Ryan Dark
Ryan has been in the GoEngineer technical support team since February 2008 where he most notably provides support for all FEA and CFD software offered by SOLIDWORKS. His most recent accolade is the title of Elite Application Engineer awarded by SOLIDWORKS Corp.
SOLIDWORKS Simulation – Frequency Analysis of Tensioned Guitar Strings
February 1st, 2016 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.
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.
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:
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% |
Learn more about SOLIDWORKS Simulation and join us for a webinar on February 12 from 12PM to 1PM PST: Troubleshooting Most Common Simulation Errors.
Related
Tags: FEA, GoEngineer, Simulation, SOLIDWORKS
Categories: SOLIDWORKS, SOLIDWORKS Simulation
This entry was posted
on Monday, February 1st, 2016 at 12:57 pm.
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