Parameter estimation of helical machine gearbox by Using SCILAB programming

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1 Parameter estimation of helical machine gearbox by Using SCILAB programming Adhirath S. Mane 1, H. P. Khairnar 2 1 Mechanical Engineering Department, VJTI, Mumbai, Maharashtra, India 2 Mechanical Engineering Department, VJTI, Mumbai, Maharashtra, India Abstract Selection of a gearbox has been discussed and studied frequently and has been crucial for researchers. In the present study, force distribution on a gear is generalized to assist in selecting the gearbox by using a programming platform. There are various influential factors that affect the operation of gearbox like speeds, range ratio, number of speed steps, dimensions of gears, standard module, pressure angles, helix angles and power transmitted by the gearbox. These parameters have been used to determine the tangential forces, radial forces and axial forces using SCILAB as a programming tool. The effect on different forces acting on gear teeth and the ultimate tensile strength required for safe operation of gears is studied by using three different pressure angle values viz. 20, 22.5 and 25 and ten standard helix angles from 0 (spur gear) to 45 with constant progression of 5. The results evidenced that normal force and radial force increases with increase in helix angle and the trend of normal force was approximated by a third degree polynomial while that of radial force by a fourth degree polynomial. It is also observed that axial forces at different helix angles have no variance over change in the pressure angle and the curve is approximated by a fourth degree polynomial. The tangential forces are constant at different helix angles and have no variance over change in the pressure angle. The ultimate tensile strength required has shown a decrement with increasing helix angle with characteristics of fourth degree polynomial curve. Keywords Normal force, tangential force, radial force, axial force, ultimate tensile strength, helix angle, pressure angle, module, beam strength, speeds. I. INTRODUCTION With the development of steam engine, the need arose to adapt the engine power available to the intended use. The first steam-powered vehicles were driven by ratchet gears. Higher ratios were required to climb gradients than to drive on the flat. Since then, lots of developments have been made in the transmission systems. The development goals for mechanical geared transmissions for different kinds of vehicles were firstly low weight, reduced noise and improved ease of use with the introduction of synchronizers. One particular requirement was long service life of up to 1 million km. Vehicle transmission components are now themselves undergoing a process of evolution. The most important component is the gearwheel. The scientific study of gear systems started in late 17 th century with the work of De la Hire, continued by Euler, Willis and Reuleaux. The law of gearing was finally formulated by Saalschutz in The precondition for gear hobbing by machine was the use of mathematical, graphical methods to create theoretically correct flank profiles. With numerically controlled tooth hobbing machines, the rotary movements and longitudinal movements necessary to produce the tooth profile are electronically controlled and synchronized. [1] aims: The design of any gearbox to be used for racing purposes must always have the following DOI : /IJRTER E8OJ3 91

2 1. Provide the maximum possible efficiency in all gears. 2. Be the minimum possible weight while being capable of coping with the requisite torque throughput. 3. Have an overall simplicity in design and, more importantly, in assembly, as ratio changing and maintenance often have to be carried out under fairly primitive conditions. 4. Require the minimum amount of time and effort for maintenance -this point is affected by the simplicity in design and assembly. 5. Reduce the number of components to be removed, when changing internal gear ratios, to an absolute minimum, so that ratio changing can be carried out as quickly as possible. 6. Provide a positive method for locking the pinion in position after completing the meshing procedure with the crown wheel, to ensure that the meshing is not disturbed when changing internal gear ratios or presetting gear selection mechanisms.[2] Selection of gear ratio is the first step towards design of the gearbox. The relationship between the ratios of two neighboring gears is known as gear step. The following aspects should be considered while selecting the gear ratios: 1. The greater the number of gears, the better the engine exploits its efficiency by adhering to the traction hyperbola. But as the number of gears increases, so does the frequency of gear shifting and the weight and the size of gearbox. 2. The proportion of distance travelled in the lower gears is low, especially in the case of passenger cars. 3. The proportion of distance travelled in each gear depends on the specific power output (kw/t), the route profile, the traffic conditions and driver behaviour. 4. The smaller the gear step, the easier and more pleasant the gearshift action.[2] This is followed by determination of centre distance, gear diameters and number of teeth. In the following study, all the above stated calculations along with determination of all the forces and the ultimate tensile strength required has been determined by programming the calculations by using SCILAB as the programming tool. The numerical results obtained have been tabulated and plotted to generalize the graphs and study the trend of the changes in forces and ultimate tensile strength required. The results evidenced that the trend of normal force was approximated by a third degree polynomial while that of radial force by a fourth degree polynomial. It is also observed that axial forces at different helix angles have no variance over change in the pressure angle and the curve is approximated by a fourth degree polynomial. The tangential forces are constant at different helix angles have no variance over change in the pressure angle. The ultimate tensile strength required has shown a decrement with increasing helix angle with characteristics of fourth degree polynomial curve. II. DESIGNING METHODOLOGY The design of any gearbox to be used for automotive purposes must always provide maximum possible efficiency in all gears. Gearboxes are designed by considering minimum possible weight while being capable of coping with the requisite torque throughput. Gearbox should have an overall simplicity in design and, more importantly, in assembly, as ratio changing and maintenance often have to be carried out under fairly primitive conditions. Design of gearbox should be such that it reduces the number of components to be removed, when changing internal gear ratios, to an absolute minimum, so that ratio changing can be carried out as quickly as possible. This is achieved by systematically designing every single parameter of gearbox. Power input has to be considered while designing the gearbox so as to estimate the output power required. Proper speed ratio selection results in improving maximum possible efficiency of the gearbox. Further range of transmission like single or multi range transmission is to be decided. More stages provide All Rights Reserved 92

3 operator with more range of output speeds. Considering all the above factors, face width and the forces acting of the gear teeth are estimated. Once the geometry is designed, the centre distance between meshing gears estimates the overall size of the gearbox. Entire procedure is followed by designing the shafts and selection of bearings. For study purpose, input power to the gearbox is assumed to be 32 kw with maximum speed of 1440 rpm. Numerical methodology is employed further to estimate the relation between the helix angle and the forces acting on the gear teeth. There has always been a systematic procedure to design a complete gearbox. The standard procedure adopted is as stated above. However, the nature of changes in forces by varying the parameters like pressure angles and helix angles is complex when approached by this method. The designing and estimation of forces on the gears can be simplified and can be displayed by using graphical interpretations. 2.1 Estimation of range ratios and speeds: The inputs required to begin with the conventional designing method for gearbox are the power input from clutch and the input speed. The gearbox of an automobile may be a 2-stage gearbox or a multistage gearbox. In this paper we are assuming a 2-stage gearbox for designing. It is assumed that the gearbox is a 4 speed gear box with maximum speed equal to the input speed. Minimum speed can be assumed as per the requirement. Range ratio of the gear box is calculated as [3] Rn = Nmax / Nmin (1) Progression ratio for the gears is estimated to determine various output speeds of the gearbox. Most commonly geometric progression is used for designing the gearbox. In mathematics, a geometric progression is a sequence of numbers where each term after the first is found my multiplying the previous one by fixed, non-zero number called the common ratio. [3] Ø = Rn (1/(z-1)) (2) With the use of progression ratio, various speeds of the gearbox are obtained. Ni = {Nmin, ø x Nmin, ø 2 x Nmin, ø 3 x Nmin...} For the purpose of the paper a 4 speed gearbox with 2 speed steps is assumed. The ray diagram is plotted after finding all the speeds. [5] Figure 1: Ray Diagram for 4 speed All Rights Reserved 93

4 2.2 Gear design methodology: The fundamental law of gearing states that the angular velocity ratio between the gears of a gear set must remain constant throughout the mesh. In order to maintain this constant angular velocity ratio between two meshing gears, the common normal of the tooth profiles, at all contact points within mesh, should always pass through a fixed point on the line of centres called pitch point. [3] ω1 R1 = ω2 R2 (3) By assuming the radius of pinion, radius of the meshing gears can be found out by law of gearing. Using this first pair of gears, the centre distance can be found out. Keeping this centre distance constant and by applying law of gearing, the dimensions of rest of the gears can be found out. By assuming a standard pressure angle (20, 22.5 or 25 ), minimum number of teeth required can be found out. [3] Z = 2 / sin(ø) 2 (4) Number of teeth on gears should be greater than the number of teeth obtained by above equation to avoid teeth interference. By assuming number of teeth on pinion, number of teeth on other gears can be found out by using a standard module. A reverse approach has been implemented in this paper than conventional approach where module is selected first and by either keeping radius constant, the number of teeth are found or vice versa. Here we assume the radius and by varying the number of teeth, a standard module is selected. Table below shows some values calculated for a gearbox with assumption of 20 mm pinion radius with a module of 3 mm. Gears RPM Number of teeth Pitch Circle Diameter (mm) Centre Distance (mm) Numerical analysis of force system at gear faces Determination of forces requires prior information about the power input to the gearbox unit. With the knowledge of the input speed, the angular velocity of the pinion can be found out. Torque acting on the pinion and the meshing gear can be found out by the relationship between the power, torque and angular velocity. [3] T = P/ω (5) Here, T is torque, P is power and ω is the angular velocity. By using the previously obtained radii, the tangential forces acting on the gear pair can be found out. This tangential force can be further used to determine the radial force and the axial force. The resultant of these forces is called as the normal force. Axial force is absent in case of 0 helix angle (spur gears). All Rights Reserved 94

5 Ft = T/r (6) Fn = Ft / (cos ø x cos ψ) (7) Fr = Fn x sin (ø) (8) Fa = Fn x cos (ø) x sin (ψ) (9) In above equations, Ft is the tangential force, Fn is normal force, Fr is radial force, Fa is axial force, ø is the pressure angle and ψ is the helix angle. Following table shows some values calculated for a gearbox with assumption of 20 mm pinion with a module of 3 mm, 20 pressure angle and 30 helix angle. Gears Tangential Force Normal force (Ft) (Fn) Radial Force (Fr) Axial Force (Fa) Estimation of Beam strength and Ultimate tensile strength required Beam strength of the gear tooth is given by Lewis equation stated as [3] Ft x cos (ψ) = m x b x σb x Y (10) Where, Ft is maximum value of tangential force on gear tooth, ψ is the helix angle, m is the module, b is the width of the tooth, σb is the permissible bending stress and Y is the Lewis s form factor. Once the permissible bending stress is calculated, the required ultimate tensile strength can be obtained which is assumed to be three times of permissible bending stress. [3] 2.5 Estimation of shaft diameter and selection of bearings By using the forces obtained by the program the reaction in vertical as well as horizontal plane can be found out. By taking the moments in both the planes, shaft will be designed against failure at the point where bending moment is the highest. By assuming a suitable material, the diameter of the shaft can be estimated [3] Where, d is diameter in mm, Mb is bending moment N-mm, Mt is torsional moment in N-mm, τmax is maximum permissible shear stress, kb is combined shock and fatigue factor applied to bending moment and kt is combined shock and fatigue factor applied to torsional moment. By calculating the equivalent reactions at the support, load can be estimated for selection of the bearings.[3] III. COMPUTATION METHODOLOGY The above stated entire procedure is programmed using SCILAB as programming tool. The entire procedure has been divided in 2 programs to avoid complexity. First program returns the value All Rights Reserved 95

6 of n number of speeds which are used in second program to determine the forces and the ultimate tensile strength required. These values are further used to plot the graphs with varying helix angles for different pressure angles. Graphs assist in determining the equations of curves obtained to study the variation of forces by changing the geometry of the gear wheel. Figure 2: Algorithm used for programming 3.1 Estimation of speeds The first program is a basic input-output program which is used for determining the speeds of the gearbox. The basic structure of the syntax of program uses the procedure explained under the point 3.1. Following table gives the information about the inputs required in program and the outputs that are obtained. Input parameter Value Maximum speed (rpm) 1440 Minimum speed(rpm) 400 z number of speed steps 4 n number of gears Estimation of forces and ultimate tensile strength Values of speeds obtained from first program are used to determine the dimensions of gears first. With the prior knowledge of input power and varying the helix angle, forces on each gear can be obtained. Different values for the Lewis form factor for each gear can be fed in the program. Program also allows the user to input different widths for different gears, in order to optimise the design. The syntax for the program follows the procedure as explained under points 3.2, 3.3 and 3.4. The inputs required and the outputs obtained are as All Rights Reserved 96

Input parameter Range n number of gears 8 n number of speeds of each gear (rpm) 400-1440 Radius of input pinion (mm) 30 Number of

standard number of teeth Width of n gears (mm) 40 3.

7 Input parameter Range n number of gears 8 n number of speeds of each gear (rpm) Radius of input pinion (mm) 30 Number of teeth on pinion 20 Input power (kw) 32 Pressure angle (degrees) 20 Helix angle (degrees) 30 Lewis form factor for n gears As per standard number of teeth Width of n gears (mm) Numerical results For pressure angle of 20, constant module of 3 mm and constant width of 40 mm for all gears, results obtained for various helix angle variations by program are as All Rights Reserved 97

Similarly values of all the forces and the ultimate tensile strengths are

5 and 25 to plot the graphs and understand the nature of variance.

determined by varying the module values.

9 Similarly values of all the forces and the ultimate tensile strengths are determined for pressure angles of 22.5 and 25 to plot the graphs and understand the nature of variance. Values for change in Forces and the ultimate tensile strength are determined by varying the module values. Following table gives values of such forces for pressure angle of 20, helix angle of 30 and tooth thickness of 40 All Rights Reserved 99

10 4.1 Variation of Normal force with helix angle International Journal of Recent Trends in Engineering & Research (IJRTER) IV. RESULTS AND DISCUSSION Figure 3: Variation of Normal force with helix angle obtained by program In the above graph the values of pressure angle is varied from 20 to 25. At a lower pressure angle, the normal force values are low at all helix angles. The module is assumed to be 3 mm and the width is taken as 40 mm. The graphical interpretation deduced that variation of the normal force is close to a fourth degree polynomial given by, y = x x x x Normal force increases exponentially with increase in helix angle. 4.2 Variation of Radial force with helix angle Figure 4: Variation of Radial force with helix angle obtained by All Rights Reserved 100

11 Similar to normal force, radial force also shows exponential growth with increase in helix angle. For the same module of 3 mm and tooth thickness of 40 mm, above graph shows the variation of radial force. It is clear that the increase in radial force is very low compared to the increase in the normal force and axial force to be discussed in next part. The graphical interpretation deduced that variation of radial force is close to polynomial given by, y= x x Variation of Axial force and tangential force with helix angle Figure 5: Variation of Axial force and Tangential force with helix angle obtained by program The tangential force for the gear pair remains constant throughout the change in helix angle. However, there is drastic and obvious exponential rise in the axial force with increase in helix angle. High axial force will make it difficult during selection of bearings as well as contribute towards increase in saturated bending moments. To make the design of shaft simpler, helix angles should be taken low. There is no change observed in both tangential force and axial force with change in pressure angle. The variation in axial force obtained from graph is given by polynomial, y = x x x 4.4 Variation of Ultimate tensile strength with helix angle and forces on gears Figure 6: Variation of UTS and forces with helix angle obtained by All Rights Reserved 101

12 Above graph shows the variation of ultimate tensile strength of a 40 mm wide gear and a module of 3 mm with helix angle. Variation of pressure angle does not affect the ultimate tensile strength of a gear wheel. It is observed that the ultimate tensile strength reduces with increase in helix angle. But more the helix angle, higher the forces acting on the gear wheel. 4.5 Variation in forces and Ultimate tensile strength with change in module Figure 7: Variation of UTS and forces with module obtained by program With increase in module, the forces as well as the ultimate tensile strength of the material required for manufacturing of the gear wheel reduces drastically. But with increase in module, the size of the gearbox will be increasing by higher rate. Ultimate tensile strength at low values of modules is very high which makes it difficult to select material for the gear design. Higher forces at the low values of modules tend to gear failure. V. CONCLUSION The relationship of influential factors in gear design like module, pressure angle and helix angle with the changes in tangential force, radial force, axial force and the ultimate tensile strength required for safe operation of the gear wheels is computed using SCILAB and the graphical interpretations are studied. 1. At lower pressure angles, the values of normal forces are low. With increase in helix angle and by keeping module and pressure angle constant, normal forces show increment similar to a fourth degree polynomial curve as observed in figure It is observed from figure 4 that increment in radial force with increase in helix angle is not as much as increase in axial force. Values of radial force are always on safer side. 3. Tangential force remains constant for a constant module gear for any values of pressure angle as well as helix angle as evident from figure Axial force shows a drastic increase with increase in helix angle. However, axial force remains constant for any pressure angle for a particular module as seen in figure 5. Higher axial forces result in higher saturated moments on the shaft. 5. However, higher helix angles have tended to show a decrement in the ultimate tensile strength required by the gear wheel as observed in figure All Rights Reserved 102

13 6. By keeping helix angle and pressure angle constant, forces as well as ultimate tensile strength reduces with increase in module to a greater extent as noticed from figure 7. But higher module results in bigger size of gearbox. The programming methodology applied in the above study can instantly estimate the forces acting on all the gears in the gearbox for any values of pressure angles, helix angles and modules. Selection and optimization of gearbox can be done easily for any machine with minimal inputs. REFERENCES 1. Alec Stokes, Manual gearbox design, SAE International, Butterworth-Heinemann Ltd Gisbert Lechner, Harald Naunheimer, Automotive Transmissions-Fundamentals, Selection, Design and Application, Springer-Verlag Berlin Heidelberg V. B. Bhandari, Design of Machine elements, Tata McGraw-Hill Pvt. Ltd., Third edition, Tenth reprint, Rahul kumar, Design and Simulation of 7 speed manual gear box, International conference on advances in Mechanical engineering, DOI: 02.AETAME Pravin S, Ghawade, Tushar B. Kathoke, Satish B, Chawale, Ravikant V. Paropate, Ranjan K. Waghchore, An Appropriate Stepwise Solution for Design of Speed Box A Practical Approach, Current Trends in Technology and Science, ISSN: , Volume: 3, Issue: 4 (June-July 2014) Nomenclature: Rn Range ratio Ø Progression ratio N Speed (rpm) Nmin Minimum speed (rpm) Nmax Maximum speed (rpm) Ω Angular velocity (rad/s) R Radius (mm) D Diameter (mm) Z Number of teeth Ø Pressure angle (degree) T Torque (N-m) P Power (kw) Ψ Helix angle (degree) Ft Tangential force (kn) Fn Normal force (kn) Fr Radial force (kn) Fa Axial force (kn) UTS Ultimate tensile strength (N/mm 2 ) M Module (mm) B Width of tooth (mm) σ b Permissible Bending Stress (N/mm 2 ) Y Lewis s form factor τmax Maximum permissible shear stress (N/mm 2 ) Mb Bending moment (N-mm) Mt Torsional moment (N-mm) kb Combined shock and fatigue factor applied to bending moment Combined shock and fatigue factor applied to torsional moment All Rights Reserved 103

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