Linear Analysis

1. Free Vibrations of Volume of Air

Free vibration modes and eigenfrequencies of an unconstrained (unbounded) volume of air are calculated, demonstrating the acoustic elements solving the Helmholtz equation. The geometry consists of a cube 0.5m by 0.5m by 0.1m (see figure below). To resolve frequencies of about 1000[Hz], one needs about 10 elements over a wavelength, which is calculated with l = c/f. With the speed of sound c=340 m/s, and f=1000Hz, one obtains l=0.34 m, a tenth of which gives l=0.034 m. The cube is meshed with 20 by 20 by 5 HE8 acoustic elements:

Free vibrations of volume of air: Mesh.

Figure 106. Free vibrations of volume of air: Mesh.


To perform the vibration analysis, B2000++ is launched with a case defining the free vibration problem parameters:

cases
   case 1
      analysis free_vibration
      nmodes 20
      shift 100.0
   end
end

Note that an arbitrary positive shift can be performed, requesting 20 eigenvalues around the shift of 100.0 [Hz]. The utility script print.py prints the frequencies (see Makefile):

Free vibration analysis, case 1 of cycle 0
Title: 'Free vibration with shift 100 Hz'
Number of computed modes: 20

Mode     Eigenvalue  Frequency        Omega  
   1   -4.267e-08
   2    4.573e+06        340.3         2138 
   3    4.573e+06        340.3         2138 
   4    9.146e+06        481.3         3024 
   5    1.841e+07        682.8         4290 
   6    1.841e+07        682.8         4290   
   7    2.298e+07        762.9         4794     
   8    2.298e+07        762.9         4794 
   9    3.681e+07        965.6         6067 
  10    4.184e+07         1029         6468 
  11    4.184e+07         1029         6468  
  12    4.641e+07         1084         6813 
  13    4.641e+07         1084         6813   
  14    6.024e+07         1235         7762     
  15    6.024e+07         1235         7762 
  16    7.545e+07         1382         8686
  17    7.545e+07         1382         8686 
  18    8.002e+07         1424         8946 
  19    8.002e+07         1424         8946 

To visualize the free vibration modes, make use of the Modes tool of baspl++ by selecting, from the Renderer part, Create Modes Tool...:

Free vibrations of volume of air: baspl++ modes tool selection.

Figure 107. Free vibrations of volume of air: baspl++ modes tool selection.


Note that, when displaying scalar modes, the scale factor has to be set to 0.0! Some of the modes (mode 1, 4, and 20) is displayed in the figures below, obtained with the modes tool.

Free vibrations of volume of air: Vibration mode 1.

Figure 108. Free vibrations of volume of air: Vibration mode 1.


Free vibrations of volume of air: Vibration mode 4.

Figure 109. Free vibrations of volume of air: Vibration mode 4.


Free vibrations of volume of air: Vibration mode 20.

Figure 110. Free vibrations of volume of air: Vibration mode 20.