## 5 Theory Analysis and Discussion
### Analysis of Incipient Sediment Movement  
As shown in Figure 13, the forces during the rolling of the sediment particles included the water flow drag force \(F_{D}\), the uplift force \(F_{L}\), the gravity \(G\), the additional mass force \(F_{m}\), the vertical motion resistance \(F_{z}\), the centrifugal force \(F_{c}\), the normal reaction force \(F_{N}\), and the shear stress \(F_{\tau}\) on the contact point \(e\). Figure 12: Comparison of non-uniform sand and uniform sand transport rate of the same median diameter under unsteady flow. Figure 13: Sediment particle force model. In the case of uniform sand, considering the conditions of unsteady flow, the expressions of each force element are listed as follows:  
\[F_{D}=C_{D}A_{1}D^{2}\frac{\rho(u_{b}-u_{x})^{2}}{2} \tag{5}\]  
\[F_{L}=C_{L}A_{2}D^{2}\frac{\rho(u_{b}-u_{x})^{2}}{2} \tag{6}\]  
\[G=A_{3}(\gamma_{s}-\gamma)D^{3} \tag{7}\]  
\[F_{m}=\frac{\pi}{6}\rho C_{m}D^{4}\frac{d^{2}\theta}{dt^{2}} \tag{8}\]  
\[F_{\tau}=\frac{I}{R}\frac{d^{2}\theta}{dt^{2}} \tag{9}\]  
\[F_{C}=mD\left(\frac{d\theta}{dt}\right)^{2} \tag{10}\]  
where \(C_{D}\) is the drag coefficient of sediment; \(C_{L}\) is the uplift force coefficient of sediment; \(C_{m}\) is the additional mass force coefficient, which can be approximated 1/2; \(A_{1}\) and \(A_{2}\) are the area correction coefficient of sediment particles respectively, \(A_{1}=A_{2}=\pi\)/4; \(A_{3}\) is the particle volume correction coefficient, \(A_{3}=\pi\)/6; \(u_{b}\) is the near-bottom flow velocity of sediment bed; \(u_{x}\) is the velocity of sediment particles along the water flow direction; \(\rho\) is the density of water; \(\gamma_{s}\) and \(\gamma\) is the bulk density of sediment and water respectively; \(D\) is the diameter of sediment particle; and \(m\) is the mass of sediment particles, which can be expressed as \(A_{3}\rho_{s}D^{3}\); \(R\) is the radius of the spherical sediment particle. For the unsteady flow, the inertial force \(F_{a}\) was generated due to the flow change. The particles were accelerated or decelerated and it would produce a horizontal inertial force \(F_{a}\) acting on the particle [46; 47], as follows:  
\[F_{a}=C_{a}m\frac{du_{b}}{dt}=C_{a}\rho V\frac{du_{b}}{dt} \tag{11}\]  
where \(C_{a}\) is inertial force coefficient, \(V\) is the volume of the particle. \(I_{0}\) is the moment of inertia of the mass to the centroid, and \(I\) is the moment of inertia of the particle to the centroid. The expressions are given by:  
\[I_{0}=\frac{2}{5}mR^{2} \tag{12}\]  
Figure 13:0  
The vertical motion resistance \(F_{z}\) is given by:  
Figure 13:1  
where \(C\) is the r\({}_{s}\)istance coefficient; \(u_{y}\) is the vertical movement velocity of the particles. The tangential equation of the centroid motion of the sediment particles is given by:  
Figure 13:2  
The normal equation of centroid motion is:  
Figure 13:3The equation of rotation of the sediment particles relative to the center of mass is:  
Figure 13:4  
In Equation (17), \(\beta\) is the angle at which the particles rotate around the centroid. For the calculation of the force arm, according to the parabolic distribution of the flow velocity, the moment of the horizontal drag force was \(R/3\). Since the lifting torque is much smaller than the pushing torque, the lifting torque is approximately \(R/3\). Assuming that no slip occurs, \(\beta=2\theta\), and \(mD\frac{d^{2}\theta}{dt^{2}}\) is eliminated by combining Equations (8) and (15). Figure 13:5  
Substituting Equation (17) into Equation (14). Figure 13:6  
In order to obtain the rolling equation of the sediment particles, the Equations (7)-(9) and (14) are substituted into the Equation (19), and the solution is obtained by:  
\[