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1 MLR Institute oftechnology Dundigal, Hyderabad MECHANICAL ENGINEERING Assignment Questions DYNAMICS OF MACHINERY Course Title Course Code Regulation R13 Course Structure Lectures Tutorials Practicals Credits Course Coordinator M.Venkateswar Reddy, Associate Professor, G.Ananda Rao, Associate Professor S.Navya Sree, Assistant Professor PART-A (2 Mark Questions) Unit-I 1. Derive a formula for gyroscopic couple 2. Write a short note on gyroscope 3. Describe the gyroscopic effect on sea going vessels 4. Describe the gyroscopic effect on an aeroplane 5. Briefly explain the stability of a two wheel vehicle taking a turn 6. List the type of friction 7. State the laws of static friction 8. Sketch the friction circle in a Journal bearing 9. Write the torque equation for n flat collars under uniform pressure 10. Write the equation for the efficiency of a screw jack Unit-II 1. Write the torque equation for a single plate clutches under uniform pressure having outer radius r1 and inner radius r2 2. Classify the friction clutches 3. Sketch a single plate friction clutch 4. a) The frictional torque transmitted by a single plate clutch is same as that bearing b) The frictional torque transmitted by a cone clutch is same as that bearing 5. Sketch a cone clutch 6. Distinguish between brakes and dynamometers. 7. a) The brakes commonly used in railway trains are brakes. b) The brakes commonly used in Motor cars are brakes. 8. List the types of Dynamometers. 9. Sketch an internal expanding shoe brake. 10. Classify the Transmission Dynamometers. Unit-III 1. What is a turning moment diagram? 2. Define the term Co-efficient of Fluctuation of energy. 3. What is the function of a flywheel?

2 4. a) The ratio of max.fluctuation of speed to the is called Co-efficient of Fluctuation of energy. b) Write the Equation for max. Fluctuation of energy in a flywheel. 5. Draw the Turning moment diagram for a single cylinder double acting steam engine. 6. What is the function of a Governor? 7. State the different types of governors. 8. Define the following terms relating to governors. a) Sensitiveness b) Isochronism 9. a) Hartnell governor is a governor. b) Height of a Watt governor(in mts) Is given by 10. Sketch a Watt Governor. Unit-IV 1. Why is balancing of speed necessary for high speed engines? 2. Derive the expression for variation of tractive force. 3. Derive the expression for hammer blow. 4. What is static balancing? 5. What is the condition for complete balance of several rotating masses in different planes? 6. What are direct and reverse cranks in radial engines? 7. Differentiate between primary and secondary cranks. 8. What are the conditions to be satisfied for primary balancing. 9. Write the formulae to calculate the resultant primary and secondary forces in a v-engine. 10. In partial balancing of locomotives, the maximum variation of tractive effort is given by. Unit-V 1. Define Vibrations? How are they caused? 2. What are free and damped vibrations? 3. What are forced vibrations? 4. What do you understand by Whirling of a shaft? 5. What is the basic difference between Dunkerley s and Raleigh;s method in vibrations? 6. What is torsional Vibrations? 7. Define degrees of freedom with an example. 8. Define critical speed of a rotating shaft. 9. What do you mean by period of vibration and resonance as applied to vibratory motions? 10. Write the expression for natural frequency of transverse vibration of a simply supported beam of span L carrying an uniformly distributed load m kg per meter length. PART-B (5 Mark Questions) Unit-I 1. A boat is propelled by a steam turbine. The moment of inertia of the rotor shaft and propeller is 60 kg-m 2. The turbine runs at 3000rpm in clockwise direction looking from the front.the boat describes a circular path towards the right making one revolution in 10 seconds. Find the magnitude and direction of the couple acting on boat hull. 2. An Aero plane makes a complete half circle radius towards left when flying at 210 km/hr. The rotary engine and the propeller of the plane is of 50 kg mass having a radius of gyration of 300mm.The engine rotates at 2400 rpm clockwise as seen from the rear. Find the Gyroscopic couple on the aircraft and its effect on the plane. 3. The moment of inertia of each wheel of a motorcycle is 1.5kg-m 2.The rotating parts of the engine of the motorcycle have moment of inertia of 0.28kg-m 2.The speed of the engine is six times the speed of the wheels and is in same direction. The mass of the motorcycle is 250 kg and its centre of gravity is 0.6m above the ground level. Find the angle of Wheel if the motor cycle is travelling at 45km/hr

3 and is taking a turn of 30m radius. The Wheel diameter is 0.6m. 4. A Racing car of mass 2000kg has a wheel base of 2m and track width of 1m.The C.G. lies midway between the front and rear axles and is 0.4m above the ground. The engine of the car has a flywheel rotating in a clockwise direction when seen from the front at 6000 rpm. The moment of inertia of the flywheel is 50 kg-m 2.If the car takes a curve of 15m radius towards right, while running at 45 km/hr, find the reaction between the wheels and the ground considering the gyroscopic and centrifugal effect of the fly wheel and the weight of the car respectively. 5. Explain the application of gyroscopic principles to aircrafts. 6. Derive an expression for the frictional moment of a flat collar bearing in terms of the inner radius r 1 outer radius r 2, axial thrust and coefficient of friction μ. Assume uniform intensity of pressure. 7. Determine the horse power lost in overcoming friction at the bearing and the number of collars required for a bearing whose contact surfaces are 200mm external radius and 20mm internal radius. The coefficient of friction is The total axial load is 3000 Kg. Intensity of pressure is 305 Kg/cm 2.Shaft speed is 420 rpm. Assume uniform pressure intensity at contact surfaces. 8. A bolt with a square threaded screw has a mean diameter of 25mm and a pitch of 3mm.It carries an axial thrust of 10kn on the bolt head of 25mm mean radius.if μ =0.12, find the force required at the end of a spanner 450mm long,in tightening up the bolt. 9. A thrust shaft of a ship has 6 collars of 600mm external diameter and 300mm internal diameter. The total thrust from propeller is 100kn. If the coefficient of friction is 0.12 and speed of engine 90 rpm., find the power absorbed in friction at the thrust block assuming i. Uniform pressure and ii. Uniform wear 10. Neglecting collar friction, derive an expression for mechanical advantage of a square threaded screw moving in a nut, in terms of helix angle of the screw and friction angle. 11 `An aircraft makes a half circle of 100m radius towards left when flying at 400 km/hr. The engine and the propeller of the plane weigh 4.9 kn having a radius of gyration of 50 cm. The engine rotates at 3000 rpm clockwise when viewed from the rear. Find the gyroscopic couple and its effect on the aircraft. 12 A rear engine automobile is travelling along a track of 100 m mean radius. Each of the four road wheels has a moment of inertia of 2 kg-m 2 and an effective diameter of 60 cm. The rotating parts of the engine have a moment of inertial of 1 kg-m 2. The engine axis is parallel to the rear axle and the crank shaft rotates in the same sense as the road wheels. The gear ratio, engine to back axle, is 3: 1. The vehicle weighs kn and has its centre of gravity 50 cm above the road level. The width of the track of the vehicle is 1.5 m. Determine the limiting speed of the vehicle around the curve for all the four wheels to maintain contact with the road surface if this is not cambered. 13 The turbine rotor weighing 9.8 kn rotates at 2000 rpm clockwise when looking from stern. The vessel pitches with an angular velocity of 0.5 rad/s. Calculate the gyroscopic couple during the rise of bow. Assume radius of gyration of the rotor as 25.4 cm 14 A turbine rotor of a ship weighs 196 kn and has a radius of gyration of 75 cm. Its speed is 2000 rpm.the ship pitches 6above and below the mean position. A complete oscillation takes place in 20 seconds and the motion is simple harmonic. Determine: a) the maximum couple tending to shear the holding down bolts of the turbine. b) the direction in which the bow will tend to turn while rising, if the rotation of the rotor is clockwise, when looking from the aft.

4 15 Discuss a naval ship usually subjected to what type of motions and associated gyroscopic effects. 16 A turn buckle, with right and left hand single start threads, is used to couple two wagons. Its thread pitch is 12 mm and the mean diameter 40 mm. The coefficient of friction between the nut and the screw is i. Determine the work done in drawing the wagons together a distance of 240 mm, against a steady load of 2500 N. ii. If the load increases from 2500 N to 6000 N over a distance of 240 mm. What is the work to be done? 17 A screw jack has a screw thread, 7.5 cm mean diameter and 1.5 cm pitch. The load on the jack revolves with the screw. The coefficient of friction at the screw threads is i. Find the tangential force to be applied to the jack at 36 cm radius so as to lift a load of 600 N. ii. State whether the jack is self-locking. If it is, find the torque necessary to lower the load. If not, find the torque which must be applied to keep the load from descending. 18 Derive the expression for the friction torque in flat collar bearing considering uniform wear. 19 TheThrust on the propeller shaft of a marine engine is taken up by 8 collars whose external and internal diameters are 660 mm and 420 mm respectively. The thrust pressure is 0.4 MN/mm2and may be assumed uniform. The coefficient of friction between the shaft and collars is If the shaft rotates at 90 r.p.m.: find I toti i Thrust on the collars: and ii.power absorbed by friction on the bearing. 20 The spindle of a screw jack has single start square threads with an outside diameter of 45 mm and pitch of 10 mm. The spindle moves in a fixed nut. The load is carried on a swivel head but is not free to rotate. The bearing surface of the swivel head has a mean diameter of 60 mm. The coefficient of friction between the nut and the screw is 0.12 and that between the swivel head and the spindle is 0.10.Calculate the load which can be raised by efforts of 100 N each applied at the end of two levers each of effective length of 350mm. also determine the velocity ratio and the efficiency of the lifting arrangement. Unit-II 1. A leather faced conical clutch has a cone angle of 30.If the intensity of pressure between the contact surfaces is limited to 0.35N/mm 2 and the breadth of the conical surface is not to exceed onthird of the mean radius, find the dimensions of the contact surface to transmit 22.5 kw at 2000rpm.Assume uniform wear and take μ= A dry single plate is to be designed for an automobile whose engine has a maximum torque 500N-m. The outer radius of the friction plate is 25% more than the inner radius. The intensity of pressure between the plates is not to exceed 0.07 N/mm 2. Take μ= Describe with a neat sketch the working of a single plate friction clutch. 4. Determine the maximum and minimum and average pressure in a plate clutch, when the axial Force is 4kN, the inside radius of the contact surface is 50mm and the outer radius is 100mm. Assume uniform wear. 5. A Multidisc clutch has three discs on the driving shaft and two on the driven shaft. The outside diameter of the contact surfaces is 240mm and inside diameter 120 mm. Assuming uniform wear and coefficient of friction as 0.3, find the maximum axial intensity of pressure between the discs for

5 transmitting 25kW at 1575 rpm. 6. Describe with the help of a neat sketch the principles of operation of an internal expanding shoe brake. 7. The simple band brake, as show in fig.1 is applied to a shaft carrying a flywheel of mass 400 kg. The radius of gyration of the flywheel is 450mm and runs at 300rpm. If the co-efficient of friction is 0.2 and the brake drum diameter is 240, find the torque applied due to a hand load of 100N 8. With a neat sketch explain the working of a centrifugal clutch. 9. Explain the principle of working of an absorption dynamometer with a neat sketch. 10. An engine developing 50 kw at 1200rpm is fitted with a cone clutch. The cone angle is 12 0 with a mean diameter of 500mm. The coefficient of friction is 0.25.The normal pressure on the clutch face is not to exceed 0.1MPa.Determine the axial spring force to engage the clutch. 11 A cone clutch with cone angle 200 is to transmit 7.5 kw at 750 r.p.m. the normal intensity of pressure between the contact faces is not to exceed 0.12N/mm 2. The coefficient of friction is 0.2. if face width is 1/5th of mean diameter, find :1. The main dimensions of the clutch, and 2. Axial force required while running. 12 A centrifugal clutch has 4 shoes which slide radially in a spider keyed to the driving shaft and make contact with internal cylindrical surface of a rim keyed to the driven shaft. When the clutch is at rest, each shoe is pulled against a stop by a spring so as to leave a radial clearance of 5 mm between the shoe and the rim. The pull exerted by the spring is then 500 N. The mass centre of shoe is 160mm from the axis of the clutch. If the internal diameter of the rim is 400 mm, the mass of each shoe is 8 kg, the stiffness of each spring is 50 N/mm and the coefficient of friction between the shoe and the rim is 0.3. Find the power transmitted by the clutch at 500 rpm. 13 A cone clutch with a semi-cone angle of 15 transmits 10kW at 600 rpm. The normal pressure intensity between the surfaces in contact is not to exceed 100kN/m 2. The width of the friction surfaces is half of the mean diameter. Assume μ=0.25. Determine (i) the outer and inner diameters of the plate (ii) width of the cone face (iii) The axial force to engage the clutch. 14 Describe with neat A car engine has its rated output of 12 kw. The maximum torque developed is 100 N-m. The clutch used is of single plate type having two active surfaces. The axial pressure is not to exceed 85 kn/m 2. The external diameter of the friction plate is 1.25 times the internal diameter. Determine the dimensions of the friction plate and the axial force exerted by the springs. Coefficient

6 of friction = Derive a relation for the frictional torque acting on a centrifugal clutch. D 16 A single block brake is shown in figure the diameter of drum is 250mm and the angle of contact is 120 degrees. If the operating force of 900N is applied at the end of a lever and the coefficient of friction between the drum and the lining is 0.45 determine the torque that may be transmitted by the block brake. 17 A Simple band brake is applied to a drum of 560mm diameter, which rotates at240 rpm. Angle of contact of band is 270 degrees. One end of the band is fastened to a fixed pin and the other end of the brake lever 140mm from the fixed pin. The brake lever is 800mm long and is placed perpendicular to the diameter that bisects the angle of contact. The coefficient of friction is 0.3; determine the necessary pull at the end of the lever to stop the drum if 40 kw of power is being absorbed. Also find the width of the band if its thickness is 3mm and the maximum tensile stress is limited to 40N/mm A simple band brake is fitted on a crane having the diameter of barrel as 50cm. The band embraces 3/4 th of the circumference of the brake drum of diameter 70 cm. The tight end of the band is attached to the fulcrum of the brake lever while the slack end of the band is attached to a pin which is 10 cm away from the fulcrum. Calculate the braking torque acting on the drum shaft if the operating force of N is acting on the lever at a distance of 65 cm from the fulcrum. The coefficient of friction is 0.3. Instead of attaching the tight end of the band to the fulcrum of the lever, if it is attached to a pin 2 cm away from the fulcrum of the lever and on opposite side of the pin to which the slack end of the band is attached, what will be the increase of braking torque acting on the drum shaft? 19 What is the function of a flywheel? How does it differ from that of a governor 20 With neat sketch explain epicycle gear dynamometer. Unit-III 1. Prove that the maximum fluctuation of energy. E = E x 2C S Where E= Mean K.E of the flywheel and Cs= coefficient of friction of speed. 2. The mass of flywheel of an engine is 6.5 tones and the radius of gyration is1.8 meters. It is found from the turning moment diagram that the fluctuation of energy is 56KN-m.If the mean speed of the engine is 120rpm,find the maximum and minimum speeds. 3. The turning moment diagram for a petrol engine is drawn to the following scale. Turning moment 1mm=5N-m; crank angle 1mm=1 0. The turning moment diagram repeats itself at every half revolution of the engine and the areas above and below the mean turning moment line taken in order, are 295, 685, 40,340,960,270mm 2. The rotating parts are equivalent to amass of 36kg at a radius of gyration of 150mm. Determine the coefficient of fluctuation of speed when the engine runs at 1500rpm.

7 4. An engine flywheel has a mass of 6500 kg and a radius of gyration of 2m. If the maximum and minimum speed is 120 rpm and 118rpm respectively, find maximum fluctuation of energy. 5. A vertical double acting steam engine develops 75 kw at 250rpm. The maximum fluctuation of energy is 30 percent of the work done per stroke. The max and min 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 0.6m. 6. Define the following terms relating to governors. (a) Stability (b) Sensitiveness (c) Isochronism (d) Hunting 7. Prove that the Sensitiveness of a Proell Governor is greater than that of a Porter Governnor. 8. Calculate the vetical height of a Watt Governnor when it rotates at 60 rpm. Also find the change in vertical height when its speed increases to61 rpm. 9. The arms of a Porter governor are each 250mm long and pivoted on the governor axis. The mass of each ball is 5kf and the mass of the central sleeve is 30kg.The radius of rotation of the balls is 150mm when the sleeve begins to rise and reaches a value of 200mm for max.speed. Determine the speed range of the governor. 10. In a spring loaded Hartnell governor the extreme radii of rotation of the balls are 80mm and 120 mm. The ball arm and the sleeve arm of the bell crank lever are equal in length. The mass of each ball is 2kg.If the speeds at the two extreme positions are 400 and 420rpm,find the initial compression of the central spring. 11 In an engine governor of the porter type, the upper and lower arms are 20 cm and 25 cm long respectively and pivoted on the axis of rotation. The central load is N the weight of each ball is 19.6 N and friction of sleeve together with the resistance of the operating gear is equal to the weight of 24.5 N at the sleeve. If the limiting inclinations of the upper arms to the vertical are 30 0 and 40 0, find range of speeds of the governor. 12 A governor of the Proell type has each arm 250 mm long. The pivots of the upper and lower arms are 25 mm, from the axis. The central load acting on the sleeve has a mass of 25 kg and each rotating ball has a mass of 3.2 kg when the governor sleeve is in mid position, the extension link of the lower arm is vertical and the radius of path of rotation of the masses is 175 mm. the vertical height of the governor is 200 mm. if the speed of governor is 160 rpm, when in mid position, find: (a) Length of the extension link and (b) Tension in the upper arm. 13 A Proell governor has all the four arms of length 25 cm. The upper and lower ends of the arms are pivoted on the axis of rotation of the governor. The extension arms of lower links are each 10 cm long and parallel to the axis when the radius of the ball path is 15 cm. The weight of each ball is N and the central weight of N. Determine the equilibrium speed of the governor. 14 Describe the function of a simple watt governor. What is its limitation?

8 15 The mass of each ball of a proell governor is 3 kg and the weight on the sleeve is 20 kg each arm is 220 mm long and the pivots of the upper and the lower arms are 20 mm form the axis. For the mid position of the sleeve the extension links of the lower arms are vertical, the height of the governor 180 mm and the speed 150 rpm. Determine the lengths of the extension links and the tension in the upper arms. 16 What is meant by effort and power of a governor? Find the expressions for the same in a porter governor. 17 The crank of a three- cylinder single acting engine are set equally at 1200 the engine speed is 540 rpm. The turning-moment diagram for each cylinder is a triangle for the power stroke with a maximum torque of 100 N-m at 600 after deadcentre of the corresponding crank. On the return stroke, the torque is sensibly zero. Determine (i) the power developed (ii) the coefficient of fluctuation of speed if the flywheel has a mass of 7.5 kg with a radius of gyration of 65 mm.(iii) the coefficient of fluctuation of energy (iv) the maximum angular acceleration of the flywheel Explain precisely the uses of turning moment diagram of reciprocating engines. 20 The following data relate to a Hartnell governor M= 1.5 kg; a= 100 mm; b = 40 mm; r1 = 70 mm; r2 =110 mm; N1 =260 rpm; and N2= 275 rpm. The axis of rotation is 80 mm form the fulcrum. Calculate the rate of the spring and the equilibrium speed when the radius of the balls is 80 mm. Unit-IV 1. Four masses A, B, C, and D are attached to a shaft and revolve in the same plane. The masses are 12kg,10kg,18kg and 15kg respectively and their radii of rotations are 40mm,50mm,60mm and 30mm.The angular position of the masses B,C, and D are and from the mass A. Find the magnitude and position of the balancing of mass at a radius of 100mm. 2. Discuss how a single revolving mass is balanced by two masses revolving in different planes. 3. Explain briefly static and dynamic balancing of rotating masses. 4. Explain how masses m i (i=1..n) rotating in the same plane with radii r i (i=1..n) with same angular speed rad/sec can be balanced. 5. What is hammer blow? Write an equation for maximum magnitude of hammer blow. 6. Explain briefly Primary balancing and Secondary balancing as applied to balancing of reciprocating masses. 7. Explain the term Partial balancing of primary forces. Why is it necessary? 8. What do you understand by inside cylinder locomotives and outside cylinder locomotives? 9. Prove that max. swaying couple is equal to

9 Where a= distance between the centre lines of the two cylinders and C= fraction of the mass of reciprocating parts which is to be balanced. 10. Prove that the limiting condition for the angular speed when the wheel does not lift from the rails is given by, 11 Four masses m1, m2, m3 and m4 having 100, 175, 200 and 25 kg are fixed to cranks of 20 cm radius and revolve in places 1, 2, 3 and 4. The angular position of the cranks in planes 2, 3 and 4 with respect to the crank in plane 1 are 750, 1350 and 2000 taken in the same sense. The distance of planes 2, 3 and 4 from plane 1 are 60 cm, 186 cm and 240 cm respectively. Determine the position and magnitude of the balance mass at a radius of 60 cm in plane L and M located at middle of the plane 1 and 2 and the middle of the planes 3 and 4 respectively. 12 Two weights of 8 kg and 16 kg rotate in the same plane at radii of 1.5 and 2.25 m respectively. The radii of these weights are 600 apart. Find the position of the third weight of the magnitude of 12 kg in the same plane which can produce static balance of the system. 13 Discuss static and Dynamic balancing. 14 A shaft carries four masses A, B, C and D placed in parallel planes perpendicular to the shaft axis and in this order along the shaft. The masses of B and C are 353 N and 245 N respectively and both are assumed to be concentrated at a radius of 15 cm, while the masses in planes A and D are both at a radius of 20 cm. The angle between the radii of B and C is 1000 and that between B and A is 1900, both angles being measured in the same sense. The planes containing A and B are 25 cm apart and those containing B and C are 50 cm apart. If the shaft is to be in complete dynamic balance, determine

10 i) Masses of A and D ii) distance between the planes containing C and D iii) angular position of the mass D. 15 A mass is attached to a shaft which is rotating at an angular speed of w rad/s. Describe the procedure of balancing this mass by (i). A single mass only (ii). Two masses in different planes. 16 A shaft carries four rotating masses A, B, and C 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 900 and 1950 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 mass A and its angular position (ii). The position of planes of A and D. 17 Four masses (m 1, m 2, m 3, and m 4 ) 300kg, 450 kg, 360 kg and 390 kg are attached to a shaft. These masses are revolving at radii 135 mm, 105 mm, 150 mm and 180 mm respectively in planes measured from A at 135 mm, 210 mm and 360 mm respectively. The angle measured anti-clockwise are m1 to m 2 450, m 2 to m 3 750, m 3 to m and the distance between the planes L and M in which the balance masses are to be placed is 250 mm. the distance between planes A and L is 60 mm and M and D is 50 mm. if the balancing masses revolve at a radius of 36 mm, find their magnitudes and angular positions. 18 Explain and derive the equation for the following terms (i). Tractive force (ii). Swaying couple (iii). Hammer blow. 19 Derive the condition to the limiting speed for which wheels of the locomotive are not lifted from rails. 20 A two cylinder uncoupled locomotive has inside cylinders 60 cm apart. The radius of each crank is 30 cm. The cranks are at right angles. The weight of the revolving mass per cylinder is N and the weight of the reciprocating mass per cylinder is 2943 N. The whole of the revolving and 2/3rd of the reciprocating masses are to be balanced and the balanced weights are placed, in the planes of rotation of the driving wheels, at radius of 80 cm. The driving wheels are 2 m in diameter and 1.5 m apart. If the speed of the engine is 80 km/hr, find the hammer blow, maximum variation of tractive effort and maximum swaying couple. Unit-V 1. Prove that the natural frequency and angular velocity of a body having free longitudinal vibration is given by

11 = Where S= stiffness of the spring m= mass of vibrating body 2. A simply supported beam of span L carries an uniformly distributed mass m kg per meter length. Prove that the natural frequency of transverse vibration is given by 3. A vertical shaft 100mm in diameter and 1m in length has its upper end fixed at the top. At the other end it carries a disc of weight 20kn.The Young s modulus of the material of the shaft is 2x10 5 N/mm 2. Neglecting the weight of the shaft, determine the frequency of longitudinal vibrations. 4. A simply supported shaft of length 1.6m carries a mass of 120 kg placed 500mm from one end. If E=200GN/m 2 and diameter of shaft =50mm, find the Natural frequency of transverse vibrations. 5. A shaft of length 0.75m, supported freely at the ends, is carrying a body of mass 90kg at 0.25m from one end. Find the natural frequency of transverse vibration. Assume E=200GN/m 2 and shaft diameter =50mm 6. Establish an expression for the natural frequency of free transverse vibrations for a simply supported beam carrying a number of point loads by Dunkerley s method. 7. Discuss the effect of inertia of a shaft in longitudinal and transverse vibrations. 8. A shaft of diameter 10mm carries at its centre a mass of 12kg. It is supported by two short bearings with a centre distance of 400mm. Find the whirling speed neglecting the mass of the shaft. 9. An electric motor rotating at 1500 rpm, drives a centrifugal pump at 500 rpm through a single stage reduction gearing. The M.I. of the electric and the pump impeller arc 400 kg-m 2 and 1400 kg-m 2 respectively. The motor shaft is 45mm in diameter and 180mm long. The pump shaft is 90mm diameter and 450mm long. Determine the natural frequency of torsional oscillations of the system neglecting the inertia of the gears. The modulus of rigidity of the shaft material is 84 GN/m A shaft of diameter is supported in two bearings 2.5 m apart. It carries three discs of mass 250kg, 500kg, and 200kg are 0.6m, 1.5m and 2m from the left end bearing. Assuming the mass of the shaft 190 kg/m 2.Determine the critical speed of the shaft. Young s modulus of the material of shaft =211 GN/m Define: i. Free vibrations ii. Forced vibrations iii. Damping. 12 A mass weighing 85 kgf is supported on springs which deflects 1.8 cm under the weight of the mass. The vibration of the mass are constrained to be linear and vertical and are damped by a dashpot which reduces the amplitude to one-quarter of its initial value in two complete oscillations, find: (a) The magnitude of the damping force at unit speed and (b) Periodic time of damped vibrations. `13 Derive an expression for the natural frequency of free transverse vibration of simply supported beam carrying several of point loads by energy method. 14 A trailer has 1000 kg mass when fully loaded and 250 kg when empty. The spring of the suspension is 350 kn/m. The damping factor is 0.5 when the trailer is fully loaded. The speed is 100 km/hr. The road varies sinusoidally with a wave length of 5 m. Determine the amplitude ratio of the trailer when

12 fully loaded and empty What do you understand by torsionally equivalent shaft? A shaft is simply supported at the ends and is of 20 mm in diameter and 600 mm in length. The shaft carries a load of N at its centre. The weight of the shaft per metre length is N. Find the critical speed of the shaft. Take Young s modulus as 200 GN/m2. 17 Determine the frequency of the free vibrations, when a body of mass 20 kg is suspended from a spring which deflects 15 mm under the weight of the body. Also find the viscous damping force required to make the motion a period at a speed of 1m/s. If when damped to this extent, a distinguish force having a maximum value of N and vibrating at 8 Hz is made to act on the body, fine the amplitude of the ultimate motion. 18 Determine natural frequency of the pendulum system. D 19 Explain the term critical speed, prove that the critical speed of a rotating shaft is the same as the natural frequency of transverse vibration. 20 A steel shaft of 1.5 m long is 95mm in diameter for the first 0.6m of its length, 60 mm in diameter for the next 0.5 m of the length and 50 mm 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 95mm diameter end and the second having a mass of 700 kg and 0.55 m radius of gyration located at the other end. Determine the location of the node and natural frequency of free torsional vibration of the system. The modulus of rigidity of shaft is assumed as 80 GN/m2.

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