Dynamic stiffness and vibration analysis model of angular contact ball bearing considering vibration and friction state variation

2.1.1. Bearing integral force analysis

Assuming that outer ring is fixed, external (static) load $F=\left\{\begin{array}{ll}{F}_a& F_r\end{array}\right\}$ acts on the inner ring of bearing, as illustrated in Fig. 1(a), there is relative displacement $\delta =\left\{\begin{array}{ll}{\delta }_a& {\delta }_r\end{array}\right\}$ between inner ring, the contact angular changes from ${\alpha }_0$ to ${\alpha }_0^{\text{'}}$, as shown in Fig. 1(b). ${\delta }_a$ and ${\delta }_r$ is relative displacement along axial direction and radial direction, respectively.

2.1.2. Rolling element force analysis

Considering centrifugal force and gyroscopic moment, mechanical properties analysis of ball$j$is executed. As the bearing is running, ball center position changes owing to centrifugal force, thus, the contact angular between ball and inner or outer ring is different, connecting line of two raceway curvature centers turns into fold line, as depicted in Fig. 2. The location of ball center moves from $O_b$ to $O_b^{\text{'}}$. ${\alpha }_0$ is free contact angular between ball and inner or outer ring without load, ${\alpha }_{ij}$ and ${\alpha }_{oj}$ is contact angular between ball and inner or outer ring affected by centrifugal force. As for outer ring is fixed, its raceway curvature center $O_o$ keeps invariant, outer ring raceway curvature center transforms from initial location $O_i$ to $O_i^{\text{'}}$.

Fig. 1Force analysis of angular contact ball bearing

a)

b)

Fig. 2Angular position of ball j and position of ball center and raceway groove curvature center at angular position ψj with and without load

a) Azimuthal angle

b) Relative location of ball center and raceway curvature center before and after loaded

Before loaded, the distance between inner and outer ring raceway curvature center $O_i$ and $O_o$ is signed as $A_0$, as illustrated in Fig. 2(b), whose expression is depicted as follows:

The distance between inner ring raceway curvature center that is fixed and ball center that is loaded is shown as follows:

Similarly, the distance between outer ring raceway curvature center that is loaded and ball center that is loaded is shown as follows:

where, ${\delta }_{oj}$ and ${\delta }_{ij}$ is normal contact deformation of inner and outer ring raceway.

According to Fig. 1(b), ${\delta }_a$ and ${\delta }_r$ is the relative axial displacement and radial displacement between inner ring and outer ring, respectively, the axial distance $A_{1j}$ between inner ring raceway curvature center and outer ring raceway curvature center after loaded is represented as follows:

After loaded, the radial distance $A_{2j}$ between inner ring raceway curvature center and outer ring raceway curvature center is shown as follows:

According to Fig. 2(b), some expressions are drawn as follows:

Based on the Pythagorean theorem, it is illustrated as follows:

Supposing that friction is restricted in plane made up of bearing axis and ball $j$ center, the instantaneous location and load condition of the ball considering centrifugal force and gyroscopic moment are depicted in Fig. 3.

Fig. 3Instantaneous position and load of ball

a) Instantaneous location

b) Load condition

Assuming that the contact mode between inner ring and outer ring is point contact, $Q_{ij}$ and $Q_{oj}$ is Hertz contact force between inner ring and outer ring, respectively. Thus, the relationship between normal contact load and contact deformation of ball is explained as follows:

where, ${\delta }_{ij}$ and ${\delta }_{oj}$ is normal contact deformation between ball and outer ring or inner ring.

The equilibrium equations along horizontal and vertical direction are illustrated as follows:

Bringing Eq. (6) and (8) into Eq. (9):

where, in the assumption of “outer ring control”, ${\lambda }_{ij}=0$, ${\lambda }_{oj}=2$. $F_{cj}$ is the centrifugal force of ball, $M_{gj}$ is the gyroscopic moment of ball.

The centrifugal force $F_{cj}$ is:

where, ${\omega }_m$ is the orbital speed of ball, $\omega$ is the rotational speed of inner ring around bearing axis.

The gyroscopic moment $M_{gj}$ is:

where, $\beta$ is pitch angular of ball determined by Eq. (13); ${\omega }_R$ is the rotational speed around its axis; $J$ is inertia moment of ball:

where, ${\gamma }^{\text{'}}=D/d_m$.

For inner ring rotation, ${\omega }_R/\omega$ is expressed as follows:

${\omega }_m/\omega$ is expressed as follows:

2.1.3. Bearing inner ring force analysis

The force condition of bearing inner ring enduring axial load $F_a$ and radial load $F_r$ is shown in Fig. 4.

Fig. 4Loads of inner ring

The contact force between inner ring and ball in certain moment is in balance, the corresponding force equilibrium equation is as follows:

where, $n$ is the number of ball.

Submitting Eq. (6) and (8) into Eq. (16):

2.1.4. Bearing body contact stiffness analysis

The contact stiffness between ball and inner ring or outer ring is arranged in series, but it is arranged in parallel between each ball. Therefore, axial contact stiffness $K_{ca}$ influenced by axial load and radial contact stiffness $K_{cr}$ [32] affected by radial load is shown as follows:

where, $K_{c*ij}$ ($*=a,r$) is axial and radial component of normal contact stiffness between ball and inner ring; $K_{c*oj}$ is axial and radial component of normal contact stiffness between ball and outer ring. Specific expression is as follows:

where, $K_{C*j}\left(*=i,o\right)$ is normal contact stiffness between ball and inner or outer ring.

2.2.1. Angular contact ball bearing abnormal heat generation mechanism analysis

As angular contact ball bearing is under service, friction pair is wounded owing to contact fatigue accumulation, which results in increment of bearing vibration and friction moment, with the development of surface damage, bearing produces massive amount of heat. Hence, the factors having an effect on bearing abnormal heat generation are as follows:

(1) The increment of vibration load as a result of contact surface damage.

As bearing is under operation, contact load and contact frequency between bearing inner parts speed up, consequently, it enhances bearing inner heat production.

The integral vibration model of angular contact ball bearing is depicted in Fig. 5.

Fig. 5The mechanical diagram of bearing

For single-degree-of-freedom system without damp, whose vibration equation is shown as follows:

where, $x$ is the displacement vector, which is expressed as follows:

where, $A_i$ ($i=\mathrm{1,2},3$) is displacement amplitude, which represents the maximum displacement away from the equilibrium position; $A_1$, $A_2$ and $A_3$ is displacement amplitude belonging to low, medium and high frequency band, respectively.

(2) The development of surface topography.

Abrasive wear induced by spall in contact surface or abrasive contaminant in lubrication oil under angular contact ball bearing is running would accelerate the evolvement of contact surface roughness.

As surface topography decays to a certain extent, lubrication condition changes, the oil film thickness of contact surface between ball and raceway is inadequate to keep contact surface asperities of ball and raceway apart, thus, there is rub-impact between surface asperities of mating parts, which results in abnormal vibration and excessive heat production of bearing, it would cause poor lubrication, the “skid” accelerating bearing failure would appears after a further development.

Bearing lubrication state is determined by oil film lubrication parameter (film-thickness ratio) $\lambda$:

where, ${\sigma }_1$ and ${\sigma }_2$ is root-mean-square deviation of both surface roughness, respectively. They are arithmetic mean value of surface roughness $Ra$, i.e. ${\sigma }_i=1.25Ra\left(i=\mathrm{1,2}\right)$.

Lubrication oil affected by temperature in contact zone is treated as non-Newtonian fluid, whose viscosity and oil film thickness is illustrated as follows:

Owing to friction heat originated from bearing contact zone has little access to dissipate effectively, the temperature of lubrication oil would raise up. Hence, the relationship between lubrication oil viscosity and temperature is defined by viscosity-temperature equation:

where, $T_0$ is lubrication oil temperature, unit is ℃. $A_{O1}$, $A_{O2}$ and ${\beta }^{\text{'}}$ are parameters related to the lubrication oil.

Influenced by thermal effect, the lubrication oil formed in contact zone between ball and outer or inner ring raceway, whose minimum oil film thickness [33] is expressed as follows:

where, ${\phi }_t$ is oil film thermal revised parameter of contact surface. ${\phi }_{nc}$ is non-Newtonian fluid revised parameter. $h_{\mathrm{m}\mathrm{i}\mathrm{n}}$ is minimum oil film thickness of isothermal Newton fluid.

Minimum oil film thickness of isothermal Newton fluid is obtained by adopting Hamrock-Dowson equation [34]:

where, $U={\eta }_{0}u/E^{\text{'}}{R}_x$ is the non-dimensional parameter, $u$ is the surface average speed. $G={\alpha }_v{E}^{\text{'}}$ is the non-dimensional material parameter. ${\alpha }_v$ is viscosity-pressure index. $W=Q_j/E^{\text{'}}{R}_x^2$ is the non-dimensional load parameter. $R_x=1/\left(2D\left(1\mp \gamma \right)\right)$ is the equivalent curvature radius, among which, negative sign is suitable for inner ring, positive sign is suitable for outer ring. $\kappa =a/b$ is ellipticity.

Oil film thermal revised parameter is calculated utilizing Gupta equation [35, 36]:

where, $p$ is Hertz contact pressure. $L={\eta }_0{\beta }^{\text{'}}{u}^2/4k_b$, $k_b$ is thermal conductivity in inlet, ${\beta }^{\text{'}}$ is temperature-viscosity coefficient of lubrication oil.

Non-Newtonian fluid revised parameter is calculated adopting Bair equation [37]:

where, $G_r$ is shear stress of lubrication oil. ${\eta }_r$ is exponent sign. $\eta$ is dynamic viscosity of lubrication oil [38].

(3) The alteration of clearance.

The working clearance $P_d$ changes caused by thermal expansion, the corresponding expression is shown as follows:

where, ${\delta }_r^0$ is inner radial clearance affected by axial force, $\mathrm{\Delta }{\delta }_t$ is the variation of clearance due to thermal expansion, $\mathrm{\Delta }D$ is the increment of ball diameter owing to difference in temperature.

As angular contact ball bearing is in service, although ambient temperature is constant, the bearing inner and outer ring would expand affected by friction heat generation, the variation of internal radial clearance is expressed as follows:

where, $T_i$ is inner ring temperature, $T_o$ is outer ring temperature, $T_a$ is reference temperature; $d_{oi}$ is inner diameter of outer ring, $d_{io}$ is outer diameter of inner ring; ${\alpha }_t$ is coefficient of linear expansion, for bearing steel, ${\alpha }_t=1.25\times 10^{-5}$ (1/℃).

As the temperature of inner ring is higher than that of outer ring, the expansion of inner ring results in decrease of inner radial clearance, it makes normal contact force and temperature in surface of mating parts ascend. Meanwhile, bearing radial clearance declines owing to difference in temperature, $\mathrm{\Delta }{\delta }_t$ is negative.

As the temperature of outer ring is higher than that of inner ring, the expansion of outer ring results in increase of inner radial clearance, it makes vibration ascend Meanwhile, bearing radial clearance climbs up owing to difference in temperature, $\mathrm{\Delta }{\delta }_t$ is positive.

The increment of ball diameter owing to thermal expansion is:

where, $T_D$ is ball temperature.

Combining Eqs. (28)-(30) to achieve variation of bearing inner radial clearance due to thermal effect.

2.2.2. Heat generation calculation

Heat generation in virtue of friction [4, 39] is shown as follows:

where, $H$ is the heat generation, W. $M$ is the total friction moment, N⋅mm. $n_r$ is the rotational speed, r/min.

According to Palmgren method [5, 40], the total friction moment of bearing is achieved by calculating load friction moment and viscous friction moment:

where, $M_E$ is the friction moment due to elastic hysteresis. $M_D$ is the friction moment due to differential sliding motion. $M_S$ is the friction moment due to spin-sliding motion. $M_{CB}$ is the friction moment due to the contact between cage and ball. $M_{CR}$ is the friction moment due to the contact between cage and inner ring guide face. $M_{Oil}$ is the friction moment due to the loss of oil film viscosity. The corresponding expression is shown as follows:

(1) The friction moment due to elastic hysteresis $M_E$:

where, $\gamma =D\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_0/d_m$, $D$ is ball diameter, $d_m$ is pitch diameter, ${\alpha }_0$ is free contact angular. $f_E$ is elastic hysteresis coefficient, $f_E=\text{0.007}$ for bearing steel. $n$ is number of ball. $Q_{ij}$ and $Q_{oj}$ is the contact load between ball $j$ and inner and outer ring raceway, respectively, which are determined by quasi-statics equation. As illustrated in Figs. 2-3.

(2) The friction moment due to differential sliding motion $M_D$:

where, $f_{DS}$ is sliding friction coefficient determined by lubrication state:

where, ${\alpha }^{*}$ is semi-major axis of contact ellipse with the dimension is 1. $R_{*j}\left(*=i,o\right)$ is contact surface equivalent curvature radius between ball and inner or outer ring raceway, respectively. $E^{\text{'}}$ is integrated elastic modulus of mating contact body. $\sum {\rho }_{ij}$ is the curvature sum of mating contact body. Among which:

where, ${\rm E}$ is the second kind complete elliptic integral:

(3) The friction moment due to spin-sliding motion $M_S$:

where, $a_{ij}$ is the semi-major axis of contact ellipse between ball and inner ring raceway.

(4) The friction moment due to the contact between cage and ball $M_{CB}$:

where, ${\gamma }_1=d_m\left(1-\gamma \right)/2$, $W$ is cage weight, unit is N. $f_C$ is sliding friction coefficient, $f_C=0.2$ for the friction between steel and bakelite.

(5) The friction moment due to the contact between cage and inner ring guide face $M_{CR}$:

where, $E$ is elastic modulus, unit is$$N/(mm²). $n_{ic}$ is the speed of guidance ring relative to cage, unit is$$rad/s. ${\mathrm{\Delta }}_Z$ is the eccentricity of cage center relative to bearing center, unit is mm. $W$ is cage weight, unit is$$N.

(6) The friction moment due to the loss of oil film viscosity $M_{Oil}$:

where, ${\lambda }_p$ is viscosity-pressure coefficient of lubrication oil, unit is mm²/N. $\lambda$ is lubrication-sufficient factor, which is equal to oil film lubrication coefficient. $h_{*j}\left(*=i,o\right)$ is oil film thickness in contact zone between ball and inner or outer ring raceway, unit is m.

Nevertheless, rolling bearing is the component with multiple friction pairs. In the operation of bearing, the mating parts surface rough is evolving due to abrasive wear, inner friction location and severity of bearing is fluctuant, which results in the time-variant of friction moment. Consequently, the revision is required to Eq. (32), as shown in follows:

where, ${\zeta }_{*}\left(*=E,D,S,CB,CR,Oil\right)$ is called as “thermal contribution factor”, which is the parameter considering friction location and severity of bearing synthetically.

2.3.1. Thermal-conduction resistance

For angular contact ball bearing system, heat flow transfers between ball, inner and outer rings, shaft and house. The corresponding thermal resistance is classified into radial thermal-conduction resistance, axial thermal-conduction resistance and contact thermal-conduction resistance.

(1) Radial thermal-conduction resistance.

The thermal-conduction resistance along radial direction of ball is:

where, $D$ is ball diameter.

Inner ring, outer ring and house of angular contact ball bearing is regarded as thin circular ring with inner diameter $d_{int}$, outer diameter $d_{ext}$ and width $B$, respectively. Its thermal-conduction resistance along radial direction [42] is:

Shaft is equivalent to cylinder with outer diameter $d_{ao}$ and width $L$, whose thermal-conduction resistance along radial direction [42] is:

where, $k_D$ is thermal conductivity of material.

(2) Axial thermal-conduction resistance.

Shaft and house is individually treated as flat-wall with section area $S$ and width $B$ at the moment of calculating their axial thermal-conduction resistance. The corresponding thermal-conduction resistance is achieved by flat-wall thermal-conduction resistance:

where, section area of shaft is $\pi d_{ao}^2/4$. Section area of house is $\pi \left(d_{ext}^2-d_{int}^2\right)/4$.

(3) Contact thermal-conduction resistance.

Assuming that contact zone between ball and inner or outer ring is Hertz point contact, whose size is far less than that of bearing. Considering the influence of contact zone size and motion, the model proposed by Muzychka [43, 44] is utilized to determine thermal-conduction resistance of contact zone:

where, $a$ and $b$ is semi-major and semi-minor axis of contact ellipse, respectively. $Pe=V_c\cdot a/a_d$ is Peclet number. $a_d$ is thermal diffusion coefficient. $V_c$ is characteristic velocity.

In order to installation, the relationship of inner ring and shaft, outer and house is transition fit. According to literature [45, 46], the corresponding contact thermal-conduction resistance is:

where, $L_g$ is the thickness of gap between two contact surface. $A$ is nominal contact area in contact zone. $A_r^{*}$ is dimensionless actual contact area. $k_f$ is thermal-conduction coefficient of media (in general, air). $k_D^{*}=2k_{D1}{k}_{D2}/\left(k_{D1}+k_{D2}\right)$ is equivalent thermal-conduction coefficient of two contact body.

2.3.2. Thermal-convection resistance

There is a complex fluid-solid coupling zone in angular contact ball bearing, such as contact zone between inner ring and ball, contact zone between outer ring and ball, convection heat transfer would be dominant as lubrication oil is present. Besides, convection heat transfer would occur in region between house and ambient air.

(1) Convection heat transfer coefficient between shaft and ambient air.

Inner ring is connected to shaft, thus, bearing revolution is driven by shaft rotation. Taking shaft as relative fixed coordinate system, convection heat transfer process between shaft and ambient air is equivalent to ambient air skims over shaft surface with high speed. At present, this convection heat transfer asking help for shaft rotation is forced convection heat transfer. The convection heat transfer coefficient between shaft and ambient air is indicated as follows [47]:

where, ${\lambda }_{air}$ is thermal conductivity of air. $L$ is characteristic size, $L$ is shaft diameter as convection heat transfer occurs in shaft surface. $Nu$ is Nusselt number, depicted as follows:

where, $Re$ is Reynolds coefficient. $\mathrm{P}\mathrm{r}$is Prandt coefficient. $w$ is angular speed of shaft. $d$ is shaft diameter. $\alpha$ is heat diffusion coefficient. ${\upsilon }_{air}$ is air dynamic viscosity. ${\rho }_{air}$ is air density.

(2) Natural convection heat transfer between house and ambient air.

The house is cylindrical structure, the convection heat transfer process between its external surface and ambient air is treated as natural convection heat transfer due to relative lower motion speed.

Convection heat transfer coefficient [42] between external surface of house and ambient air is:

where, $T_a$ is ambient temperature of house. $T$ is house temperature.

(3) Convection heat transfer coefficient between ball or ring and lubrication oil.

There would be heat exchange between fluid and ring or rolling element surface as fluid passes through the house and rolling element surface. According to Harris [48], the convection heat transfer coefficient $\alpha$ during the heat transfer process between rolling element and lubrication, in addition, ring surface and lubrication is shown as follows [48]:

where, $k_o$ is thermal conductivity of lubrication oil. $P_r$ is Prandtl number of lubrication oil. $Re=vx/{\upsilon }_o$ is Reynolds number, ${\upsilon }_o$ is kinematic viscosity of lubrication oil, $x$ is characteristic length. In the case of calculating heat loss from ball to lubrication, $v$ is ball orbital revolution speed, $x$ is ball diameter; in the case of calculating heat loss from ring to lubrication, $v$ is ring rotation speed, $x$ is ring diameter.

Thermal-convection resistance of each convection heat transfer coupling is shown as follows:

(1) Thermal-convection resistance between shaft external surface and ambient air.

As calculating thermal-convection resistance between shaft external surface and ambient air, the shaft external surface is simplified to cylindrical surface with external diameter $d_s$ and width $L_s$. Thus, the heat exchange area is $\pi d_s{L}_s$, the corresponding radial thermal-convection resistance between shaft external surface and ambient air is:

(2) Thermal-convection resistance between shaft end face and ambient air.

As calculating thermal-convection resistance between shaft end face and ambient air, the exchange area is $\pi d_s^2/4$, the corresponding thermal-convection resistance between shaft end face and ambient air is:

(3) Thermal-convection resistance between house external surface and ambient air.

As calculating thermal-convection resistance between house external surface and ambient air, the house external surface is simplified to cylindrical surface with external diameter $d_h$ and width $L_h$. Thus, the heat exchange area is $\pi d_h{L}_h$, the corresponding thermal-convection resistance between house external surface and ambient air is:

(4) Thermal-convection resistance between house end face and ambient air.

As calculating thermal-convection resistance between house end face and ambient air, the house end face is simplified to circular ring with external diameter $d_h$ and inner diameter $d$. Thus, the heat exchange area is $\pi \left(d_h^2-d^2\right)/4$, the corresponding thermal-convection resistance between house end face and ambient air is:

(5) Thermal-convection resistance between ball and ambient air.

As calculating thermal-convection resistance between ball and ambient air, the ball is simplified to sphere with external diameter $d_j$. Thus, the heat exchange area is $\pi d_j^2$, the corresponding thermal-convection resistance between ball and ambient air is:

(6) Thermal-convection resistance between inner or outer ring and ambient air.

As calculating thermal-convection resistance between inner or outer ring and ambient air, the ring is simplified to cylindrical surface with external diameter $d_{*}\left(*=i,o\right)$ and width $L_{*}\left(*=i,o\right)$. Thus, the heat exchange area is $\pi d_{*}{L}_{*}$, the corresponding thermal-convection resistance between inner or outer ring and ambient air is:

According to Eqs. (44)-(49) and (56)-(61), thermal resistance transfer path of bearing system is established, shown in Fig. 6. The heat generation in each friction coupling delivers to external environment through inner ring, outer ring and ball, ignoring the cooling effect of lubrication oil, convection heat transfer mainly occurs in several ways: on one hand, forced convection heat exchange between inner ring end face and ambient air, go a step further, shaft and ambient air, on the other hand, natural convection heat exchange between house and ambient air.

Fig. 6Thermal resistance transfer of bearing

2.4.1. Thermal network node arrangement

Ignoring seal and edge chamfer structure, bearing system is separated into 17 nodes, as depicted in Fig. 7. Among which, bearing body has 7 nodes, shaft has 4 nodes in different position and 1 node of external environment. Node definition and meaning is illustrated in Table 1.

2.4.2. Numerical solution

When calculating steady temperature field utilizing thermal network model, supposing that the heat flow in inlet is equal to that in outlet of node $i$, as illustrated in Fig. 8. Consequently, according to heat flow balance theory, the balance equation of steady status heat transfer in 2-D system is shown as follows:

where, $T_i$ is temperature of node $i$, $T_{*}\left(j,k,m,n\right)$ is adjacent nodes, $R_{i-*}\left(j,k,m,n\right)$ is thermal resistance of adjacent nodes, $Q_i$ is heat generation of node $i$.

Fig. 7Node thermal network model

Fig. 8Diagram of node heat transfer mechanism

2.4.3. Calculation process

According to Eq. (62), the thermal balance equation of thermal network system is obtained. In order to obtain node steady temperature of bearing system, considering bearing structure, surface topography feature and load characteristic, the bearing system steady temperature model is employed to conduct multi-level and coupling analysis on structural mechanics, heat generation and heat transfer (heat conduction and heat convection). Calculation procedure is depicted in Fig. 9. Among which, radial load is referred as vibration load obtained from tested signal over the whole life of bearing.

Taking environment temperature as initial value, firstly, bearing quasi-statics is executed to calculate motion relation and mechanics relation of bearing parts according to boundary condition and working condition. Secondly, friction power loss, thermal-conduction resistance and thermal-convection resistance are obtained. Finally, node temperature is achieved by solving the steady status heat transfer equation set. Friction power loss and thermal resistance are updated after making comparison between calculation value and test value of house temperature. Nest, the system structure size, load distribution and installation tightness are revised based on the calculation result of thermal deformation of parts and bearing clearance. If relative difference percentage between calculation value and test value of house temperature is less than set one, the calculation is over.

Fig. 9Calculation process of thermal analysis considering thermal-solid coupling effects

2.5.1. Angular contact ball bearing dynamic stiffness analysis

Dynamic stiffness of angular contact ball bearing refers to instantaneous stiffness during bearing is running, which is the reflection of ball-raceway contact characteristics. Considering Hertz contact and elastohydrodynamic lubrication, dynamic stiffness model of angular contact ball bearing [1] is derived by the establishment of ball-raceway Hertz contact model and oil film stiffness model.

Bearing stiffness model is depicted in Fig. 10. Bearing totally has $n$ contact coupling, each one is consist of a ball and its mating ring. Stiffness of each contact coupling is composed of ball-raceway contact stiffness and oil film stiffness in tandem, as shown in Fig. 10(a). Hence, bearing integral stiffness is consist of $n$ contact coupling stiffness in parallel.

Stiffness of ball-ring in single contact coupling is:

Thereinto, contact stiffness $K_C$ is illustrated in Eq. (8). In accordance of elastohydrodynamic lubrication theory, oil film stiffness of angular contact ball bearing $K_E$ is determined by minimum oil film thickness [49]. Oil film stiffness in contact zone of ball and inner/outer ring is derived according to Eq. (31):

Consequently, the component stiffness along radial and axial direction of single contact pair is [50]:

The whole radial and axial stiffness of bearing is:

Fig. 10Diagram of bearing radial stiffness

In the light of above explanation, some parameters are demanded to calculate bearing stiffness, such as dynamic contact load $Q_{ij}$ and $Q_{oj}$, mean speed $u_{ij}$ and $u_{oj}$ in contact zone between ball and inner/outer ring, contact angular ${\alpha }_{ij}$ and ${\alpha }_{oj}$ between ball and inner/outer ring, oil film thickness $h_{\mathrm{m}\mathrm{i}\mathrm{n}ij}$ and $h_{\mathrm{m}\mathrm{i}\mathrm{n}oj}$ in contact zone between ball and inner/outer ring. The bearing system thermal analysis model based on elastohydrodynamic lubrication is employed to acquire parameters needed to compute bearing stiffness in accordance to working condition. The calculation routine is depicted as follows.

2.5.2. Angular contact ball bearing vibration analysis

Radial displacement $X_r$ of bearing inner ring is sum of relative radial contact deformation ${\delta }_r$ and radial working clearance $P_d$, which is shown as follows:

Thereinto:

According to Eq. (34):

Fig. 11Calculation process of bearing stiffness

3.2.1. Contact characteristics evolution rule analysis over the whole life

The contact characteristics during the whole life of bearing under excessive preload mainly includes contact deformation, contact angular and contact stiffness.

Fig. 16Contact deformation

a) Contact zone between ball and outer ring

b) Contact zone between ball and inner ring

As shown in Fig. 16, contact deformation between ball and inner/outer ring has the same evolution rule, there is a little difference in amplitude. The curve behaves itself in the form of “D” in general, which is inclined to the location of 0° and 30°. At the location of 180° and 210°, the amplitude of contact deformation begins to decrease at 14min until the minimum at 18min.

Fig. 17Contact angular

a) Contact zone between ball and outer ring

b) Contact zone between ball and inner ring

As depicted in Fig. 17, there is difference of contact angular between ball and inner/outer ring within the scope 150-240°. During the whole life of bearing, the curve of contact angular between ball and inner/outer ring has the opposite tendency, mainly including some phases: 0-6 min, 8-10 and 14 min, 16-18 min, 12, 20 and 28 min, 6 and 30 min. For example, the amplitude of contact angular between ball and outer ring is the maximum over the whole life of bearing at 0-6 min, at the same time, the amplitude of that between ball and inner ring is the minimum. The amplitude of contact angular between ball and outer ring begins to decrease at 8 minute until the minimum at 16, 18 min. After that, the amplitude of that gradually increases, the amplitude keeps steady at the location of 180° and 210° among 22-30 min. However, the amplitude of contact angular between ball and inner ring at 12min raises until it is closed to that at 20 min, on the contrary, the amplitude at 28 min is closed to that at 20 min.

Fig. 18 shows that the evolution rule of contact stiffness between ball and outer / inner ring has the same tendency, the curve shape is similar to “circle”, there is a little difference in the amplitude. The significant variation zone of amplitude is 60-300°, the amplitude with little fluctuation is minimum at 0-6 min. It begins to increase at 8 min until maximum at 16 and 18 min, among which, it moves down at 12 min. It is the degradation phase of contact stiffness at 20-30 min, which is the minimum at 30 min, nevertheless, it relatively increases at 28 min. Over the whole life of bearing, there is the maximum difference at the location of 180° and 210°.

Fig. 18Contact stiffness

a) Contact zone between ball and outer ring

b) Contact zone between ball and inner ring

3.2.2. Oil film thickness and stiffness evolution rule analysis over the whole life

The diversification of temperature have an effect on the variation of lubrication oil viscosity and thickness, furthermore, it makes instantaneous contact characteristics change. The oil film thickness, stiffness and bearing dynamic stiffness is depicted from Fig. 19 to Fig. 21.

As shown in Fig. 19, the oil film thickness between ball and inner/outer ring has the same tendency expressing gradual decrease, the amplitude of oil film thickness between ball and outer ring is bigger than that between ball and inner ring. There is maximum variation of amplitude at 0-2 min, it keeps stable at another time.

Fig. 19Oil film thickness

a) Contact zone between ball and outer ring

b) Contact zone between ball and inner ring

As shown in Fig. 20, the oil film stiffness between ball and inner/outer ring has the same tendency expressing itself as “D”, there is little difference in the amplitude. Over the whole life of bearing, it generally raise up, there is the maximum amplitude in 0° and 30°.

Fig. 21 shows the integral radial and axial stiffness evolution curves. As we can see, radial stiffness value alters more obviously than axial stiffness value, which are all rising up. At 18 minute, radial stiffness has an abrupt enlargement, next, it steps into a mild state. As shown in Fig. 18, contact stiffness in 18 minute drastically increases to maximum value, after that, the value decreases mildly, besides, the value of contact stiffness is far greater than that of oil film stiffness, therefore, the integral radial stiffness in 18 minute has an abrupt increment followed by a mild state due to the tandem relation between contact stiffness and oil film stiffness.

Fig. 20Oil film stiffness

a) Contact zone between ball and outer ring

b) Contact zone between ball and inner ring

Fig. 21Stiffness

The radial clearance and displacement over the whole life of bearing is illustrated in Fig. 22. It shows that clearance has a negative correlation with house temperature, on the contrary, displacement has a positive correlation with house temperature. Clearance value is minus, the radial clearance reduces rapidly. However, there is a larger gap between the test value and calculation value of displacement at 0-12 min and 24-30 min. The reason is that the model is built without considering the temperature effect acting on fit between shaft and inner ring, fit between house and outer ring, shaft and house.

Fig. 22Radial clearance and displacement

a) Radial clearance

b) Radial displacement

3.2.3. Contact characteristics analysis under different conditions

Based on the dynamic stiffness and vibration analysis model of bearing built in this paper, the contact characteristics and node temperature affected by preload or speed with/without EHL is illustrated as follows. Among which, the results shown in Fig. 23-Fig. 32 are calculated with EHL, the others are calculated without EHL. The speed keeps 2400 r/min as the preload increases, besides, the preload keeps 220 N as the speed adds up.

As shown in Fig. 23, the contact angular between ball and outer ring increases with the enlargement of axial preload, however, the contact angular between ball and inner ring has the opposite variation trend.

Fig. 23Contact angular

a) Ball and outer ring

b) Ball and inner ring

As shown in Fig. 24, the contact load between ball and outer/inner ring has the same evolution trend as axial preload increases, besides, the difference of their amplitude is approximate equal to zero.

Fig. 24Contact load

a) Ball and outer ring

b) Ball and inner ring

Fig. 25 illustrates that the dynamic stiffness along radial and axial direction of bearing affected by axial preload, which express themselves in the form of increasing firstly, then keeping steady gradually. With the increment of axial preload, the dynamic stiffness value of radial direction is larger than that of axial direction. In the initial growth phase of axial preload, the variation of dynamic stiffness along radial direction has a notable enlargement, then it slows down.

Fig. 26 depicts the temperature rise of nodes with the development of axial preload. The temperature of all nodes grows up in different degree. Among which, the temperature of inner ring and contact zone between ball and inner ring has the maximum expansion, whose value is similar.

Fig. 25Dynamic stiffness

Fig. 26Temperature

As indicated in Fig. 27, thermal deformation amount of inner ring and outer ring has the similar trend. The amplitude of inner ring deformation is much bigger than that of outer ring deformation in general.

Fig. 27Thermal deformation

As shown in Fig. 28, the contact angular between ball and inner ring increases with the enlargement of speed, the contact angular between ball and inner ring has the opposite variation trend, it is obvious that the contact angular variation affected by axial preload has the opposite trend with comparison to that influenced by speed.

Fig. 29 advocates that the contact load between ball and outer/inner ring has the similar evolution rule and amplitude. From 2400 r/min to 3000 r/min, the curve shape behaves itself in the form of “square”, with the development of speed, the curve grows up ranging from 150° to 240°, which likes irregular “polygon”.

Fig. 28Contact angular

a) Ball and outer ring

b) Ball and inner ring

Fig. 29Contact load

a) Ball and outer ring

b) Ball and inner ring

As shown in Fig. 30, compared with the dynamic stiffness variation rule affected by axial preload, the dynamic stiffness along radial direction gently increase, which is similar to that along axial direction.

Fig. 30Dynamic stiffness

Fig. 31 depicts the temperature rise of nodes with the development of speed, whose amplitude is lower. The temperature of all nodes grows up in different degree. Among which, the temperature of inner ring and contact zone between ball and inner ring has the maximum expansion, whose value is similar.

Fig. 31Temperature

As indicated in Fig. 32, deformation amount of inner ring and outer ring influenced by speed has the similar trend, but the incremental quantity of inner ring is higher than that of outer ring. The amplitude of inner ring deformation is much bigger than that of outer ring deformation in general.

Fig. 32Thermal deformation

The contact angular, contact load, temperature rise and thermal deformation affected by speed ignoring EHL is shown in Fig. 33-Fig. 36. Compared with that affected by speed considering EHL, the variation rule of the above results is almost identical, which indicated the effect of lubrication oil on contact characteristics and thermal characteristics is weak dry-lubricated or lightly lubricated condition.

Fig. 33Contact angular

a) Ball and outer ring

b) Ball and inner ring

Fig. 34Contact load

a) Ball and outer ring

b) Ball and inner ring

Fig. 35Temperature

Fig. 36Thermal deformation