Anna University, Chennai

**SRINIVASAN ENGINEERING COLLEGE, PERAMBALUR DEPARTMENT OF MECHANICAL ENGINEERING ME2302-DYNAMICS OF MACHINERY**

**16 MARKS**

**U****NIT-I FORCE ANALYSIS AND FLYWHEELS**

**1.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. (16)**

**2. A horizontal gas engine running at 210rpm has a bore of 220mm and a stroke of 440mm.The connecting rod is 924mm long the reciprocating parts weight 20kg.When the crank has turned through an angle of 30° from IDC, the gas pressure on the cover and the crank sides are 500KN/m****2 ****and 60KN/m****2 ****respectively. Diameter of the piston rod is 40mm.Determine,1. Turning moment on the crank shaft 2.Thrust on bearing 3. Acceleration of the flywheel which has a mass of 8kg and radius of gyration of 600mm while the power of the engine is 22KW. (16) [MU, April 2000]**

**3. A single cylinder vertical engine has a bore of 300mm and a stroke of 400mm.The connecting**

**rod is 1000mm long. The mass of the reciprocating parts is 140kg.On the expansion stroke with the crank at 30°from the top dead center, the gas pressure is 0.7MPa.If it runs at 250rpm, 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. (16)**

**4. A vertical double acting steam engine has a cylinder 300mm diameter and 450mm stroke and runs at 200rpm.The reciprocating parts has a mass of 225kg and the piston rod is 50mm diameter. The connecting rod is 1.2m long. When the crank has turned 125° from IDC the steam pressure above the piston is 30KN/m2.calculate,**

**(i)Crank-pin effort[MS U. Nov’97]**

**(ii)The effective turning moment on the crank shaft. (16) [MS U. Nov’97]**

**5. The turning moment diagram for a petrol engine is drawn to a scale of 1mm to 6N-9-9m and the horizontal scale of 1mm 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, 50,350,980and 275mm2.**

**The mass of rotating parts is 40kg at a radius of gyration of 140mm.Calculate the coefficient of**

**fluctuation of speed if the mean speed is 1500rpm.(16)**

**6. The torque delivered by a two stroke engine is represented by T= (1000+300sin2_-500cos2_) N- m where _ is the angle turned by the crank from the IDC. The engine speed is 250rpm.The mass of**

**the flywheel is 400kg and radius of gyration 400mm.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. (16)[Au May-11]**

**7. A single cylinder horizontal steam engine has a stroke of 0.7 m and a connecting rod 1.8m long. The mass of reciprocating parts is 520kg and that of the connecting rod is 230kg. Centre of gravity of the connecting rod is 0.8m from crank pin and the moment of inertia about an axis through the centre of gravity perpendicular to the plane of motion is 100kg-m****2****. For an engine speed of 90rpm and a crank position of 45° from the IDC, determine the torque on the crankshaft due to the**

**inertia of these parts.[Au Dec-11]**

**UNIT-II BALANCING**

**8. A shaft is rotating at a uniform angular speed. Four masses M1, M2, and M3and M4 of magnitudes 300kg, 450kg, 360kg, 390kg respectively are attached rigidly to the shaft. The masses are rotating in the same plane. The corresponding radii of rotation are 200mm, 150mm, 250mm and 300mm 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 200mm. (16)**

**9. Four masses M1, M2, M3, and M4 are 200kg, 300kg, 240kg and 260kg respectively. The corresponding radii of rotation are 0.2m, 0.15m, 0.25m and 0.3m respectively and the angle between successive masses45°, 75°, and135°.Find the position and magnitude of balance mass required if its radius of rotation is 0.25m. (16) [MS U. Nov’97]**

**10. The data for three rotating masses are given below:- M1=4kg r1=75mm θ1=45**

**M2=3kg r2=85mm θ2=135**

**M3=2.5kg r3=50mm θ3=240**

**Determine the amount of counter mass at a radial distance of 65mm required for their static balance (16) [MKU, Oct’97, Au Dec-11]**

**11. 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=25kg rA=150mm mB=40kg rB=200mm mC=35kg rC=100mm rD=180mm Planes B and C are**

**250mm apart. Determine (i) the mass A and its angular position (ii) the position of planes A and D. (16)**

**12. A, B, C and D are four masses carried by a rotating shaft at radii 100mm,125mm,200mm and**

**150mm respectively. The planes in which the masses revolve are spaced 600mm apart and the**

**masses of B,C and D are 10kg,5kg and 4kgrespectively.Find the required mass A and relative angular setting of the four masses so that the shaft be in complete balance. (16)[ AU, DEC-12]**

**13. Four masses A, B, C and D revolves at equal radii and equally spaced along a shaft. The mass B is 7kg and the radii of C and D make angle s 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. (16) [MU, April 2000]**

**14. A shaft caries four rotating masses A, B, C and D which are completely balanced. The masses B, C and Dare 50kg, 80kg and 70kg 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**

**75mm,100mm,50mm and 90mmrespectively.The plane of rotation of masses B and C are 250mm apart. Determine (i) the magnitude of mass A and its angular position(ii) the position of planes A and D. (16)**

**15. A four cylinder vertical engine has cranks 150mm long. The plane of rotation of the first,**

**second and fourth cranks are 400mm,200mm and 200mm respectively from that of the third crank and their reciprocating masses are 50kg,60kg and 50kg 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. (16) [MS U. Nov’97]**

**16. A four cylinder vertical engine has cranks 300mm long. The plane of rotation of the first, third and fourth cranks are 750mm,1050mm and 1650mm respectively from that of the second crank and their reciprocating masses are 10kg,400kg and 250kg 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. (16)**

**17. 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 (16) [MU, April 2000]**

**UN****IT-III FREE VIBRATION**

**18. Derive an expression for the natural frequency of the free longitudinal vibration by**

**(i)Equilibrium method (ii) Energy method (iii)Rayleigh’s method (16)**

**19. In a single degree of damped vibration system a suspended mass of 8kg 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. (16)[Au Dec-11],[ AU, DEC-12].**

**20. 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. (16)**

**21. (i)Deduce the expression for the free longitudinal vibration in terms of spring stiffness, its inertia effect and suspended mass. (8) [MU, April 2000]**

**(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 12Hz.An extra 2kg mass is coupled to ‘m’ and natural frequency reduces by 2Hz.Find the value of ‘s’ and ‘m’. (8)**

**22.Avibrating system consists of a mass of 8kg,spring of stiffness 5.6N/m and dashpot of damping coefficient of 40N/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. [MU, April 2000]**

**23. An instrument vibrates with a frequency of 1Hz when there is no damping. When the damping is provided, the frequency of damped vibration was observed to be 0.9Hz.**

**Find, (i) damping factor (ii) logarithmic decrement. (16)**

**24. 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[MS U. Nov’97]**

**25. Between a solid mass of 0kg and the floor are kept two slabs of isolates, natural rubber and felt, in series. The natural rubber slab has a stiffness of 3000N/m and equivalent viscous damping coefficient of 100 N-sec/m.The felt has a stiffness of 12000N/m and equivalent viscous damping coefficient of 330Nsec/m.Determine undamped and the damped natural frequencies of the system in vertical direction. (16) [Bharathiyar Univ. Nov’97]**

**26. (i) A cantilever shaft 50mm diameter and 300mm long has a disc of mass 100kg at its free end. The young’s modulus for the shaft material is 200GN/m2.SDetermine the frequency of longitudinal and transverse vibration of the shaft. (10)**

**(ii)Explain the sketches different cases of damped vibrations. (6)**

**27. 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 400kg and initial velocity of recoils 1m.Determine spring stuffiness and critical damping coefficient of dashpot. (16)**

**28. A steel shaft 100mm in diameter is loaded and support in shaft bearing 0.4m apart. The shaft carries three loads: first mass 12kg at the centre, second mass 10kg at a distance 0.12m from the left bearing and third mass of 7kg at a distance 0.09m from the right bearing. Find the value of the critical speed by using Dunker ley’s method. E=2X10****11****N/****m****2 ****(16)**

**29. write short notes on a. vibration isolation and transmissibility b. Whirling of rotating shafts.[Au**

**May-11]**

**UNIT-IVFORCED VIBRATION**

**30.A mass of 50kg is supported by an elastic structure of total stiffness 20KN/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 newtons. Find amplitude of the vibration and phase angle caused by the damping. (16) [MS U. Nov’97]**

**31. A mass of 50kg is supported by an elastic structure of total stiffness 20KN/m.The damping ratio ofthe system is 0.25.A simple harmonic disturbing force acts on the mass and at any time ‘t seconds, the force is 75cos12t newtons. Find amplitude of the vibration and phase angle caused**

**by the damping. (16) [Bharathiyar Univ. Nov’97]**

**32. A mass of 10kg is suspended from one end of a helical spring, the other end being fixed. The stiffness of the spring is10N/mm.The viscous damping causes the amplitude to decreases to one- tenth 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? What is its value of resonance? (16)**

**33. A harmonic exiting force of 25N is acting on a machine part which is having a mass of 2Kg and vibrating in viscous medium. The exciting force causes resonant amplitude of 12.5mm with a period of 0.2sec. (16) [MS U. Nov’97]**

**34. A body having a mass of 15kg is suspended from a spring which deflects 12mm 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 100N and vibrating at 6Hz is made to act on the body, determine the amplitude of the ultimate motion. (16) [MU, April 2000]**

**35. A single cylinder vertical petrol engine of total mass of 200kg is mounted upon a steel chassis frame. The vertical static deflection of the frame is 2.4mm due to the weight of the engine .The mass of the reciprocating parts is 18kg and stroke of piston 160mm with S.H.M. If dashpot of damping coefficient of 1N/mm/s used to damped the vibrations, calculate al steady state (i)Amplitude of vibrations at 500rpm engine speed.(ii)The speed of the driving shaft at which resonance will occurs. (16) [MKU, Oct’97]**

**36. A vertical single stage air compressor having a mass of 500kg is mounted on spring having stiffness of 1.96X10****5****N/****m and dashpot with damping factor of 0.2m.The rotating parts are completely balanced and the equivalent reciprocating parts weight 20kg.The stroke is**

**0.2m.Determine the dynamic amplitude of vertical motion of the excitation force if the compressor is operate at 200rpm. (16)**

**37. A machine 100kg has a 20kg rotor with 0.5mm eccentricity. The mounting spring have s=85x103. The operating speed is 600rpm and the unit is constrained to move vertically. Find (i) Dynamic amplitude of machine (ii) the force transmitted to the support. (16)**

**38.A single cylinder engine has an out of balance force of 500N at an engine speed of 30rpm.The total mass of engine is 150kg and its carried on a set of total stiffness 300N/cm. (i) Find the amplitude of steady motion of the mass and maximum oscillating force transmitted to the foundation. [MKU, Oct’97]**

**(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. ` (16)**

**39. 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. (16)**

**40. The mass of an electric motor is 120kg and it runs at 1500rpm.The armature mass is 35kg and its centre gravity lies 0.5mm from 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) the stiffness of the spring (ii) the dynamic force transmitted to the base at the operating speed. (iii) Natural frequency of system. (16) [MU, April 2000]**

**41. Find the stiffness of each spring when a refrigerator unit having a mass of 30kg is to be support by three springs. The force transmitted to the supporting structure is only 10% of the impressed force. The refrigerator unit operates at 420rpm. (16) [MS U. Nov’97]**

**42. A 100kg machine is mounted on a spring stiffness 1****× ****10****6****N/****m and damping factor 0.3. A 3kg piston within the machine has a reciprocating motion with a stroke of 100 mm and a speed of**

**1500 rpm, thus producing a harmonic force. Determine the amplitude of vibration of the machine and the force transmitted to the foundation.[Au dec-11]**

**43. Describe the principle of undamped dynamic vibration absorber with suitable illustrations.[Au**

**May-11]**

**UNIT-V MECHANISMS FOR CONTROL**

**44. 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. (16)**

**45. The length of the upper and lower arms of a porter governor are 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. (16) [MKU, Oct’97]**

**46. 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. (16) [MU, April 2000]**

**47. 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 respectively. The levers are pivoted at 120mm from the governor axis 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. 16) [MS U. Nov’97]**

**48. 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 % in the engine speed. (16) [AU. May 2004, Au May-11]**

**49. 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. (16)**

**50. 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. (16)**

**51. 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. (16)**

**52. (i) Explain the effect of Gyroscopic couple on a Naval ship during pitching.(8) (ii) Explain the effect of Gyroscopic couple on a aero plane. (8) [MKU, Oct’97]**

**53.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) [MU, April 2000]**

**54. The rotor of a turbine yatch 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? (16)**

**55. 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. (16) [Anna Univ. May 2004]**

**56. The controlling force F****c ****and radius of a spring controlled governor is given by the expression**

**F****c****=2000r-76. The mass of the ball is 5kg and extreme radii of rotation, of the ball are 0.1m and**

**0.175m respectively for maximum and minimum speeds. If the friction on the governor is 5N at each ball. Find the coefficient of insensitiveness of the governor at extreme radii. [Au May-11]**

**57. The hartnell governor having ball mass 1.5kg each has ball arms 55mm long and sleeve arms**

**45mm long. The total movement of sleeve is 24mm. In mid position of sleeve, sleeve arms are horizontal and perpendicular to the governor axis and ball rotates at 80mm radius. Equilibrium speed at the top stop of the sleeve is 425rpm and the bottom stop is 440rpm. Find (a) Stiffness of spring and spring compressive force at the bottom stop (b) If top stop equilibrium speed is required to be 12rpm greater than the bottom stop speed, what will be spring force at bottom stop and (c) What bottom stop spring force will make the governor isochronous? [Au Dec-11]**

**58. A ship is propelled by a turbine rotor having a mass of 6 tonnes and speed of 2400 rpm. The direction of rotation of the rotor is clockwise when viewed from the stern. The radius of gyration of the rotor is 450 mm. Determine the gyroscopic effect when (a) The ship steers to the right in a curve of 60 m radius at a speed of 18 knots(1 knot=1860 m/hr) (b) The ship pitches 7.5 degrees below the normal position and the bow is descending with its maximum velocity. The pitching motion is SHM with a periodic time of 18 seconds. (c) The ship rolls and at the instant its angular velocity is 0.035 rad/s counter clockwise when viewed from the stern. Also find the maximum angular acceleration during pitching. Explain how the direction of motion due to gyroscopic effect is determined in each case.[Au Dec-11, AU DEC-12]**

**59. The radius of rotation of the balls of a Hartnell governor is 80 mm at the minimum speed of**

**300 rpm. Neglecting gravity effect, determine the speed after the sleeve has lifted by 60 mm. Also determine the initial compression of the spring, the governor effort and the power. The particulars of the governor are given below: Length of the ball arm=150 mm, length of sleeve arm=100 mm, mass of each ball=4kg, and stiffness of the spring=25 N/mm.**