Simulation and Analysis of Torsional Vibrations in Carbon Fiber Sucker Rod Strings Induced by Helical Buckling Deformation

Carbon fiber sucker rods possess superior properties such as being lightweight, high- strength, and corrosion-resistant, making them suitable for deep, ultra-deep, and corrosive wells [1], [2]. The failure modes of carbon fiber sucker rods differ from those of steel sucker rods. According to the studied [3], [4], the primary failure modes of carbon fiber continuous sucker rods are disengagement and rod breakage, with the most common issues being splitting and broken wires, specifically transverse fractures and longitudinal splits. Among these, longitudinal split failure is unique to carbon fiber rods. Currently, the design methods for carbon fiber sucker rod strings are still based on simulations of axial vibration, axial load, and alternating axial stress, ensuring that carbon fiber rods do not experience compressive or axial stress and meet the API strength requirements, resulting in a combination of carbon fiber and steel heavy-duty rods [5]. Clearly, current designs for carbon fiber sucker rod strings focus solely on preventing transverse fractures in carbon fiber rods, without considering the strength conditions necessary for splitting. Transverse fractures are more likely to occur when subjected to alternating torque, alternating bending moment, or a combination of both. The phenomena such as the longitudinal splitting of the rod string, the twisting of the wire rope of the rope hanger and the helical scratch of the plunger of the oil well pump in the actual working process of the carbon fiber sucker rod string all indicate that the carbon fiber sucker rod string has torsional vibration [6], [7]. Therefore, establishing a torsional vibration simulation model for the carbon fiber sucker rod string can enable quantitative analysis of these phenomena, which aids in exploring methods to prevent the damage of the carbon fiber sucker rod string from a theoretical perspective.

During the downward stroke, the carbon fiber sucker rod string experiences instability deformation due to axial pressure. As the axial pressure increases, the instability process begins with sinusoidal bending and then progresses to helical buckling [8], [9], [10]. The helical bending of the sucker rod string induces a torque [11], [12], [13], [14], which changes periodically as the sucker rod bottom end bends under the forces from the reciprocating motion of the sucker pump's suspension point, thus exciting torsional vibrations in the sucker rod string. The studies [11], [12] suggest that the torsional vibration of the sucker rod string is caused by the torsion of the suspension rope and the torsional bending of the sucker rod. They define the upper boundary of the torsional vibration as the initial angle of rotation when the suspension rope experiences axial force, inducing a torsional torque, and the lower boundary as the concentrated effect of the torsional bending-induced torque on the plunger pair. They provide an analysis of the mechanism of the sucker rod string torsional vibration and a mechanical model. Zdvizhkov et al. [13] established a horizontal wellbore experimental platform. Using the experimental method, it was obtained that when the tubing reached the full helical shape, the internal torsion continued to increase with the increase of axial buckling force in the way of theoretical prediction, but the results were not given from the theoretical research. Chen et al. [14] established a mathematical model for the torsional vibration of the sucker rod string without considering damping, suggesting that analyzing the torsional state of the sucker rod can solve the torsional vibration of the sucker rod string, but no solution was provided. The studies [15], [16] investigated the torsional vibration of the rotating sucker rod string, but the rotating sucker rod string driven by surface-driven screw pumps produces continuous unidirectional active torque, while the torsional vibration of the rod pump sucker rod string system is mostly alternating or random effects, depending on the working conditions, thus there are significant differences between the two. Wang et al. [17] studied the torsional vibration of the sucker rod string under the influence of the curved wellbore trajectory in directional wells, which is significantly different from the study of the straight well sucker rod string torsional vibration in this paper. The studies [18], [19], [20] studied the buckling and post-buckling problems under torsional loads, exploring the influence of active torque on buckling, which has reference significance for this paper. However, it differs significantly from the research problem of induced torsional vibration caused by helical buckling in this paper.

The research by the aforementioned scholars provides a basis for analyzing the torsional vibration patterns of sucker rod strings. However, this paper suggests that the upper boundary should be the constrained angular displacement of the suspension device, the induced torque should be generated by the helical buckling segment, and the excitation torque should be the distributed torque applied to the helical buckling segment. Therefore, in this paper, the upper boundary of the suspension device in the mechanical model is simplified to a torsion spring, with the stiffness of the torsion spring varying with the displacement and load of the sucker pump's suspension point. The lower boundary is treated as a free end, and the helical buckling-induced torque is applied to the helical buckling segment as an excitation. A simulation model for the torsional vibration of carbon fiber sucker rod strings is established.

The angular displacement of the top cross-section of the carbon fiber sucker rod is constrained by the suspension rope, so it is simplified as a torsion spring that responds to changes in the load and displacement at the pumping unit's suspension point. The bottom cross-section of the rod string, which is the plunger, is not constrained by the pump barrel, so it is simplified as a freely rotating end. For ease of study, the following assumptions are made:

(1) Assume that the well is a vertical well, and the sucker rod string and tubing are concentric;

(2) Taking the wellhead as the coordinate origin, periodic distributed torque is applied to the helical bending section of the rod string;

(3) The influence of coupling and centralizer on the local torsional stiffness of the rod string is not considered;

(4) The sucker rod string is a combination of two stages, and only the torsional vibration of the sucker rod string is considered.

Under the above assumptions, the torsional vibration dynamic model of the carbon fiber sucker rod string shown in Figure 1(a) is established. The force on the microelement at the section $dx$ at the well depth $x$ is shown in Figure 1(b).

In Figure 1, $K_{\mathrm{e}}$ is the torsional stiffness of the suspension rope; $G_1 J_{\mathrm{p} 1}$ is the torsional stiffness of the carbon fiber rod; $G_2 J_{\mathrm{p} 2}$ is the torsional stiffness of the steel rod; $L_1$ is the length of the carbon fiber rod; $L_2$ is the length of the steel $\operatorname{rod} ; M_{\mathrm{n}}(x, t)$ is the distributed torque subjected to periodic variation in the helical buckling section.

The fluctuation equation is applied to describe the longitudinal vibration of the carbon fiber and steel hybrid sucker rod string, and the boundary condition and continuity condition are considered, so as to obtain the mathematical model of the longitudinal vibration of the hybrid rod string [21].

In the formula, $u_1(x, t)$ represents the displacement of any cross-section $x$ of the carbon fiber rod at time $t$ relative to the suspension point, $\mathrm{m} ; u_2(x, t)$ represents the displacement of any cross-section $x$ of the steel rod at time $t$ relative to the suspension point, $\mathrm{m} ; a_1$ and $a_2$ are the propagation speeds of sound waves in the carbon fiber rod and the steel rod, $\mathrm{m} / \mathrm{s}$, with $a_{\mathrm{i}}=\sqrt{E_i / \rho_i} ; c_1$ and $c_2$ are the damping coefficients of the oil well fluid on the carbon fiber rod and steel rod, $1 / \mathrm{s} ; A_1$ and $A_2$ are the cross-sectional areas of the carbon fiber rod and steel rod, $\mathrm{m}^2 ; L_1$ and $L_2$ are the lengths of the carbon fiber rod and steel rod, $\mathrm{m} ; E_1$ and $E_2$ are the elastic moduli of the carbon fiber rod and steel rod, $\mathrm{Pa} ; \rho_1$ and $\rho_2$ are the material densities of the carbon fiber rod and steel rod, $\mathrm{kg} / \mathrm{m}^3 ; u^*(t)$ is the displacement of the suspension point, $\mathrm{m} ; F(t)$ is the axial load acting on the pump end, N.

The solution method of the suspension point displacement $u^*(t)$ and the pump end axial load $F(t)$ is given in detail by the study [18], which will not be repeated in this paper.

The difference method is applied to solve Eq. (1) to obtain $u_1(x, t)$ and $u_2(x, t)$. Considering the self-weight of the sucker rod string, the load of the pumping unit suspension point is calculated by the following formula:

In the formula, $W_{\mathrm{r}}$ is the weight of the carbon fiber and steel sucker rod string, N.

The axial distributed load of the sucker rod string at any section $x$ at time $t$ is:

Figure 2 illustrates the schematic of a pumping unit suspension rope. The suspension rope consists of two steel wire ropes with opposite twists. If the suspension rope is subjected to a torsional moment $T$ and rotates through an angle $\theta$, as shown in Figure 3, the torque acting on the suspension rope can be divided into three components: the same- twist torsional moment $T_{\mathrm{s}}$, the opposite-twist torsional moment $T_{\mathrm{n}}$, and the swinging torsional moment $T_{\mathrm{BT}}$, which can be expressed as:

In the formula, $J_t$ is the section modulus of the steel wire rope $J_t=\frac{\pi a^4}{32}, \mathrm{~m}^4 ; a$ is the diameter of the steel wire rope, $\mathrm{m} ; G_s$ is the shear modulus of the same-twist direction of the steel wire rope, in $\mathrm{N} / \mathrm{m}^2 ; G_n$ is the shear modulus of the opposite-twist direction of the steel wire rope, $\mathrm{N} / \mathrm{m}^2 ; P$ is the load on the suspension point, N; $h$ is the distance between the two steel wire ropes, m; $\beta$ is the angle between the steel wire rope and the vertical line, $\beta \approx \sin \beta=\frac{\theta h}{2 L_s},{ }^{\circ} ; L$ is the length of the steel wire rope $L_s=S-x+\Delta S$, m; $S$ is the stroke length, $\mathrm{m} ; x$ is the suspension point displacement, m, and $\Delta S$ is the safety distance between the sucker rod string and the donkey head, m.

Write Eq. (4) in the form of $\frac{T}{\theta}=J_g G_g$, the torsional stiffness of the suspension device can be obtained in the form of $J_g G_g$.

The three-stage rod buckling mechanical model shown in Figure 4 is established: helical buckling bc section, spatial suspension Ab and cB section.

In the diagram, $L_1$ and $L_2$ represent the lengths of the carbon fiber rod and steel rod, $\mathrm{m} ; q_1$ and $q_2$ are the axial distributed loads on the carbon fiber rod and steel rod, respectively, $\mathrm{N} / \mathrm{m} ; E_1 I_1$ and $E_2 I_2$ are the bending stiffness of the carbon fiber rod and steel rod, respectively, $\mathrm{N} \cdot \mathrm{m}^2 ; F$ is the axial pressure at the bottom end of the sucker rod, N; $a$ is the connection point between the carbon fiber rod and the steel rod; $b$ and $c$ are the contact points between the bottom suspension section and the upper suspension section with the tubing; $L_{\mathrm{b}}$ and $L_{\mathrm{c}}$ are the lengths from the bottom of the rod to the contact points with the tubing, $\mathrm{m} ; \theta$ represents the lateral deflection $y$ and $z$ at any axial position $x$ along the suspension section in the Cartesian coordinate system, and the polar angle at any axial position $x$ along the helical section in the polar coordinate system; $\theta_{\mathrm{b}}$ and $\theta_{\mathrm{c}}$ are the polar coordinates corresponding to the Cartesian coordinates $b$ and $c$.

The differential equation of rod string helical buckling is:

The upper suspended section is a carbon fiber sucker rod string and a steel sucker rod string, and the bending equation is:

The bending equation of the bottom suspended section is:

The boundary conditions are that both the top end of the rod and the pump end are fixed ends. Continuity conditions: the geometric, moment, and shear continuity conditions are satisfied at contact point $b$; the geometric, moment, and shear continuity conditions are satisfied at contact point $c$.

The numerical simulation process of the bending configuration is given in detail by the study [22]. From the above numerical simulation, the length of the spiral buckling section can be further calculated as follows:

In the study of rod buckling, a small unit is taken as the research object. Assuming the rod undergoes elastic deformation and the material is uniform across all segments, the rod can be considered a cylinder throughout the process. Based on these assumptions, the rod that exhibits helical buckling can be simplified into a cylindrical helix formed by an elastic line, as illustrated in Figure 5.

In the figure, $\vec{K}$ represents the curvature vector of the simplified cylindrical helical space curve; $\vec{t}$ is the tangent vector of the cylindrical helical space curve; $s$ denotes the arc length of any segment of the cylindrical helix, m; $\vec{n}$ is the normal vector of the spatial curve; $\vec{b}$ is the secondary normal vector of the spatial curve; $\vec{u}$ is the spatial position vector; $r$ is the radial clearance, m.

If $M$ is the external torque in the rod string, then the equilibrium equation of torque is

In the formula, $E I \kappa \vec{b}$ is the bending moment in the rod string; $\kappa$ is the curvature of the spatial curve; $M_{\mathrm{n}}$ is the torque in the rod string.

The cylindrical helical spatial curve has the following relationship:

The tangent vector of a cylindrical helical space curve is

where, $\vec{i}, \vec{j}, \vec{k}$ respectively represent the unit vectors of the $y, z$, and $x$ axis.

The curvature vector of the space curve is

The gap $r$ between the rod and the tubing is very small compared to the helix formed by the rod, so it can be assumed that $d z$ is three $d s$. Then we get

From the above formula, we can get the torque of each axis:

It can be obtained from Eq. (5):

Assuming there is no external torque on the x-axis, the internal torque of the rod string can be expressed as follows:

The mathematical model of the torsional vibration of carbon fiber and steel hybrid rod string is obtained by micro-element force analysis:

In the formula, $\rho_1$ and $\rho_2$ represent the densities of the carbon fiber rod and steel rod, $\mathrm{kg} / \mathrm{m}^3 ; \nu_1$ and $v_2$ are the resistance coefficients of the oil well fluid to the carbon fiber rod and steel rod, Pa.s; $G_1$ and $G_2$ are the shear moduli of the carbon fiber rod and steel rod, $\mathrm{Pa} ; J_{p 1}$ and $J_{p 2}$ are the polar moments of inertia of the cross-sections of the carbon fiber rod and steel rod, $\mathrm{m}^4 ; K_e(t)$ is the torsional spring stiffness derived from the simplified suspension device, $\mathrm{N} \cdot \mathrm{m} / \mathrm{rad} ; M_n(x, t)$ is the distributed excitation torque, $\mathrm{N} \cdot \mathrm{m}$.

Assuming that the initial position of the carbon fiber and steel hybrid draw rod string is at the dead point under the suspension point and is in a natural stationary state, the initial condition of the whole system satisfying $t=t_0=0$ is:

The torsional spring stiffness varies with the displacement and load of the suspension point, making it time-varying. This paper first establishes a numerical simulation model for nonlinear vibration equations using the finite difference method, considering the time-varying torsional stiffness. To analyze the effect of stiffness on torsional vibration, we assume that the torsional stiffness is constant and take the average value of the torsional stiffness over one cycle. In this case, the torsional vibration is considered linear, and the mode superposition method is used for simulation and calculation.

The mathematical model for torsional vibration, which is a nonlinear differential equation, is used to describe the changes in torsional stiffness. The carbon fiber rod and steel rod are discretized into $I_1$ and $I_2$ units along the axis with unit lengths $\Delta x_1$ and $\Delta x_2$, respectively. The time $t$ is discretized into $J+1$ nodes with a step length of $\Delta \mathrm{t}$. $\theta_{i, j}\left(i=0,1,2, \ldots, I_1, \ldots, I_1+I_2\right) ; j=0,1,2, \ldots, J$ denotes the angular displacement of node i on the sucker rod string at time $j$.

Differential form of the differential equation at the carbon fiber rod:

In the formula, $\gamma_{s 1}=v_1 \Delta t ; \gamma_1=\sqrt{\frac{G_1}{\rho_1}} \frac{\Delta t}{\Delta x_1}$.

Differential form of the differential equation at the steel rod:

In the formula, $\varsigma=\frac{\Delta t^2}{\rho_2 J_{p 2}} ; \gamma_{s 2}=v_2 \Delta t ; \gamma_2=\sqrt{\frac{G_2}{\rho_2} \frac{\Delta t}{\Delta x_2}}$. The difference form of the differential equation at the connection point between the carbon fiber rod and the steel rod:

In the formula, $\alpha_s=\alpha_1+\alpha_2 ; \beta_s=\beta_1+\beta_2 ; \alpha_i=\frac{\Delta x_i\left(G_i J_{p i}\right)}{2\left(c_i \Delta t\right)^2}, i=1,2 ; \beta_i=\frac{\Delta x_i\left(G_i J_{p i}\right) v_i}{2 c_i^2 \Delta t}, i=1,2$; $\mu_i=\frac{G_i J_{p i}}{\Delta x_i}, i=1,2 ; \quad c_i=\sqrt{G_i / \rho_i}, i=1,2$.

The difference form of the upper boundary condition:

The difference form of the lower boundary condition:

From the initial conditions, we know that $\theta(x, 0)=0$ and $\theta(x, 1)=0$. We discretize $\theta(x, 0)$ and $\theta(x, 1)$ along the axis.

The differential form of the differential equation, boundary condition, continuity condition, and initial condition established above constitute the simulation model of torsional vibration of carbon fiber sucker rod string.

If the torsional spring stiffness $K_{\mathrm{e}}$ is taken as the average value of the torsional stiffness of a period, then the torsional vibration is linear vibration, so the super-position method of vibration modes is used to solve it.

The damping and induced torque in Eq. (17) can be removed to obtain the equation of free torsional vibration, whose solution is:

In the formula, $\Theta_i(x)$ is referred to as the mode shape; $\omega$ represents the natural frequency of the torsional vibration of the carbon fiber sucker rod string. The eight undetermined coefficients can be determined through the boundary conditions, continuity conditions, and initial conditions. Substituting Eq. (24) into the boundary and continuity conditions yields the natural angular frequency equation and the mode shape function:

In the formula, $C_1=1, D_1=\frac{G_1 J_{p 1} \omega}{K_e a_1}, C_2=\frac{G_1 J_{p 1}}{G_2 J_{p 2}}\left(\frac{a_2}{a_1} \cos \frac{\omega}{a_1} L_1-\frac{G_1 J_{p 1} \omega a_2}{K_e a_1^2} \sin \frac{\omega}{a_1} L_1\right)$, $D_2=\sin \frac{\omega}{a_1} L_1+\frac{G_1 J_{p 1}}{K_e a_1} \cos \frac{\omega}{a_1} L_1$.

The orthogonality of the moment of inertia with respect to the main vibration mode can be obtained:

In which $A_r$ is the regularization coefficient:

$ A_r=\sqrt{\frac{1}{\begin{aligned} & \rho_1 J_{p 1}\binom{\frac{L_1}{2}-\frac{a_1}{4 \omega_r} \sin \frac{2 \omega_r}{a_1} L_1+\frac{a_1 D_1}{2 \omega_r}-\frac{a_1 D_1}{2 \omega_r} \cos \frac{2 \omega_r}{a_1} L}{+\frac{D_1^2 L_1}{2}+\frac{D_1^2 a_1}{4 \omega_r} \sin \frac{2 \omega_r}{a_1} L_1} \\ & +\rho_2 J_{p 2}\binom{\frac{a_2 C_2 D_2}{2 \omega_r}+\frac{L_2}{2}\left(C_2^2+D_2^2\right)-\frac{C_2^2 a_2}{4 \omega_r} \sin \frac{2 \omega_r}{a_2} L_2}{+\frac{D_2^2 a_2}{4 \omega_r} \sin \frac{2 \omega_r}{a_2} L_2-\frac{a_2 C_2 D_2}{2 \omega_r} \cos \frac{2 \omega_r}{a_2} L_2} \end{aligned}}} $

When the carbon fiber sucker rod string is excited by induced distributed torque, the regular force is

The regular damping is

The regular coordinate ${q}_r({t})$ is introduced, and ${q}_r({t})$ satisfies the following relation:

The canonical equation is

The initial conditions give us the$\left.q_r\right|_{t=0}=\left.\dot{q}_r\right|_{t=0}=0$.

The fourth-order Runge-Kutta method is used to solve Eq. (31) to solve the torsional vibration equation of the rod string.

Basic parameters of simulation: carbon fiber rod diameter $D_1=19$ mm, carbon fiber rod length $L_1= 1400$ m, carbon fiber rod density $\rho_1=2100 \mathrm{~kg} / \mathrm{m}^3$, the shear elastic modulus of the carbon fiber rod $G_1= 0.653$ GPa, the diameter of the steel rod $D_2=22$ mm, the length of the steel rod $L_2=600$ m, the density of the steel rod $\rho_2=7800$ kg / $\mathrm{m}^3$, the shear elastic modulus of the steel roa $G_2=8$ GPa, the inner diameter of the oil pipe $D_{\mathrm{r}}=63.5$ mm, the number of strokes $n=3$, and the axial distributed load $q(x, t)$ and the hanging point load $P(t)$ are shown in Figure 6 and Figure 7. The simplified torsional spring stiffness $K_e$ of the suspension device is shown in Figure 8. The helical buckling of the rod at time $t_1$ is illustrated in Figure 9. The distributed torque is calculated using this buckling configuration. Since the movement of the sucker rod is periodic, each $t_1+n T$ moment provides an excitation to the system, as shown in Figure 10.

According to the above simulation model, a torsional vibration simulation system of carbon fiber sucker rod string under the torque excitation induced by helical buckling is developed. Figure 11 shows the vibration law of the top rod string in different strokes $n$ in the first 15 cycles.

As can be seen from Figure 11, with the increase of stroke, the maximum amplitude of the top vibration first increases and then decreases, indicating that resonance is generated. The amplitude-frequency characteristic curve of the maximum amplitude of the top rod string is drawn as shown in Figure 12.

As shown in Figure 12, the torsional vibration system of the carbon fiber sucker rod string has two wave peaks, which are the two resonance points. By averaging the torsional spring stiffness and using the superposition method of vibration modes, the first six natural angular frequencies are calculated, as shown in Table 1. Converting the first-order natural angular frequency to a natural frequency yields 0.1117 Hz, indicating that the second wave peak, or stroke, near 6.7 Hz is the resonance point. The first wave peak occurs because the induced torque is a non-sinusoidal periodic wave, which can be decomposed into a sum of a sine wave of the same frequency and many integer multiples of the frequency. When the integer multiple frequency equals the natural frequency, high-order harmonic resonance occurs.

Figure 13 and Figure 14 show the vibration response curves of the rod string top with variable torsional stiffness and average torsional stiffness. It can be seen that the angular displacement response curves calculated by the finite difference method and the mode superposition method are almost the same. The use of these two calculation methods can verify the correctness of the simulation results in this paper.

The dynamic characteristics of rod pumping systems are influenced by stroke length, frequency, damping, pump diameter, dynamic liquid level, and well depth. These parameters are of significant interest in the engineering field.

Figure 15 illustrates the effects of different parameters on the vibration at the top of the rod string. Figure 15(a) shows that as the damping coefficient increases, the torsional vibration of the rod string decreases. Figure 15(b) indicates that a longer stroke results in a greater amplitude of the stroke pressure under pump end load, leading to a higher induced torque and more intense rod string vibration. Figure 15(c) shows that a smaller pump diameter tends to increase rod string vibration. Figure 15(d) indicates that a lower dynamic liquid level height leads to a greater amplitude of the stroke pressure under pump end load, resulting in a higher induced torque and more intense rod string vibration. Figure 15(e) shows that a longer rod string results in weaker wellhead rod string vibration. The conclusions from Figure 15 are highly consistent with real-world engineering conditions.

In order to obtain the torsional angular displacement along the entire length of the rod, the influence of the suspension device is ignored, and the wellhead is simplified as a fixed end. Two methods are used for simulation. One is without applying the bending configuration excitation, and the other is to apply the bending configuration excitation according to this paper.

1. Analysis of torsion angle of rod string without helical buckling

It can be concluded from Figure 16 and Figure 17 that, under the condition of no bending configuration excitation, the torsional angular displacement difference along the entire well depth is not large, and the torsion situation is relatively uniform along the well depth, slightly larger near the bottom of the well.

2. Analysis of torsion angle of rod string under helical buckling excitation in this paper

From Figure 18 and Figure 19, it is evident that the application of bending configuration excitation significantly affects the torsional vibration of the sucker rod string. The amplitude of the torsional vibration angle displacement increases as the depth of the well approaches the bottom. At the same depth, the angular displacement changes through four stages over time: In the first stage, the elastic torsional energy of the sucker rod string accumulates, and there is no relative torsional movement; in the second stage, the rod body begins to experience relative torsional movement due to reaching the critical torque, which occupies most of the downstroke; in the third stage, the elastic torsional energy of the sucker rod string is released, and the rod stops experiencing relative torsional movement, typically occurring in the latter part of the downstroke; in the fourth stage, during the upstroke, the bending is minimal, and its effect on torsion can be disregarded. Figure 18 and Figure 19 demonstrate that helical buckling has a significant impact on the overall torsional vibration of the sucker rod string, with the torsional effect primarily concentrated in the middle and lower parts of the well. This conclusion aligns with the analysis of the torsional process described in the study [12], indicating consistency between the simulation results and previous practical experience, and can serve as an experimental validation for this study.

(1) The mechanical model and the mathematical model of torsional vibration of carbon fiber sucker rod string with top torsion spring constraint are established, and the response is calculated by the finite difference method and the mode superposition method.

(2) Through the analysis of the torsional vibration law of carbon fiber sucker rod string, it can be concluded that with the increase of stroke, the torsional vibration first increases and then decreases, so as to avoid the generation of resonance; damping coefficient, stroke, pump diameter, dynamic liquid level, and rod length have a great influence on the torsional vibration of the rod string.

(3) Through the simulation of torsional vibration by helical buckling, it is concluded that helical buckling has a significant influence on the overall torsional vibration of the rod string, and the torsion is mainly concentrated in the middle and lower part of the well depth.

(4) The torsional vibration simulation model of the carbon fiber sucker rod string under the torque excitation induced by helical buckling can further analyze the longitudinal splitting mechanism of the carbon fiber sucker rod, and provide a theoretical basis for improving the working life and design of the carbon fiber sucker rod string.