Dynamics of Machines - Part II - IFS.pdf
Dynamics of Machines - Part II - IFS.pdf Dynamics of Machines - Part II - IFS.pdf
approaches the natural frequency of the structure, the magnitude at resonance rapidly approaches a sharp maximum value, provided that the damping ratio is less than about 0.5; (b) the phase of the response between excitation force and displacement shift by 180 o as the frequency sweeps through resonance, with the value of the phase at resonance being 90 o . This physical phenomenon is used to determine the natural frequency of a structure from measurements of the magnitude and phase of the force response of the structure as the driving frequency is swept through a wide range of values. Identify 4 different 1-DOF systems around each one of the 4 natural frequencies of the building. Use your frequency domain identification procedure, which was already developed in the project 1 and is based on the Least Square Method, and obtain the experimental natural frequencies and the experimental damping factors of each one of the 4 mode shapes of the building. 8. MODEL VALIDATION (VERIFICATION) – Compare the theoretical and experimental frequency response functions of the first, second, third and fourth floors, when the structure is excited by the magnetic forces on the first floor. Justify the discrepancies between theoretical and experimental results. 9. APPLICATION OF THE MODEL – The values of the unbalance mass and eccentricity are: % Disk Unbalance m=0.045 % [kg] unbalance mass e=0.040 % [m] eccentricity Consider the 5 different angular velocities, close to the resonances of the building and among them, as following: • 225 rpm (3,75 Hz), • 495 rpm (8,25 Hz), • 615 rpm (10,25 Hz), • 900 rpm (15,00 Hz) and • 975 rpm (16,25 Hz). Use your mathematical model to predict the vibration amplitude of the top mass, i.e. acceleration of the top mass, when the rotor-disk operates unbalanced at the 5 different angular velocities. Check your results comparing with the experimental results. Explain the discrepancies between the results obtained with help of your mathematical model and the experiments. Please, download the files yyy4−unbal−3−75−HZ.txt, yyy4−unbal−8−25−HZ.txt, yyy4−unbal−10−25−HZ.txt, yyy4−unbal−15−00−HZ.txt, yyy4−unbal−16−25−HZ.txt and acc−in−time−domain.m to rebuild figure 41. 10. MODEL ADJUSTMENT – Try to adjust the natural frequencies ωi (i = 1, ...,4) of the analytical model using the experimental natural frequencies obtained via Experimental Modal Analysis. Remember that the parameter % Beam Properties E=(2.0 +- 0.1)e11 % [N/m^2] elasticity modulus 70
acc [m/s 2 ] acc [m/s 2 ] 0.8 0.6 0.4 0.2 −0.2 −0.4 −0.6 −0.8 0 1 2 3 4 time [s] 5 6 7 8 5 4 3 2 1 0 −1 −2 −3 −4 1 0 (b) Acceleration of Mass 4 (d) Acceleration of Mass 4 −5 0 1 2 3 4 time [s] 5 6 7 8 acc [m/s 2 ] 2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 (a) Acceleration of Mass 4 −2.5 0 1 2 3 4 time [s] 5 6 7 8 acc [m/s 2 ] acc [m/s 2 ] 6 4 2 0 −2 −4 (c) Acceleration of Mass 4 −6 0 1 2 3 4 time [s] 5 6 7 8 10 8 6 4 2 0 −2 −4 −6 −8 (e) Acceleration of Mass 4 −10 0 1 2 3 4 time [s] 5 6 7 8 Figure 41: Experimental forced vibration response (acceleration) of mass 4 due to a disk operating with an unbalance mass m = 0.045 Kg with an eccentricity radius e = 0.040 m at 5 rotational speeds: (a) 225 rpm (3,75 Hz); (b) 495 rpm (8,25 Hz); (c) 615 rpm (10,25 Hz); (d) 900 rpm (15,00 Hz) and (e) 975 rpm (16,25 Hz). 71
- Page 19 and 20: 1.6.6 Homogeneous Solution or Free-
- Page 21 and 22: (a) Amplitude [m/s 2 ] x Signal 10
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- Page 25 and 26: Imag(A(ω)) [m/N] 0 −1 −2 −3
- Page 27 and 28: 1.6.11 Superposition of Transient a
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- Page 33 and 34: ⎧ ⎫ ⎪⎨ ˙y1(t) ⎪⎬ ˙y
- Page 35 and 36: zini = U c + A ⇒ c = U −1 {(zin
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- Page 39 and 40: 0.7 0.6 0.5 0.4 0.3 0.2 0.1 First M
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- 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/
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- Page 51 and 52: which could be verified using Modal
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approaches the natural frequency <strong>of</strong> the structure, the magnitude at resonance rapidly<br />
approaches a sharp maximum value, provided that the damping ratio is less than about<br />
0.5; (b) the phase <strong>of</strong> the response between excitation force and displacement shift by 180 o<br />
as the frequency sweeps through resonance, with the value <strong>of</strong> the phase at resonance being<br />
90 o . This physical phenomenon is used to determine the natural frequency <strong>of</strong> a structure<br />
from measurements <strong>of</strong> the magnitude and phase <strong>of</strong> the force response <strong>of</strong> the structure as<br />
the driving frequency is swept through a wide range <strong>of</strong> values.<br />
Identify 4 different 1-DOF systems around each one <strong>of</strong> the 4 natural frequencies <strong>of</strong> the<br />
building. Use your frequency domain identification procedure, which was already developed<br />
in the project 1 and is based on the Least Square Method, and obtain the experimental<br />
natural frequencies and the experimental damping factors <strong>of</strong> each one <strong>of</strong> the 4 mode shapes<br />
<strong>of</strong> the building.<br />
8. MODEL VALIDATION (VERIFICATION) – Compare the theoretical and experimental<br />
frequency response functions <strong>of</strong> the first, second, third and fourth floors, when the structure<br />
is excited by the magnetic forces on the first floor. Justify the discrepancies between<br />
theoretical and experimental results.<br />
9. APPLICATION OF THE MODEL – The values <strong>of</strong> the unbalance mass and eccentricity<br />
are:<br />
% Disk Unbalance<br />
m=0.045 % [kg] unbalance mass<br />
e=0.040 % [m] eccentricity<br />
Consider the 5 different angular velocities, close to the resonances <strong>of</strong> the building and among<br />
them, as following:<br />
• 225 rpm (3,75 Hz),<br />
• 495 rpm (8,25 Hz),<br />
• 615 rpm (10,25 Hz),<br />
• 900 rpm (15,00 Hz) and<br />
• 975 rpm (16,25 Hz).<br />
Use your mathematical model to predict the vibration amplitude <strong>of</strong> the top mass,<br />
i.e. acceleration <strong>of</strong> the top mass, when the rotor-disk operates unbalanced at<br />
the 5 different angular velocities. Check your results comparing with the experimental<br />
results. Explain the discrepancies between the results obtained with<br />
help <strong>of</strong> your mathematical model and the experiments. Please, download the<br />
files yyy4−unbal−3−75−HZ.txt, yyy4−unbal−8−25−HZ.txt, yyy4−unbal−10−25−HZ.txt,<br />
yyy4−unbal−15−00−HZ.txt, yyy4−unbal−16−25−HZ.txt and acc−in−time−domain.m to<br />
rebuild figure 41.<br />
10. MODEL ADJUSTMENT – Try to adjust the natural frequencies ωi (i = 1, ...,4) <strong>of</strong> the<br />
analytical model using the experimental natural frequencies obtained via Experimental<br />
Modal Analysis. Remember that the parameter<br />
% Beam Properties<br />
E=(2.0 +- 0.1)e11 % [N/m^2] elasticity modulus<br />
70
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