Dynamics Course

1. Prof. A. Meher Prasad Department of Civil Engineering Indian Institute of Technology Madras email: prasadam@iitm.ac.in NUMERICAL METHODS

4. For SDOF System Let ∆t = time interval t n = n ∆t P n is the applied force at time t n Direct Integration of the Equations of Motion… P(t) ∆ t 0 1 …… t n t n+1 P n P n+1 mx + cx + kx = P(t) .. . x n , x n , x n Displacement, velocity and acceleration at time station ‘n’ . ..

5. General Expression for the time integration methods R is a remainder term representing the error given by x (m) is the value of m th differential of x at t= ξ A l , B l and C l are constants ( some of which may be equal to zero) (n-k) ∆t ≤ ξ ≤ (n+1) ∆t . ..

8. Then equations (1) & (2) reduce to x n+1 = x n + ∆t(1- γ )x n + ∆t γ x n+1 +R (3) x n+1 = x n + ∆t x n + ( ∆t) 2 (1/2 – β ) x n + (∆t) 2 β x n+1 + R (4) Third relationship m x n+1 + cx n+1 + kx n+1 = P n+1 (5) Substituting eqn.(3) and (4) in eqn.(5), we get expression for x n+1 To begin the time integration, we need to know the values of x o , x o and x o at time t=0. . . . .. .. .. .. . .. .. . .. Newmark’s β Method…

9. Acceleration, Time, t γ =0, β =0 Constant Acceleration x .. ∆ t ∆ t x n ..

10. γ =1/2, β =1/4 Average Acceleration Acceleration, x .. ∆ t Time, t x n .. x n+1 .. x = .. .. ..

11. γ =1/2, β =1/6 Linear Acceleration Acceleration, x .. ∆ t Time, t x n .. x n+1 .. x = .. .. .. .. t

12. Algorithm Enter k, m, c, β , γ and P(t) x 0 = .. . Select ∆t ^

13. x i+1 = x i + ∆x i ; x i+1 = x i + ∆x i ; x i+1 = x i + ∆x i ∆ x i = ^ ^ . .. i = 0 i = i+1 . ∆ p i = p i + a x i + b x i ^ .. . .. . .. . . .. . . .. ..

14. Elastoplastic System x 0 = .. ; x t = ; x c = Define key = 0 (elastic) key = -1 (plastic behavior in compression key = 1 (plastic behavior in tension) Newmark’s β Method -- Enter k, m, c, R t , R c and P(t) Set x 0 = 0, x 0 = 0; . Select ∆t

15. Calculate x i and x i . key = -1; R=R c x i > x c x i < x t R = R t – (x t – x i ) k x i < x c x i > x t key = 1; R=R t < 0 = (P(t i+1 ) – c i+1 – R) /m x i+1 .. x i+1 . > 0 x i . key = 0; x t = x i ; x c = x i – (R t – R c )/k R = R t – (x t – x i ) k x i . key = 0; x c = x i ; x t = x i + (R t – R c )/k R = R t – (x t – x i ) k n y y n y y y n i = 0 i = i+1

16. Central Difference Method The method is based on finite difference approximations of the time derivatives of displacement (velocity and acceleration) at selected time intervals Displacement, u Time, t x n+1 θ (n-1) ∆t (n+1) ∆t x n-1

17. . ^ Algorithm Enter k, m, c, and P(t) x 0 = .. x -1 = x 0 - ∆t x 0 + 0.5 ∆t 2 x 0 .. .

18. x i+1 = ^ ^ . .. i = 0 i = i+1 p i = p i – a x i-1 – b x i ^

20. . n=0 Algorithm Enter k, m, c, , ∆t and P(t) Specify initial conditions p n+ θ = p n (1- θ ) + p n+1 θ ^ k = a 1 m + a 3 c + k ; a 5 = a 1 x n + a 4 x n + 2x n ; a 6 = a 3 x n + 2x n + a 2 x n . .. . .. A

21. x n+ θ = ( p n+ θ +ma 5 +ca 6 ) /k ^ . .. .. A n = n+1 x n+1 = x n + (x n+ θ – x n ) / θ .. .. .. .. x n+1 = x n + (x n + x n+1 ) h /2 .. . . ..

25. Spectral radii for α -methods, optimal collocation schemes and Houbolt, Newmark, Park and Wilson methods

26. Selection of a numerical integration method Period elongation vs. ∆t/T Amplitude decay vs. ∆t/T * For the numerical integration of SDOF systems, the linear acceleration method, which gives no amplitude decay and the lowest period elongation, is the most suitable of the methods presented

33. Starting Vectors (1) When some masses are zero, for non zero d.o.f have one as vector entry. (2) Take ratio .The element that has minimum value will have 1 and rest zero in the starting vector.

35. Eqn. 2 are not true. Eigen values unless P = n If [  ] satisfies (2) and (3),they cannot be said that they are true Eigen vectors. If [  ] satisfies (1),then they are true Eigen vectors. Since we have reduced the space from n to p. It is only necessary that subspace of ‘P’ as a whole converge and not individual vectors.

36. Algorithm: Pick starting vector X R of size n x p For k=1,2,…..   k+1 – { X } k+1 -  k -   static p x p p x p Smaller eigen value problem, Jacobi

37. Factorization Subspace Iteration Sturm sequence check (1/2)nm 2 + (3/2)nm nq(2m+1) (nq/2)(q+1) (nq/2)(q+1) n(m+1) (1/2)nm 2 + (3/2)nm 4nm + 5n nq 2

38. Total for p lowest vector. @ 10 iteration with nm 2 + nm(4+4p)+5np q = min(2p , p+8) is 20np(2m+q+3/2) This factor increases as that iteration increases. N = 70000,b = 1000, p = 100, q = 108 Time = 17 hours

40. (2) Neq x m [ I ] k,k+1 - with # of rows = # of attachment d.o.f. between k and k+1 = # of columns Ritz analysis: Determine [ K r ] = [R] T [k] [R] [ M r ] = [R] T [M] [R] [k r ] {X} = [M] r +[X] [ ] - Reduced Eigen value problem Eigen vector Matrix, [  ] = [ R ] [ X ]

41. Use the subspace Iteration to calculate the eigen pairs (  1 ,  1 ) and (  2 ,  2 ) of the problem K  =  M  ,where Example