1. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -1- Chapter 11 Numerical Integration Methods in Vibration Analysis 11

2. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -2- Chapter Outline 11.1IntroductionIntroduction 11.2Finite Difference MethodFinite Difference Method 11.3Central Difference Method for Single DOF SystemsCentral Difference Method for Single DOF Systems 11.4Runge-Kutta Method for Single DOF SystemsRunge-Kutta Method for Single DOF Systems 11.5Central Difference Method for Multi-DOF SystemsCentral Difference Method for Multi-DOF Systems 11.6Finite Difference Method for Continuous SystemsFinite Difference Method for Continuous Systems 11.7Runge-Kutta Method for Multi-DOF SystemsRunge-Kutta Method for Multi-DOF Systems 11.8Houbolt MethodHoubolt Method 11.9Wilson MethodWilson Method 11.10Newmark MethodNewmark Method

3. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -3- 11.1 Introduction 11.1

4. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -4- 10.1 Introduction Numerical methods used when equation of motion cannot be integrated in closed form The solution only satisfy the equations at discrete time intervals Suitable type of variation for displacement, velocity and acceleration is assumed within each time interval. Explicit integration methods – finite difference method, Runge-Kutta method Implicit integration methods – Houbolt, Wilson, Newmark methods

5. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -5- 11.2 Finite Difference Method 11.2

6. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -6- 11.2 Finite Difference Method Use approximations to derivatives Central difference formula is used to derive finite difference equations Replace solution domain with finite number of equally-spaced grid points.

7. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -7- 11.2 Finite Difference Method Taking 2 terms only and subtracting one equation from the other, This is the central difference approx. to the first derivative of x at t = t i

8. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -8- 11.2 Finite Difference Method Taking terms up to 2 nd derivative and adding one equation to the other, This is the central difference approx. to the 2nd derivative of x at t = t i

9. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -9- 11.3 Central Difference Method for Single-DOF Systems 11.3

10. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -10- 11.3 Central Difference Method for Single-DOF Systems Viscously damped single-DOF system: Replace derivatives by central differences Recurrence formula:

11. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -11- 11.3 Central Difference Method for Single-DOF Systems Repeated application yields the complete time history of the system behavior. can be found by substituting known values of x 0 and into Eq. 11.5. Application of central difference approx for 1 st and 2 nd derivative yields the value of x -1 :

12. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -12- 11.3 Central Difference Method for Single-DOF Systems Example 11.1 Response of Single-Degree-of-Freedom System Find the response of a viscously damped single-DOF system subjected to a force with the following data: F 0 =1, t 0 =π, m=1, c=0.2 and k=1. Assume the values of the displacement and velocity of the mass at t=0 to be zero.

13. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -13- 11.3 Central Difference Method for Single-DOF Systems Example 11.1 Response of Single-Degree-of-Freedom System Solution Governing equation of motion: Solution can be found from the recurrence formula with x 0 =0, x - 1 =(Δt) 2 /2, x i =x(t i )=x(iΔt) and

14. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -14- 11.3 Central Difference Method for Single-DOF Systems Example 11.1 Response of Single-Degree-of-Freedom System Solution Thus Δt must be less than τ n /π=2.0

15. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -15- 11.4 Runge-Kutta Method for Single-DOF Systems 11.4

16. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -16- 11.4 Runge-Kutta Method for Single-DOF Systems The Taylor’s series expansion is written as Eq. (11.5) can be rewritten as

17. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -17- 11.4 Runge-Kutta Method for Single-DOF Systems Define The following recurrence formula is used to find at different grid pts which is stable and self-starting

18. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -18- 11.4 Runge-Kutta Method for Single-DOF Systems Example 11.2 Response of Single-Degree-of-Freedom System Find the solution of Example 11.1 using the Runge-Kutta method.

19. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -19- 11.4 Runge-Kutta Method for Single-DOF Systems Example 11.2 Response of Single-Degree-of-Freedom System Solution Use step size of Δt=0.3142 and define From initial conditions we have

20. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -20- 11.4 Runge-Kutta Method for Single-DOF Systems Example 11.2 Response of Single-Degree-of-Freedom System Solution The following table shows the values of

21. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -21- 11.5 Central Difference Method for Multi-DOF Systems 11.5

22. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -22- 11.5 Central Difference Method for Multi-DOF Systems Viscously damped multi-DOF system: Central difference formulae for velocity and acceleration vectors at time t i = iΔt: Thus equation of motion at time t i :

23. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -23- 11.5 Central Difference Method for Multi-DOF Systems The equation can be rearranged to obtain: Special starting procedure is needed to find Eq 11.18 to 11.20 are evaluated at i=0:

24. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -24- 11.5 Central Difference Method for Multi-DOF Systems Initial acceleration vector: Displacement vector at t 1 : Substitute into Eq 11.25: where is given by Eq 11.26

25. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -25- 11.5 Central Difference Method for Multi-DOF Systems Example 11.3 Central Difference Method for a Two-Degree-of-Freedom System Find the response of the 2-DOF system shown below when the forcing functions are given by F 1 (t)=0 and F 2 (t)=10. Assume the value of c as zero and the initial conditions as

26. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -26- 11.5 Central Difference Method for Multi-DOF Systems Example 11.3 Central Difference Method for a Two-Degree-of-Freedom System Solution Use and the equations of motion is

27. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -27- 11.5 Central Difference Method for Multi-DOF Systems Example 11.3 Central Difference Method for a Two-Degree-of-Freedom System Solution Eigenvalue problem:

28. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -28- 11.5 Central Difference Method for Multi-DOF Systems Example 11.3 Central Difference Method for a Two-Degree-of-Freedom System Solution Select time step Δt= τ 2 /10=0.24216 The equation can be applied recursively to obtain

29. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -29- 11.5 Central Difference Method for Multi-DOF Systems Example 11.3 Central Difference Method for a Two-Degree-of-Freedom System Solution

30. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -30- 11.6 Finite Difference Method for Continuous Systems 11.6

31. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -31- 11.6 Finite Difference Method for Continuous Systems Longitudinal Vibration of Bars Equation of motion Divide the bar into n-1 equal parts Each part has length h=l/(n-1)

32. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -32- 11.6 Finite Difference Method for Continuous Systems Longitudinal Vibration of Bars

33. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -33- 11.6 Finite Difference Method for Continuous Systems Longitudinal Vibration of Bars Boundary conditions: Fixed end U 1 = U n =0

34. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -34- 11.6 Finite Difference Method for Continuous Systems Longitudinal Vibration of Bars Boundary conditions: Free end (dU)/(dx) = 0

35. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -35- 11.6 Finite Difference Method for Continuous Systems Transverse Vibration of Beams Equation of motion Using central difference formula for 4 th derivative, Divide the bar into n-1 equal parts Each part has length h=l/(n-1)

36. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -36- 11.6 Finite Difference Method for Continuous Systems Transverse Vibration of Beams

37. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -37- 11.6 Finite Difference Method for Continuous Systems Transverse Vibration of Beams Boundary conditions: x=0 is fixed Introduce a fictitious node -1 x=l is fixed Introduce a fictitious node n+1

38. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -38- 11.6 Finite Difference Method for Continuous Systems Transverse Vibration of Beams Boundary conditions: x=0 Simply supported

39. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -39- 11.6 Finite Difference Method for Continuous Systems Transverse Vibration of Beams Boundary conditions: x=0 is free

40. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -40- 11.6 Finite Difference Method for Continuous Systems Example 11.4 Pinned-Fixed Beam Find the natural frequencies of the simply supported-fixed beam as shown. Assume that the cross section of the beam is constant along its length.

41. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -41- 11.6 Finite Difference Method for Continuous Systems Example 11.4 Pinned-Fixed Beam Solution Divide the beam into 4 segments. Boundary conditions at simply supported end: Boundary conditions at fixed end:

42. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -42- 11.6 Finite Difference Method for Continuous Systems Example 11.4 Pinned-Fixed Beam Solution Thus E1, E2 and E3 reduces to: Matrix form:

43. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -43- 11.6 Finite Difference Method for Continuous Systems Example 11.4 Pinned-Fixed Beam Solution

44. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -44- 11.7 Runge-Kutta Method for Multi-DOF Systems 11.7

45. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -45- 11.7 Runge-Kutta Method for Multi-DOF Systems Acceleration vector

46. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -46- 11.7 Runge-Kutta Method for Multi-DOF Systems i.e.

47. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -47- 11.7 Runge-Kutta Method for Multi-DOF Systems 4 th order Runge-Kutta Method:

48. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -48- 11.7 Runge-Kutta Method for Multi-DOF Systems Example 11.5 Runge-Kutta Method for a Two-Degree-of-Freedom System Find the response of the 2 DOF system considered in Example 11.3 using the 4 th order Runge-Kutta method.

49. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -49- 11.7 Runge-Kutta Method for Multi-DOF Systems Example 11.5 Runge-Kutta Method for a Two-Degree-of-Freedom System Solution For Δt=0.24216,

50. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -50- 11.8 Houbolt Method 11.8

51. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -51- 11.8 Houbolt Method Finite difference expansion To find solution at step i+1, consider equation of motion at t i+1 :

52. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -52- 11.8 Houbolt Method Substituting finite difference expansion into equation of motion: Compute starting with i=2 using

53. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -53- 11.8 Houbolt Method Example 11.6 Houbolt Method for a Two-Degree-of-Freedom System Rind the response of the 2-DOF system consider in Example 11.3 using the Houbolt method.

54. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -54- 11.8 Houbolt Method Example 11.6 Houbolt Method for a Two-Degree-of-Freedom System Solution At Δt=0.24216,

55. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -55- 11.9 Wilson Method 11.9

56. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -56- 11.9 Wilson Method Assume acceleration is linear from time t i =iΔt to t i+θ =t i +θΔt where θ≥1.0 We can predict at any time t i +τ:

57. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -57- 11.9 Wilson Method To find x i +θ, consider equilibrium equation at time t i +θΔt:

58. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -58- 11.9 Wilson Method Steps:  From initial conditions, find  Select suitable time step Δt and θ  Calculate effective load factor starting with i=0

59. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -59- 11.9 Wilson Method  Find  Calculate acceleration, velocity and displacement at time t i +1

60. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -60- 11.9 Wilson Method Example 11.7 Wilson Method for a Two-Degree-of-Freedom System Find the response of the system considered in Example 11.3, using the Wilson method with θ=1.4.

61. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -61- 11.9 Wilson Method Example 11.7 Wilson Method for a Two-Degree-of-Freedom System Solution At Δt=0.24216, we have the results as shown in table.

62. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -62- 11.10 Newmark Method 11.10

63. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -63- 11.10 Newmark Method Assume acceleration varies linearly with time. α and β can be chosen to obtain desired stability characteristics

64. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -64- 11.10 Newmark Method Steps:  From initial conditions, find  Select suitable time step Δt, α and β  Calculate starting with i=0  Find acceleration and velocity at time t i+1 :

65. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -65- 11.10 Newmark Method Example 11.8 Newmark Method for a Two-Degree-of-Freedom System Find the response of the system considered in Example 11.3, using the Newmark method with α=1/6 and β=1/2.

66. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -66- 11.10 Newmark Method Example 11.8 Newmark Method for a Two-Degree-of-Freedom System Solution At Δt=0.24216, results as shown in the table.