Harmonic Response of a Guitar String


Reference:R. D. Blevins, Formulas for Natural Frequency and Mode Shape, Van Nostrand Reinhold Co., New York, NY, 1979, pg. 90, tab. 7-1.
Analysis Type(s):
Static Analysis (ANTYPE = 0)
Linear Perturbed Modal Analysis (ANTYPE = 2)
Linear Perturbed Harmonic Analysis (ANTYPE = 3)
Element Type(s):3-D Spar (or Truss) Elements (LINK180)
Input Listing:vm76.dat

Test Case

A uniform stainless steel guitar string of length and diameter d is stretched between two rigid supports by a tensioning force F1, which is required to tune the string to the E note of a C scale. The string is then struck near the quarter point with a force F2. Determine the fundamental frequency, f1. Also, show that only the odd-numbered frequencies produce a response at the midpoint of the string for this excitation.

Figure 76.1:  Guitar String Problem Sketch

Guitar String Problem Sketch

Material PropertiesGeometric PropertiesLoading
E = 190 x 109 Pa
ρ = 7920 kg/m3
= 710 mm
c = 165 mm
d = 0.254 mm
F1 = 84 N
F2 = 1 N

Analysis Assumptions and Modeling Notes

Enough elements are selected so that the model can be used to adequately characterize the string dynamics. The stress stiffening capability of the element is used. Linear perturbed harmonic analysis determines the displacement response to the lateral force F2. The harmonic response is displayed with the time-history postprocessor, POST26, to show the excitation of the odd-numbered frequencies at peak displacement amplitudes. Refer to Figure 76.2: String Midpoint Displacement Amplitude.

Results Comparison

 TargetMechanical APDLRatio
Modalf, Hz322.2322.31.000
POST26f1, (322.2 Hz)ResponseResponse, 320 < f < 328-
f2, (644.4 Hz)No ResponseNo Response-
f3, (966.6 Hz)ResponseResponse, 968 < f < 976-
f4, (1288.8 Hz)No ResponseNo Response-
f5, (1611.0 Hz)ResponseResponse,1624 < f < 1632-
f6, (1933.2 Hz)No ResponseNo Response-

Figure 76.2:  String Midpoint Displacement Amplitude

String Midpoint Displacement Amplitude

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