Superiority of Fault-Caused-Speed-Fluctuation-Based Dynamics Modeling: An Example on Planetary Gearbox with Cracked Sun Gear

Abstract

A planetary gear fault generates periodic speed fluctuations, which significantly influence its vibration signal. It is a necessity to explore the vibration modulation features of gear faults to provide an effective indicator for fault detection. Therefore, a superior rigid-flexible coupling dynamics model of a planetary gearbox involving the fault-caused speed fluctuation is developed, where the meshing stiffness under the impact of fault-caused speed fluctuation is innovatively deduced utilizing the potential energy method; then, the meshing stiffness is substituted into the rigid dynamics model to calculate the excitation forces. Transfer path functions from excitation locations to the sensor installed on the housing are obtained by considering the modal parameters of the flexible housing. Finally, the excitation forces are combined with their transfer path functions to calculate the vibration signal. The fault modulation features of the cracked sun gear deduced by the superior dynamics model emerge surrounding the meshing frequency and its harmonics, as well as the resonance ranges, which can be a reliable sign for identifying faults. The experiment conducted on a single-stage planetary gearbox confirms the validity and superiority of the proposed model, which holds significant value for guiding fault detection and prognosis in planetary gearboxes.

1. Introduction

Planetary gearboxes are extensively applied in numerous pieces of mechanical equipment, such as wind turbines and helicopters, as they have exceptional power density combined with a high transmission ratio in a compact package. However, prolonged operations result in gear defects, such as tooth cracks, potentially causing severe vibrations and noise, and even catastrophic accidents. In order to ensure the reliability of planetary gear systems, it is important to supervise the condition and detect gear faults in a timely manner [1,2,3].

Vibration modulation mechanisms and fault diagnosis of planetary gearboxes have captured considerable focus over the past few years [4,5,6,7,8]. Liu et al. [9] introduced a vibration signal model of a planetary gear set, which incorporated the contributing sources and the transmission path. Here, a lumped parameter model (LPM) was applied to obtain vibration sources, and the transfer path influences were divided into two components. The vibration signal model could reveal some vibration features of a healthy planetary gear set. A wind turbine planetary gearbox dynamics model integrating rigid and flexible coupling was constructed by Zhang [10], taking into account transfer paths and variable scenarios to study the vibration amplitude modulation (AM) features. Here, the housing was regarded as an elastic body. Moreover, a vibration signal phenomenon model of the planetary gearbox was developed by Lei [11], which considered the angular shift of a planet gear and the AM effect of time-varying transfer paths. Subsequently, they derived the vibration spectrum structure under nonequal planet load conditions [12]. Vibration signal models resulting from the localized fault sun gear were built by Xu [13] to reveal the resonance AM mechanism. However, the model did not consider different transfer path functions from various ring teeth engagement positions to the sensor, and it could only explain some resonance modulation and failed to reveal the modulation sidebands surrounding the meshing frequency and its harmonics caused by the localized fault sun gear. On the whole, it is observable from the above-mentioned literature that the time-varying transfer path is a primary source of vibration AM sidebands.

Research concentrated on the dynamics modeling of planetary gear sets, with tooth cracks, is also conducted widely because gears are prone to tooth cracks under poor work conditions, especially the sun gear. As a primary excitation source in the dynamics model of a gear set, mesh stiffness can be calculated by the principle of potential energy [14,15,16] or the finite element method [17,18,19]. Using the potential energy method, the mesh stiffness of a planetary gear set was analytically calculated by Liang [20]. Meng et al. [21] first computed the meshing stiffness, influenced by different crack sizes and locations, and then constructed a six-DOF dynamic gear model to research the vibration fault features. Chen et al. [22] first researched the impact of flexible ring gear on meshing stiffness utilizing the principle of potential energy and then analyzed the spectrum structures of a planetary gearbox with tooth cracks. Moshrefzadeh et al. [23] combined the LMP of a planetary gearbox and bearing model to research the sidebands of crack gear faults and different bearing component faults. Additionally, Jiang et al. [24] established an LPM of the planetary gearbox with gear cracks when considering flexible transfer path functions, which was applied to study vibration AM sidebands. However, the proposed model was unable to provide clarification for certain modulation sidebands. Han et al. [25] first computed the meshing stiffness of cracked gear to build the LPM of a planetary gear set and finally explored vibration AM sidebands. A 3D-loaded tooth contact analysis model was developed by Han [26], taking into consideration the changes in center distances and the tooth crack defect of the planet gear. The vibration response, influenced by assembly errors and the crack fault, was then determined. A dynamic model involving a cracked sun gear in a multi-stage planetary gear set was constructed by Hu [27] to research the fault mechanisms and dynamic properties of tooth cracks through an RMS curve, phase diagram, etc.

The overwhelming majority of dynamic models only consider the vibration AM factors and ignore the frequency modulation (FM) effect. However, a gear fault causes periodic speed fluctuations, which highly affect the vibration modulation sidebands. This is because the speed fluctuation is widely perceived as contributing to the FM signal [28,29]. In the works of Yang and Ding [30,31], the vibration signal models were enhanced to include periodic speed fluctuations due to distributed gear faults and tooth cracks in planetary gear sets, explaining the presence of vibration AM-FM sidebands. Moreover, factors influencing the vibration signal, such as the meshing force, projection function, and transfer path function, were all impacted by the fault-caused speed variations. Therefore, the fault-caused speed fluctuation cannot be ignored when investigating the vibration modulation features of the cracked sun gear in the planetary gear set.

The literature review reveals that the dynamics model of a planetary gearbox, considering the meshing stiffness and transfer path function, was generally employed to analyze the vibration modulation mechanism. Here, the transfer path does not involve the influence of modal parameters of flexible housing, and the meshing stiffness does not contain the impact of fault-caused periodic speed fluctuation. However, the modal parameters of flexible housing affect the AM, and fault-caused periodic speed fluctuation is the main source of FM. Additionally, the commonly used models result in some fault modulation sidebands that are inexplicable, and the fault modulation mechanism is unclear, which hinders the fault diagnosis and condition monitoring of planetary gear systems. Hence, there continues to be a shortage of extensive research on the dynamic modeling of planetary gear sets that can more accurately analyze the vibration modulation sidebands resulting from gear faults.

Targeting the issues above, a rigid-flexible coupling dynamics model of planetary gearbox integrating the flexible transfer path and rigid LPM is proposed, of which the modal parameters of flexible housing are involved in the transfer path function, and the speed fluctuation resulting from the gear fault is taken into account the meshing stiffness to establish a rigid LPM. The transfer path function is highly associated with the vibration AM signal, while the speed fluctuation resulting from the gear fault is the primary reason for the vibration FM signal. Consequently, the proposed dynamics model can be applied to explore the vibration AM-FM mechanism of gear fault. Here, the cracked sun gear, as a typical gear fault, is used to develop the dynamics model. The major contributions are as follows.

(a)

The meshing stiffness of the cracked sun gear is deduced by the potential energy method, which is represented by the rotational angle of the sun gear. Therefore, fault-caused speed fluctuation is creatively integrated into the meshing stiffness by integrating the rotation speed to obtain the rotational angle.

(b)

Modal parameters of flexible housing are considered in the transfer path, which highly affects the transfer path functions and modulation sidebands in resonance ranges.

(c)

A rigid-flexible coupling dynamics model of the planetary gear set combines fault-caused speed fluctuation and modal parameters of flexible-housing-affected transfer paths, which perfects the previous dynamics model of the planetary gear set and clearly reveals the vibration AM-FM behavior of the cracked sun gear.

2. Rigid-Flexible Coupling Dynamics Model of Planetary Gear Set with Cracked Sun Gear

The subject of this study is a planetary gear set comprising a carrier (c), a sun gear (s), three evenly positioned planet gears (p), and an unmoving ring gear (r), where the sun gear and the carrier serve respectively as the input and output. As shown in Figure 1, the sensor mounted on the housing collects the vibration signal in the Y-direction, where the vibration signal captured by the sensor comes from multiple excitation points by several transfer paths. However, some transfer paths attenuate the vibration signal greatly due to the long distances; only three transfer paths displayed in Figure 1 are thus taken into account, which are demonstrated as follows.

(a)

Path 1: Meshing force of p-r gear pair acting on ring → flexible housing → sensor.

(b)

Path 2: Excitation force acting on the bearing of the sun gear → flexible housing → sensor.

(c)

Path 3: Excitation force acting on the bearing of the carrier → flexible housing → sensor.

Figure 1. Three main transfer paths of vibration in planetary gearbox.

Figure 1. Three main transfer paths of vibration in planetary gearbox.

According to Duhamel’s integral, the vibration response of a linear system to an arbitrary excitation can be calculated by the convolution of the time-domain unit impulse response function and its corresponding excitation. Therefore, the vibration response signal y(t) acquired by the sensor installed on the housing is transmitted by the above three paths, which can be derived from

$$y(t)={\displaystyle \sum _{i=1}^{Z_r}{f}_i(t)\ast h_i(t)+}{f}_s(t)\ast h_s(t)+f_c(t)\ast h_c(t)$$

(1)

where symbol * denotes convolution; f_i(t) and h_i(t) represent the excitation force on the ith tooth of the ring gear and the unit impulse response function from the ith tooth of the ring gear to the fixed sensor, respectively; h_s(t) and h_c(t) are respectively the unit impulse response functions of paths two and three; f_s(t) and f_c(t) are the excitation forces respectively operating on the bearings of sun gear and carrier, but projected on the Y-direction namely the measurement direction of the immobile sensor.

Frequency domain vibration signal Y(f) collected by the sensor could be expressed as

$$Y(f)={\displaystyle \sum _{i=1}^{Z_r}{F}_i(f)H_i(f)}+F_s(f)H_s(f)+F_c(f)H_c(f)$$

(2)

where F_i(f), F_s(f), and F_c(f) are the frequency domain functions corresponding to f_i(t), f_s(t), and f_c(t), respectively; H_i(f), H_s(f) and H_c(f) represent the transfer path functions of paths one, two, and three in the frequency domain, respectively. Here, the excitation forces can be calculated by LPM, and the transfer path functions are obtained by the housing finite element model.

Therefore, a rigid-flexible coupling dynamics model is proposed to calculate excitation forces and transfer path functions and, finally, to obtain the vibration signal. A flowchart of the proposed model is presented in Figure 2. Speed fluctuation resulting from the cracked sun gear influences the engaged location of the gear pair and further influences meshing stiffness, which is thus introduced into the time-varying meshing stiffness. The structure parameters of the planetary gearbox and meshing stiffness are applied to establish the LPM and calculate the excitation forces acting on the ring tooth, carrier bearing, and sun gear bearing, respectively. Meanwhile, the transfer path functions, encompassing the modal parameters of the housing, are obtained through the housing finite element model. Finally, the product of excitation forces and their transfer path functions can acquire the vibration signal in the frequency domain, which can be applied to analyze the vibration AM-FM features resulting from the cracked sun gear. The LPM with speed fluctuation and the flexible transfer path functions are introduced below.

2.1. LPM Involving Fault-Caused Speed Fluctuation

2.1.1. LPM of Planetary Gear Set

An LPM illustrated in Figure 3 is adopted to model the vibration of the planetary gear set. Each element exhibits two translational, vibrational motions in the x and y directions and a single rotational vibrational motion u. The local coordinate system of each gear is attached to its carrier, while the global coordinate system is anchored to the ground. The specific structure parameters of the planetary gear set are listed in

Table 1. Structure parameters of planetary gear set.

	Planet (p)	Sun (s)	Ring (r)	Carrier (c)
Tooth number	Zp = 28	Zs = 20	Zr = 79	—
Tooth width (mm)	28	25	25	—
Mass (kg)	mn = 0.678 (n = 1, 2, 3)	ms = 0.307	mr = 1.689	mc = 4.246
Inertia (kg∙mm2)	In = 3.381 × 10−4	Is = 7.88 × 10−5	Ir = 0.016	Ic = 0.01
Pressure angle (°)	20°
Module (mm)	2.5
Support stiffness (N/m)	ks = 9.25 × 105; kr = 1.57 × 109; kc = 3.48 × 108; kp = 1.855 × 108
Torsional stiffness (N·m/rad)	kst = 3.07 × 106; krt = 1.027 × 109; kct = 6.915 × 106

. The gear meshing interface is simulated by a spring-damper system. (k_sp, c_sp), (k_rp, c_rp) represent, respectively, the meshing stiffness and damping coefficients of s-p and r-p gear pairs. k_i and k_it, c_i and c_it (i = s, p, c, r) denote respectively the radial and torsional stiffness, radial and torsional damping coefficient of each element.

Based on Newton’s Second Law, motion differential equations of s, c, r, and the nth p are established [24]. Finally, the motion differential equations are written in matrix terms as

$$M\ddot{q}+(C_b+C_m+2w_{c}G)\dot{q}+(K_b+K_m-{w_c}^2{K}_{\Omega })q=T+E$$

(3)

where M, T, and E are the mass matrix, external torque array, and error excitation array, respectively. C_b, C_m, K_b, and K_m are the matrices of support damping, meshing damping, support stiffness, and meshing stiffness, respectively. G and K_Ω denote the matrices of gyroscopic and centripetal stiffness, respectively. The specifics of each matrix can be found in Ref. [24]. q is the local coordinate array of each component, which is represented as

$$q={[x_s{y}_s{u}_s{x}_1{y}_1{u}_1{x}_2{y}_2{u}_2{x}_3{y}_3{u}_3{x}_r{y}_r{u}_r{x}_c{y}_c{u}_c]}^T$$

(4)

Time-varying meshing stiffness under the influence of fault-caused speed fluctuation is the main excitation of LPM, which can be determined through the potential energy method. Moreover, meshing damping in the LPM is regarded to be proportional to the corresponding meshing stiffness, as mentioned in Ref. [32]. Therefore, the meshing stiffness involving the fault-caused speed fluctuation is deduced below.

2.1.2. Meshing Stiffness Considering Fault-Caused Speed Fluctuation

• The meshing stiffness of the cracked sun gear calculated by the potential energy method

Friction is neglected in this analysis due to the assumption of ideal lubrication conditions. Take the common external gear whose base circle is smaller than the root circle as the research object. The equivalent cantilever beam model of the cracked tooth on a sun gear is displayed in Figure 4, whose outline is involute. The fixed end of the equivalent cantilever beam is the vertical face of the tooth surface where the crack ends. In Figure 4, the purple line is the propagation path of the crack, which starts from the intersection N between the root circle and tooth involute profile, then extends with the angle v. When the crack does not stretch to the median line of the gear tooth, the length of crack NA is set as q₁. While the crack reaches the centerline of the gear tooth, the crack spreads at the same angle v toward the other tooth root. The length of crack DE is set as q₂. Point M is the intersection of the tooth involute profile and the base circle, where straight line NM is employed as a replacement for the transition curve between point N and point M. The line of action of meshing point B is tangential to the base circle at point F, and the line of action of arbitrary meshing point C is tangential to the base circle at point G. α₁ represents the pressure angle at engagement point B; α₂ is the half base tooth angle; α_r is the half-tooth angle at the intersection point between the fixed end of the equivalent cantilever beam and the root circle; α₃ and α are the half-tooth angle at point P and the pressure angle at arbitrary meshing point C, respectively. Moreover, d_a symbolizes the distance from the meshing point B to the fixed end of the cantilever beam; h_out and h_x represent the heights from the meshing point B, arbitrary point C on the tooth profile to the tooth centerline, respectively; x is the length between the arbitrary point C on the tooth profile and the fixed end of the cantilever beam. r_b and r_f are the radii of the base circle and root circle, respectively. d₁ and d₂ are the distances from points N and M to the fixed end of the cantilever beam, respectively. The red solid and dashed lines are the force boundaries before and after the crack reaches the gear tooth centerline, respectively, which are modeled by the more suitable parabolas [33].

Founded on the potential energy method, the bending stiffness k_b, shear stiffness k_s, and axial compression stiffness k_a can be respectively acquired by Ref. [20]:

$${\displaystyle \frac{1}{k_b}}={\displaystyle {\int }_0^{d_a}\frac{{\left[-(d_a-x)\cos {\alpha }_1+h_{out}\sin {\alpha }_1\right]}^2}{E{I}_x}dx}$$

(5)

$${\displaystyle \frac{1}{k_s}}={\displaystyle {\int }_0^{d_a}\frac{1.2\cos {{\alpha }_1}^2}{G{A}_x}}dx$$

(6)

$${\displaystyle \frac{1}{k_a}}={\displaystyle {\int }_0^{d_a}\frac{\sin {{\alpha }_1}^2}{E{A}_x}}dx$$

(7)

where E and G are the symbols for the gear material’s elastic and shear moduli, respectively; A_x and I_x are, respectively, the terms for the section’s area and area moment of inertia, where the distance from the fixed end of the cantilever beam is x.

The Hertzian-contact stiffness of a pair of mating teeth crafted from identical materials remains consistent along the line of action, which is unaffected by the meshing location. Moreover, the influence of gear tooth crack on Hertzian-contact stiffness k_h can be ignored, which is similar to that under the normal state and can be expressed as [34]

$${\displaystyle \frac{1}{k_h}}={\displaystyle \frac{4(1-{\upsilon }^2)}{\pi EL}}$$

(8)

where L is the gear tooth width; υ is the Poisson’s ratio. The bending, shear, and compression stiffness of the sun gear with small and large cracks are deduced separately as follows.

(a)

Case 1: Small crack without extending to the centerline of the sun gear tooth

The fixed end of the equivalent cantilever beam and the gear tooth centerline are respectively set as the y and x axis to build a rectangular coordinate system O’xy, which is displayed in Figure 4. The coordinates of end point A of small crack AN and point H on the tooth tip circle are $(0,r_f\sin {\alpha }_3-q_1\sin v)$ and $(q_1\cos v+\sqrt{{r_a}^2-{h_0}^2}-r_f\cos {\alpha }_3,h_0)$, respectively. Here, r_a is the radius of the tooth tip circle, and h₀ is the half-tooth thickness on the tooth tip circle. Based on the characteristic of the force boundary parabola, the distance g_x between the force boundary and centerline of the tooth can be represented as

$$g_x=k^{\prime }{x}^2+r_f\sin {\alpha }_3-q_1\sin v$$

(9)

where k′ is a coefficient. It can be obtained by substituting the coordinate of point H, and is represented as

$$k^{\prime }={\displaystyle \frac{h_0-r_f\sin {\alpha }_3+q_1\sin v}{{(q_1\cos v+\sqrt{{r_a}^2-{h_0}^2}-r_f\cos {\alpha }_3)}^2}}$$

(10)

The expressions of I_x and A_x can be further written as

$$I_x={\displaystyle \frac{{(h_x+g_x)}^{3}L}{12}}$$

(11)

$$A_x=(h_x+g_x)L$$

(12)

where h_x is represented by

$$h_x=\left\{\begin{array}{l}{r}_f\sin \alpha (0\le x\le d_1)\\ r_b\sin {\alpha }_2(d_1\le x\le d_2)\\ r_b\left[({\alpha }_2-\alpha )\cos \alpha +\sin \alpha \right](d_2\le x\le d_a)\end{array}\right.$$

(13)

According to the geometric characteristic of involute, the following relationships can be obtained.

$$d_1=r_f(\cos {\alpha }_3-\cos {\alpha }_r)$$

(14)

$$d_2=r_b\cos {\alpha }_2-r_f\cos {\alpha }_r$$

(15)

$$d_a=r_b\cos {\alpha }_1+r_b({\alpha }_1+{\alpha }_2)\sin {\alpha }_1-r_f\cos {\alpha }_r$$

(16)

$$x=\left\{\begin{array}{l}{r}_f(\cos \alpha -\cos {\alpha }_r)(0\le x\le d_1)\\ r_f({\displaystyle \frac{\sin {\alpha }_3}{\tan \alpha }}-\cos {\alpha }_r)(d_1\le x\le d_2)\\ r_b(\cos \alpha +\alpha \sin \alpha -{\alpha }_2\sin \alpha )-r_f\cos {\alpha }_r(d_2\le x\le d_a)\end{array}\right.$$

(17)

$$dx=\left\{\begin{array}{l}-r_f\sin \alpha d\alpha (0\le x\le d_1)\\ {\displaystyle \frac{\sin {\alpha }_3}{{(\sin \alpha )}^2}}(d_1\le x\le d_2)\\ r_b(\alpha -{\alpha }_2)\cos \alpha d\alpha (d_2\le x\le d_a)\end{array}\right.$$

(18)

where ${\alpha }_r=\arccos (\cos {\alpha }_3-{\displaystyle \frac{q_1\cos v}{r_f}})$.

Substituting Equations (9)–(18) into Equations (5)–(7), the bending, shear, and axial compressive stiffness of one tooth can be represented as

$$\begin{array}{l}{\displaystyle \frac{1}{k_b}}={\displaystyle {\int }_{{\alpha }_3}^{{\alpha }_r}\frac{12\sin \alpha {\left[\frac{-z\cos {\alpha }_0}{z-2(h{a}^{\ast }+c^{\ast }-x_o)}+\cos \alpha \cos {\alpha }_1\right]}^2}{EL{\left[k^{\prime }{(r_f(\cos \alpha -\cos {\alpha }_r))}^2+\sin {\alpha }_3-\frac{q_1\sin v}{r_f}+\sin \alpha \right]}^3}}d\alpha \\ +{\displaystyle {\int }_{{\alpha }_2}^{{\alpha }_3}\frac{12{\left[\frac{z\cos {\alpha }_0}{z-2(h{a}^{\ast }+c^{\ast }-x_o)}\right]}^3{\left(-1+\frac{\sin {\alpha }_2\cos \alpha \cos {\alpha }_1}{\sin \alpha }\right)}^2\sin {\alpha }_3}{EL{\left[k^{\prime }{(r_f(\frac{\sin {\alpha }_3}{\tan \alpha }-\cos {\alpha }_r))}^2+\sin {\alpha }_3-\frac{q_1\sin v}{r_f}+\sin {\alpha }_3\right]}^3{(\sin \alpha )}^2}}d\alpha \\ +{\displaystyle {\int }_{-{\alpha }_1}^{{\alpha }_2}\frac{12{\left[1+\cos {\alpha }_1(({\alpha }_2-\alpha )\sin \alpha -\cos \alpha )\right]}^2({\alpha }_2-\alpha )\cos \alpha }{EL{\left[k^{\prime }{r}_b{(\cos \alpha +\alpha \sin \alpha -{\alpha }_2\sin \alpha -\frac{r_f}{r_b}\cos {\alpha }_r)}^2+\sin {\alpha }_3-\frac{q_1\sin v}{r_b}+({\alpha }_2-\alpha )\cos \alpha +\sin \alpha \right]}^3}}d\alpha \end{array}$$

(19)

$$\begin{array}{l}{\displaystyle \frac{1}{k_s}}={\displaystyle {\int }_{{\alpha }_3}^{{\alpha }_r}\frac{2.4(1+\upsilon ){(\cos {\alpha }_1)}^2\sin \alpha }{EL{\left[k^{\prime }{(r_f(\cos \alpha -\cos {\alpha }_r))}^2+\sin {\alpha }_3-\frac{q_1\sin v}{r_f}+\sin \alpha \right]}^3}}d\alpha \\ +{\displaystyle {\int }_{{\alpha }_2}^{{\alpha }_3}\frac{2.4(1+\upsilon ){(\cos {\alpha }_1)}^2\sin {\alpha }_3}{EL{\left[k^{\prime }{(r_f(\frac{\sin {\alpha }_3}{\tan \alpha }-\cos {\alpha }_r))}^2+\sin {\alpha }_3-\frac{q_1\sin v}{r_f}+\sin {\alpha }_3\right]}^3{(\sin \alpha )}^2}}d\alpha \\ +{\displaystyle {\int }_{-{\alpha }_1}^{{\alpha }_2}\frac{2.4(1+\upsilon ){(\cos {\alpha }_1)}^2({\alpha }_2-\alpha )\cos \alpha }{EL{\left[k^{\prime }{r}_b{(\cos \alpha +\alpha \sin \alpha -{\alpha }_2\sin \alpha -\frac{r_f}{r_b}\cos {\alpha }_r)}^2+\sin {\alpha }_3-\frac{q_1\sin v}{r_b}+({\alpha }_2-\alpha )\cos \alpha +\sin \alpha \right]}^3}}d\alpha \end{array}$$

(20)

$${\displaystyle \frac{1}{k_a}}={\displaystyle \frac{\sin {{\alpha }_1}^2(\cos {\alpha }_2-\frac{z-2h{a}^{\ast }-2c^{\ast }+2x_o}{z\cos {\alpha }_0}\cos {\alpha }_3)}{2EL\sin {\alpha }_2}}+{\displaystyle {\int }_{-{\alpha }_1}^{{\alpha }_2}\frac{\sin {{\alpha }_1}^2({\alpha }_2-\alpha )\cos \alpha }{2EL(({\alpha }_2-\alpha )\cos \alpha +\sin \alpha )}}d\alpha$$

(21)

where z is the tooth number of cracked sun gear; h_a*, c*, and x_o refer to the addendum coefficient, tip clearance coefficient, and modification coefficient, respectively; α₀ is the pressure angle on the pitch circle.

Equations (19)–(21) reveal that the bending, shear, and compressive stiffness for a pair of involute-profiled meshing teeth are governed by the pressure angle α₁. This angle is responsible for the contact position along the tooth surface.

• (b) Case 2: large crack extending below the centerline of the sun gear tooth

Similarly, a rectangular coordinate system O“xy” is built, displayed in Figure 4, when the large crack extends below the tooth centerline. The coordinates of large crack end point E and point H on the tooth tip circle are $(0,-q_2\sin v)$ and $(({\displaystyle \frac{r_f\sin {\alpha }_3}{\sin v}}-q_2)\cos v+\sqrt{{r_a}^2-{h_0}^2}-r_f\cos {\alpha }_3,h_0)$, respectively. The distance g_x between the force boundary and centerline of the tooth is represented as

$$g_x=k^{\prime }{x}^2-q_2\sin v$$

(22)

The coefficient k’ is similarly obtained as

$$k^{\prime }={\displaystyle \frac{h_0+q_2\sin v}{{\left[(\frac{r_f\sin {\alpha }_3}{\sin v}-q_2)\cos v+\sqrt{{r_a}^2-{h_0}^2}-r_f\cos {\alpha }_3\right]}^2}}$$

(23)

Other parameters such as I_x, A_x, h_x, d₁, d₂, d_a, x, and dx are similar to those in Equations (11)–(18). The half-tooth angle α_r is represented as

$${\alpha }_r=\arccos (\cos {\alpha }_3-{\displaystyle \frac{q_2\cos v}{r_f}})$$

(24)

Therefore, the corresponding stiffness of one tooth is the function of pressure angle α₁ and can be expressed as

$$\begin{array}{l}{\displaystyle \frac{1}{k_b}}={\displaystyle {\int }_{{\alpha }_3}^{{\alpha }_r}\frac{12\sin \alpha {\left[\frac{-z\cos {\alpha }_0}{z-2(h{a}^{\ast }+c^{\ast }-x_o)}+\cos \alpha \cos {\alpha }_1\right]}^2}{EL{\left[k^{\prime }{(r_f(\cos \alpha -\cos {\alpha }_r))}^2-\frac{q_2\sin v}{r_f}+\sin \alpha \right]}^3}}d\alpha \\ +{\displaystyle {\int }_{{\alpha }_2}^{{\alpha }_3}\frac{12{\left[\frac{z\cos {\alpha }_0}{z-2(h{a}^{\ast }+c^{\ast }-x_o)}\right]}^3{\left(-1+\frac{\sin {\alpha }_2\cos \alpha \cos {\alpha }_1}{\sin \alpha }\right)}^2\sin {\alpha }_3}{EL{\left[k^{\prime }{(r_f(\frac{\sin {\alpha }_3}{\tan \alpha }-\cos {\alpha }_r))}^2-\frac{q_2\sin v}{r_f}+\sin {\alpha }_3\right]}^3{(\sin \alpha )}^2}}d\alpha \\ +{\displaystyle {\int }_{-{\alpha }_1}^{{\alpha }_2}\frac{12{\left[1+\cos {\alpha }_1(({\alpha }_2-\alpha )\sin \alpha -\cos \alpha )\right]}^2({\alpha }_2-\alpha )\cos \alpha }{EL{\left[k^{\prime }{r}_b{(\cos \alpha +\alpha \sin \alpha -{\alpha }_2\sin \alpha -\frac{r_f}{r_b}\cos {\alpha }_r)}^2-\frac{q_2\sin v}{r_b}+({\alpha }_2-\alpha )\cos \alpha +\sin \alpha \right]}^3}}d\alpha \end{array}$$

(25)

$$\begin{array}{l}{\displaystyle \frac{1}{k_s}}={\displaystyle {\int }_{{\alpha }_3}^{{\alpha }_r}\frac{2.4(1+\upsilon ){(\cos {\alpha }_1)}^2\sin \alpha }{EL{\left[k^{\prime }{(r_f(\cos \alpha -\cos {\alpha }_r))}^2-\frac{q_2\sin v}{r_f}+\sin \alpha \right]}^3}}d\alpha \\ +{\displaystyle {\int }_{{\alpha }_2}^{{\alpha }_3}\frac{2.4(1+\upsilon ){(\cos {\alpha }_1)}^2\sin {\alpha }_3}{EL{\left[k^{\prime }{(r_f(\frac{\sin {\alpha }_3}{\tan \alpha }-\cos {\alpha }_r))}^2-\frac{q_2\sin v}{r_f}+\sin {\alpha }_3\right]}^3{(\sin \alpha )}^2}}d\alpha \\ +{\displaystyle {\int }_{-{\alpha }_1}^{{\alpha }_2}\frac{2.4(1+\upsilon ){(\cos {\alpha }_1)}^2({\alpha }_2-\alpha )\cos \alpha }{EL{\left[k^{\prime }{r}_b{(\cos \alpha +\alpha \sin \alpha -{\alpha }_2\sin \alpha -\frac{r_f}{r_b}\cos {\alpha }_r)}^2-\frac{q_2\sin v}{r_b}+({\alpha }_2-\alpha )\cos \alpha +\sin \alpha \right]}^3}}d\alpha \end{array}$$

(26)

The small or large tooth crack hardly affects the axial compressive stiffness k_a, because the cracked gear tooth will be in contact with the main body of the gear under the radial force F_a. Hence, the axial compressive stiffness k_a under a large tooth crack is regarded as similar to that under a small tooth crack, which is represented in Equation (21).

The pressure angle α₁ at different meshing points on the input sun gear is related to its rotational angle θ, which can be represented as [34]

$$\left\{\begin{array}{l}{\alpha }_1=\theta -{\displaystyle \frac{\pi }{2z_1}}-\tan {\alpha }_0+{\alpha }_0\\ +\tan (\arccos ({\displaystyle \frac{z_1\cos {\alpha }_0}{\sqrt{a^2+b^2-2ab\cos ({\alpha }^{\prime }-\arccos (\frac{z_2\cos {\alpha }_0}{z_2+2(h{a}^{\ast }+x_o-\Delta y)}))}}}))\\ a=z_2+2(h{a}^{\ast }+x_o-\Delta y)\\ b={\displaystyle \frac{(z_2+z_1)\cos {\alpha }_0}{\cos {\alpha }^{\prime }}}\end{array}\right.$$

(27)

where ${\alpha }^{\prime }$ is the angle of engagement at the node after modification; $\Delta y$ is the top land reduction coefficient; z₁ and z₂ denote, respectively, the tooth numbers of sun and planet gears. Consequently, according to the relation of pressure angle α₁ and rotational angle θ, the bending, shear, and axial compressive stiffness can all be represented as the functions of input rotation angle θ of the sun gear, whether the tooth crack is small or large. This means that meshing stiffness is highly associated with the rotational angle of the gear and varies with the meshing location of the gear tooth surface.

• (c) Effect of fault-caused speed fluctuation on meshing stiffness of cracked sun gear

Speed fluctuation coming from the cracked sun gear will make the engagement positions of gear pairs change relative to their normal conditions, which further affects the meshing stiffness. The Fourier series using the fault feature frequency as the base frequency can express the fault-caused periodic speed fluctuation of the gear system [30,31]. Hence, the rotation speed of the cracked sun gear is expressed as

$$n_s(t)=n_{in}+\underset{\mathrm{speed}\mathrm{fluctuation}}{\underbrace{{\displaystyle \sum _{l=1}^L{A}_l\cos (2\pi l{f}_{nsc}t+{\beta }_l)}}}$$

(28)

where n_in and f_nsc are the input rotation speed and fault feature frequency of sun gear, respectively; A_l and β_l are the amplitude and phase of the lth fluctuation component, respectively.

The mathematical linkage between speed fluctuation and rotation angle θ of the input sun gear can be obtained by integrating its rotation speed, which is expressed as

$$\theta ={\displaystyle {\int }_0^t\frac{2\pi n_s(t)}{60}dt}={\displaystyle {\int }_0^t\frac{2\pi }{60}}[n_{in}+{\displaystyle \sum _{l=1}^L{A}_l\cos (2\pi l{f}_{nsc}t+{\beta }_l)}]dt$$

(29)

Therefore, the mathematical linkage between speed fluctuation and pressure angle variation can be obtained by substituting Equation (29) into Equation (27). In other words, the rotation speed of the sun gear with speed fluctuation is first integrated to obtain the fluctuant rotation angle θ according to Equation (29); then, the rotation angle θ is substituted into Equation (27) to obtain the pressure angle α₁. Finally, pressure angle α₁ is applied to calculate the bending, shear, and axial compressive stiffness of cracked sun gear, which contains fault-induced speed fluctuation.

• Meshing stiffness of s-p and p-r gear pairs with cracked sun gear

The bending, shear, and axial compressive stiffness of healthy planet and ring gears can be represented by the gear rotational angle [20] in the potential energy method, while the cracked sun gear leads to speed fluctuation of the gear system, including planet and ring gears. This means that the speed fluctuation coming from the cracked sun gear also influences the meshing stiffness of healthy planets and ring gears. Similarly, the bending, shear, and axial compressive stiffness of one healthy tooth on planet and ring gears under the impact of fault-caused speed fluctuation can be calculated by substituting the rotational angle acquired by integrating the speed fluctuation into the formulas of these meshing stiffness.

In spur gears with a contact ratio ranging from one to two, single-tooth-pair and double-tooth-pair contacts occur alternately. During the period of single-tooth-pair meshing, the overall effective meshing stiffness can be determined as [20]

$$k(\phi )={\displaystyle \frac{1}{\frac{1}{k_h}+\frac{1}{k_{b1}(\phi )}+\frac{1}{k_{s1}(\phi )}+\frac{1}{k_{a1}(\phi )}+\frac{1}{k_{b2}(\phi )}+\frac{1}{k_{s2}(\phi )}+\frac{1}{k_{a2}(\phi )}}}$$

(30)

where φ is the rotational angle of the driving gear. Subscripts 1 and 2 denote, respectively, the driving and the driven wheel. It is necessary to point out that Equation (30) is suitable for both externally and internally meshing gear pairs.

During the dual-tooth-pair engagement period, there is simultaneous contact between two gear pairs. The overall effective meshing stiffness, denoted as k_t, can be derived as

$$k_t(\phi )=k(\phi )+k(\phi +{\displaystyle \frac{2\pi }{z_1}})$$

(31)

Therefore, after obtaining the bending, shear, and axial compressive stiffness of one tooth in cracked sun gear, healthy planet, and ring gears, the meshing stiffness of s-p and p-r gear pairs considering the fault-caused speed fluctuation can ultimately be obtained using Equations (30) and (31).

• Effect demonstration of fault-caused speed fluctuation on meshing stiffness

To investigate the impact of speed fluctuation resulting from the cracked sun gear on meshing stiffness and vibration modulation signal, meshing stiffness under normal and tooth-cracked sun gear conditions is analyzed. Sun gear in the planetary gearbox with the structure parameters listed in Table 1 operates at the speed of 1800 rpm, and its fault feature frequency is 23.94 Hz. The fault feature order of the cracked sun gear and meshing order are respectively 3.95 and 79 after normalizing with the carrier rotation frequency. The meshing stiffness of the s-p gear pair under normal state in the time domain and order domain is displayed in Figure 5, which indicates alternating engagement of single-tooth and double-tooth in the time domain and harmonics of the meshing order in the order domain.

When a tooth root crack on the sun gear propagates to the centerline of the tooth at a 45-degree angle, the meshing stiffness of the s-p gear pair without the consideration of fault-caused speed fluctuation is displayed in Figure 6. Meshing stiffness in the time domain decreases periodically with a time interval of 0.0418 s as the sun gear rotation, while its order spectrum contains the meshing order and its harmonics, as well as the harmonics of fault feature order of sun gear that distribute in the full band. When taking into account the speed fluctuation resulting from the cracked sun gear, the meshing stiffness of the s-p gear pair is calculated. Here, only one order of fault feature frequency in Equation (28) is involved to facilitate the calculation, and the rotation speed of the sun gear is thus represented as

$$n_s(t)=1800+15\cos (2\pi f_{nsc}t+{\displaystyle \frac{\pi }{6}})$$

(32)

A time-domain waveform and order spectrum of the meshing stiffness of the s-p gear pair, taking into account the speed fluctuation, are illustrated in Figure 7, which also contains basically periodic decreases in the time domain. However, a comparison between Figure 6 and Figure 7 reveals that though the fault feature frequency and its harmonics emerge in the full band in the two figures, there are more dominant sun gear fault modulation sidebands surrounding the meshing order and its harmonics in Figure 7 than those in Figure 6. The main difference comes from periodic speed fluctuation. It indicates that the speed fluctuation causes FM surrounding the meshing frequency and its harmonics in the meshing stiffness, which is coincident with the research in Ref. [28].

The meshing stiffness of the p-r gear pair influenced by the speed fluctuation in Equation (32) is displayed in Figure 8. Modulation sidebands associated with the sun gear fault feature order surround the meshing order and its harmonics, in which the modulation sidebands are generated by periodic speed fluctuations.

2.1.3. Excitation Forces in Three Paths

After substituting the meshing stiffness with fault-caused speed fluctuation into Equation (3), the local coordinate array q can be calculated by the built-in function ode15 in the Matlab 2021a platform. Excitation force in path one comes from the engagement of the p-r gear pair, namely, the meshing force of the p-r gear pair, that exists only when the tooth on ring gear is in engagement with planet gear. The meshing force sequentially acts on each tooth of the ring gear with the rotation of the carrier. Hence, the meshing excitation force f_i(t) can be obtained from

$$f_i(t)={\displaystyle \sum _{n=1}^N{f}_{nl}(t)}$$

(33)

$$f_{nl}(t)=\left\{\begin{array}{l}{k}_{nl}(t){\delta }_{nr}(t)+c_{nl}(t){\dot{\delta }}_{nr}(t),l=1,2,\dots ,z_r;\mathrm{when}\mathrm{the}{l}^{th}\mathrm{tooth}\mathrm{on}\mathrm{ring}\mathrm{gear}\mathrm{engages}\\ 0,\mathrm{when}\mathrm{the}{l}^{th}\mathrm{tooth}\mathrm{on}\mathrm{ring}\mathrm{gear}\mathrm{does}\mathrm{not}\mathrm{engage}\end{array}\right.$$

(34)

where N signifies the count of planet gears; z_r is the tooth number of ring gear; k_nl and c_nl are, respectively, the meshing stiffness and damping of a single tooth when the nth planet gear meshing with the lth tooth on the ring gear.

f_s(t) is the force operating on the bearing pedestal of the sun gear, and it is related to the motion equations of the sun gear. However, the motions x_s and y_s of sun gear calculated by the LPM are in the local coordinates, which should be converted to the absolute coordinate Y, namely the measured direction of the sensor, to obtain the excitation f_s(t). Here, the relationship between the local coordinate and the absolute coordinate Y is displayed in Figure 9. Therefore, the excitation force f_s(t) acting on the Y-direction of the bearing pedestal of the sun gear can be represented as

$$f_s(t)=\left[c_s{\dot{x}}_s(t)+k_s{x}_s(t)\right]\sin [{\omega }_c(t)t]+\left[c_s{\dot{y}}_s(t)+k_s{y}_s(t)\right]\cos [{\omega }_c(t)t]$$

(35)

where w_c(t) is the rotation speed of the carrier.

Similarly, the Y-direction force f_c(t) acting on the bearing pedestal of the carrier is closely related to the motions of the carrier, as well as the support stiffness and damping of the bearing, which is represented by

$$f_c(t)=\left[c_c{\dot{x}}_c(t)+k_c{x}_c(t)\right]\sin [{\omega }_c(t)t]+\left[c_c{\dot{y}}_c(t)+k_c{y}_c(t)\right]\cos [{\omega }_c(t)t]$$

(36)

It should be mentioned that the speed fluctuation affects the carrier rotation frequency w_c(t), which is further represented as

$$w_c(t)=2\pi {\displaystyle \frac{n_s(t)}{i60}}={\displaystyle \frac{2\pi }{i60}}[n_{in}+{\displaystyle \sum _{l=1}^L{A}_l\cos (2\pi l{f}_{nsc}t+{\theta }_l)}]$$

(37)

where i is the gear ratio of the planetary gear set.

2.2. Transfer Path Functions Considering Modal Parameters of Flexible Housing

From the main three transfer paths in Figure 1, it is evident that path one is time-varying with the carrier rotation, while transfer paths two and three are time-invariant. However, three transfer paths are highly influenced by flexible housing. The dynamics analysis software LMS Virtual.Lab 13.6 is applied to simulate the flexibility of the housing, then calculate the transfer path functions from each tooth on the ring gear to the sensor. Here, the tooth of the ring gear right below the sensor is set as No.1, and the count of the remaining teeth increases in a clockwise direction. The sensor location is designated as the response point, and the excitation point is progressively placed at the points where the ring gear teeth mesh. The frequency response function is defined as the vibration amplitude at the measurement point in response to a unit impulse force applied at the excitation location, with the force direction aligned with the direction of the meshing force. Transfer path functions of path one are displayed in Figure 10. There are z_r transfer path functions in path one because each ring gear tooth corresponds to one transfer path. Moreover, it is obvious that the transfer path function varies with frequency and the meshing position on the ring gear, which includes the modal parameters of flexible housing.

Similarly, the transfer path functions of paths two and three are displayed in Figure 11, where the response point is positioned at the same location as the sensor, and the excitation point is strategically placed along the Y-direction of bearing pedestals of sun gear and carrier.

Finally, substituting the excitation forces obtained from Section 2.1.3 and the flexible transfer path functions from Section 2.2 into Equation (1) or Equation (2), the vibration AM-FM signal resulting from the cracked sun gear can be solved.

3. Resultant Vibration Signal Analysis

When the tooth crack on the sun gear propagates to the tooth centerline at a 45-degree angle, based on the proposed coupling dynamics model, the planetary gearbox with the structure parameters in Table 1 is simulated with and without the consideration of fault-caused speed fluctuation to investigate the vibration modulation features. The sun gear rotates at an input speed of 1800 rpm while the carrier experiences a load torque of 100 Nm. The sampling rate is 12,600 Hz, and the duration of sampling is 2 s. To correlate the feature frequencies with the structural parameters of the gearbox, all the feature frequencies are standardized against the carrier rotational frequency, as detailed in

Table 2. Feature orders of planetary gearbox.

Meshing Order Om	Fault Feature Order of Sun Gear Onsc	Rotation Order of Carrier Onc
79	3.95	1

.

3.1. Tooth Root Crack of Sun Gear Without Considering the Fault-Caused Speed Fluctuation

By substituting the meshing stiffness without taking into account the speed fluctuation in the proposed model, the vibration signal acquired by the sensor is calculated, whose order spectrum is shown in Figure 12. Near the harmonics of the meshing order, orders that are the integer multiples of planets number N are dominant, such as the orders (75, 78, 81, 156, 159, …), which resemble those induced by the planetary gear set under normal condition. It can be found that a resonance frequency of 2667 Hz in Figure 11a shows the order of 440 after being normalized by the rotation frequency of the carrier as 6.0606 Hz. Therefore, the order spectrum within the resonance band of 426–447 is enlarged and displayed in Figure 13, which includes four clusters of AM sideband components with the fault feature order of cracked sun gear 3.95 as the interval.

3.2. Tooth Root Crack on Sun Gear Considering the Fault-Caused Speed Fluctuation

When considering the speed fluctuation in Equation (32) that comes from the cracked sun gear, the vibration signal is calculated by the proposed model. The vibration order spectrum is presented in Figure 14, which shows obvious fault modulation sidebands of cracked sun gear centering on the meshing order and its harmonics. The third meshing order and the resonance range around 440 are enlarged in Figure 15 to exhibit the vibration AM-FM sideband features. Prominent modulation sidebands centering on the meshing order 237 with an interval of fault feature order of cracked sun gear are observed. Moreover, with these fault modulation sidebands as center frequencies, modulation sidebands with the rotation order of the carrier as an interval emerge. Hence, two clusters of modulation elements appear around the meshing order and its harmonics, namely the fault feature order of the sun gear and the rotation order of the carrier. Here, sidebands of fault feature frequency of sun gear are the FM sidebands caused by the fault-induced speed fluctuation, just as sideband components 229.1, 233.05, 240.95, and 244.9 are labeled in Figure 15. Moreover, more than one group of modulation components with the fault feature order of sun gear as an interval are dominant in the resonance region, which can be regarded as an effect diagnosis indicator of cracked sun gear. However, it should be noted that modulation components of the fault feature order within resonance regions simultaneously contain FM caused by fault-induced speed fluctuation and AM highly related to the transfer function, but FM and AM components are coupled. Comparing the order spectra in Figure 12 and Figure 14, the main difference in order spectra with and without involving the fault-caused speed fluctuation is around the meshing order and its harmonics. When taking into account the fault-caused speed fluctuation, vibration fault sidebands of sun gear present around harmonics of the meshing order, which are caused by fault-induced speed fluctuation. Moreover, modulation sidebands of a carrier rotation order that are non-integral multiples of planet numbers are more obvious, surrounding the harmonics of the meshing order. The similarity between the vibration modulation sidebands, whether taking into account speed fluctuations or not, is evident in the resonance band, where multiple sets of modulation sidebands spaced by the fault feature order of the cracked sun gear are observed. However, modulation components in resonance regions are all AM components without considering fault-induced speed fluctuation.

3.3. Vibration AM-FM Sidebands of Cracked Sun Gear

Vibration AM-FM sidebands resulting from the cracked sun gear are classified into two categories. (1) One is within the resonance range, which contains more than one group of AM-FM sidebands with the fault feature order of cracked sun gear as an interval. (2) Another is around the meshing order and its harmonics, which is different from the vibration AM signal without considering the speed fluctuation. An interval of the fault feature order of sun gear emerges around the meshing order and its harmonics to produce fault modulation sidebands, which are generated by fault-induced speed fluctuation. The second group of components spaced by the carrier rotation order exists around the fault modulation sidebands. Consequently, two sets of modulation sidebands, namely the fault feature order of sun gear and carrier rotation order, appear surrounding the meshing order and its harmonics.

4. Experimental Validation

The test planetary gear system in Figure 16 is primarily composed of one motor and load motor, as well as one single-stage planetary gearbox, which is used to verify the proposed model. Three planets are positioned uniformly within the planetary gearbox. The power input and output are, respectively, the sun gear and carrier, displayed in Figure 16c. Moreover, the ring gear is bolted to the housing, and a PCB acceleration sensor of model number 356A16 is installed on the housing to collect vibration acceleration signals. Its frequency range and non-linearity are 0.5 to 5000 Hz and ≤ 1%, respectively. The tooth root crack of the sun gear is processed artificially by wire electrode discharge machining, as displayed in Figure 16d. Data acquisition equipment, Müller-BBM, shown in Figure 17, is used to collect vibration data with sampling frequency and duration, respectively, being 96,000 Hz and 50 s.

The planetary gearbox can configure different transmission ratios. In this experiment, two transmission ratios in planetary gear sets are 6.18 and 7.

Table 3. Structural parameters and feature orders of two planetary gear sets.

Tooth Number	Planetary Gear Set 1			Planetary Gear Set 2
	Planet	Ring	Sun	Planet	Ring	Sun
	34	88	17	36	90	15
Fault feature order of sun gear	5.18			6
Transmission ratio	6.18			7
Meshing order	88			90

provides the structural parameters and feature orders for the two planetary gear sets. Experiments under the condition of tooth-cracked sun gear are conducted. In order to simplify the modulation sideband analyses, the feature frequencies are standardized using the carrier rotation frequency. Moreover, the transfer path functions of the planetary gearbox from excitation points on the ring gear to the stationary sensor are displayed in Figure 18, which can identify two resonance frequencies about 1215 Hz and 2672 Hz.

4.1. Planetary Gearbox with Gear Ratio of 6.18

The input speed on the sun gear is configured as 2000 rpm, while the load subjected to the carrier is 200 Nm. Therefore, the carrier rotation frequency is 5.39 Hz. The vibration signal spectrum acquired by the immobile sensor is presented in Figure 19a, which indicates evident modulation sidebands around harmonics of the meshing order. For the purpose of clearly analyzing the modulation sidebands feature, order spectra around the fifth meshing order and resonance range are enlarged and shown in Figure 19b and Figure 20, respectively. Around the fifth meshing order (440) in Figure 19b, the fault feature order of the sun gear modulates the fifth meshing order to produce the fault modulation sidebands, which are identified by the red dots and caused by fault-induced speed fluctuation. Another group of sidebands spaced by the rotation order of the carrier emerges surrounding the fault modulation sidebands, highlighted by the black and green dots. It can be seen from Figure 20 that there is more than one cluster of fault modulation sidebands of sun gear in the resonance ranges around the orders of 225 and 496, like the red, black, green, and purple marks, which are the coupled AM-FM sidebands. Therefore, two resonance regions with different natural frequencies and damping ratios have similar modulation sideband distribution features, and the modal parameters hardly affect the modulation sideband distribution. The significant consistency of vibration AM-FM features respectively acquired from the experiment and the proposed model verifies the effectiveness of the proposed model.

To further verify the superiority of the proposed model over existing models, the AM sidebands obtained by existing models [13,24] are displayed in Figure 21. Here, sidebands within the resonance region are the absolute frequency of the sun gear (6.18 order) and rotational frequency of the carrier, which are marked by red, black, and brown spots, respectively. Additionally, sidebands around the harmonics of the meshing frequency are N (number of planets) multiples of the carrier rotational frequency. Comparing the sideband components in Figure 19b, Figure 20a and Figure 21, it can be seen that existing models cannot explain the generation mechanisms of fault modulation sidebands of sun gear around harmonics of the meshing frequency. Additionally, some modulation components within resonance regions cannot still be predicted by existing models. However, the proposed model can clearly reveal the mechanisms of AM-FM sidebands around the harmonics of the meshing frequency and resonance regions. This is because the proposed model involves fault-induced speed fluctuation and housing modal parameters simultaneously. Modal parameters primarily affect the vibration amplitudes and the AM sideband distribution within resonance ranges, while the fault-induced speed fluctuation generates FM sidebands of fault feature frequency around the harmonics of the meshing frequency and resonance ranges. Therefore, the superiority of the proposed model over existing models in the aspect of explaining modulation sidebands is verified.

4.2. Planetary Gearbox with Gear Ratio of 7

Figure 22a shows the vibration order spectrum induced by cracked sun gear in the planetary gearbox with a gear ratio of seven, where the system operates at an input speed of 1500 rpm under a load of 100 Nm. The rotation frequency of the carrier is about 3.57 Hz. Enlarged order spectra around the meshing order and resonance range are displayed in Figure 22b and Figure 23, respectively. Obviously, similar to the theoretical derivations by the proposed model, abundant frequency components in the vibration order spectrum can also be categorized in two parts: (1) Obvious sun gear fault modulation sidebands centering on the meshing order can be seen, like the orders of 84 and 96, which are generated by fault-induced speed fluctuation. Moreover, centered around the fault modulation sidebands, there are modulation sidebands of the carrier rotation order, which are marked by the black spots in Figure 22b. (2) Dominant faults, which are more than one group of modulation components of sun gear fault feature orders, emerge within the resonance range, which are labeled in Figure 23. From these sideband components, we can see that the theoretical derivations by the proposed model considering the fault-caused speed fluctuation agree well with the experimental results.

5. Conclusions

In this paper, a superior rigid-flexible coupling dynamics model of the planetary gear set is developed considering the fault-caused speed fluctuation. Vibration AM-FM features of cracked sun gear based on the model are investigated. The essential conclusions are outlined below. (1) A fault-caused speed fluctuation generates FM sidebands in the meshing stiffness and vibration signal spectra, which emerge centering on the meshing frequency and its harmonics and cover the fault-related frequency components within the resonance range. Fault-caused speed fluctuations should be considered to explain the fault-related sidebands clearly. (2) Two sets of modulation sidebands, namely the cracked sun gear fault feature frequency and carrier rotation frequency, exist around harmonics of the meshing frequency, which is an obvious sideband behavior when taking into account the fault-caused speed fluctuation. Moreover, more than one cluster of modulation sidebands with the sun gear fault feature frequency as an interval emerges within the resonance bands, which can serve as a reliable fault detection indicator. (3) Theoretical derivations by the proposed model match well with the experiential results from a test planetary gearbox, which indicates that the proposed model is superior to the dynamics model without considering the fault-caused speed fluctuation in the aspect of revealing the vibration modulation sidebands. Consequently, the developed dynamics model of a planetary gear set can explore the true spectral features of tooth root crack sun gears, which provides insights into the vibration mechanisms and aids in the fault diagnosis of planetary gearboxes. Additionally, it should be noted that dynamics modeling research to obtain accurate vibration amplitudes is an important future topic.