# Simplified Approach to Nonlinear Vibration Analysis of Variable Stiffness Plates

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulation

#### 2.1. Kinematics and Generalized Strains

#### 2.2. Generalized Forces

#### 2.3. Variational Framework

#### 2.4. Boundary Conditions

## 3. Approximate Solution

#### 3.1. Spatial Approximation via Ritz Method

#### 3.2. Differential Quadrature Method

#### 3.3. Arc-Length Harmonic Balance Method

#### 3.4. Single-Mode Solution

## 4. Results

#### 4.1. Validation

#### 4.1.1. Linear Vibrations

#### 4.1.2. Nonlinear Vibrations

#### 4.2. Comparison between Methods

#### 4.3. Parametric Studies

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Vibration shapes of plates with layup $[\u23290|45\u232a,\u2329-45|-60\u232a,\u23290|45\u232a]$: (

**a**–

**d**) simply-supported; (

**e**–

**h**) clamped.

**Figure 4.**DQ method–comparison with [20].

**Figure 5.**HB method–comparison with [20].

**Figure 6.**Vibration shapes of plates with layup ${[\u232945|90\u232a,\u23290|-45\u232a,\u23290|45\u232a,\u2329-45|0\u232a]}_{\mathrm{s}}$.

**Figure 8.**VS laminate [<${T}_{0}$|45>, 90<${T}_{0}$|45>]${}_{\mathrm{s}}$: effect of angle ${T}_{0}$.

**Figure 9.**VS laminate ${[\mp \u2329{T}_{0}|{T}_{1}\u232a]}_{2\mathrm{s}}$: (

**a**) maximum deflection ${w}_{\mathrm{max}}/h$; (

**b**) slope ${\theta}_{\mathrm{H}}$.

${\mathit{E}}_{1}$ (MPa) | ${\mathit{E}}_{2}$ (MPa) | ${\mathit{G}}_{12}$, ${\mathit{G}}_{13}$ (MPa) | ${\mathit{G}}_{23}$ (MPa) | ${\mathit{\nu}}_{12}$ | |
---|---|---|---|---|---|

M1 | $f({E}_{1}/{E}_{2})$ | 1.00 | 0.60 | 0.50 | 0.25 |

M2 | 120,500 | 9630 | 3580 | 3580 | 0.32 |

M3 | 173,000 | 7200 | 3760 | 3760 | 0.29 |

M4 | 131,700 | 9860 | 4210 | 4210 | 0.28 |

**Table 2.**Frequency parameter ${\overline{\omega}}_{1}$ for straight-fiber plates: comparison with [29].

${\mathit{E}}_{1}/{\mathit{E}}_{2}$ | $\mathit{a}/\mathit{h}$ | n${}_{\mathbf{dof}}$ = 48 | n${}_{\mathbf{dof}}$ = 108 | n${}_{\mathbf{dof}}$ = 363 | Exact [29] |
---|---|---|---|---|---|

10 | 5 | $8.2731$ | $8.2718$ | $8.2718$ | $8.272$ |

10 | $9.8430$ | $9.8410$ | $9.8410$ | $9.841$ | |

20 | 5 | $9.5276$ | $9.5263$ | $9.5263$ | $9.526$ |

10 | $12.2205$ | $12.2180$ | $12.2180$ | $12.218$ | |

40 | 5 | $10.7885$ | $10.7873$ | $10.7873$ | $10.787$ |

10 | $15.1100$ | $15.1073$ | $15.1073$ | $15.107$ |

Ref. [21] | Ref. [43] | Ritz | Ritz | |||||
---|---|---|---|---|---|---|---|---|

n${}_{\mathbf{dof}}$ = 500 | n${}_{\mathbf{dof}}$ = 500 | n${}_{\mathbf{dof}}$ = 506 | n${}_{\mathbf{dof}}$ = 156 | |||||

Mode | SSSS | CCCC | SSSS | CCCC | SSSS | CCCC | SSSS | CCCC |

1 | 355.41 | 567.56 | 358.49 | 579.40 | 357.30 | 575.62 | 360.70 | 578.08 |

2 | 600.50 | 831.39 | 589.90 | 821.53 | 589.12 | 818.39 | 595.32 | 821.76 |

3 | 986.65 | 1253.18 | 960.36 | 1225.79 | 960.46 | 1222.74 | 980.81 | 1254.50 |

4 | 1027.55 | 1448.46 | 1075.21 | 1493.76 | 1073.03 | 1479.45 | 1086.21 | 1500.97 |

5 | 1309.92 | 1719.96 | 1327.88 | 1726.96 | 1322.84 | 1713.21 | 1350.30 | 1764.76 |

6 | 1506.90 | 1818.98 | 1474.67 | 1775.16 | 1466.67 | 1771.14 | 1690.45 | 1908.24 |

7 | 1743.33 | 2175.80 | 1726.71 | 2135.76 | 1718.08 | 2121.56 | 2265.94 | 2220.54 |

8 | 2106.31 | 2505.73 | 2137.13 | 2443.53 | 2085.50 | 2437.05 | 2324.50 | 2772.34 |

9 | 2171.03 | 2750.46 | 2262.35 | 2706.78 | 2227.60 | 2690.78 | 2591.59 | 2947.36 |

**Table 4.**Nonlinear frequency parameters ${\widehat{\omega}}_{\mathrm{nl}}={\omega}_{\mathrm{nl}}a\sqrt{\frac{\rho}{{E}_{2}}}$: comparison with [22].

$|{\mathit{w}}_{\mathbf{max}}|/\mathit{h}=$ | 0.2 | 0.6 | 1.0 | |||
---|---|---|---|---|---|---|

HBS | Ref. [22] | HBS | Ref. [22] | HBS | Ref. [22] | |

${[\mp \u232940|10\u232a]}_{\mathrm{s}}$ | 0.316 | 0.325 | 0.335 | 0.345 | 0.370 | 0.383 |

${[\mp \u232940|20\u232a]}_{\mathrm{s}}$ | 0.310 | 0.318 | 0.328 | 0.338 | 0.362 | 0.375 |

${[\mp \u232940|30\u232a]}_{\mathrm{s}}$ | 0.302 | 0.308 | 0.319 | 0.328 | 0.351 | 0.364 |

${[\mp \u232940|40\u232a]}_{\mathrm{s}}$ | 0.292 | 0.297 | 0.308 | 0.316 | 0.338 | 0.350 |

${[\mp \u232940|50\u232a]}_{\mathrm{s}}$ | 0.282 | 0.287 | 0.297 | 0.304 | 0.325 | 0.335 |

${[\mp \u232940|60\u232a]}_{\mathrm{s}}$ | 0.273 | 0.278 | 0.286 | 0.294 | 0.311 | 0.322 |

${[\mp \u232940|70\u232a]}_{\mathrm{s}}$ | 0.267 | 0.272 | 0.279 | 0.287 | 0.300 | 0.313 |

${[\mp \u232940|80\u232a]}_{\mathrm{s}}$ | 0.264 | 0.269 | 0.275 | 0.283 | 0.296 | 0.309 |

$|{\mathit{w}}_{\mathbf{max}}|/\mathit{h}$ | Time (s) | ||
---|---|---|---|

DQ | f = 0.5 N | 1.0427 | $4.3$ |

HB | 1.0451 | $8.5$ | |

HBS | 1.0875 | $<<0.1$ | |

DQ | f = 1.0 N | 1.6502 | $9.5$ |

HB | 1.6668 | $7.6$ | |

HBS | 1.8080 | $<<0.1$ |

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**MDPI and ACS Style**

Andérez González, J.; Vescovini, R.
Simplified Approach to Nonlinear Vibration Analysis of Variable Stiffness Plates. *J. Compos. Sci.* **2023**, *7*, 30.
https://doi.org/10.3390/jcs7010030

**AMA Style**

Andérez González J, Vescovini R.
Simplified Approach to Nonlinear Vibration Analysis of Variable Stiffness Plates. *Journal of Composites Science*. 2023; 7(1):30.
https://doi.org/10.3390/jcs7010030

**Chicago/Turabian Style**

Andérez González, Jorge, and Riccardo Vescovini.
2023. "Simplified Approach to Nonlinear Vibration Analysis of Variable Stiffness Plates" *Journal of Composites Science* 7, no. 1: 30.
https://doi.org/10.3390/jcs7010030