# Direct Determination of Dynamic Elastic Modulus and Poisson’s Ratio of Timoshenko Rods

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analysis

#### 2.1. Finite Element Analysis

#### 2.2. Frequency Ratio Analysis

_{1}, only a single polynomial expression is needed for A

_{1}. Equation (13) was found from the best-fit curve shown in Figure 4. Equation (13) can be used to obtain ${A}_{1}$ for any diameter/length (D/L) ratio:

## 3. Experiments

#### Experimental Data

## 4. Data Analysis

#### Sensitivity Analysis

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Values of ${\chi}_{t}$ and ${\chi}_{b}$ for different length/diameter ratios at various Poisson’s ratios

Length/Diameter | Poisson’s Ratio | ${\mathit{\gamma}}_{1}$ | ${\mathit{\gamma}}_{2}$ | ${\mathit{\chi}}_{\mathit{t}}$ | ${\mathit{\chi}}_{\mathit{b}}$ |
---|---|---|---|---|---|

2 | 0 | 3.948 | 5.476 | 2.599 | 3.699 |

0.1 | 3.938 | 5.431 | 2.626 | 3.617 | |

0.2 | 3.928 | 5.389 | 2.653 | 3.541 | |

0.3 | 3.919 | 5.348 | 2.679 | 3.469 | |

0.4 | 3.909 | 5.309 | 2.705 | 3.402 | |

0.5 | 3.900 | 5.271 | 2.731 | 3.338 | |

3 | 0 | 4.275 | 6.336 | 4.256 | 4.827 |

0.1 | 4.267 | 6.305 | 4.288 | 4.768 | |

0.2 | 4.259 | 6.275 | 4.319 | 4.712 | |

0.3 | 4.252 | 6.246 | 4.349 | 4.658 | |

0.4 | 4.244 | 6.218 | 4.379 | 4.607 | |

0.5 | 4.237 | 6.191 | 4.408 | 4.558 | |

4 | 0 | 4.439 | 6.801 | 6.508 | 5.511 |

0.1 | 4.433 | 6.776 | 6.542 | 5.460 | |

0.2 | 4.427 | 6.753 | 6.576 | 5.412 | |

0.3 | 4.422 | 6.730 | 6.608 | 5.365 | |

0.4 | 4.417 | 6.708 | 6.640 | 5.321 | |

0.5 | 4.411 | 6.686 | 6.672 | 5.278 | |

5 | 0 | 4.530 | 7.088 | 9.372 | 5.992 |

0.1 | 4.526 | 7.068 | 9.408 | 5.948 | |

0.2 | 4.522 | 7.049 | 9.443 | 5.906 | |

0.3 | 4.518 | 7.031 | 9.477 | 5.866 | |

0.4 | 4.514 | 7.013 | 9.510 | 5.827 | |

0.5 | 4.510 | 6.996 | 9.543 | 5.790 | |

8 | 0 | 4.645 | 7.496 | 21.707 | 6.782 |

0.1 | 4.643 | 7.485 | 21.745 | 6.755 | |

0.2 | 4.641 | 7.475 | 21.782 | 6.729 | |

0.3 | 4.639 | 7.465 | 21.818 | 6.704 | |

0.4 | 4.637 | 7.455 | 21.853 | 6.680 | |

0.5 | 4.636 | 7.446 | 21.888 | 6.656 | |

10 | 0 | 4.675 | 7.613 | 33.073 | 7.035 |

0.1 | 4.673 | 7.606 | 33.111 | 7.016 | |

0.2 | 4.672 | 7.598 | 33.149 | 6.997 | |

0.3 | 4.671 | 7.591 | 33.185 | 6.978 | |

0.4 | 4.669 | 7.584 | 33.221 | 6.960 | |

0.5 | 4.668 | 7.577 | 33.256 | 6.943 | |

20 | 0 | 4.716 | 7.789 | 127.726 | 7.441 |

0.1 | 4.715 | 7.786 | 127.766 | 7.435 | |

0.2 | 4.715 | 7.784 | 127.804 | 7.429 | |

0.3 | 4.715 | 7.782 | 127.841 | 7.423 | |

0.4 | 4.714 | 7.780 | 127.877 | 7.418 | |

0.5 | 4.714 | 7.778 | 127.913 | 7.412 | |

50 | 0 | 4.728 | 7.843 | 790.220 | 7.572 |

0.1 | 4.728 | 7.842 | 790.259 | 7.571 | |

0.2 | 4.728 | 7.842 | 790.298 | 7.570 | |

0.3 | 4.728 | 7.842 | 790.335 | 7.569 | |

0.4 | 4.728 | 7.841 | 790.372 | 7.568 | |

0.5 | 4.727 | 7.841 | 790.408 | 7.568 | |

100 | 0 | 4.729 | 7.851 | 3156.257 | 7.592 |

0.1 | 4.729 | 7.850 | 3156.297 | 7.592 | |

0.2 | 4.729 | 7.850 | 3156.335 | 7.591 | |

0.3 | 4.729 | 7.850 | 3156.373 | 7.591 | |

0.4 | 4.729 | 7.850 | 3156.409 | 7.591 | |

0.5 | 4.729 | 7.850 | 3156.446 | 7.591 |

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**Figure 1.**Mesh of 75 mm (D) × 300 mm (L) rod: (

**a**) 1st bending mode: 2101.5 Hz, (

**b**) 2nd bending mode: 4910.6 Hz, (

**c**) 1st torsional mode: 3622.1 Hz.

**Figure 2.**Relationship between ${\chi}_{t}$ and Poisson’s ratio with different rod dimensions (diameter × length): (

**a**) 100 mm × 400 mm, (

**b**) 20 mm × 160 mm, and (

**c**) normalized ${\chi}_{t}$ for different L/D ratios (L/D = 2, 4, 8, and 10).

**Figure 3.**Plot of B

_{1}versus the length/diameter ratio: (

**a**) $\frac{L}{D}\le 10$ and (

**b**) $\frac{L}{D}>10$.

**Figure 5.**Relationship between ${\chi}_{b}$ and Poisson’s ratio with different rod dimensions (diameter × length): (

**a**) 100 mm × 400 mm, (

**b**) 20 mm × 160 mm, and (

**c**) normalized ${\chi}_{b}$ for different L/D ratios (L/D = 2, 4, 8, and 10).

**Figure 9.**Picture of experiment: (

**a**) Impact hammer, accelerometer, and specimens; (

**b**) Experimental setup.

**Figure 10.**(

**a**) Typical acceleration time history and (

**b**) Typical response power spectrum (transverse mode).

**Figure 11.**(

**a**) Typical impact loading time history and (

**b**) Typical impact power spectrum (transverse mode).

**Figure 12.**(

**a**) Typical acceleration time history and (

**b**) Typical response power spectrum density (torsional mode).

**Figure 13.**(

**a**) Typical impact loading time history and (

**b**) Typical impact power spectrum (torsional mode).

Test | Elastic Modulus (GPa) | Poisson’s Ratio |
---|---|---|

1 | 25.0 | 0.1 |

2 | 25.0 | 0.15 |

3 | 25.0 | 0.2 |

4 | 10.0 | 0.2 |

5 | 20.0 | 0.2 |

6 | 30.0 | 0.2 |

Test | Transverse Mode (Hz) | Torsional Mode (Hz) | |||||||
---|---|---|---|---|---|---|---|---|---|

1st | 2nd | ||||||||

Exact | FEM | Error (%) | Exact | FEM | Error (%) | Exact | FEM | Error (%) | |

1 | 2103.083 | 2101.5 | 0.08% | 4914.196 | 4910.6 | 0.07% | 3626.66 | 3622.1 | 0.13% |

2 | 2100.394 | 2098.9 | 0.07% | 4896.9 | 4894.0 | 0.06% | 3546.943 | 3542.4 | 0.13% |

3 | 2097.75 | 2096.3 | 0.07% | 4879.969 | 4877.4 | 0.05% | 3472.262 | 3467.8 | 0.13% |

4 | 1326.551 | 1325.6 | 0.07% | 3085.938 | 3084.3 | 0.05% | 2195.749 | 2193.0 | 0.13% |

5 | 1876.349 | 1875.0 | 0.07% | 4364.927 | 4362.6 | 0.05% | 3105.793 | 3101.8 | 0.13% |

6 | 2297.917 | 2296.3 | 0.07% | 5345.615 | 5342.8 | 0.05% | 3803.585 | 3798.8 | 0.13% |

Length/Diameter (L/D) | |||||||||
---|---|---|---|---|---|---|---|---|---|

Coefficient | 2 | 3 | 4 | 5 | 8 | 10 | 20 | 50 | 100 |

${B}_{1}$ | 2.59893 | 4.25642 | 6.50793 | 9.37206 | 21.7075 | 33.0728 | 127.7261 | 790.2198 | 3156.257 |

${A}_{1}$ | 0.26573 | 0.30686 | 0.33084 | 0.34551 | 0.36556 | 0.37104 | 0.37898 | 0.38135 | 0.38617 |

**Table 4.**Calculation of ${\gamma}_{1}$, ${\gamma}_{2}$, and ${\chi}_{b}$ for different length/diameter ratios.

Poisson’s Ratio, v | L/D = 4 | L/D = 20 | L/D = 50 | L/D = 100 | L/D = 200 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\gamma}}_{1}$ | ${\mathit{\gamma}}_{2}$ | ${\mathit{\chi}}_{\mathit{b}}$ | ${\mathit{\gamma}}_{1}$ | ${\mathit{\gamma}}_{2}$ | ${\mathit{\chi}}_{\mathit{b}}$ | ${\mathit{\gamma}}_{1}$ | ${\mathit{\gamma}}_{2}$ | ${\mathit{\chi}}_{\mathit{b}}$ | ${\mathit{\gamma}}_{1}$ | ${\mathit{\gamma}}_{2}$ | ${\mathit{\chi}}_{\mathit{b}}$ | ${\mathit{\gamma}}_{1}$ | ${\mathit{\gamma}}_{2}$ | ${\mathit{\chi}}_{\mathit{b}}$ | |

0 | 4.3609 | 6.801 | 5.511 | 4.7108 | 7.789 | 7.441 | 4.7269 | 7.843 | 7.572 | 4.7293 | 7.851 | 7.592 | 4.7298 | 7.853 | 7.597 |

0.1 | 4.3526 | 6.776 | 5.460 | 4.7103 | 7.786 | 7.435 | 4.7268 | 7.842 | 7.571 | 4.7292 | 7.850 | 7.592 | 4.7298 | 7.853 | 7.597 |

0.2 | 4.3443 | 6.753 | 5.412 | 4.7097 | 7.784 | 7.429 | 4.7267 | 7.842 | 7.570 | 4.7292 | 7.850 | 7.591 | 4.7298 | 7.852 | 7.597 |

0.3 | 4.3362 | 6.730 | 5.365 | 4.7091 | 7.782 | 7.423 | 4.7266 | 7.842 | 7.569 | 4.7292 | 7.850 | 7.591 | 4.7298 | 7.852 | 7.597 |

0.4 | 4.3281 | 6.708 | 5.321 | 4.7085 | 7.780 | 7.418 | 4.7266 | 7.841 | 7.568 | 4.7292 | 7.850 | 7.591 | 4.7298 | 7.852 | 7.597 |

0.5 | 4.3202 | 6.686 | 5.278 | 4.708 | 7.778 | 7.412 | 4.7265 | 7.841 | 7.568 | 4.7291 | 7.850 | 7.591 | 4.7298 | 7.852 | 7.597 |

Length/Diameter (L/D) | |||||||||
---|---|---|---|---|---|---|---|---|---|

Coefficient | 2 | 3 | 4 | 5 | 8 | 10 | 20 | 50 | 100 |

${E}_{1}$ | 3.69886 | 4.282677 | 5.51093 | 5.99205 | 6.78228 | 7.03545 | 7.44104 | 7.57243 | 7.59195 |

${D}_{1}$ | −0.83478 | −0.60134 | −0.51647 | −0.44455 | −0.27264 | −0.19948 | −0.06117 | −0.01043 | −0.00263 |

${C}_{1}$ | 0.22969 | 0.12914 | 0.10186 | 0.08175 | 0.04227 | 0.02862 | 0.00748 | 0.00120 | 0.0002986 |

Size (mm × mm) | FEM | Theoretical | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Bending (Hz) | Torsional (Hz) | A_{1} | B_{1} | C_{1} | D_{1} | E_{1} | Bending (Hz) | Torsional (Hz) | |||

1st | 2nd | 1st | 2nd | ||||||||

100 × 2000 | 112.09 | 305.44 | 784.8 | 0.381786 | 127.7267 | 0.00748 | −0.06116 | 7.4433 | 112.32 | 306.079 | 787.62 |

100 × 400 | 2468.2 | 5718.0 | 3932.7 | 0.330818 | 6.50796 | 0.10188 | −0.51670 | 5.5142 | 2470.39 | 5723.82 | 3938.11 |

Material | Properties | ||
---|---|---|---|

Diameter (mm) | Length (mm) | Density (kg m^{−3}) | |

Aluminum | 63.50 | 736.60 | 2780.56 |

Steel | 46.83 | 279.4 | 8143.13 |

Concrete | 99.63 | 622.30 | 2288.75 |

Rod | Transverse Frequency (Hz) | Torsional Frequency (Hz) | $\mathit{v}$ | E (GPa) | |||
---|---|---|---|---|---|---|---|

1st | 2nd | Method 1 | Method 2 | Method 1 | Method 2 | ||

Aluminum | 531.05 ± 0.03 | 1415.85 ± 0.07 | 2,083.02 ± 0.45 | 0.434 | 0.364 | 75.12 | 75.08 |

Steel | 2589.83 ± 0.33 | 6460.95 ± 0.43 | 5686.97 ± 0.87 | 0.332 | 0.300 | 219.10 | 218.91 |

Concrete | 847.02 ± 0.20 | 2130.61 ± 0.43 | 1972.06 ± 0.77 | 0.283 | 0.241 | 35.38 | 35.35 |

Measured Properties | Method | Aluminum | Steel | Concrete | |||
---|---|---|---|---|---|---|---|

E (GPa) | v | E (GPa) | v | E (GPa) | v | ||

Mass (±0.01 kg) | 1 | 75.12 ± 0.1 | - | 219.10 ± 0.5 | - | 35.38 ± 0.03 | - |

2 | 75.08 ± 0.1 | - | 218.91 ± 0.5 | - | 35.35 ± 0.03 | - | |

L (±0.01 mm) | 1 | 75.12 ± 0.003 | 0.434 ± 0.00004 | 219.10 ± 0.02 | 0.332 ± 0.00009 | 35.38 ± 0.002 | 0.283 ± 0.00004 |

2 | 75.08 ± 0.003 | 0.364 ± 0.00008 | 218.91 ± 0.02 | 0.300 ± 0.0002 | 35.35 ± 0.002 | 0.241 ± 0.00009 | |

D (±0.01 mm) | 1 | 75.12 ± 0.05 | 0.434 ± 0.0004 | 219.10 ± 0.2 | 0.332 ± 0.0005 | 35.38 ± 0.001 | 0.283 ± 0.0002 |

2 | 75.08 ± 0.05 | 0.364 ± 0.0009 | 218.91 ± 0.2 | 0.300 ± 0.001 | 35.35 ± 0.001 | 0.241 ± 0.0005 | |

n_{1} (±0.25 Hz) | 1 | 75.12 ± 0.07 | 0.434 ± 0.001 | 219.10 ± 0.04 | 0.332 ± 0.0003 | 35.38 ± 0.02 | 0.283 ± 0.0008 |

2 | 75.08 ± 0.1 * | 0.364 ± 0.05 * | 218.91 ± 0.06 | 0.300 ± 0.004 | 35.35 ± 0.03 | 0.241 ± 0.01 | |

n_{2} (±0.25 Hz) | 1 | - | - | - | - | - | - |

2 | 75.08 ± 0.01 ** | 0.364 ± 0.02 ** | 218.91 ± 0.008 | 0.300 ± 0.001 | 35.35 ± 0.004 | 0.241 ± 0.004 | |

n” (±0.25 Hz) | 1 | 75.12 ± 0.07 | 0.434 ± 0.001 | 219.10 ± 0.001 | 0.332 ± 0.0001 | 35.38 ± 0.0003 | 0.283 ± 0.0003 |

2 | - | - | - | - | - | - |

_{1}, E = 75.08 ± 0.02 GPa and v = 0.364 ± 0.009. ** For a frequency resolution of ±0.05 Hz for n

_{2}, E = 75.08 ± 0.002 GPa and v = 0.364 ± 0.004.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Leon, G.; Chen, H.-L.
Direct Determination of Dynamic Elastic Modulus and Poisson’s Ratio of Timoshenko Rods. *Vibration* **2019**, *2*, 157-173.
https://doi.org/10.3390/vibration2010010

**AMA Style**

Leon G, Chen H-L.
Direct Determination of Dynamic Elastic Modulus and Poisson’s Ratio of Timoshenko Rods. *Vibration*. 2019; 2(1):157-173.
https://doi.org/10.3390/vibration2010010

**Chicago/Turabian Style**

Leon, Guadalupe, and Hung-Liang (Roger) Chen.
2019. "Direct Determination of Dynamic Elastic Modulus and Poisson’s Ratio of Timoshenko Rods" *Vibration* 2, no. 1: 157-173.
https://doi.org/10.3390/vibration2010010