SATHYABAMA UNIVERSITY(Established under section 3 of UGC Act, 1956)

Course & Branch: B.E – Mechanical (Part Time)Title of the paper: Dynamics of MachinerySemester: II Max. Marks: 80 Sub.Code: 615PT204 Time: 3 HoursDate: 14-05-2007 Session: FN

PART – A (10 x 2 = 20)Answer ALL the Questions

1. Define Inertia force2. Explain - turning moment diagram.3. Why balancing is necessary?4. The reciprocating masses can not be completely balanced by a rotating mass –

explain.5. Define: Degrees of freedom.6. Define: Logarithmic decrement7. What is forced vibration?8. When a forced vibration becomes a free vibration? Explain.9. What is a semi definite system?10. What do you understand by a two degrees of freedom system and multi degrees

freedom system?

PART – B (5 x 12 = 60)Answer All the Questions

11. The crank and connecting rod of a reciprocating engine are 150 mm and 600 mm respectively. The crank is rotating in clockwise direction at 120 radians per second. Find, by Klein’s construction (a) Velocity and acceleration of the piston(b)Velocity and acceleration of the mid point D of the connecting rod, and(c) Angular velocity and angular acceleration of the connecting rod at the instant when the crank is at 300 to I.D.C.

(or)12. The turning moment diagram for a multi cylinder engine has been drawn to a

scale of 1 mm = 5 N. m. vertically and 1 mm=6.50 horizontally. The intercepts between out put and mean resistance line is taken in order from one end are -28, +400, -342, +230, -245, +305, -115, +281, and -178 mm2 and the engine is running at 600 r.p.m. The engine has a stroke of 250 mm and total fluctuation of speed is not to exceed ±2% of mean speed. Determine a suitable value of diameter and cross section of the flywheel rim for a limiting value of centrifugal stress of 7 MPa. Take density = 7200 kg/m3 and width of the rim is 4 times the thickness.

13. A shaft has four masses m1 , m2 , m3 , and m4 of magnitude 150 kg, 250 kg, 350 kg and 200 kg respectively and revolving radii 85 mm, 70 mm, 60 mm and 90 mm in planes measured from m1 at 300 mm, 350 mm and 650 mm. The angles between the cranks measured anticlockwise are m1 to m2 450, m2 to m3 700, and m3 to m4 1200 . The balancing masses are to be placed in planes X & Y. The distance between the planes m1 and X is 100 mm, between X and Y is 400 mm and between Y and m4 is 200 mm. If the balancing masses revolve at a radius of 100 mm, find their magnitudes and angular positions.

(or)14. A single cylinder reciprocating engine has speed a 260 r.p.m. stroke 350 mm,

mass of reciprocating parts 60 kg, mass of revolving parts at 160 mm radius is 50 kg. If 0.7 of reciprocating parts and all the revolving parts are to be balanced, find(a) the balance mass required at radius of 250 mm (b) the unbalanced force when the crank has turned 700 from the inner dead centre.

15. A fly wheel is mounted at the end on a vertical shaft with the following details.Length of the shaft = 1.2 m.Diameter of the shaft = 8 cmMass of the fly wheel = 450 kg.Young’s modulus of material = 200 GN/ m2

Find the natural frequencies of longitudinal and transverse vibrations.(or)

16. A body of mass 500 kg is supported by an elastic structure of stiffness 70 kN/m. The motion of the body is controlled by a dashpot such that the amplitude of vibration decreases by 1/3rd of its original value after 4 complete vibrations. Determine (a) Damping force at unit velocity (b) Natural frequency (c) logarithmic decrement(d) Damping factor .

17. A reciprocating pump 200 kg is driven through a belt by an electric motor at 3000 r.p.m. The pump is mounted on isolators with total stiffness 5 MN/m and damping 4.125 kNS/m. Determine the vibratory amplitude of the pump at the running speed due to fundamental harmonic force of excitation 1.2 kN. Also determine the maximum vibratory amplitude when the pump is switched on and the motor speed passes through resonant condition.

(or)18. A machine weighing 600 N is mounted on springs of stiffness k = 11 kN/cm with

an assumed damping factor of 0.2. A piston within the machine weighing 25 N has a reciprocating motion with a stroke of 75 mm and a speed of 3000 r.p.m. Assuming the motion of the piston to be Simple harmonic, determine (a) the amplitude of motion (b) the phase angle with respect to the exciting force (c) the transmissibility and the force transmitted to the foundation (d) the phase angle of the transmitted force with respect to the exciting force.

19. Two bodies having equal masses as 100 kg each and radius of gyration 0.4 m are keyed to both ends of a shaft 1 m long. The shaft 0.08 m in diameter for 0.35 m

length, 0.12 m diameter for 0.2 m length and 0.09 m diameter for rest of the length. Find the frequency of torsional vibration. Take G= 9x1011 N/m2.

(or)20. Two equal masses of weight 500 kg each of radius of gyration 40 cm are keyed to

the opposite ends of a shaft 65 cm long. The shaft is 7.5 cm diameter for the first 25 cm length, 12.5 cm diameter for next 10 cm and 10.5 cm diameter for the remaining of its length. 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.

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SATHYABAMA UNIVERSITY(Established under section 3 of UGC Act, 1956)

Course & Branch: B.E – Mechanical (Part Time) Title of the paper: Dynamics of MachinesSemester: II Max. Marks: 80 Sub.Code: 615PT204 (2006/2007/2007 JAN) Time: 3 HoursDate: 21-05-2008 Session: FN

PART – A (10 x 2 = 20)Answer All the Questions

1. In what way a mechanism differ from a machine?

2. Explain the term lower pair and higher pair.

3. Why is balancing of rotating parts necessary for high speed engines?

4. What are in-line engines? How are they balanced?

5. What are the causes and effects of vibrations?

6. Define free vibrations, forced vibrations and damped vibrations.

7. Explain the term ‘whirling speed’ or critical speed of the shaft.

8. What do you understand by transmissibility?

9. Discuss the effect of inertia opf a shaft on the free torsional vibrations.

10. How the natural frequency of torsional vibrations for a two rotor system is obtained?

PART – B (5 x 12 = 60)Answer All the Questions

11. In a mechanism shown in the figure. The crank OA is 100mm long and rotates clockwise about O at 120 r.p.m. The connecting rod AB is 400mm long. At a point C on AB 150mm from A, the rod CE 350mm long is attached. This rod CE slides in a slot in a trunnion at D. The end E is connected by a link EF, 300mm long to the horizontally moving slider F. For the mechanism in the position shown, find (a) Velocity of F, (b) Velocity of sliding of CE in the trunnion, and (c) Angular velocity of CE.

(or)12. Sketch a quick return motion of the crank and slotted lever type and

explain the procedure of drawing the velocity and acceleration diagram, for any given configuration of the mechanism.

13. Four masses A, B, C and D as shown below are to be completely balanced.

A B C DMass (Kg) - 30 50 40

Radius (mm)

180 240 120 150

The planes containing masses B and C are 300mm apart. The angle between planes containing B and C is 90 . B and C make angles of 210 and 120 respectively with D in the same sense. Find (a) The

magnitude and the angular position of mass A, And (b) The position of planes A and D.

(or)14. The cranks and connecting rods of a 4-cylinder in-line engine

running at 1800 r.p.m. are 60mm and 240mm each respectively and the cylinders are spaced 150mm apart. IF the cylinders are numbered 1 to 4 in sequence form one end, the cranks appear at intervals of 90 in an end view in the order 1-4-2-3. The reciprocating mass corresponding to each cylinder is 1.5 kg. Determine (a) Unbalanced primary and secondary forces, if any and (b) Unbalanced primary and secondary couples with reference to central plane f the engine.

[15. Determine the natural frequency if a simple pendulum as shown in

the figure., taking the mass of the rod into consideration.

(or)16. In a single-degree damped vibrating system, a suspended mass of 8

kg makes 30 oscillations in 18 seconds. The amplitude decreases to 0.25 of the initial value after 5 oscillations. Determine (a) the stiffness of the spring, (b) The logarithmic decrement (c) The damping factor, and (d) the damping coefficient.

17. A single –cylinder vertical diesel engine has a mass of 400 kg and is mounted on a steel chassis frame. The static deflection owing to the

weight of the chassis is 2.4mm. the reciprocating masses of the engine amounts to 18 kg and the stroke of the engine is 160mm. A dashpot with a damping coefficient of 2N/mm/s is also used to damped the vibrations. In the steady state of the vibrations determine: (a) the amplitude of the vibrations if the driving shaft rotates at 500 rpm. (b) The speed of the driving of the driving shaft when the resonance occurs.

(or)18. 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 100mm with simple harmonic motion. Neglecting damping, determine the combined stiffness of the springs so that the force transmitted to the foundation is 1/20th of the impressed force. The machine crankshaft rotates at 800 rpm. If under actual working conditions, the damping requces the amplitudes of successive, vibrations by 30%, find (a) the force transmitted to the foundation at 800 rpm, (b) the force transmitted to the foundation at resonance, and (c) the amplitude of the vibrations at resonance.

19. Use the Holzer method to determine the natural frequencies fo the four mass system as shown in figure., if k = 11 b/in and m = 11 b/sec2/in.

(or)20. Use the stood method to determine the, lowest natural frequency of

the four-degree of freedom spring-mass system as shown in figurer.

SATHYABAMA UNIVERSITY Question Bank

Course : B.E–Mechanical(Part time) Sub Code : 715PT204 Subject : DYNAMICS OF MACHINERY Semester : II

Unit 1 (2 marks)

1. What do you mean by inertia force? 2. Explain : D’ Alembert’s Principle. 3. Explain how a dynamic problem could be converted into a static problem, using D’

Alembert’s principle. 4. What is a reversed effective force? 5. Define : Angular acceleration. 6. Why flywheel is preferred in rotating machines? 7. Define : Coefficient of fluctuation of Energy. 8. Define : Coefficient of fluctuation of Speed. 9. What is maximum fluctuation of Energy? 10. What is a function of flywheel? How does it differ from that of governor?

Unit 2 (2 marks)

1. Why is balancing of rotating parts necessary for high speed engines? 2. Explain : Static balancing. 3. Explain : Dynamic balancing. 4. What is complete balancing? 5. Dynamic balancing includes Static balancing also - Explain. 6. What is Primary balancing? 7. What is Secondary balancing? 8. What is Variation in tractive force? 9. What is Swaying couple? 10. Define: Hammer blow.

Unit 3 (2 marks)

1. Define: Logarithmic decrement. 2. What is viscous damping? 3. Define Simple Harmonic motion. 4. What is amplitude? 5. Define: Periodic motion. 6. What is Resonance? 7. Define: Damping. 8. Explain about degree of freedom? 9. Define: single degree of freedom. 10. Define: Time period. Unit 4 (2 marks)

1. Define: Forced vibration. 2. Define: Magnification factor. 3. Define the term Critical damping. 4. Explain the importance of vibration isolation. 5. What do you understand by transmissibility? 6. What is a characteristic curve? 7. Write the governing differential equation for forced vibration and explain the terms. 8. What is a low speed system? 9. What is a high speed system? 10. When forced vibration becomes as damped vibration?

Unit 5 (2 marks) 1. Explain Torsional vibrations with respect to two degree freedom system. 2. Define: Critical speed of a shaft. 3. What is a Semi definite system? 4. What is continuous system? 5. What is Energy method and what we evaluate from it? 6. Write the Dunkerley’s equation for natural frequency of transverse vibration? 7. Define node in torsional vibrations. 8. What is torsionally equivalent system? 9. What is a multi degree freedom system? 10. How many frequencies will exist for a two degree freedom system? Why?

PART – B Unit 1 (12 marks)

1. The following refers to horizontal Internal Combustion engine. Length of connecting rod = 22 cm, Mass of connecting rod = 1.8 kg, Mass of reciprocating parts (Piston and gudgeon pin) = 2.5 kg,

Stroke = 15 cm, Cylinder bore = 12 cm, The C.G of connecting rod from small end = 15cm, Radius of gyration = 9 cm, Crank angle from IDC = 45°. Piston is moving away from Inner Dead Centre under gas pressure of 20 bar. The engine speed is 1200 rpm. Determine the various forces acting on the reciprocating engine mechanism.

2. The following data refer to a steam engine. Diameter of piston =24 cm; stroke =60 cm; length of connecting rod =1.5 m; mass of reciprocating parts =400 kg; mass of connecting rod = 350 kg; Speed =125 rpm; Centre of gravity of connecting rod from crank pin =50 cm; radius of gyration of the connecting rod about an axis through the centre of gravity =65 cm. Determine the magnitude and direction of the torque exerted on the crank shaft when the crank has turned through 30° from IDC. Also, find other forces.

3. A horizontal cross compound steam engine develops 300 kW at 90 rpm. The coefficient of fluctuation of energy as found from the turning moment diagram is to be 0.1 and the fluctuation of speed is to be kept within + 0.5% of the mean speed. Find the weight of the flywheel required, if the radius of gyration is 2 m.

4. The turning moment curve for an engine is re presented by the equation, T= (22000 + 9500 sin 2θ – 5700 cos 2θ) N-m, where θ is the angle moved by the crank from inner dead centre. If the resisting torque is constant, find :- 1) Power developed by the engine. 2) Moment of inertia o flywheel in kg- m2, if the total fluctuation of speed is not to exceed 1% of mean speed which is 180 rpm. 3) Angular acceleration of the flywheel when crank has turned through 45° from inner

dead centre. 5. The turning moment diagram for a multi-cylinder engine has been drawn to a scale of 1 mm to

500 N-m torque and 1 mm to 5° of crank displacement. The intercepted areas between output torque curve and mean resistance line taken in order from one end, in sq. mm are -30, +410, -280, +320, -330, +250, -360, +280, -260 sq. mm, when the engine is running at 800 rpm. The engine has a stroke of 300 mm and the fluctuation of speed is not to exceed + 2% of the mea speed. Determine a suitable diameter and cross-section of the flywheel rim for a limiting value of the safe centrifugal stress of 7 Mpa. The material density may be assumed 7200 kg/m3. The width of the rim is to be 4 times the thickness.

6. A single cylinder double acting steam engine develops 150 kW at a mean speed of 80 r. p. m. The coefficient of fluctuation of energy is 0.1 and the fluctuation of speed is ± 2% of mean speed. If the mean diameter of the flywheel rim is 2 meter and the hub and spokes provide 5% of the rotational inertia of the flywheel, find the mass and cross-sectional area of the flywheel rim. Assume the density of the flywheel material (which is cast iron) as 7200 kg/m3

Unit 2 (12 marks)

1. A shaft carries four masses A, B, C and D of magnitude 200 kg, 300 kg, 400 kg and 200 kg respectively and revolving at radii 80 mm, 70 mm, 60 mm and 80 mm in planes measured from A at 300 mm, 400 mm and 700 mm. The angles between the cranks measured anticlockwise are A to B 45°, B to C 70° and C to D 120°. The balancing masses are to be placed in planes X and Y. The distance between the planes A and X is 100 mm, between X and Y is 400 mm and between Y and D is 200 mm. If the balancing masses revolve at a radius of 100 mm, find their magnitudes and angular positions.

2. A, B, C and D are four masses carried by a rotating shaft at radii 100,125, 200 and 150 mm

respectively. The planes in which the masses revolve are spaced 600 mm apart and the mass of B, C and D are 10 kg, 5 kg, and 4 kg respectively. Find the required mass A and the relative angular settings of the four masses so that the shaft shall be in complete balance.

3. A shaft carries four masses in parallel planes A, B, C and D in this order along its length. The

masses at B and C are 18 kg and 12.5 kg respectively, and each has an eccentricity of 60 mm. The masses at A and D have an eccentricity of 80 mm. The angle between the masses at B and C is 100° and that between the masses at B and A is 190°, both being measured in the same direction. The axial distance between the planes A and B is 100 mm and that between B and C is 200 mm. If the shaft is in complete dynamic balance, determine: 1.The magnitude of the masses at A and D; 2.The distance between planes A and D; and 3.The angular position of the mass at D.

4. A single cylinder reciprocating engine has speed 120 r.p.m., stroke 300 mm, mass of

reciprocating parts 60 kg, mass of revolving parts at 130 mm radius 42 kg. If 0.75 of the reciprocating parts and all the revolving parts are to be balanced, find : 1. The balance mass required at a radius of 300 mm, and 2. The residual unbalanced force when the crank has rotated 60° from inner dead centre.

5. The centre distance between the cylinders of inside cylinder locomotive is 0.8 m. It has a stroke

of 0.5 m. The rotating mass per cylinder is equivalent to 200 kg at the crank pin, and the reciprocating mass per cylinder is 240 kg. The wheel centre lines are 1.7 m apart. The cranks are at right angles. The whole of the rotating masses and 2/3 of the reciprocating masses are balanced by masses at a radius of 0.6 m. Find the magnitude and direction of the balancing masses. Find the hammer blow, variation in tractive effort and the magnitude of swaying couple at a speed of 300 rpm.

6. The following data relate to a single cylinder reciprocating engine:- Mass of reciprocating parts = 40kg Mass of revolving parts = 30kg at 180 mm radius Speed = 150 rpm, Stroke = 350 mm. If 60% of the reciprocating parts and all the revolving parts are to be balanced, determine (i) The balance mass required at a radius of 320mm. (ii) The unbalanced force when the rank has turned 45° from the top dead centre.

Unit 3 (12 marks)

1. Find the natural frequency of the system shown in the figure. k =2 X 105 N/m, m=50 kg.

2. For the system shown in Fig., if k1= 2400 N/m; k2 = 1600 N/m; k3 = 3600 N/m and k4 = k5 = 600 N/m, find the mass m such that the natural frequency fn = 10Hz.

3. Develop an expression for logarithmic decrement in terms of damping factor. Also prove that logarithmic decrement = 1/n {ln (Xo / Xn )}, where Xo = amplitude of 1st cycle and Xn = amplitude after ‘n’ cycles.

4. A vibrating system is defined by the following parameters:

m = 3 kg, k = 100 N/m, C = 3N-S/m. Determine (a) the damping factor, (b) the natural frequency of damped vibration, (c) logarithmic decrement, (d) the ratio of two consecutive amplitudes and (e) the number of cycles after which the original amplitude is reduced to 20 percent.

5. The following data are given for a vibratory system with viscous damping:

Mass = 2.5 kg; spring constant = 3 N/mm and the amplitude decreases to 0.25 of the initial value after five consecutive cycles. Determine the damping coefficient of the damper in the system.

6. A machine of mass 75 kg is mounted on springs and is fitted with a dashpot to damp out vibrations.

There are three springs each of stiffness 10 N/mm and it is found that the amplitude of vibration diminishes from 38.4 mm to 6.4 mm in two complete oscillations. Assuming that the damping force varies as the velocity, determine : 1. the resistance of the dash-pot at unit velocity ; 2. the ratio of the frequency of the damped vibration to the frequency of the undamped vibration ; and 3. the periodic time of the damped vibration.

Unit 4 (12 marks)

1. A 300kg single cylinder vertical petrol engine is mounted upon chassis frame and cause a deflection of 1.5 mm. Mass of reciprocating parts of engine is 25 kg and it has a stroke of 145 mm. A dashpot is provided whose damping resistance is 1.5KN.s/m. Determine 1) Amplitude of

forced vibration, When driving shaft rotates at 480 rpm. 2) Speed of the driving shaft at which resonance occurs.

2. A machine part having a mass of 2.5 kg executes vibration in a viscous damping medium. A

harmonic exciting force of 30 N acts on the part and causes a resonant amplitude of 14mm, with a period of 0.22 Sec. Find the damping coefficient when the frequency of exciting force is changed to 4 Hz. Determine the increase in amplitude of forced vibration upon the removal of damper.

3. A 12kg mass is suspended from of helical spring, other end is fixed. Spring stiffness is equal to

15 N/mm. Due to viscous damping, amplitude decreases to 1/10th of initial value in 4 oscillations. If a periodic force 150cos 50t N is applied at mass in vertical direction, find amplitude of force vibration. What is its value of resonance?

4. A machine of mass one tonne is acted upon by an external force of 2450 N at a frequency of

1500 rpm. To reduce the effects of vibration, isolator of rubber having a static deflection of 2mm under the machine load and an estimated damping factor of 0.2 are used. Determine a)The force transmitted to the foundation, b)The amplitude of vibration of machine, c)The phase lag.

5. In a vibrating system, the total mass of the system is 25 kg. At speed of system and eccentric

mass have a phase difference of 90° and the corresponding amplitude is 1.5 cm. The eccentric unbalanced mass of 1 kg has a radius of rotation 4 cm. Determine: i) the natural frequency of the system. ii) The damping factor, iii) the amplitude at 1500 rpm and iv) The phase angle at 1500 rpm.

6. An electric motor is supported on a spring and a dashpot. The spring has the stiffness 6400

N/m and the dashpot offers resistance of 500 N at 4 m/s the unbalanced mass 0.5 kg rotates at 5 cm radius and total mass of vibratory system is 20 kg. The motor runs at 400 rpm. Determine: 1) Damping factor. 2) Amplitude of vibration and phase angle. 3) Resonant speed and resonant amplitude. 4) Forces exerted by the spring and dashpot on the motor.

Unit 5 (12 marks) 1. Figure shows a vibrating system having two degrees of freedom. Determine the two natural

frequencies of vibrations and the ratio of amplitudes of the motion of m1 and m2 for the two modes of vibration.

Given : m1 = 1.5 kg, m2 = 1.8 kg, k1=k2 = 40 N/m

2. Calculate the natural frequency of a shaft of diameter 10 cm and length 300 cm carrying two discs of diameters 125 cm and 200 cm respectively at its ends and with masses 480 kg and 900 kg respectively. Modulus of rigidity of the shaft may be taken as 196.2x109 N/m2

3. Two equal masses 400 kg each and radius of gyration 40 cm are keyed to the opposite ends of a shaft 60 cm long. The shaft is 7.5 cm diameter for the first 25 cm of its length, 12.5 cm diameter for the next 10 cm and 8.5 cm diameter for the remaining of its length. Find the frequency of free torsional vibrations of the system and position of node. Assume G = 8.24 x 1010 N/m2

4. Determine the frequency of the system shown in figure.

Given: k1 = k2 = 40 N/m k = 60 N/m, m1=m2 = 10 kg.

5. A shaft 1.5 m long supported by inflexible bearings at the ends carries two wheels each of 50 kg mass. One wheel is situated at the centre of the shaft and the other at a distance of 375 mm from the centre towards left. The shaft is hollow of external diameter 75 mm and internal diameter 40 mm. The density of the shaft material is 7700 kg/m3 and its modulus of elasticity is 200 GN/m2. Find the lowest whirling speed of the shaft, taking into account the mass of the shaft.

6. A shaft of diameter 10 mm carries at its centre a mass of 12 kg. It is supported by two short

bearings, the centre distance of which is 400 mm. Find the whirling speed: 1. neglecting the mass of the shaft, and 2. taking the mass of the shaft also into consideration. The density of shaft material is 7500 kg/m3.