# Mathematical Model of New Type of Train Buffer Made of Polymer Absorber—Determination of Dynamic Impact Curve for Different Temperatures

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Determination of Absorber Curves

## 3. Mathematical Model

_{1}as a function of the temperature.

_{0}= 90 mm, corresponding to the initial mounting position of the supporting block in the kit.

## 4. Result and Discussion

_{0}= 1.15 m/s and v

_{0}= 3.5 m/s are given in Figure 10 and Figure 11.

_{0}= 1.15 m/s and v

_{0}= 3.5 m/s, are shown in Figure 13 and Figure 14, respectively.

_{0}= 1.15 m/s and v

_{0}= 3.5 m/s) are presented in Figure 15. By reducing the impact velocity on the polymer block, the working length and the displacement of the same are reduced. In the operation process, the initial velocity increases at impact and then drops to zero, and at higher speeds the percentage of the increase is lower.

## 5. Conclusions

- The model can be used to analyze the action of the longitudinal forces that occur during transient conditions of the movement of the carriages.
- The dynamic force $F\left(x,\dot{x}\right)$ obtained for a working temperature of +15 °C is five times greater than the dynamic force obtained for a working temperature of −60 °C. This indicates that the influence of the temperature change is of great importance.
- For a working temperature of +15 °C and a working stroke of 180 mm, it was calculated that $F\left(x,\dot{x}\right)=350\mathrm{kN}$; meanwhile, for a working temperature of −60 °C, $F\left(x,\dot{x}\right)=1750\mathrm{kN}$. For a working temperature of +15 °C and a working stroke of 150 mm, it was calculated that $F\left(x,\dot{x}\right)=180\mathrm{kN}$; for a working temperature of −60 °C, $F\left(x,\dot{x}\right)=900\mathrm{kN}$. This can be seen from the diagrams in Figure 9 and Figure 12 and this ratio is maintained for the whole working length of the polymer block.
- In the process of operation, the initial impact velocity increases and then drops to zero. At higher speeds, the percentage of this increase is lower. For example, for an initial velocity v
_{0}= 3.5 m/s, the increase is about 1%, and for v_{0}= 1.15 m/s it is about 6%. - Expensive classical experiments can be avoided, and the error of the force obtained with the mathematical model does not exceed 5% related to the maximum force, and does not exceed 1% related to the maximum displacement.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Elastic-frictional absorption device principle scheme (

**a**) with package of polymer elements (

**b**).

**Figure 4.**Dynamic force–deformation curve of the polymer pack for different impact speeds, —Static (test bench), --v

_{0}= 1.59 m/s, ···v

_{0}= 0.68 m/s.

**Figure 9.**Change of dynamic force $F\left(x,\dot{x}\right)$ at an operating temperature of +15 °C and ${\eta}_{CT}=0.4$.

**Figure 10.**Displacement (blue line) and velocity of the buffer (orange line) at an operating temperature of +15 °C, an impact speed of v

_{0}= 1.15 m/s, and ${\eta}_{CT}=0.4$.

**Figure 11.**Displacement (blue line) and velocity of the buffer (orange line) at an operating temperature of +15 °C, an impact speed of v

_{0}= 3.5 m/s, and ${\eta}_{CT}=0.4$.

**Figure 12.**Change of dynamic force $F\left(x,\dot{x}\right)$ at an operating temperature of −60 °C and ${\eta}_{CT}=0.7$.

**Figure 13.**Displacement (blue line) and velocity of the buffer (orange line) at an operating temperature of −60 °C, an impact speed of v

_{0}= 1.15 m/s, and ${\eta}_{CT}=0.7$.

**Figure 14.**Displacement (blue line) and velocity of the buffer (orange line) at an operating temperature of −60 °C, an impact speed of v

_{0}= 3.5 m/s, and ${\eta}_{CT}=0.7$.

**Figure 15.**Displacement and velocity of the buffer at different operating temperatures and different impact speeds.

Temperature T, °C | −60 | −50 | −40 | −20 | 0 | 21 | 41 | 51 |
---|---|---|---|---|---|---|---|---|

${a}_{1}$, $N/m$ | −4.4 × 10^{4} | −1.9 × 10^{4} | −6.5 × 10^{3} | −8.2 × 10^{2} | −3.8 × 10^{3} | −3.9 × 10^{3} | −1.1 × 10^{2} | −7.3 × 10^{3} |

${a}_{2}$, $N/{m}^{2}$ | −1.1 × 10^{6} | −4.7 × 10^{5} | −1.5 × 10^{5} | −1.1 × 10^{4} | −8.2 × 10^{4} | −8.8 × 10^{4} | −3.2 × 10^{4} | −1.6 × 10^{5} |

${a}_{3}$, $N/{m}^{3}$ | −1.1 × 10^{7} | −4.4 × 10^{6} | −1.3 × 10^{6} | −4.0 × 10^{5} | −6.1 × 10^{5} | −6.1 × 10^{5} | −6.3 × 10^{5} | −1.2 × 10^{6} |

${a}_{4}$, $N/{m}^{4}$ | −4.6 × 10^{7} | −1.8 × 10^{7} | −5.5 × 10^{6} | −2.6 × 10^{6} | −2.1 × 10^{6} | −1.7 × 10^{6} | −3.9 × 10^{6} | −3.7 × 10^{6} |

${a}_{5}$, $N/{m}^{5}$ | −7.1 × 10^{7} | −2.6 × 10^{7} | −7.6 × 10^{6} | −5.7 × 10^{6} | −2.2 × 10^{6} | −8.7 × 10^{5} | −8.3 × 10^{6} | −3.3 × 10^{6} |

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**MDPI and ACS Style**

Mickoski, H.; Mickoski, I.; Djidrov, M.; Zdraveski, F. Mathematical Model of New Type of Train Buffer Made of Polymer Absorber—Determination of Dynamic Impact Curve for Different Temperatures. *Machines* **2018**, *6*, 47.
https://doi.org/10.3390/machines6040047

**AMA Style**

Mickoski H, Mickoski I, Djidrov M, Zdraveski F. Mathematical Model of New Type of Train Buffer Made of Polymer Absorber—Determination of Dynamic Impact Curve for Different Temperatures. *Machines*. 2018; 6(4):47.
https://doi.org/10.3390/machines6040047

**Chicago/Turabian Style**

Mickoski, Hristijan, Ivan Mickoski, Marjan Djidrov, and Filip Zdraveski. 2018. "Mathematical Model of New Type of Train Buffer Made of Polymer Absorber—Determination of Dynamic Impact Curve for Different Temperatures" *Machines* 6, no. 4: 47.
https://doi.org/10.3390/machines6040047