Introduction
Impact energy has been used to drill rock formations for many decades now. The various tools that have been used for this purpose include percussive drills, percussive hammers, down hole hammers, and so on. Percussive drilling dynamics has been possible using drifting oscillator models. This has been used to characterize the optimum conditions of the applied forces (both static and dynamic). And in this case, the sliders provide the contact force required during interactions. This force acts on the drill-bit, and the model accounts for contact geometrics in the crashing phases of the interactions.
A series of experiments were conducted by Franca to design an in-house rotary percussive drilling. He proposed that quantitative information on rock properties, drilling efficiency and bit conditions can be obtained from the drilling data. To do this, he came up with a new bit-rock interaction model. Later, Weber experimented with resonance hammer drilling models that exhibit drift. He demonstrated the variations in symmetrical behaviour.
Succeeding that, Wiecigroch conducted extensive research to study the behaviour of ultrasonic percussive drilling. In his laboratory, he used tools coated with diamond for use with rocks such as limestone, basalt, granite, and sandstone. His main objective was to explore the applications of this methodology in down-hole drilling. He wanted to use mathematical models to support his technique, and be in a position to describe the main processes that occur during drilling.
The resulting experimental set-up is as shown in Figure 1. A PEX magneto-stricture device generates the bit’s vertical oscillations. This, along with the static weight on the bit due to the hydraulic cylinder, assists data gathering in the experiments. The vertical lathe has a rotating table that aids in the specimen’s rotary motion, which is separated from the two components previously mentioned. The entire arrangement effectively maintains the total rotary percussive action at the interface. It also simplifies the overall instrumentation involved.
Figure 1: Rig Schematic
Two models designed based on the principles explained below, have been chosen for analysis in the subsequent sections. The first one involves a new model with the rig set-up as in Figure 1. The second one is low dimension, simplified model that describes the same interaction between the bit and formation (Pavlovskaia et a l.).
New Model
Mass m1 represents the drill bit, which is placed at a distance G initially, from the rock’s surface. A visco-elastic slider represents the rock. The elastic behaviour of the rock and it damping properties are accounted for by the damper with a damping co-efficient of c1, and the spring with the stiffness or spring constant k1. This is true when the assumption that contact forces are small, is satisfied. In case the force exceeds Pr, then movement begins even though the resistive force is maintained a constant. ‘x1’ represents the displacement of the drill head, ‘xt’ represents the displacement of the rock surface, and ‘xb’ represents the displacement of the slider.
When the rock and drill head are not in contact, the slider velocity is reduced to zero. During this time, there is free motion, and the rock surface moves under the damper and spring forces. When the rock and drill head come in contact, the difference between x1 and xt is the initial gap G. The analysis for this model can be carried out from Figure 2.
Figure 2: Physical model of rig
The best progression rates are possible during periodic regime. The major assumption here is that the system under consideration is piecewise linear. This means that construction of dynamic response by joining the linear solutions at discontinuities. The major phases involved in a typical period one motion sequence, is explained below.
Phase 1: Progression takes place when the slider and mass come in contact.
Phase 2: Even though the slider and mass come in contact, the slider’s bottom does not move. In other words, there is no progression even with contact.
Phase 3: When there is no contact, the two objects move separately.
Phase 4: This is the same as Phase 2.
Low Dimension Model
This model comprises a loaded mass with static and harmonic components and a slider with dry friction. This is shown in Figure 3. The mathematical equations that describe such a system are much simplified when compared to the new model. In other words, the number of parameters required to assess the behaviour of a system, is reduced.
Figure 3: Physical analysis for low dimension model
Equations Governing the Models
New Model
Dz1=z2
Dz2=(-l3*b3*z1)-(2*e*l3*g3*z2)+(l3*b3*z3)+(2*e*l3*g3*z4)
Dz3=z4
Dz4=(l2*b2*z1)+(2*e*l2*g2*z2)-(l2*b2)-(b3*z3)-(2*e*l2*g2*z4)-(2*e*l2*g3*z4)+(l2*b2*z5)+(l2*b)
Dz5=z6
Dz6 = (acos (w*D + p)-(b2*(z5-z3))-(2*e*(z6-z4))-(p1*p2*(1-p3)*((2*e*z6) +z7-z8)-(p1*p3))
Dz7= (p1*z6)-((1-p1)*(z7-z8)/ (2*e))
Dz8= (p1*p2*p3)*((z7-z8-1)/ ((2*e) +z6))
Where l3 is α3, l2 is α2, e is ξ, b2 is β2, b3 is β, g2 is γ2, g3 is γ3, w is ω, d is tow, p is ф and ‘a’ and ‘b’ are constants.
Low Dimension Model
DZ1=Z2
DZ2= (Acos (W*t +P) +B-(P1*P2*(1-P3)*((2*E*Z2)+Z3-Z4)-(P1*P3))
DZ3=(P1*Z2)-((1-P1)*(Z3-Z4)*(1/E))
DZ8=P1*P2*P3*((Z7-Z8-1)/ ((2*E) +Z6))
Where E is ξ, W is ω, T is tow, P is ф and ‘A’ and ‘B’ are constants (Pavlovskaia et a l.).
These Ordinary Differential Equations were implemented in MATLAB, and the following results were obtained for a typical period 1 response, and a given set of constants.
Figure 4: Displacements in the new model
Figure 5: Displacements in low dimension model
Using these results, relative displacements can also be plotted. The above analysis has been done keeping the frequency and amplitude of the dynamic force, and the applied static force, as constants. It is evident that the characteristics of the models change when these parameters change. It is important to analyze the boundaries within which each model works well. For instance, in Figure 4, the ω value is taken to be 0.4. It is seen that is ω is outside the open interval (0.3, 0.5), then the model becomes chaotic.
Works Cited
Pavlovskaia E, et al. Modelling of high frequency vibro-impact drilling. Int. J. Mech. Sci. (2013), http://dx.doi. org/10.1016/j.ijmecsci.2013.08.009i
