# Symmetric Properties of Eigenvalues and Eigenfunctions of Uniform Beams

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Problem of Transverse Vibrations of a Beam Lying on a Winkler’s Type Foundation

#### 2.1. Clamped-Clamped Euler–Bernoulli Beam

**Theorem**

**1.**

- 1.
- $\sigma \left({A}_{1}\right)\equiv \sigma \left({B}_{1}\right)\cup \sigma \left({C}_{1}\right)$
- 2.
- If $\lambda \in \sigma \left({B}_{1}\right)$ or $\lambda \in \sigma \left({C}_{1}\right)$, then the eigenfunctions of problems $A-\lambda I$ corresponding to the eigenvalues λ are symmetric or asymmetric with respect to the middle of the beam at the point $x=\frac{1}{2}$ on the interval $(0,1)$, respectively.

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**1.**

**Proof**

**of**

**Theorem**

**1.**

#### 2.2. Hinged-Hinged Euler–Bernoulli Beam

**Theorem**

**2.**

- 1.
- $\sigma \left({A}_{2}\right)\equiv \sigma \left({B}_{2}\right)\cup \sigma \left({C}_{2}\right)$
- 2.
- If $\lambda \in \sigma \left({B}_{2}\right)$ or $\lambda \in \sigma \left({C}_{2}\right)$, then the eigenfunctions of problems ${A}_{2}-\lambda I$ corresponding to the eigenvalues λ are symmetric or asymmetric with respect to the middle of the beam at the point $x=\frac{1}{2}$ on the interval $(0,1)$, respectively.

**Proof**

**of**

**Theorem**

**2.**

#### 2.3. Free-Free Euler–Bernoulli Beam

**Theorem**

**3.**

- 1.
- $\sigma \left({A}_{3}\right)\equiv \sigma \left({B}_{3}\right)\cup \sigma \left({C}_{3}\right)$
- 2.
- If $\lambda \in \sigma \left({B}_{3}\right)$ or $\lambda \in \sigma \left({C}_{3}\right)$, then the eigenfunctions of problems ${A}_{3}-\lambda I$ corresponding to the eigenvalues λ are symmetric or asymmetric with respect to the middle of the beam at the point $x=\frac{1}{2}$ on the interval $(0,1)$ respectively.

**Proof**

**of**

**Theorem**

**3.**

**Proof**

**of**

**the**

**Statement**

**2.**

## 3. Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Intervals | The Physical Interpretation | The Types of the Boundary Conditions | Number of Conditions |
---|---|---|---|

$${I}_{1}=(0,\phantom{\rule{0.166667em}{0ex}}1)$$
| Clamped-clamped at the points $x=0,x=1$ | $v(\lambda ,0)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime}(\lambda ,0)=0,$$v(\lambda ,1)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime}(\lambda ,1)=0$ | (2) |

$${I}_{2}=\left(\frac{1}{2},\phantom{\rule{0.166667em}{0ex}}1\right)$$
| Sliding at the point $x=\frac{1}{2}$, Clamped at the point $x=1$ | ${v}^{\prime}\left(\lambda ,\frac{1}{2}\right)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime \prime \prime}\left(\lambda ,\frac{1}{2}\right)=0,$$v(\lambda ,1)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime}(\lambda ,1)=0$ | (3) |

$${I}_{2}=\left(\frac{1}{2},\phantom{\rule{0.166667em}{0ex}}1\right)$$
| Hinged at the point $x=\frac{1}{2}$, Clamped at the point $x=1$ | $v\left(\lambda ,\frac{1}{2}\right)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime \prime}\left(\lambda ,\frac{1}{2}\right)=0$ , $v(\lambda ,1)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime}(\lambda ,1)=0$ | (4) |

$${I}_{1}=(0,\phantom{\rule{0.166667em}{0ex}}1)$$
| Hinged-hinged at the points $x=0,x=1$ | $v(\lambda ,0)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime \prime}(\lambda ,0)=0,$$v(\lambda ,1)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime \prime}(\lambda ,1)=0$ | (5) |

$${I}_{2}=\left(\frac{1}{2},\phantom{\rule{0.166667em}{0ex}}1\right)$$
| Sliding at the point $x=\frac{1}{2}$, Hinged at the point $x=1$ | ${v}^{\prime}\left(\lambda ,\frac{1}{2}\right)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime \prime \prime}\left(\lambda ,\frac{1}{2}\right)=0,$$v(\lambda ,1)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime \prime}(\lambda ,1)=0$ | (6) |

$${I}_{2}=\left(\frac{1}{2},\phantom{\rule{0.166667em}{0ex}}1\right)$$
| Hinged at the point $x=\frac{1}{2}$, Hinged at the point $x=1$ | $v\left(\lambda ,\frac{1}{2}\right)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime \prime}\left(\lambda ,\frac{1}{2}\right)=0$ , $v(\lambda ,1)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime \prime}(\lambda ,1)=0$ | (7) |

$${I}_{1}=(0,\phantom{\rule{0.166667em}{0ex}}1)$$
| Free-free at the points $x=0,x=1$ | ${v}^{\prime \prime}(\lambda ,0)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime \prime \prime}(\lambda ,0)=0,$${v}^{\prime \prime}(\lambda ,1)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime \prime \prime}(\lambda ,1)=0$ | (8) |

$${I}_{2}=\left(\frac{1}{2},\phantom{\rule{0.166667em}{0ex}}1\right)$$
| Sliding at the point $x=\frac{1}{2}$, Free at the point $x=1$ | ${v}^{\prime}\left(\lambda ,\frac{1}{2}\right)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime \prime \prime}\left(\lambda ,\frac{1}{2}\right)=0,$${v}^{\prime \prime}(\lambda ,1)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime \prime \prime}(\lambda ,1)=0$ | (9) |

$${I}_{2}=\left(\frac{1}{2},\phantom{\rule{0.166667em}{0ex}}1\right)$$
| Hinged at the point $x=\frac{1}{2}$, Free at the point $x=1$ | $v\left(\lambda ,\frac{1}{2}\right)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime \prime}\left(\lambda ,\frac{1}{2}\right)=0$ , ${v}^{\prime \prime}(\lambda ,1)=0,\phantom{\rule{0.166667em}{0ex}}{v}^{\prime \prime \prime}(\lambda ,1)=0$ | (10) |

Clamped-Clamped at the Points $\mathit{x}=0,$ $\mathit{x}=1$ | Sliding at the Point $\mathit{x}=\frac{1}{2},$ Clamped at the Point $\mathit{x}=1$ | Hinged at the Point $\mathit{x}=\frac{1}{2},$ Clamped at the Point $\mathit{x}=1$ |
---|---|---|

500.56 | 500.56 | 3803.53 |

3803.53 | 14,617.63 | 39,943.79 |

14,617.63 | 89,135.41 | 173,881.32 |

39,943.79 | 308,208.45 | 508,481.54 |

89,135.41 | 793,403.13 | $1.184\times {10}^{6}$ |

Clamped-Clamped at the Points $\mathit{x}=0,$ $\mathit{x}=1$ | Sliding at the Point $\mathit{x}=\frac{1}{2},$ Clamped at the Point $\mathit{x}=1$ | Hinged at the Point $\mathit{x}=\frac{1}{2},$ Clamped at the Point $\mathit{x}=1$ |
---|---|---|

509.69 | 509.69 | 3811.41 |

3811.41 | 14,624.97 | 39,950.9 |

14,624.97 | 89,142.37 | 173,888.2 |

39,950.9 | 308,215.28 | 508,488.34 |

89,142.37 | 793,412.99 | $1.184\times {10}^{6}$ |

Hinged-Hinged at the Points $\mathit{x}=0,$ $\mathit{x}=1$ | Sliding at the Point $\mathit{x}=\frac{1}{2},$ Hinged at the Point $\mathit{x}=1$ | Hinged at the Point $\mathit{x}=\frac{1}{2},$ Hinged at the Point $\mathit{x}=1$ |
---|---|---|

97.63 | 97.63 | 1558.72 |

1558.72 | 7890.31 | 24,936.9 |

7890.31 | 60,880.85 | 126,242.35 |

24,936.9 | 233,879.4 | 398,987.81 |

60,880.85 | 639,101.62 | 974,094.13 |

Hinged-Hinged at the Points $\mathit{x}=0,$ $\mathit{x}=1$ | Sliding at the Point $\mathit{x}=\frac{1}{2},$ Hinged at the Point $\mathit{x}=1$ | Hinged at the Point $\mathit{x}=\frac{1}{2},$ Hinged at the Point $\mathit{x}=1$ |
---|---|---|

99.15 | 97.94 | 1558.76 |

1560.86 | 7890.49 | 24,937.01 |

7899.55 | 60,881.01 | 126,242.48 |

24,939.18 | 233,879.54 | 398,987.94 |

60,883.15 | 639,101.76 | 974,094.28 |

Free-Free at the Points $\mathit{x}=0,$ $\mathit{x}=1$ | Sliding at the Point $\mathit{x}=\frac{1}{2},$ Free at the Point $\mathit{x}=1$ | Hinged at the Point $\mathit{x}=\frac{1}{2},$ Free at the Point $\mathit{x}=1$ |
---|---|---|

3.32 | 3.32 | 5.99 |

5.99 | 505.38 | 3807.59 |

505.38 | 14,621.36 | 39,947.39 |

3807.59 | 89,138.92 | 173,884.78 |

14,621.36 | 308,211.88 | 508,484.95 |

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**MDPI and ACS Style**

Nurakhmetov, D.; Jumabayev, S.; Aniyarov, A.; Kussainov, R.
Symmetric Properties of Eigenvalues and Eigenfunctions of Uniform Beams. *Symmetry* **2020**, *12*, 2097.
https://doi.org/10.3390/sym12122097

**AMA Style**

Nurakhmetov D, Jumabayev S, Aniyarov A, Kussainov R.
Symmetric Properties of Eigenvalues and Eigenfunctions of Uniform Beams. *Symmetry*. 2020; 12(12):2097.
https://doi.org/10.3390/sym12122097

**Chicago/Turabian Style**

Nurakhmetov, Daulet, Serik Jumabayev, Almir Aniyarov, and Rinat Kussainov.
2020. "Symmetric Properties of Eigenvalues and Eigenfunctions of Uniform Beams" *Symmetry* 12, no. 12: 2097.
https://doi.org/10.3390/sym12122097