## 5 Discussion  
The experimental results revealed a positive correlation between the increase in the grip force and perceived heaviness at different vibratory frequencies in Experiment 1 (\(\rho=0.66\), \(p=1.24\times 10^{-9}\)) and at different amplitudes in Experiment 2 (\(\rho=0.77\), \(p=1.37\times 10^{-7}\)). Specifically, the vibratory stimuli felt to be heavier introduced significant increases in the grip force. The results are consistent with our hypothesis, which posited that the grip force adjustment is the main factor underlying the vibration-weight illusion. The grip force adjustment might have functioned autonomously to retain the vibrating object. The increase in the grip forces indicates an increase in the motor commands, also known as the sense of effort, which in turn leads to an increase in the perceived weight [26, 27, 28]. In Experiment 1, though we did not fully equalize the vibratory amplitudes at different frequencies, those at 100 Hz (44 \(\upmu\)m), 200 Hz (48 \(\upmu\)m), and 300 Hz (39 \(\upmu\)m) were similar. Among these three frequencies, the lowest frequency, that is, 100 Hz, led to the greatest and most significant increase in the grip force, whereas the other two frequencies exhibited smaller increases in the grip force. This suggests that the vibration-weight illusion depends on the frequency such that low vibration frequencies are effective. In Experiment 2, the greater the amplitude, the greater the increase in the grip force and perceived weight when the frequency was 30 Hz. This result suggests that the weight illusion also depends on the amplitude. One might query whether the differences in the outputs of the load cell were caused by the vibratory amplitudes and resultant finger pad restoration forces and not by the grip force. However, this factor did not have a major impact on the results. In Experiment 1, although the amplitudes of the 100, 200, and 300 Hz stimuli were nearly equal, 44, 48, and 39 \(\upmu\)m, respectively, as listed in Table 1, only the 100 Hz vibration significantly increased the grip force. Furthermore, the increases in the grip force were nearly identical for the 60 Hz and 100 Hz stimuli, despite their amplitudes differing by a factor of almost two. Therefore, the amplitudes of the stimuli did not notably affect the outputs of the load cell. Experiment 1 showed that the 30 Hz vibratory stimulus increased the grip force the most. Additionally, the lower the frequency of the vibratory stimuli, the greater the increase in the grip force. This may be due to the characteristics of the fast-adapting type 1 (FAI) mechanoreceptive units that affect the grip force adjustment [33, 39]. The FAI units play an important role in the grip force adjustment by detecting fast and subtle skin deformations, such as slippage, with a peak of activation between 30 Hz and 40 Hz [36, 40], whereas the mediation of other types of receptive units are not negated [39]. The presentation of the 30 Hz local vibrations to the fingertips, which is expected to effectively stimulate the FAI units, was reported to increase the grip force, even without actual slippage [34]. It is possible that the stimuli with lower frequencies in the present study also effectively activated the FAI units. This might have resulted in an increase in the grip force, even when there was no actual slippage between the fingertips and surfaces of the voice coil motor. Here, one question arises, namely whether the vibratory stimuli at frequencies below 30 Hz effectively cause the weight illusion. To investigate this, as a post hoc experiment, we compared the intensities of the illusion between the 15 Hz and 30 Hz stimuli. The same voice coil motor as in Experiments 1 and 2 was used, and the accelerations of the stimuli were 3 dB higher than the thresholds of the illusion (15 Hz: 1.39 m/s\({}^{2}\), 30 Hz: 5.5 m/s\({}^{2}\)). The threshold of the 15 Hz stimulus was determined by the psychophysical method of the limits with five participants, following a previous experiment [25].