Dynamics of Machines - Part II - IFS.pdf

(a) Amplitude [m/s 2 ]
(b) Amplitude [m/s 2 ]
1
0.5
0
−0.5
−1
x 10 −4 Signal (a) in Time Domain − (b) in Frequency Domain
0 5 10 15
time [s]
20 25 30
0.8
0.6
0.4
0.2
x 10−5
1
0
0 5 10 15 20 25
frequency [Hz]
Figure 14: Free vibration – Spring-mass systems with 1 D.O.F. One mass m = m1 = 0.191 Kg
fixed at the middle of the beam L1 = 0.155 m, resulting in a system ”B” natural frequency of
4.94 Hz
• Damping Factor of the system ”B” – From fig.13, one gets: yo = 0.95 · 10 −4 [m/s 2 ],
yN = y14 = 0.50 · 10 −4 [m/s 2 ] and N = 14:
ξ =
1 +
1
2π·14 ln
1
2π·14 ln
0.95·10 −4
0.50·10 −4
0.95·10 −4
0.50·10 −4
• Equivalent Viscous Damping (d)
=
2 0.007296
≈ 0.007
1.000026
d = 2 · ξ · ωn · m = 2 · 0.007 · (4.94 · 2 · π) · 0.191 ≈ 0.08 [N · s/m]
IMPORTANT CONCLUSION: THE DAMPING FACTOR IS A CHARACTER-
ISTIC OF THE GLOBAL MECHANICAL SYSTEM AND SIMULTANEOUSLY DE-
PENDS ON MASS m, STIFFNESS k AND DAMPING d COEFFICIENTS, NOT
ONLY ON THE DAMPING COEFFICIENT, AS YOU CAN SEE IN THE DEFINI-
TION:
• ξ = d
2·m·ωn
= d1
2· √ m·k
IT IS POSSIBLE TO INCREASE THE DAMPING FACTOR OF A MECHANICAL
SYSTEM EITHER BY DECREASING THE MASS m, OR BY DECREASING THE
STIFFNESS k OR BY INCREASING THE DAMPING COEFFICIENT d. THE
DAMPING FACTOR IS A VERY USEFUL PARAMETER FOR DEFINING THE
RESERVE OF STABILITY IN MACHINERY DYNAMICS.
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