**3.1 Modelling the effect of a fatigue crack in tooth fillet area of a gear**

Classical tooth root fillet fatigue fracture is the most common cause of gear tooth breakages (Figure-3.1.1). Stress raisers, such as micro cracks from the heat treatment, hob tears, inclusions and grinding burns are common causes that initiate the cracks. The cracks occurring in the gear tooth fillet region progressively grow until the whole tooth or part of it breaks away. The breakage of a tooth is a serious failure. Not only the broken part fails, but serious damage may occur to the other gears as a result of a broken tooth passing though the transmission [48].

Fig. 3.1.1. A spur gear missing two teeth. They broke away due to the propagation of fatigue cracks. (Courtesy of DANA [48])

The research conducted for NASA by Lewicki [51] sets a clear guideline for predicting the trajectory of cracks occurring at the gear tooth fillet. Lewicki predicted crack propagation paths of spur gears with a variety of gear tooth and rim configurations, including the effect of: rim and web thickness, initial crack locations and gear tooth geometry factors (Diametral pitch, number of teeth, pitch radius and tooth pressure angle). A summary of the results is presented in Figure-3.1.2.

The effect of gear tooth fillet cracks (TFC) and spalls on gear transmission error was studied in detail by using a static simulation models (FEA and LTCA (HyGears [50])). A pair of meshing gears were modelled and analysed in step incremented non-linear static environment. Note: the transmission error obtained from the static simulation models are

It was explained earlier that the interaction between two meshing gears can be expressed in the dynamic model as time-varying stiffness, damping and gear tooth topological error elements linking the two lumped mass moments of inertia. The effect of gear tooth faults can be implemented into the dynamic simulation model as changes to these parameters. The understanding gained from the detailed simulation model studies of TFCs and spalls on gear motion has lead to the method of modelling the effect of the faults in dynamic model. The relevance between the types of gear faults to the selected parameters will be explained

Further to the simulation of gear tooth faults, this chapter also briefly touches on the modelling of spalls in rolling element bearings (REB), which is also a common type of faults

Classical tooth root fillet fatigue fracture is the most common cause of gear tooth breakages (Figure-3.1.1). Stress raisers, such as micro cracks from the heat treatment, hob tears, inclusions and grinding burns are common causes that initiate the cracks. The cracks occurring in the gear tooth fillet region progressively grow until the whole tooth or part of it breaks away. The breakage of a tooth is a serious failure. Not only the broken part fails, but serious damage may occur to the other gears as a result of a broken tooth passing though

Fig. 3.1.1. A spur gear missing two teeth. They broke away due to the propagation of fatigue

The research conducted for NASA by Lewicki [51] sets a clear guideline for predicting the trajectory of cracks occurring at the gear tooth fillet. Lewicki predicted crack propagation paths of spur gears with a variety of gear tooth and rim configurations, including the effect of: rim and web thickness, initial crack locations and gear tooth geometry factors (Diametral pitch, number of teeth, pitch radius and tooth pressure angle). A summary of the results is

**3.1 Modelling the effect of a fatigue crack in tooth fillet area of a gear** 

referred to as Motion Errors (ME) here forth by following the HyGears convention.

through subsequent sections.

in geared transmission systems.

the transmission [48].

cracks. (Courtesy of DANA [48])

presented in Figure-3.1.2.

Fig. 3.1.2. Effect of backup ratio (mb) and initial crack location on propagation path. (Lewicki [51])

A set of spur gears later used in the validation of the simulation result have a "backup ratio" (rim thickness divided by tooth height) greater than mb=1.3. Therefore based on the Lewicki's prediction the cracks occurring in the tooth fillet region are most likely to propagate in the trajectory shown in Pattern Figure-3.1.2 (i); roughly 30~45° in to the tooth relative to the radial line path through the symmetric axis of the spur gear tooth profile.

Full 3D modelling of a propagating gear tooth crack is one of the actively researched areas. Some examples of simulation studies using the Boundary Element Method (BEM) are given in [52, 53, 54, 55]. The simulation studies using 3D models show complex behaviour crack growth from the small crack seeded at the middle of the gear tooth fillet. An example is shown in Figure-3.1.3 from "Modelling of 3D cracks in split spur gear", by Lewicki [52]. The crack front expands rapidly across the width of the gear tooth as it progresses into the thickness of the tooth. The tooth fillet crack (TFC) model used in this work assumes 2D

Gearbox Simulation Models with Gears and Bearings Faults 37

25Nm 50Nm 100Nm 200Nm

**LTCA MARC [Undamaged]**

0 3 5 8 10 13 16 18 21 23 **Pinion Roll Angle [Deg's]**


**ME [um]**

(a)

Fig. 3.1.5. Motion Errors of (32x32) teeth gear pairs; (a) Undamaged gear set; (b) TFC

25Nm 50Nm 100Nm 200Nm

25Nm 50Nm 100Nm 200Nm

1 4 7 9 12 15 18 21 24 26 **Pinion Roll Angle [Deg's]**

**FEA, MARC [Steel Gears (32x32), No Prof Mod]**

1 4 7 9 12 15 18 21 24 26 **Pinion Roll Angle [Deg's]**

**FEA, MARC [Steel Gears (32x32), No Prof Mod]**

(L=1.18mm); (c) TFC (L=2.36mm)




**ME [um]**

(c)


0




**ME [um]**

(b)


0

Fig. 3.1.3. 3D Crack propagation model. a) Boundary Element Model of Split Spur Gear, b) Close up of view of the gear teeth and crack section at earlier stage of development and c) more progressed crack. (Lewicki [53])

conditions which approximates the weakening of the cracked gear tooth when the crack is extended across the whole width of the tooth face.

The motion error (ME), as obtained from the finite element (FE) model of a gear pair (32x32 teeth) presented in Figure-3.1.4, is shown in Figure-3.1.5. The MEs of the gears at different

Fig. 3.1.4. Spur Gears (32x32) and its FE mesh (L), detailed view of the mesh around the gear teeth (R). [56]

(a)

Fig. 3.1.3. 3D Crack propagation model. a) Boundary Element Model of Split Spur Gear, b) Close up of view of the gear teeth and crack section at earlier stage of development and c)

NOTE: The crack front starts to straighten up as it

NOTE: Very fine mesh on the gear teeth

conditions which approximates the weakening of the cracked gear tooth when the crack is

The motion error (ME), as obtained from the finite element (FE) model of a gear pair (32x32 teeth) presented in Figure-3.1.4, is shown in Figure-3.1.5. The MEs of the gears at different

Fig. 3.1.4. Spur Gears (32x32) and its FE mesh (L), detailed view of the mesh around the gear

more progressed crack. (Lewicki [53])

(b) (c)

teeth (R). [56]

extended across the whole width of the tooth face.

Fig. 3.1.5. Motion Errors of (32x32) teeth gear pairs; (a) Undamaged gear set; (b) TFC (L=1.18mm); (c) TFC (L=2.36mm)

Gearbox Simulation Models with Gears and Bearings Faults 39

Fig. 3.1.6. Comparison of RMEs of gears with TFCs; (a) Illustrated definition of RME; (b)

RMEs of TFC sizes L=1.18mm; (c) RMEs of TFC sizes L=2.36mm.

amount of loadings (25, 50, 100 & 200Nm) are compared in the same plot. The magnitude of MEs increases with the larger loads as the deflection of the meshing teeth become greater with the larger loads. The change in the amount of ME is roughly in linear relationship with the load, which reflects the linear elastic behaviour of the gear tooth deflection. Note the square pattern of the ME, which resulted from the time (or angular position) dependent variation of gear mesh stiffness, due to the alternating single and double tooth engagement.

The plots presented in Figures 3.1.5 (b) & (c) show the MEs of the gears with tooth fillet cracks of three different sizes. The localized increase in the amount of ME over a period of ME pattern is a direct consequence of the reduced gear tooth stiffness caused by the TFC.

The plots shown in Figure-3.1.6 (a) ~ (c) are the residual ME (RME) obtained by taking the difference between the MEs of uncracked gears and the ones with TFCs. The RMEs show a "double stepped" pattern that reflects the tooth meshing pattern of the gears, where smaller step with less deflection occurs as the crack tooth enters the mesh and share the load with the adjacent tooth; the larger second step follows when the cracked tooth alone carries the load.

The RMEs of the TFCs show linearly proportional relationship between the amount of loading and the change in RME for a given crack size. The linear relationship between the loading and the amount of tooth deflection on a cracked gear tooth indicates that the effect of TFC can be modelled effectively as a localized change in the gear mesh stiffness.

The plots in Figure-3.1.7 show the transmission errors (TEs) measured from a pair of plastic gears with a root fillet cut (Figure-3.1.7 (a)), which the cut replicates a tooth fillet crack. The TEs of Figure 3.1.7 (c1-c3 and d1-d3) are compared to the simulated patterns of MEs (figure 3.1.7 (b)). Composite TEs (CTEs) and the zoomed view of the CTEs are shown in Figure-3.1.7 (c1 & d1) and (c2 & d2) respectively. The CTE combines the both long and short term components of the TE (LTC and STC) presented earlier in section 2.2 (figure 2.2.1). The resemblance between the simulated MEs and measured the TEs confirms the validity of the simulated effect of TFC. The STCs (c3 & d3) were obtained by high pass filtering the CTE. The pattern in the STCs shows clear resemblance to the simulated TFC effect (Figure3.1.7 (b)).

The simulation model used in this study does not consider the effect of plasticity. This assumption can be justified for a gear tooth with small cracks where localised effect of plasticity at the crack tip has small influence on the overall deflection of the gear tooth, which is most likely the case for the ideal fault detection scenario.

More recent work published by Mark [57, 60] explains that the plasticity can become a significant factor when work hardening effect can cause a permanent deformation of the cracked tooth. In this case, the meshing pattern of the gears changes more definitively by the geometrical error introduced in the gears by the bent tooth. In some cases the bent tooth result in rather complex meshing behaviour that involves tooth impacting. Further explanation on this topic is available from the works published by Mark [57, 60].

Within the limitation of the simulated TFC model discussed above, the approach to model the TFC as a localized variation in the gear mesh stiffness is acceptable for a small crack emerging in the gear tooth fillet area. For the purpose of developing a dynamic simulation model of a geared transmission system with an emerging TFC the model presented here offers a reasonable approach.

amount of loadings (25, 50, 100 & 200Nm) are compared in the same plot. The magnitude of MEs increases with the larger loads as the deflection of the meshing teeth become greater with the larger loads. The change in the amount of ME is roughly in linear relationship with the load, which reflects the linear elastic behaviour of the gear tooth deflection. Note the square pattern of the ME, which resulted from the time (or angular position) dependent variation of gear mesh stiffness, due to the alternating single and double tooth engagement. The plots presented in Figures 3.1.5 (b) & (c) show the MEs of the gears with tooth fillet cracks of three different sizes. The localized increase in the amount of ME over a period of ME pattern is a direct consequence of the reduced gear tooth stiffness caused by the TFC.

The plots shown in Figure-3.1.6 (a) ~ (c) are the residual ME (RME) obtained by taking the difference between the MEs of uncracked gears and the ones with TFCs. The RMEs show a "double stepped" pattern that reflects the tooth meshing pattern of the gears, where smaller step with less deflection occurs as the crack tooth enters the mesh and share the load with the adjacent tooth; the larger second step follows when the cracked tooth alone carries the load. The RMEs of the TFCs show linearly proportional relationship between the amount of loading and the change in RME for a given crack size. The linear relationship between the loading and the amount of tooth deflection on a cracked gear tooth indicates that the effect

The plots in Figure-3.1.7 show the transmission errors (TEs) measured from a pair of plastic gears with a root fillet cut (Figure-3.1.7 (a)), which the cut replicates a tooth fillet crack. The TEs of Figure 3.1.7 (c1-c3 and d1-d3) are compared to the simulated patterns of MEs (figure 3.1.7 (b)). Composite TEs (CTEs) and the zoomed view of the CTEs are shown in Figure-3.1.7 (c1 & d1) and (c2 & d2) respectively. The CTE combines the both long and short term components of the TE (LTC and STC) presented earlier in section 2.2 (figure 2.2.1). The resemblance between the simulated MEs and measured the TEs confirms the validity of the simulated effect of TFC. The STCs (c3 & d3) were obtained by high pass filtering the CTE. The pattern in the STCs shows clear resemblance to the simulated TFC effect (Figure3.1.7 (b)).

The simulation model used in this study does not consider the effect of plasticity. This assumption can be justified for a gear tooth with small cracks where localised effect of plasticity at the crack tip has small influence on the overall deflection of the gear tooth,

More recent work published by Mark [57, 60] explains that the plasticity can become a significant factor when work hardening effect can cause a permanent deformation of the cracked tooth. In this case, the meshing pattern of the gears changes more definitively by the geometrical error introduced in the gears by the bent tooth. In some cases the bent tooth result in rather complex meshing behaviour that involves tooth impacting. Further

Within the limitation of the simulated TFC model discussed above, the approach to model the TFC as a localized variation in the gear mesh stiffness is acceptable for a small crack emerging in the gear tooth fillet area. For the purpose of developing a dynamic simulation model of a geared transmission system with an emerging TFC the model presented here

explanation on this topic is available from the works published by Mark [57, 60].

which is most likely the case for the ideal fault detection scenario.

offers a reasonable approach.

of TFC can be modelled effectively as a localized change in the gear mesh stiffness.

Fig. 3.1.6. Comparison of RMEs of gears with TFCs; (a) Illustrated definition of RME; (b) RMEs of TFC sizes L=1.18mm; (c) RMEs of TFC sizes L=2.36mm.

Gearbox Simulation Models with Gears and Bearings Faults 41

Fig. 3.2.1. Examples of (a) progressive pitting and (b) severe spalling damages on gear teeth.

spalls and pitting occurring on a gear tooth. This work follows the definition of contact surface damage given by Tallian [58]: Spalling designated only as macro-scale contact fatigue, reserving the term pitting for the formation of pores and craters by processes other

1. Initial Phase: Bulk changes in the material structure take place around the highly stressed area under the contact path. Change in hardness, residual stress and some

2. A Long Stable Phase: Microscopic flow occurs in the highly stressed area changing the material structure and residual stress conditions at the microscopic level. The change in the structure brought out by the microscopic flow can be observed by eyes in the

Spalls have a distinctive appearance that is characterised by how they were formed. A fully developed spall typically has its diameter much larger than its depth. The bottom of the spall has a series of serrations caused by propagating fatigue cracks running transverse to the direction of rolling contact. The bottom of the spall parallels the contact surface roughly

The sidewalls and the wall at the exiting side of the spall (as in the exiting of rolling contact) are often radially curved as they are formed by material breaking away from the fatigued area. The entrance wall of the spall is characterised by how it was initiated. Tallian [58] explains that shallow angled entry (less than 30 inclination to the contact surface) occurs when the spall is initiated by cracks on the surface. Spalls with steep entry (more than 45)

Surface originated spalls are caused by pre-existing surface damage (nicks and scratches). It is also known that lubrication fluid could accelerate the crack propagation when contact occurs in such a way that fluid is trapped in the cracks and squeezed at extremely high

The subsurface originated spalls are caused by presence of inclusions (hard particles and impurities in the metal) and shearing occurs between the hard and the soft metal layers formed by case hardening. A spall caused by the initial breakage of the gear tooth surface continues to expand by forming subsequent cracks further down the rolling direction. Figure-3.2.2 illustrates the formation and expansion of spall damage by Ding & Rieger [59].

3. Macroscopic Cracking: This is the last failure phase instituting the crack growth.

at the depth of maximum unidirectional shear stress in Hertzian contact.

occur when the spall is initiated by subsurface cracks.

pressure as the contacting gear teeth rolls over it.

There are three distinctive phases in the development of surface fatigue damage:

microscopic changes in the grain structure of the metal.

(a) (b)

(DANA [48])

than fatigue cracking.

illuminated etched areas.

Fig. 3.1.7. Comparison of simulated vs. measured transmission errors; (a) Picture of a gear with a seeded TFC; (b) MEs from a FE model. (c1) ~ (c3) TEs of the gears with TFC loaded with 5Nm; (d1) ~ (d3) TEs of the gear with TFC loaded with 30Nm; CTE (c1, d1), Zoomed views (c2, d2) and STC (c3, d3).

#### **3.2 Modelling the effect of a spall on a tooth face of a gear**

Symptoms of surface fatigue vary, but they can generally be noticed by the appearance of cavities and craters formed by removal of surface material. The damage may start small or large and may grow or remain small. In some cases gears cure themselves as they wear off the damage: Initial pitting [47]. The terms "Spalling" and "Pitting" are often used indiscriminately to describe contact fatigue damages. Figure-3.2.1 shows some examples of

(c1) (c2) (c3)

(d3)

(b)

Fig. 3.1.7. Comparison of simulated vs. measured transmission errors; (a) Picture of a gear with a seeded TFC; (b) MEs from a FE model. (c1) ~ (c3) TEs of the gears with TFC loaded with 5Nm; (d1) ~ (d3) TEs of the gear with TFC loaded with 30Nm; CTE (c1, d1), Zoomed

(d1) (d2**)** 

Symptoms of surface fatigue vary, but they can generally be noticed by the appearance of cavities and craters formed by removal of surface material. The damage may start small or large and may grow or remain small. In some cases gears cure themselves as they wear off the damage: Initial pitting [47]. The terms "Spalling" and "Pitting" are often used indiscriminately to describe contact fatigue damages. Figure-3.2.1 shows some examples of

**3.2 Modelling the effect of a spall on a tooth face of a gear** 

views (c2, d2) and STC (c3, d3).

(a)

Fig. 3.2.1. Examples of (a) progressive pitting and (b) severe spalling damages on gear teeth. (DANA [48])

spalls and pitting occurring on a gear tooth. This work follows the definition of contact surface damage given by Tallian [58]: Spalling designated only as macro-scale contact fatigue, reserving the term pitting for the formation of pores and craters by processes other than fatigue cracking.

There are three distinctive phases in the development of surface fatigue damage:


Spalls have a distinctive appearance that is characterised by how they were formed. A fully developed spall typically has its diameter much larger than its depth. The bottom of the spall has a series of serrations caused by propagating fatigue cracks running transverse to the direction of rolling contact. The bottom of the spall parallels the contact surface roughly at the depth of maximum unidirectional shear stress in Hertzian contact.

The sidewalls and the wall at the exiting side of the spall (as in the exiting of rolling contact) are often radially curved as they are formed by material breaking away from the fatigued area. The entrance wall of the spall is characterised by how it was initiated. Tallian [58] explains that shallow angled entry (less than 30 inclination to the contact surface) occurs when the spall is initiated by cracks on the surface. Spalls with steep entry (more than 45) occur when the spall is initiated by subsurface cracks.

Surface originated spalls are caused by pre-existing surface damage (nicks and scratches). It is also known that lubrication fluid could accelerate the crack propagation when contact occurs in such a way that fluid is trapped in the cracks and squeezed at extremely high pressure as the contacting gear teeth rolls over it.

The subsurface originated spalls are caused by presence of inclusions (hard particles and impurities in the metal) and shearing occurs between the hard and the soft metal layers formed by case hardening. A spall caused by the initial breakage of the gear tooth surface continues to expand by forming subsequent cracks further down the rolling direction. Figure-3.2.2 illustrates the formation and expansion of spall damage by Ding & Rieger [59].

Gearbox Simulation Models with Gears and Bearings Faults 43

contact surface of the gears. A study was carried out to investigate the effect of spalls on the ME using a 3D gear tooth model [62]. The result of the study showed that the effect of the spalls on the ME is completely dominated by the displacement caused by the topological

Formation of spalls is a complicated process and it is one of the active research topics which have been studied for some time. Although there are several items in the literature that describe the properties and process of formation of spalls, no literature was sighted which defines the specific definition of shapes and sizes of spalls occurring on spur gear teeth.

Papers presented by Badaoui et al. [63, 64] and Mahfoudh et al. [65] show a successful example of simulating the effect of a spall on spur gear tooth by modelling the fault as prismatic slot cut into the gear tooth surface. Their results were validated by experiment. The model of the spall used in this work follows the same simplification of the general shape of the spalling fault. The

1. A spall is most likely to initiate at the centre of the pitchline on a gear tooth where maximum contact stress is expected to occur in the meshing spur gear teeth as the gears

2. The spall expand in size in the direction of rolling contact until it reaches the end of the

3. When the spall reaches the end of the single tooth pair contact zone, the spall will then expand across the tooth face following the position of the high contact stresses. The contact stress patterns shown in Figure-3.2.3 strongly support this tendency of spall

error of the gear tooth surface caused by the fault.

carry the load by only one pair of teeth.

growth.

simulation of the spall is bounded by these following assumptions:

single tooth pair contact zone as shown in Figure-3.2.4.

Fig. 3.2.4. A model of spall growth pattern and the resulting RME. [62]

Fig. 3.2.2. Formation and expansion of spalls. (Ding & Rieger [59])

The result of contact stress analysis from HyGears [50] shows the occurrence of high stress concentration at the entrance and exit walls of the spall (Figure-3.2.3). The high stress concentration at these edges implies the likelihood of damage propagation in that direction. The result of the simulation corresponds to Tallian's [58] observation that spalls tend to expand in the direction of rolling contact. In the HyGears simulation the spall was modelled as a rectangular shaped recess on the tooth surface in the middle of the pitch line.

Typical spur gear tooth surfaces are formed by two curvatures: profile and lead curvatures. The 2D models are limited to simulating the effect of the spall crater on the gear tooth profile only. 3D simulation is required to comprehend the complete effect of spalls on the

Fig. 3.2.3. HyGears analysis of contact stresses around the spall. [62]

Direction of roll

The result of contact stress analysis from HyGears [50] shows the occurrence of high stress concentration at the entrance and exit walls of the spall (Figure-3.2.3). The high stress concentration at these edges implies the likelihood of damage propagation in that direction. The result of the simulation corresponds to Tallian's [58] observation that spalls tend to expand in the direction of rolling contact. In the HyGears simulation the spall was modelled

Expanding spall

Typical spur gear tooth surfaces are formed by two curvatures: profile and lead curvatures. The 2D models are limited to simulating the effect of the spall crater on the gear tooth profile only. 3D simulation is required to comprehend the complete effect of spalls on the

as a rectangular shaped recess on the tooth surface in the middle of the pitch line.

Fig. 3.2.2. Formation and expansion of spalls. (Ding & Rieger [59])

Fig. 3.2.3. HyGears analysis of contact stresses around the spall. [62]

contact surface of the gears. A study was carried out to investigate the effect of spalls on the ME using a 3D gear tooth model [62]. The result of the study showed that the effect of the spalls on the ME is completely dominated by the displacement caused by the topological error of the gear tooth surface caused by the fault.

Formation of spalls is a complicated process and it is one of the active research topics which have been studied for some time. Although there are several items in the literature that describe the properties and process of formation of spalls, no literature was sighted which defines the specific definition of shapes and sizes of spalls occurring on spur gear teeth.

Papers presented by Badaoui et al. [63, 64] and Mahfoudh et al. [65] show a successful example of simulating the effect of a spall on spur gear tooth by modelling the fault as prismatic slot cut into the gear tooth surface. Their results were validated by experiment. The model of the spall used in this work follows the same simplification of the general shape of the spalling fault. The simulation of the spall is bounded by these following assumptions:


Fig. 3.2.4. A model of spall growth pattern and the resulting RME. [62]

Gearbox Simulation Models with Gears and Bearings Faults 45

The FEA model based study of TFCs and spalls has lead to identifying some useful properties of the faults that can be used to model the effect of the faults in the lamped

The Residual Motion Errors (RME) of TFCs have shown double stepped patterns that were load dependent. The change in the amount of deflection in the gear mesh (i.e. ME) with a cracked tooth is influenced by the size of the crack and also by the amount of loading on the gears. The simulation result showed that the linearly proportional relationship between the torque applied to the gears and the resulting RME value. The bucket shaped RMEs of spalls were not affected by the loading condition but purely driven by the change in the contact path patterns of the meshing teeth, due to the geometrical deviation of the gear tooth surface caused by a spall. It was also understood from the simulation studies that the size and the

Based on the observation above, the effect of TFC was modelled as locally reduced tooth meshing stiffness and the spalls as direct displacement due to the topological alteration of the gear tooth surface. In the gear dynamic model is shown in Figure-3.3.1, the effect of a TFC was implemented as a reduction in stiffness "Km" over one gear mesh cycle. The change in the value of Km was calculated from the FEA model mapped into an angular position dependent function in the gear dynamic model. A spall was implemented as a localized displacement mapped on the "et". An illustration of a TFC and a spall models in a

**3.3 Simulating the effect of TFCs and spalls in a gear dynamics model** 

shape of a spall affect the length and the depth of the bucket.

gear rotor dynamic model is shown in Figure-3.3.1 [61].

Signal pickup point

Fig. 3.3.1. Modelling of Gear Tooth Faults in Dynamic Model. [61]

Figures 3.3.2 (a1~a3) and 3.3.3 (a1~a3) show the simulated vibration signal (acceleration) from the LPM shown in the Figure-3.3.1. The signals were measured from the free end of the

parameter type gear dynamics models.

The models of spalls were developed by following the set of assumptions described above. Resulting plots of the RMEs are presented in Figure-3.2.4 along with the shape illustration of spalls on a gear tooth. Note that the RMEs of the spalls form bucket shapes and their length and the depth are determined by the length and width of the spall; i.e. the shape of RME and the progression of the spall is directly related. This information can be used in diagnosis and prognosis of the fault.

The plots in Figure-3.2.5 show TEs measured from a pair of plastic gears with a spall (Figure-3.2.5 (a)). The change in the pattern of the TE due to the spall is comparable to the simulated pattern of the MEs (Figure-3.2.5 (b)). Composite TEs (CTEs) and the zoomed views of the CTEs are shown in Figure-3.2.5 (c1~c3). The resemblance between the simulated MEs and measured TEs confirms the validity of the simulated effect of TFC. The STCs (d1~d3) were obtained by high pass filtering the CTE. The pattern in the STCs shows clear correlation between the simulated and measured patterns of spall motion errors.

Fig. 3.2.5. Comparison of simulated vs. measured transmission errors; (a) Picture of a gear with a seeded Spall; (b) MEs from a FE model; (c1) ~ (c3) CTEs of the gear with the spall; (d1) ~ (d3) STCs of the gear with the spall; Applied torque: 15Nm (c1 & d1), 30Nm (c2, d2), 60Nm (c3, d3).

The models of spalls were developed by following the set of assumptions described above. Resulting plots of the RMEs are presented in Figure-3.2.4 along with the shape illustration of spalls on a gear tooth. Note that the RMEs of the spalls form bucket shapes and their length and the depth are determined by the length and width of the spall; i.e. the shape of RME and the progression of the spall is directly related. This information can be used in diagnosis

The plots in Figure-3.2.5 show TEs measured from a pair of plastic gears with a spall (Figure-3.2.5 (a)). The change in the pattern of the TE due to the spall is comparable to the simulated pattern of the MEs (Figure-3.2.5 (b)). Composite TEs (CTEs) and the zoomed views of the CTEs are shown in Figure-3.2.5 (c1~c3). The resemblance between the simulated MEs and measured TEs confirms the validity of the simulated effect of TFC. The STCs (d1~d3) were obtained by high pass filtering the CTE. The pattern in the STCs shows clear correlation between the simulated and measured patterns of spall motion errors.

(b)

Fig. 3.2.5. Comparison of simulated vs. measured transmission errors; (a) Picture of a gear with a seeded Spall; (b) MEs from a FE model; (c1) ~ (c3) CTEs of the gear with the spall; (d1) ~ (d3) STCs of the gear with the spall; Applied torque: 15Nm (c1 & d1), 30Nm (c2, d2),

(c1) (c2) (c3)

(d2) (d3) (d1)

and prognosis of the fault.

Red - LTC, Blue - Composite Error

(a)

60Nm (c3, d3).

High Pass Filter

#### **3.3 Simulating the effect of TFCs and spalls in a gear dynamics model**

The FEA model based study of TFCs and spalls has lead to identifying some useful properties of the faults that can be used to model the effect of the faults in the lamped parameter type gear dynamics models.

The Residual Motion Errors (RME) of TFCs have shown double stepped patterns that were load dependent. The change in the amount of deflection in the gear mesh (i.e. ME) with a cracked tooth is influenced by the size of the crack and also by the amount of loading on the gears. The simulation result showed that the linearly proportional relationship between the torque applied to the gears and the resulting RME value. The bucket shaped RMEs of spalls were not affected by the loading condition but purely driven by the change in the contact path patterns of the meshing teeth, due to the geometrical deviation of the gear tooth surface caused by a spall. It was also understood from the simulation studies that the size and the shape of a spall affect the length and the depth of the bucket.

Based on the observation above, the effect of TFC was modelled as locally reduced tooth meshing stiffness and the spalls as direct displacement due to the topological alteration of the gear tooth surface. In the gear dynamic model is shown in Figure-3.3.1, the effect of a TFC was implemented as a reduction in stiffness "Km" over one gear mesh cycle. The change in the value of Km was calculated from the FEA model mapped into an angular position dependent function in the gear dynamic model. A spall was implemented as a localized displacement mapped on the "et". An illustration of a TFC and a spall models in a gear rotor dynamic model is shown in Figure-3.3.1 [61].

Fig. 3.3.1. Modelling of Gear Tooth Faults in Dynamic Model. [61]

Figures 3.3.2 (a1~a3) and 3.3.3 (a1~a3) show the simulated vibration signal (acceleration) from the LPM shown in the Figure-3.3.1. The signals were measured from the free end of the

Gearbox Simulation Models with Gears and Bearings Faults 47

This section discusses a several simple but an effective methods of modelling a spall in the rolling element bearing model introduced previously in section 2.5.1. The ideas behind each modelling approach was discussed by using the example of modelling a spall in the outer race of a bearing. The same method can be easily expanded in to modelling an inner race spall and ball faults. More detailed explanation on this topic is available from the work

The simplest form of a spall model can be implemented to the REB model by assuming instantaneous contact loss between the bearing races and rollig an element(s) as it pass over the spall. So, in reference to the rolling element model described above, the presence of a

 

*<sup>j</sup>* which defines the contact state of rolling elements over a defined angular

*<sup>d</sup>* ). In effect, this mothod models the spall as a step function as shown in Figure-

*d j d d*

*<sup>d</sup>* ) can be modelled by using the fault

(3.4.1)

*<sup>j</sup>* is defined as follows:

Fig. 3.3.4. Vibration of a Gearbox. [62]

**3.4 Modelling a spall in a rolling element bearing** 

published by Sawalhi and Randall [29, 30, 31, 34, 35, 66, 67].

spall of a depth (*Cd* ) over an angular distance of (

3.4.1 and further illustrated by Figure-3.4.2 (a), in which

Fig. 3.4.1. Spall definition on the outer race [66]

1, 0,

*if*

*otherwise* 

*j*

switch

position (

*y ewn h* (Note: \* represents convolution) (3.3.1)

driven shaft (see Figure-3.3.1). The residual signals shown in (b1~b3) of Figures 3.3.2 and 3.3.3 were obtained by subtracting the simulated vibration of undamaged gearbox from the damaged one. Two identical simulation models, one with a gear tooth fault (TFC or spall) and the other undamaged, were run in parallel and the difference between the two model out puts were taken to separate the effect of the gear faults. The impact like effect of the gear tooth faults is seen in both the gears with a TFC and a spall.

A comparison of the simulated signals shows that the magnitude of the TFC impulses is affected by the amount of torque applied to the gears, while the spall impulses are not. This response is consistent with the observation made in the static simulation of the gears in mesh.

Careful observation of the residual signal also reveals that the fault information is not only buried in the dominant effect of gearmesh, but also somewhat distorted by the effect of transmission path from the gearmesh to the point where the signal was measured. The effect of transmission path appears in the residual signal As the transient "tail" effect convolved over the impulse due to the gear fault (see Equation-3.3.1 and Figure-3.3.4 for illustration).

Fig. 3.3.2. Simulated gearbox vibration signal with the effect of a TFC.

Fig. 3.3.3. Simulated gearbox vibration signal with the effect of a spall.

The residual signal of the TFC and the spall provide a useful means to understand the nature of diagnostic information of the faults. In machine condition monitoring signal processing techniques are often developed to detect and quantify the symptoms of a damage buried in a background noise. By being able to see the "clear" effect of a fault, the most effective signal processing technique can be applied to target and monitor the symptoms of the damage. The idea of using the simulated fault signals to design and improve the fault detection and machine condition monitoring techniques has been put to effective uses by Randall, Sawalhi and Endo [34, 35, 62, 66].

driven shaft (see Figure-3.3.1). The residual signals shown in (b1~b3) of Figures 3.3.2 and 3.3.3 were obtained by subtracting the simulated vibration of undamaged gearbox from the damaged one. Two identical simulation models, one with a gear tooth fault (TFC or spall) and the other undamaged, were run in parallel and the difference between the two model out puts were taken to separate the effect of the gear faults. The impact like effect of the gear

A comparison of the simulated signals shows that the magnitude of the TFC impulses is affected by the amount of torque applied to the gears, while the spall impulses are not. This response is consistent with the observation made in the static simulation of the gears in mesh. Careful observation of the residual signal also reveals that the fault information is not only buried in the dominant effect of gearmesh, but also somewhat distorted by the effect of transmission path from the gearmesh to the point where the signal was measured. The effect of transmission path appears in the residual signal As the transient "tail" effect convolved over the impulse due to the gear fault (see Equation-3.3.1 and Figure-3.3.4 for illustration).

tooth faults is seen in both the gears with a TFC and a spall.

840rpm T=15Nm 840rpm T=67Nm 840rpm T=120Nm

840rpm T=15Nm 840rpm T=67Nm 840rpm T=120Nm

Fig. 3.3.2. Simulated gearbox vibration signal with the effect of a TFC.

(a1) (b1)

(a2) (b2)

(a3) (b3)

(a1) (b1)

(a2) (b2)

(a3) (b3)

Fig. 3.3.3. Simulated gearbox vibration signal with the effect of a spall.

effective uses by Randall, Sawalhi and Endo [34, 35, 62, 66].

The residual signal of the TFC and the spall provide a useful means to understand the nature of diagnostic information of the faults. In machine condition monitoring signal processing techniques are often developed to detect and quantify the symptoms of a damage buried in a background noise. By being able to see the "clear" effect of a fault, the most effective signal processing technique can be applied to target and monitor the symptoms of the damage. The idea of using the simulated fault signals to design and improve the fault detection and machine condition monitoring techniques has been put to

$$y = (e + w + n) \* h \text{ (Note: \* represents convolution)}\tag{3.3.1}$$

Fig. 3.3.4. Vibration of a Gearbox. [62]

#### **3.4 Modelling a spall in a rolling element bearing**

This section discusses a several simple but an effective methods of modelling a spall in the rolling element bearing model introduced previously in section 2.5.1. The ideas behind each modelling approach was discussed by using the example of modelling a spall in the outer race of a bearing. The same method can be easily expanded in to modelling an inner race spall and ball faults. More detailed explanation on this topic is available from the work published by Sawalhi and Randall [29, 30, 31, 34, 35, 66, 67].

The simplest form of a spall model can be implemented to the REB model by assuming instantaneous contact loss between the bearing races and rollig an element(s) as it pass over the spall. So, in reference to the rolling element model described above, the presence of a spall of a depth (*Cd* ) over an angular distance of ( *<sup>d</sup>* ) can be modelled by using the fault switch *<sup>j</sup>* which defines the contact state of rolling elements over a defined angular position (*<sup>d</sup>* ). In effect, this mothod models the spall as a step function as shown in Figure-3.4.1 and further illustrated by Figure-3.4.2 (a), in which *<sup>j</sup>* is defined as follows:

$$\begin{aligned} \{ \mathcal{B}\_j = \begin{cases} 1, & \text{if } \phi\_d < \phi\_j < \phi\_d + \Delta\phi\_d \\ 0, & \text{otherwise} \end{cases} \end{aligned} \tag{3.4.1}$$

Fig. 3.4.1. Spall definition on the outer race [66]

Gearbox Simulation Models with Gears and Bearings Faults 49

from this model appears to agree with the experimental observation, however author recommends further validation on this modelling approach and more updates are expected

For more detailed explanations on the modelling of REB fault refer to the works published

The techniques for modelling the effect of gearbox faults: tooth fillet cracks, tooth face spalls and bearing spalls, were presented and discussed in this chapter. The main purpose of the damage modelling is to simulate the effect of the faults on the dynamics of a geared transmission system that can be used in improving the understanding of the diagnostic

The fault detection and diagnostic techniques based on vibration signal analysis are the ideal non-destructive machine health monitoring method, that can be applied in a minimally intrusive manner; i.e. by attaching an accelerometer on a gearbox casing. However, the dynamic interaction amongst the machine elements of a gearbox is often complex and the vibration signals measured from the gearbox is not easy to interpret. The diagnostic information that directly related to an emerging fault in a gear or a bearing is typically buried in the dominating signal components that are driven by the mechanisms of

Traditionally, the researchers worked on development of a signal processing technique for the gearbox diagnosis have embarked on their endeavours by making educated assumptions on the properties of the diagnostic information of the faults. These assumptions are often based on their careful observation of a measured vibration signals. However, the relevance of this approach is often somewhat limited by the simple fact that it's not easy to

It was demonstrated in this chapter how simulation models can be put to effective uses for studying the properties of the fault signals in greater details. A method of isolating the fault signals from the simulated gearbox signal mix was described in the section 3.3. The residual signals obtained from this process showed how the faults manifested in the resulting vibration signals in the "cleaned" state. The observation of the simulated residual signals has led to an improved understanding of the characteristics of impulses caused by the faults and the distorting effect of the transmission path (from the origin of the fault signal to the measurement location). The improved understanding of the fault signal obtained from the simulation studies led to the development of more effective signal processing techniques

The models of the gearbox faults presented in this work require further refinement. Some of the areas of future improvement aforementioned in the main body of the chapter include; improving the understanding of; the effect of plastic deformation in gear TFC, the effect of spall shapes and the effect of non-linear dynamic interaction of the gears and bearings. Improving the correlation between the simulated and the measured signals is a good way to demonstrate the understanding of the effect of faults in a geared transmission system. This

information that manifest in the vibration signal mix from a gearbox.

the transmission system themselves: For example, gear meshing signals.

observe the key details of the fault signals from the signal mix.

based on the findings.

**4. Conclusion** 

[34, 35, 62, 66].

by Sawalhi & Randall [29, 30, 31, 34, 35, 66, 67].

Fig. 3.4.2. Modified model of a spall based on a more realistic ball trajectory. [66]

The outer race spall is fixed in location between *d* and *<sup>d</sup>* + *<sup>d</sup>* . This normally occurs in the load zone. An inner race spall rotates at the same speed as the rotor, i.e. *<sup>d</sup>* = *c do dt* (*do* : initial starting location of the spall).

This model of the spall assumes that the rolling element will lose contact suddenly once it enters the spall region, and will regain contact instantly when exiting from that area (Figure-3.4.2 (a)). The abrupt change in the rolling element positions at the entry and exit of the spall results in very large impulsive forces in the system, which is not quite realistic. An modification on the previous model was introduced in [28] in which the depth of the fault (*Cd* ) was modelled as a function of ( *<sup>j</sup>* ), Figure-3.4.2 (b). The improvement on the model is to represent more realistic trajectory of the rolling element movement based on the relative size of the rolling element and the depth of the spall. Although, the profile of the trajectory appears much less abrupt than the earlier version; and apears to have only one position that may result in impulse, it still resulted in two impulses which does not agree with the experimental observation.

Careful observation of the interaction between the rolling element and spall leads to the trajectory is shown in Figure-3.4.3. The entry path of the rolling element has been represented as having a fixed radius of curvature (equal to that of the rolling element); entry of the rolling element in to the spall is therefore somewhat smoother. The smoother change in curvature at the entry would then represent a step in acceleration. On exiting the spall, the centre of the rolling element would have to change the direction suddenly, this representing a step change in velocity or an impulse in acceleration. This has been modelled as a sudden change (i.e. similar to the original model [28]). The resulting acceleration signal

Fig. 3.4.3. A correlated model of a spall based on experimental data.

from this model appears to agree with the experimental observation, however author recommends further validation on this modelling approach and more updates are expected based on the findings.

For more detailed explanations on the modelling of REB fault refer to the works published by Sawalhi & Randall [29, 30, 31, 34, 35, 66, 67].
