Dynamic Response of Electromechanical Coupled Motor Gear System with Gear Tooth Crack

Abstract

The motor gear system (MGS) is recognized for its potential in enhancing transmission efficiency and optimizing space utilization. However, the system is subjected to challenges, notably the occurrence of abnormal vibrations. These issues stem from the dynamic interaction between the motor and gears, the presence of nonlinear factors in gear system, and the impact of gear faults, all of which contribute to complex vibration patterns. Traditional dynamic models have been found to be inadequate in effectively addressing the complexities associated with electromechanical coupling problems in MGS. To address these limitations, a comprehensive analysis approach is proposed in this paper, which is grounded in the development of an electromechanical coupling model. This method involves establishing a coupled dynamic model of the motor and gear system, integrating numerical simulations, and experimental validations to thoroughly analyze the vibration characteristics of the system. Through this multifaceted methodology, a detailed analysis of the system’s vibration characteristics is conducted. The results indicate that internal excitations from tooth root cracks not only directly affect dynamic characteristics of the gear transmission system (GTS) but also indirectly influence dynamic behavior of the motor, which offers valuable insights into modeling integrated MGS and provides significant solutions for fault diagnosis within these systems.

1. Introduction

The motor gear system (MGS) is extensively employed across industrial, transportation, and agricultural sectors, owing to its high efficiency, stability, substantial torque, and precision. As efforts to enhance the efficiency, integration, and reliability of drive systems continue, MGS is increasingly moving towards more complex structures. However, this evolution introduces new challenges [1,2]. In particular, the electromechanical coupling dynamics and fault prediction within MGS necessitate comprehensive investigation.

With the development of integrated MGS, dynamic interaction issues between mechanical and electrical systems have emerged. For instance, the performance of the mechanical system is influenced by the current and voltage signals, as well as the control systems within the Electrical Drive System (EDS). Conversely, it has been observed that vibrations, shocks, and mechanical loads exert an influence on the stability of the electrical system, which has been found to contribute to the reduction in the lifespan of electrical components [3,4]. Szolc et al. [5] investigated the dynamic electromechanical coupling between the motor and gear transmission system (GTS) of rotating machinery, revealing that electromagnetic stiffness and damping coefficients are influenced not only by the motor but also gears. Liu et al. [6] developed a dynamic model for an off-grid switched-reluctance wind turbine hydrogen production system, incorporating both mechanical and electromagnetic features. Chen et al. [7] addressed the strong coupling between gear and generator systems, which influences resonance speed and vibration signals, highlighting the importance of studying these interactions for safe drivetrain operation and design. Jiang et al. [8] studied the electromechanical coupling characteristics of coal mining machinery, considering both the electromagnetic effects and bending-torsional vibration characteristics of permanent magnet synchronous motor (PMSM). Kumar [9] introduced an electromechanical model for a single-stage gearbox with gear tooth root cracks, and developed using a modified Lagrangian approach and incorporating Rayleigh dissipative potential. Drawing upon the literature review, it is evident that both the EDS and GTS exhibit abnormal nonlinear vibrations as a result of internal electromechanical coupling.

To study the electromechanical coupling dynamic characteristics of MGS, it is essential to establish an accurate electromechanical coupling dynamic model [10,11]. Hao et al. [12] developed a mathematical model to assess the reliability of GTS and subsequently designed a digital platform to facilitate rapid reliability modeling. In a related study, Zhang et al. [13] proposed a design theory and method aimed at achieving high power density in heavy-duty GTS, particularly those operating under complex loads. To optimize gear parameter design, Zhou et al. [14] explored the effects of friction, impact, and lubrication on gear dynamics. Jedliński [15] examined eccentricity and backlash in spur gears using a 26-degree of freedom model of GTS. Bartelmus [16] explored the application of mathematical modeling and computer simulation to enhance gearbox diagnostic inference during monitoring. In a subsequent study, Sakaridis [17] investigated the impact of individual tooth inertia on spur gear dynamics, proposing a model that distinguishes tooth inertia from the gear body and introduces an innovative method for analyzing tooth contact. However, the prevalent use of local model in MGS frequently neglects the interaction between electrical and mechanical systems, leading to inaccurate simulation outcomes [18,19]. Therefore, the establishment of a comprehensive dynamic model is imperative to facilitate effective multi-domain electromechanical coupling studies.

The introduction of electromechanical coupling in MGS has been observed to exacerbate vibrational phenomena, thereby presenting both novel challenges and opportunities within the domain of fault diagnosis [20]. Addressing the challenge of real-time monitoring of equipment faults in mining machinery, Lu and Zhang [21] investigated the motor current response of mining equipment due to gear tooth root crack faults, and proposed a feasible method to predict mechanical faults by monitoring motor current. Ge [22] assessed the resonance risk by analyzing torsional vibrations at resonant speed, which are identifiable through current signals. Marzebali [23] studied how torsional vibrations affect electrical signals in a complex system, highlighting challenges in detecting gearbox frequencies. Kumar [24] modeled the electromechanical behavior of a straight bevel gear pair under various fault conditions, including healthy, chipped, and missing teeth. Han [25] proposed a motor current model for diagnosing localized defects in planet rolling bearings, which includes a dynamic analysis of a motor-driven planetary gearbox system. Ouyang [26] found that by analyzing gear characteristics in both time and frequency domains, accurate diagnosis of gear status is possible. Molaie [27] analyzed the dynamics of spiral bevel gears, accounting for torsional stiffness, time-dependent stiffness, and backlash, validating the model with a single-degree-of-freedom system. Sang et al. [28] investigated the lateral synthetic vibration signals and output torsional vibration signals of the motor-3K planetary gear system, which analyzed the feasibility of using these two vibration signals for fault diagnosis. Chai et al. [29] utilized the motor driver as an intelligent sensor, leveraging signals such as current and motor speed obtained from the drive system as fault carriers to monitor and diagnose the states of GTS. Yousfi et al. [30] effectively detected gear tooth faults using vibration and motor current signature analysis in both time and frequency domains, demonstrating satisfactory responses under faulty conditions. Sheng et al. [31] proposed a dynamic model of MGS aimed at acquiring motor current and gear fault data. Chen et al. [32] proposed a calculation method for the time-varying mesh stiffness and analyzed its coupling effects with rail transit trains. Ottewill [33] extended the application of the synchronous signal averaging method, traditionally employed for diagnosing tooth faults in parallel shaft gearboxes, to include the monitoring of tooth faults in epicyclic gearboxes. To maintain high precision in MGS, continuous monitoring of gear fault behavior is essential.

Research on the dynamics of electromechanically coupled systems in motor gear systems (MGS) has achieved significant progress. However, several persistent challenges have been identified, including the development of models that exhibit insufficient accuracy, inadequate consideration of electromechanical coupling factors, and the inherent difficulties associated with monitoring tooth root crack faults. As a result, achieving a comprehensive understanding of the dynamics and fault diagnosis within electromechanically coupled systems of mechanical gearboxes (MGS) is of paramount importance. The availability of accurate models is essential for optimizing system design, minimizing failure rates, and enhancing overall performance. Nevertheless, existing models frequently neglect critical dynamic aspects and the intrinsic complexity of the system. In response to these issues, an improved MGS model that effectively balances accuracy and computational efficiency is presented. This proposed model is intended to support design optimization and facilitate effective fault diagnosis within the system.

2. Electromechanical Coupling Modeling of MGS

Figure 1 illustrates the basic structure of the MGS, which includes a three-phase permanent magnet synchronous motor (PSMS) as drive motor, gearbox, and load. Inside the gearbox, there is a spur gear, with both ends of the parallel shafts supported by rolling bearings. The motor is connected to the input shaft of the gearbox, and the load is connected to the output shaft, both via couplings.

The power transmission and coupling relationships between the subsystems of MGS are depicted in Figure 2. The improved model encompasses the motor control, electronic, motor rotor, GTS, and load, explicitly considering the coupling interactions between these subsystems. Here, u_i represents voltage signals, i_i current, [X Y θ] displacement/vibration, ω_i the angular velocity, F_i force, and T_i torque. The modeling of these subsystems will be introduced in detail below.

2.1. Equivalent Circuit Model of PMSM

In this section, the mathematical model of three-phase PMSM is established. The voltage equations and flux linkage equations of motor in stationary coordinate are detailed as follows [34]

$$\left\{\begin{array}{l}{\mathit{u}}_{abc}=R{\mathit{i}}_{abc}+{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\mathsf{\psi }}_{abc}^s\\ {\mathsf{\psi }}_{abc}^s={\mathit{L}}_{abc}{\mathit{i}}_{abc}+{\mathsf{\psi }}_{abc}^r\end{array}\right.$$

(1)

where u_abc is voltage vector of stator windings, i_abc current vector of stator windings, ${\mathsf{\psi }}_{abc}^s$ flux linkage vector of stator windings, ${\mathsf{\psi }}_{abc}^r$ flux linkage vector of rotor windings, L_abc inductance matrix of stator windings, ϕ_f flux linkage of permanent magnet, θ_e angle of rotor electric, and R resistance of stator windings.

The model of PMSM in stationary coordinate exhibits time-varying and coupling characteristics. To facilitate the design of the subsequent controller, a mathematically decoupled model is established under the d-q frame [34]. The stator voltage equation in the d-q frame is

$$\left\{\begin{array}{l}{u}_d=R{i}_d+\left[i_d{\displaystyle \frac{\mathrm{d}{L}_d(i_d,i_q)}{\mathrm{d}t}}+{\displaystyle \frac{\mathrm{d}{\varphi }_f(i_d,i_q)}{\mathrm{d}t}}\right]\\ -{\omega }_e{L}_q(i_d,i_q)i_q\\ u_q=R{i}_q+\left[i_q{\displaystyle \frac{\mathrm{d}{L}_q(i_d,i_q)}{\mathrm{d}t}}+L_q(i_d,i_q){\displaystyle \frac{\mathrm{d}{i}_q}{\mathrm{d}t}}\right]\\ +{\omega }_e(L_d(i_d,i_q)i_d+{\varphi }_f(i_d,i_q))\end{array}\right.$$

(2)

The stator flux linkage equations in the d-q frame are

$$\left\{\begin{array}{l}{\varphi }_d=L_d{i}_d+{\varphi }_f\\ {\varphi }_q=L_q{i}_q\end{array}\right.$$

(3)

Let [u_d u_q] be the d-axis and q-axis components of stator voltage, respectively. Similarly, [i_d i_q] are the d-axis and q-axis components of stator current, [ϕ_d ϕ_q] the stator flux linkage, [L_d L_q] the inductance, ω_e the electrical angular velocity, and ϕ_f the flux linkage of permanent magnet. The electromagnetic torque equation is expressed

$$T_{\mathrm{e}}={\displaystyle \frac{3}{2}}{n}_p{i}_q\left[i_d\left(L_d(i_d,i_q)-L_q(i_d,i_q)\right)+{\varphi }_{\mathrm{f}}(i_d,i_q)\right]$$

(4)

In addition, it is important to note the following relationships

$${\omega }_e={\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\theta }_e, {\omega }_m={\displaystyle \frac{{\omega }_e}{n_p}}, n={\displaystyle \frac{30}{\pi }}{\omega }_m$$

(5)

where ω_m is the mechanical angular velocity of motor, n the rotational speed of motor, and n_p the number of motor pole pairs. The differential equation governing the torsional motion of motor rotor, which incorporates rotor eccentricity, is expressed as follows:

$$(J+m{e}^2){\ddot{\theta }}_{\mathrm{m}}=T_{\mathrm{e}}-T_{\mathrm{L}}-B{\dot{\theta }}_{\mathrm{m}}+me[\sin ({\omega }_{\mathrm{m}}t)\ddot{X}-\cos ({\omega }_{\mathrm{m}}t)\ddot{Y}]$$

(6)

where X and Y are the transverse displacements of motor rotor, θ_m is the mechanical angular displacement of rotor, m the rotor mass, T_L the load torque, B the electromagnetic damping coefficient, and e the eccentricity of rotor.

To derive the fundamental equations in the stationary reference frame, one must apply the inverse Park and Clarke transformations to the equations originally formulated in the synchronous rotating d-q reference frame [34].

2.2. Vector-Control of Motor

Research frequently overlooks the effect of motor control technology on MGS, neglecting factors such as parameter errors and external disturbances, which can lead to instability. This paper addresses these effects by incorporating motor control technologies into the analysis of GTS and EDS through an electromechanical coupling model. Figure 3 illustrates the control process flowchart, which comprises two primary modules: the speed controller and the current controller. The paper introduces various control methods for designing these controllers, including the Proportional–Integral (PI) speed controller, PI current controller, and Sliding Mode Control (SMC) speed controller.

(1)

Review of PI speed controller

The PI speed controller, a widely adopted technique in industry, reduces stability errors by adjusting proportional gain based on the deviation between actual and desired speeds [35]. Further details are below.

$$i_q^{*}=K_{P}e+K_I{\displaystyle \int edt}$$

(7)

where e = ω_ref − ω, and K_P, K_I are control parameters.

(2)

Review of PI current controller

Employing a conventional PI controller in conjunction with feedforward decoupling control, the voltage in the q-d frame is derived by analytic derivation.

$$\left\{\begin{array}{l}{u}_d^{*}=\left(K_{pd}+{\displaystyle \frac{K_{id}}{s}}\right)\left(i_d^{*}-i_d\right)-{\omega }_e{L}_q{i}_q\hfill \\ u_q^{*}=\left(K_{pq}+{\displaystyle \frac{K_{iq}}{s}}\right)\left(i_q^{*}-i_q\right)+{\omega }_e\left(L_d{i}_d+{\psi }_f\right)\hfill \end{array}\right.$$

(8)

The closed-loop transfer function of the system using PI controller is defined as follows [36].

$$\left\{\begin{array}{l}{\mathit{G}}_c(s)=\mathit{G}(s)\mathit{C}(s)\\ \mathit{F}(s)={[\mathit{I}-\mathit{C}(s)\widehat{G}(s)]}^{-1}\mathit{C}(s)\\ \mathit{C}(s)={\mathit{G}}^{-1}(s)\mathit{L}(s)\end{array}\right.$$

(9)

where $\mathit{L}(s)=\alpha \mathit{I}/(s+\alpha )$, α is the design parameter, which is positive. Then,

$$\mathit{F}(s)=\alpha \left[\begin{array}{cc}{L}_d+{\displaystyle \frac{R}{s}}& 0\\ 0& L_q+{\displaystyle \frac{R}{s}}\end{array}\right]$$

(10)

By combining Equations (9) and (10), we obtain

$${\mathit{G}}_{\mathrm{c}}(s)={\displaystyle \frac{\alpha }{s+\alpha }}\mathit{I}$$

(11)

Comparing Equation (11) with Equation (8) shows that controller tuning parameters are reduced from 2 to 1, simplifying the process. This adjustment fulfills the following relationship

$$\left\{\begin{array}{l}{K}_{pd}=a{L}_d\\ K_{id}=aR\\ K_{pq}=a{L}_q\\ K_{iq}=aR\end{array}\right.$$

(12)

The rise time (T_R), defined as the time taken for the system to transition from 10% to 90% of its step response, is approximately given by T_R = ln9/α. Lowering the value of α extends the response time, whereas increasing α hastens it. However, the value of α is constrained by the electrical time constant. In this study, α is set to 1500.

(3)

Review of SMC speed controller

Setting the i_d to zero aligns with the rotor magnetic field, which is conducive to achieving better control in surface-mounted PMSM. The mathematical model detailed in Section 2.1 is subsequently represented in the q-d reference frame.

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{i}_q}{ \mathrm{d}t}}={\displaystyle \frac{1}{L_{\mathrm{s}}}}\left(-R{i}_q-p_{\mathrm{n}}{\psi }_f{\omega }_{\mathrm{m}}+u_q\right)\hfill \\ {\displaystyle \frac{\mathrm{d}{\omega }_{\mathrm{m}}}{\mathrm{d}t}}={\displaystyle \frac{1}{J}}\left(-T_{\mathrm{L}}+{\displaystyle \frac{3p_{\mathrm{n}}{\psi }_{\mathrm{f}}}{2}}{i}_q-B{\omega }_{\mathrm{m}}\right)\hfill \end{array}\right.$$

(13)

The state variable of speed controller is

$$\left\{\begin{array}{l}{x}_1={\omega }_{\mathrm{ref}}-{\omega }_{\mathrm{m}}\hfill \\ x_2={\dot{x}}_1=-{\dot{\omega }}_{\mathrm{m}}\hfill \end{array}\right.$$

(14)

Derivation of Equation (14),

$$\left\{\begin{array}{l}{\dot{x}}_1=-{\dot{\omega }}_{\mathrm{m}}={\displaystyle \frac{1}{J}}\left(T_{\mathrm{L}}-{\displaystyle \frac{3p_{\mathrm{n}}{\psi }_{\mathrm{f}}}{2}}{i}_q+B{\omega }_{\mathrm{m}}\right)\hfill \\ {\dot{x}}_2=-{\ddot{\omega }}_{\mathrm{m}}=-{\displaystyle \frac{3p_{\mathrm{n}}{\psi }_{\mathrm{f}}}{2J}}{\dot{i}}_q-B{x}_2\hfill \end{array}\right.$$

(15)

Let $u={\dot{i}}_q,D={\displaystyle \frac{3p_{\mathrm{n}}{\psi }_{\mathrm{f}}}{2J}}$, and rewrite Equation (15) as

$$\left[\begin{array}{l}{\dot{x}}_1\hfill \\ {\dot{x}}_2\hfill \end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& -1\end{array}\right]\left[\begin{array}{l}{x}_1\hfill \\ x_2\hfill \end{array}\right]+\left[\begin{array}{r}\hfill 0\\ \hfill -D\end{array}\right]u$$

(16)

The sliding surface is given

$$\left\{\begin{array}{l}s=c{x}_1+x_2\\ \dot{s}=c{\dot{x}}_1+{\dot{x}}_2=c{x}_2+{\dot{x}}_2=(c-1)x_2-Du\end{array}\right.$$

(17)

where c > 1, and the exponential reaching law is adopted

$$\dot{s}=-K_1\mathrm{sgn}(s)-K_{2}s$$

(18)

where K₁ and K₂ are the parameters of SMC. Combining Equations (17) and (18)

$$u={\displaystyle \frac{1}{D}}\left[(c-1)x_2+K_1\mathrm{sgn}(s)+K_{2}s\right]$$

(19)

The stability of SMC controller is verified using a Lyapunov function.

$$\begin{array}{l}\dot{V}=s\dot{s}=s\left[(c-1)x_2-Du\right]\\ =s\left\{(c-1)x_2-\left[(c-1)x_2+K_1\mathrm{sgn}(s)+K_{2}s\right]\right\}\\ =-K_1\mathrm{sgn}(s)s-K_2{s}^2\le 0\end{array}$$

(20)

K₁sgn(s) and K₂s² are both positive, and $\dot{V}<0$ is negative definite. Thus, the system is asymptotically stable with SMC controller. Then, the reference current in q-axis is expressed as

$$i_q^{*}={\displaystyle \frac{1}{D}}{\displaystyle {\int }_0^t\left[(c-1)x_2+K_1\mathrm{sgn}(s)+K_{2}s\right]}\mathrm{d}\tau$$

(21)

The inclusion of an integral term in controller serves not only to attenuate oscillations but also eliminate steady-state errors, thereby enhancing the overall control quality.

2.3. Mechanical Model of Motor Rotor

Based on the equivalent circuit model of PMSM, the interaction channel is established that links the torsional motion of the motor rotor with GTS, thereby forming an electromechanical coupling model. The coupling between the torsional and lateral vibrations is highlighted by Equation (6), indicating the necessity for incorporation of lateral vibration model. The traditional torsional model neglects these effects, which can influence the air gap, output torque, and noise. To address these issues comprehensively, a lateral vibration model is integrated with the equivalent circuit and torsional models of motor, resulting in an improved electromechanical coupling model.

The study employs the Jeffcott model [37] to analyze the dynamics of the motor rotor, as depicted in Figure 4. The model incorporates lateral vibration and the support forces of rolling bearings. Here, m₁ is the mass at the left bearing support, m₂ the rotor mass (same as m in Equation (6)), and m₃ mass at the right bearing support. The bearing forces (F_x, F_y) can be calculated using the method outlined in reference [38].

The rotor vibrations are described by the mathematical model in the X-Y directions. Left bearing displacements are denoted as X₁ and Y₁, rotor center displacements as X₂ and Y₂ (same as X and Y in Equation (6)), and right bearing displacements as X₃ and Y₃.

$$\left\{\begin{array}{l}{m}_1{\ddot{X}}_1+c_1{\dot{X}}_1+k_1\left(X_1-X_2\right)=-F_x({\dot{X}}_1,{\dot{Y}}_1,X_1,Y_1)\\ m_1{\ddot{Y}}_1+c_1{\dot{Y}}_1+k_1\left(Y_1-Y_2\right)=-F_y({\dot{X}}_1,{\dot{Y}}_1,X_1,Y_1)\\ m_2{\ddot{X}}_2+c_2{\dot{X}}_2+k_1\left(X_2-X_1\right)+k_2\left(X_2-X_3\right)=m_{2}e{{\omega }_m}^2\cos \left({\omega }_{m}t+\beta \right)+F_x^{UMP}\\ m_2{\ddot{Y}}_2+c_2{\dot{Y}}_2+k_1\left(Y_2-Y_1\right)+k_2\left(Y_2-Y_3\right)=m_{2}e{{\omega }_m}^2\sin \left({\omega }_{m}t+\beta \right)+F_y^{UMP}\\ m_3{\ddot{X}}_3+c_3{\dot{X}}_3+k_2\left(X_3-X_2\right)=-F_x({\dot{X}}_3,{\dot{Y}}_3,X_3,Y_3)\\ m_3{\ddot{Y}}_3+c_3{\dot{Y}}_3+k_2\left(Y_3-Y_2\right)=-F_y({\dot{X}}_3,{\dot{Y}}_3,X_3,Y_3)\end{array}\right.$$

(22)

where c₁, c₂, c₃ represent the damping at shaft and disc, and k₁, k₂ the bending stiffness of shaft. The forces F_y^UMP and F_y^UMP denote the component forces of the unbalanced magnetic pull (UMP) in the X and Y directions, respectively. The magnetic lines passing through the stator and rotor are naturally shortened, generating a pull force. Uniform air gaps result in symmetrical pull forces on the rotor, leading to a net force of zero. However, rotor lateral displacement is observed to create non-uniform air gaps, causing radial unbalanced magnetic pull [37]. Consequently, Equation (22) integrates the UMP of the motor rotor, with the force calculation method detailed in the literature [39].

2.4. Dynamic Model of GTS

For precision equipment requiring operation with low vibrations, a more precise modeling method for GTS, named the Multibody of Gear-Rotor-Bearing System [40], is referenced. The model distinctly separates parallel shaft axes and their components, employing a gear tooth rigid body dynamics model to calculate the torque exerted on the hub by meshing gear teeth. This dynamic model, as illustrated in Figure 5, treats the hub as a disc on the shaft, with the gear meshing force represented as meshing torque.

In Figure 5, the input and output torques are denoted as T_I and T_O, respectively, where the subscript I represents the input and O represents the output. The bearing forces in the x and y directions for both the input and output shafts are labeled as F_xi and F_yi (i = 1~4), and the bearing masses are denoted as m_bi. The mass of the gear hub is represented as m_i (i = p/g), with the subscript p for pinion and g for gear, and their corresponding moments of inertia are J_i. e_i is the gear eccentricity.

Equation (23) formulates the differential equation for the GTS of input shaft, whereas Equation (24) represents the corresponding system for the output shaft within the context of lateral torsional coupled vibration. The variable θ_i signifies the torsional angle displacement at each concentrated mass location, c_bi the damping of bearing, ω_i the shaft speed, ζ_i the dimensionless position parameter of gear hub. In this context, ζ_i is set to a value of 0.5. In Equations (23) and (24), c₁ and k₁ are the torsional damping and stiffness of shaft 1, while c₂ and k₂ are the lateral damping and stiffness of shaft 2. Similarly, c₃ and k₃ denote the lateral damping and stiffness of shaft 2, while c₄ and k₄ the torsional damping and stiffness of shaft 2.

$$\left\{\begin{array}{l}{J}_I{\ddot{\theta }}_I+c_I\left({\dot{\theta }}_I-{\dot{\theta }}_p\right)+k_I\left({\theta }_I-{\theta }_p\right)=T_I\\ m_{b1}{\ddot{x}}_{b1}+c_{b1}{\dot{x}}_{b1}+{\zeta }_2{c}_p\left({\zeta }_1{\dot{x}}_{b2}+{\zeta }_2{\dot{x}}_{b1}-{\dot{x}}_p\right)+{\zeta }_2{k}_p\left({\zeta }_1{x}_{b2}+{\zeta }_2{x}_{b1}-x_p\right)=F_{x1}\\ m_{b1}{\ddot{y}}_{b1}+c_{b1}{\dot{y}}_{b1}+{\zeta }_2{c}_p\left({\zeta }_1{\dot{y}}_{b2}+{\zeta }_2{\dot{y}}_{b1}-{\dot{y}}_p\right)+{\zeta }_2{k}_p\left({\zeta }_1{y}_{b2}+{\zeta }_2{y}_{b1}-y_p\right)=F_{y1}-m_{b1}g\\ \left(J_p+m_p{e_1}^2\right){\ddot{\theta }}_p+c_I\left({\dot{\theta }}_p-{\dot{\theta }}_I\right)+k_I\left({\theta }_p-{\theta }_I\right)=m_p{e}_1[\sin ({\omega }_{p}t+{\theta }_P){\ddot{x}}_p-\cos ({\omega }_{p}t+{\theta }_p){\ddot{y}}_p]-F_n{r}_{bp}-(F_{fp1}{s}_{p1}+F_{fp2}{s}_{p2})\\ m_p{\ddot{x}}_p+c_p\left({\dot{x}}_p-{\zeta }_1{\dot{x}}_{b2}-{\zeta }_2{\dot{x}}_{b1}\right)+k_p\left(x_p-{\zeta }_1{x}_{b2}-{\zeta }_2{x}_{b1}\right)=m_p{e}_1[{({\omega }_p+{\dot{\theta }}_p)}^2\cos ({\omega }_{p}t+{\theta }_p)+{\ddot{\theta }}_p\sin ({\omega }_{p}t+{\theta }_p)]+F_{fp}\\ m_p{\ddot{y}}_p+c_p\left({\dot{y}}_p-{\zeta }_1{\dot{y}}_{b2}-{\zeta }_2{\dot{y}}_{b1}\right)+k_p\left(y_p-{\zeta }_1{y}_{b2}-{\zeta }_2{y}_{b1}\right)=m_p{e}_1[{({\omega }_p+{\dot{\theta }}_p)}^2\sin ({\omega }_{p}t+{\theta }_p)-{\ddot{\theta }}_p\cos ({\omega }_{p}t+{\theta }_p)]-m_{p}g-F_n\\ m_{b2}{\ddot{x}}_{b2}+c_{b2}{\dot{x}}_{b2}+{\zeta }_1{c}_p\left({\zeta }_1{\dot{x}}_{b2}+{\zeta }_2{\dot{x}}_{b1}-{\dot{x}}_p\right)+{\zeta }_1{k}_p\left({\zeta }_1{x}_{b2}+{\zeta }_2{x}_{b1}-x_p\right)=F_{x2}\\ m_{b2}{\ddot{y}}_{b2}+c_{b2}{\dot{y}}_{b2}+{\zeta }_1{c}_p\left({\zeta }_1{\dot{y}}_{b2}+{\zeta }_2{\dot{y}}_{b1}-{\dot{y}}_p\right)+{\zeta }_1{k}_p\left({\zeta }_1{y}_{b2}+{\zeta }_2{y}_{b1}-y_p\right)=F_{y2}-m_{b2}g\end{array}\right.$$

(23)

$$\left\{\begin{array}{l}{m}_{b3}{\ddot{x}}_{b3}+c_{b3}{\dot{x}}_{b3}+{\zeta }_4{c}_g\left({\zeta }_3{\dot{x}}_{b4}+{\zeta }_4{\dot{x}}_{b3}-{\dot{x}}_g\right)+{\zeta }_4{k}_g\left({\zeta }_3{x}_{b4}+{\zeta }_4{x}_{b3}-x_g\right)=F_{x3}\\ m_{b3}{\ddot{y}}_{b3}+c_{b3}{\dot{y}}_{b3}+{\zeta }_4{c}_g\left({\zeta }_3{\dot{y}}_{b4}+{\zeta }_4{\dot{y}}_{b3}-{\dot{y}}_g\right)+{\zeta }_4{k}_g\left({\zeta }_3{y}_{b4}+{\zeta }_4{y}_{b3}-y_g\right)=F_{y3}-m_{b3}g\\ \left(J_g+m_g{e_2}^2\right){\ddot{\theta }}_g+c_O\left({\dot{\theta }}_g-{\dot{\theta }}_O\right)+k_O\left({\theta }_g-{\theta }_O\right)=m_g{e}_2[\sin ({\omega }_{g}t+{\theta }_g){\ddot{x}}_g-\cos ({\omega }_{g}t+{\theta }_g){\ddot{y}}_g]+F_n{r}_{bg}+(F_{fg1}{s}_{g1}+F_{fg2}{s}_{g2})\\ m_g{\ddot{x}}_g+c_g\left({\dot{x}}_g-{\zeta }_3{\dot{x}}_{b4}-{\zeta }_4{\dot{x}}_{b3}\right)+k_g\left(x_g-{\zeta }_3{x}_{b4}-{\zeta }_4{x}_{b3}\right)=m_g{e}_2[{({\omega }_g+{\dot{\theta }}_g)}^2\cos ({\omega }_{g}t+{\theta }_g)+{\ddot{\theta }}_g\sin ({\omega }_{g}t+{\theta }_g)]+F_{fg}\\ m_g{\ddot{y}}_g+c_g\left({\dot{y}}_g-{\zeta }_4{\dot{y}}_{b4}-{\zeta }_4{\dot{y}}_{b3}\right)+k_g\left(y_g-{\zeta }_3{y}_{b4}-{\zeta }_4{y}_{b3}\right)=m_g{e}_2[{({\omega }_g+{\dot{\theta }}_g)}^2\sin ({\omega }_{g}t+{\theta }_g)-{\ddot{\theta }}_g\cos ({\omega }_{g}t+{\theta }_g)]-m_{g}g+F_n\\ m_{b4}{\ddot{x}}_{b4}+c_{b4}{\dot{x}}_{b4}+{\zeta }_3{c}_g\left({\zeta }_3{\dot{x}}_{b4}+{\zeta }_4{\dot{x}}_{b3}-{\dot{x}}_g\right)+{\zeta }_3{k}_g\left({\zeta }_3{x}_{b4}+{\zeta }_4{x}_{b3}-x_g\right)=F_{x4}\\ m_{b4}{\ddot{y}}_{b4}+c_{b4}{\dot{y}}_{b4}+{\zeta }_3{c}_g\left({\zeta }_3{\dot{y}}_{b4}+{\zeta }_4{\dot{y}}_{b3}-{\dot{y}}_g\right)+{\zeta }_3{k}_g\left({\zeta }_3{y}_{b4}+{\zeta }_4{y}_{b3}-y_g\right)=F_{y4}-m_{b4}g\\ J_O{\ddot{\theta }}_O+c_O\left({\dot{\theta }}_O-{\dot{\theta }}_g\right)+k_O\left({\theta }_O-{\theta }_g\right)=-T_O\end{array}\right.$$

(24)

in which, the meshing forces F_n are expressed as

$$F_n=k(t)f(r_{bp}{\theta }_p-r_{bg}{\theta }_g+y_p-y_g-e(t))+c(t)(r_{bp}{\dot{\theta }}_p-r_{bg}{\dot{\theta }}_g+{\dot{y}}_p-{\dot{y}}_g-\dot{e}(t))$$

(25)

Here, k(t) denotes the Time-Varying Meshing Stiffness (TVMS), and f(x) represents a discontinuous function of the gear along the mesh line. In addition, the frictions experienced by the pinion and gear are denoted by F_fi (where i = p for the pinion and i = g for the gear). The moment arms for these friction forces on the pinion and gear are represented by s_pi and s_gi, respectively (where i = 1, 2 corresponds to the meshing tooth pairs). Detailed descriptions of these nonlinear parameters are provided in the Appendix A.

2.5. TVMS Considering Tooth Crack

Tooth root cracks represent a common internal fault in gears and pose a significant risk to the integrity of GTS. The accurate modeling of these cracks is crucial for understanding their impact on the dynamic behaviors of system. In this study, an analytical approach is employed to determine the meshing stiffness of gears that are influenced by the presence of cracks. By altering stiffness calculation methodologies [28], the meshing stiffness of gears affected by cracks can be ascertained, facilitating analysis of their electromechanical characteristics. The calculation method for the time-varying meshing stiffness of gears without considering tooth root cracks is shown in Appendix B.

During the meshing process, the stress experienced by gears closely resembles that of a cantilever beam. This similarity allows for the application of the 30° tangent method to effectively identify the tooth root as the critical section. This method entails drawing two lines at 30° angles to the tooth’s center of symmetry, both tangent to the fillet curve at the root. The intersection points of these lines, known as the tangent points, are then used to create a section parallel to the gear axis. As illustrated in Figure 6, this section is designated as the critical section at the tooth root.

After a crack appears in a gear, the calculation of bending, shear, and axial compressive deformation of the gear tooth section under the influence of meshing force is affected. As shown in Figure 7, the effective cross-section resisting bending along the tooth thickness direction for a gear with a root crack is h_c + h_x, whereas for a normal gear without cracks, the effective cross-section is h_x + h_x. If the crack depth is q, the effective height on the side with the crack in the tooth thickness direction is given by

$$h_c=y\left(t\right)-q\sin \theta$$

(26)

The moment of inertia I_x and the cross-sectional area A_x of the selected calculation element δ_j at a distance x from the tooth root are as follows, where I_x and A_x correspond to the moment of inertia and cross-sectional area in Equation (A5), respectively.

$$I_x=\left\{\begin{array}{l}{\displaystyle \frac{1}{12}}b\left(h_x+h_c\right) y\left(t\right)\ge h_c\\ {\displaystyle \frac{1}{12}}b\left(h_x+h_x\right) y\left(t\right)\le h_c\end{array}\right.$$

(27)

$$A_x=\left\{\begin{array}{l}b\left(h_x+h_c\right) y\left(t\right)\ge h_c\\ b\left(h_x+h_x\right) y\left(t\right)\le h_c\end{array}\right.$$

(28)

This ratio is utilized to quantify the severity of the crack and its impact on the gear’s meshing stiffness. In this study, the tooth root crack is assumed to be located on the pinion, which serves as the driving gear in the gear pair. This selection is based on the observation that cracks on the pinion tend to have a more significant influence on the system’s dynamic response due to its higher rotational speed and greater load-bearing capacity. Figure 8 presents the computed meshing stiffness results for various depths of tooth root cracks, compared with those of normal gears. The gear parameters are outlined in

Table 1. Parameters of the MGS.

Motor	Symbol
Number of Pole Pairs	np	3
Flux Linkage (Wb)	ϕf	0.175
Stator Resistance (Ω)	R	2.875
Moment of Inertia (kg·m2)	J	0.008
Stator Inductance (mH)	Ld, Lq	8.5	8.5
Gear	Symbol	Pinion	Gear
Number of teeth	z1, z2	55	75
Mass (kg)	mp, mg	2.328	3.937
Moment of inertia (kg∙m2)	Jp, Jg	0.0195	0.0614
Modules (mm)	m	2
Tooth width (mm)	B	20
Pressure angle (deg)	α	20
Pitch (mm)	Pn	5.9
Contact ratio	ε	1.797
Addendum coefficient	$h_a^{*}$	1
Tip clearance coefficient	λ	0.25
Elastic modulus (Pa)	E	210 G
Poisson’s ratio	ν	0.28
Mass density (kg·m−3)	ρ	7800

. In Figure 8, the percentage of tooth root crack depth is defined as the ratio of the crack depth (q) to the tooth thickness (h_x).

When a gear contains a single crack, the cracked tooth engages once during a full rotation. In accordance with the principle of involute gear meshing, the meshing process of the cracked tooth can be described as follows:

• Double-tooth engagement: Initially, the cracked tooth engages in double-tooth engagement, maintaining contact with the preceding gear tooth.
• Transition to single-tooth engagement: As the engagement progresses, the cracked tooth transitions into a single-tooth engagement state.
• Reversion to double-tooth engagement: Upon the engagement of the subsequent gear tooth, the cracked tooth reverts to a double-tooth engagement configuration.

The cracked tooth is observed to transition from double-tooth to single-tooth engagement and then revert to double-tooth engagement within a single meshing cycle. The variation in TVMS throughout a complete shaft rotation is depicted in Figure 9.

3. Verification of the Proposed Model

To address the electromechanical coupling dynamics of MGS, a comprehensive dynamic model was introduced. To ascertain the accuracy and efficacy of this innovative model, experimental validation is conducted using a specialized test platform.

3.1. Introduction of Experimental Platform

The experimental test setup is illustrated in Figure 10, which includes a gearbox, a variable frequency AC motor, a magnetic powder brake, an acceleration sensor, a data acquisition system, controllers, and various auxiliary elements. Operating at a rated power of 1.5 kW, the motor’s speed is adjustable from 0 to 1500 rpm. The magnetic powder brake, attached to the output shaft, facilitates the application of load, with the brake torque controllable from 0 to 10 N·m via the brake controller. The gearbox houses two shafts supported by four rolling bearings. Additional gear parameters are detailed in Table 1 and

Table 2. Parameters of the GTC.

Parameter	Symbol	Input Shaft	Output Shaft
Transverse stiffness (N/m)	kpx, kgy	3 × 108	2 × 108
Transverse damping (N∙s/m)	cgx, cgy	1.1 × 103	0.9 × 103
Torsional stiffness (N∙m/rad)	kI, kO	9 × 106	7 × 106
Torsional damping (N∙m∙s/rad)	cI, cO	17	11
Outer radius of the bearing (m)	R1, R2	0.09	0.04
Inner radius of the bearing (m)	r1, r2	0.05	0.02
Bearing clearance (mm)	γ01, γ02	0.02	0.05
Number of bearing rollers	N1, N2	14	18
Contact stiffness of supports (N/m3/2)	kb1, kb2	1.334 × 1010	1.056 × 1010

. The acceleration sensor is installed on the exterior of the gearbox, close to the bearing supports of the gear shafts, to collect vibration information at the bearings. The collected data are then compared with the simulation results of the MGS model, with the comparison variable being the bearing vibration response y_b.

3.2. Model Verification

The effectiveness of the electromechanical coupling dynamic model for the MGS is validated through experiments conducted on a specialized test platform. Figure 11 and Figure 12 illustrate the comparative results under various conditions. The experimental data reveal the vertical acceleration vibration signal of the gearbox input shaft, while the simulation results represent the lateral acceleration of the gear center. Both time-domain signals and their corresponding frequency spectra are presented for detailed analysis.

The vibration acceleration signal of the gearbox, both in its healthy state and with a crack present, exhibits harmonic patterns alongside chaotic behavior. This complexity complicates the identification of fluctuation periods. Analysis of the frequency spectrum indicates the presence of three primary frequencies: rotational frequency (f_r), meshing frequency (f_m), and twice of meshing frequency (2f_m), accompanied by dense sidebands. The comparison between simulation and experimental results highlights several key findings:

• Similarity in Time-Domain Signal: Both the simulation and experimental results display harmonic vibration patterns, suggesting that the simulation model accurately represents the system dynamics.
• Difference in Fluctuation Amplitude: The experimental results exhibit larger vibration acceleration amplitudes compared to the simulation, likely due to the complexities inherent in the real-world experimental setup.
• Consistency in Frequency Components: Both the simulation and experimental results identify the same primary frequency components, including the rotational frequency (f_r), meshing frequency (f_m), and twice of meshing frequency (2f_m).
• Comparison of Sideband Components: The simulation results display fewer sidebands around the primary frequency components than the experimental results, highlighting the idealized conditions of the simulation model.
• Simulation of Tooth Root Cracks: In a healthy system, the rotational frequency components are minimal. However, with a tooth root crack, the time-domain graph exhibits periodic harmonics, and the rotational frequency becomes more pronounced in the frequency spectrum.

4. Electromechanical Coupling Dynamics Simulation

This section delves into the effects of tooth root cracks, a prevalent fault in gear transmissions, on the dynamic characteristics of MGS. Gear mesh stiffness excitation data resulting from tooth root cracks are integrated into a multi-coupled dynamic model. The simulation analysis explores the impact of varying crack depths on system dynamics.

4.1. The Effect of Cracks on Gear Dynamics

The simulation results of gear root displacement along the mesh line for varying crack depths are illustrated in Figure 13. It is observed that gear vibrations are intensified with increasing crack depth, and the frequency spectrum exhibits sidebands associated with the rotational frequency near the natural frequency. These findings corroborate the experimental results detailed in Section 3.2. It is noted that tooth root cracks significantly influence the vibration characteristics of the GTS, altering both the mean value and amplitude of vibrations. A detailed analysis reveals that deeper cracks result in more pronounced impact characteristics and more distinct fault frequencies within gear vibrations.

The dynamic simulation results of gear dynamic transmission error (DTE) with varying tooth root crack depths are depicted in Figure 14. The presence of the tooth root crack is observed to introduce noticeable periodic impact vibrations into the DTE. Additionally, the frequency spectrum indicates multiple frequency modulations caused by meshing force fluctuations during transmission, which align with the observed actual conditions.

The depth of the tooth root crack is observed to directly influence both the gear center vibrations and the DTE of GTS. An increase in crack depth results in elevated mean values and amplitudes of gear center vibrations and DTE. Detailed analysis of the DTE reveals that greater crack depth intensifies gear meshing impacts. The frequency response also indicates that deeper cracks introduce more fault frequencies. These results demonstrate the effectiveness of the proposed multiple coupled dynamic model of the MGS in studying the dynamic impact of external excitations and internal excitation disturbances on mechanical vibrations. The model is shown to accurately simulate the dynamic characteristics of the MGS under fault conditions.

4.2. The Effect of Cracks on Motor Dynamics

The alterations in mechanical vibration characteristics induced by tooth root cracks were analyzed in the aforementioned study. Subsequently, the effects of these cracks on the motor dynamic response are explored. Figure 15 illustrates the actual output speed of the motor, as predicted by the proposed model. It is observed that the presence of tooth root cracks results in fluctuations in the motor output speed, with frequencies corresponding to the fluctuations observed in the DTE response. This indicates that the meshing impacts of tooth root cracks have an indirect influence on motor performance

Figure 16 depicts the actual output torque response of the motor for varying depths of tooth root cracks. The torque fluctuations are observed to mirror those in speed, both exhibiting impact effects. These observations suggest that tooth root crack faults induce unstable motion states within the GTS, such as meshing impacts and abnormal vibrations, which in turn affect the motor output characteristics. With the evolution of electric drive products towards more integrated and compact system structures, the significance of these impacts is expected to increase.

As shown in Figure 17, the dynamic response of motor current varies with tooth root crack depth. The amplitude of motor current is observed to be increased due to the presence of tooth root cracks. Compared to Figure 11, the current spectrum with cracks shows two frequency clusters around the main frequency. The frequency difference between these clusters equals the rotational frequency. This phenomenon is recognized as a new method for diagnosing mechanical faults in electric drive systems. Monitoring motor current, abnormal operations can be effectively detected, especially in situations where gearbox signal monitoring proves to be challenging.

These results demonstrate that internal excitations from tooth root cracks exert a direct impact on the dynamic characteristics of the GTS within MGS and also have an indirect effect on the dynamic properties of motor responses. With the evolution of electric drive products towards more compact and integrated structures, it is expected that these effects will be significantly amplified.

5. Conclusions

The challenges in vibration analysis and fault diagnosis of integrated electromechanical MGS under fault conditions are addressed in this paper. A novel electromechanical coupled dynamic model is introduced, which is designed for the design and optimization of MGS. The model is characterized by significant flexibility in parameter adjustments, making it suitable for various engineering scenarios. Then, the influence of various tooth root crack depths on the dynamic behavior of MGS is simulated by the model. One key finding is that the dynamic behaviors of both GTS and motor within MGS are affected by tooth root cracks. This capability offers a new approach for system fault diagnosis by allowing for the identification and analysis of fault signatures associated with different crack conditions. Although the accuracy of the vibration response of MGS has been experimentally verified in this study, further experimental validation of the motor response is still pending and will be addressed in future work.