ME2302 – DYNAMICS OF MACHINERYTWO MARKS QUESTION AND ANSWERS 1. What are the conditions for a body to be in static and dynamic equilibrium? Necessary and sufficient conditions for static and dynamic equilibrium are 1. Vector sum of all forces acting on a body is zero. 2. The vector sum of the moments of all forces acting about any arbitrary point or axis is zero. First condition is the sufficient condition for static equilibrium together with second condition is necessary for dynamic equilibrium. 2. Define static force analysis. If components of a machine accelerate, inertia is produced due to their masses. However, the magnitudes of these forces are small compared to the externally applied loads. Hence inertia effect due to masses are neglected. Such an analysis is known as static force analysis. 3. Define force and applied force. Force is a push or pull, which acts on a body changes or tends to change, the state of rest or of uniform motion of the body. A force is completely characterized by its point of application, its magnitude and direction. The external force acting on a system of body from outside the system are called applied force. The applied forces are classified as active and reactive force. 4. Rate of change of momentum of a body is directly proportional to Force acting on it. 5. Which law helps to measure a force quantitatively? Newton’s second law helps us to measure a force quantitatively. 6. Distinguish between space diagram and free body diagram. A space diagram is a graphical description of the system. It generally shows the shape and size of the system, the weight, the externally applied loads, the connection and the supports of the system. A free body diagram is a sketch of the isolated or free body which shows all the pertinent weight forces, the externally applied loads, and the reaction from its supports and connections acting upon it by the removed elements. 7. When will the two force member is in equilibrium? The member under the action of two force will be in equilibrium if, 1. The two forces are of same magnitude, 2. The forces act along the same line, and 3. The forces are in opposite direction 8. Give any three advantages of free body diagram. 1. Free body diagram assist in seeing and understanding all aspects of problem. 2. They help in planning the approach to the problem. 3. They make mathematical relations easier to the problem. 9. When will the three force member is in equilibrium. A body or member will be in equilibrium under the action of three forces if, 1. the resultant of the forces is zero, and 2. the line of action of the forces intersect at a point. 10. Differentiate between static force analysis and dynamic force analysis. If components of a machine accelerate, inertia forces are produced due to their masses. If the magnitude of these forces are small compared to the externally applied m1 + m2 = m 2. What do you mean by crank effort or turning moment on the crank shaft? It is the product of the crank-pin effort (FT) and crank pin radius (r). 20. What do you mean by inertia? The property of matter offering resistance to any change of its state of rest or of uniform motion in a straight line is known as inertia. The mass of the rigid body must be equal to the sum of masses of two concentrated masses. The vector sum of all external moments and inertia torques acting upon a system of rigid bodies is also separately zero. Define compound pendulum or torsional pendulum. 13. and . The radial force (FR) parallel to crank. State D’Alembert’s principle. backlash. The centre of gravity of the two masses must coincide with the centre of gravity of the rigid body. 21. Give one example each for linear and non-linear system. along the line of stroke.e. D’Alembert’s principle states that the inertia forces and torques. and the external forces and torques acting on a body together result in statical equilibrium. i. Non-linear system: Example: Systems with static or coulomb friction. When the connecting rod is large the piston executes simple harmonic motion. Piston effort is defined as the net or effective force applied on the piston. 19. A rigid body suspended vertically at a point and oscillating with very small amplitude under the action of gravitational force is known as compound pendulum or torsional pendulum. 11. they can be neglected while analyzing the mechanism. i. 14. What do you mean by Equivalent offset inertia force? Equivalent offset inertia force is the force which can replace both inertia force and inertia torque. 22. Define Piston effort. 15. l1 . Principle of super position has a limitation that it cannot be applied for non-linear systems. 12. 4. i. 16. the vector sum of all external forces acting upon a system of rigid bodies is zero. we can reduce it into an equivalent problem of static equilibrium. Inertia force of the reciprocating parts (FI) acting along the line of stroke. 3. How you will reduce a dynamic analysis probloem into an equivalent problem of static equilibrium? By applying D’Alembert’s principle to a dynamic analysis problem. The sum of mass moment of inertia of two masses about their centre of gravity is equal to the mass moment of inertia of the rigid body. 18. What are the forces acting on the connecting rod? 1. 2.e. Such an analysis is known as static force analysis. Weight of the connecting rod. The principle of super position states that for linear systems the individual responses to several disturbances or driving functions can be superposed on each other to obtain the total response of the system. m1l1 = m2 l2 3. The side thrust between the cross head and the guide bars acting at P and right angles to line of stroke. In other words. What are the requirements of an equivalent dynamical system? 1. 5. Inertia force of the connecting rod (FC).e. l2 = (kG)2 23.loads. It is also known as effective driving force (or) net load on the gudgeon pin. 17. Linear system: Example: Spring system. State the principle of super position. What is meant by turning moment diagram or crank effort diagram? It is the graphical representation of the turning moment or crank effort for various position of the crank. which when acts upon a rigid body. Governor works intermittently. 3. 1. 35. The difference between the maximum and the minimum energies is known as maximum fluctuation of energy. The inertia force is an imaginary force. and 4. Define Inertia Torque. Inertia force = . What is cam dynamics? Cam dynamics is the study of cam follower system with considering the dynamic forces and torques developed in it. Flywheel works continuously from cycle to cycle. Define coefficient of fluctuation of speed.ma 26. 27. 28. Rivetting machines. vibration. Shearing machines. So balancing of machine is necessary. the turning moment is taken as the ordinate (Y-axis) and crank angle as abscissa (X-axis) 25. 3. The function of flywheel is to reduce the fluctuations of speed during a cycle above and below the mean value for constant load from prime mover. Flywheel has no influence on mean speed of the prime mover. Punching machines. Define Inertia force. The ratio of the maximum fluctuation of speed to the mean speed is called the coefficient of fluctuation of speed. wear and tear of the system. 34. 24. 2. 30. Why balancing of dynamic forces is necessary? . It is the rat6io of maximum fluctuation of energy to the work done per cycle. 1. i. What is meant by maximum fluctuation of speed? The difference between the maximum and minimum speeds during a cycle is called maximum fluctuation of speed. Crushing machines. In turning moment diagram. Governor has no influence over cyclic speed fluctuations. brings it in an equilibrium position. 33.6. The inertia torque is an imaginary torque. 32. The tangential force (FT) acting perpendicular to crank. The function of governor is to control the mean speed over a period for output load variations. 2. Differentiate the function of flywheel and governor. It is equal to the accelerating couple in magnitude by opposite in direction. 31. which when applied upon the rigid body.Accelerating force = . brings it in equilibrium position.e. List out the few machines in which flywheel is used. Define coefficient of fluctuation of energy. What do you understand by the fluctuations of energy and maximum fluctuation of energy? The variations of energy above and below the mean resisting torque line are called fluctuations of energy. 29. Write the importance of Balancing? If the moving part of a machine are not balanced completely then the inertia forces are set up which may cause excessive noise. only when there is change in the load. Balancing of single rotating mass by two masses rotating in different planes. 45. such that vehicle vibrates vigorously. Variation in tractive force along the line of stroke. 2. engine crank shafts. constitute a couple in the horizontal direction. The unbalanced force due to reciprocating mass varies in magnitude but constant in direction. But in the case of revolving masses. It is very common in cam shafts. 47. Balancing of reciprocating masses. they will cause worse effects such as war and tear on bearings and excessive vibrations on machines. 46. and centrifugal pumps. Why complete balancing is not possible in reciprocating engine? Balancing of reciprocating masses is done by introducing the balancing mass opposite to the crank. 3. a part of the reciprocating mass is balanced. 38. The resultant couple must be zero. etc. But fraction of reciprocating mass only balanced. 42. 41. the grinding wheels are properly balanced by inserting some low density materials. Balancing of rotating masses 1. and 2. Hence complete balancing is not possible in reciprocating engines. Static balancing 2. 39. Complete balancing of revolving mass can be possible. What is swaying couple? The unbalanced force acting at a distance between the line of stroke of two cylinders. The resultant unbalanced force due to the two cylinders along the line of stroke is known as tractive force. Balancing of single rotating mass by a single mass rotating in the same plane. Swaying couple. 2. 44. Dynamic balancing 2. Write the condition for complete balancing? 1. In order to reduce the Hammer blow. Define tractive force. The resultant centrifugal force must be zero and 2. steam turbine rotors. Balancing of several rotating masses in a single plane. Whether grinding wheels are balanced or not? If so why? Yes. 37. What are the effects of an unbalanced primary force along the line of stroke of two cylinder locomotive? 1. Balancing of several rotating masses in different planes. 36. Write the equation for balancing a single rotating mass by a single mass? For balancing single rotating mass by a single rotating mass. The vertical component of the dynamic force of this balancing mass gives rise to “Hammer blow”. What are the various cases of balancing of revolving masses? 1. 43. What are in-line engines? .. the unbalanced force is constant in magnitude but varies in direction. Differentiate between the unbalanced force due to a reciprocating mass and that due to a revolving masses. This couple is known as swaying couple. Hammer blow is caused due to the effect of unbalanced primary force acting perpendicular to the line of stroke. If not the required surface finish won’t be attained and the vibration will cause much noise. 1. 4. What is the effect of hammer blow and what is the cause of it? The effect of hammer blow is to cause the variation in pressure between the wheel and the rail. the equation is m1r1 = m2r2. 40.If dynamic force are not balanced. Write the different types of balancing? 1. It is the number of cycles described in one second. 54. 60. 57. and 2. are known as in-line engine. The algebraic sum of the couples due to primary and secondary forces must be zero. Such vibration is termed as steady state vibrations. What do you meant by damping and damped vibration? The resistance against the vibration is called damping. List out the various methods of finding the natural frequency of free longitudinal vibrations. Free vibrations a) Longitudinal vibration. the amplitude of vibration decays continuously because of natural damping and vanishes finally. b) Transverse vibration. 2. Cycle is defined as the motion completed during one time period. 49.Multi-cylinder engines with the cylinder centre lines in the same plane and on the same side of the centre line of the crank shaft. then the body is said to be under forced vibrations. therefore there are no unbalanced primary or secondary couples. Forced vibrations. 53. Define frequency of vibration. The algebraic sum of the primary and secondary forces must be zero. Its unit is Hz. after giving it an initial displacement. self excitations. A cantilever beam has infinite number of degrees of freedom. Define resonance. What do you mean by a degree of freedom or movability? The number of independent coordinates required to completely define the motion of a system is known as degree of freedom of the system. the free vibration continue indefinitely as there is no damping. and c) Torsional vibration. What are the condition to be satisfied for complete balance of in-line engine? 1. Why radial engines are preferred? In radial engines the connecting rods are connected to a common crank and hence the plane of rotation of the various cranks is same. In ideal systems. In real systems. When the body vibrates under the influence of external force. the amplitude of vibration becomes excessively large. 50. Hence radial engines are preferred. elastic nature of the system. Period is the time interval after which the motion is repeated itself. 51. Such vibration in real system is called transient vibration. Define steady state and transient vibrations. When there is a reduction in amplitude over every cycle of vibration. 61. 59. This phenomenon is known as resonance. 56. What is equivalent spring stiffness? Equivalent spring stiffness is the measure of overall spring stiffness of any system having more than one spring connected in series or parallel. What are the causes of vibration? The causes of vibration are unbalanced forces. and 3. 48. winds and earthquakes. When the frequency of external force is equal to the natural frequency of a vibrating body. then the body is said to be under free or natural vibration. Define Period and cycle of vibration. What is meant by free vibration and forced vibrations? When no external force acts on the body. then the motion is said to be damped vibration. How will you classify vibration? 1. 58. Damped vibration. . 52. 55. . The damping force per unit velocity is known as damping coefficient. Eccentric mountings. b) diameter of the disc. Logarithmic decrement is defined as the natural logarithm of the amplitude reduction factor. A shaft supported in long bearing is assumed to have both ends fixed for solving transverse vibration problems. The two roots k1 and k2 are knjown as complex conjugate. 65. of the rotating masses from the axis of rotation of the shaft. Define critical or whirling or whipping speed of a shaft. 73. Define logarithmic decrement.G. the mass moves back very quickly to its equilibrium position within no time. Briefly explain elastic suspension. Non-uniform distribution of rotor material. the spring transmits the force applied on the machine to the fixed support or foundation. c) span of the shaft. What are the factors that affect the critical speed of a shaft? The critical speed essentially depends on a) the eccentricity of the C. 2. 63. In other words. When do you say a vibrating system is under damped? The equation of motion of a free damped vibration is given by 2 d x + c dx + s = 0 dt2 m dt m If (s/m) > (c/2m)2 . Energy method. This is called as transmissibility. The speed at which resonance occurs is called critical speed of the shaft. 68. What is meant by transmissibility? When a machine is supported by a spring. What are the causes of critical speed? 1. 75. and d) type of supports connections at its ends. and 3.1. It is defined as the ratio of actual damping coefficient to the critical damping coefficient. then radical becomes negative. 74. 64. 72. 66. If system is critically damped. Rayleigh’s method. What is meant by dynamic magnifier or magnification factor? It is the ratio of maximum displacement of the forced vibration to the deflection due to the static force. 71. 2. 62. Distinguish between critical damping and large damping. Whereas in large damping. 70. is known as critical speed. the mass moves slowly to the equilibrium position. 69. Bending due to self weight. 67. The amplitude reduction factor is the ratio of any two successive amplitudes on the same side of the mean position. Equilibrium method and 3. Define damping ratio. the speed at which the shaft runs so that the additional deflection of the shaft from the axis of rotation becomes infinite. Define transmissibility ratio or isolation factor. What is the principle of Rayleigh’s method of finding natural frequency of vibrations? The principle of Rayleigh’s method is “the maximum kinetic energy at the mean position is equal to the maximum potential energy or (strain energy) at the extreme position. Then the vibrating system is known as under damping. The ratio of force transmitted to the force applied is known as transmissibility ratio. 79. the balls are so arranged that the inertia forces caused by an angular acceleration or retardation of the shaft tend to alter their positions. What is equilibrium speed? The speed at which the governor balls arms. When the particles of a shaft or disc move in a circle about the axis of the shaft. Governors control the throttle valve and hence the fuel supply to cater the load variation in engines. High speed engines/machines mounted on foundation and supports cause vibrations of excessive amplitude because of the unbalanced forces.. What are the methods of isolating the vibration? 1. 2. torsional shear stresses are induced in the shaft. Explain the function of Governor? The function of a governor is to maintain the speed of an engine within specified limits whenever there is a variation of load.When machine components are suspended from elastic members. the particles of the shaft move in a circle about the axis of the shaft. These are placed between the foundation and vibrating body. it produces vibration in the foundation. The kinetic energy of the equivalent system must be equal to the kinetic energy of the original system. The materials used for vibration isolation are rubber. sleeve. 2. All machine tools. But in torsional vibrations. Such a theoretically replaced shaft is known as torsionally equivalent shaft. So. Due to torsional vibrations. 2. and 2. 76. 1. the vibrational force produced by the machine components will not be transmitted to the foundation. vibration isolation is important. 86. This is called as elastic suspension. in order to prevent these vibrations or to minimize the transmission of forces to the foundation.] 82. 81. It can be minimized providing spring-damper. tensile and compressive stresses are induced. the shaft remains unaffected by the vibration. Specify any two industrial application where the transmissibility effects of vibration are important. Specify the importance of vibration isolation? When an unbalanced machine is installed on the foundation. Define torsional vibration. 87. are in complete equilibrium and there is no upward or downward movement of the sleeve on the spindle is known as equilibrium speed. What are the conditions to be satisfied for an equivalent system to that of geared system in torsional vibrations? 1. etc. At nodes. Define torsional equivalent shaft. What is the principle of inertia governors? In inertia governors. All turbo machines. Explain controlling force? . the particles of the shaft move approximately perpendicular to the axis of the shaft. etc. 77. 80. Differentiate between transverse and torsional vibration. etc. felt cork. 85. In transverse vibrations. Due to transverse vibrations. 84. Define node in torsional vibration. 1. then the vibrations are known as torsional vibrations. 83. 78. The strain energy of the equivalent system must be equal to the strain energy of the original system. A shaft having variable diameter for different lengths can be theoretically replaced by an equivalent shaft of uniform diameter such that they have the same total angle of twist when equal opposing torques are applied at their ends. Node is the point or the section of the shaft at which amplitude of the torsional vibration is zero. The fore end of the ship is called bow and the rear end is known as stern or aft. 99. Give the application of gyroscopic principle. Explain the term stability of the governor? A governor is said to be stable if there is only one radius of rotation for all equilibrium speeds of the balls within the working range. centripetal force) is termed as controlling force of a governor. 90.An equal and opposite force to the centrifugal force acting radially inwards (i.e.What is meant by lag in response? In any control system. 97. Rolling is the movement of a ship in a linear fashion. etc. there is a delay in response due to some inherent cause and it becomes difficult to measure the input and output simultaneously. Steering is the turning of a complete ship in a curve towards left or right. It is the product of the mean value of the effort and the distance through which the sleeve moves. 92. monorail cars. This occurs in over sensitive governors. Define steering. A governor with zero range of speed is known as an isochronous governor. What is meant by hunting? The phenomenon of continuous fluctuation of the engine speed above and below the mean speed is termed as hunting. Explain sensitiveness of governors? The sensitiveness is defined as the ratio of the mean speed to the difference between the maximum and minimum speeds. Define transfer function. 88. Explain gyroscopic couple? If a body having moment of inertia I and rotating about its own axis at ω rad/sec is also caused to turn at ωp rad/sec about an axis perpendicular to axis of spin. pitching and rolling. A system is an assembly of components and linkages designed to fulfill some particular function. Define power of a governor? The power of a governor is the work done at the sleeve for a given percentage change of speed. 95. 94. Explain isochronism. A transfer function is the ratio of output from the block and input to the block. If the equilibrium speed increases the radius of governor ball must also increase. 98. . 93. gyrocompasses. Explain the governor effort? The mean force acting on the sleeve for a given percentage change of speed for lift of the sleeve is known as the governor effort. 100. It is used a) in instrument or toy known as gyroscope. This delay in response is termed as lag in response. Pitching is the movement of a complete ship up and down in a vertical plane about transverse axis. then it experiences a gyroscopic couple of magnitude( ω ωp ) in an axis which is perpendicular to both the axis of spin and axis of precission. while it moves forward. b) in ships in order to minimize the rolling and pitching effects of waves. and c) in aeroplanes. 91. 89. Define the term system. 96. Assuming the resisting moment to be uniform.11. The masses are to be arranged so that the shaft is in complete balance. and b) The resisting torque is (25000 + 3600 sin θ) N-m.7 N/mm2 when it has moved 10 mm from TDC on its power stroke. Determine 1. (Ans. 2. No. Calculate. each mm2 represents 3 N-m.D. Refer Prob. No. Determine the net load on the gudgeon pin. (Ans. Page No. The scale for the turning moment is 1 mm = 500 N-m. No. Velocity and acceleration of the piston and 2. 5. Three masses are attached to a shaft as follows: 10 kg at 90 mm radius. The gas pressure on the piston is 0. No. JAYAKUMAR) 4. Determine the angular position of masses relative to 10 kg mass. The bore/stroke is 80/100 mm and mass of the reciprocating part is 1 kg. JAYAKUMAR) 3. 3. the areas of which from the line of zero pressure are as follows: Expansion stroke = 3550 mm2 .. -250. The crank rotates at 1500 rpm clockwise direction.e. 3. No.3. Refer Prob. Refer Prob. where θ radians is the crank angle from inner dead centre. Page No. Page No. 1.C. A steam engine runs at 150 rpm..36 Dynamics of Machines by V.2. 270.15 Dynamics of Machines by V. 270.75 m.5 % of the mean. (Ans. and compression stroke = 1400 mm2. No. Also determine the crank positions for the maximum and minimum speeds. 2. Exhaust stroke = 500 mm2 . -250. JAYAKUMAR) 6. 3. Page No. Refer Prob. 190. 2. The turning moment diagram for a four stroke gas engine may be assumed for simplicity to be represented by four triangles. Page No. Page No. -340. (Ans. when the piston has traveled one-fourth of its stroke from I. find the mass of the rim of a fly wheel required to keep the mean speed 200 rpm within ±2%. Its turning moment diagram gave the following area measurements in mm2 taken in order above and below the mean torque line: 500.36 Dynamics of Machines by V. Suction stroke = 350 mm2 .5.19. 15 kg at 120 mm radius and 9 kg at 150 mm radius. The hoop stress is limited to 3 Mpa and the density of the material of the flywheel is 7500 kg/m3. width of the rim) equal to 1. determine a suitable diameter and cross-section of the rim of the flywheel assumed with axial dimension (i.5 times the radial dimension (i. 3. Refer Prob.14. JAYAKUMAR) . (Ans. The ratio of the connecting rod length to crank length for a vertical petrol engine is 4:1.3.e. A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length. -390.16. The mean radius of the rim may be taken as 0. JAYAKUMAR) 2. and 2. The total fluctuation of speed of the flywheel when a) The resisting torque is constant. (Ans.SIXTEEN MARKS QUESTION AND ANSWERS 1. The engine runs at 1800 rpm At what engine speed will this load be zero. If the fluctuation of speed is not to exceed ± 1.43 Dynamics of Machines by V.3. 2.25 Dynamics of Machines by V.9 Dynamics of Machines by V. Refer Prob. All the masses are in the same plane. The power of the engine. Angular velocity and angular acceleration of the connecting rod. JAYAKUMAR) 5. The equation of the turning moment diagram for the three crank engine is given by: T(Nm) = 25000 – 7500 sin 3θ. and for crank angle is 1 mm = 5 o. The moment of inertia of the flywheel is 400 kg-m2 and the mean engine speed is 300 rpm. thickness of the rim). 6. indicating the position of No. 2. RATTAN) 12. No. Page No.2 m and the stroke of each piston 500 mm. The damping provided is only 20 % of the critical value. vertical.8. The distance of the centers from the axis of rotation are 12 mm.9. The piston stroke is 80 mm and length of each connecting rod is 180 mm.6. (Ans. Refer Prob. JAYAKUMAR) 8. The magnitude and the position of the balancing masses required at a radial distance of 500 mm. The hammer blow per wheel when the axle rotates at 350 rpm.5. find the amount of the masses and their angular positions. JAYAKUMAR) 9. 5.6. Refer Prob. If the shaft is balanced by adding two masses at a radius 75 mm and at distance of 100 mm from the central plane of the middle eccentric. Page No.31 Dynamics of Machines by V. a spring of stiffness 30 kN/m and a damper. The pitch of the cylinder is 1. (Ans.14. 80 mm and 80 mm respectively. Page No. The pitch distances between the cylinder centre lines are 80 mm. Page No.17 Dynamics of Machines by V.1 crank for maximum values. 6. The reciprocating mass per cylinder is 1. in-line engine is 1-4-2-6-3-5. 120 mm. The planes of rotation of the balance masses are 960 mm from the inside cylinder. and 2. The cranks are of 150 mm radius.5. The density of metal is 700 kg/m3. No. machined in one piece with the shaft. JAYAKUMAR) 11. determine 1. Refer Prob. (Ans. Find the amount of out-ofbalance force and couple at 600 rpm. each 75 mm diameter and 25 mm thick. Determine .2 kg and the engine speed is 2400 rpm. the connecting rods 500 mm long and the cylinder centre line 400 mm apart. The central planes of the eccentric are 60 mm apart. An air compressor has four vertical cylinders 1.S. 18. Refer Prob. No. The cranks of a three-cylinder locomotive are set at 120o. The firing order of a six cylinder. Show that there are no out-of-balance primary or secondary forces and determine the corresponding couples. The mass of the reciprocating parts for each cylinder is 22. Determine the equivalent spring stiffness and the natural frequency of the following vibrating systems when a) the mass is suspended to a spring b) the mass is suspended at the bottom of two springs in series c) the mass is fixed in between two springs d) the mass is fixed to the mid point of a spring (Ans. (Ans. No. 6. 598 Theory of Machines by S. The reciprocating masses are 450 kg for the inside cylinder and 390 kg for each outside cylinder. No.6. A vibrating system consists of a mass of 50 kg.5 kg and the speed of rotation is 400 rpm.1. 3 and 4 in line and the driving cranks at 90o intervals reach their upper most positions in this order. The central plane of the machine may be taken as reference plane. Page No. JAYAKUMAR) 10. 80 mm. four-stroke.18 mm and 12 mm and their angular positions are 120o apart. Refer Prob. If 40% of the reciprocating masses are to be balanced.7. A shaft has tree eccentrics. Determine the out-of-balance primary and secondary forces and couples on the engine taking a plane mid-way between the cylinders 3 and 4 as the reference plane.21 Dynamics of Machines by V.41 Dynamics of Machines by V. 619 Theory of Machines by S. each of stiffness 12 N/mm. 18. RATTAN) 15. ii) the ratio of frequencies of damped and undamped vibrations.4 mm. the damping factor the critical damping coefficient the natural frequency of damped vibrations the logarithmic decrement the ratio of two consecutive amplitudes. the amplitude of the vibrations if the driving shaft rotates at 500 rpm. A single cylinder vertical diesel engine has a mass of 400 kg and is mounted on a steel chassis frame.5 m long has a mass of 15 kg per meter length. Refer Prob. 18. Page No. Page No. (Ans. determine 1. It is simply supported at the ends and carries three masses 90 kg. 18. A shaft 40 mm diameter and 2.S. The machine mounted on springs and fitted with a dashpot has a mass of 60 kg.17.12.8. the damping reduces the amplitudes of successive vibrations by 30 %. find the frequency of the transverse vibrations. Refer Prob. RATTAN) 17. the amplitude of the vibrations at resonance.2 kg which move through a vertical stroke of 100 mm with simple harmonic motion.S. No. There are three springs. No. No.S. Page No. Refer Prob. (Ans. determine the combined stiffness of the springs so that the force transmitted to the foundation is 1/20 th of the impressed force. determine i) the damping coefficient. The amplitude of vibrations reduces from 45 to 8 mm in two complete oscillations. 5. 2. Page No. 2. the force transmitted to the foundation at resonance.S. Neglecting damping. 2. A shaft supported freely at the ends has a mass of 120 kg placed 250 mm from one end. Page No. A machine supported symmetrically on four springs has a mass of 80 kg. Refer Prob. Refer Prob. 1.15. No. The mass of the reciprocating parts is 2. RATTAN) 16. RATTAN) 13. and 3. E = 200 GN/m2 and shaft diameter is 40 mm. A dashpot with a damping coefficient 2 N/mm/s is also used to dampen the vibrations.5 m and 2 m respectively from the left support. 626 Theory of Machines by S. The static deflection owing to the weight of the chassis is 2. (Ans.S. In the steady-state of the vibrations. No. 624 Theory of Machines by S. 18. (Ans. 1.1. Taking E = 200 GN/m2. 3. 18. Determine the frequency of the natural transverse vibrations if the length of the shaft is 700 mm. 611 Theory of Machines by S. Assuming that the damping force varies as the velocity. 140 kg and 60 kg at 0. (Ans. the speed of the driving shaft when the resonance occurs. The machine crank shaft rotates at 800 rpm If under actual working conditions. find.5. The reciprocating masses of the engine amounts to 18 kg and the stroke of the engine is 160 mm.8 m. the force transmitted to the foundation at 800 rpm.S. 609 Theory of Machines by S. RATTAN) 14. No. and iii) the periodic time of damped vibrations. (Ans. 633 Theory of Machines by S. 18. RATTAN) . Refer Prob. Page No. 4.16. 2 m = 14. RATTAN) 20. Refer Prob. = 0.27. 660 Theory of Machines by S. Each ball has a mass of 5 kg and the sleeve mass is 50 kg. 18. RATTAN) . Determine the equilibrium speeds corresponding to extreme radii of 180 mm and 240 mm/ (Ans. The force of friction on the sleeve of the mechanism is 40 N. RATTAN) 19. The upper and lower arms are pivoted to links of 40 mm and 50 mm respectively from the axis of rotation. RATTAN) 22. Refer Prob. 543 Theory of Machines by S. No. = 1. Refer Prob. The extensions of the lower arms to which the balls are attached are 100 mm long and are parallel to the governor axis at the minimum radius. gearwheel. G = 80 G N/m2. Refer Prob. Page No. Page No. Pump speed is four times the engine speed. 656 Theory of Machines by S. 18. = 16 kg. Find the natural frequency of the torsional oscillation of the system.S. 16. A reciprocating IC engine is coupled to a centrifugal pump through a pair of gears. RATTAN) 21. 18. 540 Theory of Machines by S.4.S. No. The shaft from the pinion to the pump has a 32 mm diameter and is 280 mm long.20. The shaft from the flywheel of the engine to the gear wheel has a 48 mm diameter and is 800 mm long. pinion and pump impeller are 1000 kg m2. Each of the arms is 300 mm long. The mass of each ball of a Proell governor is 7. No. (Ans. 5 kg m2 and 18 kg m2 respectively. Page No. determine the frequency of free torsional vibrations. No.3. (Ans.28. No. Length of shaft Diameter of shaft Mass of a rotor at mid point Eccentricity of centre of mass of rotor from centre of rotor Modulus of elasticity of shaft material Permissible stress in shaft material Determine the critical speed of the shaft and the range of unsafe to run the shaft. The following data refer to the transmission gear of a motor ship: Moment of inertia of flywheel = 4800 kg m2 Moment of inertia of propeller = 3200 kg m2 Modulus of rigidity of shaft material = 80 X 109 N/m2 Equivalent MOI per cylinder = 400 kg m2 Assuming the diameter of the torsionally equivalent crankshaft to be 320 mm and treating the arrangement as a three rotor system. Refer Prob. 16. (Ans. Page No. The following data relate to a shaft held in long bearings. Moments of inertia of flywheel.5 kg and the load on the sleeve is 80 kg.18.S. Page No. Assume the shaft to be mass less. 636 Theory of Machines by S.S.S. The upper arms are pivoted on the axis of rotation whereas the lower arms are pivoted to links of 40 mm from the axis of rotation. Determine the range of speed of the governor for extreme radii of rotation of 125 mm and 150 mm. 14 kg m2.4 mm = 200 GN/m2 = 70 X 106 N/m2 speed over which it is (Ans. Each arm of a Porter governor is 250 mm long. Page No. No. 548 Theory of Machines by S.10. taking into account the obliquity of arms. No. determine the total alteration in speed before the sleeve begins to move from the mid. RATTAN) 24.S. Refer Prob. The extreme radii of rotation are 80 mm and 140 mm. 16. Plot a graph of the controlling force vs. 16. If the values of the controlling force at the above radii be respectively 200 N and 360 N and the friction of the mechanism is equivalent to 2 N at each ball.position (Ans.32. Page No. Assuming the friction at the sleeve to be equivalent to a force of 15 N. The mass of each ball is 5 kg and the load on the sleeve is 30 kg. Determine the initial compression of the spring. the mass of each ball is 4 kg and the lift of the sleeve is 40 mm. radius of rotation and set off a speed scale along the ordinate corresponding to a radius of 160 mm.23. No. The lengths of the ball and roller arms of the bell-crank lever are 100 mm and 80 mm respectively. In a Porter governor. The governor begins to float at 200 rpm when the radius of the ball path is 90 mm. each arm is 200 mm long and is pivoted at the axis of rotation. JAYAKUMAR) 25. 10. The weight of each governor ball is 40 N and the extreme radii of rotation are 120 mm and 180 mm.68 Dynamics of Machines by V. 563 Theory of Machines by S. Page No. (Ans. The mean working speed of the governor is 16 times the range of speed when friction is neglected. (Ans.6. Refer Prob. Refer Prob. In a spring loaded Hartnell type of governor.10. Find a) the extreme equilibrium speeds of the governor.S. The pivot centre and the axis of governor are 115 mm apart. b) the equilibrium speed and the coefficient of insensitiveness at a radius of 150 mm. RATTAN) . The controlling force curve of a spring controlled governor is a straight line.