The Effect of Sensor Integration on the Load Carrying Capacity of Gears

Abstract

Classical machine elements have been around for centuries, even millennia. However, the current advancement in Structural Health Monitoring (SHM), together with Condition Monitoring (CM), requires that machine elements should be upgraded from a not-simple object to an intelligent object, able to provide information about its working conditions to its surroundings, especially its health. However, the integration of electronics in a mechanical component may lead to a reduction in its load capacity since the component may need to be modified in order to accommodate them. This paper describes a case study, where, differently from other cases present in the literature, sensor integration has been developed under the gear teeth of an actual case-hardened helical gear pair to be used within an actual gearbox. This article has two different purposes. On the one hand, it aims to investigate the effect that component-level SHM/CM has on the gear load carrying capacity. On the other hand, it also aims to be of inspiration to the reader who wants to undertake the challenges of designing a sensor-integrated gear.

1. Introduction

Classical machine elements have been around for centuries, even millennia. For instance, mechanical gears can be traced back to the world-famous Antikythera mechanism [1]; one of the very first founders of modern gear theory can be identified in Euler, who in 1781, developed the mathematical formulation of the involute that later became the basis of the modern gear [2]. A similar case is that of bearings, which were studied by Leonardo da Vinci in his friction investigation [3]. However, despite being known and used for centuries, the increasing day-to-day industrial demand for a safe, reliable and lightweight mechanical system requires continuous research as a means to provide industry with a proper solution. Within this context, as a means to increase the system reliability (or at least reduce the occurrence of failure), the idea behind Structural Health Monitoring (SHM) and Condition Monitoring (CM) plays an important role [4], especially for rotating machinery [5].

Focusing on gearboxes, the current state of the art is to monitor the whole system using additional sensor equipment which has to be attached to the outside (e.g., [6,7,8,9,10,11,12,13,14,15,16,17]) rather than on a single mechanical component, since sensors are not located directly on the component. However, the advancements of SHM and CM are now moving towards the idea of a smart mechanical element that, apart from performing the task it has been designed for, is able to communicate information to the outside world about its working conditions, especially its health.

Within this context, the SPP2305 German priority program has been established in order to evaluate the development of smart machine elements [18] within 10 projects focused on different machine elements. From a mechanical point of view, the basic requirement for all these projects is that all the electronics have to be located inside the component, without requiring additional space. On the other hand, from an electrical point of view, the components have to communicate with the outside world without cabling, implying that each electronic system has to be self-powered and with a wireless data connection. Examples of smart components being investigated and developed within the program are screw, feather key, radial seals and, obviously, gears. Indeed, this paper presents the first results of the SIZA (Sensor-Integrated Gear, in German Sensorintegrierendes Zahnrad) project, in which prototypes of smart gears will be developed.

Indeed, current studies are now directing the field of condition monitoring to the mechanical component. Furthermore, it is possible to find applications of measurement systems for the SHM/CM of single machine elements. One of the early examples is the work of Holm-Hansen, B. and Gao, R. X [19,20,21,22,23,24,25,26,27], where the feasibility of smart bearings with micro-sensors is discussed from the point of view of both the bearing load carrying capacity and SHM/CM. The concepts of a smart bearing have also been discussed recently by Schirra, T. et al. [28,29]. Other examples for sensor integration in mechanical components include feather key [30], fasteners [31,32], ball screw (e.g., [33,34]) and die (seen as part of the overall stamping system) [35].

As a matter of fact, the idea of a smart gear has recently been proven to be a promising and feasible idea by Peters, J. [36,37,38] and by Sridhar, V. [39]. However, their studies were based on a spur gear geometry in an “open” gearbox (i.e., without housing), and the sensor board was directly attached to the gear body. Therefore, in order to overcome the problem of sensor integration, Binder, M. [40], focusing more on the sensor integration itself, proposed to use additive technologies as a means to embed sensors and electronics inside the gear body. Nevertheless, despite being advantageous for the sensor integration per se, additive-manufactured sensor-integrated gears present some technological difficulties. Firstly, very few data on gear load carrying capacity that can be used for additive-manufactured gear design are available in the literature (e.g., [41,42,43]). Secondly, the idea of embedding the sensor in the initial manufacturing phase implies that all electronic components must survive the mandatory heat treatment temperatures for high performance gears. As an alternative, the gear would not be heat treated, adopting a material with a low-load capacity with a detrimental effect on the gearbox dimensions.

Within this context, the implementation of SHM/CM within gears made of classical materials (e.g., case hardened steel) seems to be a rational idea. Indeed, those kinds of materials are well known for both their technological and load capacity points of view. Moreover, as the electronics will be fitted only after manufacturing, the aforementioned temperature problem is no longer an issue. However, this idea also implies that the gear body must undergo important modifications in order to accommodate all the electronics, and the effect that such modification will have on the gear load carrying capacity is unknown.

Here, in order to discuss the implementation of in-situ gear SHM/CM and the effect that sensor integration has on gears, an actual gearbox is adapted as a reference case. The aforementioned gearbox, and the related case-hardened helical gears, have already been developed and used in previous studies (e.g., [44,45,46,47,48]). Figure 1 shows the modification that the original gear geometry will undergo in order to accommodate all sensor boards inside the gear, underneath the teeth. Indeed, the gear body will be modified by creating the various ports adopting classical machine tools (e.g., lathe and milling machine). However, as the gear body is strongly changed by the sensor-integration requirements, the stress acting on the component also varies. Therefore, the load carrying capacity of the component is affected as well. Nevertheless, the said variation is not discussed in the literature.

Therefore, this article has two different purposes. On the one hand, it aims to investigate the effect that component-level SHM/CM will have on the gear load carrying capacity. On the other hand, thanks to the critical discussion about the design choices, as well as the adopted evaluation tools, it also aims to be of inspiration for the reader who wants to undertake the challenges of designing a sensor-integrated gear.

2. Definition of the Components

The gearbox that has been chosen in order to implement the component-level SHM/CM system is part of a back-to-back test rig with a center distance of 112.5 mm, whose typical maximum working conditions are 1000 [Nm] at 3000 [rpm]. The working principle of the 112.5 test rig is completely similar to the one of the more famous 91.5 test rig (shown in Figure 2), which is a part of the ISO 14635 [49]. The test rig is composed of two gearboxes with identical gear ratios; one of which is over-dimensioned (i.e., the slave gearbox) in order to localize the failure on the other one (i.e., the test gearbox). Interested readers are directed to ISO 14635-1 [49], where power-recirculating gearboxes are described. The test gearbox features a 22/24 gear pair, mounted on the shaft using a tapered interference fit.

Table 1. Main geometrical data and calculated safety factors acc. to ISO 6336 [51,52,53,54,55].

Name	Symbol	Pinion	Wheel	U.M.
Number of teeth	z	22	24	-
Normal modulus	$m_n$	4.25		Mm
Normal pressure angle	${\alpha }_n$	20		°
Helix angle	$\beta$	29		°
Profile shift coefficient	$X$	0.100	0.077	-
Centre distance	$a$	112.5		Mm
Tip diameter	$d_a$	117.70	127.30	Mm
Addendum factor of the basic rack	$h_{ap}^{\ast }$	1.55		-
Tooth root radius factor	${\rho }_{fP}^{\ast }$	0.25		-
Facewidth	$b$	27.6		Mm
Torque	$T$	1000		Nm
Rotational velocity	$n$	3000		Rpm
Working hours	$H$	2000		H
Contact stress	${\sigma }_H$	1528	1527	MPa
Pitting stress limit	${\sigma }_{HG}$	1536		MPa
Safety factor, pitting	$S_H$	1.01	1.01	-
Nominal tooth root stress	${\sigma }_F$	432	433	MPa
Allowable stress number for bending	${\sigma }_{FE}$	950		MPa
Safety factor, tooth root bending fatigue	$S_F$	2.10	2.09	-

lists the gear pair macro geometry data. The gears are made of 18CrNiMo7-6 and have been subject to classical industrial-level case hardening. More detail about the gears can be found in [44,45,46,47,48]).

Table 1 reports the safety factor for tooth root bending fatigue and macro pitting calculated according to the ISO 6336 framework (i.e., [51,52,53,54,55]). As the adopted geometry was part of a research project regarding contact fatigue, the gear geometry has been defined in order to have a high level of tooth root and flank safety, while respecting the test rig constraints (e.g., the center distance of 112.5 mm).

Moving into the electronic aspect, the implemented SHM/CM system features several sensors that have been defined based on the current state of the art, as well as on the author’s experience. Firstly, two accelerometers are adopted as a means to detect gear torsional vibration. Indeed, this peculiar system has been proven to be a valid methodology for evaluating gear dynamic behavior [56,57]. Secondly, a thermocouple is included as tooth temperature (and its sudden variation) can be related to tribological damages (e.g., [58,59]); in order to enhance this aspect, the sensor is placed as close as possible to the contact area. Finally, a Hall-effect sensor is included as a means to obtain information about the rotational speed as well as a microphone, since it can be a cost-effective alternative to an accelerometer. The sensor system is managed by a microcontroller which elaborates sensor data and then provides information about the health status of the component. During the project, the connection of the boards with the outside world will evolve gradually from a cabled connection (via slip ring) to a wireless connection. A similar approach will be followed for the power source of the sensor board, moving from an external board to an integrated board (batteries using an energy-harvesting technique). The selected sensors and microcontroller are deliberately standard-grade electronic components. All aspects of the measurement technique, SHM/CM software, data and energy management is the subject of current research and future publications.

3. Definition of the Gear Body Shape

The development of a smart machine element poses the problem of the conflict of interest between the load carrying capacity of the components and the SHM/CM system reliability. The aim is to have the best possible SHM/CM system without significantly affecting component strength.

In the case of the gears, from the SHM/CM point of view, one aim should be to fit as many sensors as possible. Then, because of the shorter signal path and the related lower signal-to-noise ratio, the sensors should be placed as close as possible to the gear meshing. However, this would lead to a significantly reduced load carrying capacity. Furthermore, the working conditions would be very demanding for the electronic components due to high temperatures and vibrations associated with that region, posing the problem of electrical component durability.

Another challenge when implementing electronics is the board dimensions. Generally, the sensor boards should be as small as possible. Nevertheless, electronics also imply some limitations from a manufacturing standpoint. Indeed, the sensor boards include not only the sensors, but all the necessary internal connections between electronic components. Therefore, their minimum dimensions are limited.

Therefore, the gear body has been modified in order to have the smallest boards, as close as possible to the gear contact, without unreasonably affecting the load carrying capacity.

A first attempt to define a new gear body geometry has been to adopt the annular board, as proposed in [36,37,38,39]. Figure 3 shows two geometries which have been designed considering an annular sensor with a 10 mm width. The geometry in Figure 3a also includes 5 mm of radial play in order to leave gaps for the spacer.

However, despite being relatively simple, this presents several problems. Firstly, the tapered interference fit is not symmetrical with respect to the gear. If the power flow enters from behind (in respect to Figure 3 view), the problem of fretting may occur at the shaft/hub connection due to the unfavorable power flow [60]. Secondly, half of the teeth are unsupported, with detrimental effects on the tooth root bending fatigue resistance (e.g., [52,61,62,63,64,65,66]). Finally, this geometry also implies difficulties when transmitting the axial load generated by the helical gear meshing.

Therefore, taking inspiration from the concept of lightweight gears (e.g., [67,68,69]), an alternative sensor configuration has been studied. This configuration is based on the idea of having several boards placed in different positions and connected by cables, instead of having one single board. The final geometry is shown in Figure 1. The gear body, as well as the related port dimensions, have been defined after several iterations in which both boards and port locations have been defined in order to reduce their effect on the gear carrying capacity. One challenge was the imbalance of the resulting gear body due to the presence of different boards, as well as the presence of some internal walls within the ports. However, such unbalancing, if not prevented, may lead to a higher load acting on the gear than the nominal load. This aspect will be addressed by adopting the industrial-level balancing technique after the gear and boards are manufactured. Figure 1 geometry also features threaded holes which are necessary for the connection with other experimental equipment.

Nevertheless, the load carrying capacity of this new geometry must also be evaluated. Considering this specific case, the new geometry will be assessed by looking at the tapered interference fit and the tooth root bending strength.

4. Evaluation of the Sensor Port Effect on the Interference Fit

Starting from the interference in the contact area, several analytical calculation methods have been proposed as means to evaluate the phenomena occurring within interference fits (e.g., [60,70,71]). Those calculation methods are based on the evaluation of the radial stress generated in the shaft-hub region for a full body gear, that is not the case with the proposed gear body. Furthermore, as observed in experimental campaigns on compound gears [72], variation in the gear body stiffness can lead to a reduction in the contact pressure, leading to the presence of undesired relative movement. As the proposed gear body geometry features important modifications, considering that numerical methods have proven their validity in evaluating the shaft-hub connection with an interference fit specifically (e.g., [70,73,74]), FE models have been adopted as a means to estimate the component behavior. All FE simulations discussed here have been performed using ABAQUS 2020.

For all the evaluated geometries, two cases with different shaft/hub interference have been simulated, aiming to evaluate two different aspects. On the one hand, the models that adopt the minimum interference estimate the contact pressure $p$ acting on the contact area $A$. This quantity is related to the maximum torque that can be achieved (e.g., [60,70,71]). On the other hand, the models with the maximum interference allow the estimation of the overall stress acting on the component.

Interference fit simulations were carried out adopting quadratic hexahedral, establishing an interference fit between the contact surfaces. Whenever possible, symmetricity has been adopted in order to reduce the computational time, without including the teeth, limiting the external diameter to the tooth root. Figure 4 illustrates the principle behind such simulations, with a particular focus on the mesh. Both the baseline case and the case with the sensor port have been simulated.

Concerning the specific case, the interference fit must be able to transmit, by means of friction, both the axial load generated by the meshing of helical gears and the torque itself. In the contact area, they are transferred by means of two different frictional stresses, one in the circumferential (i.e., torque) and one in the axial direction (i.e., the normal force) [71],

$$\{\begin{array}{c}{\tau }_{\theta }=\frac{T}{\pi dl\frac{d}{2}}\\ {\tau }_z=\frac{F_a}{\pi dl}=\frac{T}{m_{n}z/2}\frac{\tan \left({\alpha }_n\right)}{\cos \beta }\frac{1}{\pi dl}\end{array}$$

(1)

where ${\tau }_{\theta }$ is the circumferential radial shear related to the applied torque $T$, while ${\tau }_z$ is the axial shear due to the axial force $F_a$. $d$ and $l$ are the hub diameter (average) and length, respectively. This approach averages the acting load over the total contact surface, neglecting any local effects due to the acting concentrated load in the gear mesh. Nevertheless, considering the general proportionality of friction force and total acting force, it seems reasonable for the intended application. As ${\tau }_{\theta }$ and ${\tau }_z$ are in perpendicular directions, they are summed by means of Pythagoras theorem in order to find the total friction shear stress ${\tau }_{tot}$.

In order to withstand the applied forces/moments, it is necessary that the interference fit assures a contact pressure $p_{min}$, which is related to ${\tau }_{tot}$ by means of the friction coefficient $f$:

$$p_{min}=\frac{{\tau }_{tot}}{f}$$

(2)

In order to be conservative, $f$ equal to 0.1 has been chosen.

Here, as the modified case presents a nonuniform contact pressure, $p_{min}$ will be then compared with $p_{avg,FEM}$, that is the average contact pressure estimated by means of FEM. Therefore, a safety factor for the interference fit $S_{IF}$ is defined accordingly.

Table 2. Input data and calculated safety factors for the interference fit.

	$d$	$l$	T	${\tau }_{\theta }$	${\tau }_z$	$p_{min}$	$p_{avg,FEM}$	$S_{IF}$
	mm	mm	Nm	MPa	MPa	MPa	MPa	-
Baseline	50	55	1000	4.63	1.372	48.3	220	4.56
Modified							186	3.85

summarizes the results. This implies that the stiffness modification generated by the sensor integration requirements is reducing the safety factor maximum transmissible torque by approximately 16%. Considering that the maximum transmittable torque (not calculated here) is linearly related to the contact pressure in the interference fit region, the maximum transmissible torque in the shaft/hub region is reduced by 16%.

Figure 5 and Figure 6 show the von Mises stress distribution for the case with the sensor port. Neglecting the edges, where the high constant stress is due to the fact that chamfers have not been modelled, it is possible to notice that the most stressed area is the port. This peculiar situation can be related to how the component stiffness is distributed. Indeed, the more rigid parts (i.e., the one without the sensor port) try to modify their position, moving away from their undeformed position. This movement is impeded by gear sections with the port that act as a spring, keeping the gear body joined. Therefore, this section results in being highly stressed. As can be seen in Figure 6, the highest stress in that location is around 580 [MPa]. It is worth mentioning that while evaluating the feasibility of the current geometry, this aspect resulted to be the stricter one.

Furthermore, it is possible to notice that the external diameter region corresponding to the tooth root is subject to non-negligible stress. This is also confirmed by the literature, where the effect of the interference fit on the tooth root stress is investigated (e.g., [73,74,75,76]).

5. Evaluation of the Sensor Port Effect on the Tooth Root Stress

A first attempt to estimate the effect that sensor integration will have on the tooth load carrying capacity can be carried out following the classical gear standards, such as ISO 6336-1 [50] and ISO 6336-3 [52]. On the one hand, by means of the gear blank factor $C_R$, which modifies the gear stiffness, ISO 6336-1 [50] allows designers to include the effect of the gear body shape within the classical gear assessment methods. On the other hand, the effect of the rim thickness on the tooth root stress is also included in ISO 6336-3 [52] by means of the rim thickness factor $Y_B$. It is worth mentioning that according to ISO 6336-2 [51], the rim thickness has no direct effect on the contact stress.

However, [50,52] consider a simple gear geometry, in which the whole gear body presents a thin rim. That is, while it seems reasonable to evaluate the possibility of a purely standards-based assessment for a gear body with a shape similar to those shown in Figure 3, the same cannot be applied to the proposed geometry because the result outcome is quite different to the standard. Hence, the necessity of adopting a generalized evaluation method. Following the suggestion of [61,62,63,64,65,66], numerical models have been implemented.

Here, static meshing gear simulation has been evaluated by following the procedure described in previous papers [77,78], where the simulation is carried out in separate rotational steps. Two separate reference points, located at the center of each gear, are connected to the gear hub by a multi-point constraint. Tooth flank contact is defined by surface-to-surface contact. The pinion is fixed while the wheel is free to rotate. When torque is applied on the wheel, it rotates by a small quantity in order to establish the loaded contact between teeth flanks, and the load is then transmitted from one gear to the other. Figure 7 summarizes the FE procedure. All the FE simulations discussed here have been performed using ABAQUS 2020.

Several rotational steps are then simulated thanks to an external phyton code which rotates each gear on its axis by a rotational step that depends on the wheel/pinion number of teeth and the number of the simulation step. The comparison was been made by looking at the meshing position that results in a higher tooth root stress. Following the indication of Conrado, E. and Davoli, P. [65], the maximum principal component of the stress tensor is adopted as a reference stress.

Here, it is worth mentioning that the FE tooth root stress result and the result estimated according to the ISO 6336 series framework [50,51,52,53] cannot be directly compared. Indeed, even though they are both estimating a stress in the tooth root region, the estimated stress values are calculated with different assumptions and methodologies, and cannot be directly compared. The stress number predicted by ISO 6336-3 [52] is the maximum local principal stress, calculated by considering only the tangential force and evaluated at 30° tangents to the contact of the root fillets, while the stress predicted by FEM is a stress tensor, defined all over the model, which considers all the forces exchanged by the teeth.

In order to reduce the computational time, the calculation procedure was based on two models. The first, based on tetrahedral elements, estimated the load sharing between teeth. Then, a second model based on hexahedral elements exploited the sub-modeling technique in order to estimate the actual tooth root stress.

Both the original geometry, the modified geometry (i.e., with the port), as well as the geometry with the hole for the thermocouple were simulated. However, due to the complex geometry, it was not possible to proceed with the sub-modelling for the case with the hole for the thermocouple. Figure 8 and Figure 9 show an example of results for the main model with port and its sub-model, respectively. Figure 10 highlights the contact area estimated for the case in Figure 9. The calculated contact area was in agreement with the general assumption that the contact line ending at the top edge of a helical gear is relevant for maximum tooth root stress levels [79]. The calculated equivalent stresses are summarized in

Table 3. Maximum tooth root stress estimated by the FE models.

		${\sigma }_I$ [MPa]	${\sigma }_{VM}$ [MPa]
Original Geometry	Main model	396	337
	Sub model	336	290
Modified Geometry	Main model	407	353
	Sub model	349	302
Modified Geometry with thermocouple hole	Main model	440	380

.

Nevertheless, in order to properly estimate the effect of sensor integration within the tooth root, it is necessary to consider that the interference fit is loading the root region, as shown in Figure 5. However, while the meshing stress can be schematized as a pulsating stress, the same cannot be done for the interference fit because it is a constant stress. Therefore, a simple arithmetic summation of the stress is not correct from the point of fatigue, but considering the multiaxial nature of the stress, high-cycle multiaxial fatigue criteria were adopted. Among the numerous criteria available in the literature (e.g., [80]), considering the simple load case, the Sines criteria was adopted.

Hence, the stress tensors estimated by both the two FE models were extracted and combined. The equivalent stress was compared for the cases as a means to estimate the effect of the gear body modification. The result of such comparison was the ratio between the various representative stress levels. Both cases with and without the interference fit were evaluated.

Table 4. Estimated effect of the sensor integration in the tooth root bending stress.

		Without Interference Fit	With Interference Fit
Modified Geometry	Main model	1.05	1.12
	Sub model	1.04	1.12
Modified Geometry with thermocouple hole	Main model	1.13	1.22

summarizes the results. As can be seen, the effect of the interference fit was predominant. Therefore, its neglection would have led to an underestimation of the sensor port effect. In any case, if the ratios Table 4 were considered as factors multiplying the tooth root stress, thanks to proper gear design, $S_F$ always remains higher than 1.5.

6. Results and Conclusions

Within the context of the development of smart mechanical elements, the effect of sensor integration within a gear has been discussed. The focus of this article was an actual case-hardened helical gear pair to be used within an actual gearbox. This article shows that sensor integration in an industrial-grade gear is a feasible idea, however, the designer should expect a reduction in the gear load carrying capacity.

After introducing the proposed gear body modifications and their rationale, a procedure for estimating the related load carrying capacity reductions was presented. Due to the limitations of the classical analytical approach, a FE procedure was adopted. The performed simulations aimed to evaluate the influence that the addition of the sensor ports had on the load carrying capacity of the gear.

Interference fit simulations were performed in order to estimate the effect of the sensor port on the tapered interference fit. Meshing gear simulations were also conducted as a means to evaluate the impact of the sensor port on tooth root stress.

The effect of the geometry modification was estimated by comparing the results of both the modified and the unmodified gear pair, adopting the model with the same parameters (e.g., element type and dimensions). Furthermore, in order to consider the interaction of gear meshing and interference fit, the Sines criteria was adopted as a means to determine the effect of the sensor integration on the tooth root from the point of view of fatigue. The geometry proposed here is indeed the product of all the aforementioned considerations. During its development, thanks to robust gear design (i.e., see Table 1 safety factor), the real bottle neck was found to be in the interference fit rather than in the tooth root bending strength.

As a general result, it holds that the proper design of a sensor-integrated gear needs to trade part of its load carrying capacity for the additional intelligence in the form of space for sensors.

Only static behavior has been considered. Since gear stiffness is modified by the sensor integration and load sharing between the teeth, the Noise Vibration Harshness (NVH) behavior will also subsequently change. This aspect will be a part of future numerical and experimental analysis.