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114
S
EPTEMBER
2016
AR T I C L E
Advanced Machine & Engineering/AMSAW
by Willy Goellner, chairman and founder – Advanced Machine & Engineering/AMSAW
Finite element modal analysis
Calculation using Kirchhoff plate theory
Experimental modal analysis (impact test)
You hit the blade with an
impact hammer.
Accelerometers will track
the transfer function
with the help of a data
acquisition device (DAQ).
Hint: In cas you do ’t
have a DAQ available,
you can also use an
oscilloscope and do the
signal transformation
(Fast Fourier
Transformation) in Excel.
As a result, you can
see the lowest natural
frequency, for example at
23Hz in the chart below.
The equations that arise from the modal analysis are
the same that can be found when solving eigenvalue
problems.
1) Every eigenvalue (natural frequency) has a
corresponding eigenvector (mode shape).
2) The benefit when using FEA is that you do not only
get the frequency value, but you can visualise the
mode shape easily.
3) The end result is very close to the measurement of
23Hz.
4) This mode, which is represented by node diameter 2
and node circle 0, is one which causes most damage
to the teeth.
Last but not least, you calculate the natural frequency of the blade and get a feeling for the driving parameters. If you use
Kirchhoff plate theory in polar coordinates, and replace the static load with negative mass acceleration, by solving the
Bessel differential equation you will end up with the following formula for the natural frequency
f
1
.
1) Calculate the flexural rigidity, K
K
=
E
*
t
3
= 2.1 * 10
11
* (5.5 * 10
-3
)
3
= 3,267 Nm
12 * (1–
ν
2
) 12 * (1–0.33
2
)
2) Calculate the natural frequency
f
1
=
λ
1
2
*
K
=
5.253
*
K
= 23Hz
2
π *
(
D/
2)
2
ρ
* t
2
π *
2
7,900 * 0.0055
Substituting the equation for K into the equation for
f
1
leads to
f
1
=
ξ
1
t E
D
2
ρ
ξ is a combined factor which is dependent on the boundary condition, the mode you are looking for, and other constants.
This shows how theory can prove practical testing and explains that the natural frequency increases linearly with the
thickness (
t
) and decreases by the square of the diameter (
D
), as the short formula for
f
1
shows.
λ
2
= tabulated value for the
boundary condition and mode
D
= Diameter
(1,120mm)
t
= Thickness
(5.5mm)
E
= Young’s modulus of steel
ν
= Poisson’s ratio
for steel
ρ = Density for
steel
1.120
2
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