# A Time-Domain Finite-Difference Method for Bending Waves on Infinite Beams on an Elastic Foundation

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## Abstract

**:**

## 1. Introduction

## 2. Model of a Beam on an Elastic Foundation

## 3. Finite Difference Method for Infinite Beams on Foundations

## 4. Results and Discussion

^{®}Core™i5-7200U CPU at 2.50 GHz for $5\xb7{10}^{4}$ time steps.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FDM | Finite Difference Method |

FEM | Finite Element Method |

## Appendix A. Sub-Matrices

## Appendix B. Determining the Parameters for the Boundaries and Grid

**Figure A1.**Error $Er{r}_{s}$ of the simulation depending on the boundary-domain exponent $\alpha $ and the number of boundary grid points ${n}_{B}$. (The unspecified parameters are presented in Table A1.)

**Figure A2.**Error $Er{r}_{s}$ of the simulation depending on ${d}_{r,bc}$ for varying ${m}_{r}^{\prime}/\Delta t$. (The unspecified parameters are presented in Table A1.)

Beam | |||
---|---|---|---|

Beam bending stiffness | B | 6.42 | MN/m^{2} |

Beam mass per unit length | ${m}_{r}^{\prime}$ | 60 | kg/m |

Calculation end time | ${t}_{end}$ | 0.5 | s |

Time increment | $\Delta t$ | $1\xb7{10}^{-5}$ | s |

Grid constant | b | 1 | |

Damping coefficient of boundary domains | ${d}_{r,bc}$ | $4\xb7{10}^{7}$ | Ns/m^{2} |

Length of calculation domain | ${l}_{C}$ | 100 | m |

Number of grid points of the boundary domains | ${n}_{B}$ | ${10}^{3}$ | m |

Exponent | $\alpha $ | 10 | |

Force parameter | $\sigma $ | $0.7\xb7{10}^{-4}$ | s |

Force parameter | a | $0.5\xb7{10}^{2}$ |

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**Figure 7.**Deflection of a one-layer-supported beam at a distance of 10 m from the force point. $\Delta t=2.5\xb7{10}^{-5}$ s ($\Delta x=0.058$ m, ${f}_{max}=4$ kHz), $\Delta t=1\xb7{10}^{-5}$ s ($\Delta x=0.036$ m, ${f}_{max}=10$ kHz), $\Delta t=5\xb7{10}^{-6}$ s ($\Delta x=0.026$ m, ${f}_{max}=20$ kHz).

**Figure 8.**Point mobility of an undamped infinite beam on a continuous single-layer support for different time steps. Analytical mobility (Equation (25)), $\Delta t=2.5\xb7{10}^{-5}$ s ($\Delta x=0.058$ m, ${f}_{max}=4$ kHz), $\Delta t=1\xb7{10}^{-5}$ s ($\Delta x=0.036$ m, ${f}_{max}=10$ kHz), $\Delta t=5\xb7{10}^{-6}$ s ($\Delta x=0.026$ m, ${f}_{max}=20$ kHz).

**Figure 9.**Deflection of the beam’s continuous single-layer elastic foundation at $t=0.05$ s; gray areas: boundary domains, vertical red line: driving location.

**Figure 10.**Point mobility of an undamped infinite free beam. Analytical mobility (28), with boundary domains, without boundary domains.

**Figure 11.**Point mobility of an undamped infinite beam on a continuous single-layer support. Analytical mobility (25), with boundary domains, external/export without boundary domains.

**Figure 12.**Point mobilities of an undamped infinite beam (${t}_{end}=1$ s) for different types of elastic foundations. Discretely supported beams are excited mid-span. Analytical results, finite difference method results.

**Figure 13.**Transfer mobilities of an undamped infinite beam (${t}_{end}=1$ s) at a distance of 10 m from the excitation for different types of elastic foundations. Discretely supported beams are excited mid-span. Analytical results, finite difference method results.

Beam | |||
---|---|---|---|

Beam bending stiffness | B | 6.42 | MN/m^{2} |

Beam mass per unit length | ${m}_{r}^{\prime}$ | 60 | kg/m |

Viscous damping coefficient per unit length | ${d}_{r}$ | 0 | Ns/m^{2} |

Continuous Support | |||

Stiffness per unit length in the first layer | ${s}_{p}$ | 300 | MN/m^{2} |

Stiffness per unit length in the second layer | ${s}_{b}$ | 100 | MN/m^{2} |

Viscous damping coefficient per unit length in the first layer | ${d}_{p}$ | 30,000 | Ns/m^{2} |

Viscous damping coefficient per unit length in the second layer | ${d}_{b}$ | 80,000 | Ns/m^{2} |

Support mass per unit length | ${m}_{s}^{\prime}$ | 250 | kg/m |

Discrete Support | |||

Stiffness of the first layer | ${s}_{p}$ | 180 | MN/m |

Stiffness of the second layer | ${s}_{b}$ | 60 | MN/m |

Viscous damping coefficients of the first layer | ${d}_{p}$ | 18,000 | Ns/m |

Viscous damping coefficients of the second layer | ${d}_{b}$ | 48,000 | Ns/m |

Support mass | ${m}_{s}^{\prime}$ | 150 | kg |

Support distance | ${L}_{s}$ | 0.6 | m |

Layer length | ${L}_{l}$ | $\Delta x$ |

Calculation end time | ${\mathit{t}}_{\mathit{e}\mathit{n}\mathit{d}}$ | 0.5 | s |

Time increment | $\Delta t$ | ${10}^{-5}$ | s |

Grid parameter | b | 1 | |

Local step size | $\Delta x$ | 0.036 | m |

Boundary-domain exponent | $\alpha $ | 10 | |

Number of boundary grid points | ${n}_{B}$ | ${10}^{3}$ | |

Force parameter | $\sigma $ | $0.7\xb7{10}^{-4}$ | s |

Force parameter | a | $0.5\xb7{10}^{2}$ |

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**MDPI and ACS Style**

Stampka, K.; Sarradj, E.
A Time-Domain Finite-Difference Method for Bending Waves on Infinite Beams on an Elastic Foundation. *Acoustics* **2022**, *4*, 867-884.
https://doi.org/10.3390/acoustics4040052

**AMA Style**

Stampka K, Sarradj E.
A Time-Domain Finite-Difference Method for Bending Waves on Infinite Beams on an Elastic Foundation. *Acoustics*. 2022; 4(4):867-884.
https://doi.org/10.3390/acoustics4040052

**Chicago/Turabian Style**

Stampka, Katja, and Ennes Sarradj.
2022. "A Time-Domain Finite-Difference Method for Bending Waves on Infinite Beams on an Elastic Foundation" *Acoustics* 4, no. 4: 867-884.
https://doi.org/10.3390/acoustics4040052