7 Damping +IDoF.pptx Chapter seven degree of freedom damping

Chapter seven degree of freedom damping

1. L. N. Ojha Ph.D. Welcome Students DEPARTMENT Structural Engineering Chair M. Sc. (Structural Engineering) Structural Dynamics –I ( CEng 6505)

2. Aim, Outcome and Contents Structural Dynamics-I Damping Models Distributed Parameter Systems (Beams, Plates) Introduction Beams- Exact Method Thin Plates Beams-Rayleigh-Ritz Method

3. Structural Dynamics-I Damping Viscous Damping, Coulomb Damping, and Hysteretic damping Reduces the response, Greater the damping, the greater the Reduction. Energy dissipated (into heat or radiated away) The loss of energy from the oscillatory system results in the decay of amplitude of free vibration. In steady-state forced vibration, the loss of energy is balanced by the energy which is supplied by the excitation.

4. Structural Dynamics-I Energy dissipated mechanism may emanate from Friction at supports and joints Hysteresis in material Elastic Wave Propagation Air resistance, fluid resistance Cracks in concrete Exact mathematical description is quite complicated MDOF-Damping Classical (Orthogonal, Proportional, or Modal) Damping Non-classical Damping

5. Structural Dynamics-I Damping in Structures

6. MDOF-Damping Structural Dynamics-I Sometime damping can be ignored, for example, if the force acts for relatively short time, a small amount of damping will not significantly affect the response Also, damping plays an insignificant role in steady state response of periodically forced system if the forcing frequency is not near one of the system responses.

7. MDOF-Damping Structural Dynamics-I For certain problems, it is better to solve directly the coupled differential equations instead of using the modal transformation. The damping matrix [c] is then needed.

8. MDOF-Damping Structural Dynamics-I

9. MDOF-Damping Structural Dynamics-I

10. MDOF-Damping Structural Dynamics-I

11. Structural Dynamics-I Mathematical model : Series of mass and stiffness proportional dampers. Since the mass matrix and the stiffness matrix are diagonalised by the mode shapes, so then would be any linear combination of the two. If the dampers (as shown) are given arbitrary (nonproportional) values, the damping matrix will not be uncoupled. A complex (imaginary number) eigenvalue problem would then be required. MDOF-Damping

12. MDOF-Damping Structural Dynamics-I

13. Structural Dynamics-I MDOF-Damping

14. Structural Dynamics-I MDOF-Damping

15. Structural Dynamics-I MDoF Damped Structures Modal Analysis-Procedure

16. Structural Dynamics-I MDOF-Damping Caughy Damping

17. Structural Dynamics-I Caughy Damping MDOF-Damping

18. Structural Dynamics-I MDOF-Damping

19. Structural Dynamics-I MDOF-Damping

20. Structural Dynamics-I Damping Models A dashpot is commonly used to represent the mechanism of structural damping. The dashpot has been tacitly combined with the spring in parallel. However, a number of combination dashpot and spring for dynamic systems have been proposed. The following s are the important

21. Structural Dynamics-I Damping Models The combination of dashpot and spring in parallel is called Voigt model, which is sometime is referred to as Kelvin model. Let the spring constant, damping coefficient . force applied to the system. and resulting deformation be denoted by respectively. S ince equal deformations arise in the Spring and the dashpot for a parallel model, the force are

22. Structural Dynamics-I Damping Models

23. Structural Dynamics-I Damping Models

24. Structural Dynamics-I Damping Models

25. Structural Dynamics-I Damping Models

26. Structural Dynamics-I Damping Models

27. Structural Dynamics-I Damping Models

28. Structural Dynamics-I Damping Models

29. Structural Dynamics-I Damping Models

30. Structural Dynamics-I Damping Models Although this model is apparently the same as Voigt model the damping coefficient c is assumed to be inversely proportional to the frequency w. Therefore, the expansion of equations follows Voigt model with the substitution of

31. Structural Dynamics-I Damping Models

32. Structural Dynamics-I Damping Models

33. Structural Dynamics-I Measurement of Damping

34. Structural Dynamics-I Measurement of Damping

35. Example MDoF -Rayleigh Damping CEng 6505

36. Example MDoF -Rayleigh Damping CEng 6505

37. Example MDoF -Rayleigh Damping CEng 6505

38. MDoF - Rayleigh Damping CEng 6505 DAMPING RATIO IN THE First AND Third MODES WILL BE 5 PERCENT OF CRITICAL.

39. Structural Dynamics-I Continuous, or distributed parameter systems are the systems in which inertia is continuously distributed throughout the system. A continuous system's dependent kinematic properties are functions of spatial variables, as well as time. Vibrations of continuous systems are governed by partial differential equations. Distributed Parameter (DP) Structures

40. Lumped Versus DP Structures Structural Dynamics-I p v CONTINUM MODEL

41. Structural Dynamics-I A Continuum Model Extends in all direction Has infinite particles, with continuous variation of material properties, deformation characteristics and stress state A Discrete Model is of finite size and is made up of an assemblage of substructures, components and members Distributed Parameter Systems Infinite Degree of Freedom Structure Lumped Versus DP Structures

42. Structural Dynamics-I Lumped Versus DP Structures

43. Equation of Motion Structural Dynamics-I EoM can be derived by Equilibrium of a differential element (convenient for most of the structures) Hamilton’s principle Lagrange's method Energy Methods For complex structures

44. Structural Dynamics-I Beams The beams have distributed mass and as such they have infinite degrees of freedom if their distributed mass is considered. Two Approaches: Exact Method -Analytical Method- Solving PDE Approximate Method - Emprical Method, Rayleigh’s Method, and Rayleigh Ritz Method

45. Structural Dynamics-I WAVE EQUATION : Free vibrations of certain one-dimensional systems are governed by the wave equation Beams

46. Structural Dynamics-I Beams

47. Structural Dynamics-I Exact Method

48. Structural Dynamics-I Exact Method

49. Structural Dynamics-I Free Vibration

50. Structural Dynamics-I Free Vibration

51. Structural Dynamics-I Example

52. Structural Dynamics-I Example

53. Structural Dynamics-I Example

54. Structural Dynamics-I Example

55. Structural Dynamics-I Plates Moment-Shear Force Resultants

56. Structural Dynamics-I Vibration of Plates Equation of Motion

57. Structural Dynamics-I Vibration of Plates BOUNDARY CONDITIONS

58. Structural Dynamics-I Vibration of Plates BOUNDARY CONDITIONS

59. Structural Dynamics-I Vibration of Plates BOUNDARY CONDITIONS-Free Edge There are three boundary conditions, whereas the equation of motion requires only two: Kirchhoff showed that the conditions on the shear force and the twisting moment are not independent and can be combined into only one boundary condition.

60. Structural Dynamics-I Vibration of Plates BOUNDARY CONDITIONS-Free Edge

61. Structural Dynamics-I Vibration of Plates BOUNDARY CONDITIONS-Free Edge

62. Structural Dynamics-I Vibration of Plates BOUNDARY CONDITIONS-Free Edge

63. Structural Dynamics-I Vibration of Plates FREE VIBRATION OF RECTANGULAR PLATES

64. Structural Dynamics-I Vibration of Plates FREE VIBRATION OF RECTANGULAR PLATES

65. Structural Dynamics-I Vibration of Plates FREE VIBRATION OF RECTANGULAR PLATES

66. Structural Dynamics-I Vibration of Plates Solution for a Simply Supported Plate

67. Structural Dynamics-I Vibration of Plates Solution for a Simply Supported Plate

68. Structural Dynamics-I Vibration of Plates Solution for a Simply Supported Plate

69. Structural Dynamics-I Vibration of Plates Solution for a Simply Supported Plate

70. Structural Dynamics-I Rayleigh’s Method

71. Structural Dynamics-I Rayleigh’s Method

72. Structural Dynamics-I Rayleigh-Ritz Method

73. Structural Dynamics-I Rayleigh-Ritz Method

74. Example- IDoF CEng 6505

75. CEng 6505 Example- IDoF

76. CEng 6505 Example- IDoF

77. CEng 6505 Example- IDoF

78. CEng 6505 Example-Rayleigh Method

79. Example-Rayleigh Method CEng 6505

80. Example-Rayleigh Method CEng 6505

81. Example-Rayleigh Method CEng 6505

82. Example-Rayleigh Method CEng 6505

83. Example-Rayleigh Method CEng 6505

84. Example-Rayleigh Method CEng 6505

85. Example-Rayleigh Method CEng 6505

86. Example-Rayleigh Method CEng 6505

87. Example-Rayleigh Method CEng 6505

88. Example-Rayleigh Method CEng 6505

89. Example-Rayleigh Method CEng 6505

90. Example-Rayleigh Method CEng 6505

91. Example-Rayleigh Method CEng 6505

92. Structural Dynamics-I Summary

93. Example-Rayleigh Method CEng 6505

94. Example-Rayleigh Method CEng 6505

95. Example-Rayleigh Method CEng 6505

96. Example-Rayleigh Method CEng 6505

97. Structural Dynamics-I Example

98. Structural Dynamics-I Summary

99. Structural Dynamics-I Example

100. Structural Dynamics-I Example

101. Structural Dynamics-I Example

102. Structural Dynamics-I Example

103. Structural Dynamics-I PDE for LONGITUDINAL FREE VIBRATION of BARS Example

104. Structural Dynamics-I Example

105. Structural Dynamics-I Example

106. Structural Dynamics-I Example

107. Structural Dynamics-I Example

108. Structural Dynamics-I Example

109. Structural Dynamics-I Example

110. Structural Dynamics-I Schedule S. No. Topic Date Day Timings Room 1 Last Lecture + Test 5 27 Jan Monday 9:00 PM ECR 2 Final Exam (CLOSE BOOK) Feb 10 Monday 9:00-12:00 AM ECR

111. Thank you