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Abstract

The vibration acceleration signal mechanisms in normal and defect gears are studied. An improved bending-torsion vibration model is established, in which the effect of time-varying meshing stiffness and damping, torsional stiffness for transmission shaft, elastic bearing support, the driving motor, and external load are taken into consideration. Then, vibration signals are simulated based on the model under diverse sampling frequencies. The influences of input shaft's rotating frequency, the teeth number, and module of gears are investigated by the analysis of the simulation signals. Finally, formulas are proposed to calculate the acceleration signal bandwidth and the critical and recommended sampling frequencies of the gear system. The compatibility of the formulas is discussed when there is a crack in the tooth root. The calculation results agree well with the experiments.

1. Introduction

Fault mechanism analysis is fundamental for the diagnosis of faults in gear transmission system. To study the fault mechanism is to obtain the expressions of the relations between fault status signals and system parameters through theoretical analysis or a large number of experiments [1]. Vibration signal analysis is one of the important means for the diagnosis of faults in gear transmission systems. Many scholars have done a lot of research work on the vibration fault diagnosis of gear transmission system. Tian [2] has detailed the energy method calculating time-varying meshing stiffness in spur gear and applied it to the 8-DOF bending-torsional coupling dynamic model for the one-stage gear transmission system. The influence of chipped, cracked, and broken tooth fault on the system's meshing stiffness and vibration acceleration signal characteristics has been studied. Jia and Howard [3] calculated the time-varying meshing stiffness in the normal gear and in the case of chipped fault using finite element method and analyzed the torsional vibration characteristics of the system based on 26 DOF two-stage transmission gear system dynamic model. Chen and Shao [4] calculated the time-varying meshing stiffness in the normal gear and in gear with cracks, respectively, using finite element method and the improved energy method. The influence of cracks of different sizes on the time-domain statistical information and frequency-domain amplitude information of torsional vibration signals has been analyzed based on 6-DOF bending-torsional coupling dynamic model. Wang et al. [5] established the nonlinear dynamic model of single-tooth impact, single-tooth stiffness, single-tooth wear, and wear and tear of the whole tooth by studying the dynamic behaviors of nonlinear gear system under different fault parameters. Analysis was made using chaotic oscillator. Considering the influence of the time-varying meshing stiffness, backlash nonlinearity, and transmission error, Ma and Chen [6] studied the nonlinear dynamic mechanism of crack faults for the parametric dynamic model of the sub-2-DOF torsional vibration established in the single-gear transmission in the gearbox and also experimentally verified the results of theoretical analysis. All these researches above on the dynamic mechanism in gear system focus on the acquisition of simulated response features of the system having various faults, which provides theoretical support for actual fault diagnosis. However, the research on the mechanism of fault signal bandwidth, which is the link between theory and practice, is seldom seen. In the existing researches, the basis for the selection of sampling frequency when the fault signal is analyzed is rarely mentioned or it is studied based on the past experience, with little attention paid to the bandwidth of fault signals. In order to solve this problem, based on Adams multibody dynamics solver, this paper sets up a nonlinear dynamic model of the gear system by taking into account the time-varying meshing stiffness, time-varying gear meshing damping, motor, load, torsional stiffness for transmission shaft, and elastic bearing support. Using this model and the sampling theorem, a detailed study and verification has been conducted over the vibration acceleration signal bandwidth of gear system and its sampling frequency.

2. Construction of Vibration Model of Gear System

2.1. Traditional Model of Torsional Vibration in Gear Pair

The vibration model of gear system built on Adams platform is shown in Figure 1 [7]. The gear meshing relation on Adams platform is realized by the definition of “gear pair” or the application of contact force [8 –10]. Since these two definitions are simplified methods, it is difficult for them to accurately represent the important factors influencing the system vibration characteristics such as the time-varying meshing stiffness, bearing support stiffness, time-varying friction, and geometric errors.

Figure 1:

Torsional vibration model of gear pair.

2.2. An Improved Bending-Torsional Vibration Model of the Gear System

In view of the limitations of the above models, an improved bending-torsional coupling nonlinear dynamic model of the gear transmission system is proposed in this paper, which can depict the dynamic characteristics of the gear transmission system on ADAMS platform more comprehensively and real. Figure 2 shows the nonlinear dynamic model of the gear system, which considers the time-varying meshing stiffness and damping, motor, load, torsional stiffness for transmission shaft, and elastic bearing support, where K_p and C_p represent the torsional rigidity and damping coefficient of the shaft, respectively; K_t and C_t are gear meshing stiffness and meshing damping coefficient, respectively; K_x1 and C_x1 are the bearing support stiffness and damping coefficient along the x-axis of input shaft, while K_y1 and C_y1 are the bearing support stiffness and damping coefficient along the y-axis of input shaft, respectively; K_x2 and C_x2 are the bearing support stiffness and damping coefficient along the x-axis of output shaft, while K_y2 and C_y2 are the bearing support stiffness and damping coefficient along the y-axis of output shaft, respectively. The dynamic differential equations of the system are summarized as follows:

\begin{matrix} m_{1} {\overset{‥}{x}}_{1} + k_{x 1} x_{1} + c_{x 1} {\dot{x}}_{1} = 0, \\ m_{2} {\overset{‥}{x}}_{2} + k_{x 2} x_{2} + c_{x 2} {\dot{x}}_{2} = 0, \\ m_{1} {\overset{‥}{y}}_{1} + k_{y 1} y_{1} + c_{y 1} {\dot{y}}_{1} - F_{1} = 0, \\ m_{2} {\overset{‥}{y}}_{2} + k_{y 2} y_{2} + c_{y 2} {\dot{y}}_{2} - F_{2} = 0, \\ I_{1} {\overset{‥}{θ}}_{1} + k_{p} (θ_{m} - θ_{1}) + c_{p} ({\dot{θ}}_{m} - {\dot{θ}}_{1}) + R_{b 1} F_{1} = 0, \\ I_{2} {\overset{‥}{θ}}_{2} + k_{p} (θ_{b} - θ_{2}) + c_{p} ({\dot{θ}}_{b} - {\dot{θ}}_{2}) + R_{b 2} F_{2} = 0, \\ I_{m} {\overset{‥}{θ}}_{m} + k_{p} (θ_{m} - θ_{1}) + c_{p} ({\dot{θ}}_{m} - {\dot{θ}}_{1}) - M_{m} = 0, \\ I_{b} {\overset{‥}{θ}}_{b} + k_{g} (θ_{b} - θ_{2}) + c_{g} ({\dot{θ}}_{b} - {\dot{θ}}_{2}) - M_{b} = 0, \\ F_{1} = k_{t} (y_{2} - y_{1} - R_{b 2} θ_{2} - R_{b 1} θ_{1}) \\ + c_{t} (y_{2} - y_{1} - R_{b 2} θ_{2} - R_{b 1} θ_{1}), \\ F_{2} = - F_{1} . \end{matrix}

(1)

Figure 2:

Dynamic model of gear transmission system.

Equivalent transformation based on the model in Figure 2 is required to describe the characteristic of time-varying meshing stiffness on Adams platform, that is, to transform the nonlinear meshing relation between gears in a kinematic pair into the force and force couple acting on the center of the mass of gears.

The improved model after transformation is shown in Figure 3, where F₁ and F₂ represent the normal meshing force between the driving and driven gears, respectively; M₁ and M₂ represent the torque of the normal meshing force acting on the driving and driven gears, respectively, which can be expressed as

\begin{matrix} M_{1} = F_{1} \cdot R_{b 1}, \\ M_{2} = {- F}_{2} \cdot R_{b 2}, \end{matrix}

(2)

where R_b1 and R_b2 are the radii of base circle of the driving and driven gears, respectively.

Figure 3:

Improved dynamic model of gear system.

2.3. Calculation of Time-Varying Meshing Stiffness

In this paper, the energy method is used to calculate the time-varying meshing stiffness of gear pair. First of all, gear stiffness and the potential energy in the meshing gears can be divided into four components: Hertz energy U_h, bending energy U_b, radial compression energy U_a, and shear energy U_s. The types stiffness corresponding to each energy respectively, Hertz stiffness k_h, bending stiffness k_b, radial compression stiffness k_a, and shear stiffness k_s. These are stiffness combined together in series become the meshing stiffness of gear tooth. The stiffness corresponding to each energy can be, respectively, expressed as [2]

\begin{matrix} k_{h} = \frac{π E L}{4 (1 - v^{2})}, \\ \frac{1}{k_{b}} = \int_{{- α}_{1}}^{α_{2}} \frac{3 {1 + \cos α_{1} [(α_{2} - α) \sin α - \cos α]}^{2} (α_{2} - α) \cos α}{{2 E L [\sin α + (α_{2} - α) \cos α]}^{3}} d α, \\ \frac{1}{k_{s}} = \int_{- α_{1}}^{α_{2}} \frac{1.2 (1 + v) (α_{2} - α) \cos α \cos^{2} α_{1}}{E L [\sin α + (α_{2} - α) \cos α]} d α, \\ \frac{1}{k_{a}} = \int_{- α_{1}}^{α_{2}} \frac{(α_{2} - α) \cos α \sin^{2} α_{1}}{2 E L [\sin α + (α_{2} - α) \cos α]} d α, \end{matrix}

(3)

where E is the elastic modulus, L is the axial thickness of gear, and v is the Poisson's ratio. They are the four components of gear stiffness. The meshing stiffness of gear pair should also include fillet-foundation stiffness k_f, which can be expressed as follows [9]:

\frac{1}{k_{f}} = {L^{*} {(\frac{u_{f}}{s_{f}})}^{2} + M^{*} (\frac{u_{f}}{s_{f}}) + P^{*} (1 + Q^{*} \tan^{2} α)},

(4)

where the coefficients L^*, M^*, P^*, and Q^* can be represented by polynomial functions

X^{*} = \frac{A^{*}}{θ_{f}^{2}} + B^{*} h_{f}^{2} + \frac{C^{*} h_{f}}{θ_{f}} + \frac{D^{*}}{θ_{f}} + E^{*} h_{f} + F^{*} .

(5)

X^* represents the coefficients L^*, M^*, P^*, and Q^*; $h_{f} = r_{f} / r$ , where r_f is the radius of root circle; the meanings of u_f, θ_f, and S_f are shown in Figure 4; and the values of A^*, B^*, C^*, D^*, E^*, and F^* can be seen in Table 1.

Table 1:

Valuation of the coefficients.

	L ^*	M ^*	P ^*	Q ^*

A ^*	−5.574 × 10⁻⁵	60.111 × 10⁻⁵	−50.952 × 10⁻⁵	−6.2042 × 10⁻⁵
B ^*	−1.9986 × 10⁻³	28.100 × 10⁻³	185.50 × 10⁻³	9.0889 × 10⁻³
C ^*	−2.3015 × 10⁻⁴	−83.431 × 10⁻⁴	0.0538 × 10⁻⁴	−4.0964 × 10⁻⁴
D ^*	4.7702 × 10⁻³	−9.9256 × 10⁻³	53.300 × 10⁻³	7.8297 × 10⁻³
E ^*	0.0271	0.1624	0.2895	−0.1472
F ^*	6.8045	0.9086	0.9236	0.6904

Figure 4:

Geometric parameters of gear deformation.

The Hertz stiffness k_h, bending stiffness k_b, radial compression stiffness k_a, shear stiffness k_s, and fillet-foundation stiffness k_f combined in series give the meshing stiffness of gear pair:

\begin{matrix} k_{t} = k_{t, 1} + k_{t, 2} \\ = \sum_{i = 1}^{2} (\frac{1}{k_{h, i}} + \frac{1}{k_{b 1, i}} + \frac{1}{k_{s 1, i}} + \frac{1}{k_{a 1, i}} + \frac{1}{k_{f 1, i}} + \frac{1}{k_{b 2, i}} {+ \frac{1}{k_{s 2, i}} + \frac{1}{k_{a 2, i}} + \frac{1}{k_{f 2, i}})}^{- 1}, \end{matrix}

(6)

where i is the ith pair of gear teeth for a meshing gear pair.

Equations (3) and (4) are solved separately before we substitute the results into (6) to get each stiffness and meshing stiffness of gear pair. The calculation of meshing damping coefficient is not given in detail about as the calculation method has been specified in the literature [2]. The meshing stiffness and damping coefficient of the gears with the module m = 5 and the gear teeth number $z_{1} / z_{2} = 22 / 30$ are shown in Figure 5.

Figure 5:

Meshing stiffness and damping coefficient of a normal gear pair.

3. Analysis of the Simulation Results of a Faultless System and Verification

3.1. Simulation Signal Bandwidth and Sampling Frequency

Dynamic simulation of gear system is performed by running the Adams dynamic solver. The main parameters of the gear system used are as follows: the module M = 5, the tooth numbers of the big and small gears $z_{1} / z_{2} = 22 / 30$ ; the rotational frequency of input shaft f_t = 13.36 Hz; the meshing frequency of gear pair f_m = 22 × f_t = 293.9 Hz; and the interval for double-teeth meshing is 0.002099 s. Firstly, the dynamic solution of the system is conducted using the step length of 10⁻⁸ s and the vibration acceleration signals of the gear system for five meshing periods are simulated. The time-domain graph of the simulated signals is shown in Figure 6. It can be known from Figure 6 that the vibration acceleration signals of the gear system when the excitation from time-varying meshing stiffness is considered are obviously periodical high-bandwidth attenuation pulse signals, and the signal period is about T = 0.003409 s (f = 293.3 Hz), which basically conforms to f_m = 293.9 Hz, the theoretical value of meshing frequency of gear pair; the signal peak value is about 102 m/s², which corresponds to the critical position of a gear pair changing from single-tooth meshing to double-teeth meshing; trough value is about – 179 m/s², which corresponds to the critical position of a gear pair changing from double teeth meshing into single-tooth meshing; the time from peak value to trough value is the single tooth pair meshing duration T_s = 0.001308 s, while that from trough value to peak value is the double teeth pair meshing duration T_d = 0.002101 s. The comparison of the conclusion drawn from the simulated signals above and the simulation results in the literature [2] reveals that firstly, the fillet-foundation stiffness is not taken into account when the energy method is used to calculate the meshing stiffness of gear pair in the literature, so there are large differences between the amplitude of the simulated signals and the results in this paper. Secondly, the period of the simulated signals, single-tooth meshing interval, and double-teeth meshing interval in this paper and the literature [2] are all coherent with the theoretical values because the period of meshing stiffness and the mutation characteristics of the critical point of single-tooth meshing and double-teeth meshing are not influenced by fillet-foundation stiffness.

Figure 6:

The simulated vibration acceleration signals in the y direction of the center of the small gear when the calculation step length is 10⁻⁸ s.

The selection of the sampling frequency of the simulated signals is discussed by combining with Figure 6 and sampling theorem. First, signal bandwidth is estimated. According to the sampling theorem, the signal bandwidth refers to the highest harmonic frequency component contained in the signals. The formula expression of the relation between the 10%–90% rise time of the signals T_r and the signal bandwidth f_{3 dB} can be written as [10]

f_{3 dB} = \frac{0.35}{T_{r}} .

(7)

As shown in Figure 6, 10%–90% rise time of positive pulse is T_rz1 = 0.00000887 s, and the bandwidth f_z1 = 39.46 kHz. In the same way, the negative pulse bandwidth is f_z2 = 39.95 kHz. Thus, the signal bandwidth is f_z = f_z1 = 39.95 kHz. According to the sampling theorem, the sampling frequency should satisfy the following equation:

{f_{s l} = 2 f}_{z} = 79.9 kHz,

(8)

f_sl is called critical sampling frequency. In addition, the sampling frequency in actual engineering is often selected as 3–4 times that of the bandwidth [11]. In this paper, the value is selected as

{f_{s} = 4 f}_{z} = 159.8 kHz,

(9)

where f_s is the recommended sampling frequency.

For the above example, the critical sampling frequency and recommended sampling frequency of vibration acceleration signals of the fault-free transmission system are obtained based on the analysis of simulated signals and the sampling theorem. In the following, the influence of the gear module, tooth number, and the rotational frequency of input shaft on signal bandwidth is analyzed to establish the function between them. The function is used to forecast the sampling frequency required for the measurement of vibration acceleration signals when the status parameters and operating parameters of gear system are known.

3.2. Influence of Gear Module on Signal Bandwidth

The relationship between signal bandwidth and gear module is examined on the condition that other parameters remain unchanged. According to the analysis above, it is known that gear module only affects the amplitude of meshing stiffness of gear pair, but not its period and double/single tooth pair meshing duration. The signal bandwidths corresponding to different modules are shown in Table 2.

Table 2:

The influence of gear module on signal bandwidth.

M	10%–90% rise time of positive pulse (s)	Rise time of positive pulse (s)	10%–90% rise time of negative pulse (s)	Rise time of negative pulse (s)	Bandwidth (kHz)

2	0.00000885	0.00001458	0.00000878	0.00001448	39.86
3	0.00000886	0.00001460	0.00000878	0.00001449	39.86
4	0.00000886	0.00001461	0.00000878	0.00001449	39.86
5	0.00000887	0.00001463	0.00000876	0.00001449	39.95
7	0.00000886	0.00001463	0.00000877	0.00001446	39.55

From Table 2 it can be seen that no obvious changes occur to signal bandwidth. No matter module increases by several times, there is only about 1% difference between the maximum and minimum values. Therefore, the change of gear module has little influence on signal bandwidth of the gear system. It is not a main influential factor in selecting the sampling frequency measured from the vibration acceleration signals of the gear system.

3.3. Influence of Tooth Number on Signal Bandwidth

The relationship between signal bandwidth and the number of gear teeth is then explored when other parameters remain unchanged. According to the discussion above, the number of gear teeth only affects the period of time-varying meshing stiffness and double/single tooth pair meshing duration. The signal bandwidths corresponding to different numbers of gear teeth are shown in Table 3.

Table 3:

The influence of teeth number on signal bandwidth.

z₁/z₂	10%–90% rise time of positive pulse (s)	Rise time of positive pulse (s)	10%–90% rise time of negative pulse (s)	Rise time of negative pulse (s)	Bandwidth (kHz)

19/30	0.00000885	0.00001460	0.00000879	0.00001448	39.82
22/30	0.00000885	0.00001461	0.00000878	0.00001449	39.86
25/30	0.00000887	0.00001461	0.00000878	0.00001449	39.86
30/30	0.00000887	0.00001462	0.00000879	0.00001450	39.82
50/75	0.00000893	0.00001471	0.00000884	0.00001457	39.59
22/75	0.00000887	0.00001463	0.00000876	0.00001446	39.95

From Table 3 it can be known that signal bandwidth remains basically unchanged when obvious changes occur to the number of gear teeth, and there is only about 1% difference between the maximum and minimum values. It can be concluded that the changes of the number of gear teeth have little influence on signal bandwidth of the gear system. So it is not a main influential factor in selecting the sampling frequency of the gear system signals.

3.4. Influence of Rotational Frequency of Driving Gear on Signal Bandwidth

The relationship between signal bandwidth and rotational frequency of the driving gear is also studied with other parameters remaining unchanged. According to the discussion above, the rotational frequency of the driving gear would affect the period of time-varying meshing stiffness and double/single tooth pair meshing duration. According to the setting of the rotational frequency of driving gear in the common gearbox, the signal bandwidth corresponding to the rotational frequency of 1–25 Hz is calculated in this paper as shown in Table 4.

Table 4:

The influence of different rotational frequencies of driving gear on signal bandwidth.

f	10%–90% rise time of positive pulse (s)	Rise time of positive pulse (s)	10%–90% rise time of negative pulse (s)	Rise time of negative pulse (s)	Bandwidth (kHz)

1	0.00006221	0.00010890	0.00005746	0.00009478	6.09
4	0.00002276	0.00003816	0.00002188	0.00003634	16.00
5	0.00001911	0.00002997	0.00001853	0.00003073	18.89
7.0	0.00001470	0.00002439	0.00001441	0.00002384	24.29
10	0.00001101	0.00001841	0.00001099	0.00001815	31.85
13.3	0.00000885	0.00001461	0.00000878	0.00001449	39.86
15	0.00000807	0.00001331	0.00000801	0.00001321	43.70
17	0.00000730	0.00001204	0.00000726	0.00001198	48.21
20	0.00000641	0.00000874	0.00000637	0.00001050	54.95
25	0.00000533	0.00000877	0.00000526	0.00000874	66.54

From Table 4 it can be known that with the increase of rotational frequency of the driving gear, the signal bandwidth shows an increasing trend. To analyze the functional relation between them, the slopes are first compared as shown in Table 5.

Table 5:

The relationship between rotational frequency of driving gear and signal bandwidth.

Rotational frequency	The rotational frequency differential value	The bandwidth differential value	Ratio

25/20	5	11.59	2.318
20/17	3	6.74	2.247
17/15	2	4.51	2.255
15/13.3	1.7	3.84	2.256
13.3/10	3.3	8.01	2.427
10/7	3	7.56	2.52
7/5	2	5.4	2.7
5/4	1	2.89	2.89
4/1	3	9.909	3.30

From Table 5 it is known that the ratio of bandwidth difference to the rotational frequency difference fluctuates slightly. Since bandwidth is an estimated value when signals are measured and analyzed, the data fitting method should be considered to be used to obtain the function between them, which is later used in forecasting signal bandwidth when the rotational frequency of driving gear is already known. According to the computed values in Table 4, the least square method is used for curve fitting to get the second-order equation of rotational frequency and bandwidth as follows:

f_{3 dB} = - 0.018 f_{t}^{2} + {2.93 f}_{t} + 4.13 .

(10)

Figure 7 shows the data points in Table 4 and the curve defined by (10).

Figure 7:

Functional relation between the rotational frequency of driving gear and signal bandwidth.

In order to verify the accuracy of (10) obtained, The simulation method and this equation are, respectively, used to calculate the signal bandwidth when the rotational frequency is 8 Hz, 16 Hz, and 22 Hz. The results are shown in Table 6. The results calculated by (10) and those by simulation have the error of less than 2%, which satisfies the accuracy requirement when predicting the bandwidth of vibration acceleration signals in the fault diagnosis. It thus can be known that only the rotational frequency of the input shaft where the gear pair is located is required for the calculation of signal bandwidth using (10). This parameter is one of the most common as well as the most readily available parameters in actual engineering. But when the traditional equation (7) is used, all the physical parameters and operating parameters of a gearbox are required, as well as calculating time-varying meshing stiffness. The parameters and meshing stiffness are then substituted into the gear system dynamic model to obtain the simulated vibration acceleration signals of the gear system. Finally, the time-domain analysis of the simulated signals is conducted to obtain the 10%–90% rise time of pulse signals as well as the signal bandwidth. This shows that if used in the engineering practice to calculate vibration acceleration signal bandwidth in gear system, (10) can greatly improve the calculation in terms of simplicity and speed.

Table 6:

Comparison of the results calculated by simulation and the equation.

Rotational frequency (Hz)	Simulation method (kHz)	Formula method (kHz)	Error

8	26.44	26.42	0.1%
16	45.63	46.37	1.6%
22	59.12	59.78	1.1%

3.5. Experimental Verification

The previous section focuses on the simulation method proposed in this paper for calculating the vibration acceleration signal bandwidth of the gear system and sampling frequency. For the purpose of efficiency verification for this method and for proving the importance of the selection of sampling frequency to the measurement of actual signals, experimental verification and results analysis are performed. The experiment is conducted on the gearbox experiment table in Figure 8.

Figure 8:

Gearbox experiment table.

The main parameters are gear module M = 5; the number of gear teeth of big and small gears $z_{1} / z_{2} = 22 / 30$ ; the rotational frequency of input shaft f_t = 13.36 Hz; the gear meshing frequency f_m = 22 × 13.36 = 293.9 Hz. Due to the limitations of the experimental equipments, the sampling frequency is set as 10 kHz, and the time-domain and frequency-domain graphs of the sampled signals are shown in Figure 9. From the figure it is known that there is significant impact on the time domain, but its amplitude and period follow no specific variation pattern; regular meshing frequency and its frequency multiplication can be found from the frequency domain but not so obvious in its amplitude. In addition, some unknown frequencies of higher amplitude appear near the frequencies of interest. For the convenience of comparative analysis, the time domain and frequency domain of the simulated signals of different sampling frequencies under the same parameters are listed in Figures 10 to 12.

Figure 9:

The time-domain and frequency-domain graphs of experiment vibration acceleration signals of fault-free gearbox under 10 kHz sampling frequency: (a) the time domain graph, (b) local amplification graph of the time-domain, (c) the frequency domain graph, and (d) local amplification graph of the frequency-domain.

Figure 10:

The time domain and frequency domain of simulated vibration acceleration signals of fault-free gearbox under 10 kHz sampling frequency: (a) the time-domain graph, (b) local amplification graph of the time domain, (c) the frequency-domain graph, and (d) local amplification graph of the frequency domain.

According to the calculation and analysis on signal bandwidth in the previous section, the signal bandwidth in this case is f_{3 dB} = 39.95 kHz, the critical sampling frequency is f_sl = 79.9 kHz and the recommended sampling frequency is f_s = 159.8 kHz. Obviously, when the sampling frequency is lower than the critical sampling frequency, serious frequency aliasing phenomenon will occur, which is manifested as signal amplitude and waveform distortion in time domain and frequency aliasing in frequency domain. This means that the frequency components not contained in the original signal appear or the amplitudes of original frequency components have changed.

Specifically, in Figure 10, when the sampling frequency f = 10 kHz, there is f_sl > f; that is, the actual sampling frequency is lower than the critical sampling frequency, which fails to satisfy the sampling theorem, then serious frequency aliasing phenomenon will occur, causing the distortion in time domain and frequency domain. As shown in Figure 10, some peak values and trough values in the time-domain signal are obviously deficient, and signal period greatly varies from that of the original signal; a large number of unknown frequencies appear in the frequency domain, and the meshing frequency of gear pair (293.4 Hz) and its frequency multiplication are very small, almost indiscernible from the graphs.

In Figure 11, when the sampling frequency f = 100 kHz, there is f_s > f > f_sl; that is, the actual sampling frequency is higher than the critical sampling frequency, which satisfies the sampling theorem, but lower than the recommended sampling frequency; then the signals acquired can properly represent the time-domain and frequency-domain features of the original signals. As shown in Figure 11, the amplitudes of each peak value and trough value in time domain fluctuate somewhat, but the period of the pulse can be clearly identified; in frequency, domain the frequency components consisting of the meshing frequency of gear pair (293.4 Hz) and its frequency multiplication can still be explicitly seen; some unknown frequencies of lower amplitude appear near these frequencies, suggesting the occurrence of slight frequency aliasing phenomenon.

Figure 11:

The time-domain and frequency-domain graphs of simulated vibration acceleration signals of fault-free gearbox under 100 kHz sampling frequency: (a) the time-domain graph, (b) local amplification graph of the time domain, (c) the frequency-domain graph, and (d) local amplification graph of the frequency domain.

Figure 12:

The time-domain and frequency-domain graphs of vibration acceleration signals of fault-free gearbox under 1000 kHz sampling frequency: (a) the time-domain graph, (b) local amplification graph of the time domain, (c) the frequency-domain graph, and (d) local amplification graph of the frequency domain.

In Figure 12, when the sampling frequency f = 1000 kHz, there is f > f_s > f_sl; that is, the actual sampling frequency is higher than the recommended sampling frequency; then the signal acquired can be used to accurately represent the time-domain and frequency-domain features of the original signal; as shown in Figure 12, the amplitudes of each peak value and trough value in time domain are very stable and the period is identifiable; frequency components in frequency domain are very clean. That means that it consists of meshing frequency of gear pair (293.4 Hz) and its frequency multiplication, free of interferences of other frequencies.

In addition, the comparison between Figures 9 and 10 reveals that the experimental signals and simulated signals of the same sampling frequency share some similarities: (1) the period of the pulse in time-domain signals is not apparent, and signal period does not accord with the characteristics of theoretical model. (2) The amplitude of the meshing frequency of gear pair and its frequency multiplication in frequency domain are small and there are unknown frequencies of higher amplitude nearby, such that the frequency value of 263.7 Hz on the left of the meshing frequency (293.7 Hz) in Figure 9 is basically consistent with that of 265.6 Hz on the left of the meshing frequency (293.4 Hz) in Figure 10. Similarly, the frequency value of 560.4 Hz on the left of the double meshing frequency (587.1 Hz) in Figure 9 is basically consistent with that 559 Hz on the left of the double meshing frequency (586.6 Hz) in Figure 10.

Based on the comparative analyses above, the accuracy of nonlinear dynamic model of the gear transmission system and the calculation method for the vibration signal bandwidth of the gear system and its sampling frequency is verified.

4. Simulation Analysis and Experimental Verification for Gear System with Root Crack

4.1. The Signal Bandwidth of Gear Transmission System with Root Crack

When there is a crack at the root of the driving gear, the time-varying meshing stiffness of the gear pair will significantly change during the two meshing periods as the cracked tooth in meshing, as shown in Figure 13. The dynamic response of the faulty system can be solved by substituting the meshing stiffness with root crack into the established gear system model. Figure 14 shows the vibration acceleration signal acquired at the center of mass of the driving gear.

Figure 13:

Meshing stiffness and damping coefficient of gear pair with crack faults.

Figure 14:

Simulated vibration acceleration signals of the gear system with cracks on gear root.

The sampling frequency is 100000 kHz. The parameters of the gear system are module M = 2; the number of gear teeth of big and small gears $z_{1} / z_{2} = 55 / 75$ ; the crack length is L = 2 mm; the rotational frequency of input shaft is f_t = 14.50 Hz, in which the gear meshing frequency is found to be f_m = 797.5 Hz.

Figure 14 shows that there is significant change when comparing the time-domain waveform measured at the faulty position to that at the normal parts during the two meshing periods. Combining the time domain waveform and (7), the signal bandwidth of the normal part and the part with cracks can be calculated separately. The signal bandwidth of the normal part is 42.07 kHz, while the signal bandwidth calculated by (10) is 43.09 kHz. The deviation between the two is 2%. This again proves the accuracy of (10). For the faulty position, 3 peak values of pulse and 3 trough values of pulse in two meshing periods as the cracked tooth in meshing are different from the corresponding values obtained in the case of normal gear teeth. The signal bandwidths of 6 pulses calculated by (7) are 42.02 kHz, 41.92 kHz, 42.02 kHz, 41.32 kHz, 41.22 kHz, and 41.27 kHz. Therefore, the crack fault has little effect on signal bandwidth, which is smaller than that at the normal position. For gear system with root crack, no significant changes occur to the vibration acceleration signal bandwidth when comparing with that in the fault-free case. Thus, the calculation method for the vibration acceleration signal bandwidth of the fault-free gear system described in the previous section is also applicable to the gear system with root crack.

4.2. Experimental Verification

The sampling frequency of experiment is set as 10 kHz, and the time-domain and frequency-domain graphs are shown in Figure 15. From the figure it is known that there is significant impact on the time domain, but its amplitude and period follow no specific variation pattern; regular meshing frequency and frequency multiplication can be found from the frequency domain but not so obvious in its amplitude. No apparent sideband is found near the meshing frequency. In addition, some unknown frequencies of higher amplitude appear in the vicinity of the frequencies we are interested in. For the convenience of comparative analysis, the time-domain and frequency-domain graphs of simulated signal of different sampling frequencies under the same parameters are shown in Figures 16 –18. According to the calculation and analysis concerning signal bandwidth in the previous section, the signal bandwidth in this example is f_{3 dB} = 42.07 kHz, the critical sampling frequency is f_sl = 84.14 kHz, and the recommended sampling frequency is f_s = 168.28 kHz.

Figure 15:

The time-domain and frequency-domain graphs of vibration acceleration signals in gearbox with cracks on gear root under 10 kHz sampling frequency: (a) the time domain graph, (b) local amplification graph of the time domain, (c) local amplification graph of the frequency domain near meshing frequency, and (d) local amplification graph of the frequency domain near 2 times meshing frequency.

Figure 16:

The time-domain and frequency-domain graphs of simulated vibration acceleration signals in gear system with cracks on gear root under 10 kHz sampling frequency: (a) the time-domain graph, (b) local amplification graph of the time domain, (c) local amplification graph of the frequency domain near meshing frequency, and (d) local amplification graph of the frequency domain near 2 times meshing frequency.

Figure 17:

The time-domain and frequency-domain graphs of simulated vibration acceleration signals in the gear system with cracks on gear root under 100 kHz sampling frequency: (a) the time-domain graph, (b) local amplification graph of the time domain, (c) the frequency-domain graph, and (d) local amplification graph of the frequency domain.

Figure 18:

The time-domain and frequency-domain graphs of simulated vibration acceleration signals in gear system with cracks on gear root under 1000 kHz sampling frequency: (a) the time-domain graph, (b) local amplification graph of the time domain, (c) the frequency-domain graph, (d) local amplification graph of the frequency domain.

In Figure 16, when the sampling frequency f = 10 kHz, there is f_sl > f; that is, the actual sampling frequency is lower than the critical sampling frequency, which fails to satisfy the sampling theorem; then serious frequency aliasing phenomenon would occur, causing distortion of time domain and frequency domain. As shown in Figure 16, the faulty positions cannot be distinguished, since some peak values and trough values in the time domain are obviously deficient and the signal period is completely different from that of the original signal; a large number of unknown frequencies appear in the frequency domain, and the amplitude of meshing frequency of gear pair (7 Hz) as well as its frequency multiplication is very small, hardly discernible from the graphs. No apparent sidebands can be clearly observed.

In Figure 17, when the sampling frequency f = 100 kHz, there is f_s > f > f_sl that is, the actual sampling frequency is higher than the critical sampling frequency, which satisfies the sampling theorem, but lower than the recommended sampling frequency then the signals acquired can properly represent the time-domain and frequency-domain features of the original signal. As shown in Figure 17, the amplitudes of each peak value and trough value in time domain fluctuate, but the signal period is easily identifiable and the amplitude variations at the faulty positions can be also observed. The meshing frequency of gear pair (797.5 Hz) and its frequency multiplication and sidebands can still be clearly identified, and some unknown frequencies of lower amplitude appear near these frequencies, suggesting the occurrence of slight frequency aliasing phenomenon.

In Figure 18, when the sampling frequency f = 1000 kHz, there is f > f_s > f_sl that is, the actual sampling frequency is higher than the recommended sampling frequency; then the signals acquired can be used to accurately represent the time-domain and frequency-domain features of the original signal. As shown in Figure 18, the amplitudes of each peak value and trough value in time domain fluctuate very little, and the period can be easily identified; and the changes to the amplitude at the faulty position can also be seen. The spectrum indicates typical amplitude modulation; that is, there is a large area of sidebands in the vicinity of the gear pair's meshing frequency (797.5 Hz) and its frequency multiplication, with the sideband interval being the rotational frequency of driving shaft (14.5 Hz) free of interferences of other frequencies.

In addition, the comparison between Figures 15 and 16 reveals that the experimental signals and simulated signals of the same sampling frequency have something in common: (1) the period of the pulse in time domain signal is not apparent; pulse period does not accord with the characteristics of theoretical model; and the changes of the amplitude of faulty position cannot be discovered easily. (2) The amplitude of the meshing frequency of gear pair meshing and its frequency multiplication in frequency domain are small, and there are unknown frequencies of higher amplitude nearby, such that the frequency value 733.4 Hz on the left of the meshing frequency (797.2 Hz) in Figure 15 is basically consistent with that of 733.7 Hz on the left of the meshing frequency (797.5 Hz) in Figure 16. Similarly, the frequency value of 1530 Hz on the left of double meshing frequency (1597 Hz) in Figure 15 is basically consistent with that of 1531 Hz on the left of double meshing frequency (1595 Hz) in Figure 16. Most importantly, the sidebands cannot be identified, causing difficulty in the diagnosis of crack faults on gear root in practice.

5. Conclusion

An improved bending-torsion vibration model is established, in which the effect of time-varying meshing stiffness and damping, torsional stiffness for transmission shaft, elastic bearing support, the driving motor, and external load are taken into consideration. Then, vibration signals are simulated based on the model under diverse sampling frequencies. 3 dB bandwidth method is used to analyze the simulation signals, and the critical and recommended sampling frequencies are proposed for the spur gear system based on the sampling theorem. The influence of key system parameters on signal bandwidth is examined through simulation analysis, with the conclusions obtained as follows.

The gear module and teeth number basically have no influence on signal bandwidth, but the change of the rotational frequency of driving gear would affect signal bandwidth obviously. The method of least square for the curve fitting is performed to obtain the second-order equation of rotational frequency and bandwidth. A comparative verification is made with the results calculated by the traditional formula. Compared to the 3 dB method, the formula has a higher speed in the simulation and is easier to calculating the signal bandwidth in engineering.

The validity of the calculation method proposed for the vibration acceleration signal bandwidth and sampling frequency in the gear system is also experimentally verified. The results show that, when the actual sampling frequency is lower than the critical sampling frequency, serious signal distortion and aliasing would occur. Similar high-amplitude aliasing phenomena occur to the simulated and experimental signals under the same frequency. The simulated signals whose frequency is higher than the recommended sampling frequency basically coincide with the ideal signals.

The simulation analysis of the gear system with gear root cracks on a single gear tooth shows that its signal bandwidth is basically the same as that without faults. This means that the method proposed in this paper is also applicable to the gear transmission system with root cracks on a single gear tooth. The experimental results show that, when the actual sampling frequency is lower than the critical sampling frequency, serious signal distortion and aliasing would occur. As a result, both the time-domain and frequency-domain features of fault signals cannot be identified easily.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 51075006),the Beijing Municipal Natural Science Foundation (no. 3112004),and the National High-Tech Research and Development Program of China (863 Program,no. SS2012AA040702).

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