Research Project, Zach Brock

Introduction and Research Question

Due to the relatively small extensible length of traditional steel springs, we propose to use rubber surgical tubing as our spring system. Rubber’s elastic properties are unique in that it can be elastically deformed to lengths several times that of its unstretched length. Unfortunately, rubber also exhibits other unique elastic characteristics, such as nonlinear elasticity, hysteresis, the Mullins effect, and permanent set. These characteristics can alter the stiffness of a rubber spring over time, which could affect our launch system. While permanent set and nonlinear elasticity have been studied previously, it appears that no research has been conducted to see if there is a theoretical limit to the amount of permanent set that occurs in a rubber spring. Additionally, it has not specifically been shown whether permanent set increases or decreases the nonlinearity of the rubber’s elasticity. We would like to determine whether there is a theoretical limit to permanent set and whether the force-deformation curve becomes more linear with permanent set, given that we have chosen to use rubber springs as the energy storage element of our launch system.

Related Work

The properties of nonlinear elasticity, hysteresis, the Mullins effect, and permanent set have been studied extensively. Rubber is generally modeled using chain models and strain energy functions to predict stress-stretch behavior in the material [1]. Comparison of these models to test data has shown both that they can accurately predict stress-stretch behavior and that certain regions of the stress-stretch curve can be modeled as linear [2]. However, one issue with these models and their findings is that they do not account for the Mullins effect and permanent set. The Mullins effect is the phenomenon where the stress-stretch path is changed upon subsequent loading [3]; permanent set is when the stress in the rubber is reduced for equal displacement for additional load cycles due to residual strain in the material [4]. Additional mathematical models have been created that model the Mullins effect [3] and permanent set [4]. While each of these works have sought to develop models of permanent set, hysteresis, and the Mullins effect, my research's proposed solution is to use permanent set in a practical manner to make rubber springs more consistent and, potentially, more easily modeled.

Method

To perform this research, I proposed an experiment wherein a fixed length of rubber tube is fixed rigidly at one end and have its other end stretched to five times the tube’s original length with an attached fish scale. The fish scale has its measurement recorded after each extension to measure the reduction in the rubber’s stiffness over extension cycles. The rubber is then returned to its original length and allowed to rest for two seconds; then, the rubber is stretched again to five times its original length, cycled in this manner for fifty extensions and readings. During the first and last extension, the fish scale’s output is also recorded at two, three, four, and six times the original length to measure the nonlinearity of the spring stiffness and see if the spring has become more linear after the introduction of permanent set. An image of the test setup I used is shown in the accompanying figure. The data was then input into Matlab for visualization and processing.

Expected Results and Evaluation Criteria

I expected that the rubber will experience much greater permanent set during the first few cycles with minimal changes to spring stiffness after fifty cycles. To evaluate whether this method will prove useful to our robot, I compared the force output of new springs that have not been cycled to introduce permanent set against the force output of equal lengths of cycled springs to see how much the launch distance will likely be affected.

Results

The measurements of force per cycle are presented in the figure to the right. As shown, it appears that there is an initial loss in strength during the first 2-3 cycles, but that the strength then remains relatively constant for the remaining cycles up to 50. There is notable variation in the measurement data, with a standard deviation of 0.086 lbs, which I believe is due to the method of measurement, as the stretches were performed by hand and the spring’s extended length determined by eye. This setup was used due to resource limitations; the results would most likely be less noisy if a more repeatable setup had been used, such as a universal testing machine. Acknowledging the presence of human error, it appears that there is no additional effect of permanent set after the first few cycles, with Matlab’s linear regression tool reporting a slope of 0.002 lbs/strain for measurements after the first deflection. Even with human error, the slope would be more significant if permanent set did not reach a maximum over the course of these cycles.

The next figure shows the deflection of the spring at up to 6 times the original length before and after 50 cycles. Measurements were taken at integer steps of 2 to 6 times the original spring length. From the results, it is shown that permanent set seems to have reduced the spring force in the spring for the entire range of stretches, which is an expected finding. It also appears that the spring has retained its nonlinear qualities even after permanent set, though it may be more linear between 2 and 6 times the original length. Since this experiment was conducted by hand, it is difficult to say if these results are due to measurement error or due to permanent set; a more robust testing method or additional tests by hand would be required to see if this result is consistent.

These results match my expected finding; it appears that the spring has some permanent set introduced after the first few deflections, but that its spring force remains relatively constant for additional cycles. Due to the noisiness of the hand-measured data and the small sample size for the experiment, no conclusions can be drawn as to whether the spring has become more linear in a given strain region; further testing is required in that area with better, more repeatable hardware.

Addressing Evaluation Criteria

Though there was some reduction in spring force after 50 cycles, it was no greater than 5% for all strain values measured. This is a significant difference, but it is also shown that this reduction has a maximum limit for the number of cycles we tested. This result will be helpful to us in the competition in terms of setting up the springs on the robot. For a new spring, we will attach the spring to the robot and perform 5-10 test launches just to get the springs out of the changeable permanent set range. After permanent set has been introduced in these cycles, we will then be able to tune the launch distance for repeatable performance, as the springs are not expected to change their force for additional cycles at that point based on the results of this research.

In terms of contribution to outside research, these findings suggest that permanent set has a practical maximum effect on rubber springs, and that these springs’ performance becomes more consistent after the first few extension cycles. This could be useful in industry applications where rubber springs are used. Additional testing should be performed with more repeatable hardware such as a universal testing machine, to validate these findings and determine if permanent set has an effect on the linearity of rubber springs.

Actual Competition Performance

During the competition, the results of my experiment were seen to hold true. After several dozen tests, our launch distance was largely reliable for a given setting for the desired launch distance, and the best launch distance setting we determined from testing served as a good starting point for the actual competition. The rubber springs were reliable due to the previously-introduced permanent set.

Conclusion

From this experiment, we conclude that rubber springs experience a maximum permanent set within the first few deformation cycles and do not tend to have decreased performance as long as their deformations remain within the same range. Unfortunately, our test proved inconclusive in determining whether a rubber spring becomes more linear as permanent set is introduced. In terms of contribution to outside research, these findings suggest that permanent set has a practical maximum effect on rubber springs, and that these springs’ performance becomes more consistent after the first few extension cycles. This could be useful in industry applications where rubber springs are used. Additional testing should be performed with more repeatable hardware such as a universal testing machine, to validate these findings and determine if permanent set has an effect on the linearity of rubber springs. The mathematical models used to model permanent set may benefit from the findings of this research in terms of including the maximum permanent set experienced in the model.

References

E. M. Boyce, Mary C., Arruda, “Constitutive Models of Rubber Elasticity: A Review,” Rubber Chem. Technol., vol. 73, no. 3, pp. 504–523, 2000.
E. M. Arruda and M. C. Boyce, “A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials,” J. Mech. Phys. Solids, vol. 41, no. 2, 1993.
D. G. Roxburgh, “A pseudo-elastic model for the Mullins effect in filled rubber.”
A. Dorfmann and R. W. Ogden, “A constitutive model for the Mullins effect with permanent set in particle-reinforced rubber,” Int. J. Solids Struct., vol. 41, no. 7, pp. 1855–1878, 2004.

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