Dynamics of Machines - Part II - IFS.pdf

1.6.6 Homogeneous Solution or Free-Vibrations or Transient Response - Experimental
Analysis
¨y1(t) + 2ξωn ˙y1(t) + ω 2 ny1(t) = 0 (45)
1.6.7 Natural Frequency – ωn [rad/s] or fn [Hz]
fn = 1
k1
2π m1
= 1
2π
3EI
L3 1
mi
[Hz]
number Length fn fexp fexp
of masses [m] [Hz] [Hz] [Hz]
(theor.) (*) (**)
1 0.610 1.23 12/10 = 1.2 1.250
2 0.610 0.87 8.5/10 = 0.85 0.875
3 0.610 0.71 7/10 = 0.7 0.705
4 0.610 0.61 6/10 = 0.6 0.625
Table 3: Measuring the natural frequency of the mass-spring system ”A” with 1. d.o.f, (*) using
the human eyes and a watch, and (**) using an accelerometer attached to the mass, and making
a comparison to the theoretical mathematical model.
number Length fexp
of masses [m] [Hz]
(*)
2 0.610 0.875
2 0.310 1.875
Table 4: Measuring the natural frequency of the mass-spring system ”A” with 1. d.o.f, (*) using
the human eyes and a watch, when the equivalent stiffness of the system is changed, by modifying
the position (length) where the lumped mass is attached to the beam.
1.6.8 Damping Factor ξ or Logarithmic Decrement β
• Experimental identification without using sensors (OBS: Experiment carried out using a
watch, light and shadow, a calculator, and the mass-spring system oscillating from an
initial condition of displacement yo = yini until half the initial amplitude yN = yini/2.)
1
2πN
ξ =
ln
yo
yN
1 1 + 2πN ln
yo
yN
2
or β = 2π ξ
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