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MINISTRY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF TECHNICAL AND VOCATIONAL EDUCATION SAMPLE QUESTIONS AND WORKED OUT EXAMPLES FOR ME 03031 DESIGN OF MACHINE ELEMENTS B.Tech (First Year) MECHANICAL ENGINEERING SAMPLE QUESTIONS FOR PART ONE Stresses in Simple Machine Members 1. ** A steel member has a torque of 100 N m and an axial load of 9000N applied as shown in figure (1). What is the magnitude of ( a ) the maximum shear stress, ( b ) the maximum normal stress , (c ) the minimum normal stress ? Ans : (a) 12.2 MPa, (b) 23.6 MPa, (c) – 14.9 MPa (compression) φ100 150 φ50 φ50 200 500 100Nm 565Nm 65kN Fig (2) 9000N Fig (1) 2. * A short circular bar 50 mm in diameter has a couple of 565 N m and a compressive load of 65 kN applied as shown in figure 2. Determine (a) the maximum shear stress in the bar , (b) the maximum tensile stress in the bar, (c) the maximum compressive stress in the bar . Ans . (a) 28.3 MN/m 2 ,( b) 11.8 MN/m2, (c) 44.9 MN /m 2(compression) 3. * Determine the maximum shear stress in the member loaded as shown in figure3. Ans . 13.6 MPa (shear) 500 250 5000N φ100 Fig (3) 4. ** An overhung crank has a load of 10 kN applied as shown in figure 4. Determine the maximum shear stress at section AA where the diameter is 50 mm. Ans: sx = 224 MN/m2 , τ xy = 93.7 MN/m2 ,( τ (max) = 146 MN/m2 ) 10kN ⊥ to page φ100 spur gear 100 400 A 200 230 A 150 125 Fig (4) 5. *** The three components of the total force acting on the bevel gear are mutually perpendicular with the 5000N force being perpendicular to the paper and acting at the mean radius of the gear as shown in figure 5. Determine the bending moment and the maximum shear stress at section AA. Ans . M = 1020 Nm, τ (max) = 28.2 MN/m 2 125 200 150 250 M.D A φ60 A 660N Pulley 5000N 1300N Fig (5) 6. ** Determine the maximum normal and maximum shear stress at section AA for the crank shown in figure 6, when a load of 10 kN, assumed concentrated, is applied at the center of the crank pin .Neglect the effect of transverse shear in this problem . Ans .sx= 21.2 MN/m2 ,τ xy = 13.3 MN/m2 ,τ (max) =17 MN/m2 (shear), sn( max) = 27.7 MN/m2 25 10 kN 30 ° φ50 A 125 φ75 A 75 25 38 Fig (6) 7. ** The parallel side rod of a locomotive weighs 90kg/m. The crank length OP is 375mm and the radius of driver is 0.915m. If the speed of the engine is 96.6km/hr and the tractive effort per wheel is 45kN. Find the maximum normal and the maximum shear stresses in the side rod due to inertia and axial loading for the piston shown in Fig. Take into account the weight of rod. The cross section of the side rod is 75mm x 150mm. 2000 P O 8. *** A crank built up from cylindrical sections by welding required a loading of 1kN to overcome the resistance when in position shown. (a) Compute the maximum normal stress and shear stress induced in section A-A. (b) Determine the maximum shear stress induced in part I, II, III 1kN 175 100 1kN I II 150 20 dia: 25 dia: 25 dia A III 45° 50 A 9. * Determine the required thickness of the steel bracket at section A-A, when loaded as shown in the Figure below, in order to limit the tensile stress to 70MN/m2 4500N 50 50 50 A 50 100 A b A A M=225Nm 4500Nshear stress 10. * Calculate the maximum numerical normal stress and the maximum at section A-A in the member loaded as shown in the Figure 2000N 2000N 1000N 125 A A 50dia: 125 200 250 500 Curved Beams 1. **Interference of machine parts necessitated the use of a steel member as shown in fig 1 below .If a load of 1 kN is applied, determine the maximum tensile stress and the maximum shear stress and indicate the location. Ans : 24.4MN/m2, 12.2 MN/m2 both occur at point A R100 A 400 1000N φ50 Fig (1) 2. **A ring is made from a 75 mm diameter bar. The inside diameter of the ring is 100 mm. For the load shown in figure 2, calculate the maximum shear stress in the bar ,and specify its location. Ans :33.5 MPa at point A P = 20kN B P φ50 A φ100 φ75 A B R100 A 1500N Fig (3) Fig (2) 3. ** Determine the magnitude and location of the maximum tensile stress of the machine part loaded as shown in figure 3 above. Ans :18.7 MPa at point P 4. *An offset bar is loaded as shown in the Figure. The weight of the bar can be neglected. What is the maximum offset (dimension X) if the allowable stress in tension is limited to 70Mpa? Where will the maximum tensile shear stress occur. 10kN X 100 + R 100 R 100 P A A+ 5. An open S link is made from a 25mm rod. Determine the maximum tensile stress 25 dia 900N and maximum shear stress. + R 75 R 100 + 900N 6. ***The offset bar has forces applied as shown. The bar is 25mm x 50mm. The effect of the two applied forces is a pure couple which causes the same bending moment at every section of the beam. Determine the maximum tension, compression and shear stresses, and state where each occurs 25 45° 50 C R = 90 A 900N R = 115 D 100 E B 140 F 230 900N 500 7. ***The centerline of the supporting beam of the under carriage of a crane is as shown in figure 5.The beam is supported in bearings at C and D. Consider the beam as made from a 50 mm diameter bar. (a) What are the reactions at C and D (b) How does the bending moment at sections perpendicular to the axis of the beam vary between A and B ? (c) Determine the worst stressed section or sections .(d) What is the maximum stress? Ans : (a) Reaction at C = 4.5 kN , at D = 4.5 kN,( b) Bending moment is the same at every section from A to B ,1130 Nm, (c) From A to E and F to B (d) 112MPa tension. 4500N 4500N C Rc A R100 250 E R200 R200 1000 Fig (5) F B R100 250 D Rd Power transmission Shafting 1. * A 0.225m diameter solid shaft is used to drive the propeller of a marine vessel. It is necessary to reduce the weight of the shaft by 70%. What would be the dimensions of a hollow shaft made of the same material as the solid shaft ? 2. *A hollow shaft, 500mm outside diameter and 300mm inside diameter, is supported by two bearings 6m apart. The shaft is driven by a flexible coupling at one end and drives a ship's propeller at 100rev/min. The maximum thrust on the propeller is 500kN when the shaft is transmitting 6000kW. The shaft weighs 60kN. Determine the maximum shear stress in the shaft considering the weight of the shaft and the column effect. 6m 3. *A machine shaft turning at 600rev/min is supported on bearings 750 mm apart as shown in figure. 15kW is supplied to the shaft through a 450 mm pulley located 250mm to the right of the right bearing. The power is transmitted from the shaft through a 200 mm spur gear located 250 mm to the right of the left bearing. The belt drive is at an angle of 60º above the horizontal. The pulley weights 800N to provide some flywheel effect. The ratio of the belt tensions is 3:1. The gear has a 20º tooth form and mates with another gear located directly above the shaft. If the shaft material selected has an ultimate strength of 500 MN/m2 and a yield point of 310MN/m2, determine the necessary diameter using Kb = 1.5 and Kt = 1.0 T2 Fr 450mm 200mm Ft 60° 250 W = 800N 250 750 4. *A shaft 1.2m long receives 1000Nm torque from a pulley located at the center of the shaft, as shown in Fig. A gear at the left end of the shaft transmits 600Nm of this torque from the shaft while the remainder is transmitted through a gear located at the right end of the shaft. Calculate the angular deflection of the left end of the shaft with respect to the right end of the shaft if the shaft is 50mm in diameter and is made of steel. Neglect the effect of the keyways in the calculation. Power out 600Nm θ1 600 Power in 1000Nm θ2 Power out 400Nm 600 5. *A 600 mm pulley driven by a horizontal belt transmits power through a solid steel shaft to a 250 mm pinion which drives a mating gear .The pulley weighs 1000N to provide some flywheel effect. The arrangement of elements, the belt tension, and the components of the gear reaction on the pinion are as shown in figure below. (a) Sketch in order the following : vertical loading , vertical bending moment, horizontal loading, horizontal bending moment , and combined bending moment. (b) Determine the necessary shaft diameter using ASME stress values for commercial shafting and fatigue factors of Kb = 2.0 and Kt = 1.5. Ans . Mt (max)= 1050 Nm ,Mb (max) =1784 Nm , d= 71.2 mm 1500N 3000N 250 500 200 5000N 8400N 1000N 6. **Power is transmitted to a shaft , supported on bearings 900 mm apart , by a belt running on a 450 mm pulley which overhangs the right bearing by 250 mm . Power is transmitted from the shaft by a belt running on a 250 pulley located midway between the bearings . The belt drives are at right angles to each other and the belt tension s are 3 to 1 with the total pull on the tight side of either belt being limited to 2400 N. a. Draw the moment diagrams . b. Determine the necessary size of transmission shafting (ultimate tensile strength 670 MN/ m2 , tensile elastic limit 120 MN/ m2 ). Assume Kb = 1.5 and Kt = 1.0. c. Calculate the torsional deflection in degrees. 7. **A steel shaft 2 m long has applied to it a 1000Nm torque by a pulley located at the center of the shaft .A gear at the left end of the shaft applies 800Nm of torque to the shaft while a gear located 300 mm to the left of the right end of the shaft applies 200 Nm of torque .Calculate the angular deflection of the shaft if the shaft is 50 mm in diameter for a length of 1.2 m from the left end of the shaft and 40 mm in diameter in the remainder of the shaft . Neglect the effect of the keyways in the calculations. 8. **A horizontal piece of commercial shafting is supported by two bearings 1.5 m apart .A keyed gear , 20 involute and 175 mm in diameter ,is located 400 mm to the left of the right bearing and is driven by a gear directly behind it .A 600mm diameter pulley is keyed to the shaft 600 mm to the right of the left bearing and drives a pulley with a horizontal belt directly behind it . The tension ratio of the belt is 3 to 1 with the slack side on top . The drive transmits 45 kW at 330 rev/min. a. Draw moment diagrams showing values at the change points. b. Calculate the necessary shaft diameter. c. Calculate the angular deflection in degrees. 9. **A solid shaft and a hollow shaft are to be of equal strength in torsion .The hollow shaft is to be 10% larger in diameter than the solid shaft. What will be the ratio of the weight of the hollow shaft to that of the solid shaft ? Both shafts are made of the same material. 10. **A shaft is mounted between bearings located 9.5 m apart and transmits 10,000 kW at 90 rev/min .The shaft weighs 66,220 N, has an outside diameter of 450 mm and inside diameter of 300mm. Determine the stress induced in shaft and the angular deflection between bearings. Do not neglect the weight of the shaft 11. **A line shaft , 5.4m long and 40mm in diameter , is rotating at 500 rev/min and has 10kW input at one end. Six kW is taken out at a point 2.4 m from the input end and the remaining 4kW is taken out at the opposite end. Using G = 80 GN/m2 , find the angular deflection of one end relative to the other due to this loading. 12. ** A 250mm diameter solid shaft is used to drive the propeller of a marine vessel. It is necessary to reduce the weight of the shaft by 70 %. What would be the dimension of a hollow shaft made of the same material as the solid shaft ? 13. **Two bearings located 900 mm apart support a section of commercial shafting A 2000N ,750 mm diameter ,20 degree involute gears is keyed to the shaft 200 mm to the right of the right bearing . The combined weight of the sprocket and that part of the chain weight taken by the shaft is 800N downward. Assume no tension on the slack side of the chain . The gear receives 7 kW at 210 rev/min from a gear located above. Four kW is taken from the shaft at the sprocket and the remainder is taken from the shaft through a flexible coupling located 150 mm to the left of the left bearing .Figure shows an end view of the arrangement as observed from the right. (a) Draw the bending moment diagrams showing values at the change points. (b) Calculate the diameter of commercial steel shafting based upon strength. (c) Calculate the angular deflection in degrees of the right end of the shaft with respect to the left end of the shaft when under load, neglecting the effect of the keyways and stiffening effect of the pulley and sprocket hubs. 14. ***Figure shows an arrangement for a motor and exciter with a pinion on the same shaft. The pinion drives a gear with the gear directly below the pinion .The motor develops 55 kW at 200 rev/min. The exciter absorbs 5kW, the remainder going to the pinion. The motor and exciter are assembled to the shaft by means of the force fit while the pinion is keyed to the shaft. For this unit, what is the required diameter of shaft ( a constant diameter of shaft will be used.) ? The shaft is to be made of the steel which has an ultimate strength of 520MN/m2 and a yield point of 330MN/m2 .The pressure angle of the gear is 20 degrees,and the stub form of tooth is to be used .Neglect stress concentration due to force fits . Draw all moment diagrams ,showing values at change points.Kb=1.5 and Kt = 1.5 Motor Rotor Exciter Rotor 16kN 4kN 500 500 500 200 dia: pinion 250 Critical Speeds of Shaft 1. **A shaft simply supported on two bearings 500 mm apart carries a 37 Kg flywheel 175 mm to the right of the left bearing. The static deflection curve shows the following: . Distance from 0 50 100 150 200 250 300 350 400 450 500 0 25 75 125 175 200 225 200 150 50 0 left bearing, mm Deflection, mm Estimate the critical speed. Ans. 2400 rev/ min approximately 2. **A steel shaft 1m long is simply supported at the ends and has diameter 76.2 mm 1037.7N over the middle 500mm of length . The remainder of the shaft is 63.5 mm in diameter. Masses weighing 1.335 kN each are attached at the two locations where the diameter changes Neglecting shaft mass and using the Rayleigh-Ritz equation , estimate the first critical speed. Ans. δ1 = δ2 = 1.07 x10-4 m ,ωc =303 rad/s 1061N 3. **Determine the critical speed for the steel shaft shown in figure below. Neglect shaft mass. Ans. 1910 rev/min. 350 250 W=5.6kN 65 dia 50 dia 650 4. * The shaft shown in figure below is to be made of stainless steel (E = 175GPa). Determine a safe diameter to insure that the first critical speed be no less than 3600 rev/min. Ans . d= 48.3 mm 2220N 1330N d 200 125 500 5. *For the steel shaft shown in figure below estimate the first critical speed using the Dunkerley equation. Ans. 1750 rev/min 3650N 220N 75 dia: 250 375 1000 6. ***Determine the critical speed of the steel shaft shown in figure below . Ans.1440rev/min 1.4kN 38 25 200 450 7. *A shaft (bare) has a critical speed of 800 rev/min .If the shaft diameter were doubled, what would be the critical speed? Ans. 1600 rev/min 8. *A shaft carries two equal concentrated masses in locations 1 and 2 on the shaft. With only mass 1 present , the static deflections at 1 and 2 are 0.2mm and 0.18 mm respectively. With only mass 2 present the static deflections at 1 and 2 are 0.18 mm and 0.25 mm respectively. Estimate the first critical speed for the twomass system. Ans. 1410 rev/min (Dunkerley), 1483 rev/minm ( Rayleigh- Ritz) 9. *For the shaft described in problem 8 determine the first and second critical speeds by the frequency equation ( Note : m1 = m2 = m , a11 =0.2/mg , a21 = 0.18/mg = a12, a22= 0.25/mg ) Ans . 1483 rev/min, 4540 rev/min 10. **It has been determined for the shaft shown in figure that the static deflections due to shaft bending are δ1 =0.02mm, δ2 =0.08 mm, δ3 = 0.03mm . The bearing supports have a flexibility in the vertical direction equivalent to a spring constant k= 315 MN/m . In the horizontal direction the supports are essentially rigid. Investigate the first mode critical speed (or speeds) 930N 400N 530N 1 2 3 200 300 250 350 11. *The steel shaft shown below has two gears weighing 225N and 450N respectively. Neglecting the shaft mass, determine (i) over estimated value (b) exact value and (c) under estimated value of the critical speed.. 250 φ 50 250 450N 200