Dynamics of Machines - Part II - IFS.pdf
Dynamics of Machines - Part II - IFS.pdf Dynamics of Machines - Part II - IFS.pdf
(a) Amplitude [m/s 2 ] (b) Amplitude [m/s 2 ] 6 4 2 0 −2 −4 −6 x 10 −6 Signal (a) in Time Domain − (b) in Frequency Domain −8 0 5 10 15 time [s] 20 25 30 x 10−6 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 frequency [Hz] (a) Amplitude [m/s 2 ] x Signal 10−5 1.5 1 0.5 0 −0.5 −1 (a) in Time Domain − (b) in Frequency Domain −1.5 0 5 10 15 time [s] 20 25 30 (b) Amplitude [m/s 2 ] x 10−6 8 6 4 2 0 0 5 10 15 20 25 frequency [Hz] Figure 22: Beating phenomena due to transient (low damping factor) and forced vibrations with similar frequencies: 1 D.O.F. system ”A” with natural frequency of 0.87 Hz, excited by a shaker with frequencies of 0.80 Hz and 0.90 Hz - Spring-mass system ”A” with two masses m = m1+m2 = 0.382 Kg fixed at the beam length L1 = 0.600 m, resulting in a natural frequency of 0.87 Hz. 30
1.7 Mechanical Systems with 2 D.O.F. 1.7.1 Physical System and Mechanical Model (a) (b) (c) Figure 23: (a) Real mechanical system built by two turbines attached to an airplane flexible wing; (b) Laboratory prototype built by two lumped masses attached to a flexible beam); (c) Equivalent mechanical model with 2 D.O.F. for the two lumped masses attached to the flexible beam. 1.7.2 Mathematical Model M¨y(t) + D˙y(t) + Ky(t) = f(t) (47) m11 m12 m21 m22 ¨y1 ¨y2 d11 d12 + d21 d22 ˙y1 ˙y2 k11 k12 + k21 k22 31 y1 y2 = f1 f2 (48)
- 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 and 16: 1.6.4 Analytical and Numerical Solu
- Page 17 and 18: (a) y(t) [m] (b) y(t) [m] (c) y(t)
- 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 29: (a) Amplitude [m/s 2 ] (b) Amplitud
- 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 49 and 50: 1.8 Mechanical Systems with 3 D.O.F
- 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 57 and 58: (a) Amplitude [m/s 2 ] (b) Amplitud
- 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
(a) Amplitude [m/s 2 ]<br />
(b) Amplitude [m/s 2 ]<br />
6<br />
4<br />
2<br />
0<br />
−2<br />
−4<br />
−6<br />
x 10 −6 Signal (a) in Time Domain − (b) in Frequency Domain<br />
−8<br />
0 5 10 15<br />
time [s]<br />
20 25 30<br />
x 10−6<br />
2.5<br />
2<br />
1.5<br />
1<br />
0.5<br />
0<br />
0 5 10 15 20 25<br />
frequency [Hz]<br />
(a) Amplitude [m/s 2 ]<br />
x Signal 10−5<br />
1.5<br />
1<br />
0.5<br />
0<br />
−0.5<br />
−1<br />
(a) in Time Domain − (b) in Frequency Domain<br />
−1.5<br />
0 5 10 15<br />
time [s]<br />
20 25 30<br />
(b) Amplitude [m/s 2 ]<br />
x 10−6<br />
8<br />
6<br />
4<br />
2<br />
0<br />
0 5 10 15 20 25<br />
frequency [Hz]<br />
Figure 22: Beating phenomena due to transient (low damping factor) and forced vibrations<br />
with similar frequencies: 1 D.O.F. system ”A” with natural frequency <strong>of</strong> 0.87 Hz, excited by<br />
a shaker with frequencies <strong>of</strong> 0.80 Hz and 0.90 Hz - Spring-mass system ”A” with two masses<br />
m = m1+m2 = 0.382 Kg fixed at the beam length L1 = 0.600 m, resulting in a natural frequency<br />
<strong>of</strong> 0.87 Hz.<br />
30
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