# A Numerical Study of the Vibration Characteristics of a Haptic Actuator for a Dial Gear Shifter
## 3 Results and Discussion
### Discussion  
In this study, we investigated the design optimization of a dial gear shifter module integrated with a haptic actuator through numerical analyses. First, we accurately estimated the forcing condition of the haptic actuator by comparing the FEA and experiment results (Figure 8). A dial gear shifter is installed inside a vehicle and operates only under specific conditions. Therefore, this study did not incorporate the temperature effects or long-term operation issues in the numerical analysis. The comparison between the frequency response functions of the FEA and the experiment results revealed slight differences, as shown in Figure 8. For example, the frequency response in the experiment was asymmetric with respect to the resonance frequency, whereas the simulated result appeared relatively symmetric, and there was no significant variation in the frequency or vibration level. The discrepancy in the shape of the frequency response could be attributed to the nonlinearity in the actual haptic actuator. Although the haptic actuator had a rubber damper inside the oscillator, it was difficult to realize the numerical modeling of the nonlinear contact through harmonic analysis using an FE model. Therefore, rather than including the damper in the numerical model, we tuned the model damping using the experiment results. Meanwhile, we hypothesized that the nonlinear effects caused by frequency coupling and the damper would be marginal. However, since we compared the vibration characteristics of each design parameter relatively, the linearized model could still compute the optimal design parameters at the single-frequency condition of the actuator integrated into the gear shifter. Following the vibration analysis of the haptic actuator, we constructed an FE model of the dial gear shifter. The minimum acceleration at the knob grip was estimated to be approximately 2500 mm/s\({}^{2}\) in the baseline model (Figure 9). However, the acceleration level in the baseline model was not sufficient to realize the actual tactile feeling of the user; this is based on past research indicating that the minimum threshold at the given frequency range should be at least 2500 mm/s\({}^{2}\)[31]. Therefore, we conducted a design optimization of the numerical model of a dial gear shifter with the aid of an L9 orthogonal array and a sensitivity analysis (Table 4 and Figure 10). The span of the design parameters in the optimizing process was from 1 to 1.5 mm in 0.5 mm intervals. The parameter interval was greater than the typical manufacturing tolerances of plastic products (\(\sim\)0.1 mm). Therefore, the minor discrepancy caused by the production process would be negligible. The optimal combination of design parameters significantly increased the resulting vibration level by 26.7% compared with that of the baseline model (Figure 12). The minimum and average acceleration levels at the knob exceeded the known threshold by \(>\)20%. Hence, it was expected that the haptic actuator would operate at a relatively lower voltage input compared to the baseline model, thereby saving electrical energy. Lastly, we must mention that hand-grabbing effects were not addressed in this paper as we assumed that the force exerted by two or three fingers on the knob would not be too large. Moreover, it was not a  
Figure 12: Vibration response at the knob grip of the design-optimized model, observed at the height and the angle of the knob grip. simple problem to define the hand-grabbing force for average people. Though a multi-step analysis (i.e., static analysis and the following harmonic analysis) would provide a more realistic result by including such effects, an analysis addressing the hand-grabbing effect would be an interesting research topic. As such, numerical analyses could predict the vibration responses at the knob grip of a dial gear shifter through multiple trial models without requiring actual product fabrication. Furthermore, the optimal design can confirm the vibration characteristics of a product without conducting actual experiments.