## 5 Discussion
Ten out of twelve participants (eight males and four females) reported that the 30 Hz stimulus was heavier than the 15 Hz stimulus. The intensity of the illusion may peak at 30 Hz, at which frequency FAI units are sensitive [36, 40]. Nonetheless, it is possible that the vibratory stimuli caused actual slippage. During the experiment, a participant accidentally dropped the voice coil motor from his hand at the onset of the vibratory stimuli. Such an accidental case was observed only once throughout the experiment. Further, we did not observe any other evident full slippages; however, it is known that incipient slippage occurs in the finger pad's contact area [41, 42] and the tactile system permits the detection of this partial slippage [39, 43]. The vibration reduces the sliding friction, and the effect of the vibration on the friction reduction is prominent with a greater vibratory amplitude and at a greater frequency [44]. As shown in Table 1, the vibrations at lower frequencies exhibited greater amplitudes in our setup. It is possible that the vibration had reduced the friction between the finger pad and voice coil motor, and the grip force adjustment might have occurred to readjust to the surface with a lowered coefficient of friction. From this perspective, the vibration-weight illusion can be compared with the material-weight illusion, wherein an object with a smooth surface feels heavier than an object with a rough surface [10]. The material-weight illusion was explained as a large grip force required to hold an object with a slippery smooth surface that magnified the perceived weight of the object [45, 46]. The following are some limitations of the present study. In Experiment 1, to produce the weight illusion clearly, we presented vibration stimuli with accelerations of 3 dB above the threshold [25], at which amplitude the illusion barely occurred at each vibration frequency. These settings limit the generalizability of the results. Because the present study investigated the effect of the amplitude only for 30 Hz stimuli, it is not possible to conclude how the amplitude dependence of the weight illusion changes with other frequencies. In Experiment 1, there were six different stimuli, so the participants had to experience the stimuli multiple times and take time to make a decision. Therefore, the changes in the grip force (Figure 3A) and perceived heaviness (Figure 3B) were measured in different experiments, and the conditions of these two Experiments were not perfectly the same. This difference in the experimental conditions might have affected the experimental results. In Experiment 1, however, the grip force measurement was repeated several times and the participants spent time ranking the perceived heaviness. Thus, the reliability of Experiment 1 is not particularly questionable. Further, in Experiments 1 and 2, the intensities of the perceived heaviness were ranked, and the rank correlations were calculated to investigate whether the frequency or amplitudes affect the perceived heaviness. Therefore, we cannot discuss whether the effects of the frequency and amplitude on the illusion are linear or not. Finally, this study suggests a relationship between the grip force adjustment and the vibration-weight illusion. However, it does not negate other factors. It is also inconclusive as to why the vibratory stimuli increase the grip force, that is, whether the grip force is adjusted due to actual partial slippage. Some points remain to be studied in the future. In the experiments, the participants gripped the voice coil motor with their finger pads. However, the vibration-weight illusion also occurs when other holding methods are adopted. The relationship between the holding methods and the intensity of the illusion is one of the interesting topics to be studied. Furthermore, interestingly, the temporal profiles of the grip forces differed among the frequency conditions. Nonetheless, we cannot explain the reasons behind this difference at this point.