Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers

Everything about Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers

1. Dept. of Engineering Enzo Ferrari – Centre InterMech MoRe Advanced Automotive Engineering Mechanical Vibrations Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers Prof. Farhad S. Samani Visiting Professor, Univrsity of Modena and Reggio Emilia, Modena, Italy Associate Professor, Shahid Bahonar University of Kerman, Kerman, Iran Prof. Francesco Pellicano , Prof . Antonio Zippo Dept. of Engineering Enzo Ferrari, Centre InterMech MoRe Univrsity of Modena and Reggio Emilia, Modena, Italy

2. Vibration Absorbers 12/14/2022 11:11 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 2

3. Vibration Absorbers 12/14/2022 11:15 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 3

4. Absorber Building 12/14/2022 11:16 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 4

5. Tacoma Narrows Bridge 12/14/2022 11:21 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 5 1940 U.S. state of Washington It was the world's third-longest suspension bridge by main span, behind the Golden Gate Bridge and the George Washington Bridge .

6. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 6

7. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 7

8. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 8 The model of the snap-through truss vibration absorber An example of nonlinear stiffness:

9. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 9 Lateral vibration of beam A beam in bending:

10. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 10 Equation of motion:            

11. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 11   Euler-Bernoulli       Uniform beam

12. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 12 Euler Bernoulli simply supported beam under moving load Considering a flexural beam subjected to a load moving with constant velocity: The partial differential equation of motion:  

13. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 13 Dirac delta function: I’m thinking of a single square wave pulse of area 1 unit (‘unity’). What will happen if I slowly decrease the width for a constant area ?

14. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 14 What will happen as the width tends to zero? We’re left with a spike of zero width and infinite height. This is called a delta function.

15. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 15 is a spike centred at      

16. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 16 The Dirac delta function is very useful in many areas of physics. It is not an ordinary function, in fact properly speaking it can only live inside an integral.   The product of the delta function with any function is zero except where   Formally, for any function    

17. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 17 Example:    

18. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 18 Simply supported beam under transient moving load with DVA:

19. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 19     1a y   1b   1c   1d   1e        

20. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 20   2a   2b   3

21. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 21     4a   4b     4c 4d            

22. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 22   5   6 Input energy, transmitted from moving load to beam Transferred energy from beam to DVA :Total integration time    

23. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 23 Validation:

24. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 24 Maximum deflection approach: Optimization of the linear dynamic damper

25. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 25 Energy approach: Optimization of the linear dynamic damper  

26. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 26 Maximum deflection approach: Optimization of nonlinear dynamic damper

27. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 27 Energy approach:  

28. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 28 Case Beam and dynamic damper condition Stiffness Viscous damping  [Ns/m] Dynamic damper location d [m] Maximum deflection [mm] Position of maximum deflection [m] Time of maximum deflection [s] 1 Undamped beam without dynamic damper ----- 1.6279 2.11 0.1398 0% 2 Damped beam ( ) without dynamic damper ----- 1.6042 2.11 0.1401 0% 3 Damped beam with linear dynamic damper, deflection optimization approach 1795N/m 0.1 2.2 1.5054 2.12 0.1407 3.4% 4 Damped beam with linear dynamic damper, energy optimization approach 900N/m 10.5 2.2 1.5306 2.12 0.1401 88.9% 5 Damped beam with nonlinear dynamic damper, deflection optimization approach 6.7×10 9 N/m 3 0.1 2.12 1.4852 2.12 0.1402 2.3% 6 Damped beam with nonlinear dynamic damper, energy optimization approach 0.3×10 9 N/m 3 11 2.12 1.5640 2.12 0.1398 87.4% Case Beam and dynamic damper condition Stiffness Viscous damping  [Ns/m] Dynamic damper location d [m] Maximum deflection [mm] Position of maximum deflection [m] Time of maximum deflection [s] 1 Undamped beam without dynamic damper ----- 1.6279 2.11 0.1398 0% 2 ----- 1.6042 2.11 0.1401 0% 3 Damped beam with linear dynamic damper, deflection optimization approach 1795N/m 0.1 2.2 1.5054 2.12 0.1407 3.4% 4 Damped beam with linear dynamic damper, energy optimization approach 900N/m 10.5 2.2 1.5306 2.12 0.1401 88.9% 5 Damped beam with nonlinear dynamic damper, deflection optimization approach 6.7×10 9 N/m 3 0.1 2.12 1.4852 2.12 0.1402 2.3% 6 Damped beam with nonlinear dynamic damper, energy optimization approach 0.3×10 9 N/m 3 11 2.12 1.5640 2.12 0.1398 87.4%  

29. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 29 Effect of the beam length: Beam length L [m] 1 4 10 Velocity for which the max deflection occurs [m/s] 86 21.5 8.6 Dynamic damper mass [kg] 0.3519 1.4076 3.519 Optimized stiffness recpect to deflection approach C opt [N/m 3 ] 1.7(10 15 ) 6.7(10 9 ) 1.7(10 6 ) Optimized maximum deflection [mm] 0.0235 1.4852 23.53 Optimized stiffness recpect to energy approach, C opt [N/m 3 ] 80(10 12 ) 0.30(10 9 ) 80(10 3 ) Optimized viscous damping, λ opt [Ns/m] 44 11 4.4 η 87.4% 87.4% 87.4% Results for optimization of nonlinear dynamic damper with various length:

30. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 30 Effect of moving load velocity: effect of dynamic dampers optimized for V =21.5 m/s: ’- - -‘ bare beam, ’ - ● - ● - ‘ Linear damper, ’  ‘ Nonlinear damper

31. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 31 Portion of input energy absorbed and dissipated by the dynamic damper versus velocity; ’ - - - ‘ Linear dynamic damper, ’  ‘ Nonlinear dynamic damper L =4m L =10m

32. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 32 Random Optimization, linear DVA:

33. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 33 Random Optimization, nonlinear DVA with cubic stiffness:

34. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 34 Higher order nonlinear monomial stiffness Performance of the monomial stiffness:   b    

35. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 35     Higher order nonlinear monomial stiffness  

36. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 36 Performance of the polynomial stiffness:  

37. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 37  

38. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 38 Performance of the piecewise linear stiffness: The schematic stiffness force versus stiffness deflection for piecewise linear dynamic damper  

39. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 39 Maximum deflection vs. gap Δ and stiffness k of the piecewise linear stiffness; the optimum is k =37000N/m and Δ =0.58mm (1.4735mm max deflection)

40. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 40 the optimum is k =37000N/m and Δ =0.58mm (1.4735mm max deflection)

41. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 41 Case Beam and dynamic damper condition Stiffness Maximum deflection [mm] Reduction Percentage 1 Bare beam 1.6042 ---- 2 Linear dynamic damper c 1 =1.79(10 3 ) N/m 1.5054 6.159% 3 Monomial cubic dynamic damper c 3 =6.7(10 9 ) N/m 3 1.4852 7.418% 4 Monomial fifth power stiffness c 5 =19.6(10 15 ) N/m 5 1.4786 7.829% 5 Monomial seventh power stiffness c 7 =49(10 21 ) N/m 7 1.4750 8.054% 6 Polynomial seventh power stiffness c 1 =0.0158(10 3 ) N/m c 3 =0.492(10 9 ) N/m 3 c 5 =0.564(10 15 ) N/m 5 c 7 =46.58(10 21 ) N/m 7 1.4757 8.010% 7 Monomial ninth power stiffness c 9 =88(10 27 ) N/m 9 1.4737 8.135% 8 Piecewise linear dynamic damper k =37000 N/m Δ =0.58mm 1.4735 8.147% Comparison of different kinds of dynamic dampers to obtain minimum beam deflection,  =0.1 Ns/m.

42. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 42 Moving vehicle:    

43. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 43                    

44. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 44      

45. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 45 moving vehicle excitation linear DVA : Optimal location Optimal stiffness

46. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 46 DVA with piecewise linear stiffness: Maximum deflection vs. clearance Δ and stiffness k of the piecewise linear stiffness; the optimum is k =28000N/m and Δ =0.79mm (2.2874mm max deflection). moving vehicle excitation

47. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 47 The optimum is k =28000N/m and Δ =0.79mm (2.2874mm max deflection).

48. 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 48 Case Beam and dynamic damper condition Stiffness Maximum deflection [mm] Reduction Percentage 1 Bare beam 2.4980 ---- 2 Linear dynamic damper c 1 =2120N/m 2.3393 6.35% 3 Cubic stiffness dynamic damper c 3 =3.5×10 9 N/m 3 2.3031 7.80% 4 Piecewise linear dynamic damper k= 28000 N/m Δ =0.79mm 2.2874 8.43% Performance of the nonlinear dynamic damper for the beam under moving vehicle  =0.1 Ns/m

49. The end of part 1 Thank You! 12/14/2022 11:04 AM Vibration Reduction on Beams Subjected to Moving Loads by Linear and Nonlinear Vibration Absorbers 49