between the rolling element and the ring, $d$ is the diameter of the rollers, and $D$ is the inner race curvature center radius for the ball type or the pitch radius for the roller type.

The calculated bearing stiffness matrix $C$ has dimensions of 6 × 6 and presents the relation between the force and displacement of one bearing ring. In this simulation, the housing is not modeled as a rigid body. Therefore, the displacements of both the bearing inner ring and the bearing outer ring should be taken into account. Therefore, the bearing stiffness matrix becomes a 12 × 12 matrix with the following form:

(3)$C_b=\left(\begin{array}{cc}C& -C\\ -C& C\end{array}\right).$

3.4. Housing

The housing is modeled as four hollow cylinders (nodes 13–16). These nodes connect with each other through the springs, and the stiffness of these springs is equal to the housing stiffness. For the calculation of the reduced housing stiffness matrix [31], ANSYS Workbench Version 2023 R1 has been used with an ADPL script. The CAD Model of the housing with the mesh is shown in Figure 4. The mesh is generated automatically with the standard element size. There are 28,694 elements and 60,432 nodes in this mesh. It consists of 26,883 tetrahedral elements, 1723 hexahedral elements, and 88 wedge elements. There are 11,132 corner nodes, whereas 49,300 are mid nodes. A linear elastic isotropic material with the parameters of structural steel is used. Two fixed supports of the housing at the bottom are applied, as shown in Figure 5, and the different parts of the housing are held together by bonded contacts. The average surface area is 3.7185 × 10⁻³ m², and the minimum edge length is 1.7894 × 10⁻⁵ m. The mesh quality is set to 0.05. The reduced housing stiffness matrix is related to the four nodes 13–16, which are externally defined nodes located in the middle of the bearing supports. The relation between the force and displacement of the housing on these four nodes can be calculated with the finite element method. This reduced housing stiffness matrix has dimensions of 24 × 24 because four nodes of the housing are considered, and each node has six degrees of freedom. This matrix is shown in Figure 6. For the diagonal element, the translational stiffness is much larger than the rotational stiffness. Also, there are some non-diagonal elements that have values other than zero. That can be explained by the interactions between the two support positions on the same side of the housing.

3.5. Structure of System Matrices

By ignoring the gyroscopic effect, the equation of the motion of the entire system is as follows:

(4)$M·\left({\ddot{x}}_0+{\ddot{x}}_d\left(t\right)\right)+K_0·{(\dot{x}}_0+{\dot{x}}_d\left(t\right))+(C_0+C_d\left(t\right))·{(x}_0+x_d(t\left)\right)=f_0+f_d\left(t\right)$

where $M$ is the mass matrix, $K_0$ is the damping matrix, $C_0$ is the mean stiffness matrix and is time-independent, $C_d\left(t\right)$ is the time-varying part of the stiffness matrix, $x_0$ is the static displacement, $x_{\mathrm{d}}$ is the dynamic displacement, and $f_0$ and $f_d$ represent the static and dynamic external force and torque, respectively. To calculate the damping matrix, the eigenmodes $EV$ and eigenfrequencies $\omega$ of the system are calculated through Equation (5). The normalized eigenmodes ${EV}_n$ are calculated through Equation (6). Then, the modal damping matrix $\tilde{K}$ is calculated by the given damping factor $v_i=0.02$ with Equation (7). At last, the modal damping matrix is transferred back to the orthogonal coordinate system with Equation (8). This coordinate system is identical to the coordinate system of the stiffness and mass matrices [32].

(5)$\left[{\omega }^2,EV\right]=eig\left(M^{-1}·C_0\right)$

(6)${EV}_n=-EV·{\left(\sqrt{{EV}^T·M·EV}\right)}^{-1}$

(7)$\tilde{K}=\left(\begin{array}{ccc}2{\omega }_1{v}_1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & 2{\omega }_n{v}_n\end{array}\right)$

(8)$K_0={{EV}_n}^T·\tilde{K}·{EV}_n$

$M$ is the mass matrix, $C_0$ is the mean stiffness matrix and is time-independent, $EV$ is the matrix of the eigenmodes, $\omega$ is the eigenfrequencies of the system, ${EV}_n$ is the normalized eigenmodes, $v_i$ is the damping factor, $\tilde{K}$ is the modal damping matrix, and $K_0$ is the damping matrix defined in the coordinate system of the stiffness and mass matrices [32].

The static displacement of the nodes is time-independent and can be calculated through Equation (9).

(9)$C_0·x_0=f_0$

By using Equation (9) and ignoring $C_d\left(t\right){·x}_d\left(t\right)$ [33], Equation (4) can be transformed into the following:

(10)$M·{\ddot{x}}_d\left(t\right)+K_0·{\dot{x}}_d\left(t\right)+C_0·x_d\left(t\right)=f_d\left(t\right)-C_d\left(t\right)·x_0$

Equation (10) has been used to simulate the dynamic behavior of the nodes in the test gearbox. Because of the high order of accuracy and lower computational cost, the Runge–Kutta method of the fourth order with the fixed time step 1 × 10⁻⁴ s has been used to solve this differential equation numerically. For the simulation, MATLAB R2022a is used.

4. Measurement Setup

The test rig, introduced in Section 2, is used for the experimental tests. The four acceleration sensors from Hottinger Brüel&Kjaer, Virum, Denmark, of Type BK-4518 and BK-4519 are sampled continuously, and parallel with a sampling frequency of 100 kHz. All sensors have similar specifications to compare the sensor signals. The signals of the rotating sensors are transferred with shielded cables to the mercury slip ring Magtrol SA, Rossens; Switzerland, 4 MTA/T [34]. Between the slip ring and the conditioning amplifier, the signals are also transferred in the shielded cables. A from Hottinger Brüel&Kjaer, Virum, Denmark, BK-2693-A [35] conditioning amplifier with a high-pass filter at 1 Hz and a low-pass filter at 30 kHz is used to check the signal quality. The data of the four acceleration channels are then fed into a National Instruments, Austin, USA, BNC-2090A [36] connector and then processed by National Instruments, Austion, USA, PXI-8110 [37]. The position of the accelerometers has been displayed in Figure 7 below. The positions are chosen as close to the gear as possible to achieve high signal amplitudes. Multi-body simulations were conducted for the validation. Channel 1 and Channel 2 are the rotating accelerometers of BK-4518 [38] for Channel 1 and BK-4519 [39] for Channel 2. Both sensors measure in the same absolute direction. Related to the gear, Channel 1 is in the tangential direction, and Channel 2 is located in the radial direction. This measurement system is based on an earlier work by Götz [40]. For the validation, an inductive speed sensor has been equipped at the test rig, which is Channel 3 and not considered here. The sensors on the housing of type BK-4518 [38] (Channels 4 and 5) are located at the housing close to the bearing to measure as high of amplitudes as possible. The measurements were conducted over 45 s, so 45,000,000 data points per sensor were measured.

5. Results and Discussion

In this section, the experimental results are discussed first. Then, the simulation results are compared to the experimental results to find the similarities and differences. Additionally, the vibration transmission has been investigated with the simulation results.

5.1. Experimental Results

The fast Fourier transform is used to discuss the experimental results in the frequency domain. As described in Section 4, there are two rotating accelerometers mounted on a ring that is attached to the pinion. With rotating sensors, the signals are amplitude-modulated due to the rotational frequency and gravity. The amplitude modulation can be presented through Equation (11):

(11)$F\left(x\left(t\right)·\mathrm{s}\mathrm{i}\mathrm{n}\left(2\mathsf{\pi }{f}_s\mathrm{t}\right)\right)={\displaystyle \frac{1}{2·\mathrm{i}}}\left(X\left(f-f_s\right)-X\left(f+f_s\right)\right),$

where F is the Fourier transform, $x\left(t\right)$ is the original signal, $X\left(f\right)$ is the Fourier transform of $x\left(t\right)$, $f_s$ is the rotational frequency of the shaft, and $f$ is the frequency. Due to the amplitude modulation by the carrier signal with the frequency $f_s$, the frequency spectrum of the original signal shifts left and right by $f_s$.

The effect of the amplitude modulation on the real signal from the rotating accelerometer is shown in Figure 8. In this work, the first three high harmonics of the gear mesh frequencies are discussed. Hence, the upper limit of the frequency spectrum is set to 800 Hz. The high-frequency part of the spectrum will be discussed in the future. These lower frequencies are exclusively investigated, as former works showed that mainly the frequencies around the GMF and between the GMF and the first one higher are affected by gear damage. The investigations at multiple torques and speeds show that vibration levels below 8 × 10⁻³ mm/s² can be considered as noise. Several peaks can be observed in this frequency spectrum. At 500 rpm, the rotational frequency is 8.3 Hz. The Outer Race Defect frequency can be calculated in Table 2 and equals 49.8 Hz. The gear mesh frequency is 184 Hz. Additionally, the higher harmonics of these frequencies can also be observed. Due to the amplitude modulation, the influence of the gravitation can be observed at 8.3 Hz. Additionally, due to the amplitude modulation, the peaks at the Outer Race Defect frequency, the gear mesh frequency, and their high harmonics shift left and right by the rotational frequency, and these shifted peaks are called sidebands [41]. For example, the two peaks beside the peak at the gear mesh frequency (184 Hz) are located at 175.6 Hz and 192.4 Hz. The difference between these two frequencies and 184 Hz is exactly the rotational frequency (8.3 Hz). Also, the difference in the amplitude of the main excitation frequency and that of the sidebands is small because of the amplitude modulation. The post-processed signal, in which the effect of the amplitude modulation is eliminated, is shown in Figure 9. The peak in Figure 9 at the main excitation frequency is much larger than the sideband. Hence, post-processing of the signal is necessary.

To avoid the effect of the amplitude modulation, the signal from Channels 1 and 2 should be added [40]. The signal after this post-processing presents only the tangential acceleration. As shown in Figure 10, the measurement directions of these two accelerometers are opposites. The theoretically measured signals can be calculated using Equations (12) and (13).

(12)$a_1=\ddot{y}·cos\left(\alpha \right)+\ddot{z}·sin\left(\alpha \right)-\ddot{\alpha }·r$

(13)$a_2=-\ddot{y}·cos\left(\alpha \right)-\ddot{z}·sin\left(\alpha \right)$

where $a_1$ and $a_2$ are the measured accelerations from these two sensors, $\ddot{y}$ and $\ddot{z}$ are the accelerations of the center point of the pinion in the y and z directions, and $\ddot{\alpha }$ is the angular acceleration of the pinion shaft. The sum of $a_1$ and $a_2$ is $-\ddot{\alpha }·r$, which is the tangential acceleration. The gravitation and amplitude modulation cannot affect this signal. Figure 9 shows the frequency spectrum of the signal after post-processing. The amplitude at 8.3 Hz decreases from 7.8 mm/s² (Figure 8) to 1.3 mm/s² (Figure 9), which means that the effect of the gravitation is eliminated. Additionally, the peaks beside the gear mesh frequency also become smaller.

To evaluate a good position of the sensors for condition monitoring, the peak-to-mean ratios of the signals from the sensors at the pinion and at the housing are evaluated at different rotational speeds and an applied torque of 500 Nm. Further data on this test are listed in Table 3. The peak-to-mean ratio can be presented as the peak-to-mean value, calculated with Equation (1).

As shown in Table 3, the peak-to-mean value of the signal after post-processing is larger than the signal from Channel 2 because the gravitation and sideband of the gear mesh frequency are eliminated through post-processing. The peak-to-mean value from Channel 1 is larger than the post-processing signal because Channel 1 measures the combination of the radial and tangential accelerations. Hence, this signal has a larger amplitude at the gear mesh frequency. Additionally, the peak-to-mean values of the signals from both housing sensors have an increasing trend when the rotational speed increases.

The reason for this can be explained in Figure 11 and Figure 12. Figure 11 shows the frequency spectrum of the signal from the housing sensor at 500 rpm. The amplitude at the gear mesh frequency (183.9 Hz) is 0.036. This value is much smaller than 0.745, which is the amplitude at the gear mesh frequency (916.8 Hz) at 2500 rpm, as shown in Figure 12. At the same time, the vibration of the housing sensor with the Outer Race Defect frequency does not change as much as the vibration with the gear mesh frequency, when the rotational speed increases from 500 rpm to 2500 rpm. That means that, at a high rotational speed, the vibration of the housing with the gear mesh frequency can be more dominant than the other noise.

To compare the peak-to-mean ratios from the rotating sensors at the gear and housing sensors, the value r is calculated through Equation (14), and the peak-to-mean values from Table 3 are used.

(14)$r={\displaystyle \frac{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}-\mathrm{t}\mathrm{o}-\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\left(\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{f}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{p}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\right)}{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}-\mathrm{t}\mathrm{o}-\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\left(\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{f}\mathrm{t}\right)}}$

The calculated values $r$ at different rotational speeds and a constant torque are shown in Figure 13. The larger this ratio $r$ rises, the greater the peak-to-mean ratio of the signal from the rotating sensor. As shown in Figure 13, this ratio decreases when the rotational speed increases. This means that the rotating sensor at the gear for condition monitoring has an advantage at low rotational speeds. When the rotational speed is large, the housing sensor probably has better performance relative to the cost of implementation.

The influence of the torque on the peak-to-mean values of the signal after post-processing from the rotating sensors has also been investigated, as shown in Figure 14. The relation between the peak-to-mean value and the torque is unclear from these three test scenarios (1000 rpm, 1500 rpm, and 2000 rpm). In contrast, the influence of the rotational speed on the ratio of the peak-to-mean value is quite linear. This is also obvious because the rotational speed only influences the vibration excitation. At the same time, different external torques can generate different displacements of the entire gearbox. Because of the nonlinear stiffness of the bearings and gears, the displacements can affect their stiffness, which changes the dynamic behavior of the gearbox. Hence, a strong excitation because of a large external torque does not mean a better peak-to-mean ratio.

5.2. Simulation Results

Figure 15 shows the simulation and experimental results of the pinion tangential acceleration at 500 rpm and 1000 Nm. The peaks corresponding to the gear mesh frequency, the Outer Race Defect frequency, and their high harmonics can be observed both in the experiment and in the simulation. The peak at 50 Hz corresponds to the Outer Race Defect frequency and the other three peaks to the gear mesh frequency and their high harmonics. Because of the measurement errors, noise, and effect of the amplitude modulation, there are also more small peaks in the experimental results, and these peaks are difficult to model. Hence, these peaks are not considered in the simulation and, therefore, cannot be observed from the simulation results.

Figure 16 shows the radial acceleration of the housing near the pinion shaft. Similar to the result in Figure 15, the important peaks can also be observed. The experiment result shows that the difference between the peaks at the Outer Race Defect frequency and the gear mesh frequency is small. However, the simulation result presents a large difference between the amplitudes of these peaks corresponding to the gear mesh frequency and the Outer Race Defect frequency. The reason for that may be the overestimation of the vibration excitation of the bearings in the simulation. Because of the axial force generated by the helical gear mesh, the right bearing of the pinion shaft and the left bearing of the wheel shaft are loaded in such a way that all rolling elements are always loaded. Hence, the fluctuations in the stiffness of these two bearings are small. The loads on the other two bearings are reduced due to the axial force, and this can lead to a large fluctuation in bearing stiffness, as shown in Figure 17. The larger the fluctuation in the bearing stiffness is, the stronger the vibration excitation of the bearing. This phenomenon can also be observed from the experimental results.

Additionally, the vibration transmission in the transfer path of the gearbox has been investigated through the simulation results. Table 4 presents the amplitude at the gear mesh frequency of the signals from different nodes. As shown in Figure 2, the amplitude of the signal at node 3 shows the vibration intensity of the pinion, the signal at node 6 shows the vibration of the pinion shaft near the right bearing, and the signal at node 14 shows the vibration of the housing part. As presented in Table 4, the amplitude of the vibration with the gear mesh frequency reduces during vibration transmission through the gear, the shaft, and the bearing. By analyzing the differences between the amplitudes at the different nodes in Table 4, it can be observed that the largest reduction in the vibration intensity happens during transmission through the bearings. As a result, the importance of the sensor locations at the rotating components for condition monitoring, such as sensor-integrating gear wheels, can be determined to show the significance and the effect of a different sensor position.

Therefore, the simulation shows a good presentation of the vibration frequencies of interest. The simulation shows lower amplitude results at the housing position compared to the measured values, as shown in Figure 16. This is most likely caused by the stiffness model of the housing. At the gear, the amplitudes and frequencies of the simulation show a good comparison to the measurement results, as shown in Figure 15.

6. Conclusions

The effects of the amplitude modulation, such as the gravitation and sidebands, can be observed by analyzing the original signal from the rotating sensor. Hence, the post-processing of the original signal is useful to reduce the influence of the modulation so that the condition monitoring can be more accurate.

The influence of the rotational speed on the peak-to-mean ratio at different sensor positions has been investigated. The result shows that the peak-to-mean ratio of the signal from the rotating sensor is large when the rotational speed is slow. Hence, the rotating sensor provides better performance at slow rotational speeds than the housing sensor. The influence of external torque on the peak-to-mean ratio has also been investigated, but their relationship is more complex because the torque changes the stiffness of each component and, hence, the vibration transmission of the transfer path. More speed and torque, especially with particular damages, will be investigated in the future to show the capabilities of rotating sensors in condition monitoring. Damage-related frequencies can be masked by external excitation, which raises the importance of the integrated sensors.

A modeling of the gearboxes is also introduced, and the time-varying gear mesh stiffness and bearing stiffness are also considered in this simulation. The peaks at the gear mesh frequency and Outer Race Defect frequency can be observed in the frequency spectrum. The vibration transmission through different components of the gearbox has been investigated using simulations, and the results show that the greatest reduction in the vibration intensity happens during transmission through the tapered roller bearing. Hence, the sensor at the rotating component for condition monitoring can provide good performance.

Author Contributions

Conceptualization, E.K. and C.C.; methodology, E.K., C.C. and M.O.; software, C.C.; supervision, M.O. and K.S.; writing, E.K. and C.C. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1. FZG-gear test rig based on ISO 14635-1 [20], source [21] CC-BY 4.0.

Figure 2. Simulation model of test gearbox.

Figure 3. Several coordinate systems for calculation of gear stiffness matrix, modified to [27]. (a) Global representation; (b) coordinate transformation in several steps.

Figure 4. CAD Model of housing with mesh.

Figure 5. CAD Model of housing with fixed support.

Figure 6. Reduced housing stiffness matrix with dimensions of 24 × 24.

Figure 7. Positions and measurement directions of accelerometers at pinion and housing: (a) two rotating sensors at pinion—front view; (b) two housing sensors—rear view.

Figure 8. Frequency spectrum of signal from Channel 2 at 1000 Nm and 500 rpm.

Figure 9. Frequency spectrum of signal after post-processing at 1000 Nm and 500 rpm.

Figure 10. The schematic representation of the positions and measurement directions of the accelerometers on the pinion also depicted in Figure 7a.

Figure 11. Frequency spectrum of signal from housing sensor (pinion shaft) at 500 rpm and 500 Nm.

Figure 12. Frequency spectrum of signal from housing sensor (pinion shaft) at 2500 rpm and 500 Nm.

Figure 13. Ratio of peak-to-mean values [Forumla omitted. See PDF.] at different rotational speeds and 500 Nm.

Figure 14. Peak-to-mean values of signals after post-processing from rotating sensors at different torques and constant rotational speed.

Figure 15. Tangential acceleration of pinion in axial direction at 500 rpm and 1000 Nm from simulation and rotating sensors.

Figure 16. Radial acceleration at housing near pinion shaft at 500 rpm and 1000 Nm from experiment and simulation.

Figure 17. Time-varying part of bearing stiffness matrix calculated by simulation.

Table 1

Gear parameters, according to Kadach [22].

Gear Parameter	Symbol	Unit	112.5 Helical Gear Set
Center distance	a	mm	112.5
Number of teeth (pinion/wheel)	z1/z2	-	22/24
Normal module	mn	mm	4.25
Transverse module	mt	mm	4.86
Normal pressure angle	αn	In °	20
Helix angle	β	In °	29
Profile shift coefficient (pinion/wheel)	x1/x2	-	0.100/0.077
Pitch diameter (pinion/wheel)	d1/d2	mm	106.90/116.62
Tip diameter (pinion/wheel)	da1/da2	mm	117.70/127.30
Transverse contact ratio	εα	-	1.5
Overlap ratio	εβ	-	1.0
Face width	b	mm	27.6

Table 2

General bearing defect frequencies [30].

Bearing Element	Frequency Equations
Inner Race Defect Frequency	$f_{bpfi}={\displaystyle \frac{z(f_o-f_i)\left(1+\frac{d}{D}\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha \right)\right)}{2}}$
Outer Race Defect Frequency	$f_{bpfo}={\displaystyle \frac{z(f_o-f_i)\left(1-\frac{d}{D}\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha \right)\right)}{2}}$
Cage Rotational Frequency	$f_c={\displaystyle \frac{f_i\left(1-\frac{d}{D}\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha \right)\right)}{2}}+{\displaystyle \frac{f_o\left(1+\frac{d}{D}\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha \right)\right)}{2}}$
Ball or Roller Spin Frequency	$f_{bsf}={\displaystyle \frac{(f_o-f_i)D\left(1-{(\frac{d}{D}\mathrm{c}\mathrm{o}\mathrm{s}(\alpha \left)\right)}^2\right)}{2\mathrm{d}}}$

Table 3

Peak-to-mean values of signals from different sensors at 500 Nm and different rotational speeds (dimensionless ratio).

	500 rpm	1000 rpm	1500 rpm	2000 rpm	2500 rpm	3000 rpm
Channel 1	132	707	176	159	53	93
Channel 2	7	87	42	47	24	52
Signal after post-processing	89	583	147	134	32	31
Channel 4	7	156	128	190	43	170
Channel 5	16	183	158	236	154	369

Table 4

Amplitude at gear mesh frequency at different positions of test rig at 500 Nm and different rotational speeds.

	Amplitude at Gear Mesh Frequency at Node 3	Amplitude at Gear Mesh Frequency at Node 6	Amplitude at Gear Mesh Frequency at Node 14
500 rpm	$0.343\mathrm{m}\mathrm{m}/{\mathrm{s}}^2$	$0.0077\mathrm{m}\mathrm{m}/{\mathrm{s}}^2$	$4.25\times 10{-}^6\mathrm{m}\mathrm{m}/{\mathrm{s}}^2$
1000 rpm	$0.324\mathrm{m}\mathrm{m}/{\mathrm{s}}^2$	$0.0063\mathrm{m}\mathrm{m}/{\mathrm{s}}^2$	$2.25\times 10{-}^6\mathrm{m}\mathrm{m}/{\mathrm{s}}^2$
1500 rpm	$0.251\mathrm{m}\mathrm{m}/{\mathrm{s}}^2$	$0.0049\mathrm{m}\mathrm{m}/{\mathrm{s}}^2$	$3.04\times 10{-}^6\mathrm{m}\mathrm{m}/{\mathrm{s}}^2$
2000 rpm	$0.316\mathrm{m}\mathrm{m}/{\mathrm{s}}^2$	$0.0059\mathrm{m}\mathrm{m}/{\mathrm{s}}^2$	$2.22\times 10{-}^6\mathrm{m}\mathrm{m}/{\mathrm{s}}^2$

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