# Development of the Dynamic Stiffness Method for the Out-of-Plane Natural Vibration of an Orthotropic Plate

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## Abstract

**:**

## 1. Introduction

## 2. Contributions and Relevant Scope of Present Work

- The DSM was formulated to investigate the natural vibration response of thin orthotropic plates.
- The W–W algorithm was applied to compute the natural frequency of the orthotropic plate.
- The DSM results were compared with the published literature and the finite element method.
- A new set of DSM results is reported for different aspect ratios, thickness ratios, and modulus ratios, which may be used as benchmark solutions for comparison.

## 3. Mathematical Formulation

#### 3.1. Description of Geometrical Property

#### 3.2. Equations of Motion

#### 3.3. Boundary Conditions

## 4. Formulation of Dynamic Stiffness (DS) Matrix with Levy Solution

#### 4.1. Dynamic Stiffness (DS) Matrix Assembly Procedure with Boundary Conditions

- Displacement $({W}_{i})$ is penalized for simply supported (S) boundary conditions.
- Displacement (${W}_{i})$ and rotation (${\varphi}_{i})$ are penalized for clamped (C) boundary conditions.
- No penalty is implemented for the free (F) boundary condition.

#### 4.2. Application of the Wittrick and Williams (W–W) Algorithm

## 5. Results and Discussions

#### 5.1. Comparative Study

_{1}/E

_{2}= 40, whereas the minimum error encountered is 0.00% at b/L = 0.5 and E

_{1}/E

_{2}= 25 for the FF boundary condition. It can be seen from Table 1 that most of the errors lie within 2%. It is observed that the DSM results of a thin orthotropic plate in Table 1 are in excellent agreement with the published literature and the FEM results.

#### 5.2. Parameter Studies

_{1}/E

_{2}), aspect ratios (b/L), and thickness ratios (b/h). It could be observed from Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8 that the natural frequencies, in general, decrease with an increase in Young’s modulus, aspect ratio, and thickness ratio for the orthotropic plate. It is observed in Table 3, Table 4, Table 5, Table 6 and Table 7 that with boundary conditions changing from SC to SF, the natural frequencies increase, with the increase in the modulus ratio (E

_{1}/E

_{2}) from 3 to 50 because of an increase in the stiffness of the plate. Compared with Ref. [15], the relative errors of DSM are reported in brackets in Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8.

_{1}/E

_{2}= 3 for the CC plate in Table 6. The next highest percentage of errors is −1.7539 (for the SC plate at b/L = 1, b/h = 20 and E

_{1}/E

_{2}= 20) in Table 8. Thus, for all cases reported in Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8, the error is less than 2%.

_{1}/E

_{2}) and the thickness ratio (a/h) on natural frequencies for all boundary conditions are presented in Figure 6 and Figure 7, respectively. The following observations are obtained from Figure 6 and Figure 7, respectively.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 6.**Effect on the non-dimensional frequency of the orthotropic plate for the Levy-type boundary condition with Young’s modulus ratio $\left(\frac{{E}_{1}}{{E}_{2}}\right)$ and the given value of the side-thickness ratio $\left(\frac{b}{L}=0.5\right)$.

**Figure 7.**Effect on the non-dimensional frequency of the square orthotropic plate for the Levy-type boundary condition with a side-thickness ratio $\left(b/h\right)$ and the given value of Young’s modulus ratio $\left(\frac{{E}_{1}}{{E}_{2}}=40\right)$.

**Table 1.**Comparison of fundamental frequency parameters $\varpi =\omega \frac{{b}^{2}}{h}\sqrt{\frac{\rho}{{E}_{2}}}$ of the Levy-type orthotropic plate (b/h = 100).

b/L | E_{1}/E_{2} | Method | Boundary Conditions | |||||
---|---|---|---|---|---|---|---|---|

CC | SC | SS | FC | FS | FF | |||

0.5 | 10 | Ref [52] | 20.6543 | 14.3450 | 9.3421 | 3.5614 | 1.3190 | 0.7124 |

Ref [15] | 20.5603 | 14.3137 | 9.3331 | 3.5600 | 1.3190 | 0.7123 | ||

DSM | 20.5857 | 14.3116 | 9.3209 | 3.5154 | 1.3188 | 0.7123 | ||

FEM | 20.6766 | 14.3565 | 9.3455 | 3.5558 | 1.3186 | 0.7141 | ||

Error (%) | (−0.1234) | (0.0144) | (0.1309) | (1.2677) | (0.0152) | (0.0011) | ||

25 | Ref [52] | 32.4390 | 22.4259 | 14.4578 | 5.3051 | 1.3193 | 0.7123 | |

Ref [15] | 32.0795 | 22.3069 | 14.4245 | 5.3003 | 1.3192 | 0.7122 | ||

DSM | 32.4229 | 22.4148 | 14.4507 | 5.3042 | 1.3191 | 0.7122 | ||

FEM | 32.2347 | 22.3539 | 14.4430 | 5.2925 | 1.3189 | 0.7141 | ||

Error (%) | (−1.0590) | (−0.4813) | (−0.1815) | (−0.0735) | (0.0076) | (0.0000) | ||

40 | Ref [52] | 40.9633 | 28.2855 | 18.1876 | 6.6030 | 1.3194 | 0.7123 | |

Ref [15] | 40.2478 | 28.0480 | 18.1215 | 6.5937 | 1.3193 | 0.7122 | ||

DSM | 40.9620 | 28.2846 | 18.1870 | 6.6010 | 1.3190 | 0.7122 | ||

FEM | 40.4100 | 28.0814 | 18.1426 | 6.5826 | 1.3189 | 0.7141 | ||

Error (%) | (−1.7436) | (−0.8366) | (−0.3603) | (−0.1106) | (0.0227) | (0.0000) | ||

1 | 10 | Ref [52] | 21.2889 | 15.2042 | 10.4963 | 5.0586 | 3.6114 | 2.8503 |

Ref [15] | 21.2078 | 15.1747 | 10.4863 | 5.0564 | 3.6105 | 2.8496 | ||

DSM | 21.2405 | 15.1709 | 10.4750 | 5.0144 | 3.6112 | 2.8501 | ||

FEM | 21.3090 | 15.2098 | 10.4949 | 5.0538 | 3.6125 | 2.8569 | ||

Error (%) | (−0.1539) | (0.0248) | (0.1080) | (0.8376) | (−0.0194) | (−0.0175) | ||

25 | Ref [52] | 32.8464 | 22.9847 | 15.2278 | 6.4146 | 3.6118 | 2.8493 | |

Ref [15] | 32.5515 | 22.8835 | 15.1972 | 6.4100 | 3.6110 | 2.8486 | ||

DSM | 32.8303 | 22.9736 | 15.2207 | 6.4014 | 3.6115 | 2.8490 | ||

FEM | 32.6351 | 22.9073 | 15.2089 | 6.4025 | 3.6128 | 2.8558 | ||

Error (%) | (−0.8491) | (−0.3921) | (−0.1544) | (0.1343) | (−0.0138) | (−0.0140) | ||

40 | Ref [52] | 41.2866 | 28.7305 | 18.8052 | 7.5253 | 3.6121 | 2.8492 | |

Ref [15] | 40.7062 | 28.5337 | 18.7477 | 7.5178 | 3.6112 | 2.8485 | ||

DSM | 41.2853 | 28.7296 | 18.8046 | 7.5045 | 3.6120 | 2.8491 | ||

FEM | 40.7277 | 28.5213 | 18.7566 | 7.5051 | 3.6130 | 2.8557 | ||

Error (%) | (−1.4027) | (−0.6818) | (−0.3028) | (0.1772) | (−0.0221) | (−0.0211) | ||

2 | 10 | Ref [52] | 25.5184 | 20.5941 | 17.1364 | 12.9377 | 12.2379 | 11.4094 |

Ref [15] | 25.4427 | 20.5543 | 17.1129 | 12.9238 | 12.2259 | 11.3981 | ||

DSM | 25.4651 | 20.5536 | 17.1046 | 12.9145 | 12.2298 | 11.3998 | ||

FEM | 25.4958 | 20.5658 | 17.1086 | 12.9324 | 12.2395 | 11.4254 | ||

Error (%) | (−0.0879) | (0.0032) | (0.0483) | (0.0720) | (−0.0319) | (−0.0149) | ||

25 | Ref [52] | 35.7303 | 26.8537 | 20.3682 | 13.5562 | 12.2305 | 11.3993 | |

Ref [15] | 35.5237 | 26.7659 | 20.3288 | 13.5404 | 12.2186 | 11.3880 | ||

DSM | 35.7135 | 26.8417 | 20.3596 | 13.5504 | 12.2301 | 11.3895 | ||

FEM | 35.4925 | 26.7502 | 20.3265 | 13.5433 | 12.2319 | 11.4155 | ||

Error (%) | (−0.5316) | (−0.2822) | (−0.1514) | (−0.0738) | (−0.0940) | (−0.0132) | ||

40 | Ref [52] | 43.6154 | 31.9099 | 23.1622 | 14.1271 | 12.2301 | 11.3977 | |

Ref [15] | 43.2415 | 31.7630 | 23.1043 | 14.1093 | 12.2182 | 11.3864 | ||

DSM | 43.6141 | 31.9090 | 23.1616 | 14.1270 | 12.2298 | 11.3944 | ||

FEM | 43.0347 | 31.6809 | 23.0942 | 14.1072 | 12.2312 | 11.4140 | ||

Error (%) | (−0.8543) | (−0.4576) | (−0.2473) | (−0.1253) | (−0.0949) | (−0.0702) |

E_{1}/E_{2} | ||||||
---|---|---|---|---|---|---|

3 | 10 | 20 | 30 | 40 | 50 | |

DSM | 7.1504 | 10.2475 | 13.4417 | 15.7097 | 17.4294 | 19.0676 |

20 × 20 FEM | 7.0938 | 10.1434 | 13.1326 | 15.3657 | 17.1616 | 18.6632 |

40 × 40 FEM | 7.0610 | 10.1132 | 13.1070 | 15.3336 | 17.1285 | 18.6298 |

80 × 80 FEM | 7.0507 | 10.1040 | 13.0932 | 15.3250 | 17.1192 | 18.6203 |

100 × 100 FEM | 7.0494 | 10.1030 | 13.0918 | 15.3239 | 17.1179 | 18.6188 |

**Table 3.**Comparison of the fundamental frequency parameters $\left(\varpi =\omega \frac{{b}^{2}}{h}\sqrt{\frac{\rho}{{E}_{2}}}\right)$ of the orthotropic plate with the SC boundary conditions.

b/L | b/h | Method | E_{1}/E_{2} | |||||
---|---|---|---|---|---|---|---|---|

3 | 10 | 20 | 30 | 40 | 50 | |||

0.5 | 20 | RPT [15] | 8.0410 | 13.6063 | 18.2286 | 21.3634 | 23.7191 | 25.5851 |

DTM [53] | 8.0500 | - | 18.2250 | 21.3750 | 23.7500 | 25.7900 | ||

TCM [53] | 8.0418 | - | 18.2310 | 21.3647 | 23.7211 | 25.5865 | ||

DSM | 8.0368 | 13.5844 | 18.1881 | 21.3584 | 23.7235 | 25.2657 | ||

FEM | 8.0196 | 13.4668 | 17.8312 | 20.6852 | 21.7298 | 21.7850 | ||

Error (%) | 0.0523 | 0.1609 | 0.2227 | 0.0234 | −0.0184 | 1.2642 | ||

50 | RPT 15] | 8.1690 | 14.2179 | 19.7562 | 23.9265 | 27.3668 | 30.3311 | |

DTM [53] | 8.1750 | - | 19.9270 | 23.9440 | 27.4445 | 30.4350 | ||

TCM [53] | 8.1692 | - | 19.7581 | 23.9296 | 27.3706 | 30.3356 | ||

DSM | 8.1648 | 14.2078 | 19.7124 | 23.9210 | 27.3718 | 30.3117 | ||

FEM | 8.1883 | 14.2348 | 19.7344 | 23.8543 | 27.2119 | 30.0908 | ||

Error (%) | 0.0514 | 0.0711 | 0.2222 | 0.0232 | −0.0184 | 0.0639 | ||

1 | 20 | RPT [15] | 9.4219 | 14.2277 | 19.0510 | 22.2738 | 24.7632 | 26.7775 |

DTM [53] | 9.3740 | - | 19.1240 | 22.3250 | 24.8120 | 26.9144 | ||

TCM [53] | 9.4227 | - | 18.3073 | 22.2747 | 24.7638 | 26.7789 | ||

DSM | 9.4305 | 14.2204 | 19.3911 | 22.2704 | 24.7508 | 26.6002 | ||

FEM | 9.3187 | 15.0824 | 18.3699 | 21.0717 | 23.2367 | 25.3476 | ||

Error (%) | −0.0912 | 0.0513 | −1.7539 | 0.0153 | 0.0501 | 0.6665 | ||

50 | RPT [15] | 9.5898 | 14.4921 | 20.4197 | 24.5288 | 27.9597 | 30.9416 | |

DTM [53] | 9.4265 | - | 20.5705 | 24.5450 | 27.9880 | 30.8850 | ||

TCM [53] | 9.5900 | - | 20.4210 | 24.5305 | 27.9616 | 30.9441 | ||

DSM | 9.5986 | 14.5044 | 20.4134 | 24.5251 | 27.9457 | 30.7368 | ||

FEM | 9.5888 | 15.0722 | 20.3359 | 24.3373 | 27.6373 | 30.4720 | ||

Error (%) | −0.0919 | −0.0848 | 0.0310 | 0.0149 | 0.0500 | 0.6664 | ||

2 | 20 | RPT [15] | 16.3187 | 19.5797 | 23.2516 | 26.1972 | 28.6627 | 30.7812 |

DTM [53] | 16.3940 | - | 23.3000 | 26.2455 | 28.7151 | 30.8502 | ||

TCM [53] | 16.3198 | - | 23.2521 | 26.1978 | 28.6630 | 30.7814 | ||

DSM | 16.3173 | 19.5944 | 23.2510 | 26.1632 | 28.6581 | 30.7657 | ||

FEM | 16.2784 | 19.5250 | 23.1165 | 26.7783 | 28.1530 | 30.4003 | ||

50 | Error (%) | 0.0086 | −0.0750 | 0.0026 | 0.1300 | 0.0161 | 0.0504 | |

RPT [15] | 16.8218 | 20.4196 | 24.6437 | 28.1975 | 31.3099 | 34.1026 | ||

DTM [53] | 16.8825 | - | 24.6202 | 28.2062 | 31.3150 | 34.1000 | ||

TCM [53] | 16.8221 | - | 24.6442 | 28.1982 | 31.3105 | 34.1033 | ||

DSM | 16.8204 | 20.4214 | 24.6431 | 28.1609 | 31.3050 | 34.0855 | ||

FEM | 16.7497 | 20.339 | 24.462 | 27.8501 | 30.7535 | 33.3047 | ||

Error (%) | 0.0084 | −0.0088 | 0.0024 | 0.1298 | 0.0158 | 0.0502 |

**Table 4.**Comparison of the fundamental frequency parameters $\left(\varpi =\omega \frac{{b}^{2}}{h}\sqrt{\frac{\rho}{{E}_{2}}}\right)$ of the orthotropic plate with the SS boundary conditions.

b/L | b/h | Method | E_{1}/E_{2} | |||||
---|---|---|---|---|---|---|---|---|

3 | 10 | 20 | 30 | 40 | 50 | |||

0.5 | 20 | RPT [15] | 5.4685 | 9.1141 | 12.4009 | 14.7974 | 16.7105 | 18.3073 |

DTM [53] | 5.4774 | - | 12.4005 | 14.7975 | 16.7100 | 18.3070 | ||

TCM [53] | 5.4685 | - | 12.4009 | 14.7974 | 16.7105 | 18.3073 | ||

DSM | 5.4996 | 9.1045 | 12.4112 | 14.9467 | 16.6942 | 18.3017 | ||

FEM | 5.4419 | 9.0998 | 12.3710 | 14.7391 | 16.6188 | 18.1777 | ||

Error (%) | −0.5655 | 0.1054 | −0.0830 | −0.9989 | 0.0976 | 0.0306 | ||

50 | RPT [15] | 5.5126 | 9.3044 | 12.8804 | 15.6246 | 17.9239 | 19.9333 | |

DTM [53] | 5.5000 | - | 12.9060 | 15.6250 | 17.9240 | 19.9300 | ||

TCM [53] | 5.5126 | - | 12.8804 | 15.6247 | 17.9239 | 19.9333 | ||

DSM | 5.4609 | 9.3130 | 12.9564 | 15.7823 | 17.9064 | 19.9273 | ||

FEM | 5.5125 | 9.3014 | 12.8910 | 15.5475 | 17.9278 | 19.9310 | ||

Error (%) | 0.9476 | −0.0923 | −0.5867 | −0.9992 | 0.0975 | 0.0301 | ||

1 | 20 | RPT [15] | 7.2194 | 10.2349 | 13.2676 | 15.5845 | 17.4839 | 19.1002 |

DTM [53] | 7.2200 | - | 13.2500 | 15.5840 | 17.4835 | 19.1000 | ||

TCM [53] | 7.2194 | - | 13.2676 | 15.5846 | 17.4839 | 19.1002 | ||

DSM | 7.1504 | 10.2475 | 13.4417 | 15.7097 | 17.4294 | 19.0676 | ||

FEM | 7.0938 | 10.1434 | 13.1326 | 15.3657 | 17.1616 | 18.6632 | ||

Error (%) | 0.9650 | −0.1230 | −1.2952 | −0.7970 | 0.3127 | 0.1710 | ||

50 | RPT [15] | 7.3012 | 10.4530 | 13.7360 | 16.3474 | 18.5726 | 20.5377 | |

DTM [53] | 7.3000 | - | 13.7600 | 16.3772 | 18.6072 | 20.5765 | ||

TCM [53] | 7.3012 | - | 13.7360 | 16.3474 | 18.5726 | 20.5377 | ||

DSM | 7.2312 | 10.4555 | 13.7963 | 16.4788 | 18.5149 | 20.5027 | ||

FEM | 7.2778 | 10.4426 | 13.7230 | 16.3228 | 18.5285 | 20.4713 | ||

Error (%) | 0.9686 | −0.0239 | −0.4370 | −0.7972 | 0.3119 | 0.1705 | ||

2 | 20 | RPT [15] | 14.9772 | 16.5030 | 18.4742 | 20.2036 | 21.7468 | 23.1427 |

DTM [53] | 14.9795 | - | 18.4740 | 20.2036 | 21.7469 | 23.1428 | ||

TCM [53] | 14.9773 | - | 18.4742 | 20.2036 | 21.7468 | 23.1427 | ||

DSM | 14.7602 | 16.5247 | 18.5482 | 20.3326 | 21.6428 | 23.0351 | ||

FEM | 14.6185 | 16.2143 | 18.1645 | 19.7863 | 21.1685 | 22.3638 | ||

Error (%) | 1.4702 | −0.1313 | −0.3990 | −0.6344 | 0.4805 | 0.4671 | ||

50 | RPT [15] | 15.3796 | 17.0294 | 19.1992 | 21.1436 | 22.9151 | 24.5504 | |

DTM [53] | 15.3502 | - | 19.2156 | 21.1663 | 22.9120 | 24.5480 | ||

TCM [53] | 15.3796 | - | 19.1992 | 21.1436 | 22.9151 | 24.5504 | ||

DSM | 15.2813 | 17.0145 | 19.2761 | 21.2786 | 22.8056 | 24.4363 | ||

FEM | 15.2894 | 16.9572 | 19.1294 | 21.0565 | 22.7983 | 24.3920 | ||

Error (%) | 0.6430 | 0.0876 | −0.3989 | −0.6347 | 0.4803 | 0.4670 |

**Table 5.**Comparison of the fundamental frequency parameters $\left(\varpi =\omega \frac{{b}^{2}}{h}\sqrt{\frac{\rho}{{E}_{2}}}\right)$ of the orthotropic plate with the SF boundary conditions.

b/L | b/h | Method | E_{1}/E_{2} | |||||
---|---|---|---|---|---|---|---|---|

3 | 10 | 20 | 30 | 40 | 50 | |||

0.5 | 20 | RPT [15] | 1.3160 | 1.3163 | 1.3165 | 1.3166 | 1.3166 | 1.3167 |

DTM [53] | 1.3164 | - | 1.3165 | 1.3166 | 1.3168 | 1.3169 | ||

TCM [53] | 1.3161 | - | 1.3165 | 1.3166 | 1.3167 | 1.3167 | ||

DSM | 1.3162 | 1.3162 | 1.3162 | 1.3164 | 1.3164 | 1.3165 | ||

FEM | 1.2903 | 1.2903 | 1.2906 | 1.2906 | 1.2906 | 1.2906 | ||

Error (%) | −0.0152 | 0.0076 | 0.0228 | 0.0152 | 0.0152 | 0.0152 | ||

50 | RPT [15] | 1.3183 | 1.3186 | 1.3188 | 1.3189 | 1.3189 | 1.3189 | |

DTM [53] | 1.3189 | - | 1.3189 | 1.3190 | 1.3190 | 1.3190 | ||

TCM [53] | 1.3183 | - | 1.3188 | 1.3189 | 1.3189 | 1.3190 | ||

DSM | 1.3118 | 1.3154 | 1.3187 | 1.3187 | 1.3870 | 1.3880 | ||

FEM | 1.3133 | 1.3135 | 1.3137 | 1.3138 | 1.3137 | 1.3138 | ||

Error (%) | 0.4946 | 0.2433 | 0.0076 | 0.0152 | 0.0152 | 0.0076 | ||

1 | 20 | RPT [15] | 3.5900 | 3.5851 | 3.5854 | 3.5857 | 3.5858 | 3.5859 |

DTM [53] | 3.5949 | - | 3.5875 | 3.5874 | 3.5876 | 3.5877 | ||

TCM [53] | 3.5900 | - | 3.5855 | 3.5857 | 3.5859 | 3.5860 | ||

DSM | 3.5949 | 3.5901 | 3.5853 | 3.5856 | 3.5857 | 3.5885 | ||

FEM | 3.5358 | 3.5303 | 3.5300 | 3.5301 | 3.5301 | 3.5302 | ||

Error (%) | −0.1363 | −0.1393 | 0.0028 | 0.0028 | 0.0028 | −0.0725 | ||

50 | RPT [15] | 3.6121 | 3.6072 | 3.6074 | 3.6077 | 3.6079 | 3.6080 | |

DTM [53] | 3.6309 | - | 3.6080 | 3.6082 | 3.6084 | 3.6085 | ||

TCM [53] | 3.6122 | - | 3.6075 | 3.6078 | 3.6080 | 3.6081 | ||

DSM | 3.6108 | 3.6001 | 3.6001 | 3.6066 | 3.6078 | 3.6079 | ||

FEM | 3.6039 | 3.5986 | 3.5984 | 3.5991 | 3.5991 | 3.5991 | ||

Error (%) | 0.0371 | 0.1972 | 0.2028 | 0.0305 | 0.0028 | 0.0028 | ||

2 | 20 | RPT [15] | 11.9528 | 11.9012 | 11.8952 | 11.8945 | 11.8944 | 11.8946 |

DTM [53] | 12.0400 | - | 11.9100 | 11.9000 | 11.9000 | 11.9050 | ||

TCM [53] | 11.9551 | - | 11.8960 | 11.8949 | 11.8947 | 11.8947 | ||

DSM | 11.9520 | 11.8951 | 11.8950 | 11.8948 | 11.8946 | 11.8946 | ||

FEM | 11.8526 | 11.8090 | 11.7943 | 11.7919 | 11.7910 | 11.7906 | ||

Error (%) | 0.0067 | 0.0513 | 0.0017 | −0.0025 | −0.0017 | 0.0000 | ||

50 | RPT [15] | 12.2370 | 12.1817 | 12.1752 | 12.1743 | 12.1742 | 12.1742 | |

DTM [53] | 12.3926 | - | 12.1802 | 12.1792 | 12.1790 | 12.1791 | ||

TCM [53] | 12.2375 | - | 12.1755 | 12.1745 | 12.1744 | 12.1745 | ||

DSM | 12.2370 | 12.1785 | 12.1751 | 12.1744 | 12.1742 | 12.1742 | ||

FEM | 12.2274 | 12.1720 | 12.1655 | 12.1655 | 12.1632 | 12.1631 | ||

Error (%) | 0.0000 | 0.0263 | 0.0008 | −0.0008 | 0.0000 | 0.0000 |

**Table 6.**Comparison of the fundamental frequency parameters $\left(\varpi =\omega \frac{{b}^{2}}{h}\sqrt{\frac{\rho}{{E}_{2}}}\right)$ of an orthotropic plate with the CC boundary conditions.

b/L | b/h | Method | E_{1}/E_{2} | |||||
---|---|---|---|---|---|---|---|---|

3 | 10 | 20 | 30 | 40 | 50 | |||

0.5 | 20 | RPT [15] | 11.1940 | 18.6410 | 24.1770 | 27.5890 | 29.9780 | 31.7720 |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 11.0440 | 18.5440 | 24.0010 | 27.4220 | 29.8140 | 31.6450 | ||

FEM | 11.2241 | 18.5395 | 23.7936 | 26.9128 | 29.0241 | 30.5589 | ||

Error (%) | 1.3582 | 0.5231 | 0.7333 | 0.6090 | 0.5501 | 0.4013 | ||

50 | RPT [15] | 11.5350 | 20.2830 | 28.0500 | 33.7400 | 38.3200 | 42.1780 | |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 11.4450 | 20.0450 | 28.0010 | 33.4520 | 38.0150 | 42.0040 | ||

FEM | 11.5988 | 20.3688 | 28.1096 | 33.7420 | 38.2453 | 42.0120 | ||

Error (%) | 0.7864 | 1.1873 | 0.1750 | 0.8609 | 0.8023 | 0.4142 | ||

1 | 20 | RPT [15] | 12.2680 | 19.4910 | 25.2600 | 28.9750 | 31.6500 | 33.6990 |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 12.0540 | 19.3540 | 25.2600 | 28.8990 | 31.6498 | 33.6880 | ||

FEM | 12.2199 | 19.0866 | 24.1686 | 27.2132 | 29.2816 | 30.7884 | ||

Error (%) | 1.7720 | 0.7079 | 0.0000 | 0.2630 | 0.0006 | 0.0327 | ||

50 | RPT [15] | 12.6300 | 20.9640 | 28.6650 | 34.4150 | 39.1030 | 43.0940 | |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 12.6300 | 20.9620 | 28.6420 | 34.3440 | 39.0050 | 43.0920 | ||

FEM | 12.6752 | 20.9800 | 28.5476 | 34.0931 | 38.5449 | 42.2785 | ||

Error (%) | 0.0000 | 0.0095 | 0.0803 | 0.2067 | 0.2512 | 0.0046 | ||

2 | 20 | RPT [15] | 18.1620 | 23.7190 | 29.2240 | 33.2480 | 36.3950 | 38.9550 |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 18.0450 | 23.7180 | 29.2140 | 33.1450 | 36.3470 | 38.5450 | ||

FEM | 17.9076 | 22.9240 | 27.1301 | 29.7723 | 31.6051 | 32.9569 | ||

Error (%) | 0.6484 | 0.0042 | 0.0342 | 0.3108 | 0.1321 | 1.0637 | ||

50 | RPT [15] | 18.8420 | 25.1990 | 32.0380 | 37.5070 | 42.1470 | 46.2150 | |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 18.7540 | 24.9880 | 31.8970 | 37.1240 | 41.8740 | 46.0240 | ||

FEM | 18.8150 | 25.1027 | 31.6413 | 36.6844 | 40.8220 | 44.3360 | ||

Error (%) | 0.4692 | 0.8444 | 0.4420 | 1.0317 | 0.6520 | 0.4150 |

**Table 7.**Comparison of the fundamental frequency parameters $\left(\varpi =\omega \frac{{b}^{2}}{h}\sqrt{\frac{\rho}{{E}_{2}}}\right)$ of an orthotropic plate with the FC boundary conditions.

b/L | b/h | Method | E_{1}/E_{2} | |||||
---|---|---|---|---|---|---|---|---|

3 | 10 | 20 | 30 | 40 | 50 | |||

0.5 | 20 | RPT [15] | 2.3220 | 3.5230 | 4.7060 | 5.6190 | 6.3810 | 7.0410 |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 2.3210 | 3.5220 | 4.6990 | 5.6180 | 6.3740 | 7.0010 | ||

FEM | 2.3013 | 3.4995 | 4.6668 | 4.5382 | 6.2941 | 6.9261 | ||

Error (%) | 0.0431 | 0.0284 | 0.1490 | 0.0178 | 0.1098 | 0.5713 | ||

50 | RPT [15] | 2.3320 | 3.5550 | 4.7800 | 5.7450 | 6.5660 | 7.2910 | |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 2.3310 | 3.5350 | 4.7010 | 5.7440 | 6.5540 | 7.2870 | ||

FEM | 2.3266 | 3.5479 | 4.7691 | 4.6778 | 6.5441 | 7.2632 | ||

Error (%) | 0.0429 | 0.5658 | 1.6805 | 0.0174 | 0.1831 | 0.0549 | ||

1 | 20 | RPT [15] | 4.2160 | 4.9950 | 5.8970 | 6.6620 | 7.3360 | 7.9420 |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 4.2160 | 4.9920 | 5.8240 | 6.5770 | 7.2540 | 7.8450 | ||

FEM | 4.1595 | 4.9287 | 5.8066 | 6.5343 | 7.1612 | 7.7141 | ||

Error (%) | 0.0000 | 0.0601 | 1.2534 | 1.2924 | 1.1304 | 1.2365 | ||

50 | RPT [15] | 4.2500 | 5.0480 | 5.9820 | 6.7820 | 7.4940 | 8.1420 | |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 4.2450 | 5.0120 | 5.8440 | 6.4510 | 7.4240 | 8.0140 | ||

FEM | 4.2402 | 5.0346 | 5.9620 | 6.7541 | 7.4569 | 8.0942 | ||

Error (%) | 0.1178 | 0.7183 | 2.3614 | 5.1310 | 0.9429 | 1.5972 | ||

2 | 20 | RPT [15] | 12.2640 | 12.5540 | 12.9420 | 13.3020 | 13.6470 | 13.9780 |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 12.1240 | 12.4510 | 12.8450 | 13.2040 | 13.6140 | 13.8540 | ||

FEM | 11.8526 | 12.4252 | 12.7885 | 13.1213 | 13.4293 | 13.7178 | ||

Error (%) | 1.1547 | 0.8272 | 0.7552 | 0.7422 | 0.2424 | 0.8950 | ||

50 | RPT [15] | 12.5670 | 12.8740 | 13.2880 | 13.6750 | 14.0460 | 14.4050 | |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 12.4570 | 12.7450 | 13.1440 | 13.2470 | 14.0440 | 14.4020 | ||

FEM | 12.2274 | 12.5838 | 13.2596 | 13.6953 | 14.0022 | 14.3510 | ||

Error (%) | 0.8830 | 1.0122 | 1.0956 | 3.2309 | 0.0142 | 0.0208 |

**Table 8.**Comparison of the fundamental frequency parameters $\left(\varpi =\omega \frac{{b}^{2}}{h}\sqrt{\frac{\rho}{{E}_{2}}}\right)$ of an orthotropic plate with the FF boundary conditions.

b/L | b/h | Method | E_{1}/E_{2} | |||||
---|---|---|---|---|---|---|---|---|

3 | 10 | 20 | 30 | 40 | 50 | |||

0.5 | 20 | RPT [15] | 2.3250 | 2.3260 | 2.3260 | 2.3260 | 2.3260 | 2.3260 |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 2.3240 | 2.3250 | 2.3258 | 2.3258 | 2.3258 | 2.3258 | ||

FEM | 2.3252 | 2.3245 | 2.3260 | 2.3266 | 2.3260 | 2.3260 | ||

Error (%) | 0.0430 | 0.0430 | 0.0086 | 0.0086 | 0.0086 | 0.0086 | ||

50 | RPT [15] | 2.3310 | 2.3310 | 2.3320 | 2.3320 | 2.3320 | 2.3320 | |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 2.3312 | 2.335 | 2.3322 | 2.3325 | 2.3328 | 2.3320 | ||

FEM | 2.3315 | 2.3315 | 2.3325 | 2.3325 | 2.3325 | 2.3325 | ||

Error (%) | −0.0086 | −0.1716 | −0.0086 | −0.0214 | −0.0343 | 0.0000 | ||

1 | 20 | RPT [15] | 2.8380 | 2.8300 | 2.8290 | 2.8290 | 2.8290 | 2.8290 |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 2.8140 | 2.8244 | 2.8344 | 2.8344 | 2.8344 | 2.8344 | ||

FEM | 2.8440 | 2.8371 | 2.8364 | 2.8466 | 2.8466 | 2.8466 | ||

Error (%) | 0.8529 | 0.1983 | −0.1905 | −0.1905 | −0.1905 | −0.1905 | ||

50 | RPT [15] | 2.8560 | 2.8470 | 2.8460 | 2.8460 | 2.8460 | 2.8460 | |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 2.8488 | 2.8444 | 2.8442 | 2.8442 | 2.8442 | 2.8442 | ||

FEM | 2.8626 | 2.8543 | 2.8534 | 2.8549 | 2.8549 | 2.8549 | ||

Error (%) | 0.2527 | 0.0914 | 0.0633 | 0.0633 | 0.0633 | 0.0633 | ||

2 | 20 | RPT [15] | 11.1510 | 11.0960 | 11.0880 | 11.0860 | 11.0850 | 11.0850 |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 11.1500 | 11.0900 | 11.0820 | 11.0810 | 11.0810 | 11.0810 | ||

FEM | 11.3046 | 11.1211 | 11.2471 | 11.2455 | 11.2455 | 11.2455 | ||

Error (%) | 0.0090 | 0.0541 | 0.0541 | 0.0451 | 0.0361 | 0.0361 | ||

50 | RPT [15] | 11.4160 | 11.3570 | 11.3490 | 11.3460 | 11.3460 | 11.3450 | |

DTM [53] | - | - | - | - | - | - | ||

TCM [53] | - | - | - | - | - | - | ||

DSM | 11.3967 | 11.3450 | 11.3460 | 11.3460 | 11.3450 | 11.3450 | ||

FEM | 11.4429 | 11.3850 | 11.3778 | 11.3759 | 11.3759 | 11.3759 | ||

Error (%) | 0.1695 | 0.1058 | 0.0264 | 0.0000 | 0.0088 | 0.0000 |

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## Share and Cite

**MDPI and ACS Style**

Chauhan, M.; Mishra, P.; Dwivedi, S.; Ragulskis, M.; Burdzik, R.; Ranjan, V.
Development of the Dynamic Stiffness Method for the Out-of-Plane Natural Vibration of an Orthotropic Plate. *Appl. Sci.* **2022**, *12*, 5733.
https://doi.org/10.3390/app12115733

**AMA Style**

Chauhan M, Mishra P, Dwivedi S, Ragulskis M, Burdzik R, Ranjan V.
Development of the Dynamic Stiffness Method for the Out-of-Plane Natural Vibration of an Orthotropic Plate. *Applied Sciences*. 2022; 12(11):5733.
https://doi.org/10.3390/app12115733

**Chicago/Turabian Style**

Chauhan, Manish, Pawan Mishra, Sarvagya Dwivedi, Minvydas Ragulskis, Rafał Burdzik, and Vinayak Ranjan.
2022. "Development of the Dynamic Stiffness Method for the Out-of-Plane Natural Vibration of an Orthotropic Plate" *Applied Sciences* 12, no. 11: 5733.
https://doi.org/10.3390/app12115733