International Journal of Mechanical Engineering and Technology (IJMET), ISSN (Print), INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING

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1 INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND TECHNOLOGY (IJMET) ISSN (Print) ISSN (Online) Volume 3, Issue 3, September - December (2012), pp IAEME: Journal Impact Factor (2012): (Calculated by GISI) IJMET I A E M E DESIGN OF SOLID SHAFTS USING MATLAB GOPICHAND ALLAKA Head of Department, Department of Mechanical engineering, PRASAD RAJU KALIDINDI IV Year, B.Tech, Department of Mechanical engineering prasadraju377@gmail.com KOTESWARA RAO S IV Year, B.Tech, Department of Mechanical engineering MANIBABU DAADI IV Year, B.Tech, Department of Mechanical engineering ABHAY PATNALA IV Year, B.Tech, Department of Mechanical engineering ABSTRACT Shaft is a most common and important machine element. Shafts are widely used mechanical components which are used to transmit power through devices such as gears and pulleys. The shaft is generally acted upon by bending moment, torsion and axial force. Design of shaft primarily involves in determining stresses at critical point in the shaft that is arising due to aforementioned loading. The friction and other losses in this type of power transmission equipment are comparatively very low. In this paper we use a software called MATLAB to write a program to design a shaft. MATLAB is extensively used for scientific and research purposes. It is accurate and also has a number of built in functions which makes it versatile. Still MATLAB is not effectively being used in mechanical engineering field. At present the MATLAB code for design of shafts doesn t exists. Our MATLAB code works for all solid shafts mounted with number of pulleys and gears. The program is user friendly one & when executed it ask the inputs and performs the necessary design calculations and gives necessary output values. We 645

2 have taken both maximum shear stress theory and maximum normal stress theory into consideration. It also generates the diagrams for horizontal, vertical and resultant bending moments over the lengths of the shaft. As computers are used to perform the task of shaft design becomes simple, fast, friendly and error free. Keywords: shaft design, MATLAB, bending moments, torque, diameter. 1. INTRODUCTION 1.1 SHAFTS A shaft is a rotating member, usually of circular cross-section, used to transmit power or motion and to support components like gears, pulleys etc. Shafts must have adequate torsion strength to transmit torque and not to be over stressed. Components such as gears are mounted on shafts using keys. Shaft must sustain a combination of bending and torsion loads. 1.2 STANDARD SIZES OF SHAFTS Typical sizes of solid shaft that are available in the market are, Up to 25 mm 25 to 50 mm 50 to 100 mm 100 to 200 mm 0.5 mm increments 1.0 mm increments 2.0 mm increments 5.0 mm increments 1.3 MATERIAL FOR SHAFTS The ferrous, non-ferrous materials and non metals are used as shaft material depending on the application. Some of the common ferrous materials used for shaft are discussed below Hot-rolled plain carbon steel These materials are least expensive. Since it is hot rolled, scaling is always present on the surface and machining is required to make the surface smooth Cold-drawn plain carbon/alloy composition Since it is cold drawn it has got its inherent characteristics of smooth bright finish. Amount of machining therefore is minimal. Better yield strength is also obtained. This is widely used for general purpose transmission shaft Alloy steels Alloy steel as one can understand is a mixture of various elements with the parent steel to improve certain physical properties. To retain the total advantage of alloying materials one requires heat treatment of the machine components after it has been manufactured. Nickel, chromium and vanadium are some of the common alloying materials. However, alloy steel is expensive. These materials are used for relatively severe service conditions. When the situation demands great strength then alloy steels are used. They have fewer tendencies to crack, warp or distort in heat treatment. Residual stresses are also less compared to CS (Carbon Steel). 646

3 2. DESIGN OF SHAFT When the shaft, carrying heavy pulleys, gears or some loads, transmits power, it is said to be subjected to combined torque and bending moment. In this case, the shaft may be designed based on two theories. 1. Guest s theory (or) Maximum shear stress theory. 2. Rankine s theory (or) Maximum normal stress theory. Let Ss = Torsion shear stress due to pure bending moment, (T) Sb =Bending stress induced due to pure bending moment, (M) 2.1 Maximum shear stress theory: According to this theory, the equivalent torque is given by Te = Ss Where Te = is known as Equivalent torque. 2.2 Maximum normal stress theory: According to this theory, the equivalent bending moment is given by Me = Sb Where Me = is known as Equivalent bending moment. Generally, the Guest s theory will be used for ductile material and Rankine s theory will be used for brittle material. For designing of shaft, the diameter is calculated based on both theories and the larger value will be chosen. In actual practice, the torque and bending moment may not be constant because of change of power and loads due to voltage variations, and the surroundings nature like nonuniformity of roads as in case of automobiles and so onm. Hence for designing such shafts, subjected to this type of fluctuating loads, certain safety factors called shock and fatigue factors may be taken into account. Let Km = Combined shock and fatigue factor for bending. Kt = Combined shock and fatigue factor for torsion. By including the above factors, the equivalent torque Te, may be changed as Te =.. And the equivalent bending moment Me, may be changed as Me =

4 Table: Shock and Fatigue factors Nature of load Km Kt 1. Stationary shafts : (i) Gradually applied load (ii) Suddenly applied load 1.5 to to Rotating shafts : (i) Gradually applied load (ii) Suddenly applied load with minor shock 1.5 to to 1.5 (iii) Suddenly applied load with major shock 2.0 to to ABOUT MATLAB MATLAB stands for MATrix LABoratory. Hence, as the name suggests, here you play around with matrices. MATLAB is a numerical computing environment and fourth-generation programming language. Developed by Math Works, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages, including C, C++, Java and FORTRAN. There are 4 main windows: Command window: This is the main window where you write the commands, as well as see the outputs. In other words, here is your interaction with the software. Command History: As the name suggests, it shows the list of the commands recently used in chronological order. Hence, you can double click on a command to execute it again. Current directory: It is the default directory (folder) for saving your files. All the files which you make (like m-files) are saved here and can be accessed from here directly. The location of the current directory is shown in the toolbar at the top. You can change it by changing the address here. Workspace: It displays the list of the variables defined by you in the current session of MATLAB. 4. DESIGN OF SHAFT USING MATLAB In this paper we designed a MATLAB code and a MATLAB script file is developed to design a solid shaft mounted with gears and pulleys and generates bending moment diagrams. Inputs used in this work are speed, length of the shaft, diameter, weight, and power for gears and pulleys, distances of pulleys and gears to the right of the left hand bearing, allowable shear and tensile stresses for the given shaft material. 4.1 MATLAB OUTPUT WINDOW The MATLAB code we designed takes the input values as shown in below figure and the logic that we have formulated gives the output values such as torque, tangential, radial force for gears, torque, total vertical and horizontal loads for pulleys and maximum bending moments, torques and the final standard diameter of the shaft. It also generates horizontal, vertical and resultant bending moment diagrams over the length of the shafts. We are solving an example problem 4.15 in page number 4.53 from Machine Design by S.Md.Jalaludeen textbook. 4.2 PROBLEM 4.15: A solid shaft is supported on two bearings 2 meters apart and rotates at 500 rpm. Two pulleys whose diameters measuring 800 mm and 600 mm respectively are mounted on the shaft at distances 400 mm and 1200 mm respectively to the right of the left hand bearing. A 20 involute gear of 300 mm diameter is keyed to the shaft at a distance of 200 mm to the left of right hand bearing. A power of 50 kw is supplied to the gear, out of which 30 kw is transmitted by bigger pulley weighing 600 N and 20 kw is transmitted to the smaller pulley whose weight measures 450 N. The drive from bigger pulley is vertically downward and from smaller pulley is horizontal. The tension ratio for both pulleys is 2. Design the shaft for the working stress of 45 Mpa in shear and 80 Mpa in tension. Assume the combined shock and fatigue factors for bending and torsion as 2 and 1.5 respectively. 648

5 4.3 SOLUTION USING MATLAB: COMMAND WINDOW 1 649

6 COMMAND WINDOW 2 650

7 COMMAND WINDOW 3 651

8 COMMAND WINDOW 4 652

9 The inputs are to be given whenever asked in the correct units specified. The horizontal, vertical and resultant bending moment diagrams generated by MATLAB code are BENDING MOMENT DIAGRAM FOR THE ABOVE PROBLEM 5. CONCLUSION The answers generated by MATLAB code are verified with the textbook answers and are proved to be correct.our MATLAB code also works for any solid shaft simply supported on bearings carrying number of gears and pulleys. 6. REFERENCES [1] Shigley, J.E. and Uicker,J.J.,Theory of machines and mechanisms, McGraw-Hill, 1986 [2] R.S. KHURMI and J.K. GUPTA, Theory of machine, S. Chand publications, Edition 16 reprint (2008), pp [3] Rudra Pratap, Getting started with MATLAB, Oxford university Press, updated for Version 7.8 (2009). [4] Machine Design by S.Md.Jalaludeen, Anuradha Publications (2009) [5] Design Data Hand Book for Mechanical Engineers By K.Mahadevan & K..Balaveera Reddy. 653