THE EFFECTS OF STRESS CONCENTRATION FACTOR ON METALLIC RACK-PINION DRIVE

International Journal of Mechanical and Production Engineering Research and Development (IJMPERD) ISSN (P): 2249–6890; ISSN (E): 2249–8001 Vol. 10, Issue 3, Jun 2020, 2293-2300 © TJPRC Pvt. Ltd.

THE EFFECTS OF STRESS CONCENTRATION FACTOR ON METALLIC RACK-PINION DRIVE NAZIEH HASAN Assistant Professor, Mechanical Engineering Department, Faculty of Engineering, Zarqa University, Jordan ABSTRACT A finite element analysis is carried out on the static loading of the rack-pinion drive. The complexity of this problem becomes evident especially at gear teeth and rack-pinion drive due to the significant effect of kinematics and geometric conditions and fabricating precision for finding the working and critical stress which are important for checking the gear teeth working capability. The geometry is modeled by plane stress elements with variable depth of 10, 11, 12 and 15 mm to find the optimal rack depth that gives minimum stress concentration factor. Tooth failure is predicted when the tensile strength is exceeded in the gear tooth root. The results show that the stress concentration factor “Kt” variesalong the flanks of the rack and displays its maximum value at the point of contact between the flank of the rack tooth and the root of the tooth. The rack-depth ratio also significantly influences the stress concentration factor “Kt”, where depth ratio of 2.2 shows the minimum stress concentration factor.

Received: Jun 02, 2020; Accepted: Jun 22, 2020; Published: Jul 04, 2020; Paper Id.: IJMPERDJUN2020213

INTRODUCTION Due to the shape of the straight-line sides of the rack, rack-pinion drive can be considered as a self-stress raiser

Original Article

KEYWORDS: Concentration, Stress & Numerical Researches

element. Analytical, experimental and numerical researches for the purpose of determining the critical stress and the optimal shape of gears are always actual, for they represent the basis for evaluating their safety and reliability. The complexity of this problem becomes evident especially at gear teeth and rack-pinion drive due to the significant effect of kinematics and geometric conditions and manufacturing accuracy for finding the operating and critical stress which are relevant for checking the gear teeth operating capability.

Traditionally, the gear designer had a limited number of analytical tools that were developed by Wilfred Lewis for the prediction of gear tooth strength. This method predicted a safe transmitted load on the gear tooth. It should be noted that this method has been modified throughout the years and is still used in industry. Other elaborate techniques have been developed to determine the stress field in the gear teeth. Baronet and Tordion [1,2] used conformal mapping based on a transformation function given by Aida and Terauchi [3].

With the advent of the computer age, different methods have made their way into the design engineers toolkit. The major reason is that they have become more friendly and lend themselves to be very flexible in solving many complex engineering problems.

However, the use of finite element analysis methods (FEA) is mostly used on materials that have linear elastic behavior. For metals, this assumption gives satisfactory results. Unfortunately polymers do not behave linear elastically. However, they exhibit this type of behavior under special conditions. The gear designer is greatly limited by using a linear elastic approach to optimize gear design out of polymer materials [4].

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The aim of this work is determining the maximum stress concentration factor that found in rack. Also, it is aimed to search for the optimal depth of the rack that gives minimum stress concentration factor.

Problem Physics A rack was imagined as a spur gear having an infinitely large pitch diameter. Therefore the rack has an infinite number of teeth and a base circle which is an infinite distance from the pitch point. The rack flanks are straight lines making an angle to the line of centers equal to the pressure angle. It is assumed that the rack model consists of six teeth and has a constant flank width of 10 mm. Figure 1 shows a detailed sketch for the rack model. The depth “d” of the rack has different values of 10, 11, 12 and 15 mm. The tooth has 5 mm height and one tooth is considered to be in meshing with the pinion. The analysis is carried out for rack of steel with modulus of elasticity 2x10 5 N/mm2 and Poisson’s ratio of 0.3.

Figure 1: Sketch of the Rack Model.

A finite element analysis using the package of JL-Analyzer finite element code [5] was used to evaluate the generated elastic stresses at gear tooth root as well as along flank length and rack depth under quasi-static conditions.

Therefore, the finite element analysis was performed assuming such a plane stress condition (4-node solid element) to exist on the rack using a free mesh technique in JL-Analyzer code as shown in Fig. 2. Uniform distributed stress of 15 N/mm2 acts normal to the flank surface of the third tooth of the rack. All nodes at the opposite side of the teethed surface of the rack are constrained in all three directions.

Figure 2: A Free Meshed Rack (depth =10 mm) as Example to Show the Grids.

Several paths are taken to represent the determined stresses. Stresses at both tooth flanks (the Left flank І and the right flank П) are plotted along the flank length at different rack depth ratios. Also, three individual paths (A, B and C) are considered through the rack depth to illustrate the variation of the generated stresses as well as the stress concentration factor “Kt” as shown in the Fig.3.

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Figure 3: Scheme of Stress Paths Stress Concentration Factor Kt based on the Modified Lewis Equation Analysis of the stress condition at the rack teeth is carried out using Lewis equation approach for bending stress. Figure 4 shows a gear tooth loaded by the transmitted tangential load Ft, which is assumed to act at the tip of the tooth. The radial force component acting on the tooth is neglected. The Lewis formula for the stress at the tooth root can be expressed by the simple bending stress formula as follows:

 

Mc 6 Ft l  2 I bt

The values for the tooth thickness at the critical section and the beam length are obtained from inscribing a parabola inside the tooth, tangent to the fillet, with the apex at the point of load application. The procedure is based on the fact that a parabolic-shaped beam is beam of constant bending stress. For a family of geometrically similar tooth forms, a dimensionless tooth form factor “Y”is defined as follows:

Y

t2P 6h

It is clear that the form factor “Y” is a function of the diametral pitch “P”, the tooth height “h” and the root width “t”. Using the form factor “Y”, the normal bending stress can be expressed as follows:

Figure 4: Gear Tooth Load.

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where F t is the transmitted tangential load and b is face width.

It is important to notice that Lewis equation is based on the following assumptions: 

T he radial load component is neglected.



One tooth carries the full transmitted load.



Uniform line contact performs between contact teeth, which cause uniform load distribution across the face width.



The effect of stress concentration at the fillet is neglected.

A more comprehensive equation (defined by the AGMA) for bending stress number is expressed as follows [6]:



Ft p k a k s k m b k g kv

where: 

σ : The bending stress number. (MPa)



ka : Application factor



ks : Size factor



Km : Load distribution factor



Kv : Dynamic factor or velocity factor



K g : Geometry factor.

Application of AGMA-equation in rack-pinion drives is simplifier than that by gear drives due to the straight

flank form of the rack. Therefore, the calculated stress concentration factor based on the modified Lewis equation has a value of 2.2.

RESULTS AND DISCUSSIONS A tri-axial state of stress generates at the locations of tooth root. The distortion energy theory of elastic failure has been considered as an accurate description for the equivalent stress [7].

Figure 5 illustrates the equivalent stress distribution (von-Mises stress) at the loaded tooth of the rack. It can be noticed that the generated stresses centered locally around the rack root where the stress causing failure is at the point of contact between the tooth flank and tooth root.

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Figure 5: Stress Distribution at Loaded Tooth (von Mises, MPa).

Figure 6 shows the variation of the stress concentration factor “Kt” along the flanks of the loaded tooth for metallic racks with depth ratio of 2.0, 2.2, 2.4 and 3.0, where the depth ratio is defined as the ratio between the rack depth and the teeth height. It can be seen that the maximum value of the stress concentration factor varies between 1.5 and 1.65 for depth ratio of 2.2, 2.4 and 3. However, the case of depth ratio value 2 shows different trend in comparison with the other.

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Figure 8: Variation of Stress Concentration Factor with Rack Depth Ratio.

The variation of the stress concentration factor along the rack depth at the rack cross-section below the loaded tooth is illustrated in Fig.7. It can be seen that the maximal stress concentration factor takes place at the bottom corner of the flank of the loaded tooth (the point of contact between rack tooth flank and rack tooth root). The value of the stress concentration factor reaches a value of about 2.6 by rack depth ratio greater than 2.4. However, the optimal value of the stress concentration factor is obtained by depth ratio of 2.2, which has a value less than 2. Moreover, Fig. 8 illustrates the variation of the stress concentration factor with the depth ratio for metallic racks. It can be noticed that the stress

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concentration factor “Kt” takes its maximum value at depth ratio of about 2.6 and the optimal stress concentration factor value is obtained by depth ratio of 2.2.

CONCLUSIONS From the present work the following conclusions can be drawn: 

The stress concentration factor “Kt” varies along the rack flanks and displays its maximum value at the point of contact between the rack tooth flank and tooth root.



The rack depth ratio affects significantly the stress concentration factor “Kt”, where depth ratio of 2.2 shows the minimum stress concentration factor.

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