# Transverse Vibration of Clamped-Pinned-Free Beam with Mass at Free End

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## Abstract

**:**

## 1. Introduction

## 2. Frequency Calculations

## 3. Calculations of Eigenvalues

## 4. Mode Shapes

## 5. Experimental Validation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Boundary Conditions Applied to Transverse Displacement Equations

## References

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**Figure 7.**Mode shapes for pinned at $a=100$ mm and $\frac{{m}_{\mathrm{attached}}}{{m}_{\mathrm{beam}}}=0.2$.

**Figure 8.**Mode shapes for pinned at $a=200$ mm and $\frac{{m}_{\mathrm{attached}}}{{m}_{\mathrm{beam}}}=0.2$.

**Figure 9.**Mode shapes for pinned at $a=300$ mm and $\frac{{m}_{\mathrm{attached}}}{{m}_{\mathrm{beam}}}=0.2$.

**Figure 10.**Mode shapes for pinned at $a=400$ mm and $\frac{{m}_{\mathrm{attached}}}{{m}_{\mathrm{beam}}}=0.2$.

**Figure 12.**FRF for pinned at $a=50$ mm and $\frac{{m}_{\mathrm{attached}}}{{m}_{\mathrm{beam}}}=0.2$.

**Figure 13.**FRF for pinned at $a=100$ mm and $\frac{{m}_{\mathrm{attached}}}{{m}_{\mathrm{beam}}}=0.2$.

**Figure 14.**FRF for pinned at $a=150$ mm and $\frac{{m}_{\mathrm{attached}}}{{m}_{\mathrm{beam}}}=0.2$.

**Figure 15.**FRF for pinned at $a=200$ mm and $\frac{{m}_{\mathrm{attached}}}{{m}_{\mathrm{beam}}}=0.2$.

Literature [14] | Proposed Model | Difference | |

Mode | (Hz) | (Hz) | (%) |

1 | 19.64 | 19.63 | 0.051 |

2 | 123.07 | 123.02 | 0.041 |

3 | 344.64 | 344.45 | 0.055 |

4 | 675.31 | 674.97 | 0.050 |

5 | 1116.33 | 1115.79 | 0.048 |

**Table 2.**Comparison of analytical results: Clamped-free with mass at free end ($\frac{{m}_{\mathrm{attached}}}{{m}_{\mathrm{beam}}}=0.2$) (Case 2).

Literature [7] | Proposed Model | Difference | |

Mode | (Hz) | (Hz) | (%) |

1 | 14.56 | 14.58 | 0.14 |

2 | 101.47 | 101.64 | 0.17 |

3 | 298.59 | 299.00 | 0.14 |

4 | 603.06 | 604.02 | 0.16 |

5 | 1017.07 | 1018.48 | 0.14 |

Literature [6] | Proposed Model | Difference | |

Mode | (Hz) | (Hz) | (%) |

1 | 41.70 | 41.59 | 0.26 |

2 | 279.25 | 278.46 | 0.28 |

3 | 635.80 | 635.19 | 0.10 |

4 | 899.94 | 897.69 | 0.25 |

5 | 1650.85 | 1646.48 | 0.26 |

Parameter | Value |

L | 505 mm |

base | 50 mm |

height | 6.4 mm |

$\rho $ | 7970 kg/m${}^{3}$ |

E | 190 GPa |

${m}_{\mathrm{attached}}$ | 0.259 Kg |

Proposed Model | Experiment | Difference | |

Mode | (Hz) | (Hz) | (%) |

1 | 16.73 | 17.75 | 6.1 |

2 | 118.28 | 128.88 | 9.0 |

3 | 350.24 | 378.68 | 8.1 |

4 | 710.71 | 872.01 | 22.7 |

5 | 1202.46 | 1400.19 | 16.4 |

Proposed Model | Experiment | Difference | |

Mode | (Hz) | (Hz) | (%) |

1 | 19.48 | 21.37 | 9.7 |

2 | 141.29 | 157.37 | 11.4 |

3 | 423.53 | 440.11 | 3.9 |

4 | 864.86 | 996.16 | 15.2 |

5 | 1462.39 | 1680.39 | 14.9 |

Proposed Model | Experiment | Difference | |

Mode | (Hz) | (Hz) | (%) |

1 | 23.09 | 25.87 | 12.0 |

2 | 173.89 | 195.62 | 12.5 |

3 | 526.12 | 614.09 | 16.7 |

4 | 1015.61 | 1088.84 | 7.2 |

5 | 1292.51 | 1421.86 | 10.0 |

Proposed Model | Experiment | Difference | |

Mode | (Hz) | (Hz) | (%) |

1 | 28.02 | 31.39 | 12.0 |

2 | 221.20 | 259.65 | 17.4 |

3 | 595.97 | 646.99 | 8.6 |

4 | 798.79 | 950.15 | 18.9 |

5 | 1479.27 | 1741.16 | 17.7 |

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## Share and Cite

**MDPI and ACS Style**

Hong, J.; Dodson, J.; Laflamme, S.; Downey, A.
Transverse Vibration of Clamped-Pinned-Free Beam with Mass at Free End. *Appl. Sci.* **2019**, *9*, 2996.
https://doi.org/10.3390/app9152996

**AMA Style**

Hong J, Dodson J, Laflamme S, Downey A.
Transverse Vibration of Clamped-Pinned-Free Beam with Mass at Free End. *Applied Sciences*. 2019; 9(15):2996.
https://doi.org/10.3390/app9152996

**Chicago/Turabian Style**

Hong, Jonathan, Jacob Dodson, Simon Laflamme, and Austin Downey.
2019. "Transverse Vibration of Clamped-Pinned-Free Beam with Mass at Free End" *Applied Sciences* 9, no. 15: 2996.
https://doi.org/10.3390/app9152996