Stress Analysis and Material Selection for Timing Gears

Stress Analysis and Material Selection for Timing Gears

(Hasan Amjad,M.Haseeb Shahzad,M.Taimoor,Faisal Aziz)

1.    Introduction

Figure 1 Timing gears

Timing gears of engine are of great significance. They govern Cam gears, Air Compressor gears, Oil pump gears, Turbo Charger gears and other mechanisms as shown in figure no.1.These gears are subjected to heavy as well as small loadings. Our concern in this project is with timing gears which are a type of gears in accordance with their function.They are not the conventional transmission gears. We have to perform calculations about different types of stresses acting on these gears while they are in operation and we have to select a suitable material for these gears also keeping economic factors in mind so that these gears can perform their best action at minimum possible cost.This report presents Thermal and Mechanical Stress Analysis of Timing Gears and the Material used for manufacturing such gears. 

2. Background Information

2.1 Gears:

Gear is a rotating circular component used for transmission of torque. It is a mechanism that works with other parts to alter the relation between the speed of driving mechanism and the speed of driven parts. A gear can be used for the control of speed, torque or direction of source.

2.2 Gear Terminologies:

Now we shall define some basic terminologies related to gears which are helpful in stress analysis:

       

Figure 3 Gear Terminologies

·         Pitch Circle:

                                It is an imaginary circle along which the pitch of the teeth is measured. The pitch circle is concentric with the toothed wheel.

·         Dedendum Circle:

                               Dedendum circle is the inner most circle of profile.

·         Dedendum:-

                               The radial distance between pitch circle and dedendum circle is called dedendum.

·         Addendum Circle:

                              Addendum is the outer most circle of profile.

·         Addendum:

                            The radial distance between pitch circle and addendum circle is called addendum.

·         Diametral Pitch:-

                              It is the number of teeth per unit volume. Its reciprocal is called module.

·         Gear Ratio:

                              It is the ratio of number of teeth of larger gear to that of smaller gear.

·         Contact Ratio:

                              It shows the number of pairs of teeth that are in contact with each other.

·         Tooth Space:

                              It is the distance between adjacent teeth in gear.

·         Face Of Tooth:

                              It is the part of tooth which is above or outside the pitch surface.

2.3 Types of Gears:

Gears can be classified into many types according to their shape like spur, bevel, helical and many more. We will discuss principal three types of gears.

2.2.1 Spur Gears:

Figure 4 Spur ears

Spur gears have radial teeth which are used for power transmission and motion between parallel axes. They are used extensively for speed variation, high torque and also controlling relative positions. They are placed on a shaft or a hub. Spur gears come in variety of sizes, designs and shapes. We can use them depending on our requirements.

2.2.2 Helical Gears:

Figure 5 Helical Gears

Helical Gears contain teeth inclined to the axis of the shaft in the form of a helix. The inclinations between the two teeth allow smooth meshing of gears. They are basically advanced form of spur gears. They also come in different sizes and shapes depending upon customer specifications.

2.2.3 Bevel Gears:

Figure 6 Bevel Gears

It is a cone-shaped gear that transmits power between two axles.The teeth be straight, spiral or hypoid. They are widely used between the non-parallel axes and axes intersecting perpendicularly. These types of gears are used where the direction of motion is to be changed.

3. Timing Gears:

3.1 Structure of timing gears:

A timing gear is usually driven by (or consists of) a belt which is made up of rubber or other high tensile fibers and sturdy materials such as molded polyurethane, neoprene or welded urethane with various pitches (the distance between centers of two adjacent teeth on the belt).The timing gears also driven by chains in some cases which are usually made of metals such as steel. If we consider the R5 Diesel Engine as shown if fig. 7, the whole set of timing gears is located at the engine rear between engine and clutch. It includes 12 helical gears which are forming one direct branch to the camshaft and three branches to the auxiliary components.

Figure 7  Structure of Timing Gears

3.2 Function of Timing Gears:

Timing gears are connected with chains, belts or other gears to crankshaft at one end as well as camshaft at the other end. Timing gears are marked by tiny increments on their surface which correspond to degrees which show the position of the camshaft and crankshaft. The main function of timing gears is to ensure that the engine valves open and close at the right time so that to fill the cylinder with the correct amount of air- fuel mixture and to release the spent fumes from the cylinder during the exhaust cycle. The tiny increments on the surface enable the timing gears to do so. They also operate other mechanisms such as

Figure 8  Function of Timing Gears

• Coolant pump
• Vane pump for power steering
• Air conditioner compressor
• Oil pump
• Turbo charger 

3.3 Working Environment:

Timing gears are subjected to loadings which are heavy in case of heavy vehicles like Trucks, Cranes, and Railway Engines etc.

Pressure: 150-200psi

Temperature: 650-1800℃

3.4 Ways to Transfer Power:

Timing gears usually transfer the rotational movement from an engine's crankshaft to the camshaft. They can do it by three ways:

1. By direct gear meshing

2. By other forms of transmission like timing belt or timing chain or with the help of other gears

3. Timing gears can drive by other gears, belts or chains as shown in the following figures: 

4. Stress Analysis for Timing Gears

4.1 Stress:

Stress is defined as internal resistive force per unit area of the surface. Our major concern here is the stress acting on timing gears.

4.2 Stress in Timing Gears:

We will discuss following stresses acting on timing gears:

1.      Thermal Stress

2.      Mechanical Stress

  4.2.1   Thermal stress:

The stress which is produced due to temperature variation are thermal stress. As these gears are adjacent to engine so they are exposed to this thermal stress. Normally the temperature is 500℃.So the thermal stress are the prominent ones. As shown in figure 9.

Figure 9 ANSYS Model showing thermal stress acting on timing gears

4.2.2 Mechanical Stress:

This stress is due to mechanical forces acting on the gears

Mechanical are of three types:

1.       Surface stress

2.      Hertz stress (contact stress)

       3.      Lewis stress (tangential stress) 

 Surface stress:

As the gear tries to roll other gear the friction force produce resistance which induces stress.

These stress causes fatigue and creep of gear tooth. Surface stress lead to abrassion and degrades the fine surface finish. The material should have the ability to withstand these stress.

·        Hertz stress (contact stress):

These stresses are due to the normal force exerted by one gear on another meshed gear. These stresses are compressive in nature.

Tangential stress:

These stresses are due to the tangential loading on gear tooth. They can deform tooth of gears. These stresses can be analyzed using Lewis stress equation.

.2.4 Principal stresses:

Principal normal stresses:

This is the normal stress which correspond the failure of material in a specific plane. They are represented by 

Principal shear stresses:

This is the maximum tensile stresses which correspond the failure of material in a specific plane

4.2.5 Vonmises Stress:

This stress is not a real stress. This concept predicts that the yield stress is not enough for failure calculation

If Vonmises stress < yield stress: Gears will experience failure

If Vonmises stress > yield stress: Gears will not experience failure

4.5  Lewis Stress Equation for Timing Gears:

In order to analyze the stress on the gear i.e. timing gear we use Lewis equation. First of all the assumptions made in Lewis analysis is as

• Tooth is considered as Cantilever beam.
• Load is applied to the tip of single tooth.
• The load is uniformly distributed over full face width of the gear tooth.
• Only tangent component of load is considered while radial component is ignored.
• Stress concentration in the tooth fillet is negligible i.e. can be ignored in calculation.

Figure 10 Force acting on gear teeth

Let us consider gear tooth as shown is fig. with force F acting on the tip of tooth. As we assumed that radial component of force F_r is not considered thus stresses produced in gear will only due to tangential component F_t. Basic equation of stress is

...............................(1)

where M is moment about horizontal x-axis passing through point ‘a’. c is the maximum distance from the neutral center of tooth & I is the moment of inertia about centroidal x-axis.

We know that

Putting all these values in eq. (1)

..................................(2)

Now if we consider stress & thickness of tooth are constant in the eq. (2) then it mean that there is linear relation between face width and height of tooth i.e.

................(3)

Now once again if we consider stress & face width of tooth are constant in eq. (2) then it means that:                    

...............(4)

                          

The eq. (4) gives us constant strain parabola which gives ease in our calculation.

Let us consider a section passing through the tooth which cuts the tooth outer edge at point A and constant strain parabola at point B. thickness of tooth along that section up to point B is t₁ while up to the point A is t₂. Thus stress at these can be interpreted by eq. (2) as:

From the above two equation it is clearly shown that stress at the the point A is less than at point B i.e.  σ_a˂ σ_b. Thus we can say that stress at any point outside the constant strain parabola is less. Also maximum stress is at the point a where tooth surface and constant strain parabola meet.   

Now as shown in fig let us consider distance between horizontal axis passing through point a and lowest of tooth, this is given as

x = t²/4h

also consider circular pitch ’p_c’. Modified form of eq. (2) is as

here        Y = 2x/3p, So

...................(5)

In the eq. (5) Y is the Lewis form factor & the equation itself is called as Lewis stress equation for gear.

There are many other factors which should be considered in the analysis of stress of  a gear like

• Dynamic load (K_v)
• Fatigue stress concentration (K_f)
• Over load & load distribution (c & K_m)

AGMA strength Equation thus will be like

If we have Lewis equation in terms of dimeteral pitch rather than circular pitch then formulation of Lewis stress will be like this

AGMA strength Equation then will be

........(7)

Here       J = Y/K_f

............(8)

From Lewis equation we can conclude that when gear is to be designed then it should be considered that maximum stress acts at point of intersection of hypothetical constant strain parabola and the gear original surface.                                                                                                   

5. Design and Modelling

5.1 Modeling:

Modeling of timing gears is done using AutoCAD. We consider meshed gears for simplicity. The model is as follows:5.2 Finite Element Analysis (FEA):

Stress analysis of timing gears is done by CATIA V5R20 by FEA (Finite Element Analysis).We consider one of the two gears for stress analysis because stresses on the other meshed gear is approximately the same.

Figure 11 AUTOCAD MODEL FOR TIMING GEARS

The following table shows the parameters taken in this modeling:

Steps:

1. Made a CATIA model
2. Applied material on the model which is cast iron.
3. Applied constraints which were along the shaft axis and on the point of contact for thermal analysis
4. Apply pressure 415 MPa.
5. Divided the material into finer nodes.
6. Applied deformation and showed the von mises stresses.
7. Similarly for temperature, apply temperature field.
8. Results are shown in following sections.

5.2.1 Mechanical Stresses:

Figure 12 CATIA Model showing mechanical stresses on Gear tooth

Results:

The graph in model shows maximum stress in red color that is 1.1e+3 N-m². Our selected material “Grey Cast Iron” has yield strength 100 MPa which is less than the maximum allowable stress. Similar stress would be generated on the other gear also.

5.2.2 Thermal Stresses:

Figure 13 CATIA Model showing Thermal stresses on Gear tooth

Results:

The graph in model shows maximum stress in red color that is 6.24 e+7 N-m². Our selected material “GreyCast iron” has yield strength 100 MPa which is less than the maximum allowable stress. Similar stress would be generated on the other gear also.

6.  Material Selection

Material Selection is the preliminary phase in designing of a component/part of machine. Perhaps it is one of the most important tasks an engineer may encounter during a component design. Improper and Inappropriate use of material can be disastrous from both economic and safety perspectives.

The manufacturing starts with a selected material. The material is selected keeping on account all the stresses and its use. The material may vary depending on our use and functionality. A specific standard or chart is set and the material possessing all the desirable qualities is chosen. The factors that are taken into consideration while selection of a material include

Mechanical Properties                                         

• Strength
• Hardness
• Rigidity
• Hardness
• Resistance to Fatigue

Physical Properties

• Density
• Electrical Properties
• Thermal Properties
• Operational Characteristics

6.1 Material Selection for Timing Gears:

The gears present in an engine are basically the timing gears. These gears attached to cam and crankshaft and are sometimes called as the cam and crank gears.

The bending moment and fatigue of gear teeth are of great importance for the case of timing gears. The type of load of load gears are subjected is very vital i.e. whether the load is constant, gradually changing or instant. The parts of gears will have to handle bending, scoring and contact stresses. The two main types of failure a timing gears faces is Tooth Breakage (due to bending stress) and surface wear (contact stress).

Cost is also very decisive matter while selection of material. The material with low cost and high strength is chosen. The following figure shows distribution of material costs.

Figure 14 Distribution of Material Costs

Keeping into account all the other stresses on the gears the following materials are selected for the manufacturing of the gears:

6.1.1 Steel:

Steel is one of the most common type of material used for making timing gears. Because of its high strength per unit volume and low cost per kilogram make a very good choice for selection. Steel has diverse types from low carbon steels to high carbon steel. Normally Alloy Steel is preferred to low carbon steel. A variety of Steel is used in making of gears.

6.1.2 Austempered Ductile Iron (ADI):

As the name shows the Ductile Iron is Austempered(Heat treatment process applied to ferrous metals to increase strength and toughness).Alloys are added to the cast iron which improve its materialistic characteristics and make it much durable and strong enough to use for making gears. Some alloys added include are copper, graphite, silicon, manganese etc.

The carbon content added to ADI is very important. Usually, it ranges from 3.6-3.8% which facilitates the tensile strength. ADI is flexible due to its nodular graphite inclusions. Its cost relatively high due to processes performed on it. But due its exceptional qualities it is a strong competitor to steel and cast iron.

6.1.3 Cast Iron:

Cast Iron is also excessively used for making gears. Cast Iron is cheaper, brittle and easily available. The cost required in manufacturing is also relatively low which make its use common. It is alloyed with different materials which enhance its strength and properties. Table 1 shows different types of Cast Iron and their properties

Table 1 Types of Material and their Properties

6.2 Choosing the Appropriate Material

 6.2.1 Mass Equation:

In order to calculate the most appropriate material, first we calculate the performance index of the material. Firstly, we calculate mass equation for gears.

Safety Factor is given by

...............(8)

In order to get stress equation consider eq.(5)

..................(9)

Here,

F_t  is the tangential force 

B   is the face width

p_c    is the dimetral pitch

Y    is Lewis form factor                                                   

As we know,

Considering Gear as a cylinder the volume is given by

.....................(10)

Putting value of V in eq.8 we get

...................(11)

Eq. (11) can be written in terms of b as:

..........................(12)

Putting value of b in eq.(5), we get:

..........................(13)

Putting value of eq. (13) in eq. (8) we get mass equation:

...(14)

6.2.2 Performance Index:

Performance index is the reciprocal of performance parameter

......................(15)

......................(16)

Table 2 Types of Material and their Performance Indices

Keeping in view the above table 2 the appropriate material for making gears is Grey Cast Iron because it has high performance index and least product (P*C).

6. Conclusion

This analysis has presented the stress analysis and material selection for timing gears. It was discovered that the mechanical stress has the peak value of 1.1e+3 N-m² and thermal stress has 6.24 e+7 N-m². In order to accommodate these stresses we selected Grey Cast iron.

7. References

1. Theory of Machines  (Khurmi, R. Et al ) Chap 12,13
2. Analysis and synthesis of mechanisms and machines (G.S Shashidhara. Et al)  page no. 39-44
3. AGMA standards, http://www.agma.org/.
4. R.L. Norton, Design of Machinery
5. KHK Materials http://www.khkgears.net/.
6. Industrial Materials  (W.Callister)