Modal interaction

The following shows an examplel of modal interactions for a 20 kHz \( \phi \)40 mm horn (\( c_{tw} \) = 5100 m/sec; Poisson's ratio = 0.33). The horn length was tuned as needed to maintain the axial resonance (the maximum adjustment was 0.1 mm). Also see Amplitude asymmetry.

Graph - Resonant axial and nonaxial resonances for a 40 mm diameter 20 kHz ultrasonic horn
Figure 1. Frequencies for a 20 kHz \( \phi \)40 mm horn
Graph - Amplitude asymmetries due to modal interaction for a 40 mm diameter 20 kHz ultrasonic horn
Figure 2. Face amplitude asymmetries for a 20 kHz \( \phi \)40 mm horn

 

The following table shows some data from the above graphs. Note how quickly the axial mode asymmetry changes for small changes in the horn diameter and as the axial and nonaxial resonances converge. Also note how the nonaxial modeshape changes as it interacts with the axial mode.

Table notes —

  1. All displacements (as represented by the image colors) are in the axial direction.
  2. All displacements are displayed at the same scale.

 

Diameter (mm) Frequency
separation (Hz)
Axial mode Nonaxial mode
38.00 -265
0.001 amplitude asymmetry due to modal interaction for a 38.00 mm diameter 20 kHz ultrasonic horn

20005 Hz
Asymmetry = 0.001

19740 Hz bending resonance for a 38.00 mm diameter ultrasonic horn
19740 Hz
39.00 -27
0.06 amplitude asymmetry due to modal interaction for a 39.00 mm diameter 20 kHz ultrasonic horn

20000 Hz
Asymmetry = 0.06

19973 Hz bending resonance for a 39.00 mm diameter ultrasonic horn
19973 Hz
39.02 -18
0.34 amplitude asymmetry due to modal interaction for a 39.02 mm diameter 20 kHz ultrasonic horn

19998 Hz
Asymmetry = 0.34

19982 Hz bending resonance for a 39.02 mm diameter ultrasonic horn
19982 Hz
39.05 -12
0.50 amplitude asymmetry due to modal interaction for a 39.05 mm diameter 20 kHz ultrasonic horn

20000 Hz
Asymmetry = 0.50

19988 Hz bending resonance for a 39.05 mm diameter ultrasonic horn
19988 Hz
39.10 6
0.96 amplitude asymmetry due to modal interaction for a 39.10 mm diameter 20 kHz ultrasonic horn

20000 Hz
Asymmetry = 0.96

20006 Hz bending resonance for a 39.10 mm diameter ultrasonic horn
20006 Hz
39.20 27
0.18 amplitude asymmetry due to modal interaction for a 39.20 mm diameter 20 kHz ultrasonic horn

20000 Hz
Asymmetry = 0.18

20027 Hz bending resonance for a 39.20 mm diameter ultrasonic horn
20027 Hz
39.30 42
0.06 amplitude asymmetry due to modal interaction for a 39.30 mm diameter 20 kHz ultrasonic horn

20001 Hz
Asymmetry = 0.06

20043 Hz bending resonance for a 39.30 mm diameter ultrasonic horn
20043 Hz
41.00 424
0.008 amplitude asymmetry due to modal interaction for a 41.00 mm diameter 20 kHz ultrasonic horn

19996 Hz
Asymmetry = 0.008

20420 Hz bending resonance for a 41.00 mm diameter ultrasonic horn
20420 Hz