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UNIT III FREE VIBRATION Basic features of vibratory systems - Basic elements and lumping of parameters - Degrees of freedom - Single degree of freedom - Free vibration - Equations of motion - natural frequency - Types of Damping - Damped free vibration Whirling of shafts and critical speed - Torsional systems; Natural frequency of two and three rotor systems. Natural frequency of free longitudinal vibrations (undamped):

1. A mass of 10kg is connected to a free end of a spring of stiffness 6 N/mm. Another end of the spring is fixed. Calculate the natural frequency of vibration of the system.

2. A shaft of 50 mm diameter and 350 mm long carries a disc of mass 110 kg at its free end. Calculate the natural frequency of longitudinal vibration of the shaft. The Youngs modulus for the shaft material is 200 GN/m2.

3. In a spring-mass vibrating system, the natural frequency of vibration is 3.56Hz. When the amount of suspended mass is increased by 5kg, the natural frequency is lowered to 2.9Hz. Determine the original unknown mass and the spring constant.

Longitudinal vibration ( Effect of Inertia):

4. Obtain the natural frequency of vibration for the system of the combined spring shown in Fig. Take k1 = k3 = 2000 N/m;k2 = 1500 N/m and m = 10 kg.

5. Determine the equivalent spring stiffness and natural frequency of the 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. Take the stiffness of the two springs are 7 N/mm and 12 N/mm respectively and the mass suspended is 15 kg.

Frequency of free damped vibration (Viscous damping):

6. A machine weight 18 kg and is supported on springs and dashpot. The total stiffness of the springs is 12000 N/m and damping ratio is 0.2 N/mm/s. The system is initially at rest and a velocity of 120 mm/s is imparted to the mass. Determine (i) the displacement and velocity of mass as a function of time (ii) the displacement and velocity after 0.4 sec.

7. A vibrating system consists of a mass of 8 kg, spring of stiffness 5.6 N/mm and a dashpot of damping coefficient of 40 N/m/s. Find (a) the critical damping coefficient (b) the damping factor (c) the natural frequency of damped vibration (d) the logarithmic decrement (e) the ratio of two consecutive amplitudes (f) the number of cycles after which the original amplitude is reduced to 20 % .

8. In a single degree vibrating system, the suspended mass of 3.75 kg makes 12 oscillations in 7 seconds when distributed from its equilibrium position. The amplitude decreases to 0.33 of the initial value after 4 oscillations. Determine (i) the stiffness of the spring (ii) the logarithmic decrement (iii) the damping factor and (iv) damping coefficient.

9. A mass of 70 kg is suspended from a spring of stiffness 80 N/mm. A dashpot is fitted to the system and it is found that the amplitude of vibration diminishes to 10% of its original value after 5 complete oscillations. Find (i) Value of damping coefficient, (ii) Frequency of vibration and compare it with frequency of free vibration. (iii) Periodic time of damped vibrations.

10. A guitar vibrates with a frequency of 5Hz when there is no damping. When the damping is provided, the frequency of damped vibration was found to be 4Hz. Find (i) damping factor, (ii) logarithmic decrement.

11. A machine on springs and fitted with a dashpot, has a mass of 90 kg. There are three springs each of stiffness 13 N/mm. The amplitude of vibrations reduces from 42.4 mm to 8.4 mm in two complete oscillations. Assuming that the damping force varies as the velocity, determine: (i) The damping coefficient (ii) The ratio of frequencies of damped and undamped vibrations, and (iii) The periodic time of damped vibrations.

12. 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 400 Kg and initial velocity of recoil is 20 m/s. The barrel recoils 1 m. Determine spring stiffness and critical damping coefficient of dashpot.

Transverse vibrations:

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13. A shaft of 100 mm diameter and 1metre long is fixed at one end and the other end carries a flywheel of mass 1 tonne. Taking Youngs modulus for the shaft material as 200 GN/m2, find the natural frequency of transverse and longitudinal vibrations.

14. A steel bar 25 mm wide, 45 mm deep and 900 mm long is fixed at one end and the other carries a load of mass 600 kg. Find the natural frequency of transverse vibrations. Take the Youngs modulus for the shaft material as 210 GN/m2. If an additional mass of 200kg is distributed uniformly over the length of the shaft, what will be the frequency of vibration?

Dunkerleys method

15. A shaft 180 mm diameter is supported in two bearings 2.5 m apart. It carries three discs of mass 250 kg, 500 kg and 200 kg at 0.6 m, 1.5 m and 2 m from the left end. Determine the natural frequency of transverse vibrations. Take modulus of Elasticity for the shaft material as 211 GN/m2.

16. A shaft 1.5 m long is supported in flexible bearings at the ends and 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 0.4 m from the centre towards right. The shaft is hollow of external diameter 75 mm and inner diameter 37.5 mm. The density of the shaft material is 8000 kg/m3. The Youngs modulus for the shaft material is 200 GN/m2. Find the frequency of transverse vibrations.

17. A shaft of diameter 10 mm carries at its centre a mass of 12 kg. It is supported by two short bearings the distance of which is 400 mm. Find the whirling speed (i) neglecting the mass of the shaft, (ii) taking the mass of the shaft also into consideration. The density of the shaft material is 7500 kg/m3. Take E = 200 GN/m2.

18. A hollow shaft 1.8 m long is supported in flexible bearings at the ends. It carries two wheels each of 60 kg mass, one at the centre of the shaft and the other at 450 mm from the centre. The external and internal diameters of the shaft are 80 mm and 50 mm respectively. Determine the whirling speed of the shaft. The density of the shaft material is 7500 kg/m3 and the modulus of elasticity 210 GN/m2.

19. A shaft 12.5 mm diameter rotates in long bearings and a disc of mass 16 kg is secured to a shaft at the middle of its length. The span of the shaft between the bearings is 0.5 m. The mass centre of the disc is 0.5 mm from the axis of the shaft. Neglecting the mass of the shaft and taking E = 200 GN/m2, find (i) Critical speed of rotation in r.p.m. (ii) The range of speed over which the stress in the shaft due to bending stress will not exceed 120 MN/m2.

Torsional vibration:

20. A shaft of 100 mm diameter and 1 m long has one of its end fixed and the other end carries a disc of mass 500 kg at a radius of gyration of 450 mm. The modulus of rigidity for the shaft material is 80 GN/m2. Determine the frequency of torsional vibration.

(a) Single rotor system:

21. Determine the torsional frequency of a disc mass 96 kg which is on the vertical rod of diameter 40 mm. The rod is 1.8 m in length and the disc is fixed at 1 m from top end. Both ends of the rods are fixed. The radius of gyration of disc is 0.4 m. Take G = 85 GPa.

22. A stepped shaft is 0.05 m in diameter for the first 0.6 m length, 0.08 m diameter for the next 1.8 m and 0.03 m diameter for the remaining 0.25 m length. While the 0.05 m diameter end is fixed, the 0.03 m diameter end of the shaft carries a rotor of mass moment of inertia 14.7 kg-m2. If the modulus of rigidity of the shaft material is 0.83 x 1011 N/m2. Find the natural frequency of torsional oscillations neglecting the inertia effect of the shaft.

(b) Two rotor system:

23. A steel shaft 1.5 m long is 95 mm in diameter for first 0.6 m of its length, 60 mm in diameter for the next 0.5 m of the length and 50 mm in 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 95 mm diameter end and the second having a mass of a 700 kg and 0.55 m radius of gyration located at the other end. Determine the location of the node and the natural frequency of the free torsional vibration of the system. The modulus of rigidity of shaft material may be taken as 80 GN/m2.

(c) Three rotor system:

24. Three rotors A, B and C having moment of inertia of 2000, 6000 and 3500 kg.m2 respectively are carried on a uniform shaft of 0.35 m diameter. The length of the shaft between the rotors A and B is 6 m and between B and C is 32 m. Find the natural frequency of torsional vibrations. The modules of rigidity for the shaft material is 80 GN/m2.

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25. A motor generator set consists of two armatures P and R as shown in Fig. with a flywheel between them at Q. The modulus of rigidity of the material of the shaft is 84 GN/m2. The system can vibrate with one node at 106.5 mm from P, the flywheel Q being at antinode. Using the data of rotors given below. Find (i) The position of the other node. (ii) The natural frequency of free torsional vibrations for the given position of the nodes. (iii) The radius of gyration of the rotor R. Data of rotors. Rotor P Q R Mass (kg) 450 540 360 Radius of gyration (mm) 250 300, xxxx respectively.

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