PESIT

B.E Mechanical 4th Semester Course Information

DESIGN OF MACHINE ELEMENTS - I

Subject Code: AU46 Faculty: RSS No. of Hours: 52

% of Portions covered Class

# Chapter Title /

Reference Literature Topics to be covered Reference chapter

Cumulative

1 - 6

Chapter : 1 Introduction T1: page 13 – 35 T2: page 44 – 74 R1: page 3 – 29 R2: page 13-15, 70 – 72

Materials and their properties, Design considerations, codes, standards, stress- strain diagram, Definitions – stress, strain, shear stress, biaxial and triaxial loads, Stress tensor, Principal stress

10%

10 %

7 – 14

Chapter : 2 Design for static strength T1: page 182 – 212 T2: page 103 – 107 R1: page 13 – 53 R2: page 67 – 92

Static loads – Types of loads and problems, Theories of failure and problems. Members under combined loads, Stress concentration – explanation and examples, Reduction of stress concentration, Determination of stress concentration factor, combined stress concentration factor, Problems

15%

25%

15– 20

Chapter : 3 Design for fatigue strength T1: page 227 – 275 T2: page 114 – 125 R2: page 117 – 156

Introduction, S-N diagram, low cycle fatigue, high cycle fatigue, Endurance limit. Modifying factors – size effect, surface effect, stress concentration effects; Fluctuating stresses, Fatigue strength under fluctuating stresses, Goodman and Soderberg relationship; stresses due to combined loading, cumulative fatigue damaged.

10%

35%

21 – 25

Chapter : 4 Impact Loading T1: page T2: page T3: page R1: page R2: page

Derivation of instantaneous stress due to axial, bending and torsion loading, effect of inertia.

5%

40%

26 – 31

Chapter : 5 Design of Shafts T1: page 565 – 576 T2: page 465 – 473 R1: page 465 – 481 R2: page 234 – 246

Torsion of shafts, design for strength and rigidity, with steady loading, ASME and BIS codes for design of transmission shafting, shafts under fluctuating and combined loads design of rigid flange coupling and bushed pin type flexible coupling.

20%

60%

32 – 40

Chapter : 6 Fasteners T1: page 301 – 317 T2: page 493 – 498 R1: page 269 – 340 R2: page 247 – 255

Key types, Stresses in Keys, Pins and Retainers. Threaded Fasteners – Stresses, Effects of initial tension, effect of compression, effect of fatigue loading, impact loading, shear loading and eccentric loading.

15%

75%

Text Books: T1: Mechanical Engg. Design by Joseph. E Shigley & Charles R MirchKe. Tata 6th Ed

2003. Mc Graw Hill Edition 2001 T2: Design of Machine Elements by C.S.Sharma and Kamlesh Purohit, PHI 2003. Reference Books: R1: Machine Design by Maleev & Hartman, CBS Publishers & Distribution, New Delhi R2: Design of Machine Elements – V.B.Bhandari,. Tata McGraw Hill Pub. New Delhi R3: Theory and Problems of Machine Design by Hall Holowenko, (Schaum series) R4: Machine Design by Robert L Norton, Pearson Education Asia, 2001 R5: Design of Machine Elements by M.F.Spotts, PHI 2003. R6: Machine Design by Paul H- Black, D.E.Adams McGraw Hill, 2001 Design Data Hand Books: Design Data Hand Book – K.Lingaiah, McGraw Hill, 2nd Ed, 2003 1. Design Data Hand Book – K.Mahadevan & Balaveera Reddy, CBS Publication 2. Machine Design Data Hand Book by H.G.Patil, Shri Shashi Prakashan, Belgaum

41 – 48

Chapter : 7 Power Screws T1: page 291 – 300 T2: page 266 – 273 R1: page 441 – 450 R2: page 163 – 185

Mechanics of power screw, stresses in power screws, Efficiency and self locking.

15%

90%

49 – 52

Chapter : 8 Mechanical joints: T1: page 336 – 352 T2: page 171 – 227 R1: page 213 – 256 R2: page 213 – 229

Cotter and knuckle joints, Riveted Joints – Types, rivet materials, Failures of Riveted joints, Efficiency, Boiler Joints, Tank and Structural Joints, riveted brackets. Welded joints – Types, strength of butt and fillet welds, Eccentrically loaded welds.

10%

100%

QUESTION BANK Chapter – 1 Introduction 1. Explain the term ‘factor of safety’. (5) 2. Discuss the factors influencing selection of an appropriate material for a machine

element. (8)

3. Discuss the factors influencing selection of appropriate value for the factor of safety (8)

4. * List the factors which govern the selection of a material for a machine component (3)

5. *Define Standardization. State the standards used in machine design. (3) Chapter – 2 Design for Static Strength 1. Determine the maximum normal stress and maximum shear stress at section A. A for

the crank shown in figure.1. A load of 10KN is applied at the center of the crank pin. Neglect the effect of transverse shear. (10)

2. Explain the influence of stress concentration in the design of machine elements. (5)

3. Determine the critical stress in the machine component shown in the figure 2. (7)

4. A weight of 1 KN is dropped from a height of 50 mm at the free end of a cantilever beam of effective length 300 mm. Determine the cross section of the cantilever beam of square cross – section if the allowable stress in the material of the beam is limited to 80MPa. (8)

5. A flat bar shown in figure 3 is subjected to an axial load of F equal to 50KN. Assuming that the stress in the bar is limited to 200N/mm2. Determine the thickness of flat bar. (12)

6. A round steel bar having σy = 800 MPa is subjected to the loads producing the calculated stresses of P/A = 70MPa, TR/Jp = 200 MPa, My/J = 300 MPa and 4V/3A = 170MPa. (i) Determine the safety factor with respect to initial yielding according to maximum

shear stress theory and maximum distortion energy theory (ii) Draw the sketch showing the location of maximum normal stress and maximum

shear stress planes. (12)

7. Explain the maximum normal stress, maximum shear stress and von Mises theory of failures. (09)

8. A wall bracket as shown in figure 5 is subjected to a pull of 5KN at 60o to the vertical. The cross section of the bracket is rectangular having b = 3t. Determine the dimensions ‘b’ and ‘t’, if the tensile stress in the material of the bracket is limited 30 MPa. (11)

9. Write full note on stress concentration factor. (05)

10. A C–Clamp carries a load P=20000N. The cross-section (Figure 6) of the clamp at x-x is rectangular having width equal to twice thickness. Assuming that the clamp is made of steel casting with an allowable stress of 120 MPa, find its dimensions. Also determine the stress at section zz. (6)

11. A stepped shaft has maximum diameter 45 mm and minimum diameter 30 mm. The fillet radius is 6 mm. If the shaft is subjected to an axial load of 10KN, find the maximum stress induced, taking stress concentration into account. (8)

12. *A machine part is statically loaded and has yield strength of 350 MPa. For the following stresses calculate the factor of safety using the following theories of failure:

(i) Maximum normal stress theory. (ii) Maximum shear stress theory (iii) Von mires theory (a) σ1 = -70MPa, σ2 = 0MPa (b) σ1 = 70MPa, σ2 = -70MPa (c) σ1 = 70MPa, σ2 = 70MPa

13. A tension member shown in figure 7, supports an axial load P. It is necessary to replace this member by one having a 15 mm hole as shown. Determine the thickness t and radius r at the fillet of the second member, so that the maximum stress will not exceed that of the first member. (10)

14. Find the value of the max. Stress on the fillet if the stress concentration factor for the filleted flat bar in tension is 1.8 and D/d is 1.2 as shown in figure 8. Determine the factor of safety if it is made of steel having yield strength of 320N/mm2. (10)

15. A rod of circular cross section is to sustain a torsional moment of 300 KNm and a bending moment of 200 KNm. Selecting a suitable material and assuming an appropriate value for the factor of safety, determine the diameter of the rod as per the following theories of failure:

(i) Maximum shear stress theory for failure. (ii) Von Mises or distortion energy theory for failure. (iii) Total energy theory for failure.

(15) 16. Figure 9 shows a crank shaft loaded by a force Fy = 1500N

(i) Draw separate free body forces, bending moments and turning moments that act on the crank and on the shaft. Label the directions of the co ordinates axis on these diagrams.

(ii) Compute the maximum torsional stress and the maximum principal stress in the crank at a section 80mm from the pin-end.

(iii) Locate the stress element on the top surface of the shaft at A and find the principal stresses and the maximum shear stress at the same point. (15)

17. Obtain the magnitude of normal and shear stresses at the extreme fibers on the cross section AA of a clamp loaded as shown in figure 11 (12)

18. Determine the diameters of a round rod to sustain a combined torsional load of 1500 Nm and a bending moment of 100 Nm by the following theories of failure. Material selected for the rod has a value of 300 MPa and 180 MPa for the normal stress and shear stress at yield respectively. Take a value of 2.50 for the factor of safety.

(i) Maximum shear stress theory (ii) Octa hedral shearing stress theory

(12) 19. Explain six theories of failure.

(8) 20. Determine normal stresses at the extreme fibers on the cross section AA of a C-

clamp loaded as shown in figure 12. (12)

21. Explain the following theories of failure (i) Maximum principle stress theory for failure (ii) Maximum shear stress theory for failure (iii) Octahedral shear stress theory for failure

(9) 22. A round rod of diameter 30.0 mm is to sustain an axial compressive load of 20 kN

and a twisting moment of 150 Nm. The rod is made of carbon steel C40. Determine factors of safety as per following theories of failure

(i) Maximum principal strain theory for failure (ii) Maximum Elastic energy theory for failure (iii) Distortion energy theory for failure

(12) 23. * A rectangular plate 15mm thick made of brittle material is shown in fig below. Calculate the stresses at each of three holes. (8)

3 Ø 5 Ø 10 Ø

Chapter – 3 Design for fatigue strength 1. A shaft can transmit power of 20 KW at 1000 RPM. The actual torque transmitted by

shaft is ± 60% of the mean torque calculated. Shaft is also subjected to a variable bending moment of 500 N-m to 1000 N-m. The maximum bending moment occurs at the same instant as that of maximum torque. Determine the diameter of the shaft required selecting suitable material. Take factor of safety 2, size factor = 0.85, and surface factor = 0.8. (14)

2. A stepped shaft shown in figure 13 is subjected to the transverse load. The shaft is made of steel with ultimate tensile strength of 400 MPa. The shaft is machined. Determine diameter of shaft based on the factor of safety of 2. (14)

3. Determine maximum stress induced in the following cases taking stress concentration in case

(i) A rectangular plate under an axial load of 10KN. (figure 14) (3)

(ii) The circular shaft with a step under transverse load of 10KN as shown (figure 15) (3)

(iii) The shaft under a twisting moment of 50Nm. (figure16) (3)

4. Explain the significance of Goodman’s line and soderberg line in design of members subjected to reversal of stresses. (8)

5. A steel member of circular cross section is subjected to a torsional stress that varies from 0 to 35 MPa and at the same time it is subjected to an axial stress that varies from 14 MPa to 28 MPa. Neglecting stress concentration and column effect determine (i) the maximum equivalent shear stress. (ii) the design factor of safety based upon yield in shear. The material has an endurance limit = 260 MPa and a yield strength = 480 MPa.

The size factor may be taken as UNITY and the surface has a mirror polish. (12)

6. A stepped shaft with its diameter reduced from ‘1.2 d’ to ‘d’ has a fillet radius of 0.1d. Determine the diameters of the shaft and the radius of fillet to transmit a power of 60 KW at a rated of 1000 RPM limiting the maximum shear stress induced to 65MPa. (7)

7. A shaft of circular cross section is subjected to a turning moment that fluctuates

between 800 KNm and 600 KNm and also a bending moment that fluctuates between + 500 KNm and – 300KNm. The material selected for the shaft has a shear stress value of 100 MPa at endurance limit and a shear stress value of 120 MPa of the yield limit. Determine the diameter of the solid circular shaft taking a value of 2.50 for the factor of safety. Surface factor, size factor and load factor can be taken as 0.90, 0.85 and 1.0 respectively. Shear stress concentration factor is 1.80 and the notch sensitivity is 0.95.

8. A cantilever beam of rectangular cross section has a span of 800 mm. The

rectangular cross section of the beam has a depth of 200 mm. The free end of the beam is subjected to a transverse load that fluctuates between 8 KN down to 5KN up. Selecting carbon steel C 30 as material for the beam and selecting a value of 2.50 for the factor of safety, determine the width of rectangular cross section. Surface factor and size factor are respectively 0.95 and 0.90. Stress concentration factor is given to be 1.65 (14)

9. A round rod of diameter 1.2d is reduced to a diameter d with a fillet radius of 0.1d.

This stepped rod is to sustain a twisting moment that fluctuates between +2.5KNm and +1.5KNm together with a bending moment that fluctuates between +1KNm

and –1.0KNm. The rod is made of carbon steel C40. Determine a suitable value for d. (14)

10.*A cantilever beam of circular cross-section is subjected to an alternating stress at a

point on the outer fiber in the plane of the support that varies from 21 MPa (compression) to 28 MPa (tension). At the same time there is an alternative stress due to axial loading that varies from 14 MPa (compression) to 28 MPa (tension). The material has an ultimate strength of 412 MPa, yield strength of 309 MPa. Assume that actual stress concentration factor =1, size correction factor = 0.85, and surface correction factor = 0.9. Determine

i) the equivalent normal stress due to axial loading ii) the equivalent normal stress due to bending and iii) the total equivalent normal stress due to axial loading and bending

(12) Chapter – 4 Impact Loading 1. Explain the influence of stress raiser on impact strength

(06) 2. A 5 Kg block is dropped from a height of 200 mm on to a beam shown in figure 4.

The material has an allowable yield stress of 50 MPa. Determine the dimensions of the rectangular section, whose depth is 1.5 times of the width. Take E= 70MPa. (14)

3. A weight of 1400 N is dropped on to a collar at the lower end of a vertical steel shaft of 3m long and 25 mm. diameter, calculate the height of drop if the maximum instantaneous stress produced is not to exceed 120 MPa. Take E = 2.1 x 105 MPa. (5)

4. Derive an expression for shock/impact factor. (5)

5. A cantilever beam of span 800.0 mm has a rectangular cross section of depth 200.0m. The free end of the beam is subjected to a transverse load of 1KN, dropped onto it from a height of 40.0 mm. selecting a suitable material and assuming an appropriate value for the factor of safety, determine the width of the rectangular cross section. (8)

6. Find the maximum stress due to impact in the bolt and in the beam shown in figure 10. Assume the same material, namely steel, for both the members.

Chapter – 5 Design of Shafts 1. A horizontal piece of commercial shafting is supported by two bearing 1.5 m apart. A

keyed gear 20o involute and 200 mm in diameter is located 400 mm to the left of right bearing and is driven by a gear directly behind it. A 600 mm diameter pulley is keyed to the shaft 600mm to the right of left bearing and drives a pulley with a horizontal belt directly behind it. The tension ratio of the belt is 3:1, with the slack side on top. The driver transmits 50KW at 350 RPM, Kb= Kt = 1.5.

a) Draw the moment diagram b) Calculate the diameter of the solid shaft required c) Calculate the torsional deflection in degrees

(20) 2. Design a bush type flexible coupling to connect motor and centrifugal pump shafts.

Motor transmits 10 KW at 1440 RPM. Allowable stress in shear for shaft, key and bolts are 40 MPa. Allowable bearing pressure for rubber bush is 0.3 MPa. Check for stresses. (12)

3. Three identical pulleys of 500 mm diameter and weighing 500 N each are mounted on a line shaft supported on two bearings 4000 mm apart. The pulley A is mounted at 300 mm to the right of left bearing and receives 30KW at 200RPM from a pulley vertically below it. The pulley B is mounted 1000 mm to the right of left bearing and

delivers 6KW to a pulley through a belt drive inclined backward at 45o to the vertical. The remaining power is taken out through another pulley C which is mounted at 3000 mm to the right of left bearing and drives a planning machine the drive being 30o to the front of the vertical. The angle of lap for all pulleys may be taken as 180o and the coefficient of friction is 0.3. The working stress in shear for the shaft material is 80N/mm2. Determine the diameter of the shaft.(20)

4. Design a bushed pin type flexible coupling to transmit 90 KW at 1440 RPM for

connecting two shafts of diameter 60 mm. Assume bearing pressure on the bushes as 0.35N/mm2, allowable shear stress in the material of the pins as 45 N/mm2 and allowable bending stress in the material of the pin is 80 N/mm2. (14)

5. Write a brief note on materials and heat treatments used for the shaft. (06)

6. A 1.2 m hollow shaft is subjected to bending moment 900N-m and turning moment 600 N-m. The shaft is also subjected to an end thrust 1.2KN. Taking di/do = 0.7 and material of the shaft to be cold rolled steel, determine the inner and outer diameters of the shaft. Consider heavy shock condition.

7. 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 dimensions of a hollow shaft made of the same material as the solid shaft. (5)

8. A shaft is mounted between bearings located 9.5 m apart the transmits 10000 KW at 90 rev/min. The shaft weighs 66000N has an outside diameter of 450 mm and inside diameter of 300 mm. Determine the stress induced in shaft and the angular deflection between bearing. Do not neglect the weight of the shaft. (10)

9. Design a cast Iron flange coupling (protected type) to connect two shafts and transmits a torque a 5000 Nm. The following permissible stresses may be used. Permissible shear stress for shaft, bolt and key material = 50 MPa. Permissible shear stress for CI = 16MPa.

10. Compare the weight, strength and stiffness of a hollow shaft of the same external diameter as that of solid shaft. The inside dia. of the hollow shaft being half the external diameter. Both the shafts have the same material and length. (10)

11. A line shaft is to transmit 600 KW at 500RPM. The allowable shear stress for the material of the shaft is 42N/mm2 (42MPa). If the shaft carries a central load of 900N and is simply supported between bearing 3 meter apart, determine the diameter of the shaft. The maximum tensile or compressive stress in not exceed 50 MPa. (10)

12. A shaft is required to transmit 1 MW power at 240RPM. The shaft must not twist more that 1o on a length of 15 diameters. If the modulus of rigidity for the material of the shaft is 80 KN/mm2, find the diameter of the shaft and the shear stress induced.

13. Design a cast iron protective flange coupling to connect two shafts in order to

transmit 7.5 KW at 720 RPM. the following permissible stresses may be used permissible shear stress for shaft, bolt and key material = 33MPa. Crushing stress for bolt and key material = 60 MPa. Shear stress for cast iron = 15MPa.

14. The shaft of uniform diameter as shown in figure 17 carries belt pulleys at A and B

with vertical belts. It is supported in bearings at C and D. the shaft transmits 10 KW at 400 rpm. The tension on the tight side of belt A is 2000 N, and that on the slack side of belt B is 900N. Pulley A weighs 200 N and pulley B 400 N. Estimate suitable diameter for the shaft, adopting a working shear stress of 45 MPa.

15. State the advantages of hollow shafts over solid shafts in transmission of power

(5)

16. A 60 cm pulley A receives 15 KW at 500 RPM from below at angle of 45o as shown in the figure 18. A gear C with 450 mm pitch circle diameter delivers 30% of the power horizontally to the right gear D with pitch circle diameter of 300 mm delivers the remaining power downward to the left at an angle of 30o below the horizontal. Both the gears have 20o involute teeth. Assuming working stress in shear 40 MN/m2 and in tension at 80 MN/m2 and in tension as 80 MN/m2 for the shaft material. Design the shaft of uniform diameter. The ratio of tensions in the belt is 2. (20)

17. Design a protected type CI flange coupling for a steel shaft transmitting 15 KW at 1200 RPM. take the maximum torque to be 20% more than the full-load torque. Draw to scale the coupling designed giving all important dimensions. (20)

18. A power transmission shaft is supported on two bearings 2000.0 mm apart. The shaft receives a power of 40KW through a belt drive situated, at a distance of 600.0 mm to the right of the left bearing. The weight & diameter of the pulley are respectively 800N and 400.0mm. The belt moves towards the observer below the horizontal, inclined at 60o to it. The ratio of the belt tensions is 3.0. The power is transmitted out of the shaft through a gear drive located on the shaft at a distance of 500.0 mm to the left of the right bearing. The weight and pitch diameter of the gear mounted on the shaft are respectively 600N and 300.0 mm. The gear which receives from this gear is located exactly behind. The teeth are of involute profile with a pressure angle of 20o. Determine the diameter of the solid circular shaft selecting carbon steel C40 as material & assuming a value of 2.50 for the factor of safety. (20)

19. Design a protected type of CI flange coupling to connect two shafts of the same

diameters and transmit 150 KW at 100 RPM. Select suitable materials and factors of safety. Assume 25% over load. (14)

20. A power transmission shaft 1200.0 mm long receives power of 25 KW through a belt drive located at its right extreme end. The shaft is supported at two points A and B. While A is at the left extreme end, B is at a distance of 300mm from the right extreme end. The pulley on the shaft has a diameter of 500mm and weighs 800N. The belt on the pulley moves below towards the observer making an angle of 30o with the vertical. The power is taken out through a gear drive located at distance of 400mm form the left support. The gear mounted on the shaft has a pitch diameter of 250mm and weighs 500 N. The other gear which receives power form this gear is placed just above this gear. The pressure angle is 20o. The shaft operates at 750 RPM. Selecting a suitable material and assuming an appropriate value for the factor of safety, determine the diameter of the solid circular shaft.

21. A power transmission shaft 1800 mm long is supported at two points A and B. Whereas A is at a distance of 300mm from the left extreme end of the shaft, B is at the right extreme end. A power of 50 kW is received at 500 RPM through a gear drive located at the left extreme end of the shaft. The gear mounted on the shaft here has a pitch diameter of 300mm and weighs 700N. The driver gear is located exactly behind. A power of 30KW is given out through a belt drive located at a distance of 600mm from the left support. The pulley mounted on the shaft has a diameter of 400 mm and weighs 1000N. The belt is directed towards the observer below the horizontal and inclined at 45o to it. The ratio of belt tensions is 3. the remaining power is given out through a gear drive located at a distance of 400 mm from the right support. The driver gear has a pitch diameter of 200 mm and weighs 500N. The driven gear is located exactly above. Selecting appropriate material and assuming a suitable value for the factor of safety determine the diameter of a solid shaft for the purpose.

22. Design a rigid flanged coupling to transmit a power of 40 kW at a rated speed of 100RPM

(10)

Chapter – 6 Fasteners 1. Determine load capacity of the riveted joint loaded as shown in figure 19, if the shear

stress of the material of the rivet is 100 N/ mm2. (12)

2. A 100mm shaft rotating at 100 RPM transmit 300 hp power is taken off through a gear whose hub is 200 mm long. The key is made of steel having an ultimate shearing stress of 350N/ mm2. Using a factor of safety of 5, determine the width of key required. (6)

3. A bolt in a steel structure is subjected to a tensile load of 9 KN. The initial tightening load on the bolt is 5 KN. Determine the size of the bolt taking allowable stress for the bolt material to be 80 MPa. (08)

4. A flanged bearing is fastened to a frame by means of four bolts spaced equally on 400mm bolt circle as shown in figure 20. The diameter of the flange is 500 mm and a load of 200 KN acts at a distance of 250 mm from the frame. If the tensile stress in the bolt is not to exceed 63 MPa. Determine the bolt size. (12)

5. Select a rectangular parallel key for transmitting a power of 50 KW at a rated speed of 500 RPM to mount a hub of length 60mm on a solid circular shaft of diameter 50 mm. (6)

6. Figure 21 shows the cylindrical head of a pressure vessel using 10 bolts and a confined gasket. The static pressure in the cylinder is 6N/mm2. Select the size of the metric bolts for a factor of safety of 3.

7. For the system shown in figure 22 find the maximum stress in the weld. Find the size of the bolt. (20)

8. Determine the power capacity ratio of the two system: one a 24 mm diameter shaft with a 48 x 6 x 6 mm key and another a 24 mm diameter shaft with a 6 mm dia pin. The stress concentration for the key way in the shaft is 1.3, and that for the pinned shaft is 1.75. Assume only torsional load and the same material for all parts. (12)

Chapter – 7 Power screws 1. A split nut used with a lead screw is propelled at a speed of 5 m/min, against a load

of 20KN, along the spindle of a square thread (single start), having nominal diameter of 30 mm and a pitch of 6 mm. The axial thrust is absorbed by a collar of 100mm outside diameter and 70 mm inside diameter. Assuming suitable coefficient of friction, determine :

(i) Power required to drive (ii) height of the bronze nut required

if allowable bearing pressure is 17MPa. (10) 2. The following data applies to a C clamp shown in Figure 23. The screw has

trapezoidal metric thread. Pitch – 1.75mm, Outer dia. of screw – 12mm, Root Dia. – 9.853 Coefficient of thread friction – 0.12

Coefficient of collar friction – 0.25 Friction circle radius of collar – 6mm Maximum thrust on the screw – 4 KN

Determine: (i) Length of handling if the operator exerts a force of 80N at the end of the handle (ii) Maximum shear stress induced in the body of the screw and where does it exist. (iii) Bearing pressure on threads.

(14) 3. A machine weighing 20KN is to be raised by a single start square threaded 50mm

diameter, 8mm pitch screw jack at a maximum speed of 600m/min. If the coefficient of friction between the threads is 0.2, determine the power required to lift the machine. The thrust collar of the screw has inside diameter of 30mm and out side diameter of 60mm. The coefficient of collar friction is 0.1. (10)

4. A 15KN screw jack with a maximum extension of 150 mm has double square threads. Using an allowable compressive stress of 80N/mm2 and bearing pressure on the threads 17.5 MPa Find:

(ii) Size of screw (iii) Height of nut

(10) 5. A shaft straightner is designed to exert a load of 25 KN. It utilizes a square threaded

screw having outside diameter of 75mm and pitch of 6 mm determine the force required to operate the handwheel of 300 mm diameter if the coefficient of friction for threads is 0.12. Also determine efficiency of straightner. (8)

6. What are power screws? State their applications. (4)

7. Design the following parts of 20 KN screw jack selecting suitable materials and

assuming appropriate values for the factors of safety, for a travel of 200mm (iv) Screw rod (v) Nut (vi) The hand lever

8. A turn buckle is used to tighten a wire rope. The threads are single right and left hand square in section. The outside diameter of the screw is 38mm and the pitch is 8.5 mm. The coefficient of friction between the screws and nuts is 0.15. what is the maximum shear stress induced in the screw of the turn buckle if the rope is to be tightened to a tension of 8 KN.

9. Select thread proportions for the screw rod of a screw press to sustain an axial compressive load of 40 KN for an unsupported length of 350.0mm. select an appropriate material and assume a suitable value for the factor of safety. (10)

10. A weight of 500KW is raised at a speed of 6m/min by two screw rods with square threads of 50 x 8 cut on them. The two screw rods are driven through bevel gear drives by a motor. Determine:

(i) The torque required to raise the load (4)

(ii) The speed of rotation of the screw rod assuming the threads are of double start (2)

(iii) The maximum stresses induced on the cross section of the screw rod (4)

(iv) The efficiency of screw drive (3)

(v) The length of nuts for the purpose of supporting the load (vi) check for overhaul

(2) 11.Design a turnbuckle to take an axial load of 100KN. The material of which the turn

buckle is to be made has a design normal stress of 165 N/mm2 and design shear stress of 100N/mm2. (08)

Chapter – 8 Mechanical Joints 1. Design the longitudinal joint for a boiler for a steam pressure of 2 MPa. Diameter of

the boiler is 1 m. Select a double riveted butt joint with a required efficiency of 75%. Take the following allowable stresses. σt = 80 MPa, τ = 60MPa, σc = 120 MPa. (10)

2. An eccentrically loaded bracket welded to its support is loaded as shown in figure 24. Determine the size of the weld required. (10)

3. A steel bracket is welded to a structure and loaded as shown in figure 25. Calculate the size of the weld, taking the permissible stress in the weld to be 84 N/mm2. (12)

4. Design a sleeve type of cotter joint to connect two tie rods subjected to an axial pull of 60KN. The allowable stress of C – 30 material used for rods and cotters are σt = 65 N/mm2, σc = 75 N/mm2 and τ = 35 N/mm2. Cast steel material used for the sleeve has the allowable stresses σt = 70 N/mm2 and τ = 45 N/mm2

5. Design the longitudinal and circumferential joints for a boiler whose diameter in 2

meters and is subjected to a pressure of 1 MPa. The longitudinal joint is a triple riveted butt joint with an efficiency of about 85% and the circumferential joint is a double riveted lap joint with an efficiency of about 70%. The pitch in the outer rows of the rivets is to be double that in the inner rows and the width of the cover plates is unequal. The allowable stresses are σt = 70MPa, τ shear stress = 50MPa crushing stress σc = 120MPa. Assume that the resistance of the rivets in double shear is 1.875 times that of single shear. (20)

6. A bracket as shown in figure 26 carries a load of 40000N. Calculate the size of weld, if the allowable shear is not exceed 80MPa. (10)

7. Two lengths of mild steel flat tie bars 200 mm x 10mm are to be connected by a double riveted double cover butt joint using 24 mm diameter rivets. Design the joint, if the allowable working stresses are 112 MPa in tension, 84 MPa in shear and 200MPa in crushing.

8. Sketch and explain the types of riveted joint failure. (8)

9. A knuckle joint is required for a rod which has to withstand a tensile load of 100 KN. Find the diameters of the rod and the pin. safe working stress both in tension and shear are 80 MPa and 60 MPa respectively. Suggest the suitable dimensions for the entire joint.

10. Design and draw a fully dimensioned neat sketch in two view of a double riveted butt joint with double cover plates for the longitudinal seam of a boiler 1.5m in diameter when working pressure is 1 MPa. Use the following data:

Allowable stress in tension for steel plate = 80MPa Allowable stress in shear for rivets = 60 MPa Allowable stress in crushing for rivets = 120 MPa.

(12) 11. Find the difference in the diameters to be allowed for shrinkage when a compound

cylinder 200 mm external diameter, 100 mm internal diameter and 150 mm diameter at the junction of the two tubes has a radial pressure of 31 MPa at the junction. Take E = 2.1 x 105 MPa and Poisson’s ratio = 0.25. (10)

12. A triple-rivetted butt-joint with equal cover plates is used to connect two plates 16

mm thick. Design the joint if the allowable crushing stress for rivet and plates is 60 MN/m2. Find the joint efficiency. Allowable shear stress for rivets: 45 MN/m2. Draw to scale two views of the designed joint giving all dimensions. (10)

13. A bracket supporting a load is welded to a stanchion by four fillet welds of 6mm size

as shown in the figure 28. What is the maximum value of P if the normal stress on the throat section is not to exceed 98 MN/m2? (10)

14. Design a triple riveted butt joint with unequal widths of cover plates to join two plates of thickness 10mm. The extreme row of rivets, which are in single shear, have a pitch which is twice the pitch of rivets in the inner rows. The allowable stresses are as follows:

Tensile stress of the material of the plates = 80 MPa Shear stress of the material of the rivets = 60 MPa Crushing stress of the material of the rivets = 120MPa

Sketch the joint. Also determine the various strengths & efficiencies of the joint. (10) 15. Design a knuckle joint to transmit an axial load of 120 KN. The allowable stress for

the material of the joint are as follows: σt = 120 MPa and τ = 80 MPa (10)

16. Design a cotter joint to sustain an axial load of 80 KN. Material selected for the joint has the following mechanical properties.

Normal stress at yield = 300 MPa Shear stress at yield = 150 MPa Allowable bearing pressure = 40 MPa

(12) 17. Design a longitudinal butt joint with equal widths of cover plates for a pressure vessel

of diameter 1200.0 mm subjected to an internal pressure of 0.90 MPa. A joint efficiency of 75% can be assumed at this stage. For practical reasons the pitch of rivets is to be restricted a value not less than 3 d and not more than 3.5 d where d is the diameter of the rivets. Material selected for the main plate and rivets has the following safe values:

Design normal stress for material of the main plate = 120 MPa Design shear stress of the material of the rivet = 80 MPa Design Crushing stress of the material of the rivet = 160 MPa Sketch the joint and determine the various efficiencies. (14)

18. Design a socket and spigot type of cotter joint to sustain an axial load of 100kN. The material selected for the joint has the following design stresses. σt = 120 MPa, σc = 160 MPa and pb = 60 MPa (10)

19. Design a triple riveted butt joint to join two plates of thickness 10 mm. The pitch of rivets in the extreme rows, which are in single shear, is twice the pitch of rivets in the inner rows which are in double shear. The design stresses of the materials of the main plate and the rivets are as follows:

for plate material in tension σt = 12MPa for rivet material in compression σc = 160 MPa for rivet material in shear τ = 80 MPa Draw neat sketches of the joint in two views. (10)

20. Suggest a suitable weld size for a welded joint loaded as shown in figure- 29 (10)

ENGINEERING MATHEMATICS – IV – MAT41 Subject Code : MAT 41 No. of Periods : 65

Periods

Chapter title Topics to be covered

Reference Chapter

Cumulative

COMPLEX VARIABLES 1 Introduction 2. Definition of Limit, continuity, differentiability

and problems 3. Analytical functions and problems 4. Cauchy-Riemann equations in Cartesian form 5. Cauchy-Riemann equations in Polar form 6. Problems 7. Consequences on C-R equations 8. Problems 9. Conformal transformations: z2, ez and z + a2 / z 10. Bilinear transformations 11. Problems 12. Complex integration: Line integral 13. Problems 14. Cauchy’s theorem – corollaries 15. Cauchy’s integral formula 16. Problems 17. Taylor’s series and examples 18. Laurent’s series and examples 19. Singularities, poles 20. Calculation of Residues and problems 21.

Chapter 1 Complex Analysis T1,Pg#592,623 R1,Pg#651

Residue theorem and examples.

21 32.3%

21 32.3%

SPECIAL FUNCTIONS 22. Series Solution of Bessel’s Differential equation 23. Problems 24. Recurrence relations 25. Generating function 26. Problems 27. Orthogonality Property and examples 28. Bessel’s integral formula and examples 29. Series Solution of Legendre’s differential

equation 30. Problems 31. Generating functions , 32. Rodrigue’s formula 33. Recurrence relations 34.

Chapter 2 Special Functions T1, Pg#500 R1, pg#194

Problems

14 21.5%

35 53.8%

35. Orthogonality Property and problems STATISTICS AND PROBABILITY

36. Curve fitting by the method of Least squares 37. Problems 38. Correlation and problems 39. Regression 40. Probability, conditional probability 41. Problems 42. Baye’s rule and problems 43. Discrete and continuous random variables 44. PDF and CDF 45. Binomial distribution and problems 46. Poison distribution, Exponential distribution and

problems 47.

Chapter 3 Statistics and Probability T1, Pg#733,780 R1, Pg#1049 R2,59,85,119,177,245 Normal distribution and problems

12 18.5%

47 72.3%

SAMPLING DISTRIBUTION 48. Sampling, Sampling distribution, Standard error 49. Type-I and Type-II errors and problems 50. Testing of hypothesis for means large samples 51. Testing of hypothesis for means small samples 52. Level of Significance and problems 53. Confidence limits for means Large and Small

samples 54.

Chapter 4 Sampling distribution T1,Pg#822, R1,Pg#1104

Student’s t-distribution.

07 10.8%

54 83.1%

JOINT PROBABILITY DISTRIBUTION AND MARKOV CHAINS

55. Concept of Joint Probability and Joint distribution 56. Discrete and independent random variables 57. Expectation and variance 58. Problems 59. Introduction to Markov Chains 60. Probability vectors and problems 61. Stochastic Matrices and problems 62. Fixed points and regular Stochastic Matrices 63. Higher transition probabilities 64 Stationary distribution of regular Markov chains 65

Chapter 5

Joint Probability Distributio

n & Markov Chains

R2,Pg#224 282

Absorbing states

11 16.9%

65 100%

Literature Book Type Code Title & Author Edition Publisher Year

Text Book T1 Higher Engineering Mathematics; B.S. Grewal

38th Khanna 2004

R1 Advanced Engineering Mathematics; Erwin Kreyszig

8th Wiley 2001 Reference Books

R2 Schaum’s Outlines :Probability

2nd McGraw-Hill

2000

QUESTION BANK COMPLEX ANALYSIS (20 marks) Analytic Functions: 1. Show that the function zzf =)( is continuous at every point but not differentiable at

any point. 2. Show that the function ( ) 2|| zzf = is continuous at every point but is not differentiable

at any point other than origin. 3. The necessary sufficient condition for the function f(z)= u + iv to be analytic is

yv

xu

∂∂

=∂∂

, yu

xv

∂∂

−=∂∂

4. If f (z) is analytic on an open set S and ( ) 0=′ zf for all Sz∈ show that f (z) is constant.

5. Show that an analytic function with constant real part is constant. 6. Show that an analytic function with constant modulus is constant. 7. If ( ) ivuzf += is analytic and ψ is any differential function of x and y prove that

( ) 22222

|| zfvuyx

′⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂ ψψψψ

8. If ( ) ivuzf += is an analytic function, prove the following

(a) ( ) ( ) 222

2

2

2||4|| zfzf

yx′=⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+∂∂

(b) ( ) ( ) ( ) 222

|||||| zfzfy

zfx

′=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

(c) ( ) 0||log2

2

2

2=⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+∂∂ zf

yx

(d) If ( ) ivuzf += is analytic and φ is any differentiable function of x and y, prove that

( ) 22222

|| zfvuyx

′⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂ φφφφ

(e) If ( ) ivuzf += is analytic, show that ( ) ( ) 222 |||| zfzf ′=∇

9. Prove that zz

FyF

xF

∂∂∂

=∂∂

+∂∂ 2

2

2

2

24 Here F=F(x, y) z= x+ iy, iyxz −=

10. If f(z) = u +iv is analytic u and v satisfy Laplace’s equation, show that

02

2

2

2=

∂∂

+∂∂

yu

xu

2

2

xv

∂∂ +

2

2

y

v

∂

∂= 0 i.e ., u & v are harmonic functions.

11. If f(z) = u + iv is analytic then the families of curves u= c1 and v= c2 here c1& c2 are constant are orthogonal.

12. Show that an analytic function constant modulus is constant. 13. Find the analytic function f(z)=u + iv, given

(a) u =2x(1-y) (b) u = ex (x cosy – y siny) (c) x sinx cushy – ycosx sinhy (d) v=exsiny

(e) v= cosh2ycos2x

sinxsiny+

(f) 22 yxxvu+

=+

PESIT

B.E. Mechanical 4th Semester Course Information

(g) yy

y

eexexxvu−

−

−−

−+=−

cos2sincos

14. If θirez = and ( ) ( ) ( )θθ ,, rivruzf += prove that θθ ∂∂

−=∂∂

∂∂

=∂∂ u

rrvv

rru 1;1

15. ( ) ( ) ( )θθ ,, rivruzf += is analytic function, show that u and v satisfy the function

(a) 0112

2

22

2=

∂∂

+∂∂

+∂∂

θϕϕϕ

rrrr

(b) 0112

2

22

2=

∂∂

+∂∂

+∂∂

θu

rru

rru

(c) 0112

2

22

2=

∂∂

+∂∂

+∂∂

θv

rrv

rrv

16. Find the analytic function ( ) ,ivuzf += given

(a) θθ sin42cos2 −= ru (b) 0,2cos2 ≠= r

ru θ

Complex Integration

1. Prove that ( ) ( ) ( )∫ ∫ ∫ ++−=c c c

vdxudyivdyudxdzzf

2. Prove that ( )∫ =c

dzzf 0

3. If c1,c2,c3…..cn are ‘n’ non overlapping simple closed curves within C and f(z) is analytic on these curves in the region bounded by them then prove that

( ) ( ) ( ) ( )∫∫∫∫ +++=ncccc

dzzfdzzfdzzfdzzf .....21

4. Verify the Cauchy’s theorem for the function ( ) 43 2 −+= izzzf with c as the square having vertices at 1 ± i , -1 ± i

5. If f(z) is analytic within and on a simple closed curve c in the complex plane and a is

any point c then prove that ( ) ( )∫ −

=c

dzaz

zfi

afπ21

6. If f(z) is analytic within and on a simple closed curve C and a is any point within C then

( ) ( )( )∫ +−

=c

nn dz

azzf

inaf 12!π

7. Evaluate ∫+

c

dzz

z ,12 where C is a simple closed contour enclosing the origin.

8. Evaluate ∫c

zdz

ze

3 where C is the circle |z|=1

9. Evaluate ∫ −+

c

dzzz ,

11

2

2where C is a circle of unit radius with center at (i) z= 1

(ii) z=-1

10. Obtain the Taylor’s and Laurent’s series for the function f(z)= ( )( )211

2 ++ zz for (a)|Z|<1

11. (b) 1<|z|<2 (c) |z|>2

12. Obtain Laurent’s expansion for ( ) ( )( )31

2

−−=

zzzzf in the region (a) 1<|z|<3 (b) |z-

1|<2.

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B.E. Mechanical 4th Semester Course Information

13. If C is a simple closed curve and f(z) is analytic within and on simple closed curve c except at finite points a1,a2,a3…..an inside c then prove that

( ) ( )∫ +++=c

nRRRRidzzf ......2 321π here nRRRR ....,, 321 are residues of f(z) at a1,a2,a3,……an

14. Evaluate ( )( )∫ −−−

c

dzzzz

z21

43 where C: |z|=3/2

15. ∫ −+

c

dzz

zz ,1

22

2where (i) C: |z|=2 (ii) C: |z-1|=1

16. Show that the transformation w = z2 transforms the circle | z-a | = c to a cardioid or a

limacon. 17. Find the bilinear transformation that transforms the points z1 = 1, z2 = i, z3 = -1 onto

the points w1 = 2, w2 = i, w3 = -2. Find the fixed points of the transformation.

18. Find the images of (i) x-y = 1 (ii) x2 – y2 = 1 under the transformation w = z2.

BESSELS FUNCTIONS: (10 marks) 1. Find the series solution of Bessel's differential equation. 2. Show that y = c1 Jn(kx ) + c2 J-n (kx) is the solution of x2 y2 + xy1 + (k2 x2 - n2)y =0. 3. Verify that y = xn Jn(x) is the solution of x y2 +(1-2n)y1 + xy =0.

4. Show that (a) J ½ (x) =xπ2

Sinx (b) J -½ (x) =xπ2

Cosx.

5. Show that 2n J n(x) = x [ ] (x) 1 J (x) 1J ++ n n- 6. Show that J n'(x) = x [ ] (x) 1J - (x) 1 J + nn-

7. Show that [ ])(xJxdxd

nn = xn J n-1 (x) .

8. Show that [ ])(xJxdxd

nn− = x-n J n+1 (x) .

9. Show that (a) J 3/2 (x) =xπ2

{(Sinx )/x - cosx }

(b) J --3/2 (x) =xπ2

{(Cosx)/x +sinx}

10. Show that [ ]).()( 1 xJxJxdxd

nn − = x[ J2 n (x) .- J2 n-1 (x)]

11. Show that cos (x sinθ) = J0(x) +2ΣJ2n(x)cos 2nθ 12. Show that sin (x sinθ) = 2ΣJ2n-1(x)sin (2n-1)θ

13. Prove that J n(x) = θθθπ

dxn )sincos(1∫ −

14. State and prove orthogonal property of Bessel's functions.

15. Show that22

00

1)(ba

dxbxJe ax

+=∫

∞−

16. Prove that ( ) ( ) ( ),xJ1J nn

xn −=− where n is a positive integer.

LEGENDRE POLYNOMIALS: (10 marks) 1. Find the series solution of Legendre's differential function. 2. Show that (a)Pn (1) = 1 (b)Pn (-x) = (-1) n Pn (x) . Hence deduce that Pn (-1) = (-1)n

3. Express 3 - x + 2x2 + 2x3 + x4 in terms of Legendre’s polynomials. 4. By using Rodrigue’s formula verify that Pn (x) satisfies Legendre’s differential equation.

PESIT

B.E. Mechanical 4th Semester Course Information

5. Show that Pn (x) = θθπ

π

dxx∫ ⎥⎦⎤

⎢⎣⎡ −±

0

2 cos11

6. Show that [ (2n+ 1) x Pn (x)] = (n+1) Pn+1 (x) + n Pn-1 (x) 7. Show that Pn (x) = xP'n (x) - P'n-1 (x) 8. Show that Pn (x) = P'n+1 (x) - 2x P'n (x) + P'n-1 (x)

9. Show that )32)(12)(12(

1)(n2n dx (x) 1.P(x) 1.P1

1

2

++−+

=−+∫−

nnnnnx

10. Show that )14(

2n dx (x) 1.P(x) .P 2

1

1−

=−∫−

nnnx

11. Show that )12(

2n dx (x) .P'(x) .P1

1−

=∫−

nnnx

12. Prove that ( ) ( )nn

n

nn xdxd

nxP 1

!21 2 −=

13. Express 543 23 +−+ xxx in terms of Lagendre’s Polynomials.

14. Prove that ( ) ( ) ( )xnPxxPxP nnn 11''

−− +=

STATISTICS: (10 marks)

1. Fit the straight line of the form y= a + bx to the given data x: 0 5 10 15 20 25 y: 12 15 17 22 24 30 2. Fit a parabola cbxaxy ++= 2 to the following data. x: 20 40 60 80 100 120 y: 5.5 9.1 14.9 22.8 33.3 46.0 3. Fit a curve of the form y=axb for the data

x: 1 2 3 4 5 6 y: 2.98 4.26 5.21 6.1 6.8 7.5

4. The following table gives the marks obtained by a student in two subjects in ten tests.

Find the coefficient of correlation. Sub A : 77 54 27 52 14 35 90 25 56 60 Sub B: 35 58 60 40 50 40 35 56 34 42

5. Show that there is a perfect correlation between x & y . x: 10 12 14 16 18 20 y: 20 25 30 35 40 45 6. A computer while calculating the correlation coefficient bet x & y from 25 pairs of

observations got the following constants n = 25, Σ x = 125, Σ x2 = 650, Σ y = 100, Σy2 = 460& Σ xy = 508. Later it was discovered it had copied down the pairs (8, 12) & (6, 8) as (6, 14) & (8, 6) respectively. Obtain the correct value of the correlation coefficient.

7. If θ is the angle between two regression lines show that

22x

yx2

r-1 tanyr σσ

σσθ

+= and explain the significance when r = 0.

8. Find the lines of regression for the following data: x: 1 2 3 4 5 6 7 8 9 10 y; 10 12 16 28 25 36 41 49 40 50

9. If the mean of x is 65, mean of y is 67, σx = 7. 5, σx = 3.5 & r = 0.8 find the value of x corresponding to y= 75 & y corresponding to x = 70.

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B.E. Mechanical 4th Semester Course Information

10. The two regression lines are x = 4y + 5 & 16y = x + 64 find the mean values of x, y & r.

11. In a partially destroyed laboratory record of correlation data only the following results are legible. variance of y is 16, regression equations are y = x + 5, 16x = 9y - 94, find the variance of x.

12. Fit a straight line to the data: (a) x: 0 1 2 3 4 y: 1 1.8 3.3 4.5 6.3 (b) x: 1 2 3 4 5 y: 14 13 9 5 2

13. Fit a second degree parabola of the form y = ax2 + bx + c for the data: x: 1 2 3 4 5 y: 1.8 5.1 8.9 14.1 19.8 . Estimate y for x = 2.5.

14. Fit an exponential curve of the form y = abx, for the following data: x: 1 2 3 4 5 6 7 y: 87 97 113 129 202 195 193. Estimate y for x = 8.

PROBABILITY: (10 marks) 1. Define a sample space and probability of an event.. When are two events said to be

(a) mutually exclusive (b) mutually independent. 2. If A & B are events P(A) = ½, P(B) = 1/3, P(A∩B) = 1/4, find (a) P(A/B) (b) P(B/A) (c) P(A∪B) (d) P(Ac) 3. An experiment succeeds twice as often as it fails. Find the chance that in the next 6

trials there will be at least 4 successes. 4. A class consists of 6 girls & 10 boys If a committee of 3 is chosen at random find the

probability that (a) exactly 2 boys are selected (b) at least 1 boy is selected (c) exactly 2 girls are selected.

5. A certain problem in mathematics is given to 4 students for solving. The probabilities of solving the problem individually are ½, 1/3, ¼, & 1/5 respectively. Find the probability that (a) the problem is solved (b) the problem is solved exactly by one of them.

6. The chance that a doctor will diagnose a disease correctly is 60%. The chance that a patient will die after correct diagnosis is 40% and the chance of death after wrong diagnosis is 70%. If a patient dies what is the chance that his disease was not diagnosed correctly.

7. Find the probability that a leap year selected at random will contain 53 Fridays. 8. Four cards are drawn from a pack of 52 cards without replacement. Find the probability

that (a) they are all of different suits (b) no 2 cards are of equal value. 9. State & prove Baye's theorem. 10. Define (a) a random variable (b) Discrete and continuous random variable 11. Define probability mass function and probability distribution function for a discrete

random variable. 12. Define Geometrical distribution, uniform distribution, Exponential distribution. 13. 3 machines A, B & C manufacture 40%, 50% & 10% of the total production of a factory respectively. The percentage of defective items produced by A, B & C are 2, 4, & 1.5 respectively. An item is chosen at random & is found to be defective. Find the probability that it was a product of C.

4 14. There are 3 bags which contains 1 white, 2red & 3 green, 2 white, 3 red & 1 green and

3 white, 1 red & 2 green marbles respectively. 2 marbles are drawn from a bag chosen at random and they are found to be 1 white & 1 red. Find the probability that the balls came from the second bag.

15. Obtain the mean and variance for the following distributions: Binomial, Poisson, Exponential and Normal.

16. The probability of a man hitting a target is 1/3. (a) If he fires 5 times what is the probability of hitting a target at least twice. (b) How many times must he fire so that the probability of hitting a target

PESIT

B.E. Mechanical 4th Semester Course Information

at least once is more than 90%. 17. A cricket team has probability 2/3 of winning whenever it plays. If it plays 4 games, find

the probability that it wins (i) 2 games (ii) at least one game. 18. A group of 20 airplanes are sent on an operational flight. The chance that an aero plane

fails to return from the flight is 5 %. Find the probability that (a) one plane does not return (b) at the most 5 planes do not return.

19. The probability that an individual suffers a bad reaction from a certain injection is 0.001. Determine the probability that out of 2000 individuals (a) exactly 3 (b) more than 2 individuals will suffer a bad reaction.

20. Given that 2% of the fuses manufactured by a firm are defective, find the probability that a box containing 200 fuses has (a) at least 1 defective fuse (b) at most 3 defective fuses.

21. If the probability that a target is destroyed on any one shot is 0.5. What is the probability that it will be destroyed in the 6th shot only and not before.

22. If the probability of the birth of a child with a defective heart in a certain city is 0.01. What is the probability that the 8th child born is the first one to have a defective heart?

23. On a certain city transport route, buses ply every 30 minutes between 6 a.m. & 10 p.m. If a person reaches a bus stop on this route at a random time during this period, what is the probability that he will have to wait for at least 20 minutes?

24. The duration of time that an overhead tank will serve without refilling is found to follow an exponential distribution with mean 10 days. Find the probability that (i) it needs filling within 8 days & (ii) it will serve for more than 10 days.

25. Find the mean & S.D of a normal distribution of marks in an examination where 44% of candidates obtained below 55 & 6% above 80 and rest between 55 & 80.

26. The mean marks of 1000 students is 34.4 & S.D 16.5.Assuming that the marks are normally distributed find the no. of students obtaining marks (i) bet 30 & 60 (ii) bet 70 & 80.(iii) below 20 (iv)above 80.

27. A quality control engineer inspects a random sample of 3 batteries from each lot of 24 car batteries that is ready to be shipped. If such a lot contains 6 batteries with slight defects, what are the probabilities that an inspectors sample will contain

(i) none of the batteries with defects (ii) only one of the batteries with defects (iii) at least 2 of the batteries with defects. 28. Among 300 employees of a company 240 are union members while the others are not.

If 8 employees are chosen by lot to serve on a committee, find the probability that 5 of them will be union members.

29. Find E(x) & V(x) for the following probability distribution: x: 3 4 5 6 7 8 9 p: 0.05 0.12 0.20 0.24 0.17 0.14 0.008

30. A distributor makes a profit of $20 on an item. If it is shipped from the factory in perfect condition and arrives on time but it is reduced by $2 if it does not arrive on time & $12 regardless of whether it arrives on time if it is not shipped from the factory in perfect condition. If 70% of such items are shipped in perfect condition and arrive on time, 10% are shipped in perfect condition but do not arrive on time and 20% are not shipped in perfect condition what is the distributors expected profit per item.

31. If a dealers profit in units of $1000 on a new automobile can be looked upon as a

random variable X having the density function f(x) = ⎩⎨⎧ <<−

elsewhere , 01x0 ),1(2 x

Find the

average profit per automobile and also E(X2). 32. Show that (i) E(c) = c (ii) E (aX + b) = a E(X) + b (iii)V(X) = E(X2) - E(X)2 . (iv) V(c) = 0 (v) V (aX + b) =a2 V(X). 33. The distribution of 2 independent random variables X & Y are given below: X 0 1 Y 1 2 3 P(X) 0.2 0.8 P(Y) 0.1 0.4 0.5 Find the joint probability distribution of X & Y.

PESIT

B.E. Mechanical 4th Semester Course Information

34.The following table gives the joint probability distribution of 2 random variables

X &Y X/Y -1 0 1 -1 0 0.2 0

0 0.1 0.2 0.1

1 0.1 0.2 0.1

Find the conditional probability of X given Y = 0. 35. The joint distribution of two random variables X and Y is given by the following table.

X / Y -4 2 7 1 1/8 1/4 1/8 5 ¼ 1/8 1/8

Determine (i) the marginal distributions of X and Y. (ii) E (X) and E(Y) (iii) are X and Y independent random variables? Sampling Distribution: (20 marks) 1. A Sample of 5 measurements of the diameter of a sphere was recorded as 6.33, 6.37,

6.36, 6.32, 6.37mm. Find unbiased and efficient estimates of (i) the population mean (ii) the population variance.

2. For the frequency distribution given below find the unbiased and efficient estimates for the mean and variance

Xi 60 61 62 63 64 65 66 67 68 fi 02 00 15 29 25 12 10 04 03

3. The sample mean of a population was recorded as 184.67 with a probable error of

0.236. Find the 99.74% confidence limits for the true (population) mean. 4. The S.D of life time of 200 electric bulbs was computed to be 80 hours. Find (i) 95%&

(ii) 99% confidence limits for the S.D of all such bulbs. 5. How large a sample should one take in order to be (i) 99% & (ii) 99.74 % confident that

a population S.D will not differ from a sample S.D by more than 2%. 6. A die is thrown 9000 times and a draw of 3 or 4 observed 3240 times. Show that a die

cannot be regarded as an unbiased one. Also find the limits between which the probability of throw of 3 or 4 lies at 99.74% level of confidence

7. A mean of a sample of size 900 is 3.4.Can the sample be reasonably as a true random sample for a large population with means 3.25 and S.D 1.61

8. Ten screws are chosen at random from a population and their lengths are found as (in mms) 63,63,66,67,68,69,70,70,71,71. On the basis of this information can we say that the mean length in the population is 66mm at 95%confidence level?

9. Find 99% confidence limits for the correlation coefficient, which is computed to be 0.60 from a sample of size 28

TESTING OF HYPOTHESIS:

1. An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a mean of 500 hours and a S.D of 40 hours. Test the hypothesis Ho: μ ≠ 800 of a random sample of 30 bulbs has an average life of 788 hours. Use 5 % level of significance.

2. Test the hypothesis that the average content of containers of a particular lubricant is 10 liters if the contents of the random sample of 10 containers are 10.2, 9.7, 10.1, 10.3, 10.1, 9.8, 9.9, 10.4, 10.3 & Use 0.01 level of significance and assume that the distribution of contents is normal.

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3. A random sample of size n1 = 2.5 taken from a normal population with a S.D σ1 = 5.2 has a mean .811 =x A second random sample of size n2 = 36 taken from a different normal population with a S.D σ2 = 3.4 has mean 762 =x . Test the hypothesis that μ1 = μ2 against the alternative μ1 > μ2 at 5% level of significance.

4. A large automobile manufacturing company is trying to decide whether to purchase brand A or B tyres for its new models. To help arrive at a decision, an experiment is conducted using 12 of each brand. The tyres are run until they wear out. The results are

Brand A : kmsskmsx 5100,900,37 11 == Brand B : kmsskmsx 5900,800,39 22 == . Test the hypothesis at the 0.05 level of significance that there is no difference in the 2

brands of tyres. Assume the population to be approximately normally distributed. 5. Explain the following a) Tests of Hypothesis b) Type I and Type II errors find

mean and variance of the Chi square distributions.

JOINT PROBABILITY AND MARKOVCHAINS (20 marks)

1. Show that the vector (y, x) is a fixed point of the stochastic matrix P= ⎥⎦

⎤⎢⎣

⎡−

−yy

xx1

1

2. Find the unique fixed probability vector of the regular stochastic matrix ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

0102/102/14/14/12/1

3. If P= ⎥⎦

⎤⎢⎣

⎡2/12/1

1 ois the transition matrix with initial probability distribution ( ) ( )3/2,3/10 =p .

Define and compute.(a) ( )21

3p (b) ( )3p (c) ( )2

3p 4. A salesman’s territory consists of three cities A, B &C. He never sells in the same city on

consecutive days. If he sells in a city A, the next day he sells in city B. However if he sells in either B or C then the next day he is twice as likely to sell it in city A or in other city. Show that in the long run he sells 40% of the time in the city A, 45% of the time in city B and 15% of the time in the city C

5. A software engineer goes to his workplace everyday by motorbike or by car. He never goes by bike on 2 consecutive days but if he goes by car on a day then he is equally likely to go by car or by bike the next day. Find the transition probability matrix for the chain mode of transport he uses. If car is used on the first day of a week find the probability that after 4 day s (i) bike is used (ii) car is used.

6. A gambler’s luck follows a pattern. If he wins a game, the probability of winning the next

game is 0.6. However if he loses a game, the probability of losing the next game is 0.7. There is an even chance that he wins the first game. If so,

(a) Find the transition matrix M of the Markov process. (b) Find the probability that he wins the second game. (c) Find the probability that he wins the third game. (d) Find out how often, in the long run, he wins.

Define stochastic matrix. Find the unique fixed probability vector for the regular stochastic

matrix

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

01002

12

14

14

30

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APPLIED THERMODYNAMICS – ME 43

(2 hours / week for both IV ‘A’ and IV ‘B’ sections)

Faculty :Dr.T.R.Seetharam

Lecture Topics To be Covered % Coverage of syllabus 1 – 2 Review of Basic Thermodynamics – Applications of I & II law for closed and open systems, Important property relations for ideal gases, properties of pure substances ; Use of tables & charts Nil 3 – 6 Gas Power Cycles: Introduction; Air – standard cycles for IC engines – analysis of Carnot,Otto,Diesel & Dual- combustion cycles ; 08 7- 11 Gas Turbine Cycles – Simple Brayton cycle, modified Brayton cycle for improvement in work output & thermal Efficiency, deviations of practical cycles from ideal cycle 08 12 – 17 Vapour Power cycles- Carnot vapour power cycle and its limitations; Simple Rankine cycle; effects of pressure & temperature on performance of Rankine cycles; modifications of simple Rankine cycle to increase net work output & thermal efficiency – Reheat cycle, Regenerative cycle, types of feed water heaters used in regenerative cycles; Reheat-Regenerative cycle; practical vapour power cycles . 12 18 - 20 Refrigeration – Definition of Refrigeration, refrigerated space, refrigerant Refrigeration cycle, refrigeration effect; units of refrigeration effect – Ton of refrigeration; COP ; Carnot refrigerator – analysis and its limitations; Air refrigeration plant – Bell-Coleman / Reversed Brayton cycle; practical air refrigeration cycles 06 21 - 23 Vapour compression refrigeration cycle – analysis, effects of sub-cooling and superheating on the performance ; desirable properties of refrigerants for vapour compression cycle; steam jet refrigeration 06 24 - 25 Psychometrics – Thermodynamics of air-water vapour mixture – definitions Of moist air, dry air, specific humidity, relative humidity, dries – bulb and wet – bulb and dew point temperatures; adiabatic saturation temperature, saturated unsaturated air, enthalpy of moist air, construction and use of psychometric chart 04 26 - 28 Thermodynamic analysis of psychometric processes like heating, cooling Heating & humidification, cooling and de-humidification, adiabatic mixing Air streams, summer and winter air conditioning 08

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ME 43: Applied Thermodynamics

Tutorial 1: Problems on Gas Power Cycles A. Problems on Carnot Cycle 1.1. A Carnot power cycle using air as the working substance, works between the

emperature limits of 900 K and 300 K. The pressure limits are 60 bar and 1 bar. Find (i) pressure and temperature at salient points of the cycle, (ii) heat supplied and heat rejected per unit mass of air, (iii) net work output and thermal efficiency, and (iii) mean effective pressure.

1.2. The maximum pressure and temperature in a Carnot cycle is limited to 20 bar and 400

C. The volumetric ratio of isentropic compression is 6 and the volumetric ratio of isothermal expansion is 1.5. The volume of air at the beginning of isothermal expansion is 0.1 m3. Find (i) the minimum temperature in the cycle, (ii) Thermal efficxiency of the cycle, (iii) power output form the cycle if the number of cycles per minute is 200.

1.3. In an air standard Carnot cycle heat is transferred to the working fluid at 1110 k and

heat is rejected at 273 K. The heat transfer to the working fluid is 105 kJ / kg. The minimum pressure in the cycle is 1 bar. Determine (i) the thermal efficiency and (ii) the m e p.

1.4. A Carnot engine converts 1/6 of the heat input into work. When the temperature of the

sink is reduced by 70 C, the efficiency of the cycle is doubled. Determine the temperature of the source and the sink.

B. Problems on Otto Cycle 1.5. In an air standard Otto cycle the maximum and minimum temperatures are 1400 C and

15 C respectively. The heat supplied is 800 kJ / kg. Calculate the compression ratio and the thermal efficiency. Also calculate the ratio of maximum pressure to the minimum pressure in the cycle.

1.6. In an engine working on Otto cycle, the clearance volume is 50 cm3, while the stroke

volume is 350 cm3. If the temperature at the commencement of the compression process is 27 C and the maximum temperature in the cycle is 1000 C determine (i) the compression ratio, (ii) cycle efficiency, (iii) net work output per unit mass of air and (iv) mep.

1.7. From the p-v diagram of an engine working on Otto cycle, it is found that the pressure

inside the cylinder after 1/8th of the compression stroke is completed is 1.4 bar. After 5/8th of the compression stroke is completed, the pressure was found to be 3.5 bar. The maximum temperature in the cycle is limited to 1000 C. Determine (i) the compression ratio,(ii) air standard efficiency, (iii) net work output per unit mass of air, (iv) mep, (v) compression ratio corresponding to maximum work output, (vi) maximum work output and (vii) thermal efficiency corresponding to maximum work output. Assume that the minimum temperature in the cycle to be 27 C.

1.8. An engine working on Otto cycle has a volume of 0.5 m3 , a temperature of 27 C and a

pressure of 1 bar at the beginning of the compression process. At the end of the compression process the pressure is 10 bar. Heat added is 200 kJ /kg. Determine (i) percent clearance, (ii) air standard efficiency, (iii) MEP, (iv) power developed by the engine if there are 200 cycles per minute.

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1.9. Derive an expression for the air standard efficiency of a cycle similar to Otto cycle except that the compression process is isothermal, in terms of the compression ratio, maximum cycle temperature ratio and the ratio of specific heats.

1.10. The compression curve in an Otto cycle may be approximated by the equation pV1.35 =

constant. The expansion curve is isentropic. The maximum temperature in the cycle is 1000 K. If the temperature of air at entry to the engine is 27 C, find the thermal efficiency and the relative efficiency based on the air standard Otto cycle. The compression ratio is 7.5. Assume the specific heats of air to be constant and γ = 1.4.

Tutorial 2 : Gas Power Cycles (continued)

C. Problems on Diesel Cycles 2.1. The compression ratio of an air standard diesel cycle is 14 and the cut-off ratio is 2.2. At the beginning of the cycle the pressure and temperature of air are 0.98 bar and 300

K respectively. Find (i) pressure and temperatures at salient point of the cycle, (ii) the net work output per unit mass of air, (iii) thermal efficiency, (iv)MEP, and (iv) specific air consumption in kg/kWh.

2.2. An air standard diesel cycle has a compression ratio of 18 and the heat transfer to the

working fluid is 1800 kJ/kg. At the beginning of the compression process the pressure and temperatures are 1 bar and 300 K respectively. Determine (i) air standard efficiency, and (ii) MEP

2.3. An oil engine works on a diesel cycle with compression ratio of 20. Heat addition takes

place up to 10 % of the stroke. Initial pressure and temperature of air are 1 bar and 27 C. Assume that the compression process is according to the law pv1.32 = constant and the expansion process is according to the law pv1.30 = constant. The bore and stroke of the engine are 16 cm and 20 cm respectively. Find (i) the pressure and temperature at salient points of the cycle, (ii)MEP, (iii) thermal efficiency, (iv) relative efficiency

2.4. In a diesel cycle, the pressures at two points on the compression curve are 1.7 bar and

13.4 bar, respectively, corresponding to positions where 3/10th and 9/10th of the stroke have been executed. Find the compression ratio if the compression and expansion indices are 1.38 and 1.3 respectively. If the cut-off ratio is 1.8, determine the cycle efficiency and the relative efficiency based on the air standard cycle.

2.5. In an air standard diesel cycle, air is compressed isentropically from 26 C and 105 kPa

to 3.7 kPa. The entropy change during heat rejection is − 0.6939 kJ/(kg – K). Determine (i) heat supplied per kg of air, (ii) thermal efficiency, (iii) maximum temperature in the cycle, and (iv) temperature at the start of the heat rejection.

D. Problems on Dual Combustion cycles 2.6. The compression and expansion ratio of an oil engine working on a dual cycle is 9 and

5 respectively. The initial pressure and temperature are 1 bar and 30 C. The heat added at constant pressure is twice that at constant volume. The cyclider bore is 250 mm and the stroke is 400 mm. Determine (i) thermal efficiency and (ii) MEP

2.7. The maximum and the compression pressures in a dual cycle are 64 bar and 32 bar

respectively. The compression curve is polytropic with index n = 1.35. The pressure in the cycle after 1/3rd of the compression stroke is completed is 1.65 bar. If 60 % of the energy addition occurs at constant volume while 40 % occurs at constant pressure, find

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(i) the compression ratio, (ii) the suction pressure, (iii) work output from the cycle if the expansion index is 1.34, and (iv) thermal efficiency.

2.8. A diesel engine works between the temperatures of 1250 C and 25 C. The energy

addition during combustion is 550 kJ /kg. A dual combustion cycle operates between the same temperature limits, and has the same total energy addition as for diesel cycle except that this energy is equally divided between the constant volume and constant pressure processes. Compare the efficiencies of the two cycles. Hence show using T-s diagram that the diesel cycle is more efficient than the dual cycle under the same maximum and minimum temperatures as well as the same amount of heat addition.

2.9. In a dual cycle, two thirds of the total energy added occurs at constant volume.. If the

compression ratio is 15, and the maximum pressure in the cycle is 53 bar, compute(i)the temperatures at the salient points of the cycle, and (ii) thermal efficiency. Assume standard conditions of air at the start of the compression process.

2.10. The compression ratio for an engine working on dual cycle is 7. The cylinder diameter

is 25 cm and the stroke is 30 cm. The air at the start of the compression is at 101 kPa and 20 0C. At the end of the constant volume process, the pressure is 5600 kPa. If heat is added at constant pressure during 3 percent of the stroke, compute (i) the net work output from the cycle, (ii) the thermal efficiency, (iii) the amount of heat added, and (iv) the mean effective pressure

Tutorial 3: Gas Power Cycles (Gas Turbine Cycles)

3.1. An air standard Brayton cycle has air enter the compressor at 27 C and 100 kPa. The pressure ratio is 10 and the maximum allowable temperature is 1350 K. Determine (i) pressure and temperature at salient points of the cycle, (ii) compressor and turbine work per unit mass of air, (iii) net work output and work ratio, (iv) thermal efficiency and specific air consumption in kg/(kWh).

3.2.The pressure ratio of an open gas turbine cycle is 6.The compressor inlet conditions are

1 bar and 15 0C. The maximum temperature in the cycle is 800 0C. The compressor efficiency is 85 %, the turbine efficiency is 90 % and the combustion efficiency is 95 %. There is a pressure drop of 2 % of the inlet pressure in the combustion chamber. The calorific value of the fuel used is 42,000 kJ/kg.Assuming that cp and γ remains same throughout the cycle and equal to those values for air determine (i) Net work output per unit mass of air, (ii) Air-fuel ratio, (iii) thermal efficiency of the plant, (iv) specific fuel combustion in kg / kWh, and (v) power output form the plant for a mass flow rate of air of 1.0 kg / s.

3.3. The isentropic discharge temperature for the air flowing out of a compressor is 195 0C

while the actual temperature is 240 0C. The conditions of air at compressor inlet are 1 bar and 17 0C. If the air-fuel ratio in the combustion chamber is 75:1 and net power output is 650 kW, compute (i) the isentropic efficiencies of the compressor and turbine and (ii) the overall cycle efficiency. Assume that the plant consumes 5.2 kg/min of fuel supplied and the calorific value of the fuel used is 42,000 kJ/kg.Also assume that for air cp = 1.005 kJ/kg – K and γ = 1.4 and cp = 1.148 kJ/kg – K and γ = 1.333 for products of combustion.

3.4. Determine the thermal efficiency of a gas turbine cycle having two stages of

compression and two stages of expansion. The overall pressure ratio of the cycle is 4. Air enters both the stages of compression at 15 0C and enters both the stages of turbine at 900 0C. If an ideal regenerator is incorporated in the cycle to heat the air coming out of the second stage compressor, what would be the improvement in the

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thermal efficiency of the cycle. What would be the net work output and cycle efficiency if the compressor and expansion stages have efficiencies of 80 % each and the regenerator effectiveness is 75 %

3.5. Determine the specific work output, specific fuel consumption and cycle efficiency for a

gas turbine power plant using a regenerator having the following specifications.

Compressor pressure ratio 4.0 Turbine inlet temperature 1100 K Isentropic efficiency of compressor 0.85 Isentropic efficiency of turbine 0.87 Mechanical transmission efficiency 0.99 Combustion efficiency 0.98 Heat exchanger effectiveness 0.80 Pressure losses :- (i) Combustion chamber 2 % of compressor delivery pressure (ii) Heat exchanger air side loss 3 % of compressor delivery pressure (iii) Heat exchanger gas side loss 0.04 bar

3.6. An ideal gas turbine power plant operates with ‘m’ number of stages of compression and ‘n’ number of stages of exp[ansion. The maximum temperature permitted in the plant is Tmax. Pressure ratios in all the compressor stages are equal and expansion ratios in all the turbine stages are equal. The intercoolig between the compressor stages is perfect and the working fluid is reheated to Tmax in between the stages of expansion.If ‘t’ represents the maximum cycle temperature ratio in the cycle show that

t = (r a) (m+n)/mn

where r = ratio of maximum pressure to minimum pressure in the plant and a = (γ – 1) / γ.

3.7. A Brayton cycle works between 1 bar, 300 K and 5 bar, 1250 k. There are two stages

of compression with perfect intercooling and two stages of expansion. The work output from the first stage turbine is used to drive the two compressors. The air from the first stage turbine is reheated back to 1250 K and then expanded in the second stage. Calculate the power output from the plant and the cycle efficiency. What would be the power output and cycle efficiency if an ideal regenerator is incorporated in the cycle?

3.8.In a gas turbine unit with two-stage compression and two-stage expansion, the gas

temperature at the entry to both the turbines is the same. The intercooler in between the two stages of compression has an effectiveness of 83 %. The maximum and minimum temperatures in the cycle are 25 C and 1000 C. The low and high pressure limits are 1.02 bar and 7 bar. Assuming that the intermediate pressure between the two stages of compression is same as that at the exit of the first stage expansion and equal to the geometric mean of the high and low pressures of the unit, calculate (i) the air fuel ratio if the calorific value of the fuel used is 38650 kJ/kg, (ii)power output for an air flow rate of 1 kg/s. Assume that a regenerator with an effectiveness of o.80 is incorporated in the cycle. The pressure drop through the regenerator and first combustion chamber is 0.1 bar, while the pressure drop through the second combustion chamber is 0.05 bar. The exhaust pressure of the second turbine is 1.15 bar. The pressure drop through the intercooler is negligible. Assume the compressor efficiency is 0.84 for both trhe stages and turbine efficiency is 0.89 for both the stages.

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3.9. In a closed cycle gas turbine plant, the compressor inlet and exit conditions are 5 bar and 32.5 bar respectively. After passing through a regenerator with an effectiveness of 0.83, the air is heated in a nuclear reactor to 945 K. The pressure drop in the regenerator and the reactor reduces the air pressure at the turbine inlet to 31.5.bar. After expansion to 5.25 bar in the turbine with an efficiency of 0.88, the air passes through the regenerator and a cooler, before being ready to enter the compressor ( efficiency = 0.80) again at 20 C. Calculate (a) the cycle thermal efficiency, (b) the turbine and compressor power, and (c) the heat transfer at the reactor and the net air flow rate if the net power output from the plant is 650 kW.

3.10. The schematic diagram of a gas turbine power plant is shown in Fig. P1.30.The states

of the flowing gas at various points along the circuit are numbered. The following are the data referring to these states.

T1 = 20 C; Pressure ratio for each compressor = 3.5; Compressor efficiencies = 0.84;

Intercooler effectiveness = 0.95; pressure ratio across the intercooler = 0.96; pressure ratio across the regenerator = p5 / p4 = 0.95; regenerator effectiveness = 0.85; enthalpy of combustion of fuel = − 42,000 kJ/kg; combustion efficiency = 0.98; pressure ratio across the combustion chamber = p6 / p5 = 0.97; T6 = 1110 K; T8 = 1050 K; turbine efficiencies = 0.88. Pressure drop for gases leaving the turbine and flowing back through the regenerator = p10 / p9 = 0.97; pressure drop across intake = p1 / p∞ = p∞ / p10 = 0.98. Calculate (a) T2,T3,T4,T5,T7 and T9, (b) the total work required to drive the compressors and hence the output of turbine T1, (c) the air fuel ratios at the two combustion chambers,(d) the net work output from the plant, (e) the overall cycle thermal efficiency, and (f) the specific power output from the plant in kW/(kg – s) of air flow through the compressor.

Tutorial 4 : Vapour Power Cycles

C1 C2 T1 T2

CC1

CC2 Regenerator

Inter cooler

1

2 3

4 5

6

7 8 9

10

Fig. P 4.4 : Figure for problem P1.30

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4.1. A simple Rankine cycle dry saturated steam at 20 bar enters the turbine and the condenser pressure is 1 bar. Determine (i) the pump work, (ii) the turbine work, (iii) the net work out put, (iv) the thermal efficiency, (v) quality of steam entering the condenser and (vi) steam rate in kg/kWh. What would be the corresponding values when the condenser pressure is decreased to 0.1 bar other things being the same.

4.2. Steam conditions at the boiler exit are 10 bar and 300 C. In the pipe line between the

boiler exit and turbine inlet, there is an energy loss of 42 kJ/kg and a drop in pressure of 0.2 bar. The steam expands in the turbine to a pressure of 0.09 bar at the turbine exit, the efficiency of the turbine being 0.86. Find the stream conditions at the turbine inlet, the actual enthalpy drop across the turbine and the final condition at discharge from the turbine.

4.3. In a reheat cycle, the boiler exit conditions are 25 bar and 300 C. the exit pressure of

steam at the end of the first turbine is 5 bar. The steam is reheated to 300 C before it is expanded in the second turbine to 0.05 bar. Assumiing the high and low pressure turbines to have efficiencies of 0.87 and 0.85 respectively, determine (i) the thermal energy input in the reheater, (ii) the overall thermal efficiency of the cycle, (iii) specific steam consumption in kg/kWh, and (iv) net power output from the cycle for a mass flow rate of 1.0 kg/s.

4.4. Steam at 50 bar and 350 C expands to 12 bar in a high pressure stage and is dry

saturated at the stage exit. This is now reheated to 280 C without any pressure drop. The reheated steam expands in an intermediate stage and again emerges as dry saturated steam at a lower pressure, to be reheated a second time to 280 C. Finally, the steam expands in a low pressure turbine to 0.05 bar. Assuming the work output is the same for the high and intermediate stages, and the efficiencies of the high and low pressure stages are equal, find: (i) efficiency of the high pressure stage, (ii) pressure of steam at exit of the intermediate stage, (iii) total power output from the three stages for a mass flow rate of 1.0 kg/s, (iv) condition of steam entering the condenser, and (v) thermal efficiency of the cycle

4.5. Determine the improvement in thermal efficiency which will result if one stage of

regenerative feed heating is added to a simple Rankine cycle which has the boiler exit condition of 14 bar and 300 C and a condenser pressure of 0.08 bar. Steam for feed heating is to be extracted at 2.0 bar.

4.6. In a reheat – regenerative steam power plant cycle, the HP turbine receives steam at

20 bar and 300 0C. After expansion to 7 bar, the steam is reheated to 300 0C and then it expands in an intermediate stage to 1 bar. A fraction of the steam is now extracted for feed water heating, while the remaining steam expands in a LP turbine to a final pressure of 35 mm of mercury. If the efficiencies of high, intermediate and low pressure stages are respectively 0.90, 0.88, and 0.87, determine the overall cycle efficiency of the plant.

4.7. Steam expands in a turbine from 30 bar, 360 0C to a condenser pressure of 0.04 bar.

The isentropic efficiency is 0.82 and the steam condition at any point in the turbine may be assumed to be on a straight line joining the initial and final states, drawn on an h – s chart. During expansion steam is bled at two stages where the pressures are 5 bar and 0.7 bar respectively. The heaters are of closed type, the condensed steam from the high pressure heater is being led to the steam space of the low pressure heater through a steam trap.The condensed steam from the low pressure heater is fed to the intake of the feed pump through a drain cooler. Assuming the feed water in each heater to be heated to the saturation temperature corresponding to the bled steam pressure for that heater and that the temperature of the condensate from the heaters at the exit of the drain cooler is 30.2 0C, find the overall thermal efficiency of the plant and the specific steam consumption..

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4.8. In a two-stage regenerative feed heating system, the steam conditions at the turbine

inlet and exit are 20 bar, 320 0C and 0.08 bar. The low pressure feed water heater receives steam at 0.9 bar and heats the feed water from 33.5 0C to 89.5 0C. The condensed steam is cascaded back into the condenser through a steam trap.For the high pressure heater, steam is extracted at 4 bar and heats the water from 89.5 0C to 140 0C. Its condensate is pumped by a drain pump into the boiler feed line. Find the mass flow rates of steam to each feed water heater per kg of steam generated in the boiler. If the total steam generation is 18,000 kg/h, find the power output of the plant in kW. Assume the turbine efficiency to be 0.82 and the condition of steam at any point in the turbine to be on a straight line on the h – s diagram connecting the steam states at inlet and exit. Draw a schematic for the system and indicate all the salient points on the Mollier diagram. Also determine the overall cycle efficiency of the plant and specific steam consumption.

4.9. An ideal Rankine cycle with regenerative heating operates between the pressure limits

of 10 MPa and 40 kPa. Temperature of steam at turbine inlet is 500 0C. There are two open feed water heaters. On the basis of optimum design, determine (i) the pump work to turbine work, (ii) the ratio of heat rejection to heat addition, (iii) the mass of steam bled out for each feed water heater per unit mass of steam generated in the boiler, and (iv) cycle thermal efficiency.

Tutorial 5 : Refrigeration Cycles A. Problems on Air Refrigeration cycles 5.1. A reversed Carnot cycle is used for heating and cooling. The work supplied is 10 kW. If

the COP is 3.5 for cooling determine (a) the ratio of maximum temperature to minimum temperature in the cycle , (b) refrigeration effect in tons and (c) COP if the cycle is used as a heat pump.

5.2. An ideal air refrigeration cycle has the following specifications:

Pressure of air at compressor inlet = 101 kPa; Pressure of air at turbine inlet = 404 kPa; Temperature of air at compressor inlet = −6 C; Temperature of air at turbine inlet = 27 C; Determine (i) The COP of the cycle, (ii) Power required to produce 1 ton of refrigeration, and (iii) air circulation rate per ton of refrigeration.

5.3 In an air refrigerating machine, the compressor takes in air at 1 bar and 10 C. After compression to 5.5 bar, the air is cooled to 30 C before expanding it back to 1 bar. Assuming ideal conditions, determine (i) refrigeration effect per unit mass of air,(ii)heat rejected by air per unit mass in the intercooler, and (ii) COP of the cycle, In an actual plant using the above cycle, the air flow rate is 1700 kg / h and the relative COP of the actual plant is 0.65. Determine the power required for the actual plant for the same refrigeration effect

5.4:- An air refrigeration system is to be designed according to the following specifications:

Pressure of air at compressor inlet = 101 kPa; Pressure of air at compressor exit = 404 kPa; Temperature of air at compressor inlet = − 6 C; Temperature of air at turbine inlet = 27 C; Isentropic efficiency of compressor = 85 %; Isentropic efficiency of turbine = 85 %; Relative pressure drop in each heat exchanger = 3 %

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Capacity of the plant = 1 ton Determine (a) COP of the cycle, (ii) Power required in kW, and (iii) air circulation rate

5.5:-An air refrigerator unit uses a reciprocating compressor and a reciprocating expander.

5 kg / min of air at 30 C (ambient temperature is 25 C) and 4.8 bar expand behind a piston to 1 bar. The expansion is according to the law pv1.35= constant. After expansion, the air enters a cold chamber where its temperature rises to 0 C and the it is compressed back to 4.8 bar according to the law pv1.28 = constant. Determine (a) the power required to drive the unit if the mechanical efficiencies of the expander and the compressor are both equal to 85 %, (b) capacity of the refrigerator in tons, (c) energy rejected by air to the ambient during the cooling process at 4.8 bar and (d) the actual COP of the plant.

5.6:-In an ideal air refrigeration cycle, air after compression in the compressor is first

cooled in an intercooler and then passed through a regenerative heat exchanger. It is then expanded in a turbine and after expansion the air flows through the regenerative heat exchanger where it exchanges heat with the air coming from the intercooler. Then the cold air is passed through the cold chamber before it enters the compressor.(a) Draw the schematic layout of the plant.(b) obtain an expression for the COP of the cycle in terms of the pressure ratio of the compressor and the temperature ratio of the compressor inlet temperature to the turbine inlet temperature.

5.7 :- An air refrigeration unit takes in air from a cold chamber at 5 C and compresses it

from 1 bar to 6.5 bar. The index of compression is 1.25. The compressed air is cooled to a temperature which is 10 C above the ambient temperature of 30 C before being expanded isentropically in an expander. Neglecting the clearance volume of the compressor and expander find the COP and the amount of air circulation per minute if 2000 kg of ice at 0 C is to be formed per day from water at 25 C. What will be the tonnage of the unit?

B. Problems on Vapour Compression Refrigeration Cycles 5.8:-In a vapour compression refrigeration system using ammonia, the evaporator

temperature is − 15 0C and the condensation temperature is 30 0C. The vapour before entering the compressor is just dry and saturated. If there is no sub-cooling of the refrigerant after condensation, compute (i) the COP of the unit, (ii) the ideal power required to produce 1 ton of ice at 00C per day from water at 27 0C and (iii) the actual power required if the actual COP of the unit is 50 % of that of the Carnot COP working between the same evaporator and condensation temperature.

5.9:- Repeat part (i) and (ii) of the above problem if the refrigerant after condensation is

found to be subcooled by 6 0C. Assume that the compression in the compressor occurs polytropically with a compression index of 1.14.

5.9:-In a refrigerator using ammonia as the refrigerant, the condensation and the

evaporator temperatures are 330C and − 12 0C respectively. The vapour entering the condenser is 60 0C and the compressor piston sweeps a volume of 400 lit/minute. If the liquid emerging from the condenser is sub-cooled by 4 0C compute (i) the isentropic efficiency of the compressor, (ii) the mass flow rate of ammonia, assuming the clearance volume to be zero, (iii) the refrigeration effect in tons and (iv) the COP

5.10. An ideal vapour compression refrigerator uses a sub-cooling cum superheating heat

exchanger where the refrigerant vapour coming from the evaporator in dry saturated state is superheated by 10 0C by absorbing heat from the saturated liquid refrigerant coming from the condenser.The evaporator operates at − 30 0C and the condenser pressure is 1.4 MPa. There is a cooling requirement of 50 tons. If the refrigerant is R-

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B.E. Mechanical 4th Semester Course Information

12, determine (i) the mass flow rate of the refrigerant, (ii) the COP, (iii) the degree of subcooling, (iv) the power required to drive the compressor.

Tutorial 6 : Air Conditioning 6.1. Moist air at 40 0C,101.325 kPa, and a relative humidity of 60% initially is cooled at a

constant mixture pressure to 20 0C. Determine: (a) The final relative humidity. (b) Change in specific humidity.

6.2. The pressure and temperature in a room are 101.325 kPa and 25 0C. If the relative

humidity is 40% determine: (a) Saturation pressure of water vapour at the dry bulb temperature. (b) The dew point temperature. © Specific humidity. (d) Degree of saturation.

6.3. Moist air is at a temperature of 21 0C under a total pressure of 736mm of Hg. The dew

point temperature is 15 0C. Find: (a) Partial pressure of water vapour. (b) Relative humidity. © Specific humidity. (d) Enthalpy of air per kg of dry air. (e) Specific volume of air per kg of dry air.

6.4. Calculate:

(a) Relative humidity. (b) Humidity Ratio. © Dew point temperature. (d) Density. (e) Enthalpy of atmospheric air.

When the DBT is 35 0C, WBT is 23 0C and the barometer reads 750mm of Hg. 6.5. A sample of air has DBT of 35 0C and 25 0C respectively. The barometer reads 760mm

of Hg. Calculate: (a) Humidity ratio, Relative humidity and Enthalpy of the sample. (b) Humidity ratio, Relative humidity and Enthalpy if the air were adiabatically

saturated. The use of steam tables only is permitted. 6.6. Find the heat transfer rate required to warm 40 m3/min of air at 32 0C and 90 % RH to

50 0C. 6.7. Air at 15 0C and 80%RH is conditioned to 25 0C and 50%RH. Determine the amount

of water added per kg of dry air. Assuming the make up water is added at 15 0C determine the heat supplied during the process.

6.8. Warm air is to be cooled by an adiabatic humidification process. At the beginning of the

process, the air is at 45 0C and 30%RH. The final temperature is 30 0C . Determine: (a) The amount of water added to the air. (b) The final relative humidity. Solve the problem using steam tables only and compare the answers with those

obtained using psychrometric chart. Assume the total pressure of air to be 101.325 kPa.

6.9. Air is to be conditioned from a DBT of 40 0C and a RH of 50% to a final DBT of 200C

and a final RH of 40% by a dehumidification process followed by a reheat process.

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B.E. Mechanical 4th Semester Course Information

Assume that the entire process is carried out at a constant pressure of 101.325 kPa. Determine: (a) The amount of water to be removed from air. (b) The temperature of air leaving the dehumidifier. (c) Refrigeration in tons for an air flow rate of 0.47 m3/s and heating required in kW.

6.10.A Stream of air at atmospheric pressure, 20˚ C and 30% RH, flows at a rate of 15

m3/min and mixes adiabatically with another stream of air at 35˚ C and 80% RH at 20 m3/min. For the mixed stream calculate: (a) Specific Humidity. (b)Temperature. (c) Relative Humidity. (d) Specific Volume.

6.11. Atmospheric air at 12˚ C and 25 % RH is to be conditioned to a humidity ratio

of 0.005 kg of water vapor / kg of dry air as it enters an insulated room with a flow rate of 60 m3 / min. Assuming that the humidifying water is at 12˚ C. Determine RH, the Temperature of the conditioned air and heat transfer per rate for the following humidifying process. (a) Constant Dry Bulb Temperature (b) Constant Relative Humidity (c) Adiabatic evaporative process.

6.12. The appended figure shows the air condition in a central air conditioning plant,

provided with a refrigeration circuit. It is meant to supply conditioned air at 20˚ C Dry bulb temperature and 66% RH. The return air is 300 kg / min. While the make up air is 20 kg / min taken from atmosphere. Find:

(a) The heat transfer at the cooling coil. (b) The amount of the humidification per hour. (c ) The heating coil capacity and (d) The COP of the refrigeration from unit. 6.13. 39.6 m3/min of a mixture of recirculated room air and outdoor air enters a cooling coil at 31 0C DBT and 18.5 0C WBT. The effective surface temperature of the coil is 4.4 0C. The surface area of the coil is designed so as to give 12.5 kW of refrigeration with the given entering state of air. Determine the dry and wet bulb temperatures of air leaving the coil and the coil bypass factor.

FLUID MECHANICS – ME 45 Faculty : V. Krishna / Ramachandra L.

% of portions covered Class #

Chapter title/ Reference Literature

Topic to be Covered Reference Chapter

Cumulative

1-2

Introduction to fluid mechanics, properites of fluids - Mass density, specific volume, specific weight, specific gravity.

3

Chapter : 1.0 properties of fluids

T1: Page : 3-17 T2 : Page : 13- 36

R1 : Page : 1-31 Surface tension, capillarity

12% 12%

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B.E. Mechanical 4th Semester Course Information

4-5 Bulk Modulus, Compressibility, Viscosity and Newton's law of viscosity, Numerical Problems.

6

R4 : Page : 9-36 R5 : Page : 1-29 R8 : Page : 1-25

Classification of fluids & regimes of flow

7-8 Pressure variation in a static fluid, Pascal's law of pressure.

9-10 manometers: Simple and Differential U tube manometers, Numerical Problems

11-12

Hydrostatic forces and location of hydrostatic forces on submerged plane surfaces and curved surfaces, Numerical problems.

13-14

Chapter : 2.0 fluid statics

T1:Page # : 23-61 T2:Page # : 142-144

& Page # : 59-77 R1:Page # : 31-134 R4:Page # : 43-85

R5:Page # : 104-138 R8:Page # : 51-80

Buoyancy and floatation, Metacentre, Stability of floating bodies, Determination of metacentric height by experimental and analytical methods, Numercial problems

16% 28%

15-16 Fluids flow concepts, Lines of flow -path line, Stream line, Streak line, Stream tube

17-19 Continuity equation in cartesian co-ordinates. Types of fluid flow.

20-21

Chapter # : 3.0 Fluid kinematics

T1:Page# : 320-334 T2:Page# : 204-218 R1:Page# : 139-182 R4:Page# : 104-141 R5:Page# : 31-57 R8:Page# : 36-44

Stream function for 2D flow, velocity potential function for 2D flow, Relationship between stream function and velocity potential function. Flow net. Numerical problems

12% 40%

22-23

Chapter # : 4.0 Dimensional analysis T1:Page # : 156-173

T2:Page # : 138-301 R1:Page

# : 502-549 R4:Page # : 464-478 R5:Page # : 230-258 R8:Page # : 245-281

Dimensions of physical quantites, Dimensional homogenity, Buckingham pi theorem, Numerical problems.

8% 48%

24-25 Raleigh's method, Important dimensionless numbers, Similitude.

26-27

Introduction-Forces acting on a fluid mass, General energy and Momentum equation, Euler's equation of motion along stream line.

28-29

Chapter # : 5.0 Fliud Dyanmics

T1:Page # : 85-110 T2:Page # : 250-276 R1:Page # : 233-240 R4:Page # : 171-189 Page # : 212- 220 R5:Page # : 76-92 R8:Page # : 132-

172

Bernoulli's Priniciple - Derivation from fundamentals, Euler's equation and Bernoulli's equation for real fluids.

12% 60%

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B.E. Mechanical 4th Semester Course Information

30-33

Chapter # : 6.0 Fluid flow measurements T1:Page # : 337-357 T2:Page # : 602-623 R1:Page # : 241-260 R4:Page : 223- 246 R5:Page # :155 -171

Application of Bernoulli's equation, Pitot tube, Orifice meter, venturimeter, V-notch, Numerical problems.

8% 68%

34-35

Viscous flow, Reynolds number, critical Reynolds number, laminar flow through pipes - Hagen - Poisuille's equation

36-38

Chapter # : 7.0 Laminar and Viscous

Flow

T1:Page # : 410-417 T2:Page # : 443-449 R1:Page # :420-450 R4:Page # : 318-416 R5:Page # : 362-378 R8:Page # : 598-602

Laminar flow between parallel stationary plates, Numerical problems

8% 76%

39-40 Friction loss in pipe flow - minor loss in pipe flow, Energy line & Hydraulic gradient line.

41-42

Chapter # : 8.0 Flow through pipes

T1:Page # : 182-199 T2:Page # : 391-417 R1:Page # :347-366

Darcy's & Chezy's equation, numerical problems

8% 84%

43-44 Lift & Drag, skin friction & form drag. Boundary layer concept.

45-46

Chapter # : 9.0 Flow past immersed

bodies T1:Page # : 210-222

T2:Page # : 997-1000 R1:Page # :552-556 and Page

# :591-624 R4:Page # : 347-377

R5:Page # : 406 -442

Calculation of laminar boundary layers thickness, displacement & momentum thickness.

8% 92%

47-48 Sonic velocity, Mach number, Isentropic flow, speed of sound wave.

49-50

Chapter # : 10.0 Introduction to

compressible flow T1:Page # : 262-271 R1:Page # : 636-642 R4:Page # : 442-460 R5:Page # : 535-552 R8:Page # : 511-570

Numerical problems.

8% 100%

Literature

Book Type Code Title and Author Edition Text Book T1 Fliud Mechanics by Streeter 7th

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B.E. Mechanical 4th Semester Course Information

T2 Fliud Mechanics and Hydraulics by Dr. Jagadishlal

Reference Book R1 A Text Book of fluid Mechanics and Hydralics

by Dr.R.K. Bansal

R2 Fluid Mechanics by Agarwal R3 Fluid Mechanics by Binder

R4 Engineering Fluid Mechanics by K.L.Kumar

R5 Engineering Fluid Mechanics by Dr.R.J.Garde

and Dr.A.J.Mirajgaoker

R6 Introduction to fluid Mechanics and Machinery by Som and Biwas

2nd

R7 Fluid Mechanics by John F.Douglas, Janul and M.Gasiosek and John.A.Swaffied

4th

R8 Fluid Mechanics by White 5th

QUESTION BANK

Chapter I: Properties of Fluids

1. * Define a fluid. Distinguish between Real and Ideal fluids. (3) 2. * State Newton’s law of viscosity. Hence define Absolute viscosity, Give its SI unit?

(4) 3. A standard bearing 500 mm long and 151 mm in diameter encases a shaft of 150mm

outer dia. The oil film enclosed between the shaft and the bearing has a viscosity of 0.9 poise. What is the power lost in friction., if the shaft revolves at 240 rpm? Find also the torque developed. (6)

4. * Define the following fluid properties giving their SI units. (i) Pressure (ii) Specific mass (iii) Specific Weight (iv) Relative density (v) Kinematic Viscosity (vi) Surface tension (vii) Capillarity (2 marks each)

5. What do you understand by the term capillarity? (2) Mention the fluids in which this fluid phenomenon is observed (2)

6. Derive an expression for the capillary rise. (6) 7. The dynamic viscosity of an oil used for lubrication between a shaft and sleeve is 6.0

poise. The shaft is of diameter 0.4 m. and rotates at 190 rpm. Calculate the power lost in the bearing for a sleeve length of 90 mm. The thickness of the oil film is 1.5 mm. (6)

8. A flat plate 0.1 m2 area is pulled at 400mm/s relative to another plate located at 0.15 mm from it, the fluid separating the two being water with μ = 1 centi-Poise. Find the force and power required to maintain this velocity. (6)

9. Explain the terms Compressibility and Bulk Modulus of Elasticity. (2)

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B.E. Mechanical 4th Semester Course Information

10. * Distinguish between Newtonian and Non-Newtonian Fluids. by means of a graph. (4)

11. If the equation of velocity distribution of a fluid over a plate is given by v = 2y-y2 in which ‘v’ is the velocity n m/s at a distance ‘y’ measured in meters above the plate, what is the velocity gradient at the boundary and at 75 mm from it. Also determine the shear stress at these points if the absolute viscosity, μ =8.6 poise. (6)

12. Prove that the relationship between surface tension and pressure inside a droplet in excess of inside pressure is given by P=4 σ/d, where σ = Surface tension, d = dia of droplet. ( 4)

13. The Kinematic viscosity and S.G. of a certain liquid are 5.58 Stokes and 2 respectively. Calculate the absolute viscosity of the fluid in i) Ns/m2 ii)poise. (4)

14. * Explain how certain insects are able to walk on the surface of water. (4) 15. The surface tension of water drop let in contact with air at 200 . is 0.071 N / m. The

pressure inside the droplet of water is 0.0196 N / cm2. greater than the outside pressure. Calculate the diameter of the droplet. (6)

16. A Capillary tube having an inside dia of 4mm is dipped in water at atmospheric temperature of 200 C. Determine the height of water which will rise in the tube. Take surface tension of water as 0.075 N / m and α = 600. What will be the % age increase in the value of height if the dia of the glass tube is 2mm. (6)

17. * What are the characteristics of the fluid properties to which the following phenomena are attributed?

(a) Rise of sap (liquid) in a tree. (b) Spherical shape of drop a drop of a liquid. (c) Rising of a ship as it sails fresh water to sea water. (d) A needle heavier than water can float if it is placed lengthwise on the surface

of water. 18. *Differentiate between

(a) Density and relative Density (b) Absolute viscosity and Kinematics Viscosity (c) Absolute pressure and Gauge pressure (d) Simple manometer and differential manometer.

19. *Give technical reason for (a) Viscosity of liquids decreases on heating whereas viscosity of gases increases on

heating (b) Water will wet clean glass but mercury will not.

20. * Determine the bulk modulus of elasticity of a liquid if the pressure of the liquid is increased form 70N/ cm2 to 130n/cm2. The volume of the liquid decreases by 0.15%.

21. * Give technical reasons for the following :

(a) certain insects are able to walk on the surface of water. (b) Viscosity of liquids decreses on heating whereas that of gases increases. (2)

21. * At a certain point in castor oil film, the shear stress in 0.2 N/m2 and the velocity gradient is 0.216s-1. If the mass density is 959.42 kg/m3, find the kinematic viscosity of the oil. (4) 22.* State Newton’s law of viscosity. A U-tube is made up of two capillaries of bore 1mm and 2mm respectively. The tube is held vertically and is partially filled with liquid of surface tension 0.05 N/m and zero contact angle. Calculate the mass density of the liquid if the estimated difference in the level of two menisci is 12.5mm. (2+6) 23.* A bubble of air is released at a depth of 1m in tank of water. If the diameter of the bubble at the time of release is 0.2mm, calculate the gauge pressure inside the bubble( Surface tension for the air water interface is 0.073 N/m). (4) 24.* With the help of a neat plot ( stress vs. rate of strain) show the characteristics

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B.E. Mechanical 4th Semester Course Information

behaviour for the following materials. (a) Elastic solid (b) Ideal fluid (c) Newtonian fluid (d) Ideal plastic (e) dilatant

Fluid (f) Pseudo plastic fluid. (6)

Chapter II: Fluid Statics 1. * (a) Distinguish between Pressure head and Pressure intensity

(b) Convert a Pressure head of 15m of water to (i) m of oil of SG 0.75 (ii) m of Hg of SG 13.6 ( 2+ 4)

2. State and prove Pascal’s Law. Give its applications. (8) 3. * Define the terms Gauge Pressure, Vacuum Pressure and absolute Pressure. Indicate

their relative positions on a chart. (4) 4. The pressure of a fluid is recorded as 60mm of Hg vacuum. What is the absolute

pressure of that fluid in terms of the following units. (i)N/m2 (ii) m of H2O. (3) 5. Distinguish between a simple and differential manometer. Give examples. (3) 6. Two pressure points in a water pipe are connected to a manometer, which has the form

of an inverted ‘U’ tube. The space above the water in the two limbs of the manometer is filled with toluene of SG 0.875. If the difference of levels of water columns in the two limbs is equal to 0.12m. What is the corresponding difference of pressure in N/m2. (6)

7. Show that the pressure at a point depends on the head of liquid above it in a static liquid 8. *Petrol of SG 0.8 flows upwards though a vertical pipe. A and B are two points in the

pipe, B being 300 mm higher than A. Connections are led from A and B to a ‘U’ tube containing Hg. If the difference of pr. between A and B is 0.18 N/mm2. Find the reading shown by the differential Hg gauge. (8)

9. Atmospheric pressure measured at a place showed 700 mm of Hg. What is the atmospheric pr. intensity in N/m2 and in metres of H2o if the SG of Hg is 13.56. (4)

10. * Define the terms Total Pressure and Center of Pressure (2) 11. * Show that for a vertical Lamina immersed in a liquid, the center of Pressure always

lies below the centroid. (6) 12. * Derive an expression for hydrostatic force on an inclined submerged plane surface and

depth of centre of pressure. (6) 13. A trapezoidal plate having its parallel sides equal to 2a and a at distance H apart is

immersed vertically in a liquid with 2 a side uppermost and at a distance H below the surface of the liquid. Find the thrust on the surface and the depth of center of pressure. (6)

14. * Derive an expression for the total pressure and center of pressure for an inclined surface immersed in a liquid. (4)

15. A flat angular ring of 30m ext. dia and 15 m int. dia is immersed in water such that its top most edge is 1m below and the lower most edge is 2m below the free surface of water obtain the location of center of pressure of the ring and the total pressure. (6)

16. A triangular gate which has a base of 1.5m and an altitude of 2m lies in a vertical plane. The vertex of the gate is 1m below the surface of a tank which contains oil of SG 0.8. Find the force exerted on the gate and the position of center of Pressure. (6)

17. * Define Metacentre and metacentric height . (2) 18. * With usual notations, Prove BM = I / V . (6) 19. A wooden cylinder of diameter “d” and length 2d floats in water with its axis vertical

.Is the equilibrium stable ? Locate the metacentre with reference to water surface. Specific Gravity of Wood is 0.6 . (6)

20. * What are the conditions of equilibrium of a floating body .Explain with reference to the metacentric height . (4)

21. * Explain with sketches the stability of a submerged body. (6) 22. *Explain the method of determining the metacentric height of a floating body

experimentally. (6) 23. A hollow cylinder of specific gravity 0.55 has an outer diameter of 0.6m and an inner

diameter of 0.3m and has its ends open. It is required to float in oil of specific gravity

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0.84 Calculate the maximum height of the cylinder so that it shall be stable when floating with its axis vertical .Find also the depth to which it will sink . (8)

24. Explain the terms buoyancy and centre of buoyancy . (3) 25. What is the significance of Metacentric height . (2) 26. A ship 60m long and 12m broad has a displaced water of 19620kN. A weight of 294.3kN

is moved across the deck through a distance of 6.5m .The ship is tilted through 5 . The moment of inertia of ship at water surface is 75% moment of inertia of the circumscribing rectangle. the centre of buoyancy is 2.75m below waterline. Find the metacentric height and position of C.G of the ship . Take specific weight of sea-water as 10104N/m3 . (6)

27. *A wooden cylinder having a specific gravity of 0.6 is required to float in an oil of specific gravity 0.8. If the diameter of the cylinder is ‘d’ and length ‘l’, show that ‘l’ cannot exceed 0.817d for the cylinder to float with its longitudinal axis vertical. (6)

28. A wooden cylinder of specific gravity 0.6 and diameter D and length L is required to float in oil of specific gravity 0.9. Find the L / D ratio for the cylinder to float with its longitudinal axis vertical. (8)

29. *Draw a rectangle parallelopied element of a fluid at rest, indicating the pressure on the faces. For the element derive the hydrostatic equation in the form p=γ h, where ‘p’ is the pressure intensity at a depth ‘h’ from a liquid surface of specific weight ‘γ’.

(10) 30. *Derive the criterion for stability of a floating body. (6) 31. *Determine the pressure difference Pa-Pb for the system shown below. (4)

32.* Determine the minimum force F, required to keep the gate closed, in the Figure below. The gate is a square of 0.5m side and hinged in the middle as Shown. The centre of the gate is 1m below the water surface.

ρw=1000kg/m3

Diameter D

Diameter

ρ1=.8 ρw

Diameter D

ρ3=0.6 H1

H2

H1=0.2m H2=0.05

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33.* calculate the horizontal and vertical forces, due to the gauge pressure of water On the cylinder portion of the tank shown below. The radius of the cylinder, R is 2m, the level of water in the tank, H is 5m and the width of the tank, W is 5m. The tank is open at the top. (8)

Chapter III: Fluid Kinematics 1. * Distinguish between

i) Steady flow and unsteady flow ii) Uniform flow and non uniform flow iii) Compressible and incompressible flow. iv) Laminar and turbulent flow. (8)

2. What do you understand by the term continuity equation. (4) 3. Explain in brief Lagrangian method and Eulerian method of studying fluid in motion. 4. * Define the following : (i) Path line (ii) Streak line

(iii) Stream line (iv) Stream tube (8) 5. * Obtain an expression for the continuity equation for a three dimensional flow. (6) 6. * Define the terms : 1)Velocity potential 2) Stream function (4) 7. Determine whether the continuity equation is satisfied by the following velocity

components for incompressible fluid: -

D

H

F

Hing

Gate ( 0.5*0.5

Wate

H=1m D=0.5m

H

W

H=5m W=5m R=2m

R

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B.E. Mechanical 4th Semester Course Information

u = x3 - y3 - z2 x ; v =y3 - z3 ; w = -3 x2 z - 3 y2 z + z3/ 3. (4) 8. The velocity components in a two dimensional flow field for an incompressible fluid are

as follows : u = y3 / 3 + 2 x - x2 y and v = x y2 - 2 y - x3 /3. Obtain an expression for the stream function. (6)

9. * What you mean by upper critical and Lower Critical Reynolds number. (4) 10. * Compare and contrast the following:

(a) Path line vs. streak line. (2) (b) 1D vs. 3D flow. (2)

11.* The stream function for 2D incompressible flow is given by: ψ= xy3+x2y (a) Find φ if it exists. (4) (b) What is the equation of the steam line passing through (1,1). (2)

12.* Given V= (xy+2zt)I+(2y2+xyt)J+(12xy)k where x,y and z are in metres and t in seconds, determine ax the x component of the acceleration of the fluid particle at (1,1,1) at t=1s. (4) Chapter IV: Dimensional Analysis 1. What is similitude. (3) 2. * Briefly explain [a] Geometric Similarity [b] Kinematic Similarity

[c] Dynamic Similarity (3) 3. Write a note on Model studies. (3) 4. * State Buckingham’s π theorem. (2) 5. The efficiency η of a fan depends on density, dynamic viscosity, angular velocity,

diameter of rotor & discharge. Express η in terms of dimensionless parameters. (6) 6. Define the following non-dimensional numbers. [i] Euler's Number [ii] Weber Number 7. The drop in pressure due to an obstruction in a pipe depends on the pipe diameter,

average velocity, mass density, Viscosity of fluid and the characteristic length of obstruction. Express the pressure drop in terms of dimensionless parameters. (6)

8. Using Buckingham's π- theorem, show that the velocity through a circular orifice is given by V = √ 2 g H X Φ ⟨ D / H , μ / , μ / ς VH ® where H is head causing the flow, D is the diameter of the orifice , μ is the viscosity , ς is the mass density and g is the acceleration due to gravity. (8)

9. * The rate of discharge Q of a centrifugal pump is dependent upon density ρ of the fluid, pump speed N (rpm), the diameter of the impeller D, the pressure P & the viscosity of fluid μ. Derive an expression for Q by using Buckingham's π theorem.(10)

10. * By Buckingham’s π theorem, obtain an expression for the frictional torque T of a disc of diameter D, rotating at speed N, in a fluid of viscosity μ and density ρ in a turbulent flow. (8)

11. *Assuming that the rate of discharge Q of a hydraulic machine is dependent upon the mass density ‘ρ’ and viscosity μ, show using Buckingham’s π theorem that it can be represented by Q= ND3 φ [9H/N2D2, ν/ND2] H being the head and ν the kinematic viscosity of the fluid.

12.*The distance traveled by a golf ball in still air, L , is known to be a function of the following: L=ƒ (Vo,D,d,μ,ω,m) Where, Vo is the initial velocity of the ball; D is the diameter of the ball ‘d’ is the diameter of the dimples; ρ is the density of air ω is the angular speed of the ball; μ is the viscosity of air and m is the mass of the ball.

(a)using ρ, Vo and D as repeating variable find all the revelant π groups. (8) (b) Experiments are to be conducted on a model ball that is twice as large as the actual golf ball. For dynamic similarity find the ratio of the initial velocity of the model to that of the actual ball. The fluid in both cases is air at STP. (2)

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B.E. Mechanical 4th Semester Course Information

Chapter V: Fluid dynamics 1. Name the different forces present in a fluid flow. For the Euler’s equation of motion,

which forces are taken into consideration?. (3) 2. Write Bernoulli’s equation of motion. Clearly explain the meaning of the terms in it.

(4) 3. A pipe of diameter 0.3m carries water at a velocity of 20m/s. The pressure at the points

A and B are given as 350KN/m2 and 300 KN/m2 respectively, while the datum head at A and B are 25m and 28m. Find the loss of head between A and B. (6)

4. * State and prove Bernoulli’s theorem making clearly the assumption’s made. (8) 5. * Derive Euler's equation along a streamline and reduce it to Bernoulli's equation. State

the assumptions made. (10) 6. Mention the devices that work on the above principle. (2) 7. A pipe line carrying oil of S.G 0.8 changes in diameter from 300mm at position A to

500mm diameter at position B which is 5m higher. If the pressures at A and B are 200KN/m2 and 152KN/m2 respectively and the discharge is 150l/s, determine the loss of head and direction of flow. (8)

8.* A 0.25 diameter pipe carries an oil of SG 0.8 at the rate of 120 l/s and the pressure at a point A is 19.62kN/m3. If the point A is 3.5m above the datum line, calculate the total energy at point A in m of oil. 9.*Derive Bernoulli’s equation using an infinitesimal stream tube. Clearly state all the assumptions made. (6) Chapter VI: Fluid flow measurements 1. What is a venturimeter. (2) 2. Derive an expression for the discharge through a venturimeter. (4) 3. * Draw neat sketches of venturimeter and orifice meter labeling all the main parts.

Distinguish between venturimeter and orifice meter. (8) 4. * Explain with a neat sketch the working of pitot's tube with inverted U tube differential

manometer. Derive an expression for the velocity for the same pitot's tube. (6)

5. A horizontal venturimeter with a inlet diameter 200mm and throat diameter 100mm is used to measure the flow of water. The pressure at inlet is 150KN/m2 and vacuum pressure at the throat is 400mm of Hg. Find the discharge of water through the meter. Take Cd= 0.98. (6)

6. A 30 mm X 15 mm venturimeter is provided in a vertical pipeline carrying oil of S.G. 0.9. The differential U-tube manometer shows a gauge deflection of 25 cm. Calculate the discharge of the oil. Take Cd of venturimeter as 0.98. (8)

7. What do you understand by Vena-Contracta. (2) 8. Define the hydraulic co-efficients of an orifice. How are they related. (3) 9. In order to determine experimentally the co-efficients of contraction, velocity and

discharge for a 100mm dia sharp orifice in the side of a tank , the following data were collected. Dia of jet at vena-contracta =78.42mm, H= 3.6m, Q=0.0385 m3/ s. Obtain Cc, Cv and Cd. (6)

10. Obtain an expression for the coefficient of velocity for a sharp edged orifice located in the side of a tank in terms of horizontal distance x, vertical distance y, traveled by the jet and head H over the orifice . (6)

11. A jet of water issuing from a 25mm dia orifice in the side of a tank drops 0.48m in a horizontal distance of 1.39m from vena-contracta. If it discharges 0.00131m3/s under a head of 1.07m, determine Cc, Cv and Cd. (4)

12. Distinguish between orifices and notches. (2) 13. * Derive an expression for discharge over a V-notch. (4) 14. * Mention the advantages of V-notch over rectangular notch. (3)

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B.E. Mechanical 4th Semester Course Information

15. In an experiment on a 900 V-notch , the flow is collected in a vertical cylindrical tank 0.9m dia. It is found that the depth of water in the tank increases by 0.65m in 16.8s, when the head over the notch is 0.2m. Determine the Cd of the notch. (4)

16. Find the depth and top width of a V-notch capable of discharging a maximum 0.7m3/s and such that head the head shall be 75mm for a discharge of 5.6 litres

17. / s. It’s Cd is same as that of a similar (in material and sharpness of edges only) right angled V-notch for which Q=1.407 H5/2. (6)

18. *Compare a Venturimeter and an orifice plate, based on the following points (a) Cost and ease of manufacture (b) Accuracy (c) Energy loss (d) Sensitivity ( output manometer deflection per unit flow rate) (4)

19.* Derive the head vs. discharge relation equation for a V notch. State all the assumption made. (6) 20.*The inlet and throat diameters of a vertically mounted venturimeter are 300mm and 100mm respectively. The throat is below the inlet at a distance of 100mm. The mass density of the liquid is 900kg/m3. The pressure intensity at the inlet is 140 kPa while at the throat is 80kPa. Calculate the flow rate. Assume that 2% of the differential head is lost between the inlet and the throat. (8) Chapter-VII: Flow through pipes & Chapter-VIII: Laminar & Viscous Flow 1. *Define Reynolds' Number. What is its physical significance? (4) 2. For the flow of fluid through a circular pipe show that the friction factor f = 16/Re.

(6) 3. Two reservoirs are connected by three pipes laid in parallel, their diameters being d, 2d

& 3d respectively and they are of the same length l. Assuming f to be the same for all pipes, determine the discharge through each of the larger pipes if the smallest pipe is discharging 1m3/s. (6)

4. * What are minor and major losses? Derive an expression to evaluate the loss of head due to sudden contraction. (10)

5. Write the expressions for the loss of head due to the following in pipes: Entrance to the pipe, sudden contraction, sudden expansion, Exit from the pipe, Pipe fittings. (5)

6. What is a compound pipe. Derive an expression for the equivalent size of a compound pipe. (4)

7. Two reservoirs are connected by a pipe 2250m long and 0.225m in diameter the difference in water levels being 7.5m. Determine the flow through the pipe in litres / min if f = 0.03. Also find the % increase in the discharge if for the last 600m a second pipe of the same diameter is laid alongside the first. (6)

8. Derive an expression for the loss of head due to friction for the laminar flow between two parallel plates. (6)

9. A horizontal pipe 50mm diameter carrying glycerine has shear stress at the pipe boundary as 196.s N/m2. Determine the pressure gradient, mean velocity and Reynolds' number. (6)

10. * For turbulent flow through a pipe, derive Darcy’s equation for the head loss. (8) 11. Two reservoirs are connected by a pipe 2250m long and 0.225m in diameter the

difference in water levels being 7.5m. Determine the discharge through the pipe if f = 0.0075. (6)

12. Write short notes on Water hammer. (3) 13. * Distinguish between Laminar and Turbulent flow. (4) 14. * Explain the terms

(i) Critical Reynolds number (ii) Critical velocity (iii) Transition Zone (6)

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B.E. Mechanical 4th Semester Course Information

15.* Derive the Darcy Weisbach equation for the loss of head due to friction in a pipe 16.*What are hydraulic gradient and total energy lines. 17.*Water is supplied to a town having a population of 1 lakh from a reservoir 6 km away from the town and it is stipulated that half of the daily supply of 150 litres per head should be delivered in 8 hours. What should be the dia of the supply pipe. The loss of head due to friction in the pipe line is 12m. Take Chezy’s constant as 45. (8) 18.*In the Chezy equation V=C(RS)1/2, explain the physical meaning of the terms C,R and S. (2) 19.*Derive the expression for the energy loss due to a sudden expansion in a pipe from area A1 to area A2(>A1), in terms of the inlet dynamic pressure ½ ρV12 Clearly state all the assumption made. (6) 20.*Starting from an appropriate control volume derive the expression for the velocity distribution for steady, laminar, fully developed flow of an incompressible fluid in a circular pipe. Further, show that the friction factor ƒ=2gDhf/LV2 is 64/Re for this flow. (10) 21.*For the system shown below qualitatively sketch the hydraulic and energy grade lines. The value is kept half open and the entry loss into the pipe is negligible

Chapter X: Introduction to compressible flow & Chapter IX: Flow past immersed bodies 1. Derive an expression for velocity of sound in a fluid in terms of bulk modulus.

(4) 2. A projectile travels in air of pressure 0.065 N/m2 and temperature - 70C. If the Mach

angle is 300C, find the velocity of projectile. K = 1.4, R = 287 J / Kg K. (4) 3. Find the velocity of bullet in standard air if the Mach angle is 30o. Take R = 287.14 J/ kg

o K and k = 1.4 for air. Assume temperature of air as 15o C. 4. Water at 15o C flows between two parallel plates at a distance of 1.6 mm apart.

Determine the maximum velocity , the pressure loss per unit length, Shear stress at the plates, if the average velocity is 0.2 m/s. The viscosity of water at 15o C is given as 0.01 poise. (8)

5. Define Mach No. What is its significance. (4) 6. Derive an expression for the compressible flow of a fluid for (i) Isothermal process (ii)

Adiabatic process. (8)

smooth pipe Ho

Vexit

Rough pipe

Value kept half

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B.E. Mechanical 4th Semester Course Information

7. An aeroplane is flying at an altitude of 15 Km where the temperature is -500. The speed of the plane corresponds to a Mach No.1.6. Assuming R = 1.4 and R = 2875/ kg-K, determine the speed of the aeroplane. (6)

8. * Explain the effect of area variation on one dimensional compressible fluid flow. (6) 9. Define : Boundary layer thickness, Displacement thickness, Momentum thickness. (6) 10. Derive an expression for velocity of propagation of elastic wave in an isothermal

medium. (8) 11. * A jet fighter flying at Mach number 2.0 is observed directly over head at a height of 10

km. How much distance it would cover before the sonic boom is heard on the ground? 12. * Explain (i) Lift (ii) Drag (iii) Wake region (iv) Boundary layer separation (8) 13. Define [a] Drag [b] Lift [c] Drag Co-efficient [d] Lift Co-efficient 14. Define [a] Aero foil [b] Cord length [c] Angle of attack 15. *Distinguish between friction drag and pressure drag. (5) 16. *Sketch the nature of propagation of disturbance in compressible flow when Mach

number is more than one and hence define mach angle and mach cone. (6) 17.*Experiment were conducted in a wild tunnel with a wind speed of 50 km/hr on a flat plate of size 2m long and 1m wide. The specific weight of air is 11.282 N/m3 The plate is kept at such an angle that the coefficient of lift and drag are 0.75 And 0.15 respectively. Determine:

(a) Lift force (b) Drag force (c) Resultant force (d) Power excited by air stream on the plate.(10)

18.*A Supersonic plane travels at 1.8 Mach at an altitude of 20Km above the ground How far ahead the plane will be when one hears the sonic boom on the Ground? (8) 19.*The velocity distribution in a boundary-layer is given by: u=U∝sin(π/2y/δ) 0≤y≤δ u=U∝ y≥δ

(a) Determine the displacement and momentum thickness if δ=cm and U∝=10m/s (b) What would happen to the answer of part (a) above, if the shape of the

Velocity profile remains the same and δ remains at 1cm but U∝ is doubled. 20.*An aircraft is flying at a uniform speed of 1Km/s at a height of 2 Km. The density and pressure at that altitude are 0.6kg/m3 respectively.

(a) How long after it flies directly overhead will the sonic boom be heard? R for air is 287 J/kg/K and the ratio of specific heats, γ is 1.4. (6)

(b) Calculate the pressure felt at the nose of the aircraft, assuming that, in the Reference frames of the planes, the air is isentropically brought to rest at the Nose. (4)

21.*Distinguish between the following: (a) form, drag and skin friction drag. (b) Lift force and drag force (c) The physical significance of displacement thickness and momentum thickness.(6)

* These questions have appeared in past VTU question paper

KINEMATICS OF MACHINE–ME 44 Faculty: Sunith Babu. L

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B.E. Mechanical 4th Semester Course Information

Class # Chapter Title /

Reference Literature

Topics to be covered % Of Portions Covered

1.

2.

Chapter #: 1.0

Introduction T1: 1 - 19 R1: 3 – 14

Definition – Link or element, Paring of element with degrees of freedom, Grubler’s criterion (without derivation), Kinematic chain, Mechanism, Mobility of Mechanism, Inversion, Machine

4 % 4 %

3.

4.

5

6.

Chapter #: 1.0 Kinematic chain and Inversion

T1: 20 - 30 R1: 15 - 21

Kinematic Chain and Inversion: Kinematic chain with three lower pairs, Four bar chain, Single slider crank chain & Double slider crank chain and

Their inversions

8 %

12 %

7.

8.

9.

10.

11.

12.

13.

Chapter #: 2.0

Mechanism

T1: 22 - 30 R1: 95 - 123

Mechanisms: (I) Quick return mechanisms – Drag link mechanism, Whitworth mechanism and Crank & slotted lever mechanism. (II) Straight line motion mechanisms – Peacellier’s mechanism and Roberts’s mechanism. (III) Intermittent motion mechanisms – Geneva mechanism and Ratchet & Pawl mechanism. (IV) Toggle mechanism, Pantograph, Hooke’s joint and Ackerman steering mechanism

13 %

25 %

14.

15.

16.

17.

18.

19.

20.

21.

Chapter #: 3.0 Velocity and Acceleration Analysis of

Mechanisms

T1: 31 - 52 R1: 21 - 23

Velocity and Acceleration Analysis of Mechanisms (Graphical Methods): Velocity and Acceleration Analysis by vector polygons – Relative velocity and acceleration of particles in a common link, Relative velocity and accelerations of coincident particles on separate links, Coriolis components of acceleration

15 %

40 %

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B.E. Mechanical 4th Semester Course Information

22.

23.

24.

25.

26.

Chapter #: 4.0

Velocity Analysis by Instantaneous center method & Klein’s construction

T1: 53 - 73 R1: 24 – 71

Definition, Kennedy’s Theorem Determination of velocity using instantaneous center method Analysis of Velocity and Acceleration of single slider crank mechanism by using Klein’s construction

12 %

52 %

27.

28.

29.

30.

Chapter #: 5.0 Velocity Analysis by Complex No And Loop Closure Equation

R2:

Velocity and Acceleration Analysis by Complex Numbers: Analysis of A) Single slider crank mechanism by

1) Loop closure equations 2) Complex numbers

B) Four Bar mechanism 1) Loop closure equations 2) Complex numbers

8 %

60 %

31.

32.

33.

34.

35.

36.

37.

38.

Chapter #: 6.0 Spur Gears

T1: 326 - 383 R1: 163 – 187

Law of gearing, Involutometry, Definitions, Characteristics of involute action, Path of Contact, Arc of Contact, Contact ratio, Interference in involute gears, Methods of avoiding interference, Back lash, Comparison of involute and cycloidal teeth

15 %

75 %

39.

40.

41.

42.

43.

44.

Chapter #: 7.0 Gear Trains

T1: 383 - 414 R1: 201 - 221

Simple gear trains, Compound gear trains for large speed reduction, Epicyclic gear trains, Algebraic and tabular methods of finding velocity ratio of epicyclic gear trains, Tooth load and torque Calculations in epicyclic gear trains. Differential mechanism of an automobile

12 %

87 %

PESIT

B.E. Mechanical 4th Semester Course Information

Learning Resources for the Subject

Prescribed Text Books:

Code Title Author(s) Publisher / Edition / Year

T1.

Theory of Machines – S. S. Rattan Tata Mc Graw Hill / Eighteen / 1998

T2.

Mechanism and machine

J. S. Rao New Age / Second / 2003

Prescribed Reference Books:

R1.

Kinematics of Machines A.S Ravindran Sudha Publication / Fourth / 2004

Guidelines for Quick Study* [ACTUAL EXAM PATTERN MAY VARY]

Chapters 1 and 2 are theory type hence maintain descriptive answers in the exams with neat sketches

Chapters 3 and 8 are to be solved in the Drawing Sheets hence practice problems in

Drawing Sheet only

Chapters 4 5 6 7 mainly contains problems of 15 to 20 marks while theory only 5 marks

45.

46.

47.

48.

49.

50.

51.

52.

Chapter #: 8.0 Cams

T1: 195 - 238 R1: 124 - 162

Types of followers, Displacement, Velocity and Acceleration time curves for cam profiles, Disc cam with reciprocating follower having knife edge, Roller and flat faced follower, disc cam with oscillating roller follower, Follower motions including SHM, Uniform velocity, Uniform acceleration & retardation and cycloidal motion

13 %

100 %

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B.E. Mechanical 4th Semester Course Information

QUESTION BANK

LINKAGES AND MECHANISMS:

1. Explain the following terms with examples (i) Element (ii) Link (iii) Kinematic pair (iv) Mechanism (v) Inversion (vi) Machine (vii) Mobility (viii) Degree of Freedom* (16) 2. Define a Kinematic pair. Explain the various types of Kinematic pairs. * (4) 3. Distinguish between complete, incomplete and successful constraint of the relative

motion between the two links (4) 4. Define the following (i) Lower pair (ii) Higher pair* (4) 5. Define Kinematic chain and how does it differ from a mechanism * (4) 6. Differentiate between (i) Machine and Mechanism (ii) Machine and structure*(8) 7. Define mobility of a mechanism and write the Grubler’s mobility equation for planar

mechanism* (4) 8. Write short notes on Kinematic chain with three lower pairs* (6) 9. Sketch and explain the following* (12)

(i) Four Bar Chain and its inversions (ii) Single slider crank chain and its inversions

(iii) Double slider crank chain and its inversions 10. Explain the working of an Ellipse Trammel and show how is it useful in drawing an

ellipse (10) 11. Explain the construction of Oldham’s coupling and state for what purpose it is used*

(6) 12. Describe the following quick return mechanism* (10)

(i) Drag Link (ii) Witworth (iii) Crank and slotted lever mechanism 13. What are straight-line motion mechanisms? How are they classified* (6) 14. Describe Peacellier’s mechanism with a suitable sketch* (6) 15. Describe Roberts approximate straight line motion mechanism with suitable sketches*

(4) 16. What is a Pantograph and what are its uses? With neat sketches explain its working

principle * (6) 17. Describe toggle mechanism. What are its uses * (4) 19. State and prove the condition that must satisfy by the steering mechanism of a car in

order that the wheel may have pure rolling motion when rounding a curve? (10) 19. Write short notes on Ackermann steering gear * (6) 20. What is Hooke’s joint and what are its application* (4) 21. What is a double Hooke’s joint? (4)

22. State the condition to be satisfied in double Hooke’s joint in order to provide a uniform velocity ratio (8)

23. What is intermittent mechanism? Explain the following intermittent motion Mechanism (i) Geneva (ii) Ratchet* (6) 24. Two shafts are connected by a Hooke’s joint. The driving shaft rotates at a

uniform speed of 300 rpm and the angle between the shafts is 200. Find the maximum and minimum speed of the driven shaft and its max. Acceleration. c) Sketch & explain the working of a ratchet Mechanism (10)

25. A Hooke’s joint connects two shafts whose axes are inclined at 1500. The driving shaft rotates at a uniform speed of 120 rpm. The driven shaft operates against a study torque of 150 Nm and carries a flywheel whose mass is 45 kg and the radius of gyration is 140 mm. Find the maximum torque which will

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B.E. Mechanical 4th Semester Course Information

be exerted by the driving shaft. At what value of ‘α‘ will the total fluctuation of speed of the driven shaft be limited to 24 rpm (12)

26. Find the degree of freedom of the following mechanism shown in figure – 1 * (4)

27. In a with worth quick return mechanism as shown in figure – 2, OP = 240mm, OA = 150mm, AR = 165mm, RS = 430mm. If the crank OP rotates at an angular velocity of 2.5 rad/sec & has an angular acceleration of 20 rad/sec2, determine the acceleration of the slider, the angular acceleration of the link AR, RS. (10)

VELOCITY AND ACCELERATION IN MECHANISMS:

1. a) Write a brief note an instantaneous center of rotation. (4) b) A four bar chain ABCD as a fixed link AD = 1m. The driving rank AB = 0.3m. The

Follower link CD = 0.6 m and the connecting link BC = 1.2m. Find the velocity and acceleration of point ‘P’ midway between B and C When the angle BAD =1350 and AB rotates at a speed of 300 rpm. (16)

2. Derive analytical expressions for velocity and acceleration of the piston in a reciprocating engine mechanism.

b) The crank of a reciprocating engine is 60mm long and connecting rod is 240 mm long. The crank rotates at 400 rpm. Find the velocity and acceleration of both piston and the angular velocity and angular acceleration of the connecting rod when the crank is 300 from IDC by Klein’s construction method. (10)

3. In a slider crank mechanism, the crank is 50mm long and the connecting rod is 150mm long. The crank is rotating at a speed of 10 rad/s and the crank is at 450 from IDC. By kleins construction determine:

i) Velocity of piston ii) Acceleration of the piston iii) Angular acceleration of the Connecting rod. (16) 4. A four bar chain mechanism ABCD is made up of 4 links pin jointed at ends. AD is fixed

link, which is 250 mm long. The links AB, BC & CD are 90mm, 180mm & 180mm long respectively. The crank AB rotates at 100 rpm and an angular acceleration 100rad/sec2 at the instant when the crank AB makes an angle of 600 to the horizontal. Find the angular velocities and angular accelerations of links BC & CD. (10)

5. For the four bar mechanism shown in figure – 3, determine the acceleration of C and angular acceleration of link 3 when crank 2, rotates at 20 radians per second. *

(08) 6. Determine the velocity and acceleration of the piston by “Kleins Construction” for a

steam engine for the following specification: Stroke of piston = 600mm, Ratio of length o connecting rod to crank length = 5, speed

of the engine = 450rpm CW, position of the crank = 45 degree with IDC. (10) 7. Derive analytically the expression for the velocity and the acceleration of the piston in

a reciprocating engine mechanism. (10) 8. The crank of a reciprocating engine is 90mm long and the connecting rod is 360mm

long. The crank rotates at 150rpm. Find the velocity and the acceleration of the piston and angular velocity and angular acceleration of the connecting rod when the angle, which the crank makes with the IDC, is 300. (10)

9. A link ABC of a mechanism shown in figure – 4 is in motion. At the instant shown, ‘A’ moves with 0.6 m/s in the direction shown and ‘B’ moves with a speed of 0.5 m/s/ find the magnitude and direction of (i) Velocity of C and (ii) angular velocity of ABC. AB = BC = AC = 0.5cm x-x is parallel to AB, a reference line (08)

10. The crank of a slider crank mechanism rotates CW at a constant speed of 300 rpm. The crank is 150mm and the connecting rod is 600mm long. Determine (I) linear velocity and acceleration of the mid point of the connecting rod (ii) Angular velocity and acceleration of the connecting rod, at a crank angle of 450 from IDC position.

(15)

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B.E. Mechanical 4th Semester Course Information

11. An oscillatory cylinder mechanism is shown in the figure 5. The crank O2B rotates at 300rpm (ccw). Determine the magnitude & direction of

I. Angular velocity of cylinder ii. Velocity of A3 a point on the system iii. Angular acceleration of the cylinder. Also state whether the magnitude of angular velocity. Is increasing or decreasing at the instant A3 & A4 are coincident points of piston & cylinder respectively. O2B=150mm. O2O4=600mm. A3B=400mm. (15)

12. The crank of a slider crank mechanism is 480 mm long and rotates at 20 rad/sec in the counter clockwise direction. It has a connecting rod of 160mm long, Determine the following when the crank is at 60 degree from the inner dead center 1) Velocity of slider 2) Angular velocity of connecting rod 3) The Position and velocity of a point P on the connecting rod having at least absolute velocity. (15)

13. A pin jointed 4-bar mechanism ABCD is shown in figure - 6. LINK AB = 150mm, BC = 180 mm CD = 180 mm, and the fixed Link AD = 300 mm. Link AB makes 60 degree with Link AD and rotates uniformly at 100 rpm. Locate all the instantaneous centers and find the angular velocity of link BC and Linear velocity of link CD.(16)

14. Locate all the instantaneous centers for the toggle mechanism shown in figure –7 (4)

15. Determine the velocity of point K in the mechanism shown in figure –8 (6) 16. In the mechanism shown in figure – 9, crank rotates at 3000rpm. Find the acceleration

of the point c in magnitude, direction and sense. Find also the angular acceleration of link 3. (10)

17. The crank 02A of the four bar mechanism shown in figure - 10 is rotating clockwise at a constant speed of 100 rad/sec. For the phase, shown in figure, determine I) the acceleration of the point c. II) the angular acceleration of link 3 and 4. (10)

18. A four bar chain ABCD has a fixed link AD = 1m. The driving rank AB = 0.3m. The Follower link CD = 0.6 m and the connecting link BC = 1.2m. Find the velocity and acceleration of point ‘P’ midway between B and C When the angle BAD =1350 and AB rotates at a speed of 300 rpm. (10)

19. In a slider crank mechanism, the crank is 50mm long and the connecting rod is 150mm long. The crank is rotating at a speed of 10 rad/s and the crank is at 450 from IDC. By kleins construction determine:

i) Velocity of piston ii) Acceleration of the piston iii) Angular acceleration of the Connecting rod * (15)

20. A four bar chain mechanism ABCD is made up of 4 links pin jointed at ends. AD is fixed link, which is 250 mm long. The links AB, BC & CD are 90mm, 180mm & 180mm long respectively. The crank AB rotates at 100 rpm and an angular acceleration 100rad/sec2 at the instant when the crank AB makes an angle of 600 to the horizontal. Find the angular velocities and angular accelerations of links BC & CD * (20)

21. The crank of an engine is 20cm long and connecting rod length to crank radius is 4. Determine the acceleration of the piston when the crank has turned through 45 Degree from the inner dead center position and moving towards center at 240 rpm by the following methods 1) complex algebra analysis 2) Klein’s construction and compare the values and get the error % figure – 11 * (20)

22. For the four bar mechanism shown in figure – 12 determine the acceleration of C and angular acceleration of link 3 when crank 2, rotates at 20 radians / sec* (10)

23. What is Coriolis component? Derive the expression for the same * (06) 24. The crank of a reciprocating engine is 60mm long and connecting rod is 240mm long.

The crank rotates at 400rpm. Find the velocity and acceleration of the piston and the angular velocity and angular acceleration of the connecting rod, when the crank is 30 degrees from inner dead center, by KLEIN’S construction. * (12)

25. What is instantaneous center? Locate all the instantaneous center for a single slider crank mechanism and show how velocity of slider is determined? (08)

Velocity and acceleration analysis by complex method:

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B.E. Mechanical 4th Semester Course Information

1. In an internal combustion engine mechanism, the crank radius is 100mm and the length of the connecting rod is 450 mm. the crank is rotating at 10 rad/sec, in (CCW) direction. Determine the magnitude and direction of 1) Velocity of the piston 2) The angular velocity of the connecting rod when crank is at 45 degree from the inner dead center by complex algebra method verify the same by Klein’s construction. (15)

2. In a 4 bar mechanism ABCD link, AB=300mm,BC=360mm,CD=360mm and the fixed link AD=600mm. The angle of link AB with fixed is 60degree. The AB has an angular velocity of 10rad/sec and angular acceleration of 30rad/sec2 both clock wise. Determine the angular velocity and angular acceleration of link BC and CD by RAVEN’S approach. * (20)

SPUR GEAR 1. What are toothed gears? State their uses* (4) 2. What are the advantages and disadvantages of gear drives (4) 3. How are gears classified* (4) 4. State and prove “Law of gearing”* (6) 5. What are the common curves used for tooth profile* (2) 6. Discuss the merits and demerits of involute curve and cycloidal curve for the

profile of gear tooth. (6) 7. What are the general Characteristics of spur gearing* (4) 8. Draw a neat sketch of spur gear and explain the various terms (6) 9. Define the following (i) Pitch circle diameter (ii) Circular Pitch (iii) Module (iv) Addendum (v) Dedendum (vi) Pressure angle* 10. Define the following (i) Path of Contact (ii) Arc of Contact (iii) Contact Ratio*

(12) 11. Derive the formula for the length of arc of contact for two meshing spur Gears of

involute profile* (6) 12. Explain the phenomenon of interference. State the condition for no Interference

(4) 13. Describe the various methods of avoiding interference (4) 14. Derive an expression for minimum number of teeth necessary for a pinion to avoid

interference. (10) 15. Derive an expression for the length of path of contact for a pinion on rack (6) 16. What is Involutometry? Derive an expression for the tooth thickness at any point on

the involute, if the tooth thickness at some other point is known (10) 17. What is Backlash? Derive an expression for backlash if the center distance is pulled

apart at a distance Δc. (10) 18. Two equal spur gears of 48 teeth mesh together with pitch radii of 100 mm, and the

addendum is 4.25 mm. If the pressure angle is 200, calculate the length of action and contact ratio. Sketch the different types of gear trains and explain briefly (20)

19. A pair of gears having 40 and 20 teeth are rotating in mesh the speed of smaller being 1800rpm. Determine the velocity of sliding between the gear teeth faces at the point of engagement, at the pitch point, at the point of disengagement if the smaller gear is the driver. Assume the gear teeth 200 involute form, addendum length is 5mm and the module is 5mm (20)

20. The number of teeth on each of two equal spur gears is mesh each other are 40. The teeth have 200 involute pressure angle and module is 5mm. If the arc of contact is 1.75 times the circular pitch. Find the addendum (10)

21. The following are the particulars of a single reduction spur gear the gear ratio is 10: 1 and the center distance is 275mm. The pinion transmits 375 KW at 1800 rpm. The teeth's are of involute form with standard addendum of 1 module and pressure angle is 22.50. Normal tooth pressure is not to exceed 9810N/cm width. Find (I) the nearest

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B.E. Mechanical 4th Semester Course Information

standard diameter pitch if no interference is to occur (ii) the number of teeth in each wheel (iii) the width of pinion* (20)

22. A spur Pinion 100 mm in diameter has a torque of 200Nm applied to it. The spur gear in mesh with it is 250 mm in dia. the pressure angle is 20 degree. Determine the tangential force FT and the separating force FR. Show the forces action on the wheels separately. * (5)

23. Two Gear wheels mesh externally and are to give a velocity ratio of 3. The teeth are of involute form of module 6mm, and the standard addendum is one module. If the pressure angle is 180 and the pinion rotates at 90 rpm. Find i) The minimum number of teeth on each wheel to avoid interference. The length of path of contact iii) The maximum velocity of sliding iv) number of pairs of teeth in contact* (16)

24. The following data refers to two mating involute gears of 200 pressure angle: Number of teeth on pinion – 20 Module – 12mm If the center distance between the gears is increased by 2mm, find the backlash between the gears. (15)

25. Two spur gears wheels have 24 and 30 teeth and a standard addendum of 1 module. The pressure angle is 200. Calculate the path of contact and arc of contact. Derive the expression used. * (15)

26. The following data refers to two mating involute gears of 200 pressure angle. Number of teeth on the pinion is 20, gear ratio is 2, speed of the pinion is 250rpm, module 12mm. If the addendum on each wheel is such that the path of approach and the path of recess on each side are half of the maximum permissible length. Find the maximum velocity of sliding during approach and the recess and the length of arc of contact. * (14)

27. A pair of gear has 16 teeth and 18 teeth, a module 12.5 mm an addendum 12.5mm and a pressure angle 14.5 degrees. Prove that gears have interference. Determine the minimum number of teeth and the velocity ratio to avoid interference. * (08)

GEAR TRAINS 1. What do you understand by gear trains (2) 2. Explain Train Value? How is it related to velocity ratio? * (4) 3. Name different types of Gear Trains and give examples (4) 4. Explain Simple, Compounded & Reverted gear trains * (6) 5. Explain Epicyclic gear trains (2) 6. Explain Bevel gear differential* (4) 7. Explain Spur gear differential (4) 8. Obtain the expression for the length of path of contact for two involute profile gears in

mesh (10) 9. In an epicyclic gear train of ‘ SUN & PLANET’ type the sun wheel has 15 teeth and is

fixed to the motor shaft rotating at 1500 rpm. The planet P; three in number has 45 teeth's gears with fixed annulus ‘A’ and rotates on a spindle cared by an arm which is fixed to the output shaft. The planet ‘P’ also gears with the sun wheel ‘S’ Find the speed of the output shaft, if the motor transmits 2Kw. Find the torque required to fit the annulus. (20)

10. An internal wheel B with 80teeth is keyed to shaft F. A fixed internal wheel C with 82 teeth is concentric with B. A compound wheel DE gears with the two internal wheels: D has 28 teeth and gears with C while E gears with B. The compound wheel revolves freely on a pin, which projects from a disc keyed to a shaft A co-axial with F. If the wheels have the same pitch and the shaft A makes 800rpm, What is the speed of the shaft F. Sketch the arrangement. (15)

11. In a reverted epicyclic gear train the arm A carries two wheels B & C and a compound wheel DE. The wheel C gears with wheel D. The number of teeth on wheel B, C&D are

PESIT

B.E. Mechanical 4th Semester Course Information

75, 30 & 90 respectively. Find the speed and directions of the wheel C when wheel B is fixed and the arm A makes 100rpm clockwise (15)

12. In a planetary gear system of sun and planet type, the sun wheel A has 20 teeth and is fixed to the frame. The arms C with Planet gear wheel B with 40 teeth revolve about the gear wheel A. When the arm rotates at 30rpm, determine the speed planet gear B (10)

13. Figure – 13 shows an epicyclic gear train Wheel E is fixed and Wheel C and D are integrally cast and mounted on the same pin. If arm A makes 1 revolution/ sec (CCW), determine the speed and direction of the wheels B and F. (15)

14. In the gear train shown in figure – 14. The wheel C is fixed, the gear B is connected to the input shaft, & the gear F is connected to the output shaft. The arm A, carrying the compound wheels D & E, turns freely on the output shaft. If the input speed is 100rpm(ccw) when see from right. Determine the speed of output shaft. The No of teeth on each gear is indicated in the fig. Find the output torque & the holding torque to keep the wheel C fixed if the input power is 7.5kW* (20)

15. In the epicyclic gear train shown in figure - 15 A gear C, has teeth cut both internally & externally. The gear C is free to rotate on an arm driven by shaft S1 & meshes externally with the casing D & internally with pinion B.The gears have the following No. Of teeth TB=24, TC=32 & 40, TD=48. Determine the vel. Ratio between S1 & S2 when D is fixed S1 & D when D is fixed (15)

16. In an epicyclic gear train shown in figure - 16 the internal wheels A, Fand the compound wheels C, D rotate about the axis O. The wheels B and E rotate on a pin fixed to the arm L. The wheels have same pitch and the number of teeth on B and E are 18, C = 28, D = 26. If the arm L makes 150rpm clockwise, find the speed of F when I) Wheel A is fixed and II) Wheel A makes 15rpm (CCW) by tabular column method. (10)

17. In an epicycle gear train, the internal wheel A and B the compound wheel C and D rotate independently about axis O. The wheel E and F rotate on pins fixed to the arm G. E gears with A and C and F gears with B and D. All wheels have the same module and the number of teeth are Tc = 28, Td = 26, Tf = 18,

i) Sketch the arrangement ii) Find the number of teeth on A and B iii) If the arm G makes 100rpm clock wise and A is fixed, find B speed iv) If the arm G makes 100 rpm CW and wheel A makes 10rpm, CCW find

speed of B CAMS 1. Give the classification of cams and follower* (4) 2. Discuss the types of follower displacement diagrams* (4) 3. Write short notes on cams and follower * (4) 4. Why a roller follower is preferred to that of Knife edge follower (4) 5. Explain w.r.t. Cams a) Base Circle b) Pitch circle 3) Pressure angle (6) 6. What are the different types of follower motions (4) 7. Derive an expression for velocity acceleration & displacement when the circular arc

cam is operating on an flat faced follower (i) When the contact is on the circular flank (ii) When the contact is on circular nose (10) 8. Derive an expression for velocity acceleration & displacement when the tangent cam is

operating on an radial translating roller follower When the contact is on straight flank When the contact is on circular nose (4)

9. Draw the outline of a cam which will transmit motion to a roller follower in the following manner I) the follower to move outwards through a distance of 65 mm during 180 degree of cam rotation. ii) Follower to return to its initial position during 150 degree of cam rotation. iii) Follower to dwell for the remaining 30 degree of cam

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B.E. Mechanical 4th Semester Course Information

rotation. The minimum radius of the cam is 30 mm & the displacement of the follower is to take place with cycloidal motion both during outward & return strokes. The roller diameter is 10mm & the follower axis is offset by 10mm from the axis of the shaft. * (8)

10. Draw the profile of the cam to give the following motion to the follower—Follower to move through 30mm. During 180oof cam rotation with cycloidal motion. Follower to return with cycloidal motion during 180o of cam rotation. Base circle dia. of the cam is 30mm and the roller dia of the follower is 10mm.the axis of the roller is offset by 8mm to the right. Determine the maximum velocity and acceleration of the follower during the outstroke, when cam rotates at 2000rpm. * (8)

11. A disc cam is required to lift a flat-faced follower with U.A.R.M motion through 30 mm in 1/3 rd revolution. Keep it fully raised through 1/6th revolution & to lower it with S.H.M. in 1/3rd revolution and to dwell during rest of the revolution .The minimum radius of the cam is 25 mm. Draw the cam profile & also determine the maximum velocity & acceleration of the follower during and return strokes if the cam rotates at 200 rpm clockwise. * (20)

12. Draw the cam profile for the cam with reciprocating follower .The axis of the follower passes through the axis of the cam. Details of the cam & the follower motion are the following: Roller dia = 10mm, minimum radius of the cam = 22 mm, Total lift = 25 mm .The cam has to lift the follower with SHM during 180 degree of cam rotation, then allows the follower to drop suddenly halfway & further return the follower with UARM during the remaining 180 degree cam rotation. Calculate the maximum velocity & acceleration of the follower during outstroke

(20) 13. A push rod operated by a cam is to rise and fall with SHM along an inclined path. The

least radius of the cam is 30 mm and push rod is fitted at its lower end with a roller 15 mm diameter. When in its lowest position, the roller center is above the cam axis. The max. Displacement of the follower is 40 mm in a direction 30 degree to the right of the vertical .The cam rotates at 100 rpm in a clockwise direction .The time of lift is 0.15 sec & the time of fall is 0.10 sec with a period of rest of 0.05 sec at the upper position, draw the graphical cam profile. (20)

14. A knife edged follower for the fuel valve of a four stroke diesel engine has its center line coincident with the vertical center line of the cam .It rises 25 mm with SHM during 600 of cam rotation, then dwells for 30 degree of cam rotation and finally descends with UARM during 90 degree of cam rotation, the deceleration period being 1/2 the acceleration period .The least radius of the cam is 30 mm. Draw the profile of the cam to full size. (10)

15. It is required to set out the profile of a cam for the following data. i) Follower to move outward through an angular displacement of 20 degree during 900

of cam rotation ii) Follower to dwell for 45 degree of cam rotation .iii) Follower to dwell for the remaining period of the revolution of the cam .iv) Follower to return to its initial position of zero displacement in 5 degree of cam rotation. * (20)

16. The distance between the pivot center and the roller follower center is 70mm. Roller dia is 10mm, minimum radius of cam is 30mm .The location of the pivot is 70mm to the left and 50mm above the axis of the cam. The motion of the follower is to take place with SHM during outstroke and with UARM during return stroke, the acceleration during outstroke of the follower, if the cam speed is 1200rpm. * (15)

17. A cam rotating clockwise at uniform speed of 300rpm operates a reciprocating follower through a roller 10mm dia the follower motion is defined: (I) Follower to move outwards during 1200 of the cam rotation with equal uniform acceleration and de-acceleration. (ii) Follower to dwell in the lifted position for the next 300 of cam rotation. (iii) Follower to return to its starting position during 1200 of cam rotation with SHM. (iv) Follower to dwell for the rest of cam rotation* (15)

18. Draw the full size cam profile for a cam with roller of 25mm dia attached to the follower to give a lift of 35mm. Axis of the follower is offset to right of cam axis by 18mm. Ascend of the follower takes place with SHM in 0.05sec follower by a period of rest of 0.0125sec. The follower then descends with UARM during 0.125sec and the

PESIT

B.E. Mechanical 4th Semester Course Information

remaining period rest at the minimum lifted position. The acceleration being 3/5 times retardation. The cam rotates in CCW direction at a constant speed 240rpm and the base radius is 50mm*(20)

19. A translating roller follower has lift of 4cm. The follower has a uniform acceleration and retardation motion for both rise and return phases. How ever during the rise acceleration is twice the retardation. And during return the duration of acceleration is twice the duration of retardation. The angle of rotation of cam during rise and return is 1500 each. The duration of the dwell before and after the rise is 300. Draw the displacement diagram of the follower. Do not draw the cam profile* (20)

20. A cam rotates at a uniform speed of 300 rpm clock wise and gives an oscillating follower 75mm long, an angular displacement of 30 degree in each stroke. The follower is fitted with a roller of 20mm diameter, which makes contact with the cam. The outward and inward displacement of the follower each occupying 120 degree cam rotation and there is no dwell in the lifted position. The follower moves through out by SHM. The axis of fulcrum is 80mm from the axis of cam and the least distance of roller axis from cam axis is 40mm* (20)

21. A roller follower is offset to the left by 1.2cm.the base circle radius of the cam is 3cm.the desired displacement of the follower Y for any cam rotation θ is listed in the table given below. Layout the cam profile if the radius of roller follower is 1cm. The cam rotates in clockwise direction. * (20)

Cam rotation θo

0 30 60 90 120 150 180 210

240 270 300 330

Follower (cm) Y displacement

0 0.25 0.92 1.87 2.83 3.50 3.75 3.5 2.83 1.87 0.92 0.25

22. Draw the full size cam profile for a cam with roller of 25mm diameter attached to the

follower to give a lift of 35mm. Axis of the follower is offset to right of cam axis is 18mm. Ascent of the follower takes place with SHM 0.05sec followed by a period of set 0.125sec. The Follower then descends with UARM during 0.125sec and the remaining period as rest in the minimum lifted position, the acceleration being ¾ times retardation. The cam rotates in CCW direction at a constant speed of 240rpm and the base radius is 50mm (20)

* Questions appeared in VTU examinations

METROLOGY & MEASUREMENTS – IP42 B Faculty: S.V.Satish/ Mukesh Patil

Class # Chapter title / Reference Literature

Portions to be covered

% Covered

PART-A 1 – 4 Standards of

Measurement: R1:34-40 R2:311

1.1-Definition and objectives of metrology 1.2-Standards of length - international prototype meter 1.3-Imperial standard yard, wave length standard 1.4-Subdivision of standards 1.5-Line and end standards 1.6-Comparison 1.7-Transfer from line standard to end standard 1.8-Calibration of end bars (numericals) 1.9-Slip gauges 1.10-Wringing phenomena 1.11-Indian Standards (M-87, M-112)

8

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B.E. Mechanical 4th Semester Course Information

1.12-Numerical problems on building of slip gauges 5 – 13 System of Limits,

Fits, Tolerances and gauging: R1: 73-110 R2: 312-446

2.1-Definition of tolerance 2.2-Specification in assembly 2.3-Principle of interchangeability and selective assembly 2.4-Limits of size 2.5-Indian standards 2.6-Concept of limits of size and tolerances 2.7-Compound tolerances 2.8-Accumulation of tolerances 2.9-Definition of fits 2.10-Types of fits and their designation (IS 919-1963) 2.11-Geometrical tolerance 2.12-Positional tolerances 2.13-Hole basis system and shaft basis system 2.14-Classification of gauges 2.15-Brief concept of design of gauges (Taylor’s principle) 2.16-Wear allowance on gauges 2.17-Types of gauges -Plain plug gauge, ring gauge, snap gauge, limit gauge 2.18-Gauge materials

26

14 – 18 Comparators: R1 : 63-70 R2: 447 - 519

3.1-Introduction to comparator 3.2-Characteristics and classification of comparators 3.3-Mechanical comparators - Johnson mikrokator

- Sigma comparators - Dial indicator

3.4-optical comparators-principles 3.5-Zeiss ultra optimeter 3.6-Electric and electronic comparators-principles 3.7-LVDT 3.8-pneumatic comparators 3.9-Back pressure gauges 3.10-Solex comparators

38

19 – 22 Angular measurements and Interferometer R1 : 111 – 127 R2 : 654 - 724

4.1-Bevel protractor 4.2-Sine principle, Sine bar, Sine center, Angle gauges (numericals on building of angles) 4.3-Clinometers 4.4-Principle of Interferometry 4.5-Autocollimeter 4.6-Optical flats

44

23 – 26 Screw thread and Gear measurement R1 : 174- 190 R2 : 879 -1008

5.1-Terminology of screw threads 5.2-Measurement of major diameter 5.3-Minor diameter 5.4-Pitch 5.5-Angle and effective diameter of screw threads by 2-wire & 3-wire methods 5.6-Best size wire 5.7-Tool makers microscope 5.8-Gear terminology 5.9-Use of gear tooth Vernier caliper 5.10-Use of gear tooth micrometer

52

PART-B 26 – 31 Measurements &

Measurement systems R1: 268-270

6.1-Definition and Significance of measurement 6.2-Generalised measurement system 6.3-Definition and concept of accuracy 6.4-Precision, sensitivity, calibration, threshold,

62

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B.E. Mechanical 4th Semester Course Information

-hysteresis, repeatability linearity, loading effect -System response, time delay -Errors in measurements, classification of errors

32 – 35 Transducers R1:271-314

7.1-Transfer efficiency 7.2-Primary and secondary transducers 7.3-Mechanical, Electrical, electronic transducer 7.4-Advantages of each type transducers

70

36 – 38 Intermediate modifying devices R1: 315-326

8.1-Mechanical systems 8.2-Inherent problems 8.3-Electrical intermediate modifying devices 8.4-Input circuitry 8.5-Ballast circuit 8.6-Electronic amplifiers and telemetry

76

39 – 40 Terminating Devices

9.1-Mechanical 9.2-Cathode Ray Oscilloscope 9.3-Oscillographs 9.4-X-Y Plotters

80

41 – 43 Measurement of force & torque R1:368-379

10.1-Principle 10.2-Analytical balance 10.3-Platform balance 10.4-Proving ring 10.5-Torque measurement:

- Prony brake dynamometer - Hydraulic dynamometer

86

44 – 46 Pressure measurements R1:380-418

11.1-Principle 11.2-Use of elastic members 11.3-Bridgeman gauge 11.4-Mcleod gauge 11.5-Pirani gauge

92

47 – 49 Temperature measurement R1:419-455

12.1-Resistance thermometers 12.2-Thermocouple 12.3-Laws of thermocouple 12.4-Materials used for construction of thermocouple 12.5-pyrometer 12.6-Optical pyrometer

98

50– 52 Strain measurement R1:275-312

13.1-Strain gauge 13.2-Preparation and mounting of strain gauges 13.3-Gauge factor 13.4-Methods of strain measurement

100

Text Books: 1. Mechanical Measurements - Beckwith ,Marangoni & Lienhard 2. Engineering Metrology- I.C.Gupta Reference Books: 1. Mechanical Measurements – Holeman 2. Mechanical Measurements – Sirohi & Radhakrishna 3. Mechanical measurements-Doblin 4. Metrology for engineers-J.F.Galyer & C.R.Shotbolt 5. Industrial Instrumentation-Alsutko & Jerry.D.Faulk 6. Engineering Metrology - R.K.Jain

Scheme of examination:

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B.E. Mechanical 4th Semester Course Information

Four questions to be set from Metrology – Part A Four questions to be set from Measurements – Part B Answer any five questions taking at least TWO questions from each part.

QUESTION BANK

PART-A Chapter 1: Standards of Measurement 1. Define Metrology. Explain various standards of length. 2. What do you understand by line and end standards? Discuss their relative

characteristics.* 3. Enumerate the advantages of using wavelength standard as a basic unit to define

primary standards.* 4. Explain the NPL method of deriving end standards from line standards.* 5. What are Airy points? How do they differ from the points of minimum deflection.* Chapter 2: System of Limits, Fits, Tolerances and Gauging 1. What is the difference between unilateral and bilateral tolerances? Why unilateral

tolerance is preferred over bilateral tolerance?* 2. Explain what is meant by:

i) Interchangeable part ii) Universal interchangeability iii) Local interchangeability*

3. Determine the type of fit, after deciding the fundamental deviations and tolerances in the following:

Fit Ø 70H9e7, Diameter step (50-80) Fundamental deviation for e shaft = -11D0.41 IT7=16i, IT9=40i i=0.453√D+0.001D*

4. Calculate the dimensions of plug and ring gauges to control the production of 50mm shaft and hole pair of H7d8.The following assumptions may be made.50mm falls in the diameter step of 30-50mm.Upper deviation for “d”shaft is –16D0.44. IT7=16i,IT8=25i, i=0.453√D+0.001D *

5. Explain different types of fits. 6. Write neat sketches and explain 'go' and 'Nogo' gauges. 7. Briefly explain interchangeability. 8. Briefly explain selective assembly. 9. Explain Taylor's principle for 'go' and 'Nogo' gauges. 10. Differentiate between:

i) Tolerance and allowance ii) Hole basis system and shaft basis system *

11. Explain Taylor’s principle for the design of limit gauges. * 12. Illustrate with examples:

i) Geometrical tolerance ii) Dimensional tolerance iii) Positional tolerance

13. A shaft-hole pair is designated as H7d8.The standard tolerance is given by i=0.453√D+0.001D,where D=Diameter(mm)falling in the step 18-30mm.The fundamental deviation for fit d is given by –16D0.44 Take wear allowance as 10% of the gauge tolerance. Determine the shaft and hole dimensions, their tolerances, clearance, interference and the class of fit. Sketch the fit and mark the dimensions clearly. *

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B.E. Mechanical 4th Semester Course Information

Chapter 3: Comparators 1. What are the required characteristics of comparators?* 2. What are the advantages of electrical comparator over mechanical comparator? 3. Explain the working of Sigma comparator with neat sketches. 4. Explain the working of Jhonson's Microkrater with neat sketches.* 5. Explain the working of Brook level comparator with neat sketches. 6. Explain the working of Optical comparator with neat sketches. 7. What is the difference between a comparator and a measuring instrument? * Chapter 4: Angular Measurements 1. Explain why it is preferred not to use a sine bar for generating angles larger than 45oif

high accuracy is desired.* 2. Explain how sine bars are used for measurement of angle. Show the arrangement of

gauges. i)57o 34’9” ii)12o20’36” *

3. With a neat sketch explain Bevel protractor. 4. With a neat sketch explain Universal Protactor. 5. Give the significance of Clinometer in angular measurement. 6. Write neat sketch and explain the principle of working of Auto Collimeter. 7. How is setup of angular gauges different from simple gauges? Explain with an example. 8. Sketch and label the parts of a Vernier bevel protractor. * 9. Distinguish between:

i) Sine bar and sine center ii) Angle gauges and slip gauges. *

10. Select the sizes of angle gauges required to build the following angles: i)31deg29min24sec ii)102deg8min42sec *

Chapter 5: Screw thread and Gear measurement 1. Give the procedure to measure major and pitch diameter using 3 wire method. 2. Distinguish between 2 wire and 3 wire methods of measuring and suggest the best

one. 3. What are the two corrections applied in the measurement of effective diameter by the

method of wires?* 4. Explain the working principle of Tool Makers Microscope. 5. Describe screw thread terminology with sketch. 6. Compare profile projector with tool makers microscope. 7. What are the various types of pitch errors on threads and explain the reasons for the

same. * 8. How do you find effective diameter of a screw thread using two-wire method. * 9. How do you measure the following in case of a spur gear:

i) Runout ii) Tooth thickness iii) Backlash *

10. Explain the terminology of a simple Spur gear 11. Describe a Gear Tooth Vernier caliper and show how this is used for checking gear. * 12. Explain how Gear Tooth Vernier is used for gear measurement.

PART-B

Chapter 6: Measurements and measurement systems

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B.E. Mechanical 4th Semester Course Information

1. What do you mean by static calibration? Sketch the calibration curve of an instrument and explain how it is obtained?*

2. Explain the following: i) Zero drift and sensitivity drift ii) Threshold, Resolution and Hysteresis* 3. What are the sources of errors in instruments? Explain* 4. What is a measurement? 5. Draw a block diagram of generalized measurement systems and explain the salient

features of each stage. 6. With the help of examples, distinguish between the two fundamental methods of

measurement. 7. State the three basic elements of a measuring system and give an example to each of

the basic elements. 8. Draw the displacement time characteristics for damped motion and explain the

importance of damping. 9. Draw a block diagram of a generalized measurement system and explain the salient

feature of each stage. 10. Distinguish between:

i) Digital and Analog measurement ii) Direct reading and null balancing *

11. For Oscillating systems, show that, a more sensitive instrument oscillates more slowly than a less sensitive instrument.

12. Draw a neat block diagram of measurement system employed for measuring acceleration.

13. Explain the following: i) Accuracy ii) Sensitivity iii) Precision 14. Explain briefly the different types of errors encountered during measurement. 15. Write a brief note an treatment of multisampling data. 16. What are the requirements and objectives of measurement? 17. State and explain the various forms of input to the instrument. 18. Explain with an example the various stages of a generalized measurement system. * 19. Explain with sketches:

i) Hysteresis ii) Threshold iii) Repeatability iv) Sensitivity drift *

Chapter 7: Transducers 1. Define the word Transducer. What do you understand by active and passive

transducers? Give examples.* 2. Explain the following with neat sketches: i)Mutual inductance transducer ii)Piezo electric transducer*

3. Explain any one type of elastic transducer with a neat sketch.* 4. Mention the advantages of electrical primary detector transducer elements over other

types. 5. Describe with a neat sketch the ionization transducer. 6. Define transfer efficiency. 7. Mention six mechanical elements used as detector transducers and indicate the

operations, which they perform. 8. Discuss the relative merits and demerits of mechanical and electrical transducers. 9. What are the parameters on which capacitive transducers are developed. Explain the

working of a pickup used for determining the level of liquid nitrogen with a neat sketch. 10. With a neat sketch explain the working of a transducer using electro kinetic

phenomenon and indicate its applications. 11. Explain the principle of working of a linear variable differential transducer with a neat

sketch and illustrate its characteristics.

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B.E. Mechanical 4th Semester Course Information

12. What are the relative merits and demerits of electrical transducers over mechanical transducers?

13. Explain the principle of working of a piezoelectric transducer with a neat sketch. 14. What are primary and secondary transducers? Explain with examples. 15. Explain the working of a electronic transducer with a neat sketch. 16. What are active and passive transducers, give examples. 17. Explain the principle of variable resistance transducers with a neat sketch. 18. What are photoelectric transducers? Explain any one type with a neat sketch. 19. Explain the principle of variable inductance transducer with a neat sketch. 20. Distinguish between active and passive transducers. * 21. Explain with neat sketches the following:

i) Piezo electric transducer ii) Ionization transducer *

11. Discuss briefly with sketches two types of elastic pressure transducers. * Chapter 8: Intermediate modifying devices 1. Explain in brief inherent problems encountered in mechanical systems as intermediate

modifying devices. 2. Give the advantages of electrical modifying devices compared to mechanical ones. 3. Explain with a neat sketch ballast circuit. 4. Write short notes on:

i) Electronic amplifiers ii) Telemetry

Chapter 9:Terminating Devices 1. With a neat sketch explain the construction and working of Cathode ray oscilloscope. 2. Write short notes on:

i) Oscillographs ii) X-Y Plotters

Chapter 10: Measurement of force and torque 1. With the help of a neat sketch explain the working of a prony brake dynamometer.* 2. Explain with a neat sketch the analytical balance. * 3. Write a note on hydraulic dynamometer 4. How is electric dynamometer different from mechanical ones? 5. Explain with a neat sketch, the method of torque measurement of rotating shafts using

strain gauges. *

Chapter 11:Pressure measurements 1. Explain with sketches the proper orientation of strain gauges for measurement of (i)

Bending strain (ii) Torsional strain (iii) Axial strain. 2. How do you define high pressure range and low pressure range. 3. Explain with neat sketch the working of any one device used for measurement for high

pressure. 4. Explain with a neat sketch the working and application of a Bridgeman gauge.* 5. With neat sketches explain Mcleod and Pirani gauges. 6. With neat sketches, explain the working principle of :

i) Mcleod gauge ii) Knuolsm gauge *

Chapter 12: Temperature measurement 1. What are pyrometers? Explain any one. 2. How resistance thermometer is used to measure temperature with advantages. 3. Explain the thermocouple way of measuring temperature.* 4. Differentiate between radiation and pressure thermometer. 5. With a neat sketch explain bi-metal strip thermometer. * 6. With a neat sketch explain resistance thermometer. *

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B.E. Mechanical 4th Semester Course Information

7. What is temperature sensitivity? Explain how is it compensated. * Chapter 13: Strain measurement 1. Explain with respect to strain gauges: i) Cross sentivity ii) Temperature compensation iii) Positioning of gauges to measure torsional strain 2. Write a note on bonding materials of strain gauges. * 3. Write a note on thermocouple materials and some forms of thermocouple construction.

* 4. How do you calibrate the given strain gauge? * 5. Enumerate the necessary precautions to be taken while mounting a strain gauge on

a test piece. *

*Appeared in VTU exam papers