Vibration control analysis of the power head for a rotary drilling rig

Abstract

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

Introduction

The Kelly bar, which is usually more than 10 tons, is designed as a telescopic and multilayer pipes structure. The pipes’ weight is loaded on the power head in the form of instantaneous impacting when the pipes stretch out. The impacting repeats frequently when the rig works. Usually, the pipes fall freely at any time, and the impacting is extremely destructive to the power head. The vibro-impact after the impacting may cause deformation, weld cracking, and oil leakage. In order to reduce losses, it is necessary to check the quality of the power head.¹ It is very important to prevent and control the vibro-impact.

The vibro-impact usually results in an uncertain response. Some special excitations make the vibration system uncertain and chaotic, and the response usually has the bifurcation.^2–5 Luo et al.⁶ and Yue et al.⁷ analyzed periodic motion and bifurcation phenomenon of the dynamic response by Poincaré map based on the vibro-impact system with 2 or 3 degree of freedom. Liu et al.^8,9 studied the non-continuous grazing bifurcation behavior based on the periodic motion of a 3-degree-of-freedom vibro-impact system. Liu et al.¹⁰ studied the vibro-impact in a tank laying on water and established the attractor to avoid bifurcation by the means of displacement feedback control, so as to achieve the control of forward and backward motion of the tank. For the purpose of vibration control, the absorber could be added in the process of vibration transmission by passive control to obtain the reasonable response.¹¹ The characteristic parameters could be changed by active control to improve the response performance in the vibro-impact system.¹² Harvey et al.¹³ applied both isolation and absorption methods to control the vibro-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 constraint, as well as a huge change of vectors such as velocity and acceleration.^14–21 In a multi-degree-of-freedom vibro-impact system, the response behaviors were correlated and coupled, and the solution of the motion equations were more complicated. Xue and Fan²² and Fan and Yang²³ 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 attention to solving major engineering problems in specific applications. The study provides a further development of the vibro-impact system with multi-degree of freedom. In this work, the dynamic model of vibro-impact is established with the constraint of the steel rope especially, and the influence of related parameters is analyzed 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 about steel rope velocity.

Problem statement, dynamic model, and formulation

Figure 1 shows the rotary drilling rig assembly and the vibration control system of the power head. When the rig is working, the first layer pipe extends downward with the steel rope at a certain speed. At last, the first layer pipe moves with the power head. If the hole is deep enough, the subsequent impacting of each pipe takes place at the bottom. What we expect is to suppress the periodicity or to reduce the maximum displacement in the vibro-impact so as to reduce the sliding friction displacement and reduce wear. What is more is that the expectation is realized by making the structure compact, lightweight, and material saving in the case of reasonable reduction of the displacement response. Figure 2 shows the dynamic models about the vibro-impact system. The differential equation of motion for the vibro-impact can be expressed as

m \overset{\cdot\cdot}{x} + f (x, \overset{\cdot}{x}, c, k, P) = F (t)

(1)

where $f$ is a certain function, and its specific form is related to each parameter variables. $F (t)$ is any function depending on the excitation related to time $t$ . In the vibration control system, the mass is $m$ , the equivalent elastic coefficient is $k$ , the equivalent damping coefficient is $c$ , the dry friction is P, and the displacement response of the vibration system is $x$ .

The piecewise analysis is used to establish the differential equation. Before the impacting, the first pipe with the mass $M_{1}$ falls at a uniform speed of $ν_{0}$ under the traction of the steel rope. In the instant of the impacting, the initial condition at time $t = 0$ is obtained. In a short time after the impacting, the system is loaded by the step excitation. Each piecewise process will act in the global response.

First, considering the vibro-impact between the first layer pipe and the power head according to Figure 2(a). The displacement response without constraint is expressed as follow

x_{h} (t) = e^{- ζ ω_{n} t} (x_{0} \cos ω_{d} t + \frac{{\overset{\cdot}{x}}_{0} + ζ ω_{n} x_{0}}{ω_{d}} \sin ω_{d} t)

(2)

where $ω_{n} = \sqrt{k / m}$ is the undamped natural circular frequency of the system, $ω_{d} = \sqrt{1 - ζ^{2}} ω_{n}$ is the damping circular frequency, $ζ = c / c_{c} = c / (2 m ω_{n})$ is the damping ratio.

The weight of the first layer pipe is considered as a step excitation with $x_{0} = {\overset{\cdot}{x}}_{0} = 0$ , the step-excitation response is obtained as follow

x_{p} (t) = \frac{1}{m ω_{d}} \int_{0}^{t} e^{- ζ ω_{n} (t - τ)} \cdot \sin ω_{d} (t - τ) \cdot F (τ) d τ

(3)

where $F (τ) = M_{1} g$ . The total response is the sum of the free-vibration response and the step-excitation response with the initial condition. Then the total response is

x (t) = x_{h} (t) + x_{p} (t)

(4)

In order to correct the model, the constraint of the rope is introduced into the equations with the correction parameter $α$ . Then the modified step-excitation is given by

F (t) = M_{1} g + α t

(5)

The new equation of the step-excitation response becomes

x_{pc} (t) = \frac{1}{m ω_{d}} \int_{0}^{t} e^{- ζ ω_{n} (t - τ)} \cdot \sin ω_{d} (t - τ) \cdot (M_{1} g + α τ) d τ

(6)

The new total response of the vibration control system is obtained

x_{m} (t) = x_{h} (t) + x_{pc} (t)

(7)

Then the $n th$ layer pipe of the Kelly bar in the vibro-impact is considered $(n \geq 2)$ according to Figure 2(b). The simplified dynamic model has 2 degrees of freedom, and the differential equation of motion is established with another new final static stable position about $M_{1} + M_{2} + M_{3} + \dots + M_{n - 1} + m$ as the coordinate origin

{\begin{matrix} (\sum_{i = 1}^{n - 1} M_{i} + m) {\overset{\cdot\cdot}{x}}_{1} = - k x_{1} - c {\overset{\cdot}{x}}_{1} + k_{n - 1} (x_{2} - x_{1}) + c_{n - 1} ({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1}) \\ M_{n} {\overset{\cdot\cdot}{x}}_{2} = F_{n} (t) - k_{n - 1} (x_{2} - x_{1}) - c_{n - 1} ({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1}) \end{matrix}

(8)

The initial conditions are given by

\begin{matrix} x_{1} (0) = x_{2} (0) = 0 \\ {\overset{\cdot}{x}}_{1} (0) = 0; {\overset{\cdot}{x}}_{2} (0) = v_{0} \end{matrix}

(9)

where $M_{n}$ is the mass of the $n th$ layer pipe, $k_{n - 1}$ is the equivalent elastic coefficient of the $(n - 1) th$ layer pipe, and $F_{n} (t)$ is the function depending on the step excitation related to the $n th$ layer pipe.

Numerical investigation

Setting the value of some parameters, $M_{1} = 2000 kg$ , $m = 100 kg$ , $k = 200 kN / m$ , $v_{0} = 2 m / s$ , $ζ = {0, 0.1}$ , and $α / k = {0, - 1, - 2, - 3}$ . Figure 3 shows the relationship between the total response and the steel rope displacement without $α$ . In normal conditions, the solid line should be below the dashed line to ensure that the response is not constrained by the steel rope.

After introducing the correction parameter $α$ into the system, the relationship between the total response and the steel rope displacement is shown in Figure 4.

As shown in Figure 4, the curve of the total displacement response is significantly correlated with the curve of the rope displacement as the parameter $α$ changes. When $α / k > - 2$ , the curve of the total response is above the curve of the rope displacement and it still cannot meet the model requirement. When $α / k = - 2$ , the curve of the total response has the best approximation to the curve of the rope displacement, which is approximately tangent at a point. When $α / k < - 2$ , the curve of the total displacement response is always under the curve of the rope displacement, which means that no response exceeds the rope displacement, and it contradicts the premise analysis. Therefore, only $α / k = - 2$ meets the model requirement. The value of $α / k = 2$ is not accurate enough. Try again and the result of $α / k = - 2.2$ is obtained finally.

Now considering whether $α / k = - 2.2$ satisfies all common conditions. Setting the value of some parameters such as in Table 1. Then the role of different parameters in total response is analyzed one by one. The result is obtained by numerical investigation in Figure 5.

Table 1.

The six groups of data for numerical investigation and the value of $α / k$ .

No.	$α / k$	$M_{1} (kg)$	$m (kg)$	$k (kN / m)$	$v_{0} (m / s)$	$ζ$	FSSP (mm)	Figure 5
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.

It shows that the value of $α / k$ is related to all parameters and is not linear. According to the graphs in Figure 5, (a) and (f) show that the peak of the displacement response is close to the final static stable position; (b) and (c) show that the peak of the displacement response is significantly lower than the final static stable position; and (d) and (e) show that the peak of the displacement response is significantly higher than the final static stable position. These three types of displacement response are defined in this article, respectively, for further investigation.

The first type of displacement response

Generally, the value of $α / k = - 2.2$ is substituted into equation (7) and the velocity is obtained by taking the first derivative as follow

{\overset{\cdot}{x}}_{m} (t) = {\overset{\cdot}{x}}_{h} (t) + {\overset{\cdot}{x}}_{pc} (t)

(10)

The flows of the velocity about the total response are shown in Figure 6, which corresponds to the first type of displacement response.

An important result is gained when the displacement response reaches the final static stable position (the peak) for the first time according to Figure 6(a). The velocity is zero, and its path presents a bifurcation point. Obviously, constrained by the steel rope, the velocity bifurcation point is exactly the final static stable position, and the velocity no longer changes when no external force is in the system, and as well as the displacement response will be kept at zero. Figure 6(b) shows the actual flow of the velocity with the constraint of the steel rope. Therefore, in the first type of displacement response, the direction of the displacement does not change, nor does the direction of the velocity, and the vibration control system eventually approaches the final static stable position. So the vibro-impact has no cyclic motion, less frictional loss, and shorter motion path.

The second type of displacement response

The second type of displacement response is carried out in Figure 7(a) according to the third group of data in Table 1. The velocity is faster, and the displacement response is easier to catch up with the rope displacement. The velocity is 2 m/s where the displacement curve is tangent to the rope displacement curve, which is the same as the initial value of the rope velocity. Then the vibration control system moves synchronously with the rope to the final static stable position, and the vibration control system has a non-zero initial condition ${\overset{\cdot}{x}}_{0} = v_{0}$ . At this time, the free vibration appears with the final static stable position as the coordinate origin, where the vibration control system eventually stays at.

The third type of displacement response

The third type of displacement response is carried out in Figure 7(b) according to the fifth group of data in Table 1. The velocity is slower, and the displacement response will no longer catch up with the rope displacement after the first impacting. Then the free-falling impact without the constraint of the rope appears. Due to the increase of initial value $v_{0}$ , the total displacement response amplitude continues to increase. Therefore, in the third type of displacement response, the vibration control system is affected by the free-fall of the pipe, and its maximum amplitude is more than twice of the final static stable position.

The total displacement response and the type of displacement response are obtained in Table 2 and Figure 8 according to Figure 2(b).

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.	$M_{1}$ $(kg)$	$M_{2}$ $(kg)$	$m$ $(kg)$	$k$ $(kN / m)$	$k_{1}$ $(kN / m)$	$v_{0}$ $(m / s)$	$α$	FSSP $(mm)$	Type
1	2000	1890	100	200	$432 \times 10^{3}$	2	0	92	Third
2	2000	1890	100	200	$432 \times 10^{3}$	2–0.41	0	92	Third
3	2000	1890	100	200	$432 \times 10^{3}$	0.41	−85,000	92	First
4	2000	1890	100	200–18	$432 \times 10^{3}$	2	0	92–1029	Third
5	2000	–	100	200	–	0.41	0 to $- 1 \times 10^{6}$	98	Second

FSSP: final static stable position.

The displacement responses of $x_{1}$ and $x_{2}$ are almost completely coincidental in Figure 8(a). Therefore, it is considered that the pipe in the model can be regarded as a rigid body, whose equivalent elastic coefficient $(k_{1})$ is much larger than that of the vibration control system. The conclusion is very useful in the impacting of the nth layer pipe $(n \geq 2)$ . In this study, only the performance of vibration control system is concerned, so $x_{1}$ is needed for further analysis. In the undamped condition, the motion is periodic with $(256 - 72) / 2 = 92 mm$ as the final static stable position. The $x_{1}$ displacement never catches up with the rope displacement as shown in Figure 8(b). Therefore, it is a free-falling impact without the rope constraint, and the maximum displacement response is 256 mm. In order to improve the performance, the variables that can be considered are the rope velocity $v_{0}$ and the elastic coefficient $k$ , which are further analyzed by setting different value.

Figure 9 is obtained by changing the value of $v_{0}$ only. The curve of the displacement $x_{1}$ also changes with the decrease of $v_{0}$ from $2$ to $0.41 m / s$ . When $v_{0} = 0.41 m / s$ , the displacement $x_{1}$ has the tendency to exceed the rope displacement $x_{r}$ before it reaches the final static stable position $(92 mm)$ . Now the constraint of the steel rope should be considered. Trying to make the two curves $(x_{1} and x_{r})$ tangent with $α = - 85, 000 N / s (α / k = - 0.425)$ , then the peak of $x_{1}$ reaches the final static stable position $(92 mm)$ , and the velocity is zero. The vibration control system has the shortest path of one-way motion.

Figure 10 is obtained by changing the value of $k$ only. The final static stable position increases from $92$ to $1029 mm$ as the $k$ value decreases from $200$ to $18 kN / m$ . The vibration control system will always be subject to free-falling impacting if increasing the k value. It is unnecessary to introduce the correction parameters $α$ in all cases because the system is not constrained by the rope during the analysis.

On the whole, decreasing the value of $v_{0}$ is the most effective way to get the first type of displacement 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 $k$ enables the vibration control system to get the third type of displacement response all the time, and the system always subject to free-falling impact.

Results and discussion

Various parameters need to be reasonably selected and predicted. After putting forward the definition of three types of displacement response, in the same condition about the final static stable position $M_{1} g / k = 98 (mm)$ , the first type of displacement response has the least motion path, the third type has the most, and the second type is between them. Considering the vibration control again, the control method in the first type of displacement response is best. When the displacement reaches the final static stable position for the first time, the motion stops. The vibration amplitude of the whole 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 the final static stable position for the first time, the system has initial condition $v_{0}$ and starts free vibration. The motion path is larger than that in the first type. There is cyclic periodic motion, and the vibration control performance is not as good as the former. The third type of displacement response is equivalent to the free-fall impact. The initial maximum vibration amplitude exceeds two times of the final static stable position. 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 configuration 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-pipe Kelly bar. Each pipe is $14 - m$ long and can provide the maximum drilling depth and diameter of $51 m$ and $Φ 2 m$ , respectively. Then the predictable parameters in the design are shown in Table 3. It is required to propose a reasonable design method for the vibration control system on the power head.

Table 3.

The predictable parameters in the design about the vibration control system.

Pipes	D $(mm)$	d $(mm)$	Length $(m)$	Mass $(kg)$	$k_{eq}$ $(kN / m)$	$m$ $(kg)$
$M_{1}$	$Φ 460$	$Φ 430$	14	2600	$300 \times 10^{3}$	80
$M_{2}$	$Φ 390$	$Φ 350$	14	2550	$332 \times 10^{3}$	80
$M_{3}$	$Φ 310$	$Φ 260$	14	2460	$320 \times 10^{3}$	80
$M_{4}$	$Φ 220$	$Φ 160$	14	2000	$255 \times 10^{3}$	80
$M_{b}$	–	–	–	5000	–	80

The analysis is carried out in two cases, namely impacting of one by one and impacting of total weight on the power head.

Impacting of one by one

The purpose of the optimization is to obtain the best first type of displacement response, which is related to the final static stable position.

Table 4 shows that $v_{0}$ is divided into three intervals according to the $k$ value which corresponds to different type of displacement response. There is unique $v_{0}$ to make the system to get the first type of displacement response.

Table 4.

Conclusion about the type of the displacement response in the impacting.

$k (kN / m)$	FSSP $(mm)$	$v_{0} (m / s)$			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.

The Kelly bar includes four pipes. If the rig works normally, the innermost fourth pipe with the drilling tool will never provide the impacting on the power head. Only three pipes are loaded on the vibration control system when the fourth pipe stretches out. The maximum elastic motion range of the vibration control system is $δ = 120 mm$ (allowable $δ_{\max} = 160 mm$ ). So it yields the equation as follow

k \geq \frac{M_{1} + M_{2} + M_{3} + m}{δ} = 630 kN / m

Taking $k = 700 kN / m$ , the final static stable position is $108 mm < δ$ , and the corresponding types of displacement response are shown in Table 4. The conclusions can be drawn according to the analysis. The optimal $v_{0}$ configuration of each pipe is $v_{0} = 6.65 m / s$ for the first pipe, $v_{0} = 0.26 m / s$ for the second pipe, and $v_{0} = 0.185 m / s$ for the third pipe.

Impacting of the total weight on the power head

This situation is fault prevention, and it often occurs in the case that the pipes are stuck and the pipes suddenly fall without any constraint. The most serious situation in the impacting is that the total weight of all pipes with drilling tool affects the system. According to the analysis of the third type of displacement response with the worst vibration control performance, the system needs to bear the total weight of H height free-falling impact, and the displacement response is shown in Figure 12.

Figure 12 shows that the displacement response varies from $408$ to $475 mm$ , which does not satisfy $δ = 120 mm$ (allowable $δ_{\max} = 160 mm$ ). The $k$ value can be increased to achieve the goal. However, $k = 700 kN / m$ has just met the requirements. From the economical point of view, the buffer $k_{s}$ with a larger equivalent elastic coefficient is added, and its motion range is $δ_{\max} - δ = 40 mm$ . The displacement response, which is shown in Figure 13, is in the range of $40 mm$ .

When $H \leq 5 m$ , the response has satisfied the requirements. The result of $k_{s} = 1.2 \times 10^{4} kN / m$ is obtained.

Result of the example

These conclusions are drawn through the verification of this example. (1) $k = 700 kN / m$ meets the requirements and then the optimal $v_{0}$ configuration of each pipe is $v_{0} = 6.65 m / s$ for the first pipe, $v_{0} = 0.26 m / s$ for the second pipe, $v_{0} = 0.185 m / s$ for the third pipe. (2) The impacting response can be clearly and simply judged and solved using the method of dividing the displacement response into three types. (3) In the situation of fault prevention, the buffer $k_{s}$ with a larger equivalent elastic coefficient is needed.

Figure 14 shows the practical example. The structure uses 10 parallel springs (total equivalent elastic coefficient $k = 700 kN / m$ ) and two parallel elastic buffers (total equivalent elastic coefficient $k_{s} = 1.2 \times 10^{4} kN / m$ ). They constitute the vibration control system for the power head.

Figure 15 shows the mechanism of intelligent control for the first type of displacement response which is the best vibration control method depending on the rope velocity and bars weight. This method function is realized by programming.

From the application point of view, more working conditions and parameters may appear. Using the research method proposed in this article, similar results can be obtained by changing the parameters. It can guide us to design an optimal vibration control system and better intelligent control of steel rope speed.

Conclusion

In this article, the influence of the various relevant parameters in the vibration source and the vibration transmission process is considered comprehensively. The dynamic model of the vibro-impact is proposed. The correction parameter $α$ is introduced into the model when the response is constrained by the steel rope. Three types of displacement response are defined. We find that the first type of displacement response, which has no cyclic motion, minimum frictional loss, and the shortest motion path, is the optimal vibration control method for the vibro-impact system. Three expectations are realized. First, the vibro-impact system can reasonably withstand the impacting from each pipe of the Kelly bar and can play a role of fault prevention to ensure the stable performance of a rotary drilling rig. Second, suppressing the periodicity of the vibro-impact, or decreasing the maximum displacement amplitude, we can reduce the damage and energy consumption caused by sliding friction between the parts. Finally, the structure can be made more compact, lighter, and more material saving by reasonably decreasing the magnitude of the displacement response. The study provides a good basis for the design by introducing the correction parameter into the dynamic model so as to better consider the vibration control problem from the global parameters by reducing the motion path and controlling the periodicity. The result of steel rope velocity could provide foundation for its intelligent control method.

Footnotes

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Lei Qin

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