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Free vibration

Unit 3<br>DYNAMICS OF MACHINES

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Free vibration

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  1. Free vibrations

  2. Unit Outcome • At the end of this unit, the students should be able to compute the frequency of free vibrations. SYLLABUS Basic features of vibratory systems – Degrees of freedom – single degree of freedom – Freevibration– Equations of motion – Natural frequency – Types of Damping – Damped vibration– Torsional vibration of shaft – Critical speeds of shafts – Torsional vibration – Two and threerotor torsional systems.

  3. Basic features of vibratory system • Vibrations: When elastic bodies such as a spring, a beam and shaft are displaced from the equilibrium position by the application of external forces, and then released, they execute a vibratory motion.

  4. Basic features of vibratory system • Period of vibration or time period: It is the time interval after which the motion is repeated itself. The period of vibration is usually expressed in seconds. • Cycle: It is the motion completed during one time period. • Frequency: It is the number of cycles described in one second. In S.I. units, the frequency is expressed in hertz (briefly written as Hz) which is equal to one cycle per second.

  5. Basic features of vibratory system • Causes of Vibrations: • Unbalanced forces: Produced within the machine due to wear and tear. • External excitations: Can be periodic or random • Resonance: • When the frequency of the external or applied force is equal to the natural frequency resonance occurs. Vertical Shaking Accident and Cause Investigation of 39-story Office Building

  6. Basic features of vibratory system Components of vibratory system: • Spring/Restoring element: • Its denoted by k or s; • SI unit – N/m • Dashpot/Damping component • Its denoted by c; • SI unit – N/m/s • Mass/Inertia component • Its denoted by m; • SI unit – kg

  7. Degrees of Freedom 1 DOF 2 DOF • The minimum number of independent coordinates required to determine completely the position of all parts of a system at any instant of time defines the degree of freedom of the system.

  8. Types of Vibrations • Free or Natural Vibrations: When no external force acts on the body, after giving it an initial displacement, then the body is said to be under free or natural vibrations. The frequency of the free vibrations is called free or natural frequency. • Forced vibrations When the body vibrates under the influence of external force, the the body is said to be under forced vibrations.The vibrations have the same frequency as the applied force • Damped vibrations: When there is a reduction in amplitude over every cycle of vibration, due to frictional resistance, the motion is said to be damped vibration.

  9. Types of Vibrations Longitudinal Transverse Torsional • Longitudinal Vibrations: Parallel to axis of shaft • Transverse Vibrations: Approx. Perpendicular to axis of shaft • Torsional Vibrations: Moves in circles about axis of shaft

  10. Spring - Stiffness d = Static deflection of spring in meters.

  11. Natural frequency of free vibrations Equilibrium method – Longitud. Vibrations • Restoring force • W – (sd +sx) • - sx • Accelerating force • m(d2x/dt2) • Equating both • m(d2x/dt2) + sx = 0 • d2x/dt2 + (s/m) x = 0 • SHM equation • d2x/dt2 + w2 x = 0 • Angular Velocity: W = mg = sd SF = 0

  12. Natural frequency of free vibrations Equilibrium method • Time period • tp = 2p/w • Natural frequency • A • B • Deflection • s= W/A = Ee = E x (d/l) • d = Wl/ AE

  13. Energy method

  14. Rayleigh’s method In this method, the maximum kinetic energy at the mean position is equal to the maximum potential energy (or strain energy) at the extreme position. Assuming the motion executed by the vibration to be simple harmonic, then

  15. Natural frequency of transverse vibrations Transverse vibrations <= Same as Longitudinal Vibrations

  16. Formula • Natural frequency of longitudinal and transverse vibrations: Where, fn = Natural frequency. (Hz) tp = Time period (s) s = Stiffness (N/m) m = Mass. (kg) g = acceleration due to gravity (m/s2) d = Static deflection. (m)

  17. A cantilever shaft 50 mm diameter and 300 mm long has a disc of mass 100 kg at its free end. The Young's modulus for the shaft material is 200 GN/m 2 . Determine the frequency of longitudinal and transverse vibrations of the shaft

  18. Formula 2. Static deflection in beams, Where, fn = Natural frequency. (Hz) tp = Time period (s) s = Stiffness (N/m) m = Mass. (kg) g = acceleration due to gravity (m/s2) d = Static deflection. (m)

  19. Formula 2. Static deflection in beams, Where, fn = Natural frequency. (Hz) tp = Time period (s) s = Stiffness (N/m) m = Mass. (kg) g = acceleration due to gravity (m/s2) d = Static deflection. (m)

  20. Formula 2. Static deflection in beams,

  21. Formula 2. Static deflection in beams,

  22. A shaft of length 0.75 m, supported freely at the ends, is carrying a body of mass 90 kg at 0.25 m from one end. Find the natural frequency of transverse vibration. AssumeE = 200 GN/m2 and shaft diameter = 50mm. Given l = 0.75 m ; m = 90 kg ; a = AC = 0.25 m ; E = 200 GN/m2 = 200 × 109N/m2 d = 50 mm = 0.05 m

  23. Moment of inertia of shaft • Static deflection • Natural frequency

  24. A flywheel is mounted on a vertical shaft as shown in Fig. 23.8. The both ends of the shaft are fixed and its diameter is 50 mm. The flywheel has a mass of 500 kg. Find the natural frequencies of longitudinal and transverse vibrations. Take E = 200 GN/m2 .

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