Finite Element Analysis of Stress Relief in Teeth of Spur Gear by Varying Key Shaft’s Size and Location

Spur gears are essential in mechanical power transmission systems due to their simplicity, efficiency, and reliability. They are crucial in various industrial applications, including automotive transmissions and heavy machinery. However, the distribution of stress in their teeth during operation can affect their performance and lifespan. If stresses are high in focal points, there is a higher failure rate of the system, which consequently lowers reliability and efficiency. It is extremely critical for the gear design optimization to be done in such manner that stress concentrations are minimized. First, a key shaft can be used to transfer torque from the gear to the axle. The dimensions and position of the keyway can in general have a large effect on the stress distribution inside the gear teeth [1].

One effective way for preventing the problem of energy loss is by using key shafts for transmitting torque from the gear to the axle. It is found that shape, size and position of the key way is quite critical to affect the stress distribution within the gear teeth. Besides such material improvements and surface treatments as carburization and shot peening, the provision of stress-relieving features in gear design can reduce stress concentrations still further and increase gear durability.

This is where Finite Element Analysis (FEA) has become an important tool for studying stresses and defining the best designs. Computational tools allow for experimentations with stress distribution patterns and with modifications in the key shaft’s dimensions and positions for optimal stress distribution. This research utilizes FEA to determine stress distributions in spur gear teeth with the objective of enhancing key shaft parameters for better reliability.

To avoid fatigue failure, it is necessary to restrict the stresses and select the material with the maximum stress concentration. The design improvements can include enhancing material characteristics, treatment to heat hardness and various surface finishing options as well as the addition of stress relief features to the stress zone. Some simulation packages can be employed for stress checking; however, simulations cannot provide exact results. Through computational stress analysis methods, researchers can also conduct research on stresses. In summary, the principle of minimizing stresses at the maximum stress-concentrated area is the key factor in the extension of the service life and the reliability of spur gears installation. In the review paper the effects of various parameters on the bending stress at the critical section of an asymmetric spur gear was discussed. Bending stress is an essential factor in gear design, which in turn reduces the load capacity to be increased, extended gear life, cost savings, reliability, noise and vibration reductions and less maintenance charges. The drive side pressure angle of the best magnitude is mainly affected by factors such as contact ratio, top land tip thickness, pressure angle, asymmetry factor, number of teeth, interference, undercut, center distance, gear ratio, critical section thickness, gearing profile shift, module, bending stress at critical section, optimal fillet radius, and balance of the gear [2].

Low-quality gears tend to break under high stress, but a reduction in the root tensile stress may be an answer to increase their life. The design of gears has been advanced with the use of materials, heat treatments, and the modifications of the curvature of root fillets. A work addressed the issue of redistribution of stress by the addition of stress-relieving features in the stressed section, which in turn resulted in decrease of root fillet stress in spur gear. Circular elasticity features had been employed and the improvement of the product was obtained. The FE model taken in consideration was made of 3 discrete elements and different geometry parameters were taken in to account with varied dimensions [3].

From all the Norms and requirement of American Gear Manufacturing Association (AGMA) the gear design requires such that tip radius and tooth width are key specification players. By these standards, the stressed root and the mating zones are determined by total strain from the gear teeth. The Finite-Element Model was created using either HYPERMESH or ANSYS, based on the Lewis Equation and AGMA Standards to compute stresses and optimize results [4]. The research paper examines the stress levels of mating teeth of spur gears by means of FEA and theoretical values of the Hertzian equation. When steel and the mould cast iron was used as the materials The gears inputted to ANSYS DesignModeler were based on the design and the assembling of ANSYS DesignModeler. The Finite Element Method (FEM) was a stress analysis tool, and the deformation patterns were based on the results. The result of the minimum contact stresses from Hertz equation and FEA is almost the same. The deformation patterns from steel and gray cast iron gears are also indifferent [5]. Despite advancements in gear design, such as using better materials, surface hardening, carburization, and shot penning, these techniques do not guarantee interchangeability. A work explored stress concentration methods by inserting stress-relieving features at the most stressed area to reduce root fillet or bending stress in spur gear. The study employs circular and elliptical stress-relieving holes, achieving better results than circular holes used in previous research. The optimum size and location of stress relief features for spur gear were proposed, reducing fatigue failure in gears [6].

Research analyzed the stress and reduction of a Spur gear, a common power transmission device, to reduce fatigue failure. The study used a finite element model with three teeth and introduces stress concentration reducing holes of different sizes. Static analysis using ANSYS 10.0 revealed that aero-fin-shaped holes, introduced along the stress flow direction, yielded better results. These holes redistributing force lines of force, regulating stress flow and reducing displacement. The best results were obtained by introducing aerofin holes at (38.7653, 65.4083, and 0) with a 0.6 scaling factor, resulting in a stress reduction of 50.23% and displacement reduction of 45.34% [7]. Continuous meshing of gears often leads to gear failure. Proper geometrical design is crucial to avoid this issue. In spur gears, the tooth surface of the gears meets the other gear. The study focused on stress reduction between gears using stress reduction analysis using geometrical features [8]. A paper on the simulation process of the gear synchronization is presented using the flexible quasi-static and dynamic FEA models to compute contact principal and shear stresses. The 3D full-sized spur gears are simulated in the boundary conditions of different constraints. A static analysis of the results showed the concentration of the highest level of stress at the points of the tooth contact and under the contacting surface, which was confirmed by a dynamic analysis and indicated the highest stress level at different gear engagement points along the line of action. The results agreed, and the use of the new simulation model was discussed [9].

Advanced computational models were increasingly crucial for powertrain design, enabling smart structural design that addresses seemingly incompatible goals. Modelling plays a crucial role in enhancing the competence of inverse engineering solutions in various aspects of powertrain design. By incorporating non-dimensional analytical formulation and multi-scale modelling, powertrains can be designed smarter, making them compact and low-vibration. High-precision modelling of gear meshing is essential for compact gear drives, minimizing clearance between gear teeth and ensuring robustness. Nondimensionalisation methodology reduces the number of independent parameters, quantifying the influence of design parameters on interference risk and gear mesh compliance [10]. This essay focuses on spur gears’ service life in industry; with an aim of implementing creative approaches towards gear failure, which results from pitting of the land surfaces and tooth breakage. The methods include theoretical calculations, FEA, hardness testing, and the use of a right material. In this article, the author discusses the mechanics of C45 and 19MnCr5 materials, comparing their fatigue strength, tensile strength, and yield point, which indicate superiority of the latter. The equipment made by 19MnCr5 reports headings and runs in a reasonable timeline for stipulated service. Research that can be done next will concentrate on stresses distribution variance, hidden tensions, fluids compatible with the leg, and different values of hardness at the rim and tooth of the gear [11].

A study presented a modeling for investigation of the bending and contact stresses of the involute teeth of a meshing spur gear in meshing. It carries out the ANSYS Workbench 16.2 software calculations to simulate stress levels and deformations. This analysis shows 2D prediction is more accurate than 3D prediction. The recommended is that engineers be careful when selecting the contact conditions. The viscosity of the material noticeably determines the point pressure. The major reason, which induces inaccuracy in bending and contact stress, is that the analysis software is not suited for determining such stresses [12]. A new study performed an investigation of the FEA of the spur gear pairs by ANSYS software. In the first instance, the design of spur gear is to be done and it will include the study of possible weight reduction and stress distribution between both cast steel and composite materials. The second example was the use of static analysis for the determination of the deformation and Von-Mises effects of various materials such as steel, cast iron, aluminum, and Epoxy E Glass UD [13]. Tooth breakage that is related to gearing is a result of bending and contact stress of the tooth. A work was focused on the FEM and numerical investigation of the stresses that arise from the tooth during a gear meshing for the 15Ni2Cr1Mo28 material that is made of steel. The model takes into account the index like face width and module. Models of different gear modules were made by software and ANSYS simulations were conducted for an estimate of bending stress. According to the Lewis calculation, the stress in bending deformation was determined [14].

Another advantage of Additive manufacturing technology is the capability to design the machine parts with smaller mass but still with high stiffness and weight capacity. A study aimed at comparing various infill types and densities for spur gear teeth and ascertaining which one can provide maximum rigidity and work load capacity. The study used numerical FEM analysis and proposes two new infill structures: triangular infill with a total of 5 different densities and another infill which is a 2D cantilever designed using topology optimization. Stress analysis, displacement, and bending stiffness of the gear teeth in full body and shell bodies are carried out for the gear teeth of full body and shell bodies [15]. Research has investigated the bending stress on a stone crusher machine's spur gear using a Lewis equation theory and FEA approach. It is iron/carbon steel C15 that is substituted by C45 steel/CI 30, which is more mechanical robust. The reality demonstrates that C45/CI30 is capable to exert a lower degree of bending stress compared to the currently used material. The module parameter is also made variable, and the bending stress drops by the enhanced module. The future work will look into how the module variation and shape of gear shape change the bending stress using FEA in spur gear meshing [16].

An experimental scrutinized the effect of rim thickness, profile altering, module and fit tolerance on the bending stress at the gear's foundation. Finite element models were in line with the results provided by analytical solutions. Conclusions have revealed that the main feature for the tangential stress at 12 o'clock position of the gear is a rim thickness. The shifting of a positive profile proves to be less stressful with a larger diameter to be maintained for pitch. The lower class can balance out stress through thin ring thickness modification [17]. Another work analyzed the major structures of the mass reliefs on spur gears. The system with a pinion and gear was engineered as a spur and fillet while the tooth design was varied with different relief shape and thicknesses. The FEA was performed on those geometries of the investigation, and the stresses of the tooth were compared with the integral gear. Showing these results clearly stated that these stresses on the tooth were reduced in some cases and some considerable stresses may occur in the core instead of the tooth. Another important attribute of core thickness is its influence upon the regional stress [18]. The goal of other study was to drop the mass of spur gear while simultaneously conserving its useful properties. It used ANSYS software to design, model, and simulate a three-dimensional spur gear using five materials: steel alloy, copper, and composite material that consists of 50% carbon fibers in the epoxy resin matrix with grapheme reinforced acetal and glass-filled polyamide. There are specially developed finite element programs implemented in ANSYS 14.0, which are used to estimate the stresses in the gear teeth. The result would be a notable difference when compared to Hertz analysis that was conducted using 50% carbon fibers reinforced in epoxy resin matrix with the and value reducing to 152.13 MPa in FEM. The study therefore recommends 50% carbon fibers reinforced in epoxy resin matrix as the best material for spur gear fabrication since it is strong and lightweight [19].

A research paper examined the characteristics of helical gears, a widely used type in transmission mechanics, and their ability to withstand transmission operations and rigidity. The study uses three main variables: pressure angle, helix deflection angle, and module number. The study also considers pressure angle, which affects wind turbine gear dimensions, diameter, stiffness, and tensile strength. The study focused on helical gears, as their angles increase contact area between gears. The deformation value is 4.26×10-6 m when the helix angle is 20 degrees [20]. A research paper investigated the impact of fatigue on composite materials using carbon fibers in heating and cooling technologies. It examined the bearing capacity of a stress test sample with carbon fibers added at different angles. The results showed that the best arrangement of carbon fibers was a triple layer with 45, 0, 0, with the lowest fatigue life cycle of 2349 cycles. This case reached a stress of 6.1×108 Pa compared to other cases studied [21]. Involute spur gears, which are used for power transmission in different industries, are prone to high stresses because of the number mismatch between the numbers of the teeth of the pinion and the gear. Stresses of gear-tooth contact, such as pitting, can be lowered through design modifications, such as increasing the drive side pressure angle and profile shifting factor, changing both contact ratio and center distance, and changing tooth thickness, which are used to reduce the stress of gear-tooth contact. Sophisticated research carried out using CATIA software, and then the FEA was performed to obtain the tooth thickness values that are equal to the root stresses of the gears of different tooth numbers [22].

A study examined the impact of helix angle on the reaction force and evolution of helical gears in mechanical engineering applications. 30-degree helix angle is the best option since it minimizes the stress effect on the shaft. The smallest helix angle of 5 degrees caused gear 3 to be displaced the furthest along the x-axis when it ran at 590 radiant/s and it reached 0.15 micrometers at most. The minimum error measured was at a 5-degree angle with maximum reaction force of 1080 N. This study also determined the force applied to the shaft and its temporal variation, which could be used to describe the dynamic stress of high-speed helical gears exploration [23]. A study investigated the impact of sample thickness on crack growth and fatigue in aluminum alloys 2024 and 7085. It compares samples with different thicknesses and alloy types, adjusting temperature and adding an additional processor to the Ansys program. Results showed maximum deformation at 5mm thickness and aluminum alloy 7085, with the highest deformation at 0.66 mm. Aluminum alloy 7085 has a crack growth of 7.5 mm during 2777 cycles, while alloy 2024 has the same growth during 32784 cycles. Comparing the mechanical properties of different alloys, helps understand their differences [24].

This study uses FEA to analyze stress distribution in spur gear teeth influenced by key shaft size, number, and locations. Parametric analysis aims to understand how changes in key shaft parameters affect stress levels and identify design configurations to minimize stress concentrations. The research aims to optimize spur gear designs, improving performance, reliability, and longevity in mechanical power transmission systems. The findings may also have implications for other gear types and applications requiring stress relief in gear teeth.

A finite element model with of teeth with (m = 2 and m = 7) is considered for analysis. A point load of 4000 N is applied at the pitch diameter then meshed with two-dimensional element (Plane182). The following steps are used for plotting the geometry of the gear tooth:

Divide the pitch circle, draw the root circle and draw the outside circle, draw a vertical line from the center of the circles to a point outside the circles. Illustrate the pressure line, the pressure line is drawn through the pitch point having an angle of 20 degrees from a line tangent to the pitch circle’s top, and then draw a line that divides the circular thickness angle. This line may either intersect or extend beyond the outside circle and then, locating the tooth form circle, draw circle (A) whose radius is 1/8 of the pitch diameter, with the pitch point as the center. Take another circle with the same radius as the first and the center at the vertex of circle (A) and the base one, this circle will build the top right curve of the gear tooth. Eliminate circle (A) and crop it to precisely the tooth shape, an arc that starts in pitch point and ends on the outside circle. Erase the initial round circle after you have already shaped the tooth form. Then, erase all the lines except for the circular thickness bisector, the tooth lines, and the initial round circle. Duplicate the form of molar or premolar with the central elongation line. Figure 1 is shown the steps above.

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Figure 1. Steps of drawing gear tooth (a) Initial construction, (b) Layout design, (c) Tooth profile formation, (d) Final tooth design

After that, the gear is meshed by element Plane182, the boundary condition is applied using pilot node in the gear center, which contact the shaft circle. Key is contact with the gear using Element targe169 and conta172 with MPC algorithm. Figure 2 illustrated the mesh and boundary condition of the gear [12].

The material properties used in the FE model are:

Materials: Steel (carbon steels C45). Standard Material Properties: Young's Modulus (E): 200 GPa. Poisson’s Ratio (ν): 0.28. Density (ρ): 7850 kg/m³. Yield Strength: (250–600 MPa). Ultimate Tensile Strength (UTS): (500–900 MPa).

Meshing process is essentially an important phase in FEA model since it disjoints the geometry into small elements to mimic the behavior of the analyzed structural entity. Mesh density plays crucial role in attaining a high-quality representation stress gradient, while element type is very crucial for the successful simulation. Solids elements are deployed for modeling gear teeth and key shafts contact elements being utilized for interaction points. Mesh quality is a decisive factor for attaining realistic results, and for that purpose well-formed elements are needed and for the distortion to be at the minimum. Boundary conditions serve as a connection between the simulated model and its actual environment. They are important for the accurate replication of stress distributions, so should be appropriately encoded to represent real world restrictions. The different visualisation tools of FEA software can help engineers look over the screens and detect any issues or mesh irregularities, which need to be taken care of before the analysis, can be carried out. Possibility of reproducing the numerical results is provided by the documentation as well as very important for validation of FEA. The documentation is an integral part of the analysis framework giving a full spectrum of the analysis setup. The teeth in the torque mesh, as in the spur gear, represent an important role for the overall accuracy of the results and the reliability of FEA results. This, in turn, helps engineers make better design choices and put more effort into optimization efforts, see Figure 2.

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Figure 2. Mesh of the spur gear tooth with keys