What is the speed of diamond wire saw cutting?

In this study, a mechanistic approach is used. A mathematical model of sawing stone blocks with a diamond wire has been proposed and compared with a corresponding model for circular sawing, previously presented in Ref. [7].

The application environment of diamond wires is very complex and poorly controlled. Real-time quantitative data collection systems to date have not been fitted to production equipment. Therefore, the theoretical considerations have been supported by scanty literature reports and limited experimental data acquired by the author.

2.1

Force analysis in wire sawing

The diamond-impregnated beads are subjected to various forces resulting from the cutting action, wire tension and weight, and arched shape of cut travelled by the diamond wire with a high linear speed. The dynamics of block cutting by diamond wire saw can be well-explained by the simplistic model shown in Fig. 2.

Bead arrangement in the cut (a) and array of forces acting on an individual bead (b)

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As the wire exit angle β can be directly measured on the sawing machine, it becomes possible to estimate the radius of curvature R and α from equations

$$ \frac{L_{bl}}{2R}=\mathit{\sin}\beta \kern0.5em \mathrm{and}\kern0.5em \frac{L_k}{2R}=\frac{1}{2R{n}_L}=\mathit{\sin}\alpha $$

where Lbl is the length of block and nL is the number of beads per unit length of diamond wire saw. For small angles

$$ \sin \alpha \cong \alpha \kern0.5em \mathrm{and}\kern0.5em \sin \beta \cong \beta $$

Hence, the above equations can be written as

$$ \frac{L_{bl}}{2R}\cong \beta $$

(1)

$$ \frac{1}{2R{n}_L}\cong \alpha $$

(2)

As the linear speed of beads (vs) remains constant, the vector sum of all forces acting on a bead is zero as shown in Fig. 2b. Therefore, the normal force FN can be calculated as follows:

$$ {F}_N=\left(2{F}_T+{\Delta F}_T\right)\sin \alpha +{F}_G-{F}_C\cong \left(2{F}_T+{\Delta F}_T\right)\frac{\beta }{L_{bl}{n}_L}+{F}_G-{F}_C $$

(3)

where FT, FG and FC are wire tension, weight of wire per bead and centrifugal force per bead, respectively.

As the wire tension force FT and wire linear speed vs are directly set on the sawing machine, the other two forces can be calculated from the following equations:

$$ {F}_G=\frac{m_Lg}{n_L} $$

(4)

and

$$ {F}_C=\frac{m_L{v}_s^2}{n_LR}\cong \frac{2{m}_L{v}_s^2}{n_L{L}_{bl}}\beta $$

(5)

where mL is mass per unit length of diamond wire and g is standard gravity.The wire tension increment ΔFT can be expressed as

$$ \Delta {F}_T=\frac{F_F}{\cos \alpha } $$

where FF is friction force per bead.

Putting

$$ \cos \alpha ={\cos}^2\frac{\alpha }{2}-{\sin}^2\frac{\alpha }{2}\cong 1-\frac{\alpha^2}{4} $$

and using Eqs. (2) and (1) in order to substitute α and R, respectively, yields

$$ \Delta {F}_T={F}_F/\left(1-\frac{\beta^2}{4{n}_L^2{L}_{bl}^2}\right) $$

For Lbl, nL and β, ranging from 2 to 3 m, 36 to 40 beads/m and 2 to 8 deg, as known from the industrial practice, it can be assumed that

$$ \Delta {F}_T\cong {F}_F $$

(6)

As proposed in Ref. [8], the friction force FF can be approximated empirically by measuring the power drawn by the motor while cutting

$$ P=\sqrt{3} UI\ \cos\ \varphi $$

where U is the electric potential and cosφ is the power factor. I is calculated as a difference between current consumption readings recorded during sawing and idle running.

Assuming that

$$ P={F}_F{v}_s{L}_{bl}{n}_L\kern0.5em \mathrm{and}\ \mathrm{putting}\kern0.5em \cos \varphi =1 $$

The maximum friction force per bead (FF)max is then

$$ {\left({F}_F\right)}_{\mathrm{max}}=\frac{\sqrt{3} UI\ }{v_s{L}_{bl}{n}_L} $$

(7)

From Fig. 2a and Eq. (6), it becomes evident that the tension of the steel rope gradually increases over the cutting zone from its set value FT on the tension side to FT + FF∙Lbl∙nL on the motor side of the granite block (see Fig. 1). Using the data given in Ref. [8] for sawing class 1 granite at 0.66 m2/h with 7.4 mm diameter diamond wire saw, i.e. U = 380 V, vs = 28.8 m/s, Lbl∙nL = 108 and I = 3.31 A, the maximum increase in tension is 75.6 N, whereas (FF)max = 0.7 N. Since FT typically ranges from 2000 to 2400 N, the total increase in tension reaches approximately 3.5% of its pre-set value.

This latter variable can be a major concern with wire sawing of granite. The preceding equations ignore torque (τ) produced by friction at the bead-workpiece interface. As presented in Fig. 3a, the friction force (FF) is offset relative to bead centre. Therefore, it tends to rotate the bead at an angle (ϕ), thus increasing the lever arm (r), as shown in Fig. 3b.

Bead torque produced by friction (a) and torque balance due to bead rotation (b)

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The torque balance requires

$$ {F}_F\frac{d_b\ }{2}\cong {F}_Tr $$

(8)

where db is the bead diameter and r is the lever arm directed perpendicularly to the tension force vector.

Precise assessment of ϕ is practically impossible. However, as r is proportional to ϕ, by decreasing friction forces and bead diameter, it is possible to prevent visible conical bead wear. In practical terms, faster bead wear at its leading edge that is observable by the naked eye, as demonstrated in Fig. 4, is unacceptable and indicates faulty bead formulation or improper sawing conditions or both.

Excessive friction (torque) leading to conical wear of beads

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Table 1 provides information on sawing class 1 granite block on MDW machine which enables evaluation of the normal force (FN).

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The data included in Table 1 indicates that during sawing class 1 granite at 0.98 m2/h (163 cm2/min), the normal force (FN) and friction force (FF) per bead are 3.28 and 0.35 N, respectively, which yields a mean bead surface pressure of 58 kPa. Interestingly, the total increase in rope tension is only 1.5% of the pre-set value, which effectively prevents conical wear of beads.

The above data well coincides with both field and laboratory test results recorded on class 1 granite [9]. Interestingly, no meaningful effect of diamond quality on 12-mm diameter wire/bead performance was found in Ref. [9] for normal loads ranging between 3 and 9 N. At the lowest applied load (3 N), the beads containing higher-grade diamonds showed noticeable reduction in cutting rates. This was associated with progressive wear flatting of diamond crystals on the surfaces of the beads.

2.2

Wire vibrations

Contrary to circular and frame diamond saw blades, where the cutting segments are attached to rigid steel holders, the diamond wire has no clearly defined shape and is considered to be completely flexible. Therefore, it is particularly prone to detrimental vibrations resulting from a variety of tool-, workpiece- and machining-related factors. A number of these factors were studied while sawing a medium hard granite on a laboratory test machine and have been reported in Ref. [10]. Although the sawing conditions differ from those commonly used in a production environment, some relationships are evident.

First, the vibration frequency well corresponds to the product of the number of beads per unit length of diamond wire and its linear speed (nL∙vs). This implies that beads are subjected to successive impact loads as they enter the stone block.

Second, the vibration amplitude is higher at the ends of block (0.6–1.6 mm) than at its centre (0.4–0.8 mm). It has also been found that the amplitude markedly decreases with increasing the wire linear speed, down-feed rate, tension and length of block (shorter distance between block and guide wheels).

2.3

Wire sawing versus circular sawing

As presented in Fig. 5, a slab of granite is sawn with a circular diamond saw in many passes with a reciprocating movement of the blade, which alternately operates in the up-cutting and down-cutting modes.

Schematic representation of slab cutting with a diamond circular saw blade

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The blade rotates in a constant direction at a peripheral (linear) speed ranging from 25–30 to 30–40 m/s for classes 4–5 and classes 1–3 granite, respectively. The depth of cut ranges between 1 and 20 mm, whereas the feed rate is chosen according to the rock sawability class, machine rigidity and power, surface finish, etc. [6]. High feed rates (15 m/min) combined with shallow depths of cut (1 mm) result in a short length of contact between the tool and the workpiece and favour a free-cutting action of the tool, which becomes a priority in sawing granite on multi-blade machines for the production of modular tiles.

For shallow cuts, the length of contact can be calculated from the following equation:

$$ {l}_c\cong {R}_p\sqrt{aD} $$

(9)

where Rp is the rim partition ratio calculated as the total length of diamond segments divided by the blade circumference.

Diamond loading conditions change dramatically when reversing the saw blade rotation as illustrated in Fig. 6.

Kinematic diagram of diamond crystals that come in contact with a workpiece in down-cutting mode and up-cutting mode

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In down-cutting, diamond crystals penetrate into the workpiece to full depth while coming in contact with it, whereas in the up-cutting mode, diamonds gradually increase the depth of penetration to achieve the maximum values while leaving the kerf. Consequently, the diamond breakdown is facilitated by downward rotation of the blade, and the maximum chip thickness, quantifying diamond loading conditions, can be calculated as follows [7]:

$$ {h}_{\mathrm{max}}\cong \frac{v_fa}{w{C}_s{v}_s}\sqrt{\frac{1}{aD}-\frac{1}{D^2}} $$

(10)

where w is width of a diamond crystal and Cs is surface concentration of diamonds.

In contrast to circular sawing, in wire sawing, the beads rotate around the steel rope at an angular speed ω, changing angular position of working diamonds in the cut (γ) as demonstrated schematically in Fig. 7.

Kinematic diagram of a diamond bead cutting the stone block (a) and effect of diamond angular position on its penetration depth (b)

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At γ=0 deg (see Fig. 7b), a sufficiently protruding diamond engages the workpiece increasing the depth of penetration until γ=90 deg, when

$$ {h}_{\mathrm{max}}\cong \frac{v_f}{v_s{n}_L} $$

(11)

Equation (11) holds for statistically uniform distribution of crystals over the surface of beads [6].

For 0 < γ < 180 deg, when the surface diamonds are able to cut, the average chip thickness may be approximated as

$$ {h}_{avg}\cong {h}_{\mathrm{max}}\frac{\int_0^{\pi}\sin \gamma d\gamma}{\int_0^{\pi } d\gamma}={h}_{\mathrm{max}}\frac{2}{\pi } $$

Hence,

$$ \mathrm{and}\kern0.5em {\displaystyle \begin{array}{lll}{h}_{avg}\cong 0.64\frac{v_f}{v_s{n}_L}& \mathrm{for}& 0<\gamma <180\ \deg \\ {}{h}_{avg}=0& \mathrm{for}& 180\le \gamma \le 360\ \deg \end{array}} $$

(12)

Because the length of contact between the diamond wire and stone is

$$ {l}_{\mathrm{c}}={L}_{bl}{n}_L{l}_b $$

(13)

it becomes evident that due to a huge difference in the length of contact, for the same cutting rate, tool linear speed and tool specification (diamond size and concentration), diamond crystals involved in circular sawing must deeper penetrate the workpiece in order to remove more material compared to diamonds involved in wire sawing. This has been exemplified in Table 2.

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Although it is unfeasible to find exact values of w and C in Eq. (10), a rough estimate of hmax for circular sawing can be obtained by taking w = 360 μm and Cs = 63 [6] for 40/50 mesh (297–420 μm) diamond at 25 concentration (6.25% by volume). Bearing in mind, however, that the empirical figures provided in Ref. [6] represent the total number of diamond and pullouts per unit area of wear surface of the tool, the hmax figure given for circular sawing in Table 2 seems underestimated.

By analogy with diamond penetration depths, there is also a marked difference in forces acting on wire beads and blade segments. Forces involved in circular sawing class 4 granite have been reported in great detail in Ref. [11]. The sawing tests were performed using 203-mm blade containing segments impregnated with medium-grade 40/50 mesh diamond at 20 concentration.

The results are presented in Table 3.

Full size table

High Light:

10.5mm Cutting Diamond Wire, 

Concrete Cutting Diamond Wire, 

10.5mm Stone Cutting Wire

Vortex Diamond Concrete sawing wire has three different bond, CO/01 is designed for general purpose cutting with high speed, CO/02 bond is designed for general purpose wiring with longer life, and CO/03 bond is designed for the tough project like the bridge, dam which contains heavy reinforced.

All diamond tipped cutting tools work best at a given surface feet per minute range, diamond wire operates best at a speed ranging between 4800 to 5500SFM. At this speed, material removal rate, cut time, power requirements and diamond bead wear are all optimized. Slower wire speeds are suggested at the beginning and end of cuts to reduce stress on the wire and wire sawing equipment and to allow for better control of the wire. 

 2. Specifications of Concrete Sawing Diamond Wire

 Code No.

VDW-CO/01

VDW-CO/02

VDW-CO/03