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Theoretical background of gear design
Gears are mainly used to transmit rotary motion between two shafts. Among different means of mechanical power transmission, gears are most rugged, durable and efficient. Their power transmission efficiency is as high as 98 percent [14]. In addition, as expected the manufacturing cost is also high and increases with the required precision level, which combines high speed, heavy loads and less noise. Incorrect modelling will result to power loss and ultimately damage the components. Therefore, before starting to design it is important to have some basic knowledge of the theory of gear pair teeth in mesh.
Geometry and nomenclature
Before designing any gear pair, it is important to know the basic terminologies. The nomenclature will be described for the spur gear. Following figure is according to Shigley [15].
Pitch circle is the most important is the pitch circle on which all the calculations are based on. When any gear pair comes into contact, the pitch circle of both the gears is tangent to each other. Circular pitch p, is the distance between the corresponding points of adjacent teeth on pitch circle. Addendum and dedendum are the radial distance respectively to top land and bottomland from the pitch circle. Clearance c is the amount by which the dedendum in a given gear exceeds the addendum of the mating gear. The backlash is the amount by which the width of a tooth space exceeds the thickness of the engaging tooth measured on the pitch circle.
Module m is the ratio of pitch diameter to the number of teeth. It is the index of tooth size in SI unit. Diametral pitch P is the ratio of the number of teeth in gear to the pitch diameter. Pressure angle is defined as the angle between line perpendicular to the line of action and the shared centerline of both gears.
Conjugate action
The requirement of gear tooth geometry is to maintain constant angular velocity ratios in each meshing positions, which is not possible in reality because of manufacturing inaccuracy and tooth deflection due to the load applied. The action theoretically satisfies the requirement is called conjugate gear-tooth action. According to Juvinall and Marshek [14] the basic law of conjugate action states that “as the gear rotates, the common normal to the surfaces at the point of contact must always intersect the line of centers at the same point P, called the pitch point.”
Involute profile
The shape of gear tooth can be generated using this. Involute curve can be thought of as a curve created on a straight line, which rolls on a circle without sliding. According to Mägi and Melkersson [16], the advantage of using such a profile is primarily the fact that it fulfills the requirement of constant speed ratio.
Line of action
According to Lee [17] When gear pair mates each other, it will generate a rotary motion. When two curved surfaces are pushed against one another, the contact point appears where the two surfaces are tangent. An imaginary line can be formed If the point can be traced starting from coming to contact and leave the contact. This line is called the line of action or pressure line. at any instance, force will be directed along this line. The intersecting point between line of action and the centerline is called pitch point.
Length of line of action
It can be calculated using tip and base radius, center distance and operating transverse pressure angle.
Contact ratio
This is one of the most important design aspects that indicates the average number of teeth in contact during the period of engagement and leaving gear pair. Higher contact ratio results to higher tooth stiffness and lower contact and bending stress. For a helical gear pair, total contact ratio is the sum of transverse contact ratio and overlap ratio. Transverse contact ratio is calculated from length of line of action divided by the transverse base pitch.
Overlap ratio is the action in axial direction which spur gear does not have.
This ratio increases with the width and helix angle increment.
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Different diameters
Following figure shows different diameters used to define gears geometry. Tip diameter is the highest part of gear tooth and root is the lowest in tooth geometry. Bas diameter is where the involute profile starts to form. There should not be any contact below this diameter and line of action is tangent to this. If the gear can be thought of two rolling cylinders in contact, pitch diameter can be assumed as the diameter of those cylinders[16].
Figure 3.2 different diameters of a gear tooth
Addendum modification coefficient
It is a dimensionless factor to evaluate addendum modification, also known as profile shift coefficient or rack shift coefficient. It quantifies the relationship between the distance from the datum line on the tool to the reference diameter of gear Dabs (radial displacement of the tool) and module m. this coefficient can be defined separately for pinion and gear [18]. positive addendum modification (x > 0) results in a greater tooth root width and thus in an increase tooth root strength and possibility to avoid or reduce undercut. A negative addendum modification (x < 0) will have reverse effect. This is more effective in case of gears with the lower number of teeth.
Gear Types
Different types of gears are used in industry depending on the purpose, application and space. Among all spur gear and helical gears, bevel gear and worm gears are most common types. In this research, the scope is limited to external helical gears used in automotive industry.
Helical gear
In helical gear, the teeth are inclined to the axis of rotation. The inclination is expressed in an angle called helix angle ϐ. The meshing gear should have the same helix angle but hand of helix must be the opposite. The initial contact in the helical gear pair is a point which extends into a line as the teeth come into more engagement. Unlike the spur gear, the line of contact is diagonal across the face of the teeth. Helical gears can be used for the same application as spur gear but Compared to spur gear, helical gears are less noisy because of more gradual engagement of the teeth during meshing and smooth transfer of heavy load in high speed. Sometimes they are also used to transmit motion in non-parallel shafts.
Microgeometry
To achieve favorable contact and reduce transmission error, material is intentionally removed from the tooth profile and the flank line which is called microgeometry modification in gears. according to Kissling [20], this is the last phase of designing gears and bad choice of macro geometry can never be compensated with good microgeometry modification. But it is not possible to achieve one specific design which will fulfill all the objective like noise, scuffing, micro pitting, service life etc. This kind of modification is introduced to compensate the manufacturing allowances and shaft misalignment due to assembly.
Three types of modification including profile, lead and bias can be included in the design. First two types will be discussed here according to Zhao [21] as those were used in the research.
Removal of material in tip and root of the teeth profile tip and root relief to eliminate high stresses in the tip and root areas. The relief can be linear or parabolic. It is important where the relief should be started from. That diameter is called tip or root relief start diameter. It can be expressed with rolling angle also.
Profile crowning modification is also expressed as barreling. This allows to gradually remove maximum material at start of active profile (SAP)and end of active profile (EAP). for profile slope modification, no material is removed from SAP, and the specified amount is removed at the EAP. Lead modification is the modification done to the gear in axial direction. Lead crowning makes the tooth surface slightly convex. it is applied to avoid high local contact at tooth end. Lead slope is to remove defined material from the end of face width and no modification at start of face width. Microgeometry modifications are commonly used by the designers to avoid corner contact and to optimize transmission error and the contact pattern on the gear teeth.
Misalignment
An involute gear pair without manufacturing error should provide uniform load distribution. However, in the real world, there will always be some manufacturing errors and shaft, bearing, housing will be deflecting under load, which will hinder the conjugate action of gear pair flanks. Because one side of the face width will depart from other teeth and not engage when the teeth mesh each other. Misalignment can be expressed in different ways. according to Houser [22] misalignment can be divided into three categories. Parallel misalignment or change in center distance, angular misalignment to the plane of action and angular misalignment perpendicular to plane of action. According to jones [6] misalignment can be decomposed into two linear and two angular forms. Axial and radial are forms of linear and pitch and yaw are the forms of angular.
Involute cylindrical gear sets are very sensitive to misalignment. It leads to discontinuous linear function of transmission error. This results to noise and vibration as well as edge contact and leads to high contact stresses and premature failure because of moving the peak bending stress to the edge of face width [23]. In this study, misalignment in the line of action will be considered which is combination of the angular misalignments.
Table of contents :
1 Notations
2 Introduction
2.1 Background
2.2 Literature review
2.3 Research questions and hypotheses
3 Theoretical background of gear design
3.1 Geometry and nomenclature
3.1.1 Conjugate action
3.1.2 Involute profile
3.1.3 Line of action
3.1.4 Length of line of action
3.1.5 Contact ratio
3.1.6 Different diameters
3.1.7 Addendum modification coefficient
3.1.8 Gear Types
3.1.9 Helical gear
3.2 Microgeometry
3.3 Misalignment
3.4 Gear model data
3.5 Challenges in gear design
3.5.1 Contact stress
3.5.2 Bending or tooth root stress
3.5.3 Transmission error
4 Gear modelling and simulation
4.1 Load distribution program (LDP)
4.1.1 About LDP
4.1.2 Theory and calculation method
4.1.3 Gear modelling using Windows LDP
4.1.4 Including microgeometry and misalignment
4.2 Contact Analysis using KISSsoft
4.2.1 About KISSsoft
4.2.2 Calculation method
4.2.3 Tooth root stress calculation in KISSsoft
4.2.4 Gear modelling using KISSsoft
4.2.5 Including microgeometry and misalignment
4.3 Gear modelling using FE software
4.3.1 Parts and material definition
4.3.2 Assembly
4.3.3 Steps
4.3.4 Interaction module definition
4.3.5 The mesh model definition
4.4 Verification of the model
5 Results and discussion
5.1 Studied cases
5.2 Results from Abaqus
5.2.1 The mesh sizes
5.2.2 The contact and root stress from FEM
5.2.3 Transmission error
5.3 Results from LDP
5.3.1 Contact stress from LDP
5.3.2 Tooth root stress from LDP
5.4 Results from KISSsoft
5.5 Comparison and discussion
6 Conclusion and Future work
6.1 Conclusions
6.2 Future work
7 References
8 Appendix
8.1 Comparison table with real value
8.2 Abaqus set up and results
8.2.1 Abaqus Set up
8.2.2 Results for case 1
8.2.3 Results for case 2
8.2.4 Results for case 3
8.3 Detailed figures from LDP
8.3.1 Results for Case 1
8.3.2 Results for Case 2
8.3.3 Results for Case 3
8.4 Detailed figure from KISSsoft
8.4.1 Results for Case 1
8.4.2 Results for Case 2
8.4.3 Results for Case 3
8.5 MATLAB code to calculate TE from FE result
