Estimation of the fundamental natural frequency, damping
ratio and equivalent mass
Austin Wilson
Introduction:
For the following lab the fundamental natural frequency of a beam was measured
with a small accelerometer for two different DOF scenarios: cantilever and fixedfixed support. The accelerometer was mounted to the beam for both scenarios
using super glue and an impulse hammer was used to excite the beam with an initial
impulse. After the beam was excited special software was used to collect the time
response. Two peaks were measured to find amplitude and period, with this the
natural frequency, damping coefficient, mass and stiffness was determined.
Aluminum Beam:
Thickness: 7.14mm
Width: 25.58mm
Length: 560 mm
The equation below shows the perturbed wave function of the beam
Time response = Transient response + Forced response
𝐾
Where 𝜔𝑛 = √𝑚 is the undamped natural frequency and 𝜉 = 𝑐/√2𝑚𝐾 is the
damping ratio and 𝜔𝑑 = 𝜔𝑛√1 − 𝜉 2 is the damped natural frequency.
Case 1: Fixed – Fixed- Fixed Support
Below shows the experimental set up of the fixed-fixed beam with the
accelerometer at a know position in the middle of the bar. Table 2 shows the
measured and calculated values for the fixed-fixed set up. It can be seen that the
stiffness was determined to be 60600 N/m with a natural frequency of 802 rad/s.
Figure 1: fixed support
#
1
2
3
4
Item
Time @ peak 1
Time @ peak 2
Amplitude @ peak 1
Amplitude @ peak 2
5 Time between 1 and 2
Units
ms
ms
g
g
ms
6 # of periods between 1 and 2
7 Period of oscillations E/F
8
9
10
11
12
Damped natural frequency
Natural frequency
Equivalent mass
Stiffness
Damping
Value
111.23
455.762
6.057
2.862
344.532
44
ms
rad/s
rad/s
g
N/m
N/(m/s)
7.83
802
802
94
60600
0.286
13 Natural frequency estimation rad/s
754
Table 2: Calculated and measured values (fixed-fixed beam)
Case 2: Cantilever Support
Below shows the experimental set up of the cantilever beam with the accelerometer
at a know position at the end of the bar. Table 2 shows the measured and calculated
values for the cantilever set up. It can be seen that the stiffness was determined to
be 947 N/m with a natural frequency of 91.5 rad/s.
Figure 2: cantilever support
#
1
2
3
4
Item
Time @ peak 1
Time @ peak 2
Amplitude @ peak 1
Amplitude @ peak 2
5 Time between 1 and 2
Units
ms
ms
g
g
ms
6 # of periods between 1 and 2
7 Period of oscillations E/F
8
9
10
11
12
Damped natural frequency
Natural frequency
Equivalent mass
Stiffness
Damping
Value
91.113
845.801
1.268
1.176
754.688
11
ms
rad/s
rad/s
g
N/m
N/(m/s)
68.6
91.5
91.5
113
947
0.016
13 Natural frequency estimation rad/s
123
Table 2: Calculated and measured values (cantilever beam)
Conclusion:
In conclusion it was determined that the fixed-fixed beam had a much larger
stiffness as well as damping coefficient and natural frequency then the cantilever
beam as was expected. Also the natural frequency estimation by Rayleigh method
was fairly close for both methods of support.