Research on the dynamic characteristics of multi-cylinder crankshaft considering crack and engine variable conditions

Abstract

As the core component of the engine power output, the crankshaft is prone to fatigue cracks in the complicated working process of the engine. It is necessary to analyze the vibration characteristics of the crankshaft under different engine operating conditions to prevent the serious hidden trouble caused by crankshaft fractures. At the same time, the effect of crack on crankshaft should also be considered. In this work, based on the finite element method (FEM), a four-cylinder diesel engine coupled with crack crankshaft model is established to analyze the dynamic characteristics of crankshaft considering different engine operating conditions coupled with crankshaft crack failure. The excitations including in-cylinder combustion and friction are also considered. Based on the validated model, the vibration characteristics of the crack crankshaft considering idling speed in the flameout state, normal ignition, single cylinder flame-out and abnormal firing sequence conditions are analyzed respectively. The results show that the different types of engine faults including cracks can be distinguished effectively by waterfall diagrams. It also provides theoretical guidance for fault diagnosis based on vibration signals.

1 Introduction

Since entering the twenty-first century, Marine fisheries are facing new problems and challenges. In order to protect the sound development of the offshore ecological environment, marine fisheries are required to expand far away from the direction of land, and more stringent requirements are put forward for the performance and emission of fishing boats and supporting main engines, which will have a far-reaching impact on the development of marine diesel engines [1]. The crankshaft, as the core of diesel engine transmission components, played an extremely important role in the field of industrial manufacturing [2]. However, there are always some unexpected problems with the crankshaft in the actual diesel driving engineering. The crankshaft crack, single cylinder flame-out and the abnormal firing sequence are several common instability factors [3]. Most crankshaft crack failures of diesel engines are fatigue failures caused by bending and torsional loads. The main source of load is associated with high-pressure gases during combustion, which are applied to the crank pin through the connecting rod [4,5,6].

Furthermore, in recent years, the deterioration of crankshafts has escalated alongside the rise in diesel engine power. Complex external excitations from the cylinder exert forces on the crankshaft, exacerbating crack propagation [7]. Therefore, conducting dynamic behavior analysis of crankshaft models is imperative to proactively prevent various fault conditions, including crankshaft cracks.

In the field of engines, researchers have endeavored to delve into crack propagation in crankshafts. For instance, Dong et al. [8] utilized the Constructure-Kinematics-Materials-Craftsmanship method to track crankshaft crack failures. Their findings revealed that high cycle fatigue predominantly causes crankshaft fatigue failure. Furthermore, a comprehensive approach incorporating fractographic, metallographic, and numerical analyses was employed in Ref. [9] to comprehend the crankshaft failure mechanism. Results indicated that notches at crack initiation points or crankshaft misalignment contribute to crack propagation during operation. However, the aforementioned literature primarily focuses on post-recognition of cracks and does not address preventing crack generation proactively. Therefore, in recent years, some scholars have identified common engine failure conditions using deep learning, neural networks, or acoustic emission technology [10,11,12,13]. While fault diagnosis methods based on big data enable online identification and diagnosis of faults, they often lack comprehensive analysis of dynamic mechanisms due to insufficient detailed mathematical modeling. To address this gap, Bovsunovsky [14] developed a relatively simple comparative evaluation procedure to assess the effectiveness of damage vibration diagnosis, applying it to the stepper rotors and shafting of steam turbines. These analyses highlight the necessity of establishing and analyzing dynamic models of engines and crankshafts to underpin fault diagnosis methods based on vibration signals.

In fact, in the field of rotary equipment, numerous scholars have extensively investigated rotor shaft cracks. Super-harmonic resonance within the system's subcritical speed range has been identified as an indicator of crack emergence [15,16,17,18,19,20]. Some researchers have also examined the coupling of crack faults with other common rotor system faults to understand their combined effects [21, 22]. Consequently, stability theory has been incorporated into rotor systems to analyze the impact of crack introduction on stable operation [23,24,25]. Deeper cracks and proximity to critical speeds typically undermine rotor system stability, with various bifurcation phenomena, including chaos, documented in the literature [26, 27]. Additionally, scholars have conducted thorough investigations into different crack types. For instance, Liu et al. [28, 29] developed a model involving a cantilever beam with slant edge and closed, embedded horizontal cracks, studying its vibration characteristics for early detection and diagnosis. Crescent-shaped cracks, among the most prevalent in rotating shafts, have been extensively studied [30,31,32,33,34].

However, in the field of crankshafts, the complexity of engine operating conditions and crankshaft structure poses challenges in establishing mathematical models and understanding engine excitations. Consequently, there is limited literature focusing on the dynamic behavior of crankshafts, particularly concerning engine operating conditions and crack faults. Addressing this gap, this paper establishes a comprehensive mathematical model encompassing combustion in the cylinder, friction, crack, and crankshaft dynamics. The modeling approach employs a stepped shaft model to simulate the crankshaft, while also considering breathing cracks. Based on verified models, the dynamic behavior of normal crankshaft and cracked crankshaft under four engine working conditions: idling speed in the flameout state, normal operation, single cylinder flameout and abnormal firing sequence are analyzed respectively.

2 The establishment of each part model

2.1 Modeling of the crankshaft

Crankshaft is an important part of engine power output, once it is broken, it will cause great damage to the engine and seriously endanger the safety of personnel. Therefore, it is necessary to model the cracked crankshaft and analyze its dynamic behavior. Figure 1 is a schematic diagram of the cracking process of the crankshaft of a four-cylinder engine. The exciting forces of the combustion process in the cylinder are transferred to the crankshaft through the crank connecting rod mechanism. Under the influence of complex alternating loads, the crankshaft gradually produces fatigue cracks. As shown in Fig. 1, the crankshaft model with cracks constructed in this work is composed of six parts: main journal, oil seal journal, gear journal, crank, bearing and crack. In addition, the crankshaft is also affected by various external excitations from the cylinder. The seven parts of the diagram (green circle) will be modeled separately in the following subsections.

Fig. 1

Schematic diagram of crack propagation of engine crankshaft

2.1.1 Modeling of journals

The cross-sectional area of each journal of the crankshaft is uniform, and all journals are combined into a stepped shaft model. At the same time, the FEM is used to construct each journal model. In this method, each journal model, including the main journal, oil seal journal and gear journal, is constructed using Timoshenko beams [26]. As shown in Fig. 2, the displacement coordinates at the node can be expressed as q_i = [x_i, y_i, z_i, θ_xi, θ_yi, θ_zi] when considering six freedom degrees. A complete shaft element includes 2 nodes (A and B), its generalized coordinates can be written as q_e = [x_A, y_A, z_A, θ_xA, θ_yA, θ_zA, x_B, y_B, z_B, θ_xB, θ_yB, θ_zB]. According to the theory of elastic mechanics [35], the mass matrix (M_e), stiffness matrix (K_e) and gyroscopic matrix (G_e) of the shaft element can be acquired. Their expressions are shown in “Appendix 1”.

Fig. 2

FEM schematic of a shaft element

2.1.2 Modeling of crack

2.1.2.1 Establishment of the crack element

The establishment of the crack model depends on the strain energy release rate method, which has good adaptability under complex external excitation conditions [36]. The core of this approach is the Griffith criterion. According to the change of strain energy generated by crack propagation, the compliance matrix of the shaft element with crack can be obtained. As shown in Fig. 3, x_c is defined as the length from the crack growth center to the left end of the shaft element, θ_c is defined as the angle between the crack and the horizontal direction, and d_c is defined as the crack depth. The detailed process of calculating the compliance matrix of the shaft element with crack is as follows:

Fig. 3

Schematic diagram of the shaft element with crack; a Front view of axial segment with crack b top view of an axial segment with crack c crack section diagram d crack stress state of thin plate

Firstly, the strain energy of the crack in a thin plate is calculated, followed by integration along the crack plane (z’ direction). Subsequently, the strain energy of the crack surface is determined. Finally, based on the strain energy of the crack plane, the local compliance matrix of the shaft element containing the crack can be derived. This method finds extensive application in engineering practice, with a detailed derivation of the flexibility matrix provided in “Appendix 2”.

2.1.2.2 Simulation of breathing cracks

During shaft rotation, the stress, strain, and displacement fields near the crack undergo constant change due to external system excitation, resulting in periodic alterations in the crack opening area, known as the breathing effect. As depicted in Fig. 4, as the shaft element rotates from (a) to (f), the crack transitions from fully closed to half-open-half-closed, fully open, half-closed-half-open, and finally fully closed again. Throughout this process, each stiffness value within the stiffness matrix of the cracked shaft element varies with the angle. In this study, it is assumed that during crankshaft rotation, the crack's opening and closing states change uniformly with the crankshaft angle. A total of 432 (144 * 3) stiffness values in the stiffness matrix corresponding to three different cracks are considered to have time-varying properties. The calculation process for determining the time-varying stiffness of the crack is outlined in “Appendix 3”, while a detailed explanation of the breathing crack introduction is provided in Ref. [37].

Fig. 4

Schematic diagram of the opening and closing state of the breathing crack

2.1.3 Modeling of bearings

According to the Ref. [38], the bearing element of the crankshaft can be simplified as a spring-damping element applied at the corresponding node position. For radial bearings, radial stiffness and damping are considered. For the j (j = 1,2,3) bearing element, the differential equation of motion in the static coordinate system is shown in Eq. (1):

$$\begin{aligned} {\varvec{K}}_b &= \left[ {\begin{array}{*{20}l} 0 & {\,} & {\,} & {\,} & {\,} & {\,} \\ {\,} & {K_{by} } & {\,} & {\,} & {\,} & {\,} \\ {\,} & {\,} & {K_{bz} } & {\,} & {\,} & {\,} \\ {\,} & {\,} & {\,} & 0 & {\,} & {\,} \\ {\,} & {\,} & {\,} & {\,} & {K_{b\theta y} } & {\,} \\ {\,} & {\,} & {\,} & {\,} & {\,} & {K_{b\theta z} } \\ \end{array} } \right]\\ {\varvec{C}}_b &= \left[ {\begin{array}{*{20}l} 0 & {\,} & {\,} & {\,} & {\,} & {\,} \\ {\,} & {C_{by} } & {\,} & {\,} & {\,} & {\,} \\ {\,} & {\,} & {C_{bz} } & {\,} & {\,} & {\,} \\ {\,} & {\,} & {\,} & 0 & {\,} & {\,} \\ {\,} & {\,} & {\,} & {\,} & {C_{b\theta y} } & {\,} \\ {\,} & {\,} & {\,} & {\,} & {\,} & {C_{b\theta z} } \\ \end{array} } \right] \\ \end{aligned} $$

$$ {\varvec{K}}_b {\varvec{q}}_j + {\varvec{C}}_b {\varvec{q}}_j^{\prime} = 0 $$

(1)

where K_b and C_b are the besaring stuffiness matrix and bearing damping matrix respectively. Abtained by Ref. [39], the stiffness coefficients along the y and z directions (K_by and K_bz) are taken to be 1 × 10¹⁰ and the damping coefficients is assumed to be 0.1

2.1.4 Modeling of cranks

Each crank is assumed to be rigid and subjected to concentrated mass treatment. The mass m_x = m_y = m_z corresponds to the three translational directions of x, y, and z. The moment of inertia about the three rotational directions of x, y, and z is I_xk, I_yk, and I_zk. k is for the k-th crank.

$$ \begin{aligned} {\varvec{M}}_d &= \left[ {\begin{array}{*{20}l} {m_x } & 0 & 0 & 0 & 0 & 0 \\ 0 & {m_y } & 0 & 0 & 0 & 0 \\ 0 & 0 & {m_z } & 0 & 0 & 0 \\ 0 & 0 & 0 & {I_x } & 0 & 0 \\ 0 & 0 & 0 & 0 & {I_y } & 0 \\ 0 & 0 & 0 & 0 & 0 & {I_z } \\ \end{array} } \right] \\ {\varvec{J}}_d &= \left[ {\begin{array}{*{20}l} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {I_x } \\ 0 & 0 & 0 & 0 & { - I_x } & 0 \\ \end{array} } \right] \\ \end{aligned} $$

For a crank at point k, the differential equation of motion in static coordinates is shown in Eq. (2).

$$ {\varvec{M}}_d {\varvec{q}}_k^\prime \prime + \omega {\varvec{J}}_d {\varvec{q}}_k^{\prime} = {\varvec{F}}_{gk} + {\varvec{F}}_{ik} + {\varvec{F}}_{fk} $$

(2)

in which, F_gk, F_ik and F_fk are the vector of gravity, the combustion force and the friction force on the k-th crank respectively. ω is crankshaft speed, M_d is crank's generalized mass matrix, and J_d is crank's gyro matrix.

2.1.5 Modeling of excitations

As illustrated in Fig. 5(a), F_i is the combustion pressure in the combustor. F_f is the friction resistance including the valve seat percussion (F_fv), piston ring friction (F_fr) and piston skirt friction (F_fs). The above forces can be converted into the T (tangential moment) and R (radial force) in the crankshaft by transfer functions (f_t1 and f_r1) respectively. Force transfer process is shown in Fig. 5(b), the relationship between θ (crank angle) and βs (swing angle of the connecting rod) can be obtained from the triangle sine law:

$$ \frac{\sin \beta_s }{{\sin \theta }} = \frac{r}{l_n } = h_b $$

(3)

$$ \sin \beta_s = h_b \sin \theta $$

(4)

Fig. 5

Schematic diagram of various excitations and force transfer process in the engine cylinder. a Various excitations b force transfer process

The resultant force p acting on the piston pin at point A can be decomposed into two component forces: p_H and p_C:

$$ p_H = p \times \tan \beta_s $$

(5a)

$$ p_C = \frac{p}{\cos \beta_s } $$

(5b)

Substituting Eq. (4) into Eqs. (5a–5b) yields:

$$ p_C = \frac{p}{{\sqrt {1 - h_b^2 \sin^2 \theta } }} $$

(6a)

$$ p_H = p \times \frac{\sin \theta }{{\sqrt {{\frac{1}{h_b^2 } - \sin^2 \theta }} }} $$

(6b)

The p_H force acts perpendicular to the cylinder's centerline, exerting pressure that pushes the piston against the cylinder wall, resulting in what is known as piston side thrust. The magnitude of p_H reflects, to some extent, the degree of wear between the piston and the cylinder liner. On the other hand, p_C represents the force along the centerline of the connecting rod, termed the connecting rod thrust. This force reflects the compression and tensile loads experienced by the connecting rod.

The connecting rod thrust p_C is transferred to the crank pin center point B along the central line of the connecting rod, and can be decomposed into two vertical components p_T and R:

$$ p_T = p_C \sin \left( {\theta + \beta_s } \right) = p\frac{{\sin \left( {\theta + \beta_s } \right)}}{\cos \beta_s } $$

(7a)

$$ R = p_C \cos \left( {\theta + \beta_s } \right) = p\frac{{\cos \left( {\theta + \beta_s } \right)}}{\cos \beta_s } $$

(7b)

The action direction of p_T is tangent to the crank circle, which is called tangential force. R is perpendicular to the crank circle and is the centripetal force along the direction of the crank, called the radial force.

Substituting Eq. (4) into Eq. (7a) yields:

$$ \begin{aligned} p_T & = p\left( {\sin \theta + \frac{h_b }{2}\sin 2\theta - \frac{1}{{\sqrt {1 - h^2 \sin^2 \theta } }}} \right) \\ & \approx p\left( {\sin \theta + \frac{h_b }{2}\sin 2\theta } \right) \\ {\rm{ }} & = p\left[ {\sin \left( \theta \right) + \cos \left( \theta \right)\sqrt {{\frac{1 - \lambda }{\lambda }}} } \right] \\ \end{aligned} $$

(8)

λ is the geometric function, which is shown in Eq. (9).

$$ \lambda = 1 - \left( {\frac{{r\sin \left( {\omega t} \right)}}{l}} \right)^2 $$

(9)

According to Eq. (7), the indicated torque (T_i) and the radial indicated force (R_i) are expressed in Eqs. (10a–10b).

$$ T_i = rp_i \frac{\pi }{4}d^2 f_{t1} $$

(10a)

$$ R_i = p_i \frac{\pi }{4}d^2 f_{r1} $$

(10b)

where,

$$ f_{t1} = \sin \left( {\omega t} \right) + \cos \left( {\omega t} \right)\sqrt {{\frac{1 - \lambda }{\lambda }}} $$

(11a)

$$ f_{r1} = \frac{{\cos \left( {\omega t + \beta } \right)}}{\cos \beta } $$

(11b)

Equations (11a–11b) are the transfer functions derived by Eqs. (7–9).

Obtained by Ref. [40], to simplify the analysis, the valve train is considered as a system represented by the cam/follower and is assumed to operate in the boundary lubrication mode. The valve train friction torque and radial force are formulated to depend on the load and the engine speed and is given by Eq. (12). d_i is wire diameter of the intake or exhaust valve spring, d_o is mean coil diameter of the intake or exhaust valve spring, G_v is nominal modulus of rigidity of valve train spring, N_t is number of active coils in the intake or exhaust valve spring, V₁ is valve lift, N_v is number of intakes of exhaust valves and c₃ is empirical coefficients.

$$ T_{{\text{fv}}} = \left( {\frac{{d_{\text{i}}^4 G_{\text{v}} }}{{8d_{\text{o}}^3 N_{\text{t}} }}} \right)V_1 N_{\text{v}} \left( {1 - c_3 } \right)r\omega \left| {f_{t1} } \right| $$

(12a)

$$ R_{{\text{fv}}} = \left( {\frac{{d_{\text{i}}^4 G_{\text{v}} }}{{8d_{\text{o}}^3 N_{\text{t}} }}} \right)V_1 N_{\text{v}} \left( {1 - c_3 } \right)\omega \left| {f_{r1} } \right| $$

(12b)

In reciprocating internal combustion engines, the design of the piston skirt profile serves multiple purposes. Firstly, it aims to prevent local contact resulting from thermal expansion, thereby mitigating friction losses and piston slap. This is achieved by ensuring the maintenance of an adequate lubricating oil film thickness to maximize the wedge effect, along with an appropriate piston pin offset. The piston skirt friction torque and radial force can be derived by applying Newton’s law for viscous friction. Mathematically, these can be expressed as Eq. (13). In which, μ is dynamic viscosity of the oil, O_c is oil clearance, L_s is skirt length.

$$ T_{{\text{fs}}} = \frac{{dL_{\text{s}} \mu \omega r^2 f_{t1}^2 }}{{O_{\text{c}} }} $$

(13a)

$$ R_{{\text{fs}}} = \frac{{dL_{\text{s}} \mu \omega rf_{r1}^2 }}{{O_{\text{c}} }} $$

(13b)

The ring assembly lubrication mode is assumed to be hydrodynamic, the ring assembly friction torques can be found by assuming that the friction force is equal to the product of the normal load between the ring assembly and the liner, and the friction coefficient. The tangential and the radial components of the piston ring friction force F_fr are expressed in Eq. (14). E_i is nominal modulus of elasticity of piston ring, g is gap closure of the piston ring, d_r is piston ring diameter, B_i is width of the ring in the direction of motion. f₂ and f₃ are cylinder torque transfer functions.

$$ T_{{\text{fr}}} = \gamma_r r\left| {f_{t1} } \right|\left\{ {\sum_{i = 1}^N {\left[ {\frac{E_i g}{{7.07d_{\text{r}} \left( {d_{\text{r}} /B_i - 1} \right)^3 }}} \right]} \pi d_{\text{r}} B_i + \sum_{i = 1}^N {a_i } \left| {p_{\text{i}} } \right|\pi d_{\text{r}} B_i + \frac{{\left| {p_{\text{i}} } \right|(\pi /4)d^2 - m_j f_2 \omega^2 }}{\gamma_r + f_3 }} \right\} $$

(14a)

$$ R_{{\text{fr}}} = \gamma_r \left| {f_{r1} } \right|\left\{ {\sum_{i = 1}^N {\left[ {\frac{E_i g}{{7.07d_r \left( {d_r /B_i - 1} \right)^3 }}} \right]} \pi d_r B_i + \sum_{i = 1}^N {a_i } \left| {p_i } \right|\pi d_r B_i + \frac{{\left| {p_i } \right|(\pi /4)d^2 - m_j f_2 \omega^2 }}{\gamma_r + f_3 }} \right\} $$

(14b)

where,

$$ f_2 = r\left\{ {\cos \left( {\omega t} \right)\left[ {1 + \frac{{\left( {r/l} \right)\cos \left( {\omega t} \right)}}{{\lambda^{3/2} }}} \right] - \sqrt {{\frac{1 - \lambda }{\lambda }}} \sin \left( {\omega t} \right)} \right\} $$

(15)

$$ f_3 = \frac{{l\sqrt {{1 - \left\{ {\left[ {r\sin \left( {\omega t} \right)} \right]^2 } \right\}^2 }} }}{{r\left| {\sin \left( {\omega t} \right)} \right|}} $$

(16)

The coefficient of friction for hydrodynamic lubrication, denoted as γ_r, exhibits a direct proportionality to the piston speed and the viscosity of the lubricating oil. Conversely, it shows an inverse proportionality to the ring load. Further details can be found in reference [41]. Thus

$${\gamma _r} = \left\{ {\begin{array}{*{20}l} {{c_1} - \left( {{c_1} - {z_h}} \right)\left| {\sin \theta } \right|,} & {{\text{for}}\;1.5\pi \leqslant \theta \leqslant 2.5\pi } \\ {{z_h},} & {{\text{otherwise}}} \\ \end{array}} \right.$$

(17)

where z_h is the hydrodynamic friction coefficient given by:

$$ z_h = \sqrt {{\frac{{\mu \omega r\left| {f_{t1} \left( {\omega t} \right)} \right|}}{L_r }}} $$

(18)

and L_r is the load per unit length given by:

$$ L_r = \sum_{i = 1}^N {\left[ {\frac{E_i g}{{0.07d_r \left( {{{d_r } / {B_i - 1}}} \right)^3 }} + \left| {p_i } \right|} \right]} B_i $$

(19)

2.1.6 Assembly and construction of the crankshaft mathematical model

Mathematical models including journal, bearing, crank, crack and external excitation have been constructed in the previous section. As shown in Fig. 6, the crankshaft is broken into a stepped shaft consisting of 12 elements including 13 nodes based on the FEM. The assembly process of each part (the mass matrix, stiffness matrix, damping matrix, and gyroscopic matrix) is shown in Fig. 7. As shown in Fig. 7(b), the local stiffness matrix (K_e-i) should be replaced at the crack location (K_ce-i) due to the local influence of crack on stiffness matrix. The damping matrix will also be updated after this process. The dynamics equation of the crankshaft model is shown in Eq. (20) after the combination of each element.

$$ {\varvec{M}}{\varvec{q}}^\prime \prime + \left( {{\varvec{C}} + \omega {\varvec{J}}} \right){\varvec{q}}^{\prime} + {\varvec{K}}{\varvec{q}} = {\varvec{F}} $$

(20)

where,

$$ {\varvec{C}} = \alpha_d {\varvec{M}} + \beta_d {\varvec{K}} $$

(21)

Fig. 6

Schematic diagram of crankshaft model constructed using FEM

Fig. 7

Matrix construction diagram of the crankshaft; a mass matrix b stiffness matrix c gyroscopic matrix d damping matrix

The damping matrix C is the Rayleigh damping, α_d and β_d are damping scaling coefficients. F is the external excitations. The gravity distribution of the entire crankshaft is assumed to be concentrated at four crank positions. Therefore, the expression of F is shown in Eq. (22),

$$ {\varvec{F}} = \left[ {\begin{array}{*{20}l} 0 & 0 & 0 & {{\varvec{F}}_g + {\varvec{F}}_i + {\varvec{F}}_f } & 0 & {{\varvec{F}}_g + {\varvec{F}}_i + {\varvec{F}}_f } & 0 & {{\varvec{F}}_g + {\varvec{F}}_i + {\varvec{F}}_f } & 0 & {{\varvec{F}}_g + {\varvec{F}}_i + {\varvec{F}}_f } & 0 & 0 & 0 \\ \end{array} } \right]^T $$

(22)

2.2 Model validation

Figure 8(a) and (b) present comparisons of the first three order free vibration modes of the crankshaft obtained from finite element software and a model comprised of Timoshenko beam elements, respectively. In the finite element analysis, these modes are identified as the first-order bending vibration on the y–x plane, the first-order bending vibration in the z–x plane, and the second-order bending vibration in the y–x plane. Cracks in diesel crankshafts can occur in the crank arm, crank pin, and nearby main journal due to factors such as their proximity to the big end of the crankpin and the transmission of combustion pressure from the cylinder to the crankshaft through the crank connecting rod mechanism, particularly through the big end of the connecting rod to the crankpin [31, 42]. These cracks can vary in complexity, exhibiting different leading-edge shapes, angles of inclination, and depths [43, 44]. This study primarily focuses on discussing straight cracks and slant cracks that develop on the main journal. Three crack states, as outlined in Table 1, are considered in this work. Notably, the crack depth does not exceed the radius of the shafting, the inclination angle of the crack does not surpass π/2, and the crack front remains straight and parallel to the z' axis, as depicted in Fig. 3(c). x_t represents the distance between the crack center line and the left end face of the crankshaft. Three kinds of crack models with different crack depth, crack inclination angle and crack location are introduced respectively, and compared with the finite element software for verification. Their natural frequencies are compared in Table 2, which displays a good consistency.

Fig. 8

Comparison of the first three free modes of a crack-free crankshaft: a finite element software b the method in this work; 1–3 respectively represent 1–3 mode shapes

Table 1 The list of different types of cracks

Table 2 Comparison of crankshaft natural frequency between the proposed method and finite element software under different types of cracks

When the crankshaft is running, the breathing crack model is introduced to calculate. The Newmark-β method is used to calculate the crankshaft dynamic response. The correctness of the calculation method has been verified in the previous work [45].

3 Engine operating conditions and breathing crack stiffness characteristics

3.1 Engine operating conditions

The main parameters of the model are shown in “Appendix 4”. The cylinder pressure curves of the engine at 1000 rpm and 55 kw power under six different fuel injection advance angles are shown in Fig. 9. Take the average value of six cylinder pressure curves as input. Idling speed in the flameout state, normal operation, single-cylinder flameout and abnormal firing sequence are four common engine behavior conditions. Figure 10 is a schematic diagram of radial force and tangential moment at four axial diameters changing with crankshaft angle under the later three working conditions. Figure 10(a), (b) and (c) respectively corresponds to normal operation, single cylinder flameout and abnormal firing sequence three operating conditions. At the idling speed in the flameout state, the external excitation is much less than the other three conditions, which is caused by the stop of the oil injection in the cylinder. And the influence of crankshaft dead weight and frictional torsional torque transmitted by the cylinder is considered in this situation. Numbers 1–4 in Fig. 10 indicate the firing sequence of the 4 cylinders.

Fig. 9

The cylinder pressure curves within 720° crank angle

Fig. 10

Diagram of the external force on the crank pin varying with the angle: a normal operation b single cylinder flame-out c abnormal firing sequence (1) moment of torque (2) radial force

3.2 The effect of breathing crack on stiffness

Each stiffness matrix with crack is composed of 144 stiffness values. In this work, three different cracks are considered, so that a total of 432 varying stiffness values are considered. The change of stiffness on the main diagonal has the most obvious influence on the stiffness matrix [37]. Therefore, the effects of different types of respiratory cracks on the principal stiffness will be discussed in this section. As shown in Fig. 11, the blue, orange and yellow solid lines represent crack 1, crack 2 and crack 3 respectively. It can be seen that the stiffness value varies with the angle of the crankshaft. The stiffness of crack 3 is significantly higher than that of crack 1 and crack 2. This is because crack 3 is located on the main journal, which has a larger diameter compared with other journals. In addition, it can be inferred that different types of cracks have different sensitivity to stiffness changes. The slant crack has great influence on k₁₁, k₂₂, k₃₃, k₅₅ and k₆₆ but little influence on k₄₄. However, the effect of straight crack on k₄₄ is more obvious than that of slant crack. At the same time, the stiffness curve of the straight crack is symmetrical, but the slant crack has no such characteristic. This is because the stress state of slant crack is more complex than that of straight crack under the same loading condition.

Fig. 11

The main diagonal stiffness varies with the crankshaft angle in three different crack states: a–f K₁₁–K₆₆

3.3 The effect of breathing crack on crankshaft lateral vibration in the time domain

As shown in Fig. 12, it can be found that the vibration response of the system is affected by different crack types, and this effect is most obvious when the system is running in idling speed condition. When the engine is in idling state, the introduction of the breathing crack changes both the frequency and amplitude of the steady-state response, which indicates that the breathing crack has the most obvious effect on the lateral vibration response of the crankshaft. By comparing the three figures in Fig. 12(a), it can be found that the response of breathing crack and static crack to transverse vibration of crankshaft has different sensitivity when different crack types (crack 1, crack 2 and crack 3) are considered. Slant cracks and deep cracks have more distinct dynamic characteristics. However, when considering normal firing, single cylinder flameout and abnormal firing sequence, the introduction of variable stiffness crack only changes the amplitude of the system response but does not change the frequency of the system response, which means that the effects of breathing crack and static crack on lateral vibration response of crankshaft are not obvious in these conditions. The longitudinal observation of the transverse vibration of the crankshaft under different working conditions shows that, compared with the normal firing sequence, single cylinder flameout weakens the transverse vibration amplitude of the crankshaft, while the abnormal firing sequence strengthens the transverse vibration amplitude of the crankshaft. The analysis in this section is performed under the condition of a single speed, and only the signal in the time domain is considered. In the next section, the influence of crack on the transverse vibration of the crankshaft under varying working conditions will be further discussed.

Fig. 12

The effects of breathing crack and static crack on transverse vibration of the crankshaft in time domain: a idling speed in the flameout state b normal firing c single cylinder flame-out d abnormal firing sequence (1) Crack-1 (2) Crack-2 (3) Crack-3

4 The effect of engine operating condition coupled with crack condition on crankshaft lateral vibration

4.1 The effect of four different conditions on the lateral vibration of the crankshaft without considering crack

As shown in Fig. 13 subgraphs (1–4) respectively represent four operating conditions of the engine: idling speed in the flameout state, normal firing, single cylinder flame-out and abnormal firing sequence. Figure 13(a) and (b) show the y and z direction transverse vibration of node 1 respectively.

Fig. 13

The comparison of the waterfall diagrams of the crankshaft transverse vibration under different running states: a y-direction b z-direction

First of all, transverse vibration in y direction is considered. It can be seen from Fig. 13(a) that there is no higher harmonic when the engine is at idling speed in the flameout state. This is because when there is no crack, the frictional torsional moment in the torsional direction is not reflected in the translational direction (y–z direction). When the engine fires normally, harmonic vibrations of 1/2X, 3/2X and 5/2X appear in the waterfall diagrams. When engine single cylinder flameout, 1X, 2X, 3X and other integral multiple harmonics are obviously enhanced. It should be pointed out that changing the firing sequence of the engine will further enhance the 1X, 2X, 3X harmonic amplitudes. It is worth noting that the frequency response function always reaches its highest peak near 286 Hz. This is because the system response is close to the first natural frequency, and the first natural frequency is the first bending vibration in the y–x plane, so the amplitude in the y direction is more obvious. Secondly, transverse vibration in z direction is considered. z direction vibration has a higher frequency than y direction vibration, and mostly concentrates around 426 Hz. This is because the mode of the second order natural frequency is mainly z–x plane bending vibration.

At the same time, similar to the previous analysis, 1X, 2X, 3X and other integral harmonics are significantly enhanced under single cylinder flameout and abnormal-ignition sequence fault conditions. However, unlike y direction vibration, this enhancement is only present in the speed range above 3000 RPM. Therefore, it can be said that the vibration response in the y direction is more sensitive than that in the z-direction. At the same time, the enhancement of integral harmonics can be regarded as the criterion of single cylinder flout and ignition sequence change of four-cylinder engine, no matter in y or z direction response.

4.2 The effect of crack on the lateral vibration of crankshaft under idling condition

Under idling speed in the flameout state, the vibration response of node 1 is shown in Fig. 14. Where, Fig. 14(a), (b) and (c) respectively represent cracks 1–3. Subscripts (1) and (2) in Fig. 14 represent the vibration response in the y direction and the z direction, respectively. By observing Fig. 14(a–c), it can be found that the appearance of cracks is accompanied by the generation of resonance peaks. The integer multiples of harmonics, such as 1X, 2X and 3X, become more obvious with the increase of crack inclination and depth no matter in y or z direction. This is because crack propagation deepens the bent-torsional coupling so that the response generated by the alternating torque in the torsional direction can be reflected in the translational direction. It can be seen from the observation of Fig. 14(c) that the 1X harmonic component is significantly enhanced by crack 3. Besides, an umbrella-like order with 426HZ as the central frequency is produced in the waterfall diagram of crack 3, and this phenomenon is more obvious in the z direction vibration. The frequency of umbrella order is mainly composed of f₂− X and f₂− 2X, f₂− 3X, f₂− 4X, f₂ + X, f₂ + 2X, f₂ + 3X and f₂ + 4X. f₁ and f₂ are the first and second natural frequencies of crankshaft under elastic support respectively. This indicates that the crack located on the crankshaft main journal is easier to be identified. By comparing the responses in y direction and z direction, it can be found that the harmonic component in y direction is significantly better than the harmonic component in z direction (y direction has a higher amplitude), which is especially obvious in Fig. 14(b). This phenomenon occurs because y direction vibration primarily consists of low-frequency components, whereas z direction vibration mainly comprises high-frequency components. Consequently, harmonic components in the z direction are more prominently displayed only in high-speed regions.

Fig. 14

The comparison of the waterfall diagrams of the crankshaft transverse vibration under idling speed condition considering different types of cracks: (1) y-direction (2) z-direction a crack-1 b crack-2 c crack-3

4.3 The effect of crack on the lateral vibration of crankshaft under normal condition

Under the normal running state, the vibration response of node 1 is shown in Fig. 15. Where, Fig. 15(a), (b) and (c) respectively represent cracks 1–3. Subscripts (1) and (2) in Fig. 15 represent the vibration response in the y direction and the z direction, respectively. By comparing Figs. 13(a-2), (b-2) and 15, it can be found that when the engine runs normally, no matter in y direction or z direction, the fractional harmonic signal generated by the engine will cover up the integer harmonic signal generated by the crack. Under the fault condition, the characteristic signal of crack is also covered. It can be seen that the detection of crankshaft cracks is easy to be obscured by other interference signals, so it is better to identify cracks in the idling speed condition of the engine.

Fig. 15

The comparison of the waterfall diagrams of the crankshaft transverse vibration under normal running states considering different types of cracks: (1) y-direction (2) z-direction a crack-1 b crack-2 c crack-3

To sum up, the crankshaft will show different vibration states under different engine working conditions. According to the above analysis, the harmonic component and resonance state of crankshaft transverse vibration response are distinguishing under different working conditions. This provides a theoretical basis and guidance for identifying crankshaft cracks and engine faults.

5 Conclusions

In this work, a four-cylinder crankshaft model with breathing cracks is established. The stepped shaft model is assembled by Timoshenko beam elements of different sections to fit the crankshaft model. The influences of multiple external excitations including combustion in the cylinder, friction and gravity are considered. Based on the verified model, Newmark- β method is used to solve the dynamic response of the crankshaft numerically. Finally, the transverse vibration of the crankshaft with cracks is analyzed by using the waterfall diagrams under the four-engine operating conditions, including idling speed in the flameout state, normal firing, single-cylinder flameout and abnormal firing sequence. The results show that:

• (1) At idling speed, the high-order harmonic component is not obvious without considering cracks. At this time, if the crack appears, the 1X, 2X and 3X harmonic and the umbrella-like order with 426HZ as the central frequency will be enhanced.

• (2) Engine single-cylinder flameout and abnormal firing sequence fault conditions will also increase the integer multiple harmonic amplitudes. Therefore, in engine firing operation, the single-cylinder flameout and abnormal firing sequence of the engine can be judged according to the degree of integer multiple harmonic increase.

• (3) When the engine fires, the higher harmonics are enhanced obviously, and the harmonic components caused by the crack are covered by the interference signal of the engine itself. Therefore, in engineering practice, crack identification of the crankshaft under idling speed condition is recommended.

Appendices

Appendix 1

The element mass, stiffness and gyro matrix of the system without considering crack condition.

$$ {\varvec{M}}_e = \rho Al\left[ {\begin{array}{*{20}l} \frac{1}{3} & 0 & 0 & 0 & 0 & 0 & \frac{1}{6} & 0 & 0 & 0 & 0 & 0 \\ 0 & {a_m } & 0 & 0 & 0 & {c_m } & 0 & {b_m } & 0 & 0 & 0 & { - d_m } \\ 0 & 0 & {a_m } & 0 & { - c_m } & 0 & 0 & 0 & {b_m } & 0 & {d_m } & 0 \\ 0 & 0 & 0 & {\frac{I_0 }{{3A}}} & 0 & 0 & 0 & 0 & 0 & {\frac{I_0 }{{6A}}} & 0 & 0 \\ 0 & 0 & { - c_m } & 0 & {e_m } & 0 & 0 & 0 & { - d_m } & 0 & { - f_m } & 0 \\ 0 & {c_m } & 0 & 0 & 0 & {e_m } & 0 & {d_m } & 0 & 0 & 0 & { - f_m } \\ \frac{1}{6} & 0 & 0 & 0 & 0 & 0 & \frac{1}{3} & 0 & 0 & 0 & 0 & 0 \\ 0 & {b_m } & 0 & 0 & 0 & {d_m } & 0 & {a_m } & 0 & 0 & 0 & { - c_m } \\ 0 & 0 & {b_m } & 0 & { - d_m } & 0 & 0 & 0 & {a_m } & 0 & {c_m } & 0 \\ 0 & 0 & 0 & {\frac{I_0 }{{6A}}} & 0 & 0 & 0 & 0 & 0 & {\frac{I_0 }{{3A}}} & 0 & 0 \\ 0 & 0 & {d_m } & 0 & { - f_m } & 0 & 0 & 0 & {c_m } & 0 & {e_m } & 0 \\ 0 & { - d_m } & 0 & 0 & 0 & { - f_m } & 0 & { - c_m } & 0 & 0 & 0 & {e_m } \\ \end{array} } \right] $$

(23)

$$ {\varvec{K}}_e = \left[ {\begin{array}{*{20}l} \frac{AE}{l} & 0 & 0 & 0 & 0 & 0 & { - \frac{AE}{l}} & 0 & 0 & 0 & 0 & 0 \\ 0 & {h_k } & 0 & 0 & 0 & {i_k } & 0 & { - h_k } & 0 & 0 & 0 & {i_k } \\ 0 & 0 & {h_k } & 0 & { - i_k } & 0 & 0 & 0 & { - h_k } & 0 & { - i_k } & 0 \\ 0 & 0 & 0 & {\frac{G_e I_0 }{l}} & 0 & 0 & 0 & 0 & 0 & { - \frac{G_e I_0 }{l}} & 0 & 0 \\ 0 & 0 & { - i_k } & 0 & {j_k } & 0 & 0 & 0 & {i_k } & 0 & {k_k } & 0 \\ { - \frac{AE}{l}} & 0 & 0 & 0 & 0 & 0 & \frac{AE}{l} & 0 & 0 & 0 & 0 & 0 \\ 0 & { - h_k } & 0 & 0 & 0 & { - i_k } & 0 & {h_k } & 0 & 0 & 0 & { - i_k } \\ 0 & 0 & { - h_k } & 0 & {i_k } & 0 & 0 & 0 & {h_k } & 0 & {i_k } & 0 \\ 0 & 0 & 0 & { - \frac{G_e I_0 }{l}} & 0 & 0 & 0 & 0 & 0 & {\frac{G_e I_0 }{l}} & 0 & 0 \\ 0 & 0 & { - i_k } & 0 & {k_k } & 0 & 0 & 0 & {i_k } & 0 & {j_k } & 0 \\ 0 & {i_k } & 0 & 0 & 0 & {k_k } & 0 & { - i_k } & 0 & 0 & 0 & {j_k } \\ \end{array} } \right] $$

(24)

$$ {\varvec{G}}_e = 2\rho Al\left[ {\begin{array}{*{20}l} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {p_g } & 0 & {q_g } & 0 & 0 & 0 & { - p_g } & 0 & {q_g } & 0 \\ 0 & { - p_g } & 0 & 0 & 0 & {q_g } & 0 & {p_g } & 0 & 0 & 0 & {q_g } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & { - q_g } & 0 & 0 & 0 & {s_g } & 0 & {q_g } & 0 & 0 & 0 & {w_g } \\ 0 & 0 & { - q_g } & 0 & { - s_g } & 0 & 0 & 0 & {q_g } & 0 & { - w_g } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & { - p_g } & 0 & { - q_g } & 0 & 0 & 0 & {p_g } & 0 & { - q_g } & 0 \\ 0 & {p_g } & 0 & 0 & 0 & { - q_g } & 0 & { - p_g } & 0 & 0 & 0 & { - q_g } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & { - q_g } & 0 & 0 & 0 & {w_g } & 0 & {q_g } & 0 & 0 & 0 & {s_g } \\ 0 & 0 & { - q_g } & 0 & { - w_g } & 0 & 0 & 0 & {q_g } & 0 & { - s_g } & 0 \\ \end{array} } \right] $$

(25)

where

$$ a_{\text{m}} = \left[ {13/35 + 7\phi /10 + \phi^2 /3 + 6\left( {r_{\text{g}} /l} \right)^2 /5} \right]/(1 + \phi )^2 $$

(26)

$$ b_{\text{m}} = \left[ {9/70 + 3\phi /10 + \phi^2 /6 - 6\left( {r_{\text{g}} /l} \right)^2 /5} \right]/(1 + \phi )^2 $$

(27)

$$ c_{\text{m}} = l\left[ {11/210 + 11\phi /120 + \phi^2 /24 + (1/10 - \phi /2)\left( {r_{\text{g}} /l} \right)^2 } \right]/(1 + \phi )^2 $$

(28)

$$ d_{\text{m}} = l\left[ {13/420 + 3\phi /40 + \phi^2 /24 - (1/10 - \phi /2)\left( {r_{\text{g}} /l} \right)^2 } \right]/(1 + \phi )^2 $$

(29)

$$ e_{\text{m}} = l^2 \left[ {1/105 + \phi /60 + \phi^2 /120 + \left( {2/15 + \phi /6 + \phi^2 /3} \right)\left( {r_{\text{g}} /l} \right)^2 } \right]/(1 + \phi )^2 $$

(30)

$$ f_{\text{m}} = l^2 \left[ {1/140 + \phi /60 + \phi^2 /120 + \left( {1/30 + \phi /6 + \phi^2 /6} \right)\left( {r_{\text{g}} /l} \right)^2 } \right]/(1 + \phi )^2 $$

(31)

$$ i_{\text{k}} = 6EI_z /\left[ {l^2 (1 + \phi )} \right] $$

(32)

$$ j_{\text{k}} = (4 + \phi )EI_z /[l(1 + \phi )] $$

(33)

$$ k_{\text{k}} = (2 - \phi )EI_z /[l(1 + \phi )] $$

(34)

$$ p_{\text{g}} = 6r_{\text{g}}^2 /\left[ {5l^2 (1 + \phi )^2 } \right] $$

(35)

$$ q_{\text{g}} = - (1/10 - \phi /2)r_{\text{g}}^2 /\left[ {l^2 (1 + \phi )^2 } \right] $$

(36)

$$ s_{\text{g}} = \left( {2/15 + \phi /6 + \phi^2 /3} \right)r_{\text{g}}^2 /(1 + \phi )^2 $$

(37)

$$ w_{\text{g}} = - \left( {1/30 + \phi /6 - \phi^2 /6} \right)r_{\text{g}}^2 /(1 + \phi )^2 $$

(38)

where Φ is transverse shear parameter, k is form factor and r_g is radius of gyration. Their expression is shown as:

$$ \phi = \frac{12EI_z }{{kAG_e l}}\quad k = \frac{{6\left( {1 + v} \right)}}{7 + 6v}\quad r_g = \sqrt {{\frac{I_z }{A}}} $$

(39)

Appendix 2

T_t is the transfer matrix. It can be written as:

$$ {\varvec{T}}_t = \left[ {\begin{array}{*{20}l} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & {\cos \varphi } & {\sin \varphi } & 0 & 0 & 0 \\ 0 & { - \sin \varphi } & {\cos \varphi } & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\cos \varphi } & {\sin \varphi } \\ 0 & 0 & 0 & 0 & { - \sin \varphi } & {\cos \varphi } \\ \end{array} } \right] $$

(40)

$$ {\varvec{T}}_g = diag\left\{ {{\varvec{T}}_t ,{\varvec{T}}_t } \right\} $$

(41)

$$ {\varvec{K}}_{ce} = {\varvec{T}}_g \left( {{\varvec{S}}^0 + {\varvec{S}}^c } \right){\varvec{T}}_g^T $$

(42)

where φ is the rotation angle of the shaft element. S⁰ is the flexibility matrix of the intact shaft element; S^c is the flexibility matrix of the crack shaft element.

$$ {\varvec{S}}^0 = \left[ {\begin{array}{*{20}l} \frac{l}{EA} & {\,} & {\,} & {\,} & {\,} & {\,} \\ 0 & {\frac{\alpha_s l}{{G_e A}} + \frac{l^3 }{{3EI_z }}} & {\,} & {\,} & {sym} & {\,} \\ 0 & 0 & {\frac{\alpha_s l}{{G_e A}} + \frac{l^3 }{{3El_y }}} & {\,} & {\,} & {\,} \\ 0 & 0 & 0 & {\frac{l}{G_e I_o }} & {\,} & {\,} \\ 0 & 0 & {\frac{l^2 }{{2EI_y }}} & 0 & {\frac{l}{El_y }} & {\,} \\ 0 & { - \frac{l^2 }{{2EI_z }}} & 0 & 0 & 0 & {\frac{l}{EI_z }} \\ \end{array} } \right] $$

(43)

$$ {\varvec{S}}^c = \left[ {\begin{array}{*{20}l} {g_{11}^c } & {\,} & {\,} & {\,} & {\,} & {\,} \\ {g_{21}^c } & {g_{22}^c } & {\,} & {\,} & {sym} & {\,} \\ {g_{31}^c } & {g_{32}^c } & {g_{33}^c } & {\,} & {\,} & {\,} \\ {g_{41}^c } & {g_{42}^c } & {g_{43}^c } & {g_{44}^c } & {\,} & {\,} \\ {g_{51}^c } & {g_{52}^c } & {g_{53}^c } & {g_{54}^c } & {g_{55}^c } & {\,} \\ {g_{61}^c } & {g_{62}^c } & {g_{63}^c } & {g_{64}^c } & {g_{65}^c } & {g_{66}^c } \\ \end{array} } \right] $$

(44)

where

$$ g_{11}^c = \frac{2}{{E^{\prime} \pi R^4 }}\iint {\left( {\sin^4 \theta_c F_1^2 + m_c \sin^2 \theta_c \cos^2 \theta_c F_{III}^2 } \right)\alpha_c }d\alpha_c d\beta_c $$

(45)

$$ g_{22}^c = \frac{2}{{E^{\prime} \pi R^8 }}\iint {\left( {4x_c^2 h\sin^4 \theta_c F_2^2 + 4m_c x_c^2 h^2 \sin^2 \theta_c \cos^2 \theta_c F_{III}^2 + k^2 R^4 \sin^2 \theta_c F_{II}^2 } \right)\alpha_c }d\alpha_c d\beta_c $$

(46)

$$ g_{33}^c = \frac{2}{{E^{\prime} \pi^8 }}\iint {\left( {\begin{array}{*{20}l} {k^2 R^4 \sin^2 \theta_c F_1^2 + 8kx_c \beta_c R^2 \sin 2\theta_c \sin^3 \theta_c F_1^2 } \\ { + 16m_c x_c^2 \beta_c^2 \cos^2 \theta_c \sin^4 \theta_c F_{III}^2 + m_c k^2 R^4 \cos^2 2\theta_c F_{III}^2 } \\ { + 8kx_c \beta_c R^2 \cos \theta_c \sin^2 \theta_c F_{III}^2 + 16x_c^2 \beta_c^2 \sin^6 \theta_c F_1^2 } \\ \end{array} } \right)\alpha_c }d\alpha_c d\beta_c $$

(47)

$$ g_{44}^c = \frac{2}{{E^{\prime} \pi R^8 }}\iint {\left( {h^2 \sin^2 2\theta_c F_2^2 + m_c h^2 \cos^2 2\theta_c F_{III}^2 + 4\beta_c^2 \sin^4 \theta_c F_{II}^2 } \right)}\alpha_c d\alpha_c d\beta_c $$

(48)

$$ g_{55}^c = \frac{2}{{E^{\prime} \pi R^8 }}\iint {\left( {16\beta_c^2 \sin^6 \theta_c F_1^2 + 16m_c \beta_c^2 \cos^2 \theta_c \sin^4 \theta_c F_{III}^2 } \right)\alpha_c }d\alpha_c d\beta_c $$

(49)

$$ g_{66}^c = \frac{2}{{E^{\prime} \pi R^8 }}\iint {\left( {4h^2 \sin^4 \theta_c F_2^2 + 4m_c h^2 \cos^2 \theta_c \sin^2 \theta_c F_{III}^2 } \right)\alpha_c }d\alpha_c d\beta_c $$

(50)

$$ g_{21}^c = \frac{1}{{E^{\prime} \pi R^6 }}\iint {\left( {4x_c h\sin^4 \theta_c F_1 F_2 + 4m_c x_c h\cos^2 \theta_c \sin^2 \theta_c F_{III}^2 } \right)\alpha_c }d\alpha_c d\beta_c $$

(51)

$$ g_{31}^c = \frac{1}{{E^{\prime} \pi R^6 }}\iint {\left( {\begin{array}{*{20}l} {2kR^2 \sin 2\theta_c \sin^2 \theta_c F_1^2 + 8x_c m_c \beta_c \cos^2 \theta_c \sin^3 \theta_c F_{III}^2 } \\ { + 2kR^2 \sin \theta_c \cos \theta_c \cos 2\theta_c F_{III}^2 + 8x_c \beta_c \sin^5 \theta_c F_1^2 } \\ \end{array} } \right)\alpha_c }d\alpha_c d\beta_c $$

(52)

$$ g_{41}^c = \frac{1}{{E^{\prime} \pi R^6 }}\iint {\left( {2h\sin 2\theta_c \sin^2 \theta_c F_1 F_2 + 2m_c h\sin \theta_c \cos \theta_c \cos 2\theta_c F_{III}^2 } \right)\alpha_c }d\alpha_c d\beta_c $$

(53)

$$ g_{51}^c = \frac{2}{{E^{\prime} \pi R^6 }}\iint {\left( {8\beta_c \sin^5 \theta_c F_1^2 + 8m_c \beta_c \sin^3 \theta_c \cos^2 \theta_c F_{III}^2 } \right)\alpha_c }d\alpha_c d\beta_c $$

(54)

$$ g_{61}^c = \frac{ - 1}{{E^{\prime} \pi R^6 }}\iint {\left( {4h\sin^4 \theta_c F_1 F_2 + 4m_c h\sin^2 \theta_c \cos^2 \theta_c F_{III}^2 } \right)\alpha_c }d\alpha_c d\beta_c $$

(55)

$$ g_{32}^c = \frac{1}{{E^{\prime} \pi R^8 }}\iint {\left( {\begin{array}{*{20}l} {4x_c khR^2 \sin 2\theta_c \sin^2 \theta_c F_2^2 + 16m_c h\beta_c x_c^2 \cos^2 \theta_c \sin^3 \theta_c F_{III}^2 } \\ { + 4x_c m_c khR^2 \sin \theta_c \cos \theta_c \cos 2\theta_c F_{III}^2 + 16h\beta_c x_c^2 \sin^5 \theta_c F_1 F_2 } \\ \end{array} } \right)\alpha_c }d\alpha_c d\beta_c $$

(56)

$$ g_{42}^c = \frac{1}{{E^{\prime} \pi R^8 }}\iint {\left( {\begin{array}{*{20}l} {4x_c h^2 \sin 2\theta_c \sin^2 \theta_c F_2^2 + 4k\beta R^2 \sin^3 \theta_c F_{II}^2 } \\ { + 4x_c m_c h^2 \sin \theta_c \cos \theta_c \cos 2\theta_c F_{III}^2 } \\ \end{array} } \right)\alpha_c }d\alpha_c d\beta_c $$

(57)

$$ g_{52}^c = \frac{1}{{E^{\prime} \pi R^8 }}\iint {\left( {16x_c h\beta_c \sin^5 \theta_c F_1 F_2 + 16x_c m_c h\beta_c \cos^2 \theta_c \sin^3 \theta_c F_{III}^2 } \right)\alpha_c }d\alpha_c d\beta_c $$

(58)

$$ g_{62}^c = \frac{ - 2}{{E^{\prime} \pi R^8 }}\iint {\left( {4x_c h^2 \sin^4 \theta_c F_2^2 + 4x_c m_c h^2 \cos^2 \theta_c \sin^2 \theta_c F_{III}^2 } \right)\alpha_c }d\alpha_c d\beta_c $$

(59)

$$ g_{43}^c = \frac{1}{{E^{\prime} \pi R^8 }}\iint {\left( {\begin{array}{*{20}l} {8x_c h\beta_c \sin 2\theta_c \sin^2 \theta_c F_1 F_2 + 8x_c m_c h\beta_c \cos \theta_c \cos 2\theta_c \sin^2 \theta_c F_{III}^2 } \\ { + 2khR^2 \sin^2 2\theta_c F_1 F_2 + 2m_c khR^2 \cos^2 2\theta_c F_{III}^2 } \\ \end{array} } \right)\alpha_c }d\alpha_c d\beta_c $$

(60)

$$ g_{53}^c = \frac{1}{{E^{\prime} \pi R^8 }}\iint {\left( {\begin{array}{*{20}l} {8k\beta_c R^2 \sin 2\theta_c \sin^3 \theta_c F_1^2 + 32x_c m_c \beta_c^2 \cos^2 \theta_c \sin^4 \theta_c F_{III}^2 } \\ { + 8m_c k\beta_c R^2 \cos \theta_c \cos 2\theta_c \sin^2 \theta_c F_{III}^2 + 32x_c \beta_c^2 \sin^6 \theta_c F_1^2 } \\ \end{array} } \right)\alpha_c }d\alpha_c d\beta_c $$

(61)

$$ g_{63}^c = \frac{ - 1}{{E^{\prime} \pi R^8 }}\iint {\left( {\begin{array}{*{20}l} {4khR^2 \sin 2\theta_c \sin^2 \theta_c F_1 F_2 + 16x_c m_c h\beta_c \sin^3 \theta_c \cos^2 \theta_c F_{III}^2 } \hfill \\ {4m_c khR^2 \sin \theta_c \cos \theta_c \cos 2\theta_c F_{III}^2 + 16x_c h\beta_c \sin^5 \theta_c F_1 F_2 } \hfill \\ \end{array} } \right)\alpha_c }d\alpha_c d\beta_c $$

(62)

$$ g_{54}^c = \frac{1}{{E^{\prime} R^8 }}\iint {\left( {8h\beta_c \sin 2\theta_c \sin^3 \theta_c F_1 F_2 + 8m_c h\beta_c \cos \theta_c \cos 2\theta_c \sin^2 \theta_c F_{III}^2 } \right)\alpha_c }d\alpha_c d\beta_c $$

(63)

$$ g_{64}^c = \frac{ - 1}{{E^{\prime} \pi R^8 }}\iint {\left( {4h^2 \sin 2\theta_c \sin^2 \theta_c F_2^2 + 4m_c h^2 \sin \theta_c \cos \theta_c \cos 2\theta_c F_{III}^2 } \right)\alpha_c }d\alpha_c d\beta_c $$

(64)

$$ g_{65}^c = \frac{ - 1}{{E^{\prime} \pi R^8 }}\iint {\left( {16h\beta_c \sin^5 \theta_c F_1 F_2 + 16m_c h\beta_c \cos^2 \theta_c \sin^3 \theta_c F_{III}^2 } \right)\alpha_c }d\alpha_c d\beta_c $$

(65)

where

$$ m_c = 1 + v $$

(66)

$$ E^{\prime} = E/\left( {1 - v^2 } \right) $$

(67)

The correction coefficients of various stress intensity factors are as follows:

$$ F_1 (\alpha_c /h) = \sqrt {{\frac{2h}{{\pi \alpha_c }}\tan \left( {\frac{\pi \alpha_c }{{2h}}} \right)}} \frac{0.752 + 2.02(\alpha_c /h) + 0.37[1 - \sin (\pi \alpha_c /2h)]^3 }{{\cos (\pi \alpha_c /2h)}} $$

(68)

$$ F_2 (\alpha_c /h) = \sqrt {{\frac{2h}{{\pi \alpha_c }}\tan \left( {\frac{\pi \alpha_c }{{2h}}} \right)}} \frac{0.923 + 0.199[1 - \sin (\pi \alpha_c /2h)]^4 }{{\cos (\pi \alpha_c /2h)}} $$

(69)

$$ F_{{\text{II}}} (\alpha_c /h) = \frac{1.122 - 0.561(\alpha_c /h) + 0.085(\alpha_c /h)^2 + 0.18(\alpha_c /h)^3 }{{\sqrt {1 - (\alpha_c /h)} }} $$

(70)

$$ F_{III } (\alpha_c /h) = \sqrt {{\frac{2h}{{\pi \alpha_c }}\tan \left( {\frac{\pi \alpha_c }{{2h}}} \right)}} $$

(71)

Appendix 3

Algorithm 1

Variable stiffness codes algorithm

Appendix 4

See Table 

Table 3 Main parameters of the crankshaft model

3.