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General cutting equations
Relations between shear forces and normal forces:
`S1N1`
`S2N2`
Horizontal equilibrium of forces:
`HEQ`
Vertical equilibrium of forces:
`VEQ`
This gives for the unknown grain forces `K_1` and `K_2`:
`K1general`
`K2general`
This gives for the unknown normal forces `N_1` and `N_2`:
`N1general`
`N2general`
The horizontal and vertical forces on the cutting blade can now be calculated according to:
`Fhgeneral`
`Fvgeneral`
Cutting forces in dry sand
This gives for the unknown grain forces `K_1` and `K_2`:
`K1drysand`
`K2drysand`
This gives for the unknown normal forces `N_1` and `N_2`:
`N1drysand`
`N2drysand`
The horizontal and vertical forces on the cutting blade can now be calculated according to:
`Fhdrysand`
`Fvdrysand`
Cutting forces in water saturated sand
This gives for the unknown grain forces `K_1` and `K_2`:
`K1wetsand`
`K2wetsand`
This gives for the unknown normal forces `N_1` and `N_2`:
`N1wetsand`
`N2wetsand`
The horizontal and vertical forces on the cutting blade can now be calculated according to:
`Fhwetsand`
`Fvwetsand`
With given values for the pore pressure forces `W_1` and `W_2` and distinguishing the non-cavitating and the cavitating cutting process, this results in:
No cavitation
`Fhnc`
`Fvnc`
Cavitation
`Fhc`
`Fvc`
Cutting forces in clay
This gives for the unknown grain forces `K_1` and `K_2`:
`K1clay`
`K2clay`
This gives for the unknown normal forces `N_1` and `N_2`:
`N1clay`
`N2clay`
The horizontal and vertical forces on the cutting blade can now be calculated according to:
`Fhclay`
`Fvclay`
Conditions
Curling type: `ClayCurlingType`
Tear type: `ClayTearType`
Cutting forces in rock
This gives for the unknown grain forces `K_1` and `K_2`:
`K1rock`
`K2rock`
This gives for the unknown normal forces `N_1` and `N_2`:
`N1rock`
`N2rock`
The horizontal and vertical forces on the cutting blade can now be calculated according to:
`Fhrock`
`Fvrock`
Conditions
Tear type atmospheric: `RockTearType1`
Tear type hydrostatic: `RockTearType2`
List of symbols used |
||
`A` | The adhesive force exerted by the blade on the layer cut. | kN |
`c_1, c_2` | Coefficients for the cavitating cutting process | - |
`C` | The cohesive force exerted by the situ sand on the layer cut. | kN |
`d_1, d_2` | Coefficients for the cavitating cutting process | - |
`e` | Dilatation | - |
`F_h` | Horizontal cutting force | kN |
`F_v` | Vertical cutting force | kN |
`F_hydr` | Force resulting from hydrostatic pressure | kN |
`g` | Gravitational constant (approximately 9.81 m/sec^{2}) | m/sec^{2} |
`G` | The gravitational force on the layer cut. | kN |
`h_i` | The thickness of the layer cut | m |
`h_b` | The blade height | m |
`I` | The inertial force exerted by the situ sand on the layer cut. | kN |
`k_m` | Permeability of the sand | m/sec |
`K_1` | The grain force exerted by the situ sand on the layer cut. | kN |
`K_2` | The grain force exerted by the blade on the layer cut. | kN |
`N_1` | The normal force exerted by the situ sand on the layer cut. | kN |
`N_2` | The normal force exerted by the blade on the layer cut. | kN |
`S_1` | The shear force exerted by the situ sand on the layer cut. | kN |
`S_2` | The shear force exerted by the blade on the layer cut. | kN |
`v_c` | Cutting velocity | m/sec |
`w` | Width of the blade | m |
`W_1` | The force resulting from pore pressures exerted by the situ sand on the layer cut. | kN |
`W_2` | The force resulting from pore pressures exerted by the blade on the layer cut. | kN |
`z` | Waterdepth | m |
`alpha` | The angle of the blade with the horizontal | ° |
`beta` | The angle of the shear plane with the horizontal | ° |
`phi` | The angle of internal friction (sand-sand, clay-clay, rock-rock) | ° |
`delta` | The angle of external friction (sand-steel, clay-steel, rock-steel) | ° |
`rho_w` | Density of water | tons/m^{3} |
The original
ASCIIMathML and
ASCIIsvg
scripts have been developed by by
Peter Jipsen,
Chapman University (jipsen@chapman.edu) LaTeXMathML has been developed by Douglas Woodall (and exteded by Jeff Knisley), based on ASCIIMathML The version of ASCIIMathML used here, is a modified and extended version, developed by Dr.ir. S.A. Miedema Other sources: An ASCIIsvg manual by Robert Fant. An ASCIIsvg manual by Peter Jipsen. An ASCIIMathML manual by James Gray.
Plugins and fonts required (depending on your browser):
MIT MathML
font packages,
MathPlayer,
Adobe SVGviewer Department of Marine & Transport Technology, The Chair of Dredging Engineering |