Introduction

The gear systems were widely employed in mechanical transmission equipment, including wind turbines, vehicle transmission, aero-engine cases, etc. [1]. However, because of the nonlinear factors and excitation loads in the gear transmission system, the meshing shock load can be intensified. Eventually, it leads to the gradual appearance of the gear tooth damage-type failures such as cracks, pitting, or wear [2, 3]. If cracks appear in the root of the gear teeth, significant sideband features will appear in the frequency domain of the system [4, 5]. However, this characteristic is not unique to the crack defect, but the fault that causes the meshing stiffness reduction will lead to the appearance of the sidebands. In addition, the shock of the system is further amplified and the vibration signal is made more complex by the nonlinear factors. This leads to a situation where it is difficult to diagnose the damage extent and recognize the failure mechanism, especially for gear systems affected by complex nonlinear factors. To investigate the characteristics and mechanisms of failures, it is urgent to establish an accurate coupling model of a gear system with a fault including the effect of nonlinear factors.

The vibration signal of a faulty system is seen as the basis for the study of fault characteristics. Substantial research was concentrated on the TVMS of different tooth defects and vibration characteristic analysis of a simplified gear system with a fault. Significant modeling and analysis on the calculation of TVMS for a cracked gear were proposed by Tian [6], and this research provided a strong foundation for calculating meshing stiffness by an analytical method. A critical review study by Ma et al. [7,8,9,10] examined in detail the different methods to calculate TVMS with a crack and analyzed the vibration responses from different dynamic models of the gear systems. Yang et al. [11] presented a vibration model of the gear system with a crack defect for fault detection analysis, and the backlash from the gear tooth profile and the clearance from the bearing were considered in this detection model. Chen et al. [12] established a locomotive track-coupled vibration model considering the effect of the tooth root crack, and crack characteristics were revealed by statistical indicator M8A. A study by Chen et al. [13] described in detail the effect of the complex foundation and crack propagation path on the meshing stiffness. Karpat et al. [14] proposed a new crack detection model for spur gear. Liang et al. [15, 16] established a vibration model of a six degree-of-freedom gear system with pitting, and the statistical indicators were used to describe the pitting characteristics. Lei et al. [17] presented a probability distribution model of pits, and a calculation method of TVMS with pitting fault was also proposed. Wei et al. [18] discussed the cause of pitting pit formation under the elastohydrodynamic lubrication condition. Huangfu et al. [19] presented a new topographical updating approach to investigate the pitting fault evolution and considered tooth surface friction in this model. Feng et al. [20] proposed a wear detection model for spur gear, and the pitting density and profile change were employed for vibration prediction of a worn gear. Ouyang et al. [21] investigated the crack-pitting coupling faults and established a dynamic model of a gear system to study the characteristics of the time-domain and frequency domain. Luo et al. [22, 23] researched the effect of the pitting and spalling defects on a spur gear system, and its experimental validation also was described. Not only can tooth defects affect the stability of a gear system, but also the nonlinear factors of the teeth themselves can lead to fluctuation.

The nonlinear factors between the tooth interface, such as the backlash, friction, profile, and so on, play an important role in the meshing process of the gear teeth. Various nonlinear dynamic theories are employed for simulating the effect of nonlinear factors. Considering the dynamic backlash caused by the dynamic center distance between two gears, Chen et al. [24] investigated the dynamic behaviors of a six-degree-of-freedom gear model. Chen et al. [25] proposed a random function to simulate a dynamic backlash and described the system stability based on this random backlash model. Although this random number model is simple, it can reflect the variation of the real backlash. One study by Liu et al. [26] presented a nonlinear interaction relationship between the tooth backlash and bearing clearance and researched the vibration responses of a spur gear pair system. Margielewicz et al. [27] studied the modeling of the gear backlash and carried out a model numerical test. A critical study by Shi et al. [28] proposed a new time-varying backlash model of a spur gear considering the elastic, thermal, and film thickness deformation. Huang et al. [29] investigated an indirect relationship between the bifurcation and chaos of a gear system and the fractal backlash and researched the differences between fixed backlash and fractal backlash. Wang [30] established a helical gear pair model to investigate the effect of friction and tooth contact on the gear system. Zhao et al. [31] established a micro contact model between the teeth of gears based on the fractal theory and discussed the TVMS with fixed friction coefficient considering the relationship between the contact model and friction. Liu et al. [32] established a friction model of a spur gear pair with a starved lubricated state to calculate the friction coefficient. Li and Kahraman [33] proposed a new tribo-dynamic model of a spur gear pair.

As mentioned above, the influences of nonlinear factors from a system are investigated seldom in damage defect of a gear tooth, because the signal without them is purer than with them. Although this makes the simulation signal more focused on the fault characteristics, it runs counter to the actual signal and cannot reflect the actual situation of the gear with a damaged defect. This paper proposed a nonlinear dynamic model of a spur gear system with a crack defect considering time-varying friction and dynamic backlash. Based on the actual rough tooth surface in engineering, the effect of the tooth surface microstructure on the dynamic backlash is considered adequately. Moreover, the time-varying friction coefficient and TVMS are calculated accurately. The vibration characteristics of the model are obtained by solving vibration equations. The effect of different roughness on the vibration characteristics of the system is discussed by the time-domain waveforms, phase plane, Poincaré map, and time–frequency domain. Crack defect research in this condition is investigated. Under the same parameters and conditions, the experimental platform is set up to verify the correctness of the model results. Besides, the effect of a root crack on the fault features is also analyzed in the actual roughness condition of gears.

Modeling of a Dynamics Model

Model Assumptions

  • (1) The shaft length in this system shall be defined as the short shaft. The gyroscopic effect of the shaft is ignored.

  • (2) The tooth surfaces contact is discussed without microstructure. The microstructure only affects the size of the backlash as can be seen in Fig. 1. Because the time-varying center distance caused by the vibration is ignored, the effect of the distance center on the backlash is also skipped.

  • (3) To simplify the backlash calculation model, the morphology of each gear tooth meshing position is spliced into a continuous fractal curve.

  • (4) The oil inlet temperature of the gearbox is set to a fixed value. The influence of heat on the system is not considered, especially in the calculation of the friction coefficient.

Fig. 1
figure 1

Backlash of the gear pair

Full size image

Modeling of a Spur Gear System

A nonlinear dynamic model of a spur gear system with 16-degree-of-freedom is shown in Fig. 2. The 16 generalized coordinates are shown as a vector:

$$\chi = [\theta_{i} \, \theta_{o} \, x_{g} \, y_{g} \, \theta_{g} \, x_{p} \, y_{p} \, \theta_{p} \, x_{b1} \, y_{b1} \, x_{b2} \, y_{b2} \, x_{b3} \, y_{b3} \, x_{b4} \, y_{b4} ],$$
(1)

where \(\theta_{{\text{i}}}\) and \(\theta_{{\text{o}}}\) represent vibration angular displacements of the input and output rotors in the torsion direction. \(x_{{\text{g}}}\) and \(y_{{\text{g}}}\) are the vibration displacement of the horizontal direction (x-direction) and vertical direction (y-direction) of the driving gear. \(x_{{\text{p}}}\) and \(y_{{\text{p}}}\) represent the vibration displacements of the driven pinion. \(\theta_{{\text{g}}}\) and \(\theta_{{\text{p}}}\) are vibration angular displacements of the gear and pinion. \(x_{{{\text{b}}i}} (i = 1, \, 2, \, 3, \, 4)\) are the displacement of bearings in the horizontal direction. \(y_{{{\text{b}}i}} (i = 1, \, 2, \, 3, \, 4)\) represent the displacement of bearings in the vertical direction.

Fig. 2
figure 2

Structure diagram of the gear transmission system

Full size image

Based on the Lagrange equation, the nonlinear vibration model of a spur gear system can be deduced. The elastic deformations of the driving and driven shafts are used as a link, and the coupling relationships between the gears and bearings are established by this link [34, 35].

The differential equations of the bearings are:

$$\left\{ {\begin{array}{*{20}l} {m_{{{\text{b1}}}} \ddot{x}_{{{\text{b1}}}} + c_{{{\text{s1}}}} \xi_{2} \left( { - \dot{x}_{{\text{g}}} + \xi_{2} \dot{x}_{{{\text{b1}}}} + \xi_{1} \dot{x}_{{{\text{b2}}}} } \right) + c_{{{\text{x1}}}} \dot{x}_{{{\text{b1}}}} + k_{{{\text{s1}}}} \xi_{2} \left( { - x_{{\text{g}}} + \xi_{2} x_{{{\text{b1}}}} + \xi_{1} x_{{{\text{b2}}}} } \right) = 0} \hfill \\ {m_{{{\text{b1}}}} \ddot{y}_{{{\text{b1}}}} + c_{{{\text{s1}}}} \xi_{2} \left( { - \dot{y}_{{\text{g}}} + \xi_{2} \dot{y}_{{{\text{b1}}}} + \xi_{1} \dot{y}_{{{\text{b2}}}} } \right) + c_{{{\text{y1}}}} \dot{y}_{{{\text{b1}}}} + k_{{{\text{s1}}}} \xi_{2} \left( { - y_{{\text{g}}} + \xi_{2} y_{{{\text{b1}}}} + \xi_{1} y_{{{\text{b2}}}} } \right) = - m_{{{\text{b1}}}} g} \hfill \\ {m_{{{\text{b2}}}} \ddot{x}_{{{\text{b2}}}} + c_{{{\text{s1}}}} \xi_{1} \left( { - \dot{x}_{{\text{g}}} + \xi_{2} \dot{x}_{{{\text{b1}}}} + \xi_{1} \dot{x}_{{{\text{b2}}}} } \right) + c_{{{\text{x2}}}} \dot{x}_{{{\text{b2}}}} + k_{{{\text{s1}}}} \xi_{1} \left( { - x_{{\text{g}}} + \xi_{2} x_{{{\text{b1}}}} + \xi_{1} x_{{{\text{b2}}}} } \right) = 0} \hfill \\ {m_{{{\text{b2}}}} \ddot{y}_{{{\text{b2}}}} + c_{{{\text{s1}}}} \xi_{1} \left( { - \dot{y}_{{\text{g}}} + \xi_{2} \dot{y}_{{{\text{b1}}}} + \xi_{1} \dot{y}_{{{\text{b2}}}} } \right) + c_{{{\text{y2}}}} \dot{y}_{{{\text{b1}}}} + k_{{{\text{s1}}}} \xi_{1} \left( { - y_{{\text{g}}} + \xi_{2} y_{{{\text{b1}}}} + \xi_{1} y_{{{\text{b2}}}} } \right) = - m_{{{\text{b2}}}} g} \hfill \\ {m_{{{\text{b3}}}} \ddot{x}_{{{\text{b3}}}} + c_{{{\text{s2}}}} \xi_{4} \left( { - \dot{x}_{{\text{p}}} + \xi_{4} \dot{x}_{{{\text{b3}}}} + \xi_{3} \dot{x}_{{{\text{b4}}}} } \right) + c_{{{\text{x3}}}} \dot{x}_{{{\text{b3}}}} + k_{{{\text{s2}}}} \xi_{4} \left( { - x_{{\text{p}}} + \xi_{4} x_{{{\text{b3}}}} + \xi_{3} x_{{{\text{b4}}}} } \right) = 0} \hfill \\ {\begin{array}{*{20}c} {m_{{{\text{b3}}}} \ddot{y}_{{{\text{b3}}}} + c_{{{\text{s2}}}} \xi_{4} \left( { - \dot{y}_{{\text{p}}} + \xi_{4} \dot{y}_{{{\text{b3}}}} + \xi_{3} \dot{y}_{{{\text{b4}}}} } \right) + c_{{{\text{y3}}}} \dot{y}_{{{\text{b3}}}} + k_{{{\text{s2}}}} \xi_{4} \left( { - y_{{\text{p}}} + \xi_{4} y_{{{\text{b3}}}} + \xi_{3} y_{{{\text{b4}}}} } \right) = - m_{{{\text{b3}}}} g} \\ {\begin{array}{*{20}l} {m_{{{\text{b4}}}} \ddot{x}_{{{\text{b4}}}} + c_{{{\text{s2}}}} \xi_{3} \left( { - \dot{x}_{{\text{p}}} + \xi_{4} \dot{x}_{{{\text{b3}}}} + \xi_{3} \dot{x}_{{{\text{b4}}}} } \right) + c_{{{\text{x4}}}} \dot{x}_{{{\text{b4}}}} + k_{{{\text{s2}}}} \xi_{3} \left( { - x_{2} + \xi_{4} x_{{{\text{b3}}}} + \xi_{3} x_{{{\text{b4}}}} } \right) = 0} \hfill \\ {m_{{{\text{b4}}}} \ddot{y}_{{{\text{b4}}}} + c_{{{\text{s2}}}} \xi_{3} \left( { - \dot{y}_{{\text{p}}} + \xi_{4} \dot{y}_{{{\text{b3}}}} + \xi_{3} \dot{y}_{{{\text{b4}}}} } \right) + c_{{{\text{y4}}}} \dot{y}_{{{\text{b4}}}} + k_{{{\text{s2}}}} \xi_{3} \left( { - y_{{\text{p}}} + \xi_{4} y_{{{\text{b3}}}} + \xi_{3} y_{{{\text{b4}}}} } \right) = - m_{{{\text{b4}}}} g} \hfill \\ \end{array} } \\ \end{array} } \hfill \\ \end{array} } \right.,$$
(2)

where \(m_{{{\text{b}}i}} (i = 1,{ 2)}\) are the masses of the bearings on the gear shaft; \(m_{{{\text{b}}j}} (j = 3,{ 4)}\) are the masses of the bearings on the pinion shaft; \(c_{xi} (i = 1,{ 2, 3, 4})\) are the bearing damping in the x-direction, and \(c_{yi} (i = 1,{ 2, 3, 4})\) are the bearing damping in the y-direction.

The differential equations of the gear and pinion are:

$$\left\{ {\begin{array}{*{20}l} \begin{gathered} m_{{\text{g}}} \ddot{x}_{{\text{g}}} + c_{{{\text{s1}}}} \left( {\dot{x}_{{\text{g}}} - \xi_{2} \dot{x}_{{{\text{b1}}}} - \xi_{1} \dot{x}_{{{\text{b2}}}} } \right) + k_{{{\text{s1}}}} \left( {x_{{\text{g}}} - \xi_{2} x_{{{\text{b1}}}} - \xi_{1} x_{{{\text{b2}}}} } \right) \hfill \\ = m_{{\text{g}}} \rho_{{\text{g}}} \ddot{\theta }_{{\text{g}}} \sin \left( {\omega_{{\text{g}}} t + \theta_{{\text{g}}} } \right) + m_{{\text{g}}} \rho_{{\text{g}}} \left( {\omega_{{\text{g}}} + \dot{\theta }_{{\text{g}}} } \right)^{2} \cos \left( {\omega_{{\text{g}}} t + \theta_{{\text{g}}} } \right) - F_{f} \hfill \\ \end{gathered} \hfill \\ \begin{gathered} m_{{\text{g}}} \ddot{y}_{{\text{g}}} + c_{{{\text{s1}}}} \left( {\dot{y}_{{\text{g}}} - \xi_{2} \dot{x}_{{{\text{b1}}}} - \xi_{1} \dot{x}_{{{\text{b2}}}} } \right) + k_{{{\text{s}}1}} \left( {x_{{\text{g}}} - \xi_{2} x_{{{\text{b1}}}} - \xi_{1} x_{{{\text{b2}}}} } \right) \hfill \\ = m_{{\text{g}}} \rho_{{\text{g}}} \ddot{\theta }_{{\text{g}}} \sin \left( {\omega_{{\text{g}}} t + \theta_{{\text{g}}} } \right) + m_{{\text{g}}} \rho_{{\text{g}}} \left( {\omega_{{\text{g}}} + \dot{\theta }_{{\text{g}}} } \right)^{2} \cos \left( {\omega_{{\text{g}}} t + \theta_{{\text{g}}} } \right) - F_{{\text{m}}} - m_{{\text{g}}} g \hfill \\ \end{gathered} \hfill \\ \begin{gathered} \left( {J_{{\text{g}}} + m_{{\text{g}}} \rho_{{\text{g}}}^{2} } \right)\ddot{\theta }_{{\text{g}}} + c_{{{\text{t1}}}} \left( {\dot{\theta }_{{\text{g}}} - \dot{\theta }_{{\text{i}}} } \right) + k_{{{\text{t1}}}} \left( {\theta_{{\text{g}}} - \theta_{{\text{i}}} } \right) \hfill \\ = m_{{\text{g}}} \rho_{{\text{g}}} \sin \left( {\omega_{{\text{g}}} t + \theta_{{\text{g}}} } \right)\ddot{x}_{{\text{g}}} - m_{{\text{g}}} \rho_{{\text{g}}} \cos \left( {\omega_{{\text{g}}} t + \theta_{{\text{g}}} } \right)\ddot{y}_{{\text{g}}} - F_{m} r_{{{\text{b1}}}} + T_{{{\text{f1}}}} \hfill \\ \end{gathered} \hfill \\ \end{array} } \right.,$$
(3)
$$\left\{ {\begin{array}{*{20}l} \begin{gathered} m_{{\text{p}}} \ddot{x}_{{\text{p}}} + c_{{{\text{s2}}}} \left( {\dot{x}_{{\text{p}}} - \xi_{4} \dot{x}_{{{\text{b3}}}} - \xi_{3} \dot{x}_{{{\text{b4}}}} } \right) + k_{{{\text{s2}}}} \left( {x_{{\text{p}}} - \xi_{4} x_{{{\text{b3}}}} - \xi_{3} x_{{{\text{b4}}}} } \right) \hfill \\ = m_{{\text{p}}} \rho_{{\text{p}}} \ddot{\theta }_{{\text{p}}} \sin \left( {\omega_{{\text{p}}} t + \theta_{{\text{p}}} } \right) + m_{{\text{p}}} \rho_{{\text{p}}} \left( {\omega_{{\text{p}}} + \dot{\theta }_{{\text{p}}} } \right)^{2} \cos \left( {\omega_{{\text{p}}} t + \theta_{{\text{p}}} } \right) + F_{{\text{f}}} \hfill \\ \end{gathered} \hfill \\ \begin{gathered} m_{{\text{p}}} \ddot{y}_{{\text{p}}} + c_{{{\text{s2}}}} \left( {\dot{y}_{{\text{p}}} - \xi_{4} \dot{y}_{{{\text{b3}}}} - \xi_{3} \dot{y}_{{{\text{b4}}}} } \right) + k_{{{\text{s2}}}} \left( {y_{{\text{p}}} - \xi_{4} y_{{{\text{b3}}}} - \xi_{3} y_{{{\text{b4}}}} } \right) \hfill \\ = m_{{\text{p}}} \rho_{{\text{p}}} \left( {\omega_{2} + \dot{\theta }_{2} } \right)^{2} \sin \left( {\omega_{2} t + \theta_{2} } \right) - m_{{\text{p}}} \rho_{{\text{p}}} \ddot{\theta }_{{\text{p}}} \cos \left( {\omega_{{\text{p}}} t + \theta_{{\text{p}}} } \right) - F_{m} - m_{{\text{p}}} g \hfill \\ \end{gathered} \hfill \\ \begin{gathered} \left( {J_{{\text{p}}} + m_{{\text{p}}} \rho_{{\text{p}}}^{2} } \right)\ddot{\theta }_{{\text{p}}} + c_{{{\text{t2}}}} \left( {\dot{\theta }_{{\text{p}}} - \dot{\theta }_{{\text{o}}} } \right) + k_{{{\text{t2}}}} \left( {\theta_{{\text{p}}} - \theta_{{\text{o}}} } \right) \hfill \\ = m_{{\text{p}}} \rho_{{\text{p}}} \sin \left( {\omega_{{\text{p}}} t + \theta_{{\text{p}}} } \right)\ddot{x}_{{\text{p}}} - m_{{\text{p}}} \rho_{{\text{p}}} \cos \left( {\omega_{{\text{p}}} t + \theta_{{\text{p}}} } \right)\ddot{y}_{2} - F_{{\text{m}}} r_{{{\text{b2}}}} - T_{{{\text{f2}}}} \hfill \\ \end{gathered} \hfill \\ \end{array} } \right.,$$
(4)

where \(t\) is the time; \(m_{{\text{g}}}\) and \(m_{{\text{p}}}\) are, respectively, the masses of the gear and pinion; \(J_{{\text{g}}}\) and \(J_{{\text{p}}}\) represent their moment of inertia; \(\rho_{{\text{g}}}\) and \(\rho_{{\text{p}}}\) are their eccentricity values; \(\omega_{{\text{g}}}\) and \(\omega_{{\text{p}}}\) are their rotational angular speeds;\(c_{{{\text{s}}i}} (i = 1,{ 2)}\) represent the bending damping of the gear and pinion shafts, and \(k_{{{\text{s}}i}} (i = 1,{ 2)}\) are their bending stiffness; \(c_{{{\text{t}}i}} (i = 1,{ 2)}\) represent their torsion damping, and \(k_{{{\text{t}}i}} (i = 1,{ 2)}\) are their torsion stiffness. \(\xi_{i} (i = 1, \, 2, \, 3, \, 4)\) represent the deformation coefficient of the shafts, and \(\xi_{i}\) equal 0.5 in this paper; \(F_{{\text{m}}}\) is the mesh force; g represents the acceleration of gravity; \(T_{{{\text{f}}i}} (i = 1, \, 2)\) [31] represents the friction moment of the gear and pinion pairs; \(F_{{\text{f}}}\) is the friction force, which can be obtained by the following equation:

$$F_{{\text{f}}} = \mu \lambda F_{{\text{m}}} ,$$
(5)

where \(\mu\) is the friction coefficient; \(\lambda\) is the direction coefficient of friction, and it is defined to be equal to 1 when the position of the mesh point does not exceed the position of the pitch line on the tooth surface, equal to 0 when they coincide, and equal to −1 in other cases. The mesh force \(F_{{\text{m}}}\) can be calculated by the following equation:

$$F_{{\text{m}}} { = }k_{{\text{m}}} f(\sigma (t),b) + c_{{\text{m}}} \dot{\sigma }(t),$$
(6)

where \(k_{{\text{m}}}\) is the mesh stiffness; \(c_{{\text{m}}}\) is the mesh damping; \(f(\sigma (t),b)\) is a nonlinear function in terms of the backlash b, and it can be expressed by [35]:

$$f(\Omega (t),b) = \left\{ {\begin{array}{*{20}l} {\Omega (t) - (1 - \psi )b} \hfill \\ {\psi b} \hfill \\ {\Omega (t) + (1 - \psi )b} \hfill \\ \end{array} } \right.\begin{array}{*{20}c} {\Omega (t) > b} \\ { - b < \Omega (t) < b,0 \le \psi \le 1} \\ {\Omega (t) < - b} \\ \end{array} ,$$
(7)

where \(\psi\) is the coefficient, and the range of its values is [0, 1); \(\Omega (t)\) represents the total deformation along the mesh action line between a pair of teeth, and its expression is:

$$\Omega (t) \, = (r_{{{\text{b}}1}} \varphi_{{\text{g}}} - r_{{{\text{b}}2}} \varphi_{{\text{p}}} ) + (y_{{\text{g}}} - y_{{\text{p}}} )\sin \alpha { + }(x_{{\text{g}}} - x_{{\text{p}}} )\cos \alpha - e,$$
(8)

where \(r_{{{\text{b}}i}} (i = 1,{ 2)}\) are the radii of the base circle; \(e\) represent the transmission error; \(\varphi_{{\text{g}}}\) and \(\varphi_{{\text{p}}}\) are the angular displacements of the gear and pinion, and their expressions can be written by:

$$\varphi_{{\text{g}}} = \omega_{{\text{g}}} t + \theta_{{\text{g}}} , \, \varphi_{{\text{p}}} = \omega_{{\text{p}}} t + \theta_{{\text{p}}} .$$
(9)

The differential equations of the input rotor and output rotor are:

$$\left\{ {\begin{array}{*{20}l} {J_{{\text{i}}} \ddot{\theta }_{{\text{i}}} + c_{{{\text{t1}}}} \left( {\dot{\theta }_{{\text{i}}} - \dot{\theta }_{{\text{p}}} } \right) + k_{{{\text{t1}}}} \left( {\theta_{{\text{i}}} - \theta_{{\text{p}}} } \right) = T_{{\text{i}}} } \hfill & {} \hfill \\ {J_{{\text{o}}} \ddot{\theta }_{{\text{o}}} + c_{{{\text{t2}}}} \left( {\dot{\theta }_{{\text{o}}} - \dot{\theta }_{{\text{g}}} } \right) + k_{{{\text{t2}}}} \left( {\theta_{{\text{o}}} - \theta_{{\text{g}}} } \right) = - T_{{\text{o}}} } \hfill & {} \hfill \\ \end{array} } \right.,$$
(10)

where \(J_{{\text{i}}}\) and \(J_{{\text{o}}}\) stand for the moment of inertia of the input and output; \(T_{{\text{i}}}\) and \(T_{{\text{o}}}\) denote the load torque, and they are set to a fixed value.

Time-Varying Friction Coefficient and Backlash

The problem of time-varying friction persists during the meshing of a gear pair. Because the relative sliding speed \(v_{{{\text{ss}}}}\), slip–roll ratio \(\eta_{{{\text{sr}}}}\), tooth contact stress \(P_{{{\text{cs}}}}\), and other parameters will cause the tooth surface friction coefficient to change. Friction forces of teeth surface have a time-varying characteristic. They can be calculated by a meshing model of a pair of teeth as shown in Fig. 3.

Fig. 3
figure 3

Tooth meshing diagram

Full size image

According to the gear pair meshing characteristics, it is known that the velocities of the driving and driven gears are equal along the common normal direction, and the expression of the relative sliding speed \(v_{{{\text{ss}}}}\) can be written by:

$$v_{{{\text{ss}}}} = \left| {\frac{{m_{{\text{g}}} z_{{\text{g}}} \omega_{{\text{g}}} \left( {z_{{\text{g}}} + z_{{\text{p}}} } \right)\cos \alpha_{0} }}{{2z_{{\text{p}}} }}\left( {\tan \alpha_{0} - \tan \alpha_{{{\text{k1}}}} } \right)} \right|,$$
(11)

where \(z_{{\text{g}}}\) and \(z_{{\text{p}}}\) are the tooth number of the gear and pinion; \(\alpha_{0}\) is the pressure angle, \(\alpha_{0} = \pi /9\); \(\alpha_{{{\text{k1}}}}\) is the meshing angle of the gear, and the range of it is from \(\alpha_{{{\text{b1}}}}\) to \(\alpha_{{{\text{a1}}}}\) as shown in Fig. 3; Their expressions can be given by:

$$\begin{array}{*{20}c} {\alpha_{{{\text{b}}1}} = \arctan \frac{{\left( {z_{{\text{g}}} + z_{{\text{p}}} } \right)\tan \alpha_{0} - z_{{\text{p}}} \tan \alpha_{{{\text{a2}}}} }}{{z_{{\text{g}}} }}} \\ {\alpha_{{{\text{a1}}}} = \arccos \frac{{z_{{\text{g}}} \cos \alpha_{0} }}{{z_{{\text{g}}} + 2h_{{\text{a}}}^{ * } }},\alpha_{{{\text{a2}}}} = \arccos \left( {r_{{{\text{b2}}}} /r_{{{\text{a2}}}} } \right) \, } \\ \end{array} ,$$
(12)

where \(\alpha_{{{\text{a1}}}}\) and \(\alpha_{{{\text{a2}}}}\) are the pressure angles of the addendum circles of the gear and pinion; \(h_{a}^{ * }\) represents the addendum coefficient; \(r_{{{\text{a2}}}}\) is the addendum circle radius of the gear.

The slip–roll ratio \(\eta_{{{\text{sr}}}}\) is given by:

$$\eta_{{{\text{sr}}}} = 2\left| {\frac{{\left( {z_{{\text{g}}} + z_{{\text{p}}} } \right)\left( {\tan \alpha_{0} - \tan \alpha_{{{\text{k1}}}} } \right)}}{{\left( {z_{{\text{g}}} + z_{{\text{p}}} } \right)\tan \alpha_{0} + \left( {z_{{\text{p}}} - z_{{\text{g}}} } \right)\tan \alpha_{{{\text{k1}}}} }}} \right|.$$
(13)

Tooth contact stress \(P_{{{\text{cs}}}}\) is given by:

$$P_{{{\text{cs}}}} = \sqrt {\frac{{F_{{\text{M}}} }}{{\pi R_{{\text{k}}} \left( {\frac{{1 - \upsilon_{{\text{g}}}^{2} }}{{E_{{\text{g}}} }} + \frac{{1 - \upsilon_{{\text{p}}}^{2} }}{{E_{{\text{p}}} }}} \right)}}} ,$$
(14)

where \(E_{{\text{g}}}\) and \(E_{{\text{p}}}\) are the Young's modulus of the gear and pinion; \(\upsilon_{{\text{g}}}\) and \(\upsilon_{{\text{p}}}\) are their Poisson's ratio; \(F_{{\text{M}}}\) represents the unit load on the meshing point M. Based on assumption of uniform load on tooth surface, unit load coefficient equals 1/2 when double teeth mesh and it equals 1 when a single tooth meshes. The expression of the unit load \(F_{{\text{M}}}\) can be written by:

$$\left\{ \begin{gathered} F_{{\text{M}}} = \frac{{F_{{\text{m}}} }}{{2{\text{L}}}},{\text{ if double teeth}} \hfill \\ F_{{\text{M}}} = \frac{{F_{{\text{m}}} }}{{\text{L}}},{\text{ if single tooth }} \hfill \\ \end{gathered} \right.,$$
(15)

where L is the tooth width.

Time-varying friction coefficient can be given by [35]:

$$\left\{ \begin{gathered} \mu = e^{{f\left( {\eta_{{{\text{sr}}}} ,P_{{{\text{cs}}}} ,v_{0} ,S_{{\text{f}}} } \right)}} P_{{{\text{cs}}}}^{{b_{2} }} \left| {\eta_{{{\text{sr}}}} } \right|^{{b_{3} }} V_{{\text{M}}}^{{b_{6} }} v_{0}^{{b_{7} }} R_{k}^{{b_{8} }} \hfill \\ f(\eta_{{{\text{sr}}}} ,P_{{{\text{cs}}}} ,v_{0} ,S_{{\text{f}}} ) = b_{1} + b_{4} \left| {\eta_{{{\text{sr}}}} } \right|P_{{{\text{cs}}}} \log_{10} \left( {v_{0} } \right) + b_{5} e^{{ - \left| {\eta_{{{\text{sr}}}} } \right|P_{{{\text{cs}}}} \log_{10} \left( {v_{0} } \right)}} + b_{9} e^{{s_{{\text{f}}} }} \hfill \\ \end{gathered} \right.,$$
(16)

where \(v_{{0}}\) represents the absolute viscosity; \(b_{i} (i = 1,2, \cdots ,9)\) represent regression coefficients, and their values are dependent on the lubricant types [36]; \(S_{{\text{f}}}\) represents the surface roughness. The surface roughness of the gear tooth often has different microstructure characteristics by machining accuracy, processing method, and surface wear. Since the backlash is close in magnitude to the roughness, this leads to a change in the backlash b between the gear teeth as shown in Fig. 1. Numerous studies have shown that the fractal theory can respond to the gear tooth morphology. A modified Weierstrass–Mandelbrot fractal backlash model is established to calculate a backlash increment \(\Delta {\text{b}}_{{\text{f}}}\), and this function model expression of the calculation random amplitude of the tooth profile can be given by [37]:

$$z(h)\, = \,G^{D - 1} \sum\limits_{{n = n_{\min } }}^{{{}^{n}\max }} {\frac{{\cos \left( {2\pi \gamma^{n} h + \kappa_{n} } \right)}}{{\gamma^{(2 - D)n} }}} ,(1 < D < 2,\gamma = 1.5),$$
(17)

where G is the fractal characteristic scale coefficient, and it can reflect the scale of random amplitude; D represents the fractal dimension which accurately reflects the complexity and irregularity of the random amplitude of the tooth profile; \(\gamma^{n}\) stands for the spatial frequency, and the value of \(\gamma\) is taken as 1.5 because it is adapted to the high-density spectra and the randomness of the phase in general; \(\kappa_{n}\) denotes the random phase; n is frequency index of asperities. h denotes the horizontal coordinate of a tooth profile. When two gears mesh, because each pair of teeth alternately touches, and the meshing position has a time-varying characteristic, the microscopic height of the surface profile is always different at each moment. The time series t is introduced into Eq. (16) to simplify the calculation. The backlash increment \(\Delta {\text{b}}_{{\text{f}}}\) can be expressed as:

$$\Delta b_{f} = z(h,t) = G^{D - 1} \sum\limits_{{n = n_{\min } }}^{{^{n} \max }} {\frac{{\cos \left( {2\pi \gamma^{n} h + \kappa_{n} ,t} \right)}}{{\gamma^{(2 - D)n} }}} ,(1 < D < 2,\gamma = 1.5).$$
(18)

Eventually, the time-varying backlash b with the initial value b0 is written as:

$$b = b_{0} + \Delta b_{{\text{f}}} .$$
(19)

TVMS with Friction and Crack

Based on the potential energy methods by Tian [6], a calculation model of the TVMS was established as shown in Fig. 4, and the stiffness of gear fillet-foundation deflection \(k_{{\text{f}}}\) was considered in this model. Thus, the total effective TVMS of a single tooth meshing can be written as:

$$k_{{\text{m}}} = \frac{1}{{1/k_{{\text{h}}} + 1/k_{{{\text{bg}}}} + 1/k_{{{\text{sg}}}} + 1/k_{{{\text{ag}}}} + 1/k_{{{\text{bp}}}} + 1/k_{{{\text{sp}}}} + 1/k_{{{\text{ap}}}} + 1/k_{{\text{f}}} }},$$
(20)

where \(k_{{\text{h}}}\) stands for the Hertzian stiffness; \(k_{{{\text{b}}i}} (i = {\text{g}},{\text{p}})\) denotes the bending stiffness; \(k_{{{\text{s}}i}} (i = {\text{g}},{\text{p}})\) are the shear stiffnesses; \(k_{{{\text{a}}i}} (i = {\text{g}},{\text{p}})\) represents the axial compressive meshing stiffness; subscripts p and g, respectively, denote the driving and driven gears.

Fig. 4
figure 4

Diagram of the calculation parameters of a gear tooth [5, 6]

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When the double teeth mesh, the total effective TVMS is written as:

$$k_{{\text{m}}} = \frac{1}{{1/k_{{{\text{h, }}i}} + 1/k_{{{\text{bg, }}i}} + 1/k_{{{\text{sg, }}i}} + 1/k_{{{\text{ag, }}i}} + 1/k_{{{\text{bp, }}i}} + 1/k_{{{\text{sp, }}i}} + 1/k_{{{\text{ap, }}i}} + 1/k_{{{\text{f, }}i}} }}(i = 1,{ 2),}$$
(21)

where i = 1 stands for the first pair tooth; i = 2 represents the second pair tooth.

According to the Hertzian contact theory, the Hertzian stiffness \(k_{{\text{h}}}\) can be calculated by:

$$k_{{\text{h}}} = \frac{{{\uppi }E_{{\text{g}}} L}}{{4(1 - \upsilon_{{\text{g}}}^{2} )}}.$$
(22)

The stiffness of gear fillet-foundation deflection \(k_{{\text{f}}}\) is calculated by [38]:

$$\frac{1}{{k_{{\text{f}}} }} = \frac{{\cos^{2} \alpha_{1} }}{{E_{{\text{g}}} L}}\left[ {L^{*} \left( {\frac{{u_{f} }}{{S_{f} }}} \right)^{\infty } + M^{*} \left( {\frac{{u_{f} }}{{S_{f} }}} \right) + P^{*} \left( {1 + Q^{*} \tan^{2} \alpha_{1} } \right)} \right].$$
(23)

A schematic diagram of the friction forces during gear meshing is shown in Fig. 5. \(F_{{{\text{fg}}}}\) and \(F_{{{\text{fg}}}}\) stand for the friction force of the driving and driven gears, respectively. \(F_{{{\text{ng}}}}\) and \(F_{{{\text{np}}}}\) are the normal contact forces. The change of friction force during the whole meshing process can be divided into three stages. Firstly, the approach process stage is shown in Fig. 5a. Secondly, the no-friction stage is shown in Fig. 5b, which represents that meshing point M coincides with the pitch point. Finally, the recess process stage is shown in Fig. 5c. The above three states can be determined by the relative positions of the meshing point and pitch point.

Fig. 5
figure 5

Schematic diagram of the friction forces

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Based on the analysis of Fig. 5, the values of radial force \(F_{{\text{r}}}\) and tangential force \(F_{{\text{t}}}\) on the gear tooth are shown in Table 1 and Fig. 6.

Table 1 Calculation of radial force \(F_{{\text{r}}}\) and tangential force \(F_{{\text{t}}}\)
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Fig. 6
figure 6

Schematic diagram of the friction force direction of a gear tooth

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\(F_{{{\text{fg}}}} = \mu F\).

According to the potential energy method, the shear, bending, and axial compressive meshing stiffness are, respectively, deduced to [39]:

$$\frac{1}{{k_{{\text{s}}} }} = \left\{ {\begin{array}{*{20}l} {\int_{{ - \alpha _{1} }}^{{\alpha _{2} }} {\frac{{1.2(1 + v)\left( {\cos \alpha _{1} - \mu \sin \alpha _{1} } \right)^{2} \left( {\alpha _{2} - \alpha } \right)\cos \alpha }}{{EL\left[ {\sin \alpha + \left( {\alpha _{2} - \alpha } \right)\cos \alpha } \right]}}{\text{d}}\alpha } ,{\text{ if approach - process}}} \hfill \\ {\int_{{ - \alpha _{1} }}^{{\alpha _{2} }} {\frac{{1.2(1 + v)\left( {\cos \alpha _{1} + \mu \sin \alpha _{1} } \right)^{2} \left( {\alpha _{2} - \alpha } \right)\cos \alpha }}{{EL\left[ {\sin \alpha + \left( {\alpha _{2} - \alpha } \right)\cos \alpha } \right]}}{\text{d}}\alpha } ,{\text{ if recess - process}}} \hfill \\ \end{array} } \right.$$
(24)
$$\frac{1}{{k_{{\text{b}}} }} = \left\{ {\begin{array}{*{20}l} {\int_{{ - \alpha _{1} }}^{{\alpha _{2} }} {\frac{{3\left[ {\left\{ {1 + \cos \alpha _{1} \left\{ {\left( {\alpha _{2} - \alpha } \right)\sin \alpha - \cos \alpha } \right\}} \right\} - \mu \left\{ {\alpha _{1} + \alpha _{2} + \sin \alpha _{1} \left\{ {\left( {\alpha _{2} - \alpha } \right)\sin \alpha - \cos \alpha } \right\}} \right\}} \right]^{2} \left( {\alpha _{2} - \alpha } \right)\cos \alpha }}{{2EL\left[ {\sin \alpha + \left( {\alpha _{2} - \alpha } \right)\cos \alpha } \right]^{3} }}{\text{d}}\alpha } ,{\text{ if approach - process}}} \hfill \\ {\int_{{ - \alpha _{1} }}^{{\alpha _{2} }} {\frac{{3\left[ {\left\{ {1 + \cos \alpha _{1} \left\{ {\left( {\alpha _{2} - \alpha } \right)\sin \alpha - \cos \alpha } \right\}} \right\} + \mu \left\{ {\alpha _{1} + \alpha _{2} + \sin \alpha _{1} \left\{ {\left( {\alpha _{2} - \alpha } \right)\sin \alpha - \cos \alpha } \right\}} \right\}} \right]^{2} \left( {\alpha _{2} - \alpha } \right)\cos \alpha }}{{2EL\left[ {\sin \alpha + \left( {\alpha _{2} - \alpha } \right)\cos \alpha } \right]^{3} }}{\text{d}}\alpha } ,{\text{ if recess - process}}} \hfill \\ \end{array} } \right.$$
(25)
$$\frac{1}{{k_{{\text{a}}} }} = \left\{ {\begin{array}{*{20}l} {\int_{{ - \alpha _{1} }}^{{\alpha _{2} }} {\frac{{\left( {\alpha _{2} - \alpha } \right)\cos \alpha \left( {\sin \alpha _{1} + \mu \cos \alpha _{1} } \right)^{2} }}{{2EL\left[ {\sin \alpha + \left( {\alpha _{2} - \alpha } \right)\cos \alpha } \right]}}{\text{d}}\alpha } ,{\text{ if approach - process}}} \hfill \\ {\int_{{ - \alpha _{1} }}^{{\alpha _{2} }} {\frac{{\left( {\alpha _{2} - \alpha } \right)\cos \alpha \left( {\sin \alpha _{1} - \mu \cos \alpha _{1} } \right)^{2} }}{{2EL\left[ {\sin \alpha + \left( {\alpha _{2} - \alpha } \right)\cos \alpha } \right]}}{\text{d}}\alpha } ,{\text{ if recess - process}}} \hfill \\ \end{array} } \right.$$
(26)

As a crack generates in the gear system, the expressions of the above meshing stiffness can be written as:

$$\frac{1}{{k_{s} }} = \left\{ {\begin{array}{*{20}l} {\int\limits_{{ - \alpha_{1} }}^{{\alpha_{2} }} {\frac{{2.4(1 + v)\left( {\alpha_{2} - \alpha } \right)\cos \alpha \left( {\cos \alpha_{1} - \mu \sin \alpha_{1} } \right)^{2} \cos^{2} \theta }}{{EL\left[ {\sin \alpha_{2} - \frac{q}{{R_{b2} }}\sin \vartheta + \sin \alpha + \left( {\alpha_{2} - \alpha } \right)\cos \alpha } \right]}}} d\alpha ,} \hfill \\ {\int\limits_{{ - \alpha_{1} }}^{{\alpha_{2} }} {\frac{{2.4(1 + v)\left( {\alpha_{2} - \alpha } \right)\cos \alpha \left( {\cos \alpha_{1} + \mu \sin \alpha_{1} } \right)^{2} \cos^{2} \theta }}{{EL\left[ {\sin \alpha_{2} - \frac{q}{{R_{b2} }}\sin \vartheta + \sin \alpha + \left( {\alpha_{2} - \alpha } \right)\cos \alpha } \right]}}} d\alpha } \hfill \\ \end{array} } \right.,$$
(27)
$$\frac{1}{{k_{{\text{b}}} }} = \left\{ {\begin{array}{*{20}c} {\int _{{ - \alpha _{1} }}^{{\alpha _{2} }} \frac{{12\left[ \begin{gathered} \left\{ {1 + \cos \alpha _{1} \left\{ {\left( {\alpha _{2} - \alpha } \right)\sin \alpha - \cos \alpha } \right\}} \right\} - \mu \left\{ {\alpha _{1} + \alpha _{2} + \sin \alpha _{1} \left\{ {\left( {\alpha _{2} - \alpha } \right)\sin \alpha - \cos \alpha } \right\}} \right\} \hfill \\ - 2\sin ^{2} \theta {\text{/}}2\left\{ {\sin \alpha _{1} \cos \alpha _{1} \left( {\alpha _{1} + \alpha _{2} {\text{ - }}\mu } \right){\text{ + }}\cos ^{2} \alpha _{1} {\text{ - }}\mu \left( {\sin ^{2} \alpha _{1} \left( {\alpha _{1} + \alpha _{2} } \right)} \right)} \right\} \hfill \\ \end{gathered} \right]^{2} \left( {\alpha _{2} - \alpha } \right)\cos \alpha }}{{EL\left[ {\sin \alpha _{2} - \frac{q}{{R_{{b2}} }}\sin \vartheta + \sin \alpha + \left( {\alpha _{2} - \alpha } \right)\cos \alpha } \right]^{3} }}d\alpha ,{\text{ if approach - process}}} \\ {\int _{{ - \alpha _{1} }}^{{\alpha _{2} }} \frac{{12\left[ \begin{gathered} \left\{ {1 + \cos \alpha _{1} \left\{ {\left( {\alpha _{2} - \alpha } \right)\sin \alpha - \cos \alpha } \right\}} \right\} + \mu \left\{ {\alpha _{1} + \alpha _{2} + \sin \alpha _{1} \left\{ {\left( {\alpha _{2} - \alpha } \right)\sin \alpha - \cos \alpha } \right\}} \right\} \hfill \\ - 2\sin ^{2} \theta {\text{/}}2\left\{ {\sin \alpha _{1} \cos \alpha _{1} \left( {\alpha _{1} + \alpha _{2} + \mu } \right){\text{ + }}\cos ^{2} \alpha _{1} + \mu \left( {\sin ^{2} \alpha _{1} \left( {\alpha _{1} + \alpha _{2} } \right)} \right)} \right\} \hfill \\ \end{gathered} \right]^{2} \left( {\alpha _{2} - \alpha } \right)\cos \alpha }}{{EL\left[ {\sin \alpha _{2} - \frac{q}{{R_{{b2}} }}\sin \vartheta + \sin \alpha + \left( {\alpha _{2} - \alpha } \right)\cos \alpha } \right]^{3} }}d\alpha ,{\text{ if recess - process}}} \\ \end{array} } \right.$$
(28)

where q is the crack length; \(\vartheta\) stands for the crack angle.

Results and Analysis

Effect of Roughness

The parameters of the gear and pinion are shown in Table 2. Based on the friction and fractal backlash theory, the roughness of the tooth surface is used as a link, and the relationship between the time-varying friction coefficient and the dynamic backlash can be established. Meanwhile, it can be used as the controlled variation value to investigate the effect of both on the dynamic characteristics of the spur gear model.

Table 2 Gear parameters
Full size table

Firstly, the roughness of the tooth surface is set, respectively, to 0.45 μm, 0.84 μm, and 1.40 μm. Based on Eq. (16), the time-varying friction coefficient can be shown in Fig. 7. It is seen that the friction coefficient μ is changed by the roughness \(S_{{\text{f}}}\). The variation degree is small when the range of \(S_{{\text{f}}}\) is [0, 0.84], and it is great when \(S_{{\text{f}}}\) > 0.84 because of Eq. (16). If \(S_{{\text{f}}}\) equals to 0 μm, the friction coefficient will be set to 0 and the backlash between the two teeth also is set to \(2 \times 10^{ - 5}\) mm in this section. Then, according to the fractal theory and the roughness, the fractal characteristic scale coefficient \(G\) and the fractal dimension \(D\) can be determined and they can be given as shown in Table 3. Because of the self-similarity of microscopic morphology and the difference of meshing contact point at each time, the dynamic backlash caused by microscopic morphology can be calculated by Eq. (19) as shown in Fig. 8. Three dynamic backlash curves vary with the roughness \(S_{{\text{f}}}\). It can be found that the tooth profile curves and microtopography of a gear tooth yielded through the fractal method agree well. Meanwhile, other scholars have confirmed this [40].

Fig. 7
figure 7

Time-varying friction coefficient

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Table 3 Comparison chart between the fractal parameters and the roughness [41]
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Fig. 8
figure 8

Dynamic backlash

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After obtaining the time-varying friction coefficient, the simulated TVMS with time-varying friction coefficient can be calculated by Eqs. (26), (27) and (28) as shown in Fig. 9. \(\mu_{i}^{j} (i = A,B,C;j = {\text{double,single}})\) denotes the time-varying friction coefficient of a gear pair. A, B, and C represent the three adjacent gear pairs. In Fig. 9, it can be found that the TVMS decreases with the increase of the roughness at the double teeth meshing region, and this phenomenon is the most obvious when the roughness is equal to 1.40 μm. This is because the friction coefficient is positively correlated with \(S_{{\text{f}}}\) and is large at \(S_{{\text{f}}}\) = 1.40 μm. In the single tooth meshing region, the TVMS goes up first and then down. This is because of the difference in the friction direction in front of and behind the pitch point.

Fig. 9
figure 9

Time-varying meshing stiffness and friction coefficient with different roughness

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The spur gear system dynamic model is employed for investigating the vibration characteristics of the system with time-varying friction and dynamic backlash. The input element is the pinion. The input torque is set to 110 Nm, and the rotational speed of the driving pinion equals 500r/min. The vibration differential equations of the gear system can be solved by a fixed step fourth-order Runge–Kutta numerical calculation method.

The vibration characteristics of the system in the \(y_{1}\)-direction are shown in Figs. 10 and 11. The time-domain responses are given in Fig. 10a, b. It can be seen from Fig. 10a that the time-domain waveform shows non-periodic characteristics with the change of roughness. The drastic fluctuation of amplitude is due to both effects caused by the dynamic backlash and the nonlinear function of the deformation alone gear mesh action line, whereas, as shown in Fig. 10b, a slight difference at \(S_{{\text{f}}} \le\) 0.84 μm is due to the small change of the friction coefficient and backlash. Because it is difficult to analyze the influence of the two time-varying nonlinear factors on the vibration characteristics of the system from the time domain, the frequency domain, phase plane diagram, Poincaré map, and time–frequency-domain are used to investigate both effects.

Fig. 10
figure 10

Vibration displacement in yhe \(y_{1}\)-direction and its locally enlarged image

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Fig. 11
figure 11

a1–a4 frequency spectra; b1–b4 phase plane diagram and Poincaré map; c1–c4 time–frequency figure. The numbers are explained as follows: 1 represents \(S_{{\text{f}}}\) = 0μm; 2 represents \(S_{{\text{f}}}\) = 0.45μm; 3 represents \(S_{{\text{f}}}\) = 0.84μm; 4 represents \(S_{{\text{f}}}\) = 1.4μm

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Figure 11 compares the vibration responses of the system under different roughness. The abscissa of Fig. 11(a1–a4) ranges from 0 to 2000 Hz to show the mesh frequency, harmonics frequency, sideband, and so on. In Fig. 11(a1), the dominant frequency component is the meshing frequency \(f_{{\text{m}}}\) (\(n_{{\text{p}}} z_{{\text{p}}} /60 = 458.3\) Hz) and its harmonics frequency \(2f_{{\text{m}}}\), \(3f_{{\text{m}}}\), \(4f_{{\text{m}}}\). The frequency component is relatively simple. This is because the backlash is fixed value and the friction is ignored. As can be seen from Fig. 11(b1), the closed curve and a few points reveal the dynamical behavior of the system. The system is in periodic motion. In addition, the meshing frequency \(f_{{\text{m}}}\) and its harmonics frequency \(2f_{{\text{m}}}\), \(3f_{{\text{m}}}\) can be seen clearly in Fig. 11(c1). Their amplitudes regularly decrease. Although, in this system, the low-frequency shock is slight, the shocking phenomenon is obvious in the high-frequency region. This is because the stiffness excitation caused by single and double teeth alternately cause meshing change. If the design strength is reasonable, the gear will eventually fail due to tooth surface pitting caused by fatigue in this theoretical condition, and it will only occur later than the design life.

In Fig. 11(a2–a4), both dynamic backlash and time-varying friction coefficient can cause the frequency components in the spectra to become more luxuriant. It can be found that, in addition to the dominant frequency \(f_{{\text{m}}}\) and its harmonics frequency, \(f_{i} (i = 1,2, \cdots ,12)\) appear, their amplitudes enhance with the increase of the roughness, and their frequency values are 17.9 Hz, 26.9 Hz, 39.4 Hz, 59.1 Hz, 89.6 Hz, 133.4 Hz, 200.5 Hz, 300.8 Hz, 451.2 Hz, 667.7 Hz, 1016 Hz and 1524 Hz. These are the same as the frequency components from the spectrum of the dynamic backlash. As shown in Eq. (18), the same sampling frequency as the displacement signal can be used to obtain its spectrum as shown in Fig. 12 Although the frequency components are the same, the differences in the amplitudes are due to the modulation of the system and function. In addition, the frequency sidebands appear near above frequency components caused by the increase of the friction coefficient. The frequency spectrum from the friction force signal is shown in Fig. 13. As can be seen in Fig. 12(b2–b4), it can be found that a number of of closed curves overlap each other and the red points diverge as the surface roughness increases. The system develops from multi-periodic motion to chaotic motion with the increase of the roughness.

Fig. 12
figure 12

Frequency spectrum of the dynamic backlash at \(S_{{\text{f}}} = 1.40\) μm

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Fig. 13
figure 13

Frequency spectrum of the time-varying friction force at \(S_{{\text{f}}} = 1.40\) μm

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Figure 11(c2–c4) compares the time–frequency domain characteristics under different roughness. It can be found that meshing frequency and its low multiplication components are changed from a stable stage to a fluctuating shock stage. This is because the dynamic backlash plays an important role in the calculation of the meshing force. The degree of fluctuation in the low-frequency region enhances with the increase of the roughness. High-amplitude shock frequencies are dispersed in the high-frequency region instead of being concentrated around the harmonic frequency. This is because the time-varying friction coefficient surges at \(S_{{\text{f}}} = 1.40\) μm. Therefore, the enhancement of nonlinear factors makes the time–frequency graph of the system appear as fish-scale fluctuation. The fluctuation is sensitive to the dynamic backlash, whereas the richness of the frequency component is sensitive to the time-varying friction coefficient.

Effect of Crack

A crack defect is an easy fault that occurs [7]. According to the precision class of gears, lubricating oil parameters, and friction theory, the tooth surface roughness is set to 0.45 μm, and we only consider that a crack appears in the tooth root and the crack angle is 45 degrees. Figure 14 shows the TVMS at the different crack lengths. Combined with the small figure, the TVMS of the single meshing region is influenced by the time-varying friction obviously. Figure 15 shows an FEM model of a crack gear pair and crack grid condition. By comparing Fig. 15a and b, the correctness of the calculated crack stiffness is verified.

Fig. 14
figure 14

TVMS at different crack length:s (a) theoretical results, (b) FEM results

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Fig. 15
figure 15

The FEM model of the root crack

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A dynamic model of a spur gear system with a root crack is established to study the vibration characteristics of the crack system, and the dynamic backlash and time-varying friction are considered in this model to simulate the signal realistically. The input torque is set to 110 Nm, and the rotational speed of the driving pinion equals 500r/min. To further investigate the transient frequency features of the system involved crack faults, the synchrosqueezed wavelet transform [42, 43] method was used to obtain the time–frequency response with good resolution and high aggregation as shown in Fig. 16. It can be seen that an obvious shock is caused by a crack in the high-frequency region in Fig. 16(a1) and (a2). Combined with the time-domain diagrams, the waveforms are similar. But the low-frequency regions in the time–frequency domain are different. As can be seen in Fig. 16(b1), (b2), (c1), and (c2), the low-frequency fluctuation can be enhanced by the crack defects, and the longer the crack, the stronger is the effect. Therefore, the time-varying nonlinear factor can make the vibration signals more complicated than without the nonlinear factor. It is difficult to diagnose the damage extent because of them. However, by comparing Fig. 17(a1), (a2), and (a3), it can be found that the richness of the frequency component at the crack location can be used as the basis to diagnose the fault degree of the crack based on the pure simulation signal. But low frequency is very easy to be affected by external factors, but in actual engineering may not be able to have such a strong degree of differentiation. In “Experimental verification”, the actual experimental signal can be seen and their responses will be analyzed.

Fig. 16
figure 16

Time and time–frequency responses from the simulation model

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Fig. 17
figure 17

Experimental equipment and arrangement

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In addition, in the early crack failure response, the shock caused by the crack can quickly revert to the steady-state vibration, as shown in Fig. 16(a1, a2). The region affected by the crack is three mesh cycles. When the crack length is set to 3 mm or 4 mm. This region is increased to nine mesh cycles, as can be seen in Fig. 16(b1–2, c1–2). It illustrates that cracks can disrupt the stability of the system. In the actual project, the vibration and noise of the equipment are amplified by the increase of crack length.

Experimental Verification

The effect of a crack on the gear system was investigated by the above nonlinear dynamic model involving time-varying friction and dynamic backlash. In this section, the vibration response results from the model can be verified by the experimental methods.

Experimental Equipment

A gear-rotor transmission system equipment is shown in Fig. 17. The motor provides power to the pinion through a belt drive. An acceleration sensor was set in the vertical direction and the location is shown in the figure. Two data acquisition cards NI9234 were used. A self-developed program using Labview was used to record the data. In addition, gears were lubricated with oil, and the lubrication oil grade was 32#. The surface roughness of the gear tooth was 0.8 μm. The inter-temperature was 50 ℃, and the variation was ignored. A crack was set at the tooth root of the pinion at the output shaft, and the crack angle was 45°. The rotation speed of the input shaft was set to 1000r/min. The cracks were, respectively, set to 1 mm and 4 mm.

Crack Defect

To verify the accuracy of the gear system model with a crack, the time-domain responses of the pinion in the y-direction from simulation and experiment are compared in Figs. 18 and 19. It can be seen that the impact caused by the crack is obvious in the acceleration signal, and with the increase of cracks, the impact is also enhanced. The period of the shock \(\Delta t\) is the reciprocal of the frequency of rotation, which can be calculated by 60/n. The time-domain responses of the two are in good agreement. The correctness of the proposed model is verified.

Fig. 18
figure 18

Time-domain responses from simulated signals

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Fig. 19
figure 19

Time-domain responses from experimental signals

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The time–frequency responses are given in Figs. 20 and 21. Because the original signal of the experiment is noisy, the resolution and feature expression of the time–frequency response are seriously affected. We performed VMD (Variational mode decomposition) [44] on the original signals of the experiment to obtain IMF (intrinsic mode functions) components with clear feature expression. The penalty factor is set to 2000. IMF is analyzed in the time–frequency domain as shown in Fig. 21. It can be seen that the fault characteristics of the simulation signal and experimental signal are consistent in the time–frequency domain.

Fig. 20
figure 20

Time–frequency diagram from simulated signals

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Fig. 21
figure 21

Time–frequency diagram from experimental signals

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According to the above analysis, the simulation results are basically consistent with the experimental results, which verifies the correctness of the theoretical model and directly explains the accuracy of the simulation results.

Conclusion

In this study, the effect of a root crack on the vibration characteristics of a coupled spur gear system is researched. The main conclusions are as follows:

  • (1) The change in tooth surface roughness directly affects the friction coefficient and backlash. The system is affected by two kinds of time-varying nonlinear factors, which change both in time and frequency. But the effects of both are quite different. The large fluctuation, as can be seen from the time-domain figure, is generated due to the effect of the dynamic backlash. Meanwhile, it can also add new frequency components in the frequency domain caused by microscopic fluctuations in the tooth surface. The time-varying friction causes a disordered sideband frequency to appear near the meshing frequency and its harmonic frequency. If the time-varying nonlinear factors are considered, the time–frequency response of the system will appear as fish-scale fluctuation. The greater the roughness, the stronger is the fluctuation. This is confirmed by the response experimental signals.

  • (2) Although an early crack (1 mm) is not apparent in the time-domain waveforms, amplitudes can surge in the high-frequency region in the time–frequency-domain. In low-frequency regions, the amplitude and frequency are both changed and the change is complex due to the time-varying nonlinear factors. The extent of fluctuation in the high-frequency region can be used to diagnose the damage degree of the crack. The same conclusion can be obtained by experimental vibration responses.

  • (3) In the experimental and simulation signals, the response of two adjacent crack shock regions may not be the same. This is caused by the different meshing positions of the gear teeth, which are exactly simulated by the fractal dynamic backlash in the simulation model. In addition, the setting of experimental parameters, data acquisition, measurement and experimental arrangement provides some references for the study of gear nonlinear dynamics.