5.5: The Influence of Inertial Forces

In the previous chapter the shear angle and the cutting forces are given for the influence of the weight only. This will be appropriate for very small cutting velocities, but the question is of course; what is a very low cutting speed.

Analyzing the equations for the influence of the weight (gravity) and the influence of the inertial forces shows a significant difference. The gravitational forces are proportional to the density of the soil ρ_s, the gravitational constant g, the thickness of the layer cut h_i squared and the width of the blade w. The inertial forces are proportional to the density of the soil ρ_s, the cutting velocity v_c squared, the thickness of the layer cut h_i and the width of the blade w. This implies that the ratio between these two forces does not only depends on the geometry, but even stronger on the layer thickness h_i and the cutting velocity v_c. The thicker the layer cut, the higher the influence of gravity and the higher the cutting velocity, the higher the influence of inertia. One cannot say simply the higher the cutting velocity the higher the influence of inertia.

\[\ \text{Gravitation : }\mathrm{F} \propto \rho_{\mathrm{s}} \cdot \mathrm{g} \cdot \mathrm{h}_{\mathrm{i}}^{\mathrm{2}} \cdot \mathrm{w}\tag{5-18}\]

\[\ \text{Inertia :} \quad \mathrm{F} \propto \rho_{\mathrm{s}} \cdot \mathrm{v}_{\mathrm{c}}^{2} \cdot \mathrm{h}_{\mathrm{i}} \cdot \mathrm{w}\tag{5-19}\]

The contribution of the inertial forces is determined by the following dimensionless parameter:

\[\ \lambda_{\mathrm{i}}=\frac{\mathrm{v}_{\mathrm{c}}^{2}}{\mathrm{g} \cdot \mathrm{h}_{\mathrm{i}}}\tag{5-20}\]

In dredging a layer thickness of the magnitude of centimeters is common, while for a bulldozer a layer thickness of a magnitude of a meter is not strange. At the same cutting velocity, the relative influence of inertial forces will differ between dredging applications and the operation of bulldozers. If inertial forces dominate the cutting process, the cutting forces can be expressed as:

\[\ \begin{array}{left} \mathrm{F}_{\mathrm{h}}=\rho_{\mathrm{s}} \cdot \mathrm{v}_{\mathrm{c}}^{\mathrm{2}} \cdot \mathrm{h}_{\mathrm{i}} \cdot \mathrm{w} \cdot \lambda_{\mathrm{H I}}\\
\text{With :} \quad \lambda_{\mathrm{H I}}=\frac{\sin (\alpha)}{\sin (\alpha+\beta)} \cdot \frac{\cos (\varphi)}{\sin (\alpha+\beta+\delta+\varphi)} \cdot \sin (\alpha+\delta)\end{array}\tag{5-21}\]

\[\ \begin{array}{left}\mathrm{F}_{v}=\rho_{\mathrm{s}} \cdot \mathrm{v}_{\mathrm{c}}^{\mathrm{2}} \cdot \mathrm{h}_{\mathrm{i}} \cdot \mathrm{w} \cdot \lambda_{\mathrm{V I}}\\
\text{With : }\quad \lambda_{\mathrm{V I}}=\frac{\sin (\alpha)}{\sin (\alpha+\beta)} \cdot \frac{\cos (\varphi)}{\sin (\alpha+\beta+\delta+\varphi)} \cdot \cos (\alpha+\delta)\end{array}\tag{5-22}\]

These equations are derived from equations (5-6), (5-7), (5-9) and (5-10). The shear angle β can be derived analytically for the inertial forces, giving:

\[\ \beta=\frac{\pi}{2}-\frac{2 \cdot \alpha+\delta+\varphi}{2}\tag{5-23}\]

Figure 5-13 shows the percentage of the contribution of the inertial forces to the horizontal cutting force for a layer thickness h_i of 1.0 m at a cutting velocity of 0.5 m/sec, giving λ_i=0.025. Figure 5-14 shows the percentage of the contribution of the inertial forces to the horizontal cutting force for a layer thickness h_i of 0.1 m at a cutting velocity of 15.7 m/sec, giving λ_i=250.

Table 5-1 shows the inertial effect for the dimensionless inertial effect parameter λ_i ranging from 0.025 to 250. The percentage contribution of the inertial effect on the horizontal force is given as well as the shear angle, both horizontal and vertical cutting force coefficients based on equations (5-21) and (5-22) and both horizontal and vertical cutting force coefficients based on equations (5-13) and (5-14) for the case where the blade height h_b equals the layer thickness h_i. The table shows that the inertial effect can be neglected at very small values of the dimensionless inertial effect parameter λ_i, while at large values the gravitational effect can be neglected. The shear angle β decreases with an increasing dimensionless inertial effect parameter λ_i. Since the inertial forces are not influenced by the blade height h_b, the cutting forces are not dependent on the blade height at high cutting velocities. At low cutting velocities there will be an effect of the blade height.

Figure 5-11: The horizontal cutting force coefficient λ_HD as a function of the blade angle α for h_b/h_i=2.

Figure 5-12: The vertical cutting force coefficient λ_VD as a function of the blade angle α for h_b/h_i=2.

The contribution of the inertial effect only depends on the dimensionless inertial effect parameter λ_i and not on the cutting velocity or layer thickness individually. The dimensionless inertial effect parameter λ_i in fact is a Froude number of the cutting process. Figure 5-15, Figure 5-16 and Figure 5-17 show the shear angle and both horizontal and vertical cutting force coefficients at very high values of the dimensionless inertial effect parameter λ_i (λ_i=250). The shear angles are considerably smaller than in the case where inertial forces can be neglected. Also in the case where the inertial forces dominate, the cutting forces become very high at large blade angles (close to 90°). Nature will find an alternative cutting mechanism in this case which has been identified as the wedge mechanism. At which blade angle the wedge mechanism will start to occur depends on the internal and external friction angles, but up to a blade angle of 60° the model as described here can be applied. See Chapter 11: A Wedge in Dry SandCutting for detailed information on the wedge mechanism.

Figure 5-13: The percentage inertial force for a layer thickness h_i=1.0 m, blade height h_b=1.0 m and a cutting velocity v_c=0.5 m/sec.

Figure 5-14: The percentage inertial force for a layer thickness h_i=1.0 m, blade height h_b=1.0 m and a cutting velocity v_c=15.7 m/sec.

Figure 5-15: The shear angle β, including the effect of inertial forces.

Figure 5-16: The horizontal cutting force coefficient λ_HI.

Figure 5-17: The vertical cutting force coefficient λ_VI.

Table 5-1: The inertial effect.

λi	%	β	λHI	λVI	λHD	λHV
0.025	0.74	31.6	187.68	27.49	4.78	0.70
0.250	6.65	30.8	20.26	2.97	5.09	0.75
2.500	37.69	26.0	3.12	0.46	7.95	1.16
25.00	78.98	16.9	1.25	0.18	31.40	4.60
250.0	94.98	9.6	0.96	0.14	245.36	35.94