Dynamic Analysis and Fault Diagnosis for Gear Transmission of a Vibration Exciter of a Mine-Used Vibrating Screen under Different Conditions

Abstract

The helical gear pair of a box-type vibration exciter of a mine-used linear vibrating screen is subjected to complex excitation and prone to broken tooth failures. At present, investigation regarding the difference and particularity between gear transmission in vibrating screens (i.e., vibration machinery) and that in rotating machinery is still a challenge, which is the key to revealing the performance and failure mechanism of gear transmission in the premise of application to vibrating screens. In order to intuitively display the peculiarity of gear transmission on the exciter, an innovative virtual prototype model of a gear pair of a vibrating screen exciter is proposed. This model considers the effects of internal and external excitation, such as the friction and lubrication of the gear, strong alternating load produced by a large eccentric block, the reciprocating motion of the screen body, the large clearance of bearing and so on, and its correctness is verified. Based on the comparison, the inducement for the high fatigue rate of exciter gears is revealed. Models of a vibrating screen’s excitation system with different degrees of broken teeth are also established, and tooth fault features are proposed for fault detection. Sensitive indicators for the degradation degree of tooth damage are put forward, and the monitoring strategy is presented that with the increase of damage degree, the waveform index and pulse index of axial vibration acceleration increase. The analysis results provide powerful support for the optimal design of the vibrating screen’s exciter gears and fault diagnosis.

1. Introduction

A box-type vibration exciter [1] can provide a reciprocating power source for a mine-used linear vibrating screen [2], and the helical gear pair is the key component. It plays an important role in forcing the driving and driven shafts to rotate synchronously and reversely. However, under the coupled excitation of elastic support of the spring seat under the screen body, strong alternating load caused by eccentric block and large clearance of C3 level self-aligning roller bearing [3], the gear center distance of vibration exciter fluctuates violently. As a result, frequent switching is observed between teeth squeezing and large clearance in the process of gear transmission, which increase the fatigue damage of the gear and even causes serious faults such as tooth breakage [4]. Therefore, analyzing the dynamic characteristics of the vibration exciter without and with faulty gear of gear transmission is greatly significant.

Clearance is a non-negligible factor in the study of the dynamic characteristics of the gear drive system of the vibration exciter. Many scholars have committed to studying the influence of tooth backlash [5,6,7], bearing clearance [8,9] and their coupled excitation [10,11,12,13] on the dynamic characteristics of gear transmission. Kahraman and Blankenship [5] carried out research through experiments. Guo and Parker [8] analyzed the non-linear dynamic phenomenon of planetary gear transmission systems caused by bearing clearance. Liu et al. [11] found that there is a coupling relationship between bearing clearance and tooth backlash. A shorter center distance can cause teeth squeezing and bearing clearance can delay the occurrence of teeth squeezing. At present, research on tooth backlash and bearing clearance concerns rotating machinery. Generally, bearings in rotating machinery have clearance less than C3, and eccentricity is not considered in the rotating machinery model. However, the vibrating screen belongs to vibrating machinery. There are large eccentric blocks and bearings with large clearance in the vibrator rotor system, producing more complex dynamic characteristics and stronger coupling vibration.

Excitation is another significant factor in gear transmission. Rotor systems with gear transmission under the working conditions of marine ships [14,15,16,17], wind turbines considering tower elasticity [18,19], vehicles traveling on the road [20], aeroengines considering stator [21,22] and so on belong to the typical rotor system under foundation excitation. Han and Chu [14,15] studied the dynamic response of the rotor-bearing-gear system with cracks under three periodic foundation angular motions of the ship. Yin et al. [18] and Yin et al. [19] discussed the influence of flexible tower support on a wind turbine’s helical gear system and found that the vibration displacement amplitude under flexible support is generally higher. Based on the aeroengine rotor–stator system, Yan et al. [21] found that the influence of foundation vibration on dynamic rotor characteristics is orthogonal. Those works are examples of rotors on moving bases. The foundation of the rotor system of the vibration exciter (i.e., screen body) cannot vibrate by itself, and its working principle is fundamentally different from the rotor system excited by the rotor foundation. In addition, the vibration form of the vibrator rotor is a periodic reciprocating linear motion with a small amplitude in the direction of excitation force, which is also very different from the excitation form generated by waves and wind.

Gear mating dynamic analysis has been extensively investigated and reported. Zhang et al. [23] built a finite element (FE) model to determine the fault features in the time domain of the ring gear strain signal caused by abnormal mesh force. Wu et al. [24] proposed an analytical-FE method to calculate the time-varying mesh stiffness (TVMS) of gear tooth crack. The TVMS and crack propagation behavior in spur gear tooth were, respectively, studied using the extended FE method by Govind Verma et al. [25] and Mahendra Singh et al. [26]. Even though these studies concern the premise of traditional rotating machinery, the FE method has been proven to be an efficient tool for fault feature analysis of gears. However, it also has a problem with accurate modeling, especially in the case of handling multiple coupled excitation and complex boundary conditions, such as a vibrating screen.

At present, there are two main ways to determine gear fault diagnosis. One is model-based gear dynamic analysis, which was discussed in the above paragraph, and fault features are revealed. The other way is signal-based gear vibration diagnosis. In order to enhance the fear fault diagnosis, some methods have been proposed, such as an artificial bee colony (ABC) algorithm for the optimal parameters selection of SVM’s kernel function [27], a new semi-supervised method based upon the deep convolutional generative adversarial network (DCGAN) [28] and genetic mutation particle swarm optimization VMD and probabilistic neural network (GMPSO-VMD-PNN) [29]. However, the former method can better reveal the interaction rules of different influencing factors and more easily discover hidden fault features for diagnosis.

Fatigue fracture of gear teeth is one of the most dangerous failure forms of gears. Existing research on gear tooth breakage mainly focuses on the influence of fault on gear parameters [23] and system dynamic characteristics [24,25,26]. Chaari et al. [23] put forward the calculation method of broken tooth gear stiffness and obtained the influence of a broken tooth on gear meshing stiffness. Ma and Chen [24] considered the influence of slight imbalance caused by broken teeth on dynamic gear characteristics. However, there is no research on the dynamic characteristics of the gear transmission fault of box-type vibration exciters with gear tooth damage.

In summary, there are no special model for box-type vibration exciters of vibrating screens to reflect the combined action of the gear pair under internal and external excitation and fault excitation. The dynamic meshing characteristics and complex vibration response have not been studied. In view of this, this paper mainly focuses on solving the above problems by making use of virtual prototype technology [30,31,32].

The rest of this paper is summarized as follows. Section 2 establishes the dynamic model of the vibrating screen excitation system and verifies the correctness of the model. Section 3 and Section 4 discuss the dynamic characteristics of normal gear pair and faulty gear pair from three aspects: the trajectory change of gear rotation center, normal meshing force of gears and the time–frequency domain characteristics of vibration acceleration in vertical, horizontal and axial directions (i.e., x, y and z directions, unless otherwise specified, keeping the same definitions throughout the paper). The conclusion is presented in Section 5.

2. Dynamic Model of Vibrating Screen Excitation System

2.1. Construction of Dynamic Model of Excitation System

The purpose of building the overall model of the virtual prototype of the vibration excitation system is to simulate the boundary conditions of the gear pair and accurately reflect dynamic characteristics. The model should first be able to complete the normal gear mesh movement; secondly, it should be able to correctly reflect the reciprocating movement in the direction of the excitation force; finally, it must be able to accurately simulate the coupling effect of various components inside the exciter. Therefore, the overall model must consider the following components: gear pair, shaft, bearing, eccentric block, screen body, and spring. The spatial position relationship between the components is shown in Figure 1a. The rotating shaft connects the gear, bearing, and eccentric block. Fixed pair constraints are used between the rotating shaft and the gear, as well as between the rotating shaft and the bearing’s inner race. Speed is applied to the inner race of the driving bearing, and torque is applied to the driven shaft to complete the drive of the gear. The action of the exciting force is simulated using the particle and the force with a constant magnitude that changes direction with the rotation axis. The bearing’s outer ring is fixed on the screen body. The screen body is considered a rigid body, and only its mass characteristics are considered. The springs in the x and y directions are set at the bottom of the screen body. The excitation force, spring stiffness, and mass of each component are comprehensively considered to make the whole system reciprocate in the direction of excitation force [33]. Considering the shaft flexibility and bearing clearance, it can more truly reflect the coupling effect between components. This paper uses ADAMS software to establish the virtual prototype dynamic model of a mine-used linear vibrating screen with a processing capacity of 260 t/h. The main modeling contents include the following aspects listed in Figure 1b.

(1)

Eccentric blocks: four eccentric blocks of the vibration exciter are equivalent to the mass point fixed at the four ends of the rotating shaft. The centrifugal force generated by the movement of the eccentric block is considered a force (i.e., 7800 N) whose direction changes continuously with the rotating shaft.

(2)

Bearings: the 22232-e1-xl-k bearing model provided in ADAMS is adopted, the bearing clearance grade is C3 (i.e., 0.195 mm), and the whole screen body is directly set as the bearing support.

(3)

Shaft: considering the flexibility of the shaft, the shaft support is an elastic support with a fixed inner circumference, and the MNF format file of the shaft generated by ABAQUS is imported into the ADAMS assembly.

(4)

An accurate gear pair model is established by parametric modeling with Pro/E software: gear diameter $d_r$ is 330.002 mm; normal modulus $m_n$ is 3; number of teeth $Z$ is 93; reference circle pressure angle ${\alpha }_n$ is 23.29°; addendum coefficient $ha{n}^{*}$ and backlash coefficient $c^{*}$ are 1 and 1.25; helix angle of pitch circle $\beta$ and normal pressure angle ${\alpha }_n$ are 32.28°and 20°; toot width $B$ is 100 mm; center distance $d_0$ is 330.0024 mm. Considering the problem of operation efficiency, the rigid model of helical gear pair is adopted in ADAMS.

(5)

Gear contact coefficient: the meshing force and friction force of gears are calculated by impact function and Coulomb friction in ADAMS. According to Hertz elastic collision theory, the gear meshing stiffness coefficient $K$ is 9.47e + 05 N/mm and the force index $e$ is 1.5; the damping $C$ is 947 N·s/mm, and the penetration depth $d$ is 0.1 mm. The lubrication of the gear is considered in the setting of the friction coefficient. The static friction coefficient $f_{st}$ is 0.08, the dynamic friction coefficient $f_{dy}$ is 0.05, the static translation speed $V_s$ is 100 mm/s, and the friction translation speed $V_f$ is 1000 mm/s. It is considered that the gear contact parameters are constant throughout the meshing process.

(6)

Load: speed of 750 r/min is applied on the gear driving gear, and torque $T_L$ of 229,200 N·mm is applied on the driven gear.

(7)

Screen body and spring: the screen body is simplified as a rigid body. The spring models in the x and y directions in ADAMS are used to replace the equivalent vibrating screen spring.

(8)

Coordinate system: the helical gear pair of the vibration exciter is installed at an angle of 45° to the horizontal ground. For the convenience of analysis, the coordinate system ox’y’z can be obtained by rotating the x-axis and y-axis 45° counterclockwise around the o point on the xoy plane.

2.2. Verification of Dynamic Model of Excitation System

Firstly, it is known that the meshing force of helical gears can be calculated as Formula (1) [34]. An individual gear transmission virtual prototype model is used to calculate the meshing force.

$$\{\begin{array}{c}{F}_t=\frac{{2T}_L}{d_1}=\frac{2·\mathrm{229,200}}{\frac{3}{\cos 32.28}·93}=1389.08N\\ F_r=\frac{F_t{\tan \alpha }_n}{\cos \beta }=\frac{1389.08·\tan 20}{\cos 32.28}=598.01N\\ F_{\mathrm{a}}{=F}_t\tan \beta =1389.08·\tan 32.28=877.46N\\ F_n=\frac{F_t}{\cos \beta ·{\cos \alpha }_n}=\frac{1389.08}{\cos 32.28·\cos 20}=1748.45N\end{array},$$

(1)

where $F_n$, $F_r$, $F_t$, and $F_a$ are, respectively, normal meshing force, the radial meshing force, the tangential, and the axial component of meshing force, $T_L$ is driven gear torque, $d_1$ is pitch diameter, ${\alpha }_n$ is the pressure angle of the pitch circle, and $\beta$ is the helical angle of the pitch circle.

ADAMS mainly uses two methods, i.e., the restitution method and impact method, to calculate the contact force, of which the impact function method is more suitable for the helical gear transmission of vibration exciter. As a result, the impact function model is adopted for the normal meshing force $F_n$ calculation, i.e., Formula (2) [35]. It can be seen from the formula that when the distance between two objects at the time of collision minus the distance between objects at the initial time is greater than or equal to 0, the two objects do not contact, and the $F_n$ is 0. In other cases, the specific value of $F_n$ can be calculated by Formula (2). Using the same parameter setting as shown in Figure 1b, the meshing force obtained by simulation calculation and theoretical calculation is shown in

Table 1. Comparison between theoretical value and simulation average value of meshing force.

Meshing Force	Theoretical Value (N)	Simulation Value (N)	Relative Error
Tangential force $F_t$	1389.08	1390.54	0.11%
Radial force $F_r$	598.01	616.42	3%
Axial force $F_a$	877.46	905.22	3.1%

. It can be seen that the meshing force value calculated by simulation is bigger than the theoretical value calculated by Formula (1), and the main reason is that the former considers friction. The maximum deviation between the simulated value and the theoretical value is 3.1%, which is within a reasonable range according to references [36,37,38], so the setting of gear contact parameters is reasonable.

Impact method model:

$$F_n=\{\begin{array}{c}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} q\ge q_0\hfill \\ K{\left(q_0-q\right)}^e-C·\frac{dq}{dt}·step\left(q,q_0-d, q_0 \right) q\le q_0\hfill \end{array},$$

(2)

where $K$ is the stiffness coefficient of the collision object, $q_0$ is the distance between objects at the initial time, $q$ is the distance when the object collides, $e$ is the non-linear index, $C$ is the damping coefficient, and $d$ is the penetration depth.

Secondly, the correctness of the virtual prototype model of the vibration system in Figure 1a can also be verified from the point of view of the motion trajectory of the mass center (CM) of the vibrating screen. The curve in Figure 2 is obtained by running the model in Figure 1a. It can be seen in Figure 2 that after the system enters a stable working state, the displacement curves in the x and y directions (CM_x and CM_y) change into a simple harmonic curve, which is in line with the mass center motion of the vibration exciter deduced in reference [30]. The vibration amplitudes of CM_x and CM_y are 3.32 mm and 3.34 mm. By merging, the synthesized amplitude (CM_t) is 4.7 mm, which is approximately consistent with that in the actual working condition of 4.8 mm. The motion trajectory of the vibrating screen is approximately a straight line (Figure 2b), which is also consistent with the real motion trajectory of the vibrating screen. The model can fully implement the requirements of a real coal mine-used vibrating screen. As a result, the correctness of the model is verified.

2.3. Setting of Control Group Model

Different degrees (i.e., 1/4, 1/2, 3/4, and complete broken) of broken tooth are set on a specific tooth of the driven gear of the exciter, which can introduce fault excitation into the excitation system.

In addition, in order to show the dynamic characteristics of the gear transmission of the vibrating screen exciter, a gear transmission virtual prototype model under ordinary rotating machinery dispose of screen body, eccentric blocks, and spring in Figure 1a is established as the control group. This model includes three parts: gear pair, rotating shafts, and bearings. The gear contact parameters, bearing type, speed, and load torque are the same as those of the excitation system in Figure 1a, and the bearing support is set as the earth.

3. Comparison of Dynamic Response Characteristics under Fault-Free State

The rotor rotation frequency $F_{\mathrm{r}}$ is 12.5 Hz. The gear meshing frequency (GMF) can be calculated at 1162.5 Hz. Furthermore, 2 GMF, 3 GMF, and 4 GMF represent, respectively, two, three and four times the meshing frequency.

3.1. Dynamic Response of Gear Transmission of Ordinary Rotating Machinery

In Figure 3 and Figure 4, G1 and G2 represent the driving and driven gears, respectively; $F_n$ represents the normal meshing force; AX, AY and AZ represent the vibration acceleration of the gear rotation center in the x, y, and z directions, respectively; FFTMAG(Fn) and FFTMAG(AX), respectively, represent the Fourier transform of normal meshing force magnitude and vibration magnitude in the x-direction.

Figure 3a,b shows the displacement variation of the rotation center of the driving and driven gears in the x and y directions. The driving gear and driven gear are set in the same coordinate system; the rotation centers of the driving gear and the driven gear are, respectively, set at the origin of coordinates and the position with the coordinate value of 330.0024. That is, the position of the blue line in Figure 3b. The disorganized trajectories are caused by the large clearance of the bearing and meshing impact caused by eccentric force, especially in the start-up stage. There is a violent fluctuation process within the range of 0.05 mm, but it is much smaller than the bearing clearance of 0.195 mm. During stable operation, the rotation center position of the driving gear is far away from the initial installation position and the increase of the center distance leads to tooth backlash.

Figure 4a shows that the normal meshing force fluctuates within 0–5000 N. The gear transmission is relatively stable, and no obvious impact can be seen except for position 1, resulting from tooth backlash and a slight difference in the speed of the two gears. Similarly, the vibration acceleration can also reflect a slight impact. After removing the trend part during time–frequency transformation, the main frequency components of ordinary rotating machinery gears are gear meshing frequency, i.e., GMF, and its multiples, of which the amplitude of 2 GMF is the largest. The vibration features in the y and z directions are basically consistent with that in the x direction, so only vibration acceleration in the x direction is shown.

3.2. Dynamic Response of Gear Transmission of Vibration Exciter

Two local coordinate systems are established at the geometric center of the driving wheel and the driven wheel. The position curve can be obtained by measuring the position changes of the two local coordinate systems in the global coordinate system of the system. As shown in Figure 5, the displacement range of the rotation center reaches the maximum of the bearing clearance in the x’ direction. The dynamic fluctuation of gear center distance is strong because of the combined action of big clearance and eccentric block, which is easy to lead to frequent teeth squeezing and large tooth backlash.

In Figure 6a, the red line “$F_n$” represents the normal meshing force of the gear, and the blue line “ZXJ” represents the difference between the actual gear center distance and theoretical center distance. The gear meshing force fluctuates periodically with the change of center distance. The main reason for the “wave crest” at position I is gear squeezing. Figure 6b shows the gears meshing in and out clearly. Compared with the spectrum of Figure 4a, except for GMF and its multiples, Figure 7a shows the obvious rotational frequency $F_r$, which is mainly caused by periodic teeth squeezing. It is noteworthy that the side frequency modulated by $F_r$ is symmetrically distributed on both sides of GMF and its multiples, resulting in significant amplitude-modulated vibration. It is indicated that exciter gears are subjected to more complex alternating excitation.

In Figure 8a, the vibration acceleration of the gear in three directions shows obvious periodic vibration, in which the vibration in the x and y directions is similar to simple harmonic vibration. Figure 8b has the same frequency components and distribution as in Figure 7. Compared with ordinary rotating machinery, gear transmission of the excitation system has more complex dynamic response characteristics. The stronger vibration and shock further explain the reason for the high failure rate of the exciter gear.

The statistical results in

Table 2. Vibration response of helical gear pair of ordinary rotating machinery and normal excitation exciter.

Indicators	Ordinary Rotating Machinery			Normal Exciter
	AVG	MAX	RMS	AVG	MAX	RMS
$F_n$ (N)	1776.14	5354.12	2145.71	3207.27	14,521.25	4539.12
AX (mm/s2)		7973.86	2891.07		29,040.31	11,086
AY (mm/s2)		19,984.13	6889.29		42,882.64	15,656.82
AZ (mm/s2)		11,090.18	3647.63		21,534.7	7506.97

quantitatively show the differences in meshing force and vibration response between the exciter gears and the rotating machinery gears. It can be concluded that the fault mechanism of the exciter gears is the strong alternating excitation and complex vibration mode resulting from the different working principles.

4. Dynamic Response Characteristics of Gear Transmission with Gear Breakage

The vibrating system originally had strong excitation deriving from large eccentric blocks and large bearing clearance. In addition, broken teeth can also introduce stiffness excitation and eccentric excitation at the same time. Therefore, in order to accurately diagnose the broken tooth, the dynamic response of a vibration exciter with a broken tooth is more worthy of study.

4.1. With Full Broken Tooth

As shown in Figure 9, the red curve represents fault-free, and the blue curve represents a fault state with a fully broken tooth. It can be seen that, due to the change in internal force balance, tooth breakage causes more serious fluctuation of the rotation center. The most serious impact is concentrated in positions ① and ②, especially in the x’ direction.

As shown in Figure 10, due to the loss of the entire tooth, the decentration causes additional inertial and centrifugal forces when rotating. The incidence of amplitude fluctuation of meshing force is significantly smaller at position II during one rotation as a result of the change of contact distance between two gears due to the coupling effects of dynamic gear eccentricity and other nonlinear excitation of the excitation system; in the zoom magnified image of position I, the waveform of gear meshing in and out is not clear, and the waveform caused by obviously broken teeth can be seen. Figure 11 (1)–(4) also display $F_r$, GMF, and its multiples, but the amplitudes of $F_r$, GMF, 2 GMF, and 3 GMF decrease, and the amplitude of 4 GMF increases significantly; the sidebands of the main components are more obvious with asymmetric characteristics due to the change of vibration signal formation caused by interference of broken tooth; 3 GMF is covered.

Compared to Figure 8, Figure 12 shows that the vibration acceleration curve at positions 6, 7, and 8 in Figure 12 are “thinner” because of the smaller fluctuation range of meshing force at position II in Figure 10a. The acceleration impact is caused by the sudden change of acceleration because of the broken tooth; the main frequency components are consistent with those of meshing force in Figure 11.

4.2. With 1/2 Broken Tooth or with 1/4 Broken Tooth

As shown in Figure 13, 1/2 broken tooth means that half of the tooth is missing. Figure 13 exhibits a normal meshing force spectrum and frequency spectrum with 1/2 broken tooth. By comprehensively comparing Section 3.2 and Section 4.1, it can be concluded that: with the increase of gear damage, the waveform of the broken tooth at position I becomes clearer, but the waveform of meshing in and out becomes more blurred in the zoomed-in diagram, the amplitude fluctuation also decreases at position II, and vibration acceleration has no 3 GMF. The 1/4 broken tooth reveals the same rules and has a similar spectrum which is not shown here due to space limitations.

4.3. Summary of Dynamic Characteristics of Gear Pair with Broken Tooth

Faulty gears with broken teeth at the vibration exciter have significant vibration characteristics. Some key parameter values in the time domain and frequency domain are, respectively, listed in

Table 3. Parameters in time domain in different fault states.

Indicators	Full Broken Tooth			1/2 Broken Tooth			1/4 Broken Tooth
	AVG	MAX	RMS	AVG	MAX	RMS	AVG	MAX	RMS
$F_n$ (N)	2851.5	15,815.4	3718.7	2846.3	13,822.5	3757.0	2880.6	15,413.3	3773.4
AX(mm/s2)		43,630.2	10,630.5		46,774.4	10,999.2		46,058.5	10,574.2
AY(mm/s2)		52,317.3	11,565.0		49,027.1	11,439.4		52,572.6	11,575.2

and Figure 14. $F_n\_n$, $F_n\_c$, $F_n\_3/4$, $F_n\_1/2$, and $F_n\_1/4$ are defined as the normal meshing force of normal gear, faulty gears with a full broken tooth, 3/4 broken tooth, 1/2 broken tooth, and 1/4 broken tooth, respectively. This definition rule is also applicable to other parameters. Some conclusions are summarized as follows:

(1)

As shown in Table 3, compared with the fault-free state, the AVG and RMS of the normal meshing force with a broken tooth are smaller, and the internal impact of the gear pair is generally weakened.

(2)

As shown in Table 3, compared with the fault-free state, the MAX of the vibration acceleration in the x and y directions with a broken tooth is greater; however, the RMS of the acceleration in the x, y, and z directions is smaller, indicating that the broken tooth leads to the weakening of the overall vibration intensity of the gear transmission. It can be seen that it is difficult to diagnose the broken tooth directly through the vibration amplitude.

(3)

Eight eigenvalues are calculated from meshing force and vibration acceleration in the x, y, and z directions under different states, respectively: mean value (AVG), root mean square (RMS), peak value (Xp), peak index (C), waveform index (S), pulse index (I), kurtosis index (K), and margin index (L). Statistical results show that the eight eigenvalues of meshing force and vibration acceleration in the x and y directions have no obvious regularity relative to the damage degree. However, some regularity can be discovered from vibration acceleration in the z direction, as shown in Figure 14a: the S and I of AZ increase with the increase of damage degree. After the introduction of 3/4 broken teeth, the following laws still hold: S_c > S_3/4 > S_1/2 > S_1/4 > S_n, I_c > I_3/4 > I_1/2 > I_1/4 > I_n, which demonstrates that the law is universal. As a result, the conclusion can be drawn that S and I of vibration acceleration in the z direction are sensitive to damage degree, which can be used as identification indicators.

(4)

Figure 14b reveals that the amplitudes of Fr, GMF, 2 GMF, and 3 GMF of the meshing force of the faulty gear are less than those components of normal gears, while the amplitude of 4 GMF is greater than that of normal gear. Among them, 3 GMF is difficult to find because it is covered by side frequency. In addition, this paper also analyzed the fault state with 3/4 broken tooth and proves that these conclusions are still valid, which are not listed in detail. These fault features can be used for fault diagnosis.

(5)

Figure 14b also shows that the amplitudes of GMF and 3 GMF of vibration acceleration in the x, y, and z directions of faulty gears are less than those with normal gears, among which 3 GMF is also covered; the amplitude of 4 GMF is greater than that in the fault-free state. These regularities are still valid for 3/4 broken teeth. These fault features can also judge whether the gear breaks.

5. Conclusions

First, the helical gear fault mechanism is investigated based on an innovative virtual prototype model. The influence of internal and external factors on the gear pair of the vibrating screen exciter is emphasized in the model. By comparing the dynamic characteristics of gear with that of traditional rotating machinery, the mechanism of why the exciter gear pair is more prone to fatigue failure is revealed: ① The center distance of gear pair changes violently and the teeth squeezing appears periodically. ② The AVG of gear meshing force of vibration exciter is larger, so the exciter gear meshing impact is larger and fatigue damage is easy to occur. ③ Rotating frequency has a great influence on the frequency distribution of meshing force, including strength, its multiples, and modulated sidebands, which demonstrates exciter gear teeth are subjected to more complex alternating excitation. ④ The vibration acceleration of the helical gear in x and y directions fluctuates in a simple harmonic curve in the time domain, and its frequency components are basically consistent with those of the meshing force, which generates a more complex vibration mode. ⑤ Larger MAX and RMS of the acceleration in the three directions give rise to greater vibration intensity. These conclusions can also guide the improvement design of the gear of the vibrating screen exciter.

Based on analyzing dynamic response under the condition of tooth break, fault features are presented for tooth fault detection: ① The vibration amplitude increases in the x’ and y’ directions, especially in the x’ direction. ② The following parameter values decrease: RMS of meshing force and vibration acceleration in three directions, amplitudes of GMF, and 3 GMF (3 GMF is easy to be annihilated); however, the amplitude of 4 GMF increases. ③ The sidebands distribution of meshing force presents asymmetry.

Finally, the monitoring strategy for the degradation degree of tooth damage is proposed. Waveform index S and pulse index I of vibration acceleration in the z direction are sensitive to the damage degree, which can be regarded as identification indicators.

The virtual prototype model and analysis method in this paper can also be used to verify whether the optimal design (such as surface modification) of the vibrating screen’s exciter gears can really improve the dynamic characteristics of the gear transmission. By updating the model with the improved gear and making use of the dynamic simulation analysis method of the paper, the position curve of the gear rotation center, changes in meshing force, and vibration response can be obtained. As a result, the optimal design can be testified. It is also our current work to investigate the influence of surface modification on the gear transmission performance of the vibration exciter.

Realistic condition is more complicated, such as interference of time-varying load, temperature, and unbalanced supporting force of the spring pack. Studies that consider the impact of more external actual incentives will be performed in the future. It also needs to be verified by experiments in combination with various internal and external incentive factors. Although the helical gear considered herein is a fundamental gear transmission, this virtual prototype model cannot be employed in the preliminary verification of structural improvement design, but also, through appropriate modification, it can obtain the dynamic load distribution at the shaft ends, which can be used to study the fatigue life of self-aligning roller bearing of vibration exciter under the combined action of contact fatigue and fretting fatigue with appropriate modifications.