Anna University 05th Semester Regulation 2008 - ME2302/Dynamics of Machinery Question Bank.
PART – A (2 Marks Questions)
1. Why is flywheel necessary in a punching press?
2. Define crank effort and crank pin effort.
3. Distinguish between the unbalanced force caused due to rotating and reciprocating masses.
4. State and explain the principle of superposition.
5. When the crank is at the inner dead center, in a horizontal reciprocating steam engine, then the velocity of the piston will be ______________(zero / minimum / maximum).
6. Define co-efficient of fluctuation in speed for a flywheel.
7. Find the work done per cycle for an engine whose torque curve is given by T = (25000 + 5000 sin 2θ – 10000 cos 2θ) Nm.
8. Why are the cranks of a locomotive, with two cylinders, placed at 90° to each other?
9. Sketch the Time Vs Displacement plot for under damped and over damped systems.
10. Define logarithmic decrement.
11. What type of motion is exhibited by a vibrating system when it is critically damped?
12. Define Dynamic magnifier.
13. The engine of an aeroplane rotates in clockwise direction when seen from the tail end and the aeroplane takes a turn to the left. What will be the effect of gyroscopic couple on the aeroplane?
14. When is a governor said to be ‘hunt’?
15. Define D’Alembert’s principle.
16. What is meant by balancing of rotating masses?
17. Define the term swaying couple.
18. What is meant by degrees of freedom in a vibrating system?
19. Define critical speed.
20. Define Magnification factor.
21. Explain the term ‘vibration isolation’.
22. Write a note on ‘Hunting of Governors’.
23. Write the expression for Gyroscopic Couple.
24. Define angular velocity of precession.
25. Why single cylinder needs large size flywheel?
26. Mention any two methods to avoid derailment of the locomotive.
27. Compare the magnitude and direction of the unbalanced forces in the case of rotating masses and reciprocating masses.
28. What is the angle between excitation frequency and actual frequency at resonance?
29. Give an expression for inertia and damping force.
30. What is the role of transmission ratio?
31. Give an application of critical damping.
32. Define the meaning of stability of a governor.
33. Which part of the automobile is subjected to the gyroscopic couple?
34. What is free body diagram?
35. Define static force analysis.
36. Differentiate between static and dynamic equilibrium.
37. Define applied and constraint forces.
38. Differentiate between static force analysis and dynamic force analysis.
39. Define inertia force.
40. Define inertia torque.
41. State D’Alemberts principle.
42. State the principle of superposition.
43. Define piston effort.
44. Define crank effort and crank-pin effort.
45. What is meant by turning moment diagram or crank effort diagram?
46. Explain the term maximum fluctuation of energy in fly wheel.
47. Define coefficient of fluctuation of energy.
48. Define coefficient of fluctuation of speed.
49. Define coefficient of steadiness.
50. Why flywheels are needed in forging and pressing operations?
51. What is cam dynamics?
52. Define unbalance and spring surge.
53. Define windup. What is the remedy for camshaft windup?
54. What are the effect and causes of windup?
55. What is meant by balancing of rotating masses?
56. Why rotating masses are to be dynamically balanced?
57. Define static balancing.
58. Define dynamic balancing.
59. State the conditions for static and dynamic balancing.
60. State the conditions for complete balance of several masses revolving in different planes of a shaft.
61. Why complete balancing is not possible in reciprocating engine?
62. Can a single cylinder engine be fully balanced? Why?
63. Differentiate between the unbalanced force caused due to rotating and reciprocating masses.
64. Why are the cranks of a locomotive, with two cylinders, placed at 90˚ to each other?
65. List the effects of partial balancing of locomotives.
66. Define swaying couple.
67. Define hammer blow with respect to locomotives.
68 What are the effects of hammer blow and swaying couple?
69. Define direct and reverse cranks.
70. what for the balancing machines are used?
71. What are different types of balancing machines?
72. What are the causes and effect of vibration?
73. Define frequency, cycle, period and free vibration.
74. What are the different types of vibrations?
75. State different method of finding natural frequency of a system.
76. What is meant by free vibration and forced vibration?
77. Define resonance.
78. What is meant by degrees of freedom in a vibrating system?
79. What is the natural frequency of simple spring mass system?
80. Determine the natural frequency of mass of 10 kg suspended at the bottom of two springs (of
stiffness: 5N/mm and 8N/mm) in series.
81. What is the effect of inertia on the shaft in longitudinal and transverse vibrations?
82. State the expression for the frequency of simple pendulum.
83. Give the expression for natural frequency of water, which oscillates in a ‘U’ tube manometer?
84. What are the different types of damping?
85. Draw the schematic diagram of a free damped vibration system and write the governing differential
equation of the system.
86. Sketch the Time Vs Displacement for under-damped and over-damped systems.
87. What is the limit beyond which damping is detrimental and why?
88. What is meant by critical damping?
89. What type of motion is exhibited by a vibrating system when it is critically damped?
90. Define critical or whirling speed.
91. What are the factors that affect the critical speed of a shaft?
92. What are the causes of critical speed?
93. Differentiate between transverse and torsional vibrations.
94. Define damping ratio or damping factor.
95. Define logarithmic decrement.
96. Give equation for damping factor ζ and damped frequency ωd.
97. What is meant by harmonic forcing?
98. What is the relationship between frequencies of undamped and damped vibration?
99. What is meant by dynamic magnifier or magnification factor?
100. Define transmissibility.
101. Define transmissibility ratio or isolation factor.
102. What is vibration isolation?
103. Sketch the graph for (ω/ωn) Vs Transmissibility for different values of damping factor.
104. What is the function of governor?
105. How governors are classified?
106. Differentiate between governor and fly wheel.
107. What is meant by sensitiveness of a governor?
108. What is the effect of friction on the governor?
109. Define coefficient of sensitiveness.
110. What is meant by hunting?
111. What is meant by isochronous conditions governor?
112. Give application of gyroscopic principle.
113. What is gyroscopic torque?
114. What is the effect of gyroscopic couple on rolling of ship? Why?
115. Define gyroscopic couple.
116. Write expression for gyroscopic couple.
PART – B (16 Marks Questions)
1. The variation of crankshaft torque of a four cylinder petrol engine may be approximately represented by taking the torque as zero for crank angles 0° and 180° and as 260 Nm for crank angles 20° and 45°, the intermediate portions of the torque graph being straight lines. The cycle is being repeated in every half revolution. The average speed is 600 rpm. Supposing that the engine drives a machine requiring constant torque, determine the mass of the flywheel of radius of gyration 250 mm, which must be provided so that the total variation of speed shall be one percent. (AU, NOV 06)
2. A single cylinder vertical engine has a bore of 300 mm and a stroke of 400 mm. The connecting rod is 1 m long and the mass of the reciprocating parts is 140 kg. on the expansion stroke, with the crank at 30° from the top dead center, the gas pressure is 0.7 MPa. If the engine runs at 250 rpm, determine (i) net force acting on the piston (ii) resultant load on the gudgeon pin (iii) thrust on the cylinder walls, and (iv) the speed above which, other things remaining the same, the gudgeon pin load would be reversed in direction. (AU, NOV 06)
3. A shaft carries four rotating masses A, B, C and D which are completely balanced. The masses B, C and D are 50 kg, 80 kg and 70 kg respectively. The masses C and D make angles of 90° and 195° respectively with mass B in the same sense. The masses A, B, C and D are concentrated at radius 75 mm, 100 mm, 50 mm and 80 mm respectively. The plane of rotation of masses B and C are 250 mm apart. Determine (i) the magnitude of mass A and its angular position and (ii) the position planes A and D. (AU, NOV 06)
4. The cranks of a two cylinder, uncoupled inside cylinder locomotive are at right angles and are 325 mm long. The cylinders are 675 mm apart. The rotating mass per cylinders are 200 kg at crank pin and the mass of the reciprocating parts per cylinder is 240 kg. The wheel center lines are 1.5 m apart. The whole of the rotating and two thirds of the reciprocating masses are to be balanced and the balance masses are to be placed in the planes of the rotation of the driving wheels at a radius of 800 mm. Find (i) the magnitude and direction of the balancing masses. (ii) the magnitude of hammer blow (iii) variation in tractive force and (iv) maximum swaying couple at a crank speed of 240 rpm. (AU, NOV 06)
5. a) A spring mass system has spring stiffness of “k” N/m and a mass of “M” kg. It has the natural frequency of vibration as 12 Hz. An extra 2 kg mass is coupled to M and the natural frequency reduces by 2 Hz. Find the values of” k and “M”.
b) A stepped shaft of 0.05 m in diameter for the first 0.6 m length, 0.08 m diameter for the next 1.8 m and 0.03 m diameter for the remaining 0.25 m length. While the 0.05 m diameter end is fixed, the 0.03 m diameter end of the shaft carries a rotor of mass moment of inertia 14.7 kg-m2. If the modulus of elasticity of the shaft material is 0.83 x 1011 N/m2, find the natural frequency of torsional oscillations, neglecting the inertia effect of the shaft. (AU, NOV 06)
6. Between a solid mass of 10 kg and the floor are kept two slabs of isolators, natural rubber and felt, in series. The natural rubber slab has a stiffness of 3000 N/m and an equivalent viscous damping coefficient of 100 N.sec/m. The felt slab has a stiffness of 12000 N/m and equivalent viscous damping coefficient of 330 N.sec/m. Determine the undamped and the damped natural frequencies of the system in vertical direction, neglecting the mass of isolators. (AU, NOV 06)
7. A mass of 10 kg is suspended from one end of a helical spring, the other end being fixed. The stiffness of the spring is 10 N/mm. The viscous damping causes the amplitude to decrease to one tenth of the initial value in four complete oscillations. If a periodic force of 150 cos 50 t N is applied at the mass in the vertical direction, find the amplitude of the forced vibrations. What is its value at resonance? (AU, NOV 06)
8. A machine supported symmetrically on four springs has a mass of 80 kg. The mass of the reciprocating parts is 2.2 kg which move through a vertical stroke of 100 mm with simple harmonic motion. Neglecting damping, determine the combined stiffness of the spring so that the force transmitted to foundation is 1/20th of the impresses force. The machine crank shaft rotates at 800 rpm.
If under working conditions, the damping reduces the amplitudes of successive vibrations by 30%, find (i) the force transmitted to the foundation at resonance and (ii) the amplitude of vibration at resonance. (AU, NOV 06)
9. A ship is propelled by a turbine rotor which has a mass of 5 tonnes and a speed of 2100 rpm. The rotor has a radius of gyration of 0.5 m and rotates in a clockwise direction when viewed from the stern. Find the gyroscopic effect in the following conditions. (i) the ship sails at a speed of 30 km/hr and steers to the left in curve having 60 m radius. (ii) the ship pitches 6° above and 6° below the horizontal position. The bow is descending with its maximum velocity. The motion due to pitching is simple harmonic and the periodic time is 20 seconds. (iii) the ship rolls at a certain instant it has an angular velocity of 0.03 rad/sec clockwise when viewed from stern.
(AU, NOV 06)
10. The length of the upper and lower arms of a porter governor are 200 mm and 250 mm respectively. Both the arms are pivoted on the axis of rotation. The central load is 150 N, the weight of each ball is 20 N and the friction on the sleeve together with the resistance of the operating gear is equivalent to a force of 30 N at the sleeve. If the limiting inclinations of the upper arms to the vertical are 30° and 40°, determine the range of speed of the governor.
(AU, NOV 06)
11. In a reciprocating engine mechanism, if the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm. Determine analytically,
(i) The crank angle at which the maximum velocity occurs and
(ii) Maximum velocity of the piston. (iii) Derive the equations. (AU, NOV 07)
12. A vertical double acting steam engine has a cylinder 300 mm diameter and 450 mm stroke and runs at 200 rpm. The reciprocating parts has a mass of 225 kg and the piston rod is 50 mm diameter. The connecting rod is 1.2 m long. When the crank has turned through 125° from the top dead center the steam pressure above the piston is 30 kN/m2 and below the piston is 1.5 kN/m2. Calculate
(i) Crank-pin effort and
(ii) The effective turning moment on the crank shaft. (AU, NOV 07)
13. (i) Four masses m1, m2, m3 and m4 attached to a rotating shaft on the same plane are 200 kg, 300 kg, 240 kg and 260 kg respectively. The corresponding radii of rotation are 0.2 m, 0.15 m, 0.25 m and 0.3 m respectively and the angles between successive masses are 45°, 75° and 135°. Find the position and magnitude of the balance mass required, if the radius of rotation is 0.2 m.
(ii) Explain with neat sketches, balancing of single revolving mass, by masses in two different planes in a rotating system. (AU, NOV 07)
14. A four cylinder vertical engine has cranks 150 mm long. The planes of rotation of the first, second and fourth cranks are 400 mm, 200 mm and 200 mm respectively from the third crank and their reciprocating masses are 50 kg, 60 kg and 50 kg respectively. Find the mass of the reciprocating parts for the third cylinder and the relative angular positions of the cranks in order that the engine may be in complete primary balance. (AU, NOV 07)
15. (i) A cantilever shaft 50 mm diameter and 300 mm long has a disc of mass 100 kg at its free end. The young’s modulus for the shaft material is 200 GN/m2. Determine the frequency of longitudinal and transverse vibrations of the shaft.
(ii) Explain with sketches different cases of damped vibrations. (AU, NOV 07)
16. A steel shaft 1.5 m long is 95 mm in diameter for the first 0.6 m of its length, 60 mm in diameter for the next 0.5 m of the length and 50 mm in diameter for the remaining 0.4 m of its length. The shaft carries two flywheels at two ends, the first having a mass of 900 kg and 0.85 m radius of gyration located at the 95 mm diameter end and the second having mass of 700 kg and 0.55 m radius of gyration located at the other end. Determine the location of the node and the natural frequency of free torsional vibration of the system. The modulus of rigidity of the shaft material may be taken as 80 GN/m2. (AU, NOV 07)
17. A mass of 10 kg is suspended from one end of a helical spring, the other end being fixed. The stiffness of the spring is 10 N/mm. The viscous damping causes the amplitude to decrease to one-tenth of the initial value in four complete oscillations. If a periodic force of 150 cos 50t N is applied at the mass in the vertical direction, find the amplitude of the forced vibrations. What is the value of resonance? (AU, NOV 07)
18. The mass of an electric motor is 120 kg and it runs at 1500 rpm. The armature mass is 35 kg and its CG lies 0.5 m from the axis of rotation. The motor is mounted on five springs of negligible damping so that the force transmitted is one-eleventh of the impressed force. Assume that the mass of the motor is equally distributed among the five springs. Determine: (i) Stiffness of each spring. (ii) Dynamic force transmitted to the base at the operating speed. (iii) Natural frequency of the system. (AU, NOV 07)
19. A porter governor has equal arms each 250 mm long and pivoted on the axis of rotation. Each ball has a mass of 5 kg and the mass of the central load on the sleeve is 25 kg. The radius of rotation of the ball is 150 mm when the3 governor begins to lift and 200 mm when the governor is at maximum speed. Find the minimum and maximum speeds and range of speed of the governor. (AU, NOV 07)
20. (i) Explain the effect of Gyroscopic couple on a Naval ship during pitching.
(ii) Explain the effect of Gyroscopic couple on a Aeroplane. (AU, NOV 07)
21. A vertical double acting steam engine develops 75 kW at 250 rpm. The maximum fluctuation of energy is 30 percent of the work done per stroke. The maximum and minimum speeds are not to vary more than 1 percent on either side of the mean speed. Find the mass of the flywheel required, if the radius of gyration is 0.6 m.
22. The length of crank and connecting rod of a vertical reciprocating engine are 300 mm and 1.5 m respectively. The crank is rotating at 200 rpm clockwise. Find analytically, (i) Acceleration of piston, (ii) velocity of piston and (iii) angular acceleration of the connecting rod when the crank has turned through 40 degree from the top dead center and the piston is moving downwards.
23. A two cylinder uncoupled locomotive has inside cylinders 0.6 m apart. The radius of each crank is 300 mm and are at right angles. The revolving mass per cylinder is 250 kg and the reciprocating mass per cylinder is 300 kg. The whole of the revolving and two-third of reciprocating masses are to be balanced and the balanced masses are placed, in the planes of rotation of the driving wheels, at a radius of 0.8 m. The driving wheels are 2 m in diameter and 1.5 m apart. If the speed of the engine is 80 km.p.h., find the hammer blow, maximum variation in tractive effort and maximum swaying couple.
24. A four cylinder engine has the two outer cranks at 120° to each other and their reciprocating masses are each 400 kg. The distance between the planes of rotation of adjacent cranks are 400 mm, 700 mm and 500 mm. Find the reciprocating mass and the relative angular position for each of the inner cranks, if the length of each crank is 350 mm, the length of each connecting rod 1.7 m and the engine speed 500 rpm.
25. A body of mass of 50 kg is supported by an elastic structure of stiffness 10 kN/m. The motion of the body is controlled by a dashpot such that the amplitude of vibration decreases to one –tenth of its original value after two complete cycles of vibration. Determine (i) the damping force at 1 m/s; (ii) the damping ratio; and (iii) the natural frequency of vibration.
26. Two parallel shafts A and B of diameters 50 mm and 70 mm respectively are connected by a pair of gear wheels, the speed of A being 4 times that of B. The mass moment of inertia of the flywheel is 3 kg-m2. Is mounted on shaft A at a distance of 0.9 m from the gears. The shaft B also carries a flywheel of mass moment of inertia 16 kg-m2 at a distance of 0.6 m from the gears. Neglecting the effect of the shaft and gear masses, calculate the fundamental frequency of free torsional oscillations and the positions of node. Assume modulus of rigidity as 84 GN/m2.
27. A mass of 500 kg is mounted on supports having a total stiffness of 100 kN/m and which provides viscous damping, the damping ratio being 0.4. The mass is constrained to move vertically and is subjected to a vertical disturbing force of the type F cos ωt. Determine the frequency at which resonance will occur and the maximum allowable value of F if the amplitude at resonance is restricted to 5 mm.
28. A machine of mass 75 kg is mounted on springs of stiffness 1200 kN/m and with an assumed damping factor of 0.2. A piston within the machine of mass 2 kg has a reciprocating motion with a stroke of 80 mm and a speed of 3000 cycles/min. Assuming the motion to be simple harmonic, Find (i) the amplitude of motion of the machine, (ii) its phase angle with respect to the existing force, (iii) the force transmitted to the foundation, and (iv) the phase angle of transmitted force with respect to the exciting force.
29. In a Porter governor, the mass of the central load is 18 kg and the mass of each ball is 2 kg. The top arms are 250 mm while the bottom arms are each 300 mm long. The friction of the sleeve is 14 N. If the top arms make 45° with the axis of rotation in the equilibrium position, find the range of speed of the governor in that position.
30. A disk with radius of gyration 60 mm and mass of 4 kg is mounted centrally on a horizontal axel of 80 mm length between the bearings. It spins about the axle at 800 rpm counter-clockwise when viewed from the right-hand side bearing. The axle precesses about vertical axis at 50 rpm in the clockwise direction when viewed from above. Determine the resultant reaction at each bearing due to the mass and gyroscopic effect
31. A porter governor has equal arms each 250mm long and pivoted on the axis of rotation. Each ball has
a mass of 5kg and mass of the central load on the sleeve is 25kg.The radius of rotation of the ball is
150mm when governor is at maximum speed. Find the maximum and minimum speed and range of
speed of the governor.
32 . (i) Explain the effect of Gyroscopic couple on a Naval ship during pitching. (8)
(ii) Explain the effect of gyroscopic couple on an Aero plane. (8)
33. The length of the upper and lower arms of a porter governor is 200mm and 250mm respectively. Both
the arms are pivoted on the axis of rotation. The central load is 150N, the weight of the each ball is
20N and the friction of the sleeve together with the resistance of the operating gear is equivalent to a
force of 30N at the sleeve. If the limiting inclinations of the upper arms to the vertical are 30˚ and 40˚
taking friction in to account. Find the range of speed of the governor.
34. Calculate the rage of speed of a porter governor which has equal arms of each 200mm long and pivoted
on the axis of rotation .The mass of each ball is 4kg and the central load of the sleeve is 20kg.The
radius of rotation of the ball is 100mm when the governor being to lift and 130mm when the governor
is at maximum speed.
35. A hartnell governor having a central sleeve spring and two right angled bell crank lever operates
between 290rpm and 310rpm for a sleeve lift of 15mm.The sleeve and ball arms are 80mm and
120mm repectively.The levers are pivoted at 120mm from the governoraxis and mass of the ball is
2.5kg.The ball arms are parallel at lowest equilibrium speed.Determine (i) load on the spring at
maximum and minimum speeds and (ii) Stiffness of the spring.
36. A governor of hartnell type has equal balls of mass 3kg, set initially at a radius of 200mm.The arms of
the bell- crank lever are 110mm vertically and 150mm horizontally. Find (i) the initial compressive
force on the spring at a radius of 200mm at240rpm and (ii) the stiffness of the spring required to
permit a sleeve movement of 4mm on a fluctuation of 7.5 percent in the engine speed.
37. The controlling force in a spring controlled governor is 1500N when radius of rotation is 200mm and
887.5N when radius of rotation is 130mm.The mass of each ball is 8kg.If the controlling force curve is
a straight line, then find (i) Controlling force at 150mm radius of rotation (ii) Speed of the governor at
150mm radius.(iii)Increase in initial tension so that governor is isochronous. (iv) Isochronous speed.
38. In a spring controlled governor, the controlling force curve is a straight line. When the balls are
400mm apart, the controlling force is 1200N and when 200mm apart, the controlling force is
450N.Determine the speed at which the governor runs when the balls are 250mm apart. When initial
tension on the spring would be required for isochronisms and what would be the speed. Take mass of
each ball to be 10kg.
39. Calculate the minimum speed of a Proell governor, which has equal arms each of 200mm and are
provided on the axis of rotation. The mass of each ball is 4kg and the central mass on the sleeve is
20kg.The extension arms of the lower links are each 60mm long and parallel to the axis when the
minimum radius of the ball is 100mm.of load.
40. Each paddle wheel of a steamer have a mass of 1600kg and a radius of gyration of 1.2meters.The
steamer turns to port in a circle of 160meters radius at 24Km/hr.The speed of the paddle is
90rpm.Find the magnitude and effect of the gyroscopic couple acting on the steamer. (16)
41. The rotor of a turbine yacht rotates at 1200rpm clockwise when viewed from stern. The rotor has a
mass of 750 kg and radius of gyration of 250mm.Find the maximum gyroscopic couple transmitted to
the hull when yacht pitches with a maximum angular velocity of 1 rad/s. What is the effect of this
couple?
42. The turbine rotor of a ship has a mass of 20 tonnes and a radius of gyration 0.75.Its speed is
2000rpm.The ship pitches 6˚ above and below the horizontal position .One complete oscillation takes
18 seconds and the motion is simple harmonic. Determine (i) the maximum couple tending to shear the
holding down bolt of the turbine. (ii)The maximum angular acceleration of the ship during pitching.
(iii) The direction in which the bow will tend to turn while, if the rotation of the rotor is clockwise
when locking from rear.
43. A mass of 50 kg is supported by an elastic structure of total stiffness 20 KN/m. The damping ratio of
the system is 0.2. A simple harmonic disturbing force acts on the mass and at any time ‘t ’
seconds, the force is 60sin10t N. Find amplitude of the vibration and phase angle caused by the
damping.
44. A mass of 50 kg is supported by an elastic structure of total stiffness 20KN/m. The damping ratio of
the system is 0.25. A simple harmonic disturbing force acts on the mass and at any time‘t seconds,
the force is 75cos12t N. Find amplitude of the vibration and phase angle caused by the damping.
45. A mass of 10 kg is suspended from one end of a helical spring, the other end being fixed. The
stiffness of the spring is10 N/mm. The viscous damping causes the amplitude to decreases to 1/10th
of the initial value in four complete oscillations. If a periodic force of 150cos50t N is applied at the
mass in the vertical direction. Find the amplitude of the forced vibrations and the value of resonance.
46. A harmonic exiting force of 25 N is acting on a machine part which is having a mass of 2 Kg and
vibrating in viscous medium. The exciting force causes resonant amplitude of 12.5 mm with a period
of 0.2sec.
47. A body having a mass of 15 kg is suspended from a spring which deflects 12 mm under the weight
of the mass. Determine the frequency of the free vibrations. What is the viscous damping force
needed to make the motion a periodic at a speed of 1mm/s? If, when damped to this extend a
disturbing force having a maximum value of 100 N and vibrating at 6 Hz is made to act on the body,
determine the amplitude of the ultimate motion.
48. A single cylinder vertical petrol engine of total mass of 200 kg is mounted upon a steel chassis
frame. The vertical static deflection of the frame is 2.4 mm due to the weight of the engine. The
mass of the reciprocating parts is 18 kg and stroke of piston 160 mm with S.H.M. If dashpot of
damping coefficient of 1 N/mm/s used to damped the vibrations, calculate al steady state (i)
Amplitude of vibrations at 500 rpm engine speed. (ii) The speed of the driving shaft at which
resonance will occurs.
49. A vertical single stage air compressor having a mass of 500 kg is mounted on spring having
stiffness of 1.96x105 N/m and dashpot with damping factor of 0.2 m. The rotating parts are
completely balanced and the equivalent reciprocating parts weight 20 kg. The stroke is 0.2 m.
Determine the dynamic amplitude of vertical motion of the excitation force if the compressor is
operates at 200 rpm.
50. A machine 100 kg has a 20 kg rotor with 0.5 mm eccentricity. The mounting spring have s = 85x103.
The operating speed is 600 rpm and the unit is constrained to move vertically. Find (i) Dynamic
amplitude of machine (ii) the force transmitted to the support.
51. A single cylinder engine has an out of balance force of 500 N at an engine speed of 30rpm.The total
mass of engine is 150kg and its carried on a set of total stiffness 300 N/cm. (i) Find the amplitude of
steady motion of the mass and maximum oscillating force transmitted to the foundation. (ii)If a
viscous damping is interposed between the mass and the foundation the damping force 1000N at
1m/s of velocity, find the amplitude of force damped oscillation of the mass and its angle of lag with
disturbing force.
52. An industrial machine weighting 445kg is supported on a spring with a statical deflection of
0.5cm.If the machine has rotating imbalance of 25kg-cm.Determine the force transmitted at
1200rpm and the dynamic amplitude at the speed.
53. Derive an expression for the natural frequency of the free longitudinal vibration by (i) Equilibrium
method (ii) Energy method (iii) Rayleigh’s method.
54. In a single degree of damped vibration system a suspended mass of 8 kg makes 30 oscillations in 18
seconds. The amplitude decreases in 18 seconds. The amplitude decreases to 0.25 of the initial value
after 5 oscillations. Determine (i) the spring stiffness (ii) logarithmic decrement (iii) damping factor
(iv) Damping coefficient.
55. Determine equation of motion when a liquid column vibrating in a ‘U’tube by (i) Newton’s method
(ii) Energy method and hence find its natural frequency.
56. (i) Deduce the expression for the free longitudinal vibration in terms of spring stiffness, its inertia
effect and suspended mass. (ii) A spring mass system has spring stiffness‘s’ N/m and has a mass of
‘m’ kg. It has the natural frequency of vibration as 12 Hz. An extra 2 kg mass is coupled to ‘m’ and
natural frequency reduces by 2 Hz. Find the value of‘s’ and ‘m’.
57. Avibrating system consists of a mass of 8 kg, spring of stiffness 5.6 N/m and dashpot of damping
coefficient of 40 N/m/s. Find (i) Critical damping coefficient (ii) the damping factor (iii) the natural
frequency of damped vibration (iv) the logarithmic decrement(v)the ratio of two consecutive
amplitude (vi) the number of cycle after which the original amplitude is reduced to 20 percent.
58. An instrument vibrates with a frequency of 1 Hz when there is no damping. When the damping is
provided, the frequency of damped vibration was observed to be 0.9 Hz. Find, (i) damping factor ( ii)
logarithmic decrement.
59. Find the equation of notion for the spring mass-dashpot system for the cases when (i) ζ = 2 (ii) ζ = 1
and (iii)ζ = 0.3. The mass ‘m’is displaced by a distance of 30mm and released.
60. Between a solid mass of 0 kg and the floor are kept two slabs of isolates, natural rubber and felt, in
series. The natural rubber slab has a stiffness of 3000 N/m and equivalent viscous damping
coefficient of 100 N-sec/m. The felt has a stiffness of 12000 N/m and equivalent viscous damping
coefficient of 330N-sec/m. Determine undamped and the damped natural frequencies of the system in
vertical direction.
61. (i) A cantilever shaft 50 mm diameter and 300 mm long has a disc of mass 100 kg at its free end.
The young’s modulus for the shaft material is 200 GN/m2. Determine the frequency of longitudinal
and transverse vibration of the shaft. (ii) Explain the sketches different cases of damped vibrations.
62. The barrel of a large gun recoils against a spring on firing. At the end of the firing, a dashpot is
engaged that allows the barrel to return to its original position in minimum time without
oscillation. Gun barrel mass is 400 kg and initial velocity of recoils 1m.Determine spring stuffiness
and critical damping coefficient of dashpot.
63. A steel shaft 100 mm in diameter is loaded and support in shaft bearing 0.4 m apart. The shaft
carries three loads: first mass 12 kg at the centre, second mass 10 kg at a distance 0.12 m from the
left bearing and third mass of 7 kg at a distance 0.09 m from the right bearing. Find the value of the
critical speed by using Dunkerley’s method. E = 2 x 1011 N/m2.
64. A shaft is rotating at a uniform angular speed. Four masses M1, M2, and M3and M4 of magnitudes
300 kg, 450 kg, 360 kg, 390 kg respectively are attached rigidly to the shaft. The masses are
rotating in the same plane. The corresponding radii of rotation are 200 mm, 150 mm, 250 mm and
300 mm respectively. The angle made by these masses with horizontal are 0°, 45°, 120° and 255°
respectively. Find (i) The magnitude of balancing mass (ii) The position of balancing mass if its
radius of rotation is 200 mm.
65. Four masses M1, M2, M3, and M4 are 200 kg, 300 kg, 240 kg and 260 kg respectively. The
corresponding radii of rotation are 0.2 m, 0.15 m, 0.25 m and 0.3 m respectively and the angle
between successive masses 45°, 75°, and 135°. Find the position and magnitude of balance mass
required if its radius of rotation is 0.25 m.
66. The data for three rotating masses are given below:- M1 = 4 kg, r1 = 75 mm, θ1 = 45°, M2 =3 kg,
r2 = 85 mm, θ2 = 135°, M3 = 2.5 kg, r3 = 50 mm, θ3 = 240°. Determine the amount of counter mass at a
radial distance of 65mm required for their static balance.
67. Four masses A, B, C, and D are completely balanced masses C and D makes angles of 90°
and 195° respectively with B in the same sense. The rotating masses have the following properties:
MA = 25 kg, RA = 150 mm, MB = 40 kg, RB = 200 mm, MC = 35 kg RC = 100 mm, RD = 180 mm
Planes B and C are 250mm apart. Determine (i) the mass A and its angular position (ii) the position of
planes A and D.
68. A, B, C and D are four masses carried by a rotating shaft at radii 100 mm,125 mm, 200 mm
and 150 mm respectively. The planes in which the masses revolve are spaced 600 mm apart and the
masses of B, C and D are 10 kg, 5 kg and 4 kg respectively. Find the required mass A and relative
angular setting of the four masses so that the shaft be in complete balance.
69. Four masses A, B, C and D revolves at equal radii and equally spaced along a shaft. The mass B is
7 kg and the radii of C and D make angles of 90° and 240° respectively with the radius of B. Find the
magnitude of masses A, C and D and angular position of A so that the system may be completely
balanced.
70. A shaft caries four rotating masses A, B, C and D which are completely balanced. The masses B, C
and D are 50 kg, 80 kg and 70 kg respectively. The masses C and D make angles of 90° and 195°
respectively with mass B in the same sense. The masses A, B, C and D are concentrated at radius
75 mm,100 mm,50 mm and 90 mm respectively.The plane of rotation of masses B and C are 250 mm
apart. Determine (i) the magnitude of mass A and its angular position (ii) the position of planes A and
D.
71. A four cylinder vertical engine has cranks 150 mm long. The plane of rotation of the first, second
and fourth cranks are 400 mm, 200 mm and 200 mm respectively from that of the third crank and
their reciprocating masses are 50 kg, 60 kg and 50 kg respectively. Find the mass of the
reciprocating parts for the third cylinder and relative angular position of the cranks in order that the
engine may be in complete balance.
72. A four cylinder vertical engine has cranks 300 mm long. The plane of rotation of the first, third and
fourth cranks are 750 mm, 1050 mm and 1650 mm respectively from that of the second crank and
their reciprocating masses are 10 kg, 400 kg and 250 kg respectively. Find the mass of the
reciprocating parts for the second cylinder and relative angular position of the cranks in order that the
engine may be in complete balance.
73. Derive the following expression of effects of partial balancing in two cylinder locomotive
engine (i) Variation of tractive force (ii) Swaying couple (iii) Hammer blow
74. In a reciprocating engine mechanism, if the crank and connecting rod are 300 mm and 1m long
respectively and the crank rotates at a constant speed of 200 rpm. Determine analytically, (i) The crank
angle at which the maximum velocity occurs (ii) Maximum velocity of piston. (iii) Derive the relevant
equations.
75. (i) Deduce the expression for the inertia force in the reciprocating force neglecting the weight of the
connecting rod. (ii) A vertical petrol engine with cylinder of 150mm diameter and 200mm strokes
has a connecting rod of 350mm long. The mass is 1.6kg and the engine speed is 1800 rpm. On the
expansion stroke with crank angle 30˚ from TDC, the gas pressure is 750KPa. Determine the net
thrust on the piston.
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3. (i) Define coefficient of fluctuation of speed and coefficient of fluctuation of energy. (ii) The radius
of gyration of a fly wheel is 1meter and fluctuation of speed is not to exceed 1% of the mean
speed of the flywheel. If the mass of the flywheel is 3340 kg and the steam develops 150KW at 135
rpm, then find 1. Maximum fluctuation of energy 2. Coefficient of fluctuation of energy.
76. The length of crank and connecting rod of a horizontal reciprocating engine are 100 mm and
500 mm respectively. The crank is rotating at 400rpm.When the crank has turned 30˚ from the IDC,
f i n d analytically 1.Velocity of piston 2. Acceleration of piston 3. Angular velocity of connecting
rod 4. Angular acceleration of connecting rod.
77. The length and connecting rod of a horizontal reciprocating engine are 200mm and 1meter
respectively. The crank is rotating at 400rpm.When the crank has turned 30˚ from the inner dead
center, the difference of pressure between cover end and piston rod is 0.4 N/mm2. If the mass of the
reciprocating parts is 100Kg and a cylinder bore is 0.4meters. Calculate (i) Inertia force (ii) Force on
piston (iii) Piston effort (iv) Thrust on the side of the cylinder walls (v) Thrust in the connecting rod
( vi) Crank effort.
78. A horizontal gas engine running at 210 rpm has a bore of 220 mm and a stroke of 440 mm. The
connecting rod is 924 mm long the reciprocating parts weight 20 kg. When the crank has turned
through an angle of 30° from IDC, the gas pressure on the cover and the crank sides are 500 KN/m2
and 60 KN/m2 respectively. Diameter of the piston rod is 40 mm. Determine, 1. Turning moment on
the crank shaft 2.Thrust on bearing 3. Acceleration of the flywheel which has a mass of 8 kg and
radius of gyration of 600 mm while the power of the engine is 22 KW.
79. A single cylinder vertical engine has a bore of 300 mm and a stroke of 400 mm. The connecting rod
is 1000 mm long. The mass of the reciprocating parts is 140 kg. On the expansion stroke with the
crank at 30° from the top dead center, the gas pressure is 0.7 MPa. If the runs at 250 rpm, determine;
1. Net force acting on the piston 2. Resultant load on the gudgeon pin 3. Thrust on cylinder walls 4.
The speed above which other things remaining same, gudgeon pin loads would be reversed in
direction.
80. A vertical double acting steam engine has a cylinder 300 mm diameter and 450 mm stroke and runs at
200 rpm. The reciprocating parts have a mass of 225 kg and the piston rod is 50 mm diameter. The
c o n n e c t i n g rod is 1.2 m long. When the crank has turned 125° from IDC the steam pressure
above the piston is 30 KN/m2. Calculate, (i) Crank-pin effort (ii) The effective turning moment on the
crank shaft.
81. The turning moment diagram for a petrol engine is drawn to a scale of 1mm to 6 N-9-9m and the
horizontal scale of 1 mm to 1°.The turning moment repeat itself after every half revolution of the
engine. The area above and below the mean torque line are 305, 710, 50,350,980 and 275 mm2. The
mass of rotating parts is 40 kg at a radius of gyration of 140 mm. Calculate the coefficient of
fluctuation of speed if the mean speed is 1500 rpm. (16)
82. The torque delivered by a two stroke engine is represented by T= (1000+300sin2θ-500cos2θ) N-m
where θ is the angle turned by the crank from the IDC. The engine speed is 250 rpm. The mass of the
flywheel is 400kg and radius of gyration 400 mm. Determine, (i) The power developed (ii) The total
percentage fluctuation of speed (iii) The angular acceleration of flywheel when the crank has rotated
through an angle of 60° from the IDC. (iv) The maximum angular acceleration and retardation of the
flywheel.
83. For reciprocating engine, derive the expression for (i) Velocity and acceleration of the piston.
(ii) Angular velocity and angular acceleration of the connecting rod.
