Dynamics of Machines - Part II - IFS.pdf

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1.6.5 Comparison between the Analytical and Numerical Solutions of Equation of Motion (a) y(t) [m] 0.01 0.005 0 −0.005 (a) Exact Solution − (b) Numerical Solution (delta T = 0.3605 s) − (c) Comparison −0.01 0 1 2 3 4 5 time [s] 6 7 8 9 10 0.4 (b) y(t) [m] (c) y(t) [m] 0.2 0 −0.2 −0.4 0 1 2 3 4 5 time [s] 6 7 8 9 10 0.4 0.2 0 −0.2 −0.4 0 1 2 3 4 5 time [s] 6 7 8 9 10 Figure 5: Comparison between the analytical and numerical solutions when a time step of 0.3605 [s] is used – Divergence and numerical instability. (a) y(t) [m] (b) y(t) [m] (c) y(t) [m] 0.01 0.005 0 −0.005 (a) Exact Solution − (b) Numerical Solution (delta T = 0.3 s) − (c) Comparison −0.01 0 1 2 3 4 5 time [s] 6 7 8 9 10 0.02 0.01 0 −0.01 −0.02 0 1 2 3 4 5 time [s] 6 7 8 9 10 0.02 0.01 0 −0.01 −0.02 0 1 2 3 4 5 time [s] 6 7 8 9 10 Figure 6: Comparison between the analytical and numerical solutions when a time step of 0.3 [s] is used – Divergence between the numerical and analytical solution due to a roof time step. 16
(a) y(t) [m] (b) y(t) [m] (c) y(t) [m] 0.01 0.005 0 −0.005 (a) Exact Solution − (b) Numerical Solution (delta T = 0.1 s) − (c) Comparison −0.01 0 1 2 3 4 5 time [s] 6 7 8 9 10 0.01 0.005 0 −0.005 −0.01 0 1 2 3 4 5 time [s] 6 7 8 9 10 0.01 0.005 0 −0.005 −0.01 0 1 2 3 4 5 time [s] 6 7 8 9 10 Figure 7: Comparison between the analytical and numerical solutions when a time step of 0.1 [s] is used – Accumulation of errors with the number of iterations and divergence between the numerical and analytical solutions. (a) y(t) [m] (b) y(t) [m] (c) y(t) [m] 0.01 0.005 0 −0.005 (a) Exact Solution − (b) Numerical Solution (delta T = 0.05 s) − (c) Comparison −0.01 0 1 2 3 4 5 time [s] 6 7 8 9 10 0.01 0.005 0 −0.005 −0.01 0 1 2 3 4 5 time [s] 6 7 8 9 10 0.01 0.005 0 −0.005 −0.01 0 1 2 3 4 5 time [s] 6 7 8 9 10 Figure 8: Comparison between the analytical and numerical solutions when a time step of 0.05 [s] is used – Accumulation of errors with the number of iterations and divergence between the numerical and analytical solutions. 17
Page 1 and 2: DYNAMICS OF MACHINES 41614 PART I -
Page 3 and 4: 1 Introduction to Dynamical Modelli
Page 5 and 6: 1.3 Data of the Mechanical System
Page 7 and 8: 1.5 Calculating Stiffness Matrices
Page 9 and 10: 1.6 Mechanical Systems with 1 D.O.F
Page 11 and 12: Demanding (λ 2 + 2ξωnλ + ω 2 n
Page 13 and 14: 1 yini − A det λ1 vini − A C2
Page 15: 1.6.4 Analytical and Numerical Solu
Page 19 and 20: 1.6.6 Homogeneous Solution or Free-
Page 21 and 22: (a) Amplitude [m/s 2 ] x Signal 10
Page 23 and 24: (a) Amplitude [m/s 2 ] (b) Amplitud
Page 25 and 26: Imag(A(ω)) [m/N] 0 −1 −2 −3
Page 27 and 28: 1.6.11 Superposition of Transient a
Page 33 and 34: ⎧ ⎫ ⎪⎨ ˙y1(t) ⎪⎬ ˙y
Page 35 and 36: zini = U c + A ⇒ c = U −1 {(zin
Page 37 and 38: 1.7.4 Modal Analysis using Matlab e
Page 39 and 40: 0.7 0.6 0.5 0.4 0.3 0.2 0.1 First M
Page 41 and 42: %__________________________________
Page 43 and 44: ||y 1 (ω)|| [m/N] Phase [ o] Excit
Page 45 and 46: ||y i (ω)|| [m/N] Phase [ o] 0.8 0
Page 47 and 48: Imag(y i (ω)/f 1 (ω)) (i=1,2) [m/
Page 51 and 52: which could be verified using Modal
Page 53 and 54: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Page 55 and 56: 1.8.4 Theoretical Frequency Respons
Page 59 and 60: 4. Vary the number of masses attach
Page 61 and 62: 1.10 Project 0 - Identification of
Page 63 and 64: (a) (b) REAL(Acc/force) [(m/s 2 )/N
Page 65 and 66: 6. Model application - As explained
Page 67 and 68:
changeable unbalanced mass for simu
Page 69 and 70:
%Modal Matrix u with mode shapes %M
Page 71 and 72:
acc [m/s 2 ] acc [m/s 2 ] 0.8 0.6 0
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