# Vibration Analysis of a Guitar considered as a Symmetrical Mechanical System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Guitar as a Mechanical System Subjected to Vibration

- In case of plate with one free edge (where $x={x}_{0}$), we assume that the bending moment ${M}_{y}$ and shear force ${T}_{zx}$ are zero. Thus, the condition where the bending moment along the free edge is zero, is:$$\begin{array}{c}{\left(\frac{{\partial}^{2}w\left(x,y\right)}{\partial {x}^{2}}+\nu \frac{{\partial}^{2}w\left(x,y\right)}{\partial {y}^{2}}\right)|}_{x={x}_{0}}=0,\text{}\\ {\left(\frac{{\partial}^{3}w\left(x,y\right)}{\partial {x}^{3}}+(2-\nu )\frac{{\partial}^{3}w\left(x,y\right)}{\partial x\partial {y}^{2}}\right)|}_{x={x}_{0}}=0,\end{array}$$
- In case of the plate being supported at all edges $\left(x={x}_{0}\right)$, the displacement are null and along the all edges, the bending moment is zero, too. In Equation (8), the boundary conditions are readily seen to be satisfied exactly:$$\begin{array}{l}{w\left(x,y\right)|}_{x={x}_{0}}=0\\ {\left(\frac{{\partial}^{2}w\left(x,y\right)}{\partial {x}^{2}}+\nu \frac{{\partial}^{2}w\left(x,y\right)}{\partial {y}^{2}}\right)|}_{x={x}_{0}}=0\end{array}$$
- For a plate with one clamped edge $\left(x={x}_{0}\right)$, the displacement and rotation in a perpendicular plan to the side are null:$$\begin{array}{l}{w\left(x,y\right)|}_{x={x}_{0}}=0\\ {\left(\frac{\partial w}{\partial x}\right)|}_{x={x}_{0}}=0\end{array}$$

## 3. Properties of a System Consisting of Two Identical Parts

_{1}) and (S

_{r}) (Figure 3 and Figure 4). We denote by Δ

_{a}the common nodes of the two structures (S

_{1}), with Δ

_{l}the nodes of the left structure (S

_{1}), different from Δ

_{a}and with Δ

_{r}the nodes of the right structure (S

_{r}), different from Δ

_{a}. The equations of the undamped free vibrations of the entire structure (S), the left substructure (S

_{1}) (Equation (13)) and the right substructure (S

_{r}) (Equation (14)) are, respectively [4,28,29,30]:

**P1**—the eigenvalues for the subsystem (S

_{l}) (for the differential, Equations (1) or (2)) are eigenvalues for the system (S)).

_{l}) verified the eigenvalues problem for the whole system (S). That means that the solutions of Equations (16) are also solutions of the algebraic Equation (17):

**P2**—for the common eigenvalues of the system presented in Figure 3a and of the system presented in Figure 3b, the eigenvectors are of the form:

**Proof:**

**P3**—for the other eigenvalues, (not obtained from (S

_{l})), the eigenvectors are of the form (symmetric eigenmodes) [32,33]:

**Proof**:

## 4. Finite Element Analysis (FEA) of Guitar Plates

^{3}), system of beams; keeping Poisson’s coefficient constant (ν = 0.36) and shear’s modulus (G = 5000 MPa). The values of the Young’s modulus E and the density ρ were taken from the literature and based on the results of the analytical modeling [35,36,37].

## 5. Discussion: Use of the Symmetry in the Calculus of the Eigenvalues of the Guitar

- The first eigenmode of vibration (1,1) presents the same shape, regardless the stiffening pattern. Compare to plates with transversal bars, the plates without them (Figure 6, Figure 7 and Figure 8) are characterized by a more extended shape along the longitudinal axis. Additionally, the plates without transverse stiffening elements have the second eigenmode of the type (1,2) compared to the other plates whose eigenmode is of the shape (2,1);
- The stiffening bars change the eigenmodes and the order on the plates. Additionally, the increases of number of bars leads to the natural frequency also increased (Table 2);
- The eigenmodes for the plates with the complete stiffening bars: radial, oblique and transverse bars, regardless of their number, are similar;

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The main parts of a guitar: 1—top plate; 2—back plate; 3—guitar’s neck; 4—heel; 5—bridge; 6—strings.

**Figure 5.**Different reinforcement solutions for the guitar plates: (

**a**) simple plate with acoustic hole (denoted PSah) without stiffening elements; (

**b**) plate with three transversal reinforcing bars (denoted P3BT); (

**c**) plate with three radial resonance bars, symmetric disposed related to longitudinal axis of plate (denoted P3BR); (

**d**) plate with five radial resonance bars, symmetric disposed related to longitudinal axis of plate radial (denoted P5BR); (

**e**) plate with three radial and two transversal bars (denoted P3BR2T); (

**f**) plate with five radial and two transverse bars (denoted P5BR2T); (

**g**) plate with three radial bars, two transverse and two oblique (denoted P3BR2V); (

**h**) plate with five radial bars, two transverse and two oblique bars (denoted P5BR2V); (

**i**) plate with seven radial bars two transverse and two oblique bars (denoted P7BR2V) [35].

**Figure 6.**Guitar plate without acoustic hole PS: (

**a**–

**d**) symmetrical eigenmodes; (

**e**,

**f**) skew symmetrical eigenmodes.

**Figure 7.**Guitar plate with acoustic hole: (

**a**–

**d**) symmetrical eigenmodes PSah; (

**e**,

**f**) skew symmetrical eigenmodes.

**Figure 8.**Guitar plate with three radial bars: (

**a**–

**c**), symmetrical eigenmodes P3BR; (

**e**,

**f**) skew symmetrical eigenmodes.

**Figure 9.**Guitar plate with acoustic hole and three transversal bars, P3BT: (

**a**–

**c**), symmetrical eigenmodes; (

**d**–

**f**) skew symmetrical eigenmodes.

**Figure 10.**Guitar plate with acoustic hole, three symmetrical radial bars and two transversal bars, P3BR2T: (

**a**–

**d**) symmetrical eigenmodes; (

**e**,

**f**) skew symmetrical eigenmodes.

**Table 1.**Influence of the acoustic hole on the eigenvalues of the guitar plates for: E = 13,000 MPa, G = 2300 MPa, $\nu $ = 0.4, ρ = 500 kg/m

^{3}, h =2.5 mm.

All Eigen-Mode | Eigen Frequency of Plate without Acoustic Hole, (Hz) | Eigen Frequency of Plate with Acoustic Hole, (Hz) | Difference between the Two Cases (%) | Symmetrical/ Skew Symmetrical Modes | Eigen Frequency of Plate without Acoustic Hole, (Hz) | Eigen Frequency of Plate with Acoustic Hole, (Hz) |
---|---|---|---|---|---|---|

1 | 191.64 | 191.65 | +0.00052 | Symmetric | ||

2 | 295.79 | 310.84 | +4.5 | Symmetric | 296.21 | 310.23 |

3 | 405.56 | 404.99 | −0.14 | Skew (1) | ||

4 | 437.14 | 428.00 | −2.09 | Symmetric | ||

5 | 638.70 | 598.35 | −6.8 | Skew (2) | 638.22 | 598.43 |

6 | 646.94 | 619.39 | −4.258 | Symmetric | ||

7 | 702.26 | 695.79 | −0.921 | Symmetric | ||

8 | 727.18 | 698.61 | −3.92 | Skew (3) | 727.67 | 698.35 |

9 | 931.33 | 914.99 | −1.754 | Skew (4) | 931.73 | 914.46 |

10 | 943.61 | 944.98 | +0.145 | Symmetric |

**Table 2.**The eigenvalues of the guitar plates with different stiffening pattern, for: G = 2300 MPa, $\nu $ = 3.6, ρ = 450 kg/m

^{3}, h = 2.5 mm (Legend: PS – simple plate without acoustic hole; PSah - simple plate with acoustic hole, without stiffening bars; P3BR – plate with three radial resonance bars, symmetric disposed related to longitudinal axis of plate; P5BR – plate with five radial resonance bars, symmetric disposed related to longitudinal axis of plate radial; P3BT – plate with three transversal reinforcing bars; P3BR2T – plate with three radial and two transversal bars; P5BR2T – plate with five radial and two transverse bars; P3BR2V – plate with three radial bars, two transverse and two oblique; P5BR2V – plate with five radial bars, two transverse and two oblique bars; P7BR2V - plate with seven radial bars two transverse and two oblique bars.

Frequency (Hz) | |||
---|---|---|---|

Plates | 14,000 MPa | 12,000 MPa | 10,000 MPa |

PS | 101 | 92 | 87 |

PSah | 106 | 98 | 89 |

P3BR | 207 | 175 | 155 |

P5BR | 209 | 193 | 176 |

P3BT | 210 | 194 | 178 |

P3BR2T | 262 | 242 | 221 |

P5BR2T | 264 | 243 | 223 |

P3BR2V | 261 | 241 | 220 |

P5BR2V | 264 | 243 | 224 |

P7BR2V | 265 | 245.72 | 224.43 |

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**MDPI and ACS Style**

Stanciu, M.D.; Vlase, S.; Marin, M.
Vibration Analysis of a Guitar considered as a Symmetrical Mechanical System. *Symmetry* **2019**, *11*, 727.
https://doi.org/10.3390/sym11060727

**AMA Style**

Stanciu MD, Vlase S, Marin M.
Vibration Analysis of a Guitar considered as a Symmetrical Mechanical System. *Symmetry*. 2019; 11(6):727.
https://doi.org/10.3390/sym11060727

**Chicago/Turabian Style**

Stanciu, Mariana D., Sorin Vlase, and Marin Marin.
2019. "Vibration Analysis of a Guitar considered as a Symmetrical Mechanical System" *Symmetry* 11, no. 6: 727.
https://doi.org/10.3390/sym11060727