Undamped Free Vibration

This Is for G.T.U. Students.
This will helps to understand undamped free vibrations and for Active Learning Assignment.

1. INTRODUCTION If the external force is removed after giving the initial displacement to the system, such vibrations are known as free vibrations, if there is no external resistance(damping) to the vibrations then such vibrations are known as Undamped free vibrations. When frequency of external exciting force is equal to natural frequency of vibrating body, the amplitude of vibration becomes excessively large. Such state is known as Resonance . Resonance is dangerous and it may lead to the failure of part. Free vibration means that no time varying external forces act on the system.  The pendulum will continue to oscillate with the same time period and amplitude for any length of time.

2. The natural frequency of any body or a system depend upon the geometrical parameters and mass property of the body. It is independent of the forces acting on the body or a system. There are various method to obtain the equation of a vibrating systems, which can be used to find the natural frequency of the given vibratory system. Equilibrium Method(D’Alemberts’s Principle) Energy Method Reyleigh’s Method INTRODUCTION

3. GENERAL EQUATION

4. The simplest mechanical vibration equation occurs when γ = 0, F(t) = 0. This is the undamped free vibration. The motion equation is mu″ + ku = 0. The characteristic equation is mr ² + k = 0. Its solutions are r = + √(K/m) i or - √(K/m) i GENERAL EQUATION

5. The general solution is then U(t) = C1 cos ω˳ t + C2 sin ω˳ t Where ω˳ = √(K/m) is known as natural frequency of system. frequency at which the system tends to oscillate in the absence of any damping. A motion of this type is called simple harmonic motion. It is a perpetual, sinusoidal, motion. GENERAL EQUATION

6. Example of Simple harmonic Motion

7. NATURAL FREQUENCY OF UMDAMPE FREE VIBRATION BY EQUILIBRIUM METHOD A body or structure which is not in static equilibrium due to acceleration it possesses can be brought to static equilibrium by introducing the inertia force on it. The inertia force is equal to the mass times the acceleration direction is opposite to that of acceleration. The principle is used for developing the equation of motion for vibrating system which is further used to find the natural frequency of the vibrating system.  

8. NATURAL FREQUENCY OF UMDAMPED FREE VIBRATION BY EQUILIBRIUM METHOD

9. The gravitational force must be equal to zero. mg=k δ -------  (1) The force acting on the mass are : 1. inertia force : mẍ (upwards) 2. spring force : K(x+ δ ) (upwards) 3. gravitational force : mg NATURAL FREQUENCY OF UMDAMPED FREE VIBRATION BY EQUILIBRIUM METHOD

10. We know that the fundamental equation of SHM ẍ + ω² x = 0 -------  (3) Comparing equation 2 & 3 , ω² = K̲ rad/s -------(4) m According to D’Alembert’s principle , mẍ + K(x+ δ ) – mg = 0 mẍ + K δ + Kx – mg = 0 mẍ + Kx = 0 ẍ + K ̲ x = 0 ------  (2) m

11. The natural frequency f of vibration is , f = ω /2 ∏ or f = ½ ∏ √(K/m) Hz ----- (5) also from eq. (1), mg = K δ → K/m = g/ δ ------ (6) substituting eq (5) in eq (2) , we get f = ½∏ √(g/ δ ) H ------ (7) the time period t is , t = 1/f = 1/ (1/2∏)(√(K/m)) or t = 2∏ √(m/K) s. ------ (8)

12. NATURAL FREQUENCY OF UMDAMPED FREE VIBRATION BY ENERGY METHOD According to lao of conservation of energy , the energy can neither be created nor be destroyed, it can be converted from one form to another form. In free undamped vibrations, no energy is transferred to the system or from the system. Therefore the total mechanical energy i.e. the sum of kinetic energy and potential energy remains constant.

13. Kinetic energy is due to motion of the body or system. Potential energy consist of two parts Gravitational Potential Energy : Due to position of body or system with respect to equilibrium or mean position. Strain Energy : Due to elastic deformation of body or system NATURAL FREQUENCY OF UMDAMPED FREE VIBRATION BY ENERGY METHOD

14. At mean position, the kinetic energy is maximum and potential energy is zero; whereas at extreme positions. The kinetic energy is zero and potential energy is maximum. E = K.E + P.E = Constant _______(9) NATURAL FREQUENCY OF UMDAMPED FREE VIBRATION BY ENERGY METHOD

15. E = ½MV ² + ½Kx ² ______(10) E = ½M ẍ ² + ½Kx ² ______(11) NATURAL FREQUENCY OF UMDAMPED FREE VIBRATION BY ENERGY METHOD

16. ( dE / dt ) = ½M.2 ẍ. ẋ + ½K.2x . ẋ _______(12) 0 = ½.2 ẋ(M ẍ + K ẋ) _______(13) M ẍ + K ẋ = 0 _______(14) We know that the fundamental equation of SHM ẍ + ω² x = 0 _______(15) ẍ + (k/m) ẋ = 0 _______(16) Comparing equation 2 & 3 , ω n = √(K/M) _______(17) f = ½ ∏ √(K/m) Hz NATURAL FREQUENCY OF UMDAMPED FREE VIBRATION BY ENERGY METHOD

17. M ẍ + K ẋ = 0 _______(18) ẍ + (k/m) ẋ = 0 _______(19) We know that the fundamental equation of SHM ẍ + ω² x = 0 _______(20) By comparing above Equation with equation of natural frequency ω n = √(K/M) _______(21) NATURAL FREQUENCY OF UMDAMPED FREE VIBRATION BY ENERGY METHOD

18. f = ½ ∏ √(K/m) Hz _______(22) The (natural) period of the oscillation is given by T = 2 ∏ / ω n (seconds). NATURAL FREQUENCY OF UMDAMPED FREE VIBRATION BY ENERGY METHOD

19. This is extension of energy method. According to principle of conservation of energy, (Total Energy) mean position = (Total Energy) extreme position (K.E + P.E) 1 = (K.E + P.E) 2 (K.E) 1 + (P.E) 1 = (K.E) 2 + (P.E) 2 At Position 1 P.E is zero and K.E is Maximum. At position 2 K.E is zero and P.E is Maximum NATURAL FREQUENCY OF UMDAMPED FREE VIBRATION BY RAYLEIGH METHOD

20. But at mean position K.E is maximum and at extreme position P.E. is maximum. (K.E) MAX = (P.E) MAX Therefore, According to Lord Reyleigh’s, the maximum energy is at mean position is equal to maximum potential energy which is at extreme position. NATURAL FREQUENCY OF UMDAMPED FREE VIBRATION BY RAYLEIGH METHOD

21. (P.E)max = (K.E) max P.E = ½Kx ² ; x= Xsin ( ω n t) _______(23) (P.E)max = ½KX ² ; ω n t = 90° _______(24) NATURAL FREQUENCY OF UMDAMPED FREE VIBRATION BY RAYLEIGH METHOD

22. NATURAL FREQUENCY OF UMDAMPED FREE VIBRATION BY RAYLEIGH METHOD K.E = ½M ẋ ² ; ẋ = X ω n Cos( ω n t) _______(25) (K.E) max = ½M ω n ² X ² _______(26) (P.E) max = (K.E) max _______(27) ½KX ² = ½M ω n ² X ² _______(28) - By simulating the above equation we get next equation,

23. NATURAL FREQUENCY OF UMDAMPED FREE VIBRATION BY RAYLEIGH METHOD ω n = √(K/M) _______(29) f = ½ ∏ √(K/m) Hz The (natural) period of the oscillation is given by T = 2 ∏ / ω n (seconds).

24. Due to gravitation force ‘mg’, the cantilever beam is deflected by ‘ δ ’. At Equilibrium position mg = K δ . Let the system is subjected to one time external force due to which it will displaced by ‘x’ from equilibrium position. Undamped Free Transverse Vibration

25. Undamped Free Transverse Vibration

26. Forces acting on mass beyond mean position are, Inertia Force, m ẍ (upward) _________(30) Resisting Force, Kx (upward) According to D’amberte’s principle, Ʃ(Inertia Force + External Force) = 0 m ẍ + Kx = 0 ẍ + (K/M)x = 0 _____(31) Undamped Free Transverse Vibration

27. Comparing Eq. 31 with Eq. of S.H.M., ω n ² = (K/M) rad/s ω n = √(K/M) rad/s or f = ½ ∏ √(K/m) Hz From Eq. 30, (K/M) = (g/ δ ) Substituting above values, F n = (0.4985/ √ δ ) Hz Undamped Free Transverse Vibration

28. Consider a disc having mass moment of inertia ‘I’ suspended on shaft with negligible mass, as shown in fig. If the disc is given a angular displacement about a axis of shaft, it oscillates about that axis, such vibrations are known as Torsional vibrations. Undamped Free Transverse Vibration

29. Undamped Free Torsional Vibration

30. For angular displacement of disc ‘Ɵ’ in clockwise direction, the torques acting on the disc are: According to D’amberte’s principle, Ʃ(Inertia Force + External Force) = 0 I Ɵˊˊ + Kt. Ɵ = 0 Ɵˊˊ + (Kt/I).Ɵ = 0 _______(32) Undamped Free Transverse Vibration

31. The fundamental Eq. of S.H.M. Ɵˊˊ + ω n Ɵ = 0 _____(33) By Comparison of above Eq. 32 & 33 ω n = √( Kt /I) rad/s _____(34) f = ½ ∏ √( Kt /I) Hz ______(35) Undamped Free Torsional Vibration

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