裝配圖胡軍雄,裝配,胡軍雄 Experimental and numerical study of a new resonance hammer drilling model with drift Luiz Fernando P. Franca and Hans Ingo Weber , Mechanical Engineering Department, Pontifical Catholic University of Rio de Janeiro, Rua Marquês de S?o Vicente 225, Rio de Janeiro, RJ 22453-900, Brazil Accepted 11 December 2003.? Available online 28 January 2004. Abstract New drilling techniques have been studied to increase the penetration in hard rock formations. These techniques use harmonic loads and, in some cases, also impacts to generate a greater penetration rate. Analyzing only a percussive penetration phenomenon, the new model presented in this paper allows the forward motion (with a drift) in stick–slip condition with and without impact. Numerical and experimental investigations are presented and are qualitatively and quantitatively compared. Otherwise looking for other parameter ranges in the simulation, it is shown that the behavior may vary from periodic to chaotic motion. From the engineering side, the main interest in a study like that in this paper is if the rate of penetration can be improved. The simulation may help a lot in that, and in a short conclusion one should look for regions with period-1 behavior. Article Outline 1. Introduction 2. Experimental apparatus 3. Mathematical model 4. Parameter identification 5. Numerical simulation 6. Experimental and numerical results––case without impact 7. Experimental and Numerical Results–Case with impact 8. Optimal Parameter Choice 9. Conclusion Acknowledgements References 1. Introduction There has been an increasing effort looking for oil resources in deep seas water. Special drilling techniques have been developed due to the large amount of specific wells that are being done. Drilling may be considerably delayed if there is need to change the drillbits or to stop the procedure, to attend project conditions and improve drilling efficiency. Looking for an increase in productivity, recently, attention has been paid in improving drilling efficiency by imposing dynamic loading at the bit–rock interface [1, 3, 4 and 9]. To date, this has been applied only in the restrictive circumstances of shallowness. The various modes of the drillstrings vibrations (axial, torsional, and bending which in their most severe forms lead, respectively, to bit bouncing, stick–slip oscillations, and bit whirling) are generally regarded as detrimental. However, as shown in this work, it appears possible to control some of these vibrations modes in such a way as to enhance drilling performance. A new drilling technique called resonance hammer drilling has been studied by us, as an alternative to increase the rate of penetration (ROP) in hard rocks drilling, Fig. 1. This technique has as premise to use the already existent vibrations in the drillstring, in fact the axial vibration due to the cutting process, to generate a harmonic load on the bit and an excitation in a steel mass (hammer). When this excitation frequency is near to the steel mass resonance frequency and, since the steel mass displacement is limited in positive direction by the gap, impacts on the bit may occur. (2K) Fig. 1. Resonance hammer drilling. Therefore, besides the rotative penetration, where the teeth of the bit penetrate in the rock when the drillstring rotates, a percussive penetration happens due to the harmonical load or due the impact, increasing the ROP. Although, the harmonic force should never be larger than the preload (WOB––weight on bit), due to the possibility of the bit bounce effect, and also the hammer resonance frequency should not coincide with any drillstring natural frequency. Actually, when the exciting frequency, which equals the frequency of displacement at the bit, due to the bit rolling over the lobed bottom hole, corresponds to an axial natural frequency of the drillstring then, resonance occurs and the drillstring can bounce. Consequently, one way to minimize the response to vibrations is to avoid drillstring rotations that induce natural frequency vibrations [2]. The proposition of the present investigation is restricted to the percussive penetration phenomenon. A simple model for the longitudinal behavior of the bit–rock interface is proposed and the drilling resistance is modeled by a dry friction element. Moreover, the model presented in this work moves forward (a drift) in stick–slip phase with or without impact. Since impact and dry friction are present, usually these systems are nonsmooth. During the past decade, new analytical and numerical tools have been developed for the study of nonsmooth systems [6, 11 and 14]. The dynamics of physical systems, whose components can suffer impact or present dry friction, is very important in practical applications. Nevertheless, very few works have considered systems, which associate these conditions with progressive drift [3, 10 and 12]. In this context, this work has the objective to investigate the dynamics of the proposed nonsmooth percussive drilling model numerically and validate by experiments the numerical model. For that, some nonlinear tools are used like: phase space, Poincar maps and bifurcation diagrams. 2. Experimental apparatus The experimental data related to the new percussive response is obtained from the apparatus depicted in Fig. 2a. Two elements (two cars, A and B) are connected by two springs (2), and can move almost without friction on linear guides (4). The movement of the system occurs along the drift (5) due to the existence of the preload (6) and of the harmonic force generated in the shaker (1). The drift represents the penetration depth of the bit. The preload and the shaker are fixed at the first car. The dry friction (3) is produced by friction shoes on springs, fastened with a screw. This device is able to change the friction force. The data acquisition is done through encoders (7). The impact device (8) has the capacity to change the gap between elements. The encoders are connected in the Universal Serial Bus (USB) to the computer. The sensors measure angular position and angular velocity and their specifications are: ACCURACY = +/?0.09°; maximum SPEED=30 revolutions/s and maximum sample RATE=1000 Hz. The photograph of the whole experimental apparatus is seen in Fig. 2b. (37K) Fig. 2. (a) Experimental apparatus; (b) photograph of overall view. 3. Mathematical model In this work we considered a physical model according to Fig. 3. The dynamics of the system corresponds to a progressive oscillatory movement along the drift. The mass m1 is connected to the mass m2, through a spring K and a viscous damper C. A preload B and a harmonic force are present in the external excitation applied to m1. (3K) Fig. 3. Physical model. As in the stick–slip phenomenon, the progressive movement occurs when the sum of the elastic and viscous forces applied on m2 overcomes the friction force, Fat, between mass m2 and the drift. Therefore, the system presents two different motion phases: stick-phase, without progression and slip-phase, with progression. Then, for , we don’t have the progression and the system just corresponds to a spring-damper-mass (stick-phase) (1) where B is the static load or preload, A the dynamic load amplitude and ω the excitation frequency. The phase angle, , is a phase shift between the overall drilling force and the progressive stationary motion. On the other hand, when , we have progression and the system equations are written as (slip-phase) (2) The dry friction force is defined through a simple continuous model described as (3) where FN is the normal force, μe and μd are, respectively, the static and kinetic friction coefficients and γ is the decay parameter. However, the m1 displacement is limited in positive direction by the gap R. Thus, when numerically Xb?Xs>R, we have impact, contact or collision and the structure of the system has again to be changed. The contact force is modeled by a non-linear spring-dashpot [5, 7 and 8] (4) where, δ is the penetration, is the penetration velocity, Kc is the contact spring constant, λ the damping constant and the exponent n, which is often close to one, depends on the contact surface geometry [7 and 8]. Then, if there is impact and we don’t have progression (5) On the other hand, if there are impacts and progression (6) The ordinary differential equations of second order ((5) and (6)) can be transformed in a first order system of differential equations in an autonomous system, 5-Dim, through the following change of variables: (7) Therefore, (8) where, (9) Here, P and P2 are Heaviside functions with respect to progression and impact, respectively, which characterize the variable structure of the problem. 4. Parameter identification Physical parameters of the experimental components are: m1=25.3 Kg; m2=6.7 Kg; K=2170 N/m. Applying an impulsive force on m1 and maintaining m2 fixed, the natural frequency of the free (non impacting) system can be obtained, Fig. 4a and b. Trying to identify the value of the experimental viscous damping rate, a sinusoidal excitation is applied on m1 and some experimental points of the maximum amplitude versus frequency are presented in Fig. 4c. The value of damping ratio, ξ1, can be identified [9], and equals ξ1=0.074. (9K) Fig. 4. (a) Displacement of the mass A due to impulsive force; (b) FFT; (c) maximum amplitude versus frequency. Several steps are needed for the identification of the dry friction parameters (μe,μd,γ). Firstly, the normal force for different compressions of the spring (in the dry friction apparatus (3) in Fig. 2a) is measured. After that, the static friction force is identified, applying a growing force on m1 until m2 slips. The maximum force registered in the load cell is exactly the static friction force. The variation of normal force with static friction force is shown in Fig. 5a. Applying a linear fit among points, the kinetic friction coefficient is identified, μe=0.62. A similar procedure is adopted in the identification of dynamic friction. However, an accelerometer is used to obtain, by integration, the velocity of m2. Therefore, the velocities are obtained for three different normal forces (30.27, 62.13, 93.2 N) and plotted with the friction force in Fig. 5b. The dynamic friction coefficient and the decay parameter are identified and correspond to 0.56 and ?400, respectively. Using the dry friction parameters identified in the model, the variation of the dry friction for FN=62.13 N is presented in Fig. 5c. (9K) Fig. 5. (a) Fat static versus normal force; (b) dry friction force versus velocity; (c) numerical end experimental dry friction force. Now, the impact parameters are identified (Kc,λ,n). In this case, a force sensor (ISOTRON force transducer), placed in the point of the impact element, is used to measure the contact force. As the impact occurs between the cars, it is necessary to use relative parameters [13]. The equation of motion of these masses during the impact becomes (10) where the relative mass is m=5.06 Kg=(mAmB/(mA+mB)), and the relative velocity is equal to the m1 velocity if m2 is maintained fixed. The variation of force for different initial impact velocities is shown in Fig. 6a. It can be observed that the contact is partially elastic [5], i.e., there is loss of energy but no permanent deformation. Furthermore, the time of contact depends on the initial impact velocity, justifying the use of the nonlinear model of contact. Varying the parameters and integrating numerically the Eq. (10), the parameters are identified: Kc=2.1×108 N/mn, n=1.3 and λ=0.95 s/m. Fig. 6b presents the numerical contact force with the identified parameters together with experimental results. Finally, the relationship between volts peaks to peak (Vpp), applied by the wave generator, and maximum harmonic force measured in the load cell, can be observed in Fig. 6c. (14K) Fig. 6. (a) Contact force versus time; (b) experimental and numerical contact force; (c) Vpp versus harmonic load. 5. Numerical simulation Numerical simulations are done using the fourth order Runge-Kutta method for numerical integration with fixed step. However, when a progressive movement is identified, a sub-routine (bisection) determines the exact instant of the progression start, τp (error=10?6). At this instant, P is altered. The same procedure is applied when a contact is identified. In this case, the contact force profile (Fig. 6b) is determined and applied in the system, with P2=1. In all simulations there are considered: Δτ=2π/1000η; =π/2; ab (to avoid the bit bounce), as well as, a+bμeFN (drilling resistance). In addition, all the initial conditions are 0.0. 6. Experimental and numerical results––case without impact The comparison of numerical and experimental results is now focused. A typical steady-state time history, without impact, is presented in Fig. 7. In this case, ω=1.7 Hz, amplitude 5.0 Vpp (10.5 N), preload 14.7 N and a sample rate of 100 Hz. Because of the drift, the displacements of Xb and Xs are oscillatory with progression and stick–slip, respectively. Although, when we change the data to ω=1.9 Hz, amplitude 2.4 Vpp (4.8 N), same preload and sample rate, the behavior of the system changes, Fig. 7b. Now, the progression occurs after two periods of the harmonic force. This behavior is called period-2 or P2. It is important to explain that the velocity and the displacement are measured with the encoder, therefore, the respective units are rad and rad/s. (10K) Fig. 7. (a) ω=1.7 Hz, amp. 5.0 Vpp, preload 14.7 N (period-1); (b) ω=1.9 Hz, ampl. 2.4 Vpp, preload of 14.7 N (period-2). One way to perform the analysis in nonlinear dynamic systems is through the phase space. Two phase spaces of the relative motions (x1?x3, x2?x4) obtained numerically and by experiments are shown in Fig. 8a and b. In this case, the two dynamic behaviors are period-2 (P2). For a quantitative comparison, the numerical results are changed for a dimensional condition (m, m/s). Consequently, it can be seen the results are qualitatively and quantitatively similar. The experimental results are for: ω=1.33 Hz, preload 27 N and 5.04 Vpp in Fig. 8a, and ω=1.5 Hz, preload 32.4 N and 3.7 Vpp in Fig. 8b. Numerical results are for: η=0.95, b=0.5; a=0.15, Fig. 8a, and η=1.05, b=0.6; a=0.11, Fig. 8b. (17K) Fig. 8. (a) Experimental ω=1.33 Hz, preload 27 N, 5.04 Vpp, numerical η=0.95, b=0.5, a=0.15; (b) experimental ω=1.5 Hz, preload 32.4 N, 3.7 Vpp; numerical η=1.05, b=0.6, a=0.11. Qualitative analysis of the changes in the nonlinear dynamical system behavior can be accomplished through the bifurcation diagram. Defining a (amplitude of the harmonic load) as the control parameter and using η=1.0 and b=0.55, the bifurcation diagram, obtained with the model, is represented in Fig. 9. In this diagram the occurrence of period doubling bifurcation, or Flip bifurcation can be observed, without any further complex behavior. It is interesting to observe also that increasing a, the number of periods decreases. (7K) Fig. 9. Bifurcation diagram: η=1.0 and b=0.55. With values of a according to the sections of Fig. 9, the respective phase spaces of the relative motions and Poincar maps obtained numerically are presented in Fig. 10b, d, f and h. Applying the same parameters to the experiment, i.e., ω=1.47 Hz (η=1.0), preload 29.7 N (b=0.55), the numerical results may be compared to the experimental ones. Varying the shaker amplitude according to the section of Fig. 9, the respective phase spaces obtained experimentally are shown in Fig. 10a, c, e and g. Now however, only qualitative comparisons are made, since the nondimensional numeric results are not changed, notice that there is a good qualitative agreement between the responses. This similarity leads us to the conclusion that this simple model can be used to study the percussive drilling without impact. The experimental results obtained with: ω=1.47 Hz; preload 29.7 N use the amplitudes 3.15, 3.2, 3.7, 5 Vpps and sample rate of 100 Hz. The numerical results are obtained with: η=1.0 b=0.55 and a=0.0069, 0.083, 0.11, 0.2, respectively. (38K) Fig. 10. Experimental phase space––ω=1.47 Hz, preload 29.7 N: (a) P5––3.15 Vpp; (b) a=0.069; (c) P3––3.2 Vpp; (d) a=0.083; (e) P2––3.7 Vpp; (f) a=0.11; (g) P1––5 Vpp; numerical phase space––η=1.0, b=0.55 and (h) a=0.2. 7. Experimental and Numerical Results–Case with impact The comparison of numerical and experimental results with impacts is now focused. The steady-state time history with impact is not very different, but the initial progression is due to the impact. A period-2 behavior for: ω=1.47 Hz, amplitude 3.7 Vpp, preload 29.7 N, GAP=4 mm and sample rate of 50 Hz is presented in Fig. 10a. Again the displacements of Xb and Xs are oscillatory with progression and stick–slip, respectively. Nevertheless, the displacement of Xb and Xs are nonsmooth, because of the initial velocity after impact. The change of the velocities after impact can be seen in Fig. 11b. (9K) Fig. 11. (a) Progression versus time with: ω=1.47 Hz, amp. 3.7 Vpp, preload 29.7 N (period-2), GAP=4 mm; (b) relative velocity versus time. Defining again a (amplitude of the harmonic load), as the control parameter and using η=1.0, b=0.55 and r=0.84, the bifurcation diagram, obtained with the model is represented in Fig. 12. As in the case without impact, the occurrence of period doubling bifurcation can be observed, without any further complex behavior. (5K) Fig. 12. Bifurcation diagram: (a) η=1.0, r=0.81 and b=0.55. With values of a according to the section of Fig. 12, the phase spaces of the relative motions and the Poincar map obtained numerically, together with the phase spaces obtained experimentally, are compared in Fig. 13a–h. The experimental results of P4, P3, P2 and P1 are obtained for: ω=1.47 Hz, preload 29.7 N, GAP=4 mm and amplitudes 2.8, 3.0, 3.4, 4.7 Vpp, respectively. The numerical results are obtained with: a=0.044 (P4); a=0.05 (P3); a=0.06 (P2) and a=0.11 (P1). As it can be seen again, there is a good qualitative agreement between the responses and the study of the percussive drilling with internal impact can be done with the proposed model. (40K) Fig. 13. Experimental phase space––ω=1.47 Hz, preload 29.7 N, GAP=4 mm: (a) P4––2.8; (b) a=0.044; (c) P3––3.0 Vpp; (d) a=0.05; (e) P2––3.15 Vpp; (f) a=0.06; (g) P1––4.7 Vpp; numerical phase space––η=1.0, b=0.55, r=0.84; (h) a=0.11. As a conclusion of these comparisons, the simple model proposed in this paper to study the percussive drilling with or without internal impact is validated experimentally and can be extended for other ranges of the parameters. 8. Optimal Parameter Choice The main interest in a study like that in this paper is the answer of the question if the rate of penetration (ROP) can be improved. We called that optimal parameter choice. For the same experimental conditions of Fig. 9 and Fig. 12, the penetration after 100 iterations versus amplitude (a) is shown in Fig. 14. Firstly, when a is increased, the ROP increases, i.e., when the behavior occurs at a condition of period-1, a large drift results in both cases, with impact and without impact, Fig. 14a and b, respectively. Moreover, the penetration is practically the same, but in the case with impact, the progression starts earlier. (12K) Fig. 14. Penetration versus amplitude, a, for: ω=1.47 Hz, preload 29.7 N: (a) case without impact; (b) case with impact, r=0.81. Expanding for other parameter ranges that were outside of the range of our equipment, numerical simulations show that also a complex behavior can be obtained with this simple model. Examples of strange attractors in the Poincar map are given in Fig. 15 for: μeFN=1.0; μdF