TPT September 2016

AR T I C L E

Advanced Machine & Engineering/AMSAW

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).

Finite element modal analysis

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.

by Willy Goellner, chairman and founder – Advanced Machine & Engineering/AMSAW

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.

Calculation using Kirchhoff plate theory

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 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

5.253

K

K

= λ 1 2

=

= 23Hz

f 1

*

*

2 π *

7,900 * 0.0055

2 π * ( D/ 2) 2 ρ * t

2

1.120 2

⎧ ⎩

⎧ ⎩

t E

leads to f 1

= ξ

Substituting the equation for K into the equation for f 1

1

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.

114

www.read-tpt.com

S EPTEMBER 2016