MECHANICAL ENGINEERING DEPARTMENT, OITM

Semester : 6th

Subject : Mechanical Vibrations

Paper : ME – 306 - E

Unit - 1

1. (a) A harmonic motion has amplitude of 0'05 m a1d a frequency of 25 Hz. Find the time

period and the maximum acceleration. (2)

(b) Represent the following in exponential form: (4)

a. 3+i4

b. -3+j4.

(c) Represent the following in rectangular form: (4)

a. 9 e j0.3

b. 14 e - j1.1

(d) Represent the periodic motion given in the Fig' I by harmonic series. (10) (May 11)

2. (a) A mass of 10 kg when suspended from spring causes a static deflection of 1 cm. Find

the natural frequency of the system. (2)

(b) Find the natural frequency of oscillation for the system shown in Fig. 2 assuming the

bell crank lever to be light and stiff and the mass ‘m’ to be concentrated. (4)

MECHANICAL ENGINEERING DEPARTMENT, OITM

(c) A torsion pendulum has to have a natural frequency of 5 Hz. What length of steel wire

of diameter 2 mm should be used for this pendulum ? The inertia of the mass fixed at the

free end is 0.0098 kg-m2. Take G = 0.83 x 10

11 N/m

2. (4)

(d) Determine the natural frequency of the spring – mass – pulley system shown in Fig. 3

(10) (May 11)

3. (a) What is beats phenomenon? Explain it. (10)

(b) Determine the equivalent length of the shaft 2.5 cm diameter which will have the

same torsional stiffness as the stepped shaft shown in the Figure 4. (10) (June 10)

MECHANICAL ENGINEERING DEPARTMENT, OITM

Figure 4

4. (a) An instrument has a natural frequency of l0 Hz. It can stand a maximum acceleration

of l0 m/sec2. Find the maximum amplitude of displacement. (10)

(b) Find the natural frequency of the system shown in figure 5. The rod AC may be assumed rigid

and weightless. (10) (June 10)

Figure 5

5. (a) Define Arnplitude, Natural frequency, Resonance, Time period and Damping? (5)

(b) Differentiate between:

(i) Free Vibrations and Forced Vibrations,

(ii) Continuous and Discreet Systems,

(iii) Periodic Motion and Non-periodic Motions. (6)

(c) Split up the harmonic motion:

into two harmonic motions, one of them is having a phase angle of zero and the other having a

phase angle of 600. (4) (May 09)

6. (a) A flywheel having a mass of 35 kg was allowed to swing as Pendulum about a knife-

edge at inner side of the rim as shown in figure 5. If the measured period of oscillation

was 1.22 seconds, determine the moment of inertia of the flywheel about its geometric

axis.

(7)

MECHANICAL ENGINEERING DEPARTMENT, OITM

Figure 5

(b) A cylinder of mass ‘M’ and radius ‘r’ rolls without slipping on a cylinder surface of

radius 'R’ Find the natural frequency for small oscillations about the lowest point.

(8) (May 09)

7. (a) A body is subjected to two harmonic motions as given below

What harmonic motion should be given to the body to bring it to equilibrium? (10)

(b) A force P0 sinωt acts on a displacement x0sin ( ωt + π/3 )

If P0 = 100N x0 = 0.02 m ω = 2π rad/s

(10) (May 06)

8. (a) A spring-mass system has spring stiffness of k N/m and a mass of M kg. It has natural

frequency of vibration as 12 Hz. An extra 2 kg mass is coupled to M and the natural

frequency reduces by 2 Hz. Find k and M. (10)

(b) A sphere of diameter D floats half submerged in water. If the sphere is depressed

slightly and released, determine the period of vibration. What is this period if D = 1

meter? (10) (June 08)

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Unit - 2

9. (a) A mass of I kg is attached to a spring having a stiffness of 3920 N/m. The mass slides

on a horizontal surface, the coefficient of friction between the mass and the surface

being 0.l. Determine the frequency of vibrations of the system and the amplitude after

one cycle if the initial amplitude is 0.25 cm. (10) (May 11)

(b) A horizontal mass system with coulomb damping has mass of 5kg attached to spring of

stiffness 980 N/m, if coefficient friction is 0.025. Calculate ;

(i) Frequency of free oscillations .

(ii) The number of cycles corresponding to 50% reduction in amplitude if the initial amplitudies

5.0cm.

(iii) The time taken to achieve this 50% reduction.

(10) (May 09)

10. A machine ofmass one tonn is acted upon by an external force of 2450 N at a frequency of

500rpm. To reduce the effects of vibrations, isolator of rubber having a static deflection of 2mm

under the machine load and an estimated damping ε = 0.2 are used. Determine

(a) Force transmitted to foundation.

(b) The amplitude of vibration of machine.

(c) The phase lag. (20) (June 10)

11. The weight of an electric motor is 125 N and its run at 1500 rpm. The armature weighs 35N and its

centre of gravity lies 0.05 cm from the axis of rotation. The motor is mounted on five springs of

negligible damping so that the force transmitted is one- eleventh of the impressed force. Assume

that the weight of the motor is equally distributed among five springs. Determine;

(a) Stiffness of each spring.

(b) Dynamic force transmitted to the base at operating speed.

(c) Natural frequency of the system.

. (20) ( VP Singh)

12. Determine the two natural frequencies and the corresponding mode shapes for the

system shown in the figure 7. The string is stretched with a large tension T.

(20) (May 12)

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13. A mass weighing 1.93 kg is suspended in a box by vertical spring Whose constant k = 10

kg/cm. The box is placed on the top of a Shake table producing a vibration x = 0'09 shift.

Find the absolute amplitude of mass. Assume do damping. (15) (June 08)

14. Two equal masses of weight 400n each and radius of gyration 40cm are keyed to the opposite ends

of a shaft 60cm 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 vibration of system and position of node.

Assume G = 0.84 x 1011

N/m2.

(15) (June 08)

15. A system Shown in figure is subjected to a harmonic force F = 500 sin 13.2t. The value of spring

stiffness is 50000 N/m and damping factor is 0.2. For steady state vibration of system , determine

(a) Amplitude of motion of system

(b) Phase angle

(c) Transmissibility

(d) Max. Dynamic force transmitted to the foundation.

(e) Max. Velocity of motion.

(20) (J.S.Mehta)

16. A motor of mass 60 kg is mounted on a simple beam that has a stiffness of 39200 N/m at

that point. The rotor of the motor has a mass of 10 kg and has an eccentricity of 0.1 mm.

What will be amplitude of vibration of the motor when it runs at 1460 rpm? Neglect

damping, the weight of the beam and the deflection of the motor shaft. (20) (May 06)

Unit - 3

17. Find by Holzer’s method the natural frequency of the torsional system shown in figure 9

when the right end is fixed.

(20) (May 11)

Figure 9

MECHANICAL ENGINEERING DEPARTMENT, OITM

18. Find the first natural frequency of the right pendulum shown in figure 10 by the method of matrix

iteration using influence coefficients. Do at least four iterations.

(20) (May 11)

Figure 10

19. Estimate the lowest frequency of transverse vibrations for the system shown in figure 11 by

Stodola’s method or Rayleigh method. (20) (June 08)

Figure 11

20. (a) Explain Dunkerley’s method. (8)

(b) Using Dunkerley’s method, estimate the fundamental natural frequency of a simply

supported beam carrying three identical equally spaced masses. (12) (June 08)

21. Determine the natural frequencies and corresponding mode shapes using the method of influence

numbers for a taut string having tension 'T' and three concentrated masses as shown in figure 12.

(20) (June 10)

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22. A four rotor system schematically represented in figure 4 has the physical quantities shown a long

with the figure. Find the amplitudes of vibration when the external torque acts on the first rotor, as

shown in the figure 13. Find also the maximum twist in each section of the shaft.

(20) (June 10)

Figure 13

23. Determine the natural frequency of the system shown in Figure 14 by influence coefficient method

:

m1= 100kg, m2 = 200kg, m3 = 300kg (15) (June 08)

k1 = 2 kN/m, k2 = 1.5 kN/m, k3 = 2 kN/m.

Figure 14

24. A free bar of uniform section and length l is compressed on the two sides so as to give a total

compression e. The compressive forces are released suddenly, simultaneously. Derive an

expression for the resultant free vibrations. (15) (June 08)

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Unit - 4

25. A cantilever consists of uniform bar of length L. At mid-point along the length a force P which

acts away from the fixed end is applied and released at time t = 0, suddenly. Find the ensuing

motion. (20) (May 11)

26. Write short notes on followings :- (20) (May 11)

(i) Whirling of shafts.

(ii) Phase plane method.

27. Determine the response of an under-damped spring system to a unit impulse using the Laplace

transform method. Assume zero initial conditions. (20) (June 10)

28. Derive the equation for the natural frequency of a uniform cord of length ‘l’ fixed at the two ends.

The cord is stretched to a tension T and its mass per unit length is ρ.

(20) (June 08)

29. Determine the frequency equation in a transverse vibration for a uniform beam of length L having

one end fixed and the other simply supported. (20) (May 12)

30. Write short notes on following:-

(a) Response of a second order undamped system to a pulse-input.

(b) Critical speeds of shafts.

(20) (May 12)

31. (a) What are the continuous systems? Give three examples. (5)

(b) Explain role of boundary conditions in continuous in continuous systems. (5)

(c) Derive expression for longitudinal vibration of rods. (10) (May 09)

32. Write short notes on any two of the following:-

(a) Whirling of the shafts.

(b) Vibration absorbers.

(c) Transverse Vibrations of strings. (20) (May 06)