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Impacting between the multilayer Kelly bar and the power head has a detrimental effect on the drilling system and may cause huge losses to the construction engineering and machinery. This study is a further development of the vibro- impact mechanism, and especially, the study considers the constraint of the steel rope on the multilayer Kelly bar. Efficient analytical models are presented, and the total responses are obtained. Three types of displacement response are found by changing the matching relationship of global parameters. The first type has the optimal vibration control performance with the shortest path and no periodicity. The second type is inferior to the first type. The third type has the worst vibration-control performance with the longest path and periodicity. Then this article proposes the general feasibility to get the first type of displacement response and the most effective way is controlling the steel rope velocity. The research provides the foundation for the intelligent control of steel rope velocity on the rig machine. Keywords Vibration control, vibro-impact, intelligent control, rotary drilling rig, multilayer Kelly bar Date received: 25 January 2019; accepted: 5 November 2019 Handling Editor: James Baldwin 8,9 et al. studied the non-continuous grazing bifurcation Introduction behavior based on the periodic motion of a 3-degree- The Kelly bar, which is usually more than 10 tons, is 10 of-freedom vibro-impact system. Liu et al. studied the designed as a telescopic and multilayer pipes structure. vibro-impact in a tank laying on water and established The pipes’ weight is loaded on the power head in the the attractor to avoid bifurcation by the means of dis- form of instantaneous impacting when the pipes stretch placement feedback control, so as to achieve the con- out. The impacting repeats frequently when the rig trol of forward and backward motion of the tank. For works. Usually, the pipes fall freely at any time, and the purpose of vibration control, the absorber could be the impacting is extremely destructive to the power added in the process of vibration transmission by pas- head. The vibro-impact after the impacting may cause 11 sive control to obtain the reasonable response. The deformation, weld cracking, and oil leakage. In order characteristic parameters could be changed by active to reduce losses, it is necessary to check the quality of control to improve the response performance in the the power head. It is very important to prevent and 12 13 vibro-impact system. Harvey et al. applied both iso- control the vibro-impact. lation and absorption methods to control the vibro- The vibro-impact usually results in an uncertain response. Some special excitations make the vibration School of Mechanical Engineering, Tongji University, Shanghai, China system uncertain and chaotic, and the response usually 2–5 6 7 has the bifurcation. Luo et al. and Yue et al. ana- Corresponding author: lyzed periodic motion and bifurcation phenomenon of Lei Qin, School of Mechanical Engineering, Tongji University, Shanghai the dynamic response by Poincare´ map based on the 201804, China. vibro-impact system with 2 or 3 degree of freedom. Liu Email: qinl@tongji.edu.cn Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 2 Advances in Mechanical Engineering impact at the same time. The study was verified on a device that constrained displacement and acceleration. In a constrained vibro-impact system, the force and the energy were transferred to the object providing the con- straint, as well as a huge change of vectors such as velo- 14–21 city and acceleration. In a multi-degree-of-freedom vibro-impact system, the response behaviors were cor- related and coupled, and the solution of the motion equations were more complicated. Xue and Fan and Fan and Yang studied the 2-degree-of-freedom vibro- impact system with multiple constraints and analyzed the periodic response of the system after establishing the boundary conditions. Previous studies focused on the theoretical basis of generalized abstraction. This article pays more atten- tion to solving major engineering problems in specific applications. The study provides a further development of the vibro-impact system with multi-degree of free- dom. In this work, the dynamic model of vibro-impact is established with the constraint of the steel rope espe- cially, and the influence of related parameters is ana- lyzed according to the source and transmission process of vibration. The study focuses on the total response considering the global parameters. The analysis is undertaken to improve the optimum design. The result could provide foundation for the intelligent control Figure 1. Rotary drilling rig assembly and vibration control about steel rope velocity. system of the power head for vibro-impact. elastic coefficient is k, the equivalent damping coeffi- Problem statement, dynamic model, and cient is c, the dry friction is P, and the displacement formulation response of the vibration system is x. The piecewise analysis is used to establish the differ- Figure 1 shows the rotary drilling rig assembly and the ential equation. Before the impacting, the first pipe with vibration control system of the power head. When the the mass M falls at a uniform speed of n under the 1 0 rig is working, the first layer pipe extends downward traction of the steel rope. In the instant of the impact- with the steel rope at a certain speed. At last, the first ing, the initial condition at time t = 0 is obtained. In a layer pipe moves with the power head. If the hole is short time after the impacting, the system is loaded by deep enough, the subsequent impacting of each pipe the step excitation. Each piecewise process will act in takes place at the bottom. What we expect is to sup- the global response. press the periodicity or to reduce the maximum displa- First, considering the vibro-impact between the first cement in the vibro-impact so as to reduce the sliding layer pipe and the power head according to Figure friction displacement and reduce wear. What is more is 2(a). The displacement response without constraint is that the expectation is realized by making the structure expressed as follow compact, lightweight, and material saving in the case of reasonable reduction of the displacement response. x + zv x 0 n 0 zv t x ðÞ t = e x cosv t + sinv t ð2Þ Figure 2 shows the dynamic models about the vibro- h 0 d d impact system. The differential equation of motion for pffiffiffiffiffiffiffiffiffi the vibro-impact can be expressed as where v = k=m is the undamped natural circular fre- pffiffiffiffiffiffiffiffiffiffiffiffiffi quency of the system, v = 1 z v is the damping d n m€x +fxðÞ , x _, c, k, P =FtðÞ ð1Þ circular frequency, z = c=c = c=(2mv ) is the damping c n where f is a certain function, and its specific form is ratio. related to each parameter variables. F(t) is any function The weight of the first layer pipe is considered as a depending on the excitation related to time t. In the step excitation with x = x _ = 0, the step-excitation 0 0 vibration control system, the mass is m, the equivalent response is obtained as follow Wu et al. 3 Figure 2. Dynamic models of the vibro-impact system: (a) impacting of the first layer pipe and (b) impacting of the nth layer pipe. Then the nth layer pipe of the Kelly bar in the vibro- zvðÞ tt x ðÞ t = e sinvðÞ t t FðÞ t dt ð3Þ p d impact is considered (n ø 2) according to Figure 2(b). mv The simplified dynamic model has 2 degrees of free- dom, and the differential equation of motion is estab- where F(t)= M g. The total response is the sum of the 1 lished with another new final static stable position free-vibration response and the step-excitation response about M + M + M + + M + m as the coordi- 1 2 3 n1 with the initial condition. Then the total response is nate origin n1 M + m €x = kx cx _ + k ðÞ x x + c ðÞ x _ x _ i 1 1 1 n1 2 1 n1 2 1 ð8Þ i = 1 M €x = F ðÞ t k ðÞ x x c ðÞ x _ x _ n 2 n n1 2 1 n1 2 1 The initial conditions are given by xtðÞ = x ðÞ t + x ðÞ t ð4Þ h p x ðÞ 0 = x ðÞ 0 = 0 1 2 ð9Þ x _ ðÞ 0 = 0; x _ ðÞ 0 = v 1 2 0 In order to correct the model, the constraint of the rope is introduced into the equations with the correc- where M is the mass of the nth layer pipe, k is the n n1 tion parameter a. Then the modified step-excitation is equivalent elastic coefficient of the (n 1)th layer pipe, given by and F (t) is the function depending on the step excita- tion related to the nth layer pipe. FtðÞ = M g + at ð5Þ The new equation of the step-excitation response Numerical investigation becomes Setting the value of some parameters, M = 2000 kg, m = 100 kg, k = 200 kN=m, v = 2 m=s, z = f0, 0:1g, zvðÞ tt x ðÞ t = e sinvðÞ t t ðÞ M g + at dt pc d 1 and a=k = f0, 1, 2, 3g. Figure 3 shows the rela- mv tionship between the total response and the steel rope displacement without a. In normal conditions, the solid ð6Þ line should be below the dashed line to ensure that the The new total response of the vibration control sys- response is not constrained by the steel rope. tem is obtained After introducing the correction parameter a into the system, the relationship between the total response x ðÞ t = x ðÞ t + x ðÞ t ð7Þ m h pc and the steel rope displacement is shown in Figure 4. 4 Advances in Mechanical Engineering Figure 3. Relationship between total response and steel rope displacement: (a) z = 0, a = 0 and (b) z = 0:1, a = 0. Figure 4. Relationship between total response x (t) and steel rope displacement x with correction parameter a: (a) z = 0 and (b) m r z = 0:1. As shown in Figure 4, the curve of the total displace- curve of the total displacement response is always ment response is significantly correlated with the curve under the curve of the rope displacement, which means of the rope displacement as the parameter a changes. that no response exceeds the rope displacement, and it When a=k. 2, the curve of the total response is contradicts the premise analysis. Therefore, only above the curve of the rope displacement and it still a=k = 2 meets the model requirement. The value of cannot meet the model requirement. When a=k = 2, a=k = 2 is not accurate enough. Try again and the the curve of the total response has the best approxima- result of a=k = 2:2 is obtained finally. tion to the curve of the rope displacement, which is Now considering whether a=k = 2:2 satisfies all approximately tangent at a point. When a=k\ 2, the common conditions. Setting the value of some Wu et al. 5 Table 1. The six groups of data for numerical investigation and the value of a=k. No. a=kM (kg) m (kg) k (kN=m) v (m=s) z FSSP (mm) Figure 5 1 0 1 –2.2 2000 100 200 2 0 98 (a) 2 –13 4000 100 200 2 0 196 (b) 3 –6 2000 50 200 2 0 98 (c) 4 –0.4 2000 100 400 2 0 49 (d) 5 1.1 2000 100 200 4 0 98 (e) 6 –1.6 2000 100 200 2 0.1 98 (f) FSSP: final static stable position. parameters such as in Table 1. Then the role of differ- The second type of displacement response ent parameters in total response is analyzed one by The second type of displacement response is carried out one. The result is obtained by numerical investigation in Figure 7(a) according to the third group of data in in Figure 5. Table 1. The velocity is faster, and the displacement It shows that the value of a=k is related to all para- response is easier to catch up with the rope displace- meters and is not linear. According to the graphs in ment. The velocity is 2 m/s where the displacement Figure 5, (a) and (f) show that the peak of the displace- curve is tangent to the rope displacement curve, which ment response is close to the final static stable position; is the same as the initial value of the rope velocity. Then (b) and (c) show that the peak of the displacement the vibration control system moves synchronously with response is significantly lower than the final static sta- the rope to the final static stable position, and the vibra- ble position; and (d) and (e) show that the peak of the tion control system has a non-zero initial condition displacement response is significantly higher than the x _ = v . At this time, the free vibration appears with the 0 0 final static stable position. These three types of displa- final static stable position as the coordinate origin, cement response are defined in this article, respectively, where the vibration control system eventually stays at. for further investigation. The third type of displacement response The first type of displacement response The third type of displacement response is carried out Generally, the value of a=k = 2:2 is substituted into in Figure 7(b) according to the fifth group of data in equation (7) and the velocity is obtained by taking the Table 1. The velocity is slower, and the displacement first derivative as follow response will no longer catch up with the rope displace- ment after the first impacting. Then the free-falling x _ ðÞ t = x _ ðÞ t + x _ ðÞ t ð10Þ m h pc impact without the constraint of the rope appears. Due to the increase of initial value v , the total displacement The flows of the velocity about the total response are response amplitude continues to increase. Therefore, in shown in Figure 6, which corresponds to the first type the third type of displacement response, the vibration of displacement response. control system is affected by the free-fall of the pipe, An important result is gained when the displacement and its maximum amplitude is more than twice of the response reaches the final static stable position (the peak) for the first time according to Figure 6(a). The final static stable position. velocity is zero, and its path presents a bifurcation The total displacement response and the type of dis- point. Obviously, constrained by the steel rope, the placement response are obtained in Table 2 and Figure velocity bifurcation point is exactly the final static sta- 8 according to Figure 2(b). ble position, and the velocity no longer changes when The displacement responses of x and x are almost 1 2 no external force is in the system, and as well as the dis- completely coincidental in Figure 8(a). Therefore, it is placement response will be kept at zero. Figure 6(b) considered that the pipe in the model can be regarded shows the actual flow of the velocity with the constraint as a rigid body, whose equivalent elastic coefficient (k ) of the steel rope. Therefore, in the first type of displace- is much larger than that of the vibration control sys- ment response, the direction of the displacement does tem. The conclusion is very useful in the impacting of the nth layer pipe (n ø 2). In this study, only the perfor- not change, nor does the direction of the velocity, and mance of vibration control system is concerned, so x is the vibration control system eventually approaches the final static stable position. So the vibro-impact has no needed for further analysis. In the undamped condi- cyclic motion, less frictional loss, and shorter motion tion, the motion is periodic with (256 72)=2 = 92 mm path. as the final static stable position. The x displacement 1 6 Advances in Mechanical Engineering Figure 5. Total response corresponding to the parameters in Table 1 and the value of a=k: (a) a=k = 2:2, (b) a=k = 13, (c) a=k = 6, (d) a=k = 0:4, (e) a=k = 1:1, and (f) a=k = 1:6. Wu et al. 7 Figure 6. Flows of the velocity in the first type of displacement response: (a) z = 0, a=k = 2:2 and (b) z = 0, a=k = 2:2. Figure 7. Relationship between velocity and displacement: (a) in the second type of displacement response, z = 0, a=k = 6 and (b) in the third type of displacement response, z = 0, a=k = 1:1. Table 2. The data of some parameters for numerical investigation in the impacting between the second pipe and the power head and the conclusion about the type of the displacement response. No. M (kg) M (kg) m (kg) k (kN=m) k (kN=m) v (m=s) c a FSSP (mm) Type 1 2 1 0 1 2000 1890 100 200 432310 2 0 0 92 Third 2 2000 1890 100 200 432310 2–0.41 0 0 92 Third 3 2000 1890 100 200 432310 0.41 0 –85,000 92 First 4 2000 1890 100 200–18 432310 2 0 0 92–1029 Third 5 2000 – 100 200 – 0.41 0 0 to 1310 98 Second FSSP: final static stable position. 8 Advances in Mechanical Engineering Figure 8. (a) Total displacement response of x and x without damping and (b) decision about the type of the displacement 1 2 response x . Figure 9. (a) Displacement response according to different values of v without correction parameter (a = 0) and (b) making the two curves tangent when v = 0:41 m=s by introducing the correction parameter (a = 85, 000). never catches up with the rope displacement as shown final static stable position (92 mm). Now the constraint in Figure 8(b). Therefore, it is a free-falling impact of the steel rope should be considered. Trying to make without the rope constraint, and the maximum displa- the two curves (x and x ) tangent with a = 1 r cement response is 256 mm. In order to improve the 85, 000 N=s(a=k = 0:425), then the peak of x performance, the variables that can be considered are reaches the final static stable position (92 mm), and the the rope velocity v and the elastic coefficient k, which velocity is zero. The vibration control system has the are further analyzed by setting different value. shortest path of one-way motion. Figure 9 is obtained by changing the value of v Figure 10 is obtained by changing the value of k only. The curve of the displacement x also changes only. The final static stable position increases from 92 with the decrease of v from 2 to 0:41 m=s. When to 1029 mm as the k value decreases from 200 to v = 0:41 m=s, the displacement x has the tendency to 18 kN=m. The vibration control system will always be 0 1 exceed the rope displacement x before it reaches the subject to free-falling impacting if increasing the k r Wu et al. 9 free-fall impact. The initial maximum vibration ampli- tude exceeds two times of the final static stable posi- tion. The motion path is the largest, and there is cyclic periodic motion. The vibration control performance is the worst. In the first type of displacement response, the best vibration control system and steel rope velocity can be obtained. This article gives a method for determining the type of displacement response, as shown in Figure 11. According to this method, the design of the power head vibration control system and the best con- figuration of the steel rope velocity are guided. Engineering example The example is proposed according to the engineering requirements. The rotary drilling rig is equipped with 4- Figure 10. Decision about the type of displacement response pipe Kelly bar. Each pipe is 14m long and can provide by changing the value of k only (a = 0). the maximum drilling depth and diameter of 51 m and F2 m, respectively. Then the predictable parameters in the design are shown in Table 3. It is required to pro- value. It is unnecessary to introduce the correction pose a reasonable design method for the vibration con- parameters a in all cases because the system is not con- trol system on the power head. strained by the rope during the analysis. The analysis is carried out in two cases, namely On the whole, decreasing the value of v is the most impacting of one by one and impacting of total weight effective way to get the first type of displacement on the power head. response. Decreasing the value of k increases the final static stable position and does not obtain the first type of displacement response. Increasing the value of k Impacting of one by one. The purpose of the optimization enables the vibration control system to get the third is to obtain the best first type of displacement type of displacement response all the time, and the sys- response, which is related to the final static stable tem always subject to free-falling impact. position. Table 4 shows that v is divided into three intervals according to the k value which corresponds to different Results and discussion type of displacement response. There is unique v to make the system to get the first type of displacement Various parameters need to be reasonably selected and response. predicted. After putting forward the definition of three The Kelly bar includes four pipes. If the rig works types of displacement response, in the same condition normally, the innermost fourth pipe with the drilling about the final static stable position M g=k = 98 (mm), tool will never provide the impacting on the power the first type of displacement response has the least head. Only three pipes are loaded on the vibration con- motion path, the third type has the most, and the sec- trol system when the fourth pipe stretches out. The ond type is between them. Considering the vibration maximum elastic motion range of the vibration control control again, the control method in the first type of system is d = 120 mm (allowable d = 160 mm). So max displacement response is best. When the displacement it yields the equation as follow reaches the final static stable position for the first time, the motion stops. The vibration amplitude of the whole M + M + M + m 1 2 3 k ø = 630 kN=m system is minimized and the motion path is least with less frictional loss. There is no cyclic periodic motion. When the second type of displacement response reaches Taking k = 700 kN=m, the final static stable posi- the final static stable position for the first time, the sys- tion is 108 mm\d, and the corresponding types of dis- tem has initial condition v and starts free vibration. placement response are shown in Table 4. The The motion path is larger than that in the first type. conclusions can be drawn according to the analysis. There is cyclic periodic motion, and the vibration con- The optimal v configuration of each pipe is trol performance is not as good as the former. The v = 6:65 m=s for the first pipe, v = 0:26 m=s for the 0 0 third type of displacement response is equivalent to the second pipe, and v = 0:185 m=s for the third pipe. 0 10 Advances in Mechanical Engineering Figure 11. Overall research methodology of this article. Table 3. The predictable parameters in the design about the vibration control system. Pipes D (mm) d (mm) Length (m) Mass (kg) k (kN=m) m (kg) c eq M F460 F430 14 2600 300310 80 0 M F390 F350 14 2550 332310 80 0 M F310 F260 14 2460 320310 80 0 M F220 F160 14 2000 255310 80 0 M – – – 5000 – 80 0 Table 4. Conclusion about the type of the displacement response in the impacting. k (kN=m) FSSP (mm) v (m=s) Type First pipe Second pipe Third pipe 200 125 \11:1 \0:48 \0:34 Second = 11:1 = 0:48 = 0:34 First .11:1 .0:48 .0:34 Third 500 50 \9:3 \0:31 \0:22 Second = 9:3 = 0:31 = 0:22 First .9:3 .0:31 .0:22 Third 700 35 \6:65 \0:26 \0:185 Second = 6:65 = 0:26 = 0:185 First .6:65 .0:26 .0:185 Third FSSP: final static stable position. Wu et al. 11 Figure 12. The response about total weight of H height free-falling impact: (a) H = 0, (b) H = 1 m, (c) H = 5 m, and (d) H = 14 m. Figure 13. The response about total weight of H height free-falling impact with the buffer k : (a) H = 0, (b) H = 1 m, and (c) H =5m. Impacting of the total weight on the power head. This situa- k = 700 kN=m meets the requirements and then the tion is fault prevention, and it often occurs in the case optimal v configuration of each pipe is v = 6:65 m=s 0 0 that the pipes are stuck and the pipes suddenly fall with- for the first pipe, v = 0:26 m=s for the second pipe, out any constraint. The most serious situation in the v = 0:185 m=s for the third pipe. (2) The impacting impacting is that the total weight of all pipes with drill- response can be clearly and simply judged and solved ing tool affects the system. According to the analysis of using the method of dividing the displacement response the third type of displacement response with the worst into three types. (3) In the situation of fault prevention, vibration control performance, the system needs to bear the buffer k with a larger equivalent elastic coefficient is needed. the total weight of H height free-falling impact, and the Figure 14 shows the practical example. The structure displacement response is shown in Figure 12. uses 10 parallel springs (total equivalent elastic coeffi- Figure 12 shows that the displacement response var- cient k = 700 kN=m) and two parallel elastic buffers ies from 408 to 475 mm, which does not satisfy (total equivalent elastic coefficient k = 1:23 d = 120 mm (allowable d = 160 mm). The k value s max 10 kN=m). They constitute the vibration control sys- can be increased to achieve the goal. However, tem for the power head. k = 700 kN=m has just met the requirements. From the Figure 15 shows the mechanism of intelligent control economical point of view, the buffer k with a larger for the first type of displacement response which is the equivalent elastic coefficient is added, and its motion best vibration control method depending on the rope range is d d = 40 mm. The displacement response, max velocity and bars weight. This method function is rea- which is shown in Figure 13, is in the range of 40 mm. lized by programming. When H ł 5 m, the response has satisfied the From the application point of view, more working requirements. The result of k = 1:23 10 kN=mis conditions and parameters may appear. Using the obtained. research method proposed in this article, similar results can be obtained by changing the parameters. It can Result of the example. These conclusions are drawn guide us to design an optimal vibration control system through the verification of this example. (1) and better intelligent control of steel rope speed. 12 Advances in Mechanical Engineering Figure 14. Engineering application example of vibration control system for the power head: (a) vibration control module: 10 Figure 15. Mechanism of intelligent control for the first-type parallel springs, (b) power head assembly, (c) two parallel elastic displacement response. buffers, and (d) application of the rotary drilling rig. parameter into the dynamic model so as to better con- Conclusion sider the vibration control problem from the global In this article, the influence of the various relevant para- parameters by reducing the motion path and control- meters in the vibration source and the vibration trans- ling the periodicity. The result of steel rope velocity mission process is considered comprehensively. The could provide foundation for its intelligent control dynamic model of the vibro-impact is proposed. The method. correction parameter a is introduced into the model when the response is constrained by the steel rope. Declaration of conflicting interests Three types of displacement response are defined. We The author(s) declared no potential conflicts of interest with find that the first type of displacement response, which respect to the research, authorship, and/or publication of this has no cyclic motion, minimum frictional loss, and the article. shortest motion path, is the optimal vibration control method for the vibro-impact system. Three expectations Funding are realized. First, the vibro-impact system can reason- ably withstand the impacting from each pipe of the The author(s) received no financial support for the research, Kelly bar and can play a role of fault prevention to authorship, and/or publication of this article. ensure the stable performance of a rotary drilling rig. Second, suppressing the periodicity of the vibro-impact, ORCID iD or decreasing the maximum displacement amplitude, Lei Qin https://orcid.org/0000-0001-9232-7059 we can reduce the damage and energy consumption caused by sliding friction between the parts. 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Advances in Mechanical Engineering – SAGE
Published: Nov 25, 2019
Keywords: Vibration control; vibro-impact; intelligent control; rotary drilling rig; multilayer Kelly bar
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