# Kane’s Formalism Used to the Vibration Analysis of a Wind Water Pump

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## Abstract

**:**

## 1. Introduction

## 2. Finite Element Kinematics

## 3. Kane’s Equations Used in Conjunction with FEA

## 4. Study of a Planar Mechanism Used in a Wind Water Pump

## 5. Conclusions and Discussions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 6.**Acceleration field of the five points analyzed at an engine angular velocity of 140 rpm. The value of the maximum acceleration was 14.33 m/s

^{2}.

**Figure 8.**Four eigenfrequencies (eigenfrequencies 3, 4, 5, and 6) of the beam discretized in 10 finite elements for an angular velocity at 140 rpm.

**Figure 9.**The section from the procedure of deriving the motion equations differences using Kane’s or Lagrange’s formalism. The use of Lagrange’s formalism.

**Figure 10.**The section from the procedure of deriving the motion equations differences using Kane’s or Lagrange’s formalism. The use of Kane’s formalism.

$\frac{\partial {E}_{c}}{\partial {\dot{q}}_{k}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\frac{\partial {E}_{c}}{\partial {q}_{k}}$ | $\left(\frac{\partial {\left\{{v}_{{M}^{\prime}}\right\}}_{G}}{\partial {\left\{\dot{\delta}\right\}}_{G}}\right)$ | ||||

Number of elements | Number of elements | ||||

1 | 10 | 30 | 1 | 10 | 30 |

Number of operations | |||||

78 | 1008 | 9486 | 6 | 33 | 93 |

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**MDPI and ACS Style**

Mitu, G.L.; Chircan, E.; Scutaru, M.L.; Vlase, S.
Kane’s Formalism Used to the Vibration Analysis of a Wind Water Pump. *Symmetry* **2020**, *12*, 1030.
https://doi.org/10.3390/sym12061030

**AMA Style**

Mitu GL, Chircan E, Scutaru ML, Vlase S.
Kane’s Formalism Used to the Vibration Analysis of a Wind Water Pump. *Symmetry*. 2020; 12(6):1030.
https://doi.org/10.3390/sym12061030

**Chicago/Turabian Style**

Mitu, Gabriel Leonard, Eliza Chircan, Maria Luminita Scutaru, and Sorin Vlase.
2020. "Kane’s Formalism Used to the Vibration Analysis of a Wind Water Pump" *Symmetry* 12, no. 6: 1030.
https://doi.org/10.3390/sym12061030