Week 1 Spur Gear Challenge

STATIC STRUCTURAL ANALYSIS ON SPUR GEAR USING ANSYS

OBJECTIVE

To carry out a static structural analysis on spur gear with following type of materials,

Case-1: Cast Iron

Case-2: Cast Bronze

Case-3: Cast Steel

To find and compare the results of Equivalent stress, Total deformation, Stress intensity for all the cases.

To recommend the choice of material based on the results.

1. THEORY

1.1 Spur Gear:

Gears are defined as toothed wheels or multilobed cams, which transmit power and motion from one shaft to another by means of successive engagement of teeth. Gears are broadly classified into four groups, viz., spur, helical, bevel and worm gears. A pair of spur gears is shown in Fig. 1.1.

Fig. 1.1 Spur Gear.

In case of spur gears, the teeth are cut parallel to the axis of the shaft. As the teeth are parallel to the axis of the shaft, spur gears are used only when the shafts are parallel. The profile of the gear tooth is in the shape of an involute curve and it remains identical along the entire width of the gear wheel. Spur gears impose radial loads on the shafts.

1.1.1 Gear Tooth Failures:

There are two basic modes of gear tooth failure breakage of the tooth due to static and dynamic loads and the surface destruction. The complete breakage of the tooth can be avoided by adjusting the parameters in the gear design, such as the module and the face width, so that the beam strength of the gear tooth is more than the sum of static and dynamic loads. The surface destruction or tooth wear is classified according to the basis of their primary causes. The principal types of gear tooth wear are as follows:

1. Abrasive Wear
2. Corrosive Wear
3. Initial Pitting
4. Destructive Pitting
5. Scoring

1.1.2 Selection of Material:

The desirable properties of gear material are as follows:

(i) The load carrying capacity of the gear tooth depends upon the ultimate tensile strength or the yield strength of the material. When the gear tooth is subjected to fluctuating forces, the endurance strength of the tooth is the deciding factor. The gear material should have sufficient strength to resist failure due to breakage of the tooth.

(ii) In many cases, it is ‘wear rating’ rather than ‘strength rating’ which decides the dimensions of the gear tooth. The resistance to wear depends upon alloying elements, grain size, percentage of carbon, and surface hardness. The gear material should have sufficient surface endurance strength to avoid failure due to destructive pitting.

(iii) For high-speed power transmission, the sliding velocities are very high and the material should have low coefficient of friction to avoid failure due to scoring.

(iv) The amount of thermal distortion or warping during the heat treatment process is a major problem in gear applications. Due to warping, the load gets concentrated at one corner of the gear tooth. Alloy steels are superior to plain carbon steels in this respect, due to consistent thermal distortion.

Gears are made of cast iron, steel, bronze and phenolic resins. Large size gears are made of grey cast iron of Grades FG 200, FG 260 or FG 350. They are cheap and generate less noise compared with steel gears. They have good wear resistance. Their main drawback is poor strength. Case-hardened steel gears offer the best combination of a wear resisting hard surface together with a ductile and shock- absorbing core. The plain carbon steels used for medium duty applications are 50C8, 45C8, 50C4 and 55C8. For heavy duty applications, alloy steels 4OCrl, 30Ni4Cr1 and 4ONi3Cr65Mo55 are used.

Although steel gears are costly, they have higher load carrying capacity. Bronze is mainly used for worm wheels due to its low coefficient of friction and excellent conformability. It is also suitable where resistance to corrosion is an important consideration in applications like water pumps. Their main drawback is excessive cost.

1.2 The Fracture Mechanics Approach to Design

Fig.1.2 Comparison of the fracture mechanics approach to design with the traditional strength of materials approach: (a) the strength of materials approach and (b) the fracture mechanics approach.

Figure 1.2 contrasts the fracture mechanics approach with the traditional approach to structural design and material selection. In the latter case, the anticipated design stress is compared with the flow properties of candidate materials; a material is assumed to be adequate if its strength is greater than the expected applied stress. Such an approach may attempt to guard against brittle fracture by imposing a safety factor on stress, combined with minimum tensile elongation requirements on the material. The fracture mechanics approach (Figure 1.2b) has three important variables, rather than two as shown in Figure 1.2a. The additional structural variable is flaw size, and fracture toughness replaces strength as the critical material property. Fracture mechanics quantifies the critical combinations of these three variables.

There are two alternative approaches to fracture analysis: the energy criterion and the stress intensity approach.

1.2.1 The Energy Criterion

The energy approach states that crack extension (i.e., fracture) occurs when the energy available for crack growth is sufficient to overcome the resistance of the material. The material resistance may include the surface energy, plastic work, or other type of energy dissipation associated with a propagating crack.

The energy release rate, G, which is defined as the rate of change in potential energy with crack area for a linear elastic material. At the moment of fracture, G = G_c, the critical energy release rate, which is a measure of fracture toughness.

Fig.1.2.1 Through-thickness crack in an infinite plate subject to a remote tensile stress. In practical terms, “infinite” means that the width of the plate is >>2a.

For a crack of length 2a in an infinite plate subject to a remote tensile stress (Figure 1.2.1), the energy release rate is given by

`G=(πσ^2 a)/E ........................................(1.1)`

where E is Young’s modulus, σ the remotely applied stress, and a is the half crack length. At fracture, G = G_c, and Equation 1.1 describes the critical combinations of stress and crack size for failure:

`G_c=(πσ_f^2 a_c)/E.....................................(1.2)`

Note that for a constant G_c value, failure stress, σ_f, varies with . The energy release rate, G, is the driving force for fracture, while G_c is the material’s resistance to fracture.

1.2.2 The Stress Intensity Approach

Fig.1.2.2 Stresses near the tip of a crack in an elastic material.

Figure 1.2.2 schematically shows an element near the tip of a crack in an elastic material, together with the in-plane stresses on this element. Note that each stress component is proportional to a single constant, K_I. If this constant is known, the entire stress distribution at the crack tip can be computed with the equations in Figure 1.2.2. This constant, which is called the stress intensity factor, completely characterizes the crack tip conditions in a linear elastic material. If one assumes that the material fails locally at some critical combination of stress and strain, then it follows that fracture must occur at a critical value of stress intensity, K_Ic. Thus, K_Ic is an alternate measure of fracture toughness.

For the plate illustrated in Figure 1.2.1, the stress intensity factor is given by

`K_I=σ√(πa) .................................(1.3)`

Failure occurs when K_I = K_Ic. In this case, K_I is the driving force for fracture and K_Ic is a measure of material resistance.

2. ANALYSIS SETUP

2.1 Geometry:

Fig.2.1 3D model of spur gear.

The given 3D model of spur gear is imported into ANSYS Workbench for static structural analysis system. The spur gear has 13 teeth with driver gear being named as right gear and driven gear being named as left gear.

2.2 Material Properties:

Case-1: Cast Iron

Case-2: Cast Bronze

Case-3: Cast Steel

Fig.2.2 Material property details.

The materials used for spur gear analysis is as shown in Fig.2.2. The analysis is carried out for each cases of material individually i.e.,

1. Case-1: Cast Iron
2. Case-2: Cast Bronze
3. Case-3: Cast Steel

Note: The analysis setup for case-1 is demonstrated.

2.3 Connection Details:

2.3.1 Contact details:

Fig.2.3.1 Frictional contact of spur gear.

The type of contact between the faces of driver and driven gear teeth is defined as frictional contact with contact bodies being right gear and target bodies being left gear. Augmented Lagrange formulation is chosen to minimize the interference errors. Interface treatment is chosen as adjust to touch to avoid penetration.

2.3.2 Joint details:

a. Left gear                                                                                 b. Right gear

Fig.2.3.2 Revolute Joint of spur gear .

The left and right gears are specified with revolute type of joint having rotation about Z-axis with connection type being body-ground.

2.4 Meshing:

Fig.2.4 Meshing details of spur gear.

Since, the academic version of software has the problem size limit of 128k nodes or elements. The 3D model of spur gear is meshed with minimum element size of 0.5 mm. From the mesh statistics of the model, the number of nodes and elements generated are 88850 and 17872 respectively.

2.5 Boundary Conditions:

2.5.1 Analysis settings:

Fig.2.5.1 Analysis settings.

The total number of steps for analysis is specified as 6 with auto time stepping being ‘On’. The initial, minimum and maximum time step is specified as 0.2 s, 5e-2 s and 0.5 s respectively. In the solver controls, the large deflection is set to ‘On’.

2.5.2 Joint loads details:

a. Left gear

b. Right gear

Fig.2.5.2 Joint loads details.

The left gear is the driven gear hence, rotation (clockwise) for 180⁰ is specified with an increment value of 30⁰ for each step.

The right gear is the driver gear hence, moment (anti-clockwise) of -10 N-m is specified for all the steps.

 

3. RESULTS AND DISCUSSIONS

3.1 Total Deformation:

Case-1: Cast Iron

Case-2: Cast Bronze

Case-3: Cast Steel

Fig.3.1 Total Deformation

From the deformation contour plot for all the cases, it is observed that the maximum deformation of 30 mm has occurred at the outer edges of the teeth of spur gear.

3.2 Equivalent stress distribution:

Case-1: Cast Iron

Case-2: Cast Bronze

Case-3: Cast Steel

Fig.3.2 Von-Mises stress distribution.

From the v-m stress contour plot for all the cases, it is observed that the maximum stress is developed at the interface of face and flank of the spur gear teeth.

3.3 Stress Intensity:

Case-1: Cast Iron

Case-2: Cast Bronze

Case-3: Cast Steel

Fig.3.3 Stress Intensity.

From the stress intensity contour plot for all the cases, it is observed that the maximum stress is developed at the interface of face and flank of the spur gear teeth.  

3.4. Comparison of Results:

The maximum and minimum values of total deformation, v-m stress and stress intensity is tabulated as shown below.

Materials	Total Deformation (mm)		Equivalent Stress (MPa)		Stress Intensity (MPa)
	Max.	Min.	Max.	Min.	Max.	Min.
Case-1: Cast Iron	30	18.5	534.46	5.0817e-9	605.24	5.7049e-9
Case-2: Cast Bronze	30	18.5	505.06	1.8192e-8	563.31	2.0613e-8
Case-3: Cast Steel	30	18.5	562.38	2.0891e-8	634.5	2.3948e-8

From the table, it is observed that the maximum and minimum value of total deformation is 30 mm and 18.5 mm respectively for all the cases of spur gear. The maximum value of Equivalent stress developed in cast bronze spur gear is least i.e., 505.06 MPa whereas, the maximum Equivalent stress developed in cast steel spur gear is highest i.e., 562.38 MPa. Hence, based on traditional strength of materials approach i.e., from the results of Equivalent stress and Total deformation for all the cases of spur gear, case-2: Cast bronze is the preferred choice of material.

The maximum value of stress intensity developed in cast steel spur gear is highest i.e., 634.5 MPa whereas, the maximum stress intensity developed in cast bronze spur gear is least i.e., 563.31 MPa. The stress intensity is nothing but failure stress due to fracture. The higher value of stress intensity indicates that the material has more service life. Based on fracture mechanics approach i.e., from the results of Stress intensity for all the cases of spur gear, case-3: Cast steel is the preferred choice of material.

Case-1: Cast Iron spur gear are cheap and generate less noise compared to other gears. They have good wear resistance. From the analysis results it has moderate values of stress intensity and equivalent stress. Hence, the recommended choice of material is Cast Iron.

4. Animation of Results:

4.1 Total Deformation:

Case-1: Cast Iron

Case-2: Cast Bronze

Case-3: Cast Steel

4.2 Equivalent stress distribution:

Case-1: Cast Iron

Case-2: Cast Bronze

Case-3: Cast Steel

4.3 Stress Intensity:

Case-1: Cast Iron

Case-2: Cast Bronze

Case-3: Cast Steel

CONCLUSION

Static structural analysis was carried out successfully on spur gear having following type of materials,

1. Case-1: Cast Iron
2. Case-2: Cast Bronze
3. Case-3: Cast Steel

Based on traditional strength of materials approach i.e., from the results of Equivalent stress and Total deformation for all the cases of spur gear, case-2: Cast bronze is the preferred choice of material.

Based on fracture mechanics approach i.e., from the results of Stress intensity for all the cases of spur gear, case-3: Cast steel is the preferred choice of material.

The recommended choice of material is Cast Iron.