singledegreeoffreedomsystem-freevibrationpart-iandii-200824161124.pptx

single dof

1. UNIT-I Single Degree of Freedom Systems – Free Vibration 1

2. Introduction to Vibration When any elastic body such as spring, shaft, or beam is displaced from the equilibrium position by the application of external force and released, it commences cyclic motion. Such cyclic motion of a body or a system , due to elastic deformation under the action of external forces, is known as vibration. 2

3. Phenomenon of Vibration 3

4. Causes of Vibration Unbalance forces and couples External excitation forces Dry friction between two mating surfaces Wind load may cause vibration in certain systems such as telephone lines, electric lines, etc Earthquakes 4

5. Disadvantages of Vibration Excessive Stresses Loosening of Assembled parts Failure of machine parts Undesirable noise 5

6. Advantages of Vibration All musical instrument Vibrating screen, Shakers and conveyors Stress relieving equipment 6

7. Terminology and Basic Concept Simple Harmonic Motion 7 Let x- displacement of point from mean position after time ‘t’ X- maximum displacement of point from mean position Type equation here. Displacement of point x=X =X ………………………………………..(1)  

8. Velocity of point = =   8 Acceleration of point =   = -   = -   A motion , whose acceleration is proportional to displacement from mean position and is directed towards the mean position, is known as SHM ………………… fundamental equation of SHM  

9. Time Period( ) is the timerequired to complete one cycle (2 . Mathematically, = Frequency (f) The number of cycles per unit time is known as frequency. It is reciprocal of time period. f= Amplitude (X) It is the maximum displacement of a vibrating body from its mean position.   9

10. Stiffness of spring (K) It is the force required to produce unit displacement in the direction of applied force. K= , N/m Where, K= Stiffness of spring , N/m F= force applied on spring, N = deflection of spring, m   10 Degree of Freedom ( D.O.F) The minimum number of independent co-ordinates required to specify the motion or configuration of a system at any instant is known as DOF . ExAMPLE

11. Damping Is the resistance to the motion of a vibrating body , which causes a vibrating body to come to rest position. 11 Damping coefficient (C) Is the damping force or resisting force developed per unit velocity. Mathematically c= , N-sec/m Where, F= Force applied on damper v= Velocity of viscous fluid  

12. Resonance When the frequency of external excitation force acting on a body is equal to the frequency of a vibrating body, the body starts vibrating with excessively large amplitude. Such state is known as resonance. 12

13. Elements of Vibratory System 13

14. Equivalent Springs 14

15. Spring in Series 15 Deflection of Equivalent Spring=Deflection of Spring 1 + Deflection of Spring 2 ……………….(a)   Force on Equivalent Spring=Force on Spring 1 + Force on Spring 2 mg   The system of two springs in series is to be replaced by an equivalent spring having stiffness   Deflection of Equivalent Spring     …….(b)   But, mg   Substituting Equation ( C ) in equation (b), we get    

16. Spring in Parallel 16 Deflection of Equivalent Spring=Deflection of Spring 1 = Deflection of Spring 2 ……………….(d)   Force on Equivalent Spring=Force on Spring 1 + Force on Spring 2 mg   The system of two springs in series is to be replaced by an equivalent spring having stiffness   Deflection of Equivalent Spring     But,   Substituting Equation ( d ) in equation (f), we get Ke ……(f)   Ke   = +  

17. Equivalent Damper 17

18. Dampers in Series Dampers in Parallel 18   = +  

19. Introduction to Modeling Physical Modeling Geometric Modeling Mathematical Modeling Combination of Geometric and Mathematical Modeling 19

20. Practical Example of Mathematical Modeling Motor Bike 20

21. Practical Example of Mathematical Modeling Bicycle 21

22. Types of Vibration 22

23. Examples of Vibration 23 Longitudinal Transverse Torsional Free Vibration Forced Vibration Damped Vibration

24. Examples of Vibration 24 Linear Vibration Non- Linear Vibration Deterministic Vibration Random Vibration

25. Determination of Natural Frequency Equilibrium Method Energy Method Rayleighs Method 25

26. Equilibrium Method( D’Alembert Principal) 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. 26

27. Equilibrium Method( D’Alembert Principal) 27

28. Equilibrium Method( D’Alembert Principal) The gravitational force must be equal to zero. mg= kδ -------  (1) The force acting on the mass are : inertia force : mẍ (upwards) spring force : K( x+δ ) (upwards) gravitational force : mg 28

29. According to D’Alembert’s principle , ( + x=0 ………………………..(2) We know that the fundamental equation of SHM ………………………(3) Comparing equation of (2) and (3) rad/s………………….................(4)   29

30. The natural frequency f of vibration is , or Hz……………(5) Or from equation (1) Substitute in equation (5) Hz and time period = s   30

31. Energy Method According to law of conservation of energy, energy can neither be created nor be destroyed, but it can be converted from one form to another form. According to law of conservation of energy, Total Energy= Constant K.E+P.E= Constant ………………………………………(a) Considering Spring mass system K.E= P.E is in the form of strain energy stored in the spring. The strain Energy is given by the area under the force vs deflection diagram.   31

32. PE= Strain Energy=Area under force-deflection diagram PE= PE= (Since we know that ,Spring force (F)= Kx ) = Substitute in equation (a) + )=0 m( )+ ( )=0 +( Kx ) + x=0   32 x F Strain Energy Spring Force Deflection

33. We know that the fundamental equation of SHM Comparing above equation with equation of SHM rad/s The natural frequency f of vibration is , or Hz   33

34. Rayleigh’s Method This is the extension of energy method , which is developed by Lord Rayleigh According to principal of conservation of energy TE= Constant ………………………………………………..(a) Let body having mass m is moving with SHM , therefore the displacement of the body is given by x = Displacement of a body from the mean position after time t sec. X= Maximum displacement of a body from mean positon = Circular natural frequency = = X (at t=0)   34

35. At mean position , the maximum KE is = ) or = And PE is , PE = At extreme position ( at x=X) , the max PE is = Substituting in equation (a) = rad/s The natural frequency f of vibration is , or Hz   Prof. S. S. Patil (ME DESIGN) TSSM’s PVPIT Bavdhan Pune 35

36. Undamped Free Transverse Vibration 36 Consider a cantilever beam of negligible mass carrying a concentrated mass “m” at the free end, as shown in fig,

37. The gravitational force must be equal to zero. mg= kδ -------  (1) The force acting on the mass are : inertia force : mẍ (upwards) spring force : K( x+δ ) (upwards) gravitational force : mg 37

38. According to D’Alembert’s principle , …..( + x=0 ………………………..(2) We know that the fundamental equation of SHM ………………………(3) Comparing equation of (2) and (3) rad/s………………….................(4)   38

39. The natural frequency f of vibration is , or Hz……………(5) Or from equation (1) Substitute in equation (5) Hz and time period = s   39

40. Torsional Stiffness Torsional Stiffness of Shaft ( ) is defined as the torque required to produce unit angular deflection in the direction of applied torque. Mathematically, = G=Modulus of rigidity N/ J= Polar Moment of Inertia ( l= Length of Shaft D = Diameter of the shaft   40

41. Parameters for linear & torsional vibration 41

42. Undamped free torsional vibration 42 Consider a disc having mass moment of inertia ‘I’ suspended on a shaft with negligible mass, as shown in fig, For angular displacement of disc ‘θ’ in clockwise direction, the torques acting on the disc are : Inertia torque Restoring troque

43. Therefor according to D’Alembert’s priciple , ∑ (Inertia Torque + External Torque)= + + ……………………(1) We know that the fundamental equation of SHM ……………….(2) Comparing eqn (1) and (2) = Hz.   43

44. Problem 1) 44 Find W such that the system has natural frequency 10 cps. Solution : If is the effective spring stiffness of the top three springs in series, then Or =0.667 kg/cm. If is the effective spring stiffness of the lower two springs in parallel, then Or =1.0 kg/cm Now and are two springs in parallel, therefore effective stiffness + Therefore  

45. 45 Problem no. 2) Find natural frequency for the mathematical model as shown in figure K= N/m , M=20 kg.   Solution : K= N/m M=20 kg. We know that when two springs are in parallel and when in series From Fig (g) Hz Ans  

46. 46 Problem no. 3 ) A mass of 1 kg is suspended by a spring passing over the pulley , as shown in fig. The system is supported horizontally by a spring of stiffness 1 KN/m. Determine the natural frequency of vibration of a system . Using following data Mass of pulley M= 10 kg Radius of pulley R= 50 mm Distance of spring from centre of pulley r= 35 mm Solution: The mass and spring K are not attached on one cord or string. Therefore , consider be the displacement of mass in downward direction & be the deflection of spring. The pulley will rotate through an angle Angular displacement of pulley = Linear displacement of mass = = Linear velocity of mass = = Linear acceleration of mass = = Deflection of spring = = Energy Method K.E of the mass = = K.E of the pulley = PE of the spring  

47. 47 Total K.E Total P.E According to Energy Method   Comparing this with SHM Equation  

48.   48

49. Problem no. 4 Determine the natural frequency of oscillation of the simple pendulum shown in fig. Solution: Let, , kg- Equilibrium Method Angular motion about O: ……………………..(a)   49

50. Natural circular frequency comparing equation (a) with the equation of SHM Or , rad/s Natural Frequency : ………………………………. Ans.   50

51. Damped Free Vibration Damping Vibration: In a vibratory system , if an external resistance is provided so as to reduce the amplitude of vibrations, the vibration is known as damped vibration Damping : The external resistance which is provided to reduce the amplitude of vibrations is known as damping. Damper : The damper is a unit which absorbs the energy of vibratory system, thereby reducing the amplitude of vibration. Important Parameter : Frequency of damped vibration Rate of decay of amplitude 51

52. Types of damping 52

53. Viscous Damping When the system is allowed to vibrate in a viscous medium , the damping is called as viscous damping. Viscosity is the property of a fluid by virtue of which it offers resistance to the motion of one layer over the adjacent one. 53 t Moving plate u   Fixed Plate It can be explained from fig. where two plates are seperateed by fluid film of thickness (t) . The upper plate is allowed to move parallel to the fixed plate with a velocity ( .The next force (F) required for maintaining the velocity ( is  

54. The force can also be written as Where C= Viscous Damping Coefficient Fluid Dashpot   54

55. Eddy Current Damping Prof. S. S. Patil (ME DESIGN) TSSM’s PVPIT Bavdhan Pune 55

56. Coulomb Damping or Dry Friction Damping 56  

57. Material or Solid or Structural or Hysteresis Damping 57