Study of the Influences of Transient Crack Propagation in a Pinion on Time-Varying Mesh Stiffness

Cracks in a cracked gear may further propagate by a tiny length in a very short time for several reasons, such as material fatigue and load fluctuations. In this paper, this dynamic process is defined as transient propagation of cracks. This research aims to calculate the time-varying mesh stiffness of gears when transient propagation of cracks arises, which has not been extensively studied in existing literatures. The transient propagation of cracks is modelled. An improved potential energy method is proposed by incorporating the propagation model into the potential energy method. The improved method can also be utilised to calculate the mesh stiffness of gears when transient propagation of cracks arises. Different transient propagation models are considered to simulate the propagation of cracks in a short amount of time. Different deterioration levels of cracks before transient propagation and different lengths and models of transient propagation are also examined. The variation rules of mesh stiffness caused by the transient propagation of cracks are summarised.The influence of the deterioration level of cracks on mesh stiffness variation when transient propagation arises is obtained. Simulation results show that the proposedmethod accurately calculates time-varyingmesh stiffness when transient propagation of cracks arises. Furthermore, the method improves the monitoring of further propagation of cracks in gears from the perspective of time-varying mesh stiffness.


Introduction
Root crack, a typical fault in gear sets, may result from poor operating conditions, such as inadequate lubrication, excessive applied torque, poor specifications, material defects, and manufacturing or installation problems [1].Undesirable dynamic responses, including serious vibration and noise, may generate once a crack appears, and this occurrence may shorten the service life of a gear [2,3].Therefore, reducing catastrophic accidents by improving the detection of crack faults is highly significant.
Time-varying mesh stiffness, which is caused by the periodic change of working teeth pairs, is an important parameter in the gear system.On the one hand, it brings about inherited periodic internal excitations of the gear system [3]; on the other hand, time-varying mesh stiffness decreases when a gear crack appears [5].Therefore, both crack propagation and the resultant reduction in time-varying mesh stiffness have elicited the interest of many researchers [6,7].The current work also belongs to this research field.
In their research on crack propagation, Lewicki and Ballarini studied the influence of rim thickness on crack propagation path [8], set up different fatigue crack growth models to predict crack propagation life for thin rim teeth [9], and proposed a method to study the influence of tooth dynamic loading on crack propagation [10].Based on linear elastic fracture mechanics, Pandya and Parey [11] adopted a 2D finite element method to study crack propagation paths in gear pairs with different contact ratios.In their recent studies [3,12,13], they studied the influence of backup ratio (rim thickness-to-tooth height), fillet radius, and pressure angle on crack propagation and mesh stiffness in gear pairs with a low contact ratio.The finite element method based on linear elastic fracture mechanics (LEFM) was developed to study the crack propagation path of spur gears with a high contact ratio (HCR) and with tooth root cracks for two gear parameters (backup ratio and pressure angle) [13].Zouari et al. [14,15] adopted a 3D finite element method (FEM) to study the effects of propagation depth, orientation, and position in the blending radius of cracks on instantaneous mesh stiffness.2D FEM was used to analyse the variation in stress intensity factor on the tooth base according to crack depth, crack propagation angle, and crack position.The propagation process was divided into two or four stages in [6,7,16,17] to study the effect of severity of crack faults on mesh stiffness.In summary, crack length is assumed to be time-invariant in all these studies.As a result, the dynamic process of crack propagation has not been revealed.
In fact, according to the principle of fracture mechanics [18][19][20], crack arrest is a common phenomenon that occurs during crack propagation, given that cracks will propagate only if the crack driving force is greater than the crack resistance.During crack propagation, the crack driving force initially increases with the increase in crack length and then decreases.Crack arrest occurs when the driving force is lower than the resistance.For ductile materials, such as metal or alloy, when the initial crack propagates, a slow propagation stage occurs before rapid propagation.However, automatic crack arrest may appear in both slow and rapid propagation stages.The stress of the gear teeth changes rapidly for fast engaging-in and engaging-out of the gear teeth.This change may also result in the propagation and arrest of cracks in a short time, and the cracks rapidly propagate in tiny lengths.
Detection of crack propagation with a tiny length in a very short time (described above) is important in condition monitoring and fault diagnosis of a device.Thus, timely detection and confirmation of further propagation of cracks are highly necessary.Crack propagation by a tiny length in a short time is defined as transient propagation of cracks (TPOC).One of the contributions of this study is the investigation of TPOC for the timely detection and confirmation of further crack propagation.Only a few studies have investigated this aspect.Moreover, in this study, crack length is assumed to be time-varying to describe TPOC.Different TPOC models are proposed to simulate crack propagation instantaneously.
For the algorithm of time-varying mesh stiffness, two methods of calculating the time-varying mesh stiffness of a gear set exist [21]; these two methods are FEM and the analytical method (AM).The former is time-consuming because every gear pair must be modelled to determine the mesh stiffness of a range of gear pairs.Meanwhile, AM provides a general approach to evaluate mesh stiffness with satisfactory results [1].The elastic deformation method has been introduced previously and has been used in other related studies [1,[22][23][24].The potential energy method, a typical analytical method initially proposed by Yang and Lin, has been used to calculate the mesh stiffness of a pair of externalexternal spur gears [25].Given its simple calculation process, the potential energy method has been adopted by a large number of scholars [3, 5-7, 17, 21, 26-28].In this method, the gear tooth is usually modelled as a cantilever beam that begins at the base circle.The Hertzian energy, bending energy, and axial compressive energy corresponding to Hertzian contact stiffness, bending stiffness, and axial compressive stiffness are considered in this model.Eventually, the shear energy corresponding to shear stiffness was also considered in [26].The potential energy method has been utilised to calculate the mesh stiffness of ring and cracked gears [27,28], and both the mesh stiffness of a spur gear tooth with spatial crack propagation and a helical gear have been calculated with modified potential energy methods [29][30][31].However, these studies regarded crack length as a constant when calculating mesh stiffness.Therefore, when a gear undergoes TPOC, the time-varying mesh stiffness cannot be calculated by existing algorithms.Although a series of different crack lengths have been used to represent crack propagation, at least two defects still exist.One is that the calculation is time-consuming because of multiple times of calculations, and the other is that only a series of discrete stiffness values for certain crack lengths can be obtained.Consequently, further propagation of cracks cannot be revealed by time-varying mesh stiffness.Instead of a constant, a propagation function is used to represent crack length in our improved potential energy method.With the improved method, the time-varying mesh stiffness when TPOC arises can be calculated, and further propagation of cracks can be detected in a timely manner.
In this study, TPOC was described and defined.Different propagation models were considered and proposed to simulate TPOC.The improved potential energy method was developed by incorporating the crack propagation model into the potential energy method.Four different deterioration levels of cracked gears were assumed, which means the lengths of existing cracks before TPOC were different.The crack was assumed to further propagate by a tiny length, as represented by the propagation model in each deterioration level.Afterwards, the time-varying mesh stiffness of gears when TPOC occurred was calculated with the improved method.The mesh stiffness variations caused by different TPOCs were compared and summarised.

The Transient Propagation of Crack and Improved Potential Energy Method
Crack propagation is a gradual process in which crack length increases.If crack length is assumed to be time-invariant, the dynamic process of crack propagation cannot be revealed.
If the crack length of a cracked gear is considered constant, time-varying mesh stiffness cannot be calculated when the crack propagates by a tiny length in a very short time.Thus, crack propagation cannot be identified in a timely manner.Crack length has always been regarded as a constant when the potential energy method is adopted to calculate the timevarying mesh stiffness of gears [1, 3, 5-7, 17, 18, 24, 25].In other words, the existing potential energy method cannot be directly used to calculate time-varying mesh stiffness when TPOC occurs in a gear.Before presenting our improved potential energy method, we introduce and describe TPOC.In this paper, TPOC is defined as the situation wherein cracks further propagate by a tiny length in a very short amount of time.
Figure 1: Nonuniform cantilever beam model of cracked gear.

Transient Propagation of Crack in a Pinion (Driving Gear)
2.1.1.Description of TPOC.We assume that TPOC appears in a pinion (driving gear).A crack exists before TPOC, and the length of the existing crack is  1 .The crack is assumed to propagate along a straight line and through the entire tooth width.The intersection angle, , between the crack and the central line of the tooth is set at 45 ∘ . is the angle displacement of pinion (driving gear); the length of TPOC is represented by (), where  is time-varying; therefore, () is time-varying.The unit of () and  is mm and rad, respectively.In Figure 1(a), the blue line is the existing crack, and the red line is TPOC.When the crack further propagates by a length of (), the total length of the crack is  1 + ().
If () = 0, then the crack has not propagated and the crack length is constant, as shown in Figure 1(b).

Modelling of TPOC.
Several transient propagation functions were modelled to describe TPOC.The function is shown with ().The step function was used to represent the crack propagating by a tiny length instantaneously.A continuous step function was used to represent the crack propagating by several segments of tiny length instantaneously.A linear function was used to represent the crack propagating by a tiny length continuously in a very short time.
In the following functions,  1 () to  1 () are step functions. 2 () is a continuous step function. 3 () is a linear function.Therefore, the propagation model of  1 () to  1 () can be adopted to investigate the effects of different propagation lengths with the same propagation model on the time-varying mesh stiffness.Moreover, the propagation model of  2 () and  3 () can be adopted to study the effects of different propagation models with the same length on the time-varying mesh stiffness.All propagation models are expressed as follows.
(1) Different Propagation Lengths with the Same Propagation Model.As is shown in Figure 2(a), (2) Different Propagation Models with the Same Propagation Length.As is shown in Figure 2(b), the purple line is  2 (), the red line is  3 (), and the total lengths of TPOCs are the same, that is, 0.15 mm, but their propagation modes are different from each other, which can be expressed as, All the transient propagation models can be plotted in Figure 4.

Improved Algorithm of Mesh Stiffness When TPOC Occurs in a
Pinion.According to the potential energy method, the tooth can be regarded as a nonuniform cantilever beam shown in Figure 1(b). represents the distance between the contact point and the root of the tooth, and the corresponding q 1d () q 1c () q 1b () angle is  1 .ℎ  represents the vertical distance between the tip of the crack and the tooth's central line, which corresponds to the angle   .  is the radius of base circle. 2 is the half tooth angle on the base circle. 1 is the length of existing crack.ℎ is half of the roof chordal tooth thickness. is the intersection angle between the crack and the central line of the tooth.The total potential energy stored in the gear system includes the following parts: the Hertzian contact energy, the bending energy, the axial compression energy, and the shear energy, which can be expressed as [7] where   ,   , and   represent the bending stiffness, the axial compression stiffness, and the shear stiffness. is Young's modulus. is the shear modulus.Hence, where   and   are the two perpendicular component forces of  and the torque  represents the bending effect of   .  represents the area moment of inertia of the section where the distance from the tooth root is ;   represents the area of the section where the distance from the tooth root is .
For a perfect gear without crack,   and   can be obtained as where ℎ  is the distance between the point on the tooth's curve and the tooth's central line where the horizontal distance from the tooth root is  and  is the width of the tooth.In this case, the bending stiffness, the axial compression stiffness, and the shear stiffness can be calculated by the following [7]: Figure 3: Geometrical parameters for the fillet-foundation deflection [28].
The fillet-foundation deflection was derived by Sainsot et al. in [32] based on the theory of [33].It can be calculated as where  represents the tooth width.  and   are given in Figure 2. The coefficients , , and  can be approached by polynomial functions [32]: where   , denotes the coefficients , , , and , ℎ  =   / int ,   and  int are given in Figure 3, and the values of   ,   ,   ,   ,   , and   are given in Table 1.
The stiffness with consideration of gear fillet-foundation deflection can be obtained by For a cracked gear shown in Figure 1,   and   can be obtained as In this study, the crack further propagates by a tiny length () instantaneously based on the existing crack, as shown in Figure 1(a).Therefore, the total crack length becomes  1 + (); then, The crack only affects the bending stiffness and shear stiffness [17]; in this case, they change as follows: The time-varying mesh stiffness of the double-tooth pair can be calculated with (6) [23].Hence, The subscripts 1 and 2 represent the pinion (driving gear) and gear (driven gear), respectively. = 1 represents the first pair of meshing teeth when two pairs of teeth mesh;  = 2 represents the second pair. is the angle displacement of pinion (driving gear).

Simulation Results and Discussion
Different deterioration levels of existing cracks were considered to study the influences of TPOC on time-varying mesh stiffness.In each deterioration level, TPOCs with the same model but different lengths and with the same length but different models were considered, and the resultant stiffness variations were compared.

Different Deterioration Levels of Crack before TPOC.
Different lengths of existing cracks were considered, as shown in Figure 4.In Figure 4, cases 1 to 4 represent the length of the existing crack becoming increasingly large; that is, the deterioration level of the crack becomes increasingly serious.The blue line represents the existing crack; the red line represents TPOC.In cases 1 and 2, the length of the existing crack is  1 .In cases 3 and 4, the length of the existing crack is  1 + 2 .ℎ  before and after TPOC arises in the different cases as shown in Table 2.
According to the formula in Table 2, the bending and shear stiffness when TPOC arises can be recalculated as follows: Case 1 Case 2 Shock and Vibration Figure 4: Different deterioration levels before TPOC.

Time-Varying Mesh Stiffness Simulation When TPOC
Occurs.Based on the improved potential energy method, the time-varying mesh stiffness when TPOC occurs in a pinion was calculated.The main parameters are shown in Table 3.

Mesh Stiffness Simulation When TPOC Occurs with the Same Model but Different
Lengths.This section focuses on mesh stiffness when TPOC occurs with the model of a step function.However, the lengths are 0.05, 0.1, 0.15, and 0.2 mm.In Figure 5, () = 0 indicates that no TPOC occurs, and the corresponding stiffness represents the mesh stiffness of a cracked gear with the length of an existing crack.
Figure 5(a) shows that an instantaneous increment in the length of the crack induces a sudden decrease in mesh stiffness.Indeed, large lengths of crack propagation induce large reductions in mesh stiffness.The same phenomenon appears in the other cases.After TPOC, the total length of cracks is the sum of the existing crack length and TPOC length.Furthermore, the stiffness curve represents the timevarying mesh stiffness of gears with the total length of cracks.
From Table 4 and Figure 7(a), we can see that, for the same length of TPOC with the model of step function, the resultant stiffness reduction becomes less gradually from case 1 to case 4. In other words, as the deterioration level of crack increases, the stiffness reduction caused by TPOC will decrease.

Mesh Stiffness Simulation for TPOC with the Same
Length but Different Models.Similar to that in Figure 5, () = 0 in Figure 6 indicates the absence of TPOC.The time-varying mesh stiffness when TCOP occurs with the same length but different propagation models is shown in Figure 6.In the figure, the crack instantaneously propagates by 0.015 mm in two different models: continuous step and linear function models.For TPOC with the linear function model, the length of TPOC changes continuously, and the resultant stiffness variation is also continuous and smooth.For the continuous step function model, several continuous mutations in propagation length lead to several sudden decrements in mesh stiffness.However, regardless of the TPOC model, the stiffness curve after TPOC overlaps; that is, the stiffness is equal after TPOC, when the transient propagation length remains constant.
We mainly investigated TPOC with the model of a continuous step function symbolized by  2 ().The transient propagation process is described with three segments of the step function, denoting that the crack has propagated the same small length for three incidents.The three segments of TPOC are represented with  21 (),  22 () and  23 (), respectively.",Table 5 shows the stiffness reduction caused by each segment of crack propagation of TPOC with the continuous step function model in different cases.In the table, 0.02 − /0.02 + represents the moment before/after TPOC from 0 mm to 0.05 mm, 0.04 − /0.04 + represents the moment before/after TPOC from 0.05 mm to 0.1 mm, and 0.06 − /0.06 + represents the moment before/after TPOC from 0.1 mm to 0.15 mm. 2 (0.02 − ),  2 (0.02 + ),  2 (0.04 − ),  2 (0.04 + ),  2 (0.06 − ), and  2 (0.06 + ) represent the mesh stiffness of the corresponding moment.A schematic of the stiffness reduction is shown in Figure 7(b).
Table 5 and Figure 7(b) show that, per 0.05 mm of TPOC, the resultant stiffness reduction decreases from case 1 to case 4. Therefore, we conclude that as the crack deterioration level becomes increasingly serious, the stiffness reduction caused by TPOC decreases.

Conclusions
Several TPOC models have been proposed to study the influences of TPOC on the time-varying mesh stiffness of the q 1a () q() = 0 q 1b () q 1d () q 1c () q 1a () q() = 0 q 1b () q 1d () q 1c ()  q 1a () q() = 0 q 1b () q 1d () q 1c () gear and detect the further propagation of cracks from the perspective of time-varying mesh stiffness in a timely manner.The improved potential energy method was developed by integrating the TPOC model into the potential energy method.The simulation results show that the improved potential energy method can accurately calculate the timevarying mesh stiffness when TPOC occurs.Different deterioration levels of cracks before TPOC, different propagation models, and different lengths of TPOC were considered.The following conclusions were obtained: (1) The stiffness variation trend is similar to that in the model of TPOC which caused the variation.For example, when TPOC occurs with the model of a step function, the resultant stiffness variation is in the form of an approximate step function.When TPOC occurs with the model of a continuous step function,  the resultant stiffness variation is in the form of an approximate continuous step function.When TPOC occurs with the model of a linear function, the resultant stiffness variation is smooth and continuous.
(2) For the certain deterioration level of cracks before transient propagation, large propagation lengths are associated with large variations in stiffness.
(3) When the crack propagates by the same tiny length instantaneously (e.g., TPOC occurs with the model of a step function), the resultant stiffness reduction decreases as the deterioration level of the crack increases (cases 1 to 4).When the crack continuously propagates by several segments of equal length (e.g., TPOC arises with the model of a continuous Crack case q 1a () q 1b () q 1c () q 1d ()  step function), for every equal length of TPOC, the resultant stiffness reduction decreases gradually from case 1 to case 4. Therefore, we conclude that, for the same length of TPOC, the resultant stiffness reduction is related to the deterioration level of the crack; high levels of deterioration are associated with small reductions in stiffness.In other words, TPOC is easy to detect in the early stage of a gear fault.(4) For a cracked gear, identical transient propagation lengths result in identical stiffness levels after TPOC regardless of the type of TPOC.

Figure 2 :
Figure 2: TPOC models.(a) Different propagation lengths with the same propagation model.(b) Different propagation models with the same propagation length.

(d) Case 4 (Figure 5 :
Figure 5: The influences of TPOC with different lengths but the same propagation model on mesh stiffness. 1 (),  1 (),  1 (), and  1 () represent the crack propagating with the model of a step function, which indicates that the crack has instantaneously propagated by a tiny length.The corresponding lengths are 0.05, 0.1, 0.15, and 0.2 mm.() = 0 represents the absence of TPOC.

Figure 6 :
Figure 6: The influences of TPOC with the same length but different propagation models on mesh stiffness. 2 () and  3 () represent the crack propagating with the same length of 0.15 mm. 2 () represents the crack with the model of a continuous step function, indicating that the crack has instantaneously propagated by several continuous segments of length. 3 () represents the crack propagating with the model of a linear function, indicating that the crack has propagated continuously and smoothly.() = 0 represents the absence of TPOC.

Figure 7 :
Figure 7: Mesh stiffness reduction caused by TCOP models in all cases.

Table 3 :
Main parameters of the gear system.Mass moment of inertia of the load   = 0.0105 Kg m 2 Mass moment of inertia of the pinion  1 = 4.3659 × 10 −4 Kg m 2 Mass moment of inertia of the gear  2 = 8.3602 × 10 −3 Kg m 2As a sample of the length of existing crack,  1 (or  1 and  2 ) can be set in the following 4 cases.Case 1 to case 4 represent the notion that the deterioration level becomes more serious:

Table 4 :
Mesh stiffness reduction caused by TPOC (with the model of step function but different lengths) in different deterioration levels.

Table 5 :
Mesh stiffness reduction caused by TPOC with the model of a continuous step function in different deterioration levels.