# An Exact Analytical Solution for Thermoelastic Response of Clamped Beams Subjected to a Movable Laser Pulse

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## Abstract

**:**

## 1. Introduction

## 2. Basic Formulations for This Problem

## 3. Solutions to the Governing Equations

#### 3.1. Solution to the Heat Conduction Equation

#### 3.2. Solution of the Vibration Equation

## 4. Calculation Results and Discussions

#### 4.1. Temperature Variation

^{−1}. It is shown that the temperature increment vibrates around 0 at the beginning. After some time, it increases in peak value and then drops to near zero very fast. Moreover, because of the motion of the laser pulse, it takes more time to reach the peak value of $T$ at the locations farther away from the left edge. It is clear that the peak $T$ arrives at the maximum value at the location of 0.002 m, which is equal to $v{t}_{p}$.

#### 4.2. Deflection Variation

#### 4.3. Frequency Spectra of Deflection

#### 4.4. Vibration Behaviors and Properties of the Beam

#### 4.5. Behaviors of Stress

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Temporal variations of $T$ at different axial locations on the surface of $z=0.005\mathrm{m}$ ($v=20\text{\hspace{0.17em}}\mathrm{m}/\mathrm{s}$).

**Figure 8.**Deflection, velocity, and acceleration of the beam at the midspan point ($v=20\text{\hspace{0.17em}}\mathrm{m}/\mathrm{s}$).

**Figure 14.**Time histories of the total stress and its two components at the midspan on the top surface of the beam (${t}_{p}=0.1\text{\hspace{0.17em}}\mathrm{ms}$).

**Figure 15.**Time histories of the total stress and its two components at the midspan on the top surface of the beam (${t}_{p}=0.5\text{\hspace{0.17em}}\mathrm{ms}$).

**Figure 16.**Time histories of the total stress and its two components at the midspan on the top surface of the beam (${t}_{p}=1\text{\hspace{0.17em}}\mathrm{ms}$).

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**MDPI and ACS Style**

Yang, X.; Ma, J.; Liu, S.; Xing, Y.; Yang, J.; Sun, Y.
An Exact Analytical Solution for Thermoelastic Response of Clamped Beams Subjected to a Movable Laser Pulse. *Symmetry* **2018**, *10*, 139.
https://doi.org/10.3390/sym10050139

**AMA Style**

Yang X, Ma J, Liu S, Xing Y, Yang J, Sun Y.
An Exact Analytical Solution for Thermoelastic Response of Clamped Beams Subjected to a Movable Laser Pulse. *Symmetry*. 2018; 10(5):139.
https://doi.org/10.3390/sym10050139

**Chicago/Turabian Style**

Yang, Xianfeng, Jingxuan Ma, Shoubin Liu, Yun Xing, Jialing Yang, and Yuxin Sun.
2018. "An Exact Analytical Solution for Thermoelastic Response of Clamped Beams Subjected to a Movable Laser Pulse" *Symmetry* 10, no. 5: 139.
https://doi.org/10.3390/sym10050139