# Frequency Response Evaluation of Guitar Bodies with Different Bracing Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Mathematical Model of the Guitar as a Symmetrical System

**Theorem**

**T1.**

_{1}is the dimension of one single part, n is the total dimension of the system).

**Proposition**

**1:**

**Theorem**

**T2.**

**Theorem**

**T3.**

#### Forced Vibrations in Systems with Symmetries

**Theorem**

**T4.**

**Proof:**

_{1}vectors correspond to eigenfrequencies ${\omega}_{i}^{2}{=}_{1}{\omega}_{i}^{2}$. Then:

**Theorem**

**T5.**

**Proof:**

## 3. Finite Element Analysis (FEA) of the Resonance Body of the Guitar

#### 3.1. Geometrical and Structural Models

#### 3.2. Discretization of Models and Loading

#### 3.3. FEA Results and Discussion

## 4. The Experimental Analysis to Forced Vibrations

#### Experimental Set-Up

## 5. Discussion

## 6. Conclusions

- Each type of bracing system from guitar body generates the nodal patterns, which overlap with the patterns found from modal analysis (Table 4);
- The first resonance is noticed first, with higher amplitude, then the second, third, etc. All appear at a specific frequency, their resonant frequency and all have different patterns in accordance with bracing systems applied on soundboards;
- In case of forced vibration of 110 Hz, all analyzed structures has one vibration antinode, in a symmetric mode. For 146 Hz, the quasi skew symmetric vibration modes are recorded, has three vibration antinodes with two vertical nodal lines, with vibrating surfaces more or less extended according to bracing patterns;
- The amplitude spectra becomes more complex with increasing the frequency. With increasing the number of bars from bracing systems, the overtones increases too.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The structural symmetry of guitar body, without strings systems—case of seven fan bracing systems.

**Figure 2.**Different types of guitar bodies: (

**a**) C3BT—without fan bracing system and three transversal bars; (

**b**) C3BR2T—with three radial (fan) bars and two transversal bars; (

**c**) C5BR2T—with five radial (fan) bars and two transversal bars.

**Figure 3.**The section of wood used in guitar construction: (

**a**) the main orthogonal section (R—radial, T—tangential, L—longitudinal); (

**b**) resonance spruce wood (Picea Abies L) for the top plate of the guitar, cut in the longitudinal-radial direction; (

**c**) curly maple (Acer Pseudoplatanus) for the back plate of the guitar, cut in the longitudinal-radial direction.

**Figure 4.**The placement of the analyzed nodes: (

**a**) C3BT; (

**b**) C3BR2T; (

**c**) C5BR2T; (

**d**) the measured points of dynamic response on the soundboard of the guitar body; (

**e**) the measured points of dynamic response on the backplate of guitar body; (

**f**) the measured points of dynamic response on ribs of the guitar body.

**Figure 6.**The variation of amplitudes in the analyzed points on the front plate of the acoustic box, according to the excitation frequency: (

**a**) along the axis of the structure; (

**b**) transversally on the axis.

**Figure 7.**The variation of vibration amplitudes for the back of the acoustic box: (

**a**) along the axis of the structure; (

**b**) comparison with the top plate.

**Figure 9.**Comparison of dynamic responses of different guitar bodies: (

**a**) the captured vibration of node P1, the same point of applied force, in the case of the soundboard; (

**b**) the captured vibration of node P1, in the case of the backplates; (

**c**) the captured vibration of node P2, the near point of P1, in the case of the soundboard; (

**d**) the captured vibration of node P1, in the case of the backplates; (

**e**) the captured vibration of node P5, the point near the soundhole, in the case of the soundboard; (

**f**) the captured vibration of node P5, in the case of the backplates; (

**g**) the captured vibration of node P6, in the case of the soundboard; (

**h**) the captured vibration of node P6, in the case of the backplates.

**Figure 10.**The experimental set-up: 1—frequency generator; 2—B&K mini-shaker; 3—force transmitter; 4—supports; 5—guitar box; 6—accelerometers; 7—the Pulse 12 platform; 8—computer for data visualization and processing.

**Figure 11.**Comparisons regarding the dynamic response in the time domain of guitar bodies with different bar systems, for frequency of 110 Hz: (

**a**) body type C3BT; (

**b**) guitar body type C3BR2T;

**(c)**guitar body type C5BR2T; (

**d**) body type C7BR2T; (

**e**) body type 3BR2V.

**Figure 12.**Comparisons regarding the dynamic response in the time domain of guitar bodies with different bar systems, for frequency of 146 Hz: (

**a**) body type C3BT; (

**b**) guitar body type C3BR2T;

**(c)**guitar body type C5BR2T; (

**d**) body type C7BR2T; (

**e**) body type 3BR2V.

**Figure 13.**Comparisons regarding the dynamic response in the time domain of guitar bodies with different bar systems, for a frequency of 440 Hz: (

**a**) body type C3BT; (

**b**) guitar body type C3BR2T;

**(c)**guitar body type C5BR2T; (

**d**) body type C7BR2T; (

**e**) body type 3BR2V.

**Figure 14.**Fast Fourier Transform (FFT) for an excitation frequency of 440 Hz:

**a**) body type C3BT; (

**b**) guitar body type C3BR2T;

**(c)**guitar body type C5BR2T; (

**d**) body type C7BR2T; (

**e**) body type 3BR2V.

Physical and Mechanical Properties of Wood | Top Plate Spruce | Backplate/Sides Maple | |
---|---|---|---|

Density (kg/m^{3}) | 420 | 685 | |

Length of guitar body (mm) | 480 | 480 | |

Width of guitar body (mm) | 380 | 380 | |

Height of guitar body (mm) | 100 | 100 | |

Thickness of plate (mm) | 2.5 | 2.5 | |

Young’s moduli (MPa) | E_{L} | 14,128 | 11,000 |

E_{R} | 8310 | 6471 | |

E_{T} | 1441 | 1122 | |

Shear moduli (MPa) | G_{RT} | 5730 | 1200 |

G_{LT} | 1975 | 414 | |

G_{LR} | 1273 | 267 | |

Poisson ratio | ν_{LR} | 0.45 | 0.44 |

ν_{RL} | 0.03 | 0.09 | |

ν_{LT} | 0.54 | 0.48 | |

ν_{TL} | 0.019 | 0.036 | |

ν_{RT} | 0.56 | 0.78 | |

ν_{TR} | 0.3 | 0.38 |

Type of Guitar Body | Frequency [Hz] at Maximum Amplitudes | ||||
---|---|---|---|---|---|

C3BT | 200/220 | 320/340 | 600 | ||

C3BR2T | 240 | 280 | 400 | ||

C5BR2T | 200 | 300 | 400 | 620 | 780 |

Excitation frequency [Hz] | 110 | 146 | 196 | 246 | 329 | 413 | 440 | 588 | 720 | 980 |

Voltage of signal amplification [V] | 1.8 | 1.8 | 1.2 | 1.2 | 1.2 | 1.5 | 2.1 | 3.5 | 3.1 | 1.5 |

Amperage [A] | 0.5 | 0.5 | 0.3 | 0.3 | 0.3 | 0.3 | 0.4 | 0.7 | 0.7 | 0.3 |

Types of Strutting System | |||||
---|---|---|---|---|---|

C3BT | C3BR2T | C5BR2T | C7BR2T | C3BR2V | |

110 Hz Symm | (1,1) _{1} | (1,1) _{1} | (1,1) _{1} | (1,1) _{1} | (1,1) _{1} |

146 Hz Quasi Skew Symm | (1,1) | (1,3) _{1} | (1,3) _{1} | (1,3) _{1} | (1,3) _{1} |

196 Hz Symm. | (1,1) | (1,1) _{2} | (1,1) _{2} | (1,1) _{2} | (1,1) _{2} |

246 Hz Symm. | (1,1) | (1,1) _{3} | (1,1) _{3} | (1,1) _{3} | (1,1) _{3} |

329 Hz Skew Symm. | (0,1) | (2,3) _{1} | (1,3) _{1} | (1,2) | (1,2) |

440 Hz Skew Symm. | (2,1) _{2} | (1,4) | (1,3) _{2} | (2,3) _{1} | (1,3) |

588 Hz Quasi Skew Symm | (1,3) | (1,5) | (1,5) | (2,3) _{2} | (1,4) |

720 Hz Quasi Skew Symm. | (3,3) | (1,6) | (1,6) | (1,5) | (1,5) |

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## Share and Cite

**MDPI and ACS Style**

Mihălcică, M.; Stanciu, M.D.; Vlase, S.
Frequency Response Evaluation of Guitar Bodies with Different Bracing Systems. *Symmetry* **2020**, *12*, 795.
https://doi.org/10.3390/sym12050795

**AMA Style**

Mihălcică M, Stanciu MD, Vlase S.
Frequency Response Evaluation of Guitar Bodies with Different Bracing Systems. *Symmetry*. 2020; 12(5):795.
https://doi.org/10.3390/sym12050795

**Chicago/Turabian Style**

Mihălcică, Mircea, Mariana D. Stanciu, and Sorin Vlase.
2020. "Frequency Response Evaluation of Guitar Bodies with Different Bracing Systems" *Symmetry* 12, no. 5: 795.
https://doi.org/10.3390/sym12050795