Motion analysis of magnetic spring pendulum

In order to analyze the motion characteristics of the spring pendulum under the action of magnetic field force, the motion of the spring pendulum will be studied by applying a uniform magnetic field in the vertical direction. Firstly, a first-order approximate solution is given by studying the micro-vibration around its equilibrium point. And an approximate solution similar to the Foucault pendulum is also presented in the case of a soft spring with strong ductility. Then, according to the resonance conditions of mechanical vibration, the internal resonance phenomenon of magnetic spring pendulum is discovered, and then the conclusion that the energy of the system is cyclically transmitted between the three modes of breathing, oscillating and deflection is presented subsequently. Finally, the influence of magnetic field strength on the motion stability of the spring pendulum is explored, and not only the bifurcation phenomenon at its equilibrium point is found, but also the complex dynamic behavior including chaotic motion occurs.


Introduction
As an indisintegrable model combining vibration mode and swing mode, spring pendulum is widely used in various engineering damping systems [1,2]. The earliest research on it was found in the article written by Vitt and Gorelik in 1933 [3], and then many scholars have studied the model from many angles such as approximate solution, internal resonance, bifurcation, and chaos. Broucke and Baxa have investigated the generation conditions and stability judgment of the equilibrium point of the periodic solution of the model through the periodic orbit research method [4]. Aldoshin and Yakovlev have discussed the chaotic modes of spring pendulum vibration, their occurrence conditions and possible evolution scenarios [5]. Awrejcewicz et al. have investigated the law of the model when forced vibration is explored by applying periodic external forces in both radial and transverse directions to the spring pendulum and then have used the multi-time scale method to obtain its approximate solution for the initial value problem, and its amplitudefrequency response relationship [6] as well. Olsson has explored the vertical motion of the spring pendulum. When the natural frequency of the spring and the swing frequency are satisfied, the two modes of motion will undergo a strong coupling phenomenon. They stimulate each other, so that the energy is alternately transferred in these two motions [7]. Amer et al. have explored the nonlinear response of the spring pendulum when the suspension point moves elliptically in the vertical plane, and have explained its dynamic behavior by obtaining a timing diagram and phase plane analysis of the approximate solution [8]. As for Alicia Gonzalez-Buelga et al., from an experimental point of view, they have analyzed the motion of the spring pendulum under the condition of periodic external force applied to the suspension point by the hybrid technology of real-time dynamic substructure, and the corresponding timing diagram, phase trajectory diagram and Poincaré cross-section diagram were plotted [9].
At the same time, it is noted that many scholars have recently carried out research on the magnetic pendulum model, such as Boeck, Sanjari and Becker, who have calculated the stability limit of magnetic pendulum in strong and weak electromagnetic coupling by applying Floquet theory and harmonic balance method, and also have discovered the chaotic behavior of finite amplitude [10]. Many other scholars, such as Kitio Kwuimy, Nataraj and Belhaq, have studied the effects of inclined harmonic excitation and parameter damping on the chaotic dynamics of asymmetric pendulum systems and have concluded that the increase of the inclination angle of the excitation leads to the increase of the lower bound of the chaotic domain and produces a singularity in the vertical position of the excitation [11]. Mann has discussed the behavior which a magnetic pendulum in stable equilibrium oscillation escapes from the adjacent barrier to the neighboring attractor and has expanded the existing quasi-steady-state escape criterion based on the influence of parametric excitation and subharmonic response behavior [12]. The initial sensitivity phenomenon and mechanism of magnetic pendulum have been studied by Qin et al. The evolution law of the attraction domain of the fixed point with the movement of the magnet was clarified. And this phenomenon was proved experimentally [13]. Pili has explored the motion of a single pendulum with magnetic damping from both theoretical and experimental perspectives and thus has provided a profound demonstration experiment to explain Lenz's law [14].
This paper aims to further explore the vibration and oscillation law of charged objects in the magnetic field, so the motion law of magnetic spring pendulum can be explored based on the above two pendulum models. Not only two approximate solutions of its kinetic equations can be acquired in Sect. 2, but also a new internal resonance relationship is presented in Sect. 3. Finally, Sect. 4 explores the influence of magnetic field strength on the stability of system motion.

Approximate solution
As shown in Fig. 1, an insulating and positively charged ball with charge q and mass m in the vertical plane is connected with the origin of coordinates through a light spring with original length and stiffness coefficient k, and a uniform magnetic field B along the z direction is applied. i, j, k are unit vectors in the three coordinate directions of x, y, z, respectively.
Firstly, it can be seen from Fig. 1 that the velocity vector of the system is v =ẋ i +ẏ j +ż k, and then its magnetic vector potential is set as A = − By 2 i + Bx 2 j; the generalized potential energy function [15] and the kinetic energy function of the system can be expressed as In the generalized potential energy function represented in Eq. (1), k 2 x 2 + y 2 +z 2 −L 0 2 is the elastic potential energy term, −mgz is the gravitational potential energy term, and Bq 2 (yẋ − xẏ) is the magnetic potential energy term obtained by −q A · v. Then the dynamic equation of the system is given by Lagrangian mechanics For the convenience of the discussion below, let In the above formula, L is the length of the vertical suspension spring pendulum when it is in the equilibrium position. ω s represents the natural frequency of the spring oscillator, that is, the breathing frequency of the spring pendulum when it moves radially. ω p is the oscillation frequency when the spring pendulum keeps the length L in the vertical plane. ω m represents the deflection frequency of the spring pendulum under the influence of the magnetic field. The relative frequencies 1 and 2 represent the ratio of the strengths of ω p and ω m to ω s , respectively, and they are the core parameters of the spring pendulum moving in the magnetic field. X, Y, Z and τ are the introduced dimensionless space variables and time variables, respectively. Therefore, Eqs. (3)∼(5) can be dimensionlessed as

Low-order expansion approximation
It can be seen from Eqs. (6), (9) that when the stationary pendulum is suspended vertically on the positive half axis of the Z -axis, its X = Y =Z = 0 and Z = 1 can be deduced. Therefore, the coordinate of the pendulum at this equilibrium point is (0, 0, 1). In order to explore the micro-vibration law of the pendulum near the equilibrium position, Z = 1 + Z is set in Eqs. (7)- (9). The root term of the three equations after Z replacement is carried out Taylor expansion with the point (0, 0, 0) as the center, and the lowest order term of X, Y, Z in various forms is retained to obtain the dynamic equations in the following form When the pendulum is micro-vibrating in the region near the equilibrium point, its X, Y and Z are both small variables, resulting in that X 2 and Y 2 in Eq. (12) are high-order small quantities. Therefore, the 1 2 1 − 2 1 X 2 + Y 2 term in this equation is ignored and the following simple harmonic vibration equation can be expressed as If the initial condition is set to Z (0) = H,Ż (0) = 0, the solution of the above equation can be expressed as Then, the complex solution method [16] is used to solve Eqs. (10)-(11), after multiplying Eq. (11) by the imaginary unit I and adding it to Eq. (10), and then the formulation can be presented as The abbreviation U = X + I Y is obvious make the initial condition as U (0) = U 0 ,U (0) = 0, and substitute Eqs. (14) into Eq. (16), then solve its differential equation and then separate the real and imaginary parts to obtain an approximate solution of X and Y where MathieuC and MathieuS are the Mathieu functions [17,18], i.e., the functions MathieuC(a, q, x) and MathieuS(a, q, x) are two series solutions to Mathieu differential equation y + (a − 2q cos(2x))y = 0 [19]. Mathieu differential equation, as a well-known equation in the field of mathematical physics, represents a wide range of engineering dynamics systems [20] and has an important enlightenment role in solving engineering problems. And with the change of parameters in the equation, the periodic solution of the system described by it may lose stability, local partial bifurcation, global bifurcation, chaos, etc., so the study and control of the complex behavior of such systems is not only of great theoretical significance, but also shows important engineering practical value [21]. The following compares the approximate solution of the mag-netic spring pendulum got above with the image of the numerical solution. From Fig. 2a-b, it can be found that the magnetic spring is placed in the X, Y direction with multiple unequal peaks. The phase difference of π 2 is roughly maintained during the movement in these two directions. At the same time, in Fig. 2c-d, it is found that the approximate solution can better match the image of the numerical solution, thus illustrating the correctness of the above exploration.

Approximation in the case of a soft spring with strong ductility
For soft springs with strong ductility, the elongation of the spring is much greater than the original length L 0 of the spring, which means under the condition of mg k L 0 , 2 1 = mg k L 0 +mg → 1 can be obtained by analyzing the definitions of 1 , ω s , ω p and L in Eq. (6). Therefore, Eqs. (7)- (9) and (16) can be rewritten as Then comparing Eqs. (19)- (20) with the dynamic equations of Foucault in the X and Y directions in the Ref. [22], it can be seen that the dynamic equations of the two models have the same form, so it shows that the two models have the same law of motion at the level. Let the initial conditions in Eq. (21) be the same as Eq. (14), and then solve the equation to obtain its kinematic equation as For Eq. (22), the characteristic root method is used to solve, and the fitting solution is set as Then, after substituting the above equation into Eq. (22), extract the coefficient of e λτ and make it 0, and give According to the above formula, the general solution of differential Eq. (22) should be the combination of two linearly independent solutions where M and N are constants determined by the initial conditions. Meanwhile, since Eq. (22) is a complex equation, the constants M and N should be complex; thus, Eq. (26) can be rewritten as Finally, Eq. (27) was substituted into the above equation and the real and imaginary parts were separated to give the kinematics equation in the X and Y directions Then set its initial condition as The above two equations can be rewritten as The above two equations can be rewritten into a vector equation where let n = sin ( 2 τ ) e X +cos ( 2 τ ) e Y , which represents the unit vector rotating with angular velocity 2 in the X-O-Y plane. Then draw a schematic diagram of ρ rotation In the Fig. 3, the projected motion of the spring on the X-O-Y plane is consistent with the vector ρ. 2 = Bq 2m m k is given according to the definition and analysis of 2 , ω m and ω s in Eq. (6), which is equivalent to the component z of the earth rotation angular velocity in the Z -axis direction of the Foucault pendulum in Ref. [23]. Thus, the motion of a spring with strong ductility and weak elasticity placed in a uniform magnetic field is similar to that of a Foucault pendulum, and its swing plane deflects at a constant angular velocity 2 . At the same time, by making 6116 Y. Meng can be simplified into the standard rose curve equation Then, according to the properties of rose curve [24], when n is odd, the number of leaves of rose curve is n, and its closing period is π . Meanwhile, when n is even, the number of leaves is 2n, and the closing period becomes 2π . When n is irrational, the graphs never overlap. Therefore, different motion trajectories of the spring pendulum can be driven by changing the size of the magnetic field, and finally the motion trajectory diagram of the magnetic spring pendulum is drawn according to Eqs. (23), (30)-(34), as shown in Fig. 4 By observing Fig. 4a-b, it can be seen that when n is equal to 5 and 6, respectively, by changing the magnetic field, the projection tracks of spring placed on the X-O-Y plane show rose curves with leaves number of 5 and 12, respectively. In Fig. 4c, n = e, and its track  is an unsealed leafy rose curve. All these phenomena are consistent with the above analysis, which shows the correctness of the above analysis. In addition, it can be seen from Fig. 4d that the movement of spring placed in three-dimensional space is the superposition of the horizontal movement with the three-leaf rose curve as the track and the simple harmonic movement in the vertical direction, and it can be seen from Eq. (23) that its track period is 2π . into the standard cycloid equation According to Ref. [25], when k is a rational number, its orbit is a closed graph. The number of points of the curve is the molecular value of the simplest fraction k. When k is irrational, the trajectory curve is never closed. Finally, according to Eqs. (23), (35)-(38), the motion trajectory diagram of the magnetic spring pendulum is drawn. In Fig. 5a-b, when the magnetic field is equal to 5 and 10/3, respectively, the projection trajectories of its motion on the X-O-Y plane are closed internal cycloid patterns of 5 and 10 points, respectively. In Fig. 5c, when k = π , the projected trajectory is an unclosed endocycloid graph. In addition, in Fig. 5d, the motion of the spring placed in the three-dimensional space is the combination of the motion of the multi-cusped endocycloid in the horizontal direction and the simple harmonic motion in the vertical direction.
In addition, if the initial condition is The first matrix at the right end of the equal sign of the above equation is a rotation matrix, which causes the ellipse defined by Eqs. (41)-(42) to rotate clockwise around the coordinate origin at an angular velocity 2 . That is, the projection of the spring on the X-O-Y plane participates in both elliptical motion and uniform circular motion, and its trajectory is the superposition of these two motions. The following is for Eqs. (39)-(40) and (23), and the motion trajectory diagram of the spring pendulum is drawn, as shown in Fig. 6.
From Fig. 6a-c, it can be observed that the projection trajectory of the pendulum on the X-O-Y plane is an ellipse that rotates clockwise around the coordinate origin, which illustrates the correctness discussed above. In Fig. 6d, it can be seen that the movement of the magnetic spring pendulum in three-dimensional space is a superposition of the rotational elliptical motion in the horizontal direction and the simple harmonic vibration in the vertical direction.

New internal resonance relation
Equations (10)- (16) are analyzed again in order to explore the internal resonance phenomenon of magnetic spring pendulum. Firstly, for dynamic Eq. (16) of the system in the horizontal direction, the sum of the terms at the left end of the equal sign is 0 when the initial condition is also U (0) = U 0 ,U (0) = 0, so that the solution can be described as where 2 3 = 2 1 + 2 2 , then use Euler's formula to simplify the above formula to It can be seen from above Eq. (16) that the two natural frequencies of U direction vibration are 3 − 2 and force. Then substitute Eqs. (45), (14) into the right end of Eq. (16) to give By observing the above equation, it can be found that term contains frequencies 2 − 3 + 1 and 1 − 2 − From the left end of the equal sign of the above formula, it can be seen that the natural frequency of the Z direction is 1, and the right end 1 2 2 1 − 1 |U | 2 of the equal sign is equal to the driving force, and then Eq. (45) is substituted into the right end of the equal sign of the above equation and simplified 1 2 From the above equation, it is observed that the frequency of |U | 2 is 2 3 . In Eq. (47), when the frequency of the driving force is equal to the natural frequency, that is, when 2 3 = 1, the vibration of the system in the U direction will cause resonance in the Z direc-tion. Its resonance conditions are the same as those in the U direction, both 3 = 1/2. Therefore, when the system meets the condition of 3 = 1/2, a strong coupling occurs between the horizontal and vertical motion modes, resulting in mutual excitation between the motion modes in different directions, which is the internal resonance phenomenon unique to the vibra- This new internal resonance relation is completely different from the one in Ref. [7] and does not need to satisfy the relation of ω s = 2ω p any more. The resonance images are given by using the new internal resonance conditions described by solving Eqs. (7)-(9) numerically. As shown in Fig. 7a-c, it is obvious that the magnetic spring presents obvious resonance behavior when placed in the X ,Y and Z directions, and its amplitude changes periodically with time. Meanwhile, from the relationship between speed and time shown in Fig. 7d, it can be found that the time when the maximum speed appears in the two directions of X and Y just corresponds to the minimum value of Z , which indicates that energy is transferred alternately in the directions of X ,Y and Z . In addition, Fig. 7e shows the phase trajectory in the X direction, with the pendulum ball jumping back and forth between the two center points in an elliptical trajectory of varying size. Figure 7f shows an elliptic track with ever-changing size centered at the equilibrium point (1,0) in the Z direction. Finally, it can be found that the motion of magnetic spring pendulum is regular three-dimensional motion in the trajectory diagram shown in Fig. 7g-i. In order to further explore the energy transfer process in different directions during the internal resonance, Eqs. (3)-(5) are rewritten into spherical coordinate system as shown in the Fig. 8.
Similar to the above, let R = r L , and then dimensionless the above three equations The meanings of 1 , 2 and the dimensionless time variable τ are the same as those of Eq. (6). Then, new internal resonance relation Eq. (49) described above is used to numerically solve Eqs. (53)-(55) to present the motion image. In Fig. 9a-b, it can be observed that the displacement appears resonance in both directions R and θ , while in Fig. 9c, the azimuth increases gradually as time changes. This is interpreted as the magnetic field force represented by 2 at the right end of equal sign in Eq. (55) drives the plane of the spring pendulum to deflect. Thus, the spring pendulum produces a third motion mode, namely deflection mode, in addition to breathing mode and swing mode. Meanwhile, in the trajectory diagram showing R, θ and φ in Figs. 9d-f, the phase traces in the R direction are similar to the trajectory diagram in Fig. 7f in the Z direction, which are concentric ellipses. The phase trace in direction θ is an oblate circle with narrow left and wide right. In addition, the phase trajectory diagram of φ is similar to the resonance diagram of θ in Fig. 9b. Finally, it is observed in Fig. 9g that the peak value of the velocity amplitude in the R direction at the same time corre- sponds to the minimum value of the velocity amplitude in the θ and φ directions, while the peak value of the velocity amplitude in the θ and φ directions corresponds to the minimum value of the velocity amplitude in the R direction. This shows that the system energy is transferred from the breathing mode to the swing mode and deflection mode at the same time and then from the swing mode and deflection mode back to the breathing mode. Therefore, the energy is transferred back and forth in these three modes.
The Jacobian matrix at the equilibrium point O 1 for Eq. (56) is Then solve for its eigenvalues λ 11 = I, λ 12 = −I, λ 13 = I ( 3 − 2 ) , The definition of 3 is the same as the third section above. For the equilibrium point O 2 , the Jacobian matrix of Eq. (56) is It can be seen from Fig. 11a-b that when 1 = 0.4, 2 = 0.8 is set, its timing diagram is stable vibration near the equilibrium point O 2 , and its phase trajectory diagram is a concentric ellipse, which also indicates that the equilibrium point O 2 is the center. When

Influence of magnetic field strength on stability of spring pendulum motion
In order to continue to investigate the influence of the magnetic field strength on the motion stability of the spring pendulum, the local maximum value of X 5 is given by numerical solution in Eq. (56), and then its bifurcation diagram is presented. And the Wolf method [26] is used to analyze the largest Lyapunov exponent of the system, and the results are shown in Fig. 12. In Fig. 12a-b, considering the system motion stability in the case of hard springs ( 1 = 0.1), it can be observed from bifurcation diagram 12a: when in periodic motion. At the same time, according to the largest Lyapunov exponential curve in Fig. 12b, it can be observed that when 2 ∈ (0, 0.1) ∪ (0.4, 0.6), its indication is significantly greater than 0, indicating that the system is in a chaotic state. When in the range of 2 ∈ [0.1, 0.4] ∪ [0.6, 2], the largest Lyapunov index is approximately 0, indicating that the system is in periodic motion. As for the motion stability of the system under weak elasticity ( 1 = 0.5) explored in Fig. 12c-d, it can be seen from bifurcation diagram 12c that when 2 ∈ (0, 2.5), the system is mainly in a chaotic state and has a periodic state many times. When 2 ∈ [2.5, 4], the system is in periodic motion. In Fig. 12d, it can be observed that when approximately 0, indicating that the system is in periodic motion. The timing diagram and phase diagram of Eq. (56) are drawn according to the conclusions obtained above, and the results are shown in Fig. 13.
Combined with the largest Lyapunov exponent in Fig. 12, the analysis of the above figure can be concluded as follows: in Fig. 13a-b, e-f, its largest Lyapunov exponent is approximately 0, so the system is in stable periodic motion under the condition of 1 = 0.1, 2 = 0.26, 0.5. The largest Lyapunov exponent in Figs. 13c-d is 0.02, so the system is chaotic in the case of 1 = 0.1, 2 = 0.5. At the same time, in Fig. 13g-h, it can be observed that under the condition of 1 = 0.5, its largest Lyapunov exponent is 0.148 when 2 = 0.28, so the system is in a chaos state. The largest Lyapunov exponent in Fig. 13i-l is approximately 0, so the system is in stable periodic motion under the condition of 1 = 0.5, 2 = 1.16, 3. In summary, these timing diagrams and phase trace dia-6126 Y. Meng   Fig. 13 Timing diagrams and phase trace diagrams (X 1 (0) = 0.1, X 5 (0) = −0.82, X 2 (0) = X 3 (0) = X 4 (0) = X 6 (0) = 0) grams clearly reflect the correctness of the stability of the system discussed above.

Conclusion
This paper explores the motion problem of magnetic spring pendulum, which has rarely been studied in previous literature. Several special approximation analytical solutions are given here. In Sect. 2.2 of this paper, it can be found that from kinetic differential Eqs. (19)- (20) and the various approximate solutions, it is indeed possible to simulate the motion of the Foucault pendulum at different latitudes and under different initial conditions by adjusting the parameters 1 and 2 of the magnetic spring pendulum model and changing the initial conditions of motion. According to the internal cycloid solution, the magnetic spring pendulum has potential value for the engineering design of swing rollers, cycloidal internal combustion engines, cycloid motors, etc. At the same time, new internal resonance relationship (49) due to the magnetic field was discovered, from which it can be seen that if the magnetic field disappears( 2 = 0), the internal resonance condition returns to the equation relationship mentioned in the previous literature [7]. In addition, the internal resonance phenomenon of the spring pendulum described in the previous literature is only the alternating transfer of energy between the breathing mode and the oscillating mode. In this paper, it can be observed that from Eqs. (52), (55), the magnetic field force is used as the driving force to deflect the swing plane of the spring pendulum, so the deflection mode of the spring pendulum due to the presence of the magnetic field is corrected. This causes energy to be transferred among breathing, deflection, and swinging modes. Finally, when exploring the stability of the system motion, it is found that the stability of the equilibrium point can be changed by adjusting the size of the magnetic field. Moreover, it is presented that under the action of a strong magnetic field, the motion of the system will tend to be in a stable periodic state. However, it is worth noting that the motion of the magnetic spring pendulum under the condition that the size and direction of the magnetic field change with time has not been discussed in this paper, and this problem will be further studied in the future. Data Availability All data generated or analyzed during this study are included in this published article.

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