SOLIDWORKS Simulation – Frequency Analysis of Tensioned Guitar Strings

by Ryan Dark

SOLIDWORKS Simulation is powerful.  Using this tool, I will demonstrate the correlation between SOLIDWORKS Simulation FEA and the solution of a theoretical equation through the analysis of a guitar string.

Setup
In this setup, a single guitar string is restrained on both ends. Restraining the string allows it to reach fundamental frequency, which is 1 half wavelength along the length of the string.
GS1

The setup will utilize beam elements for the string as it is long and thin. One end will be fixed in the radial, axial, and circumferential directions, while the other end is fixed in only the radial and circumferential directions. On the free end, a variable force will be applied to observe the change in fundamental frequency on the string.

GS2

The material “Plain Carbon Steel” is applied to the string along with the properties listed below. These properties will also be used with the analytical calculations.

String Length = 0.349 m
String Diameter = 0.5 mm
Material Density = 7800 kg/m3

A tensile static analysis reveals that the string will break once ~43 N are applied.

FEA Results
The results of this analysis reveal an expected trend of increasing fundamental frequency with increasing tension.  The tabulated results are as follows:

Force (N) Frequency (Hz)
0 13.062
1 40.506
2 55.297
3 66.774
4 76.484
5 85.062
10 118.81
15 144.76
20 166.66
25 185.95
30 203.41
35 219.46
40 234.41
43.231 243.64

Analytical Solution
The analytical solution to this problem is described by the following equation:
GS3

Where:  μ = Linear density of the string

Τ = Tensile force on the string
L = Free vibrational length of the string
n = frequency harmonic

For this experiment the frequency harmonic will be calculated only for n = 1 to compare with the FEA results.  The calculated frequencies at corresponding tensions are as follows:

Force (N) Frequency (Hz)
0 0
1 36.61
2 51.77
3 63.40
4 73.21
5 81.85
10 115.76
15 141.78
20 163.71
25 183.03
30 200.50
35 216.57
40 231.52
43.231 240.69

 

Results Comparison
The results for small values of tension reveal some discrepancy. The discrepancy is due to the analytical equation using a string with no stiffness, and the SOLIDWORKS Simulation analysis using a wire with material (steel) stiffness. The results reveal that this stiffness has less impact at higher tensions and where the SOLIDWORKS Simulation analysis resembles the analytical solution within 4%.

  FEA Analytical  
Force (N) Frequency (Hz) Frequency (Hz) % Error
0 13.062 0.00 N/A
1 40.506 36.61 10.65%
2 55.297 51.77 6.81%
3 66.774 63.40 5.31%
4 76.484 73.21 4.47%
5 85.062 81.85 3.92%
10 118.81 115.76 2.64%
15 144.76 141.78 2.10%
20 166.66 163.71 1.80%
25 185.95 183.03 1.59%
30 203.41 200.50 1.45%
35 219.46 216.57 1.34%
40 234.41 231.52 1.25%
43.231 243.64 240.69 1.23%

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