GT-style rubber- berglass (RF) timing belts are designed to e ectively trans- fer rotational motion from pulleys to linear motion in small machines and mechatronic systems. One of the characteristics of belts under this type of loading condition is that the length between load and pulleys changes during operation, thereby changing their e ective sti ness. It has been shown that the e ective sti ness of such a belt is a function of a \nominal sti ness" and the real-time belt section lengths. However, this nominal sti ness is not necessarily constant; it is common to assume linear proportional sti - ness, but this often results in system modeling error. This technical note describes a brief study where the nominal sti ness of two lengths (400 mm and 760mm) of GT-2 RF timing belts was tested up to breaking point; regression analysis was performed on the results to best model the observed sti ness. The study was replicated three times, providing a total of six sti - ness curves. It was found that cubic regression models ( R2>0:999) were the best t, but that quadratic and linear models still provided acceptable representations of the whole dataset with R2values above 0 :940. Keywords: Timing belt, belt sti ness, dynamic system modeling, mechatronic systems, 3-D printers, robotics Preprint submitted to engrXiv October 20, 2018 1. Introduction Timing belts are a common means of motion transfer between rotating motors/shafts in a machine or mechatronic system. Many small-to-medium sized mechatronic systems such as 3-D printers [1], robots [2, 3], desktop CNC machines [4], and positioners [5] use such belts, typically in the GT- style [6, 7]. The GT-style belts are speci cally designed to e ectively trans- late rotating motion from pulleys into linear motion with minimal deforma- tion, slippage, and backlash. One of the fundamental characteristics of belts under this type of loading condition is a non-constant sti ness value; since the e ective length of each belt section is changing with time during opera- tion, the e ective sti ness is also time-variant. It has been shown that the sti ness of the belts can be directly calculated as a function of a \nominal sti ness" value, the belt width, and the real-time length. For a toothed belt running on two equally-sized gears, the time-dependent sti ness of a belt sectioniis [8]: ki(t) =Cspb Li(t)(1) wherekiis the e ective sti ness as a function of time, Cspis the nominal sti ness,bis the belt width, and Li(t) is the length of the belt section at timet. The most commonly-used GT-style belt is the GT-2; Figure 1 shows the fundamental geometry and speci cations for this type of belt. Figure 1: GT-2 belt (a) speci cations and (b) basic geometry Figure 2 shows a common application, where a GT-style belt is used to transfer motion from a stepper motor to drive a linear positioning system. Also shown is a 2-D dynamic model representation of such a system (Fig- ure 2b), where the di erences in sti ness, based on belt length, in the belt sections is clearly evident. The sections L1andL2change in sti ness as a function of time, while section L3stays constant during use [9, 10]. The work described in this note explored the nominal sti ness Cspand the best way to model it in dynamic systems where belt length is not con- stant. Several previous studies have assumed that rubber-based timing belts 2 Figure 2: (a) Simple positioning system which utilizes a GT-type belt to drive the table and (b) its representative dynamic model have a linear nominal sti ness [5, 11, 12, 13, 14, 15, 16, 9, 10]. However, it is vital for designers and engineers working with dynamic systems which use belts for energy transfer to understand the true e ects of the belt sti - ness [17, 18]. Therefore, experimental data was collected and used to derive conclusions on the true sti ness behavior of the GT-2 belts during use. The collected data was subjected to regression analysis to see which type of model best t, allowing the comparison of models for the same dataset. The information in this study will prove useful, both in choosing ksti ness values for dynamic models and for judging expected model error if linear sti ness assumptions are used. 2. Procedure and Results Two lengths of new GT-2 belts, 400 mm and 760mm, were subjected to a simple tensile test at a rate of 0 :8mm=s until they ruptured, taking a force-de ection reading every 1 :6mm or every 2 seconds; the purpose of the discrete time measurements was to produce a moderately-sized dataset to be used for curve tting. This was repeated twice to obtain a set of 6 di erent curves, three from each length. The ruptured belts were observed to fail suddenly and to show tearing of the glass bers inside, as shown in Figure 3. The GT-2 belts used were a composite of neoprene (synthetic 3 rubber) and glass bers, where the bers appeared to drive the failure point of the belts. Figure 3: Belt break interface, showing broken bers The collected data, in terms of force-de ection curves, is shown in Fig- ure 4a, while the equivalent stress-strain curves for the tests are shown in Figure 4b. The length of the belts clearly had an e ect on the force-de ection curves, but this largely disappeared when the length was accounted for in the stress-strain curves. Note that the most of the curves show hyper-elastic behavior, i.e. there is no region in the curve where the sti ness is constant. Figure 4: Belt sti ness curves: (a) force-de ection curve and (b) stress-strain curve As the nominal compliance of the belts was clearly found to be non- linear, a regression analysis was performed to model the curves and nd the level of unexplained variance in these curves. One of the most common polynomial regression models [19, 20] used for hyper-elastic materials is the cubic polynomial. The basic model used for this study began with the following polynomial model: belt=A"3 belt+B"2 belt+C"belt+D (2) where a cubic model includes all of the variables, a quadratic model can be generated by setting A= 0, and a linear model can be used with A=B= 4 0. These curve ts, completed using Microsoft ExcelTM(Microsoft Corp, Redmond, Washington, USA ) are shown in Figure 5a, and the ts for each of the variable and the resulting R2values are shown in the rst six cases in Tables 1 and 2. Table 1: Cases under study Case Plot Reference 1 760mm (cubic model) - R1 Figure 5a 2 760mm (cubic model) - R2 Figure 5a 3 760mm (cubic model) - R3 Figure 5a 4 400mm (cubic model) - R1 Figure 5a 5 400mm (cubic model) - R2 Figure 5a 6 400mm (cubic model) - R3 Figure 5a 7 Full dataset (cubic model) Figure 5b 8 Full dataset (quadratic model) Figure 5b 9 Full dataset (linear model) Figure 5b 10 Low strain (linear model) Figure 5c Table 2: Belt sti ness model curve t data Case A B C D R2 1 2:0010694,969 958.80 -0.5658 0.9996 2 3:00106144,204 445.86 -0.1163 0.9996 3 2:0010695,693 1,281.60 -0.0723 0.9997 4 2:0010681,219 849.99 0.2296 0.9993 5 2:0010695,332 935.37 0.2840 0.9995 6 -922,283 50,993 810.95 0.4965 0.9994 7 2:0010698,091 937.22 -0.1758 0.9672 8 - -19,340 2,408.9 -3.3656 0.9552 9 - - 1821.1 -0.6140 0.9431 10 - - 2013.8 -2.3275 0.9573 After tting the cubic models to each of the six sets of experimental data, the cubic model, a quadratic model, and a linear model were then t to entire set at once, as shown in Figure 5b. A signi cant drop in the R2value was noted for all of the models t to the dataset, but di erence between the cubic, quadratic, and linear models were observed to be small, as shown in Tables 1 and 2. It was observed that the low-strain region of the dataset (Figures 5b and c) conforms better to a linear model when the entire dataset is used. In actual use, it is most likely that the belts will not reach more than 20 30% of the belt breaking strength during normal use [21, 22, 12], so this is a valid 5 assumption for many systems; this will, of course, need to be determined by the modeler or designer before using a linear belt model. If the low- strain assumption can be used, then the data ts a linear model with a slightly greater R2value than a quadratic model for the entire dataset and is certainly superior to a linear model for the entire dataset. The linear model for this case is shown in the last row of Table 2. Figure 5: Curve ts for (a) individual belts (cubic model), (b) full sample curve t (cubic, quadratic, and linear models), and (c) low-strain linear curve t 3. Summary and Conclusions This short technical note presents the results of a brief exploratory study on modeling the nominal sti ness of GT-2 timing belts; this information can be used to more accurately determine the true, time-dependent, sti ness behavior of common GT-2 belts when the e ective length of belt sections changes with time. It was observed that these belts do not behave in a linear way, as expected for belts with a hyper-elastic base material, but that a linear model can provide a reasonable approximation of the behavior un- der some conditions, particularly low-strain conditions. When possible, the cubic sti ness model should be used, but this would often be impractical for systems with many components, as it can cause a simple model to be- come non-linear. When practical and necessary for problem tractability, a linear model may be used with a reasonable degree of accuracy. The mod- eler or designer should keep in mind that some uncertainty will exist with any belt model and choose the model which best balances accuracy with computational cost. 6