# Nominal Stiffness of GT-2 Rubber-Fiberglass Timing Belts for Dynamic System Modeling and Design

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## Abstract

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## 1. Introduction

## 2. Procedure and Results

## 3. Recommendations for Use and Applications

- Identify the nominal stiffness ${C}_{sp}$ of each belt type used in the system (e.g., if two thicknesses of belts are used, two different nominal stiffnesses will be present). This information may be collected from manufacturer datasheets or from tests on each belt type, similar to the tests done in this technical report.
- Decide if a linear or nonlinear nominal stiffness ${C}_{sp}$ model will be used for each belt type. The primary driving force for this decision will be the computational cost for analyzing the system; for a simple system, it may be practical to use a nonlinear nominal stiffness model, but a linear model would be more feasible in a system with several elements. However, the importance of the model accuracy is a serious consideration and may justify a high computational cost if high accuracy is required.
- Based on the configuration of the system and the decisions made in the first two steps, the effective stiffness k can take one of four forms:
- (a)
- If the belt length is constant and a linear model is used for ${C}_{sp}$, the effective stiffness in the equations of motion will be constant and described by$${k}_{i}={C}_{sp}\frac{b}{L}$$
- (b)
- If the belt length is constant and a nonlinear model is used to find ${C}_{sp}$, the nominal stiffness will be a function derived form a force-deflection curve. The effective stiffness in that belt section will be described by$${k}_{i}={C}_{sp}\left(x\right)\frac{b}{L}$$
- (c)
- If the belt length is time-variant and a linear model is used for ${C}_{sp}$, the effective stiffness in the equations of motion will be time-variant and described by$${k}_{i}={C}_{sp}\frac{b}{L\left(t\right)}$$
- (d)
- If the belt length is time-variant and a nonlinear model is used to find ${C}_{sp}$, the nominal stiffness will be a function derived form a force-deflection curve. In this case, the effective belt section stiffness will be described by$${k}_{i}={C}_{sp}\left(x\right)\frac{b}{L\left(t\right)}$$

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

- b = Belt width (m)
- ${\beta}_{i}$ = Belt section i damping coefficient
- ${C}_{sp}$ = Nominal belt stiffness (N/m)
- ${k}_{i}$ = Effective (true) belt section i stiffness (N/m)
- ${L}_{i}$ = Belt section i length (m)
- ${m}_{i}$ = Mass of block i (kg)
- ${\theta}_{i}$ = Pulley i angle (degrees)

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**Figure 2.**(

**a**) simple positioning system that utilizes a GT-type belt to drive the table and (

**b**) its representative dynamic model.

**Figure 5.**Curve fits for (

**a**) individual belts (cubic model); (

**b**) full sample curve fit (cubic, quadratic, and linear models); and (

**c**) low-strain linear curve fit.

Case | Plot Reference | A | B | C | D | ${\mathbf{R}}^{2}$ |
---|---|---|---|---|---|---|

760 mm (cubic model) - $R1$ | Figure 5a | $-2.00\times {10}^{6}$ | 94,969 | 958.80 | −0.5658 | 0.9996 |

760 mm (cubic model) - $R2$ | Figure 5a | $-3.00\times {10}^{6}$ | 144,204 | 445.86 | −0.1163 | 0.9996 |

760 mm (cubic model) - $R3$ | Figure 5a | $-2.00\times {10}^{6}$ | 95,693 | 1281.60 | −0.0723 | 0.9997 |

400 mm (cubic model) - $R1$ | Figure 5a | $-2.00\times {10}^{6}$ | 81,219 | 849.99 | 0.2296 | 0.9993 |

400 mm (cubic model) - $R2$ | Figure 5a | $-2.00\times {10}^{6}$ | 95,332 | 935.37 | 0.2840 | 0.9995 |

400 mm (cubic model) - $R3$ | Figure 5a | −922,283 | 50,993 | 810.95 | 0.4965 | 0.9994 |

Full dataset (cubic model) | Figure 5b | $-2.00\times {10}^{6}$ | 98,091 | 937.22 | −0.1758 | 0.9672 |

Full dataset (quadratic model) | Figure 5b | - | −19,340 | 2408.9 | −3.3656 | 0.9552 |

Full dataset (linear model) | Figure 5b | - | - | 1821.1 | −0.6140 | 0.9431 |

Low strain (linear model) | Figure 5c | - | - | 2013.8 | −2.3275 | 0.9573 |

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**MDPI and ACS Style**

Wang, B.; Si, Y.; Chadha, C.; Allison, J.T.; Patterson, A.E. Nominal Stiffness of GT-2 Rubber-Fiberglass Timing Belts for Dynamic System Modeling and Design. *Robotics* **2018**, *7*, 75.
https://doi.org/10.3390/robotics7040075

**AMA Style**

Wang B, Si Y, Chadha C, Allison JT, Patterson AE. Nominal Stiffness of GT-2 Rubber-Fiberglass Timing Belts for Dynamic System Modeling and Design. *Robotics*. 2018; 7(4):75.
https://doi.org/10.3390/robotics7040075

**Chicago/Turabian Style**

Wang, Bozun, Yefei Si, Charul Chadha, James T. Allison, and Albert E. Patterson. 2018. "Nominal Stiffness of GT-2 Rubber-Fiberglass Timing Belts for Dynamic System Modeling and Design" *Robotics* 7, no. 4: 75.
https://doi.org/10.3390/robotics7040075