### Particle vibration categories

Pebble vibration is excited by the water flow and produces a vibration response. According to the acceleration magnitude, the vibration type before pebble entrainment is divided into two forms: (1) in-situ vibration and (2) ectopic strong vibration (Fig. 3a). In situ vibration is caused by turbulence that causes the pebble to sway back and forth around the origin. Ectopic strong vibration is due to the grain meeting the high-energy turbulence event^{18:37}, the vibration acceleration increases, the tilt angle becomes larger, and a strong vibration phenomenon out of the original position is produced. However, after the high-energy turbulence event is over, the pebble is subjected to gravity and returns to its original position of vibration. Due to the low frequency of high-energy turbulence events, only a few ectopic strong vibration events occur during the vibration (t = 30 s). It is observed experimentally that the high-energy turbulence events are mainly caused by the congestion effect of the flume, the influence of surface waves, and the superposition of vortices. The vibration acceleration measured in this paper reflects the impact of rapidly fluctuating hydrodynamic forces on grain vibration. It demonstrates that not all local flow velocity fluctuations above the mean value can lead to particle entrainment^{15}.

### Time domain characteristics of particle vibration

Figure 3b shows that the pebble vibration varies at different locations for the same flow rate. This phenomenon proves that the pebble vibration characteristics are influenced by the bed geometry. Since random vibration is characterized by irregularity of vibration, any physical quantity of vibration cannot be expressed by a definite time function, so to further analyze the pebble vibration characteristics, this paper uses statistical indicators to analyze the time-domain characteristics of the vibration signal. The article mainly analyzes the pre-entrainment data. In the cases of Q = 39 L/h, 53 L/h, and 69 L/h, the number of samples per group N = 6000, and the acquisition time t = 30 s. In the case of Q = 39 L/ h, there are only four groups of vibration conditions. In the case of Q = 86 L/h, due to the pebble entrapment event, only pre-entrapment data can be collected, the number of samples collected N = 460-2300, and the effective acquisition time t = 2.3-11.5 s.

Table 2 shows the statistical parameters of 25 sets of acceleration data, and it can be found that the statistical parameters vary under the same water flow conditions without any regularity, which again proves that the pebble vibration is a random phenomenon influenced by the bed position. Table 2 shows that of the 25 sets of mean acceleration data, 24 groups are greater than zero, and 1 set is less than zero, indicating that the pebble vibrate mainly in the direction of the water flow under the impact of the current, and a few cases in the opposite direction of the water flow. The latter event was due to the raised contact surface between the pebble and the riverbed, which prevented the pebble from vibrating forward. To overcome the randomness caused by the bed shape and to analyze the pattern of statistical parameters with flow rate, the parameters under the same flow conditions are averaged in this paper. From the mean data, it can be seen that before approaching the threshold value, the mean value of mean squared difference and maximum value tends to increase, and the mean value of minimum value decreases as the flow increases, indicating that the discrete degree of pebble Vibration acceleration is enhanced. However, near the threshold value, the mean values of mean squared difference and maximum value decrease, and the mean value of minimum value increases. The mean values of skewness coefficients were more significant than 1 for different water flow conditions, indicating that the probability distribution graph of vibration acceleration was shifted to the left.

As the pebble vibration is a random phenomenon, the vibration intensity of the same water flow condition was averaged to analyze the variation of pebble vibration intensity at different flow rates. Root mean square value analysis is a common data analysis method for signal processing, which mainly analyzes the average effective energy of the signal, and its expression is

$$ A_{rms} = sqrt {frac{1}{N}sumlimits_{i = 1}^{N} {X_{i}^{2} } } $$

(1)

where ({mathrm{X}}_{mathrm{i}}) is the vibration acceleration value, and N is the number of vibration acceleration samples.

In order to analyze the pulsation intensity variation law of the flow velocity, the near-bed flow velocity signal is considered as the sum of the average flow velocity and the pulsation flow velocity, as shown in Fig. 3c. The root-mean-square value of the pulsating flow velocity is taken as the pulsation intensity of the flow velocity. As shown in Fig. 3d, before approaching the threshold value (v = 44 cm/s), the pulsation intensity of the flow velocity increased with the increase of the flow velocity, and the pebble vibration intensity also increased, however, the vibration intensity weakened when approaching the threshold value. Because when approaching the entrainment threshold, the turbulent force generated by the mean flow velocity is greater than the particle resistance. It pushes the particles to tilt, at which time the pebble lack the inertial force to restore their original position and can only vibrate slightly under the action of the current pulsation force. When the high-energy turbulence event occurs, the pebble is entrained. This result is in agreement with Williams’ observation of solid particles^{38}.

The near-bed velocity is the main water flow parameter that determines whether the sediment vibrates and the intensity of the vibration. Observing the PDF plot of the instantaneous flow velocity near the bottom (Fig. 3e), it was found that it approximately obeyed a normal distribution (consistent with the findings of related studies). The pebble vibration event inherits the randomness of turbulent fluctuations, and the vibration acceleration PDF plot (Fig. 3f) before pebble entrainment is observed to conform to a normal distribution function. It belongs to the normal distribution with large kurtosis and sloping to the left. The results indicate that the pebble vibration is strongly correlated with the current action. The probabilistic model is characterized by a left near-Gaussian function and a long right tail. The acceleration in the near Gaussian part is relatively small and is mainly caused by in-situ vibrational events. The long right tail describes the ectopic strong vibrational events, so it is most relevant to high-energy turbulence events.

### Spectral characteristics of particle vibration

The frequency domain analysis reflects the distribution of vibration energy and the components of vibration frequency. For this purpose, the acceleration data (X

(2)

The working conditions of the spectrum analysis are consistent with those explored in the time domain, all of which are the vibration data before entrainment. Among them, the effective data of the five working conditions at Q = 86 L/h are less and different, so the accuracy of the analysis is weak compared with other working conditions. However, to analyze the vibration characteristics of the pebble near the entrainment threshold, this paper makes an approximation to analyze the frequency distribution under this water flow condition, and the results show that the vibration frequency is concentrated in the range of 25 Hz. Figure 4 shows the pebble vibration spectrum curves. It can be found that 98% of the energy of the pebble vibration signal is concentrated in the range of 20 Hz, indicating that the pebble vibration signal is a low-frequency signal. The data show that the pebble vibration intensity decreases but the vibration frequency increases when approaching the entrainment threshold. Observing Fig. 4, it was found that the amplitude maximum was concentrated within 0.5 Hz, which was influenced by the high-energy turbulence event.

The near-bed flow velocity was converted from the time domain to the frequency domain (Fig. 5). It is found that 97% of the energy of the flow velocity signal is concentrated within 20 Hz. The results indicate that the near-bed flow velocity signal is a low frequency signal^{39}. Therefore, both the pebble vibration acceleration and the velocity flow are low-frequency signals with similar frequencies. The pebble is excited by the water flow to produce a vibration response, and vibrate according to the frequency of the excitation signal, in line with the pebble vibration mechanism, proving that the data measured by the measurement system are reliable.