Dr.ir. S.A. Miedema

Abstract.
Introduction.
The Two-Dimensional Cutting Theory.
The Cutting Theory Applied to a Cutterhead.
The Cutting Theory Applied to a Dredging Wheel.
The Three-Dimensional Moving Cutterhead.
Conclusions.
Bibliography.
List of Symbols used.
Tables.

In dredging it is neccessary to know about the conditions under which a dredging operation takes place. Important conditions to design a dredger or to determine the workability of a dredger (under offshore conditions) are the forces and the torque that appear on the excavating element. The paper describes the basic theory of the cutting of watersaturated sand and gives tables by which the cutting forces on a straight blade can be determined. With some assumptions the basic theory can be applied on a cutterhead and a dredging wheel. The equations found are simplified so that by means of tables given, the cutting forces and the torque can be determined. With this knowledge research has been carried out into the forces acting on a cutterhead submitted to
simulated offshore conditions. It appeared that for these conditions, a model for a disc bottom cutterhead can be derived by combining the simplified cutterhead and dredging wheel theory. A strategy for the calculation of the cutting forces and the torque on a disc bottom cutterhead is given, whilst a result of the research carried out is shown. The theory given in this paper can be used to calculate the cutting forces and the cutting torque acting on a draghead, a cutterhead or a dredging wheel by means of a pocket calculator.

During previous WODCON's there have been a number of papers on the behaviour of seagoing cutter suction dredgers. Wichers 1980, 1983, 1989 has described the program DREDSIM, developed by the Maritime Research Institute Netherlands (MARIN). Keuning & Journee 1983, Keuning, Journee & Miedema 1983, de Koning, Miedema & Zwartbol 1983 and Miedema
1986, 1987 have described the computer program DREDMO developed by the Delft University of Technology (D.U.T) and the Delft Hydraulics (D.H.), by means of which the behaviour of seagoing cutter suction dredgers can be simulated. This paper describes the theoretical models, based on Miedema 1985, 1987, for the calculation of the cutting forces, the torque and the specific cutting energy in water saturated sand and explains how to use these models. Of interest with respect to this paper are the papers of Joanknecht 1976 on the subject of cutterhead modelling and performance, van Drimmelen, van 't Hoen, Willigen & Eygenraam 1983 on the subject of the cutting forces on a dredging wheel, van Leussen & Nieuwenhuis 1984 on the subject of soil mechanics aspects of sand cutting and van Raalte & Zwartbol 1986 on the subject of cutting forces on a disc bottom cutterhead.

From literature it is known that, during the process of cutting sand, the pore volume of the sand increases. This is caused by the phenomenen dilatancy (see figure 1). With a certain cutting velocity v_c there has to be a flow of water to the shear zone, the area where the pore volume increases. This causes a decrease in the pore pressure of the pore water and because the soil stress remains constant the grain stress will increase. Van Os 1977 stated: "If it is the aim of the engineer to know the average cutting forces needed to push the blade through the soil, he can take an average deformation rate e/t to insert into the Biot equation. But it should be noted that this is purely practical reasoning and has nothing to do with Theoretical Soil Mechanics". Van Os and van Leussen published their cutting theory in 1987.
Steeghs 1985 developed a theory with a cyclic deformation in the shear zone. This means that a cyclic changing deformation rate has to be inserted in the Biot equation. Miedema 1985 uses the average deformation rate as stated by van Os 1977 but instead of inserting this in the Biot equation, the average deformation rate is modelled as a boundary condition in the shear zone. Although the cutting process is not solely dependent upon the phenomenon dilatancy, the above mentioned research showed that for cutting velocities in a range from 0.5 to 5 m/sec the cutting process is dominated by the phenomenon dilatancy, so the contributions of gravitational, cohesive, adhesive and inertial forces can be neglected.

This leads to the first two basic cutting equations for the two-dimensional cutting process in water saturated sand (no cavitation):

(1)

(2)

(2a)

Figure 1: The two-dimensional cutting process.

When the cutting velocity increases, the pore pressure will decrease until the absolute pore pressure reaches water vapour pressure, when cavitation starts to occur. With a further increase in the cutting velocity the pore pressure and thus the cutting forces remain at a constant level, which depends upon the water depth (see figure 1). This gives the following two basic cutting equations (cavitation):

(3)

(4)

The coefficients c₁, c₂, d₁ and d₂ are dependent upon the angle of internal friction of the sand f, the soil interface friction angle d, the blade angle a and the blade height-shell thickness ratio h_b/h_i. Values for these coefficients can be found in the tables 1-4 (Miedema 1987). From the above four basic cutting equations, the specific cutting energy can be calculated. The definition for specific cutting energy is: The power (kW) required to cut (loosen) 1 m soil, so:

(5)

For the non-cavitating (equation 1) and the cavitating (equation 3) cutting process this gives:

(6)

(7)

Figure 2: The definition of axis for cutterhead and dredging wheel.

When a few assumptions are made, the two-dimensional cutting theory can be made applicable for a cutterhead, with the axis defined according to figure 2 and the cutting process according to figure 3. These assumptions are:

• The coefficients c₁, c₂, d₁ and d₂ have to be constant for the case calculated, this means that an average thickness of the layer cut has to be choosen.
• The cutter head is a conical cutter head with a top angle x.
• The blades have an angle i with the axis of the cutter head.
• The equation for the thickness of the layer cut is simplified to:
  

  (8)

• The projected width of a blade on the axis of the cutterhead is:
  

  (9)

• When the swing velocity is neglected, the cutting velocity can be simplified to:
  

  (10)

• The derived model is applicable for a segment of the cutterhead with a projected width b_pr and a radius R. For a conical or a crown cutterhead the calculation has to be repeated for each segment.

Figure 3: The cutting process of a cutterhead.

For the momentary forces acting on one blade with a width b_pr, (see figure 3) the following equations can be derived for the axial force:

(11)

The force in the swing direction:

(12)

The force perpendicular to the swing direction and the axial direction:

(13)

For the momentary cutter torque on one blade with radius R, the following equation is valid:

(14)

The average cutting forces can be obtained by integration of the momentary cutting forces on a blade over the angle covered W₀ and multiplying the result with the number of blades, according to equation 15.

(15)

where:

and:

When cutting a layer with a constant thickness the non-cavitating and the cavitating cutting processes can be distinguished. When cutting with a cutterhead however, the thickness of the layer cut is not constant but varies from a thickness zero to a thickness h_imax (when W = 90). This means that cavitation may occur in part of the layer cut (see figure 3). The angle covered where cavitation starts to occur W₁ can be determined by equaliying the momentary cutter torques on one blade for both the non-cavitating and the cavitating cutting process (this is allowed because the force F_v is small in comparison to F_h), with: W₁=<W₀ this gives:

(16)

The coefficients c_nc and c_ca, which have the dimension of force (kN), can be calculated by the following equations:

(17)

(18)

The two coefficients c_nc and c_ca can be considered as a measure for the cutting loads on a cutterhead as a function of the independent process variables.

For the calculation of the forces on the cutterhead, the coefficients c₁, c₂, d₁, d₂ and the effects of the shape of the cutterhead,
by means of the angles x and i, can be put in six new coefficients g₁ - g₆. The equations 19 - 24 give the expressions for g₁ - g₆, with after the arrows the expressions for x = 0 and i = 0.

(19)

(20)

(21)

(22)

(23)

(24)

Substitution of the equations 11 - 14 for F_c in equation 15 and simplification of the resulting equations by using equations 17, 18 and 19 - 24, gives the following four equations for the average cutting forces and the average cutter torque. In these equations the notation +/- is used, whereby the upper sign is valid for the overcutting process, while the lower sign is valid for the undercutting process.

(25)

(26)

(27)

(28)

Photo 1: A cutterhead used in the laboratory experiments.

The coefficients introduced here f₁ - f₆ are dependent upon the angles covered W₀ and W₁ and can be found in the tables 5 and 6 (Miedema 1987). When the breachheight B_n is known, the total angle covered W₀ can be calculated according to:

(29)

The specific cutting energy when cutting water-saturated sand with a cutterhead can be calculated in accordance with the equations 6 and 7 as follows:

(30)

Inserting equations 10 and 28, and given values for the swing velocity v_s, the breachheight B_n, the stepsize B_a and the radius of the cutterhead R the specific cutting energy E can be calculated.

The two-dimensional cutting theory can also be made applicable to a dredging wheel with a few assumptions, see figure 4 for the cutting process of a dredging wheel, these assumptions are:

• The buckets of the dredging wheel have radial cutting edges.
• The equation for the thickness of the layer cut is simplified to:
  

  (31)

• The cutting velocity at a radius r of the dredging wheel can be simplified to (when v_s << v_cir ):
  

  (32)

As with the cutterhead (see equations 17 and 18) two coefficients can be defined with the dimension of force (kN), these coefficients are:

(33)

(34)

The average cutting forces and the average cutting torque on the entire dredging wheel can be determined by applying equation 15 to a dredging wheel. Because the integration has proven to be very difficult and the resulting equations cover about 15 pages, these equations are simplified. It is even more difficult to determine equations for a partially cavitating cutting process, so there are separate equations for the noncavitating and for the cavitating cutting processes.

The non-cavitating cutting process gives for the entire dredging wheel:

(35)

(36)

(37)

(38)

The cavitating cutting process gives for the entire dredging wheel:

(39)

(40)

(41)

(42)

Figure 4: The cutting process of a dredging wheel.

The coefficients f_xnc, f_ync, f_znc and m_nc can be found in table 7 (Miedema 1987), the coefficients f_xca, f_yca, f_zca and m_ca can be found in table 8 (Miedema 1987). The specific cutting energy when cutting water saturated sand with a dredging wheel can also be calculated with equation 30, with the exception that for a dredging wheel the non-cavitating and the cavitating cutting process have to be calculated separately, this is done by substituting equation 38 for the non-cavitating case, equation 42 for the cavitating case and equation 32 for v_ciR. The breachheight B has to be substituted for B_n and the step S for B_a. The radius where cavitation starts to occur can be calculated with the following equation (notice the similarity to equation 16):

(43)

When partial cavitation occurs the equations for the cavitating cutting process give an upper limit for the forces and the torque. Equations 39 - 42 have to be used when M_tnc is greater the M_tca.

With the models derived above for the cutterhead and the dredging wheel the forces on a 3-dimensional moving cutterhead can
be calculated. The cutterhead has to be modelled as a disc-bottom cutterhead with radial bottom blades. The figures 5 and 6 give an impression of the shape of the breach when the cutterhead undergoes a harmonic oscillation in the radial and the axial plane respectively. The forces acting on the blades at the circumference of the cutterhead can be calculated from the  cutterhead model, while the forces acting on the bottom blades can be calculated from the dredging wheel model.

Figure 5: A radial oscillation.

If the cutterhead undergoes a harmonic oscillation in the radial plane perpendicular to the swing direction with an amplitude
a_n and a frequency w_n , then the displacement and the velocity can be determined by (the displacement n(t) is positive breach outwards):

(44)

(45)

Figure 6: An axial oscillation.

The resulting radial velocity v_rad can now be calculated with:

(46)

For the angle between the momentary direction of v_rad and the s-axis:

(47)

The total angle covered W₀"(t) can be determined by superimposing the radial dispacement on the breachheight B_n, so:

(48)

Because the angle covered W₀ has to be calculated in relation to the momentary direction of the velocity v_rad of the cutterhead, the angle covered W₀" has to be corrected with the angle W₀'. With a harmonic oscillating displacement of the cutterhead, part of the breach has already been cut. This part has an angle covered W_0c, where W_0c is negative. It is also possible that part of the breach cut is not covered by W₀' - W₀' as in figure 5, this part also has an angle covered W_0c, where W_0c is positive. Figure 5 shows the angles covered, the direction of the resulting radial velocity v_rad and the direction of the forces acting on the circumference of the cutterhead. The total angle covered W₀ can now be determined by:

(49)

For the cutting forces this means, that if W_0c is negative first the forces have to be calculated with an angle covered W₀"(t)-W₀'(t), from this the force with an angle covered W_0c(t) has to be substracted, so:

(50)

The thickness of the layer cut depends upon the radial velocity v_rad of the cutterhead. Without an oscillation this velocity equals the swing velocity v_s, however with an oscillation this is not the case. The thickness of the layer cut can be found by integrating the radial velocity v_n during the time Dt since the previous blade has passed.

(51)

The effective radial velocity v_nc(t) found by this integration can be calculated with:

(52)

So there is a phase shift e between the real radial velocity v_n and the effective radial velocity v_nc according to equation 53.

(53)

For the resulting radial velocity that has to be inserted in equation 8 instead of v_s for the calculation of the thickness of the layer cut, equation 54 is valid.

(54)

The cutting force calculated with equation 25 is in the direction of v_rad, the cutting force calculated with equation 26 is perpendicular to v_rad according to figure 5. The resulting forces in the s and n direction can be calculated according to (with use of equation 50).

(55)

(56)

Figure 7: The step S and the effective axial velocity v_ac.

If the cutterhead undergoes a harmonic oscillation in the axial plane with an amplitude a_a and a frequency w_a (see figure 6), then the displacement and the velocity can be determined by (the displacement a(t) is positive breach outwards):

(57)

(58)

The projected width of the part of the blades at the circumference of the cutterhead in contact with the breach is now:

(59)

The effective axial velocity for the determination of the thickness of the layer cut can be calculated according to:

(60)

Equation 60 shows that there is a phase shift between the axial velocity and the effective axial velocity similar to the phase shift occurring with the radial velocities.

When an axial harmonic oscillation occurs, the wavelength of this oscillation is:

(61)

For the displacement in the swing direction yields:

(62)

The step S used in the calculation for the forces on the bottom blades is in this case the distance the cutterhead has travelled since the cutterhead had the same axial position a(t) (see figure 7), so:

(63)

When:

And:

(64)

This gives for the area of the cross-section cut by the bottom blades:

(65)

Photo 2: The laboratory where the research was carried out.

The strategy for the calculation of the cutting forces with an oscilliation is, to divide a period of the oscillation into time steps,
then, per time step:

First, make an estimate for the coefficients c₁, c₂, d₁ and d₂ based on an average expected thickness of the layer cut.

For the radial oscillation:

1. Assume that the forces on the bottom blades are zero, because the thickness of the layer cut by the bottom blades is zero.
2. Calculate the effective radial velocity with equation 52 and insert this velocity in equation 8 to calculate h_imax.
3. Use equation 10 to calculate v_ciR.
4. Substitute B_a for b_pr in equations 17 and 18 and calculate c_nc and c_ca.
5. Get values for the coefficients f₁ - f₆ from tables 5 and 6 for the cases W₀ = W₀" + W₀' and W₀ = W_0c.
6. Calculate the cutting forces with equations 25 and 26 and the torque with equation 28 for the two cases mentioned in point 5 and substract these forces according to equation 50.
7. Transpose the forces found according to equations 55 and 56.

Figure 8: Verification of the radial oscillation model.

For the axial oscillation:

1. Calculate the forces and the torque (use equations 17, 18, 25, 26 and 28) on the circumference-blades by inserting equation 59 for b_pr, equation 8 for h_imax and equation 10 for v_ciR.
2. Calculate the effective axial velocity with equation 60 and insert this velocity in equation 31 to calculate h_i.
3. Use equation 65 to calculate the area of the cross-section cut A(t).
4. Use equation 10 to calculate v_ciR.
5. Insert the found values of h_i, A(t) and v_ciR in equations 33 and 34 to calculate d_nc and d_ca.
6. Get values for f_xnc, f_ync, f_znc, m_nc, f_xca, f_yca, f_zca and m_ca from the tables 7 and 8 by using S_s(t) and B_n.
7. Calculate the cutting forces and the torque with equation's 35 - 38 for the non-cavitating process and with equations 39 - 42 for the cavitating cutting process.
8. If M_tnc>M_tca take the values for the cavitating process, otherwise take the values for the non-cavitating process.

Figure 9: Verification of the axial oscillation model.

The above derived models for the calculation of the forces and the torque on an oscillating disc-bottom cutterhead, have been verified with the results of model tests, mentioned by de Koning, Miedema & Zwartbol in 1983. In 1983 the research resulted in an empiric model. The disc-bottom cutterhead used in this research has a radius of 475 mm, 8 blades and a width of the blades at the circumference of 184 mm. Figure 8 shows the measured and the calculated forces and torque for an overcutting process with a radial oscillation, with: B_n =350 mm, B_a =184 mm, T=2 sec, a_n =20 mm, v_s =0.109 m/s and n_o =33 rpm. Figure 9 shows the measured and the calculated forces and torque for an overcutting process with an axial oscillation, with: B_n =350 mm, B_a =184 mm, T=2 sec, a_a =20 mm, v_s =0.110 m/s and n_o =44 rpm.
In these figures the phase shift according to equation 53 can be noticed with the axial and the radial oscillation. In general the shape of the measured and the calculated signals agree closely.
The axial force in figure 9 is theoreticaly zero, this means that the measured axial force is not caused by the cutting process but it is caused by friction and inertial forces.

The verification of the theory was "satifying", but the calculation method of the 3-dimensional model is time consuming on a computer. The main result of the research is the understanding of the cutting processes and the possibility to calculate the loads on cutterhead and dredging wheel for non-oscillating processes with the tables given. For the 3-dimensional model the occurrence of a phase shift is important in relation to the simulation of the behaviour of a seagoing cutter suction dredger.

1. Drimmelen, N.J. van, t'Hoen, J.P.T.A., Willigen, F.A. and Eygenraam J.A., "Development and First Production Unit of the IHC Beaver Wheel Dredger". Proc. WODCON X, Singapore 1983.
2. Joanknecht, L.F.W., "A Review of Dredge Cutterhead Modelling and Performance". Proc. WODCON VII, San Francisco, 1976.
3. Journee, J.M.J. and Miedema, S.A. and Keuning, J.A. "DREDMO, A Computer Program for the Calculation of the Behaviour of Seagoing Cutter Suction Dredgers". T.U. Delft & Delft Hydraulics, 1983.
4. Keuning, P.J. & Journee, J., "Calculation Method for the Behaviour of a Cutter Suction Dredger Operating in Irregular Waves". Proc. Wodcon X, Singapore 1983.
5. Koning, J de & Miedema, S.A. & Zwartbol, A., "Soil/Cutterhead Interaction under Wave Conditions". Proc. WODCON X, Sing., 1983.
6. Leussen, W. van & Nieuwenhuis J.D., "Soil Mechanics Aspects of Dredging". Geotechnique 34 No.3, pp. 359-381.
7. Miedema, S.A., "Mathematical Modelling of the Cutting of Densely Compacted Sand Under Water". Dredging & Port Construction, July 1985, pp. 22-26. "Derivation of the Differential Equation for Sand Pore Pressures". Dredging & Port Construction, September 1985, pp. 35. "Underwater soil cutting: a study in continuity", Dredging & Port Construction, June 1986, pp. 47-53.
8. Miedema, S.A., "The Application of a Cutting Theory on a Dredging Wheel". Proc. WODCON XI, Brighton 1986.
9. Miedema, S.A., "The Calculation of the Cutting Forces when Cutting Water Saturated Sand, Basical Theory and Applications for 3-Dimensional Blade Movements with Periodicaly Varieing Velocities for in Dredging Usual Excavating Elements" (in Dutch). Doctors thesis, Delft, 1987, the Netherlands.
10. Os, A.G. van, "Behaviour of Soil when Excavated Underwater". International Course Modern Dredging. June 1977, The Hague, The Netherlands.
11. Os, A.G. van & Leussen, W. van, "Basic Research on Cutting Forces in Saturated Sand". Journal of Geotechnical Engineering, Vol. 113, No.12, December 1987.
12. Raalte, G.H. van & Zwartbol, A., "The Disc Bottom Cutterhead: A Report on Laboratory and Field Tests".
    Proc. WODCON XI, Brighton, England, 1986.
13. Steeghs, H., "Snijden van zand onder water (I & II)". Ports & Dredging No. 121, June 1985. Ports & Dredging No. 123, November 1985. Rapport: GR 37-IIB * MTI-Holland, Kinderdijk, 1986.
14. Wichers, J.E.W.,"On the Forces on a Cutter Suction Dredger in Waves". Proc. WODCON IX, Vancouver, B.C., Canada, 1980.
15. Wichers, J.E.W. and Drimmelen, N.J. van, "On the forces on the cutterhead and the spud of a cutter suction dredger, operating in waves". Proc. WODCON X, Singapore 1983.
16. Wichers, J.E.W., "On the reduction of mooring forces of a cutter suction dredger operating in waves". Proc. WODCON XII, Orlando, Florida, USA, 1989.
    

Table 1: The coefficient c₁, horizontal force, no cavitation.

Table 2: The coefficient c₂, vertical force, no cavitation.

Table 3: The coefficient d₁, horizontal force, cavitation.

Table 4: The coefficient d₂, vertical force, cavitation.

Table 5: The coefficients f₁, f₂ and f₅ for the cutterhead equations 25 - 28.

Table 6: The coefficients f₃, f₄ and f₆ for the cutterhead equations 25 - 28.

Table 7: The coefficients f_xnc, f_ync, f_znc and m_nc for the dredging wheel equations 35 - 38.

Table 8: The coefficients f_xca, f_yca, f_zca and m_ca for the dredging wheel equations 39 - 42.

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Last modified Sunday April 30, 2000 by: Sape A. Miedema

Copyright April, 2000   Dr.ir. S.A. Miedema

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