Nonlinear Frictional Dynamics on Rolling Contact

2.1 Displacement dependent nonlinearity

Figure 1 shows the relationship between the friction force F of the sliding object and its translational displacement x [7]. After the mass is starting motion, the friction force increases as the displacement increases. When the friction force is saturated with the steady state friction force F _s, it is constant until the mass reaches the reversal point A. After reversing the direction of motion at point A, the friction force changes along the outer hysteresis curve ABA. If the motion is reversed at arbitrary point a on the outer hysteresis curve, the friction force will change along the inner hysteresis curve aba starting at point a. When the internal hysteresis curve is closed, the frictional force changes along ABA again. This hysteresis rule is known as the non-local memory NSB [8]. The NSB is caused by elastic deformation of the contact surface asperities [9], microslip [10], and elastic hysteresis loss [11]. In this chapter, the curve OA is called a virgin loading curve, the displacement region where the friction depends on the displacement is called the pre-rolling region, and the displacement region where the friction force is constant is called the rolling region. Furthermore, the boundary displacement between the pre-rolling region and the rolling region is called the starting rolling displacement.

Now, the sinusoidal excitation force P(t) = P ₀ sinωt acting on the mass where P ₀ is the amplitude and ω is the excitation angular frequency. Figure 2 shows the relationship between sinusoidal excitation force P and displacement x [7]. If the force amplitude P ₀ is greater than the static friction force F _s, the displacement will change significantly (such as an arc within P > F _s). This suggests that the rolling element initiates a rolling movement. F _s can be determined as P at the motion reversal point [12].

2.2 Nonlinear friction modeling

The friction model proposed based on the Masing rule can describe the NSB and hysteresis behavior of friction in non-local memory. The Masing rule simply describes a hysteresis curve from the virgin loading curve. Therefore, it is widely used in the basic model of seismic response analysis [13] and the hysteresis model [14]. The Masing rule is formulated as follows:

where f(x) is the virgin loading curve. F _r and x _r are the friction force and displacement at the motion reversal point. The hysteresis curve is determined by the geometrically similar curve of the virgin loading curve with a similarity ratio λ = 2. If λ ≠ 2, the hysteresis curve does not become axisymmetric. In the friction model, λ = 2.

Figure 3 shows the procedure for calculating the friction force [7]. First, the friction force is calculated by Eq. (1) until reaching the motion reversal point A. After reversing the direction of motion at point A, calculate the friction force using Eq. (2). At this time, the values of x _r and F _r are replaced by the displacement and frictional force at point A. Furthermore, the friction force can be calculated as well when the motion is reversed at the point B or C. The friction force after closing the internal hysteresis curve aba is calculated with Eq. (2) by replacing the values of x _r and f _r to the displacement and friction at point A.

The hysteresis characteristics of non-local memory can be explained by above mentioned calculations. If x is greater than the maximum displacement value at the previous motion reversal point, the frictional force is calculated by Eq. (1) again.

This model has fewer parameters than previous friction models such as the bristle model [15] and the generalized Maxwell slip (GMS) model [16].

The friction model based on the Masing rule can describe the effect of NSB on the dynamic characteristics by simplifying the friction behavior. Al-Bender uses the exponential and irrational functions as the virgin loading curves to calculate the friction force [17]. However, the rolling region and the starting rolling displacement x _s do not consider. The resonances caused by NSB are depended on x _s [18]. Therefore, the starting rolling displacement and steady-state friction force in the rolling region should be introduced into the friction model.

The virgin loading curve is described by Eq. (3) proposed in this study:

where A and B are constants determined by the continuity condition as below:

This virgin loading curve has only three parameters: steady-state friction force F _s, starting rolling displacement x _s, and shape factor n. The shape factor n represents the rate of change in friction of the pre-rolling region and is the most important parameter in this model [19]. These parameters determine the characteristics of the NSB. The proposed virgin loading curves for different n are shown in Figure 4 [7].

In this model, n ≠ 1 is always satisfied. Because n = 1 means that the friction does not have hysteretic behavior, it is uncommon in the friction characteristics of a rolling bearing.

3.1 Numerical analysis of dynamics considering nonlinear friction

Figure 5 shows an analytical model for calculating the dynamics in the feed direction [7]. The equation of motion is as follows:

where m is the mass of sliding object; t is the time; [·] = d/dt; P(t) is the exciting force acting on the mass; F (x) is the friction force calculated from Eqs. (1)–(3).

The equation of motion can be expressed by Eq. (7) considering the sinusoidal excitation force P (t) = P ₀sinωt acting on the mass.

By introducing the dimensionless parameters K _s = F _s/x _s, ω _s ² = K _s/m, u = x/x _s, γ ₀ = P ₀/F _s, τ = ωt, β = ω/ω _s and [‘] = d/dτ into the Eq. (7), the dimensionless equation of motion is described as Eq. (8).

As shown in Eq. (8), the motion of the sliding object is governed by the amplitude of the dimensionless excitation force γ ₀, the dimensionless excitation frequency β, and the dimensionless friction force F/F _s. F/F _s is a function of shape factor n. Ultimately, the motion of the sliding object is described by determining γ ₀, β, and n. Eq. (8) is solved by the fourth-order Runge-Kutta method with nondimensional time derivative dτ = τ/10000. The frequency response function (FRF) G (β) is calculated by Eq. (9).

where u ₀ is the displacement amplitude in the steady-state response, u _max and u _min are the maximum and minimum displacements in the steady-state response, respectively.

Figure 6 shows an FRF with dimensionless excitation force amplitude γ ₀ = 0.6 for n = 0.5 and 1.5 [7]. As shown in Figure 6, the FRF shape and resonant frequency change by n. It means that the NSB affects the dynamic characteristics of the sliding object. There are additional small resonant peaks in the FRF. The small resonance peaks are superharmonic resonances, which are typical phenomenon of nonlinear vibration caused by the nonlinearity of restoring forces [20]. Because the proposed friction model does not take into account the difference between static and dynamic friction, these resonances does not induced by the stick-slip phenomena which is occurred by the negative damping due to the difference between static and dynamic friction forces. In the following, the main resonance peaks are called harmonic resonances and the other resonances are called superharmonic resonances.

By calculating the steady-state response of a specific excitation frequency, the effect of the excitation frequency on the steady-state motion in the feed direction is clarified. Figures 7–9 show the phase planes of the limit cycles of n = 0.5 and 1.5 at frequencies A, B, and C shown in Figure 6, respectively [7]. Frequency A is lower than the frequency of the superharmonic resonance of the adjacent harmonic resonance. Frequency B lies between the harmonic resonance and its adjacent superharmonic resonance. Finally, frequency C is higher than the frequency of harmonic resonance.

According to the results as shown in Figure 7, when a sliding object is excited by a sinusoidal force of frequency A, velocity reduction and recovery (VRAR) occurs after reversing the direction of movement. When the sliding object is excited at frequency B, VRAR does not occur after reversing the direction of movement, as shown in Figure 8. However, the shape of the phase plane is distorted. On the other hand, as shown in Figure 9, when the sliding object is excited at frequency C, the phase plane becomes circular. If the sliding object is excited by a force with a frequency higher than the harmonic resonance, the non-linearity due to NSB is negligible.

Next, the effect of excitation and friction conditions on the time history of steady-state motion is examined. Figures 10 and 11 show the dimensionless acceleration u″, dimensionless velocity u′, dimensionless displacement u, and dimensionless friction force F/F _s at frequencies A and B with n = 0.5 and 1.5, respectively [7].

Comparing Figures 10 and 11, the variation of acceleration direction (VOAD) unrelated to sinusoidal displacement motion with the excitation frequency and VRAR are clear when excited with frequency A. Hence, the VOAD causes VRAR. The displacement spike is clearly observed in Figure 10, but it does not observe in Figure 11. Thus, the displacement spike is caused by the VOAD. Furthermore, the displacement spike sharpens when the VOAD is caused discontinuously. On the other hand, it loosens when the VOAD is caused continuously. The VOAD is affected by NSB of friction and becomes discontinuous when n is small.

Displacement spikes, known as quadrant glitches, are one of the causes of poor feed drive operation accuracy. Previous studies have concluded that quadrant glitches are produced due to motion delays caused by the difference between static and dynamic friction [21]. The results of this study show that displacement spikes (quadrant glitches) are generated without modeling the difference between static and dynamic frictional forces. It means that the quadrant glitches are caused not only by the difference between static and dynamic friction, but also by NSB friction. Also, as shown in Figure 10, the quadrant glitches occur even at zero speed. Previous studies have concluded that quadrant glitches are produced by reduced acceleration [22]. These conclusions correspond to our result that quadrant glitches are caused by VOAD.

Figure 12 shows the FRF for n = 0.5 and 1.5. The figure also contains the results of various dimensionless excitation force amplitudes γ ₀ [7]. As shown in Figure 12, as γ ₀ increases, the resonant frequency decreases. Furthermore, resonance does not occur even if γ ₀ increases. The compliance at frequencies of superharmonic and harmonic resonance increases with higher values of n. For small n, not only the nonlinearity but also the damping increases. Therefore, when n is small, the compliance at resonance is not high than the case with large n.

Then, the results of the proposed friction model are compared with the experimental results using commercially available roller guideways for verifying the validity.

Figure 13 shows the relationship between the excitation force amplitude and the frequency and compliance at the harmonic resonance [7]. Figure 13 shows the measurement results of the compliance in the feed direction of the linear rolling bearing obtained in the vibration test conducted under the same conditions as the numerical analysis. The analytical results for n = 1.5 conform well to the experimental results. The comparison results prove that the proposed simple analytical model can qualitatively predict the force dependency of the dynamic characteristics of a rolling bearing in the feed direction.

4.1 Experiment for evaluating the dynamics of linear rolling bearing

In a previous study, Yi et al. evaluated the vibration characteristics of a carriage with a steel block (additional mass). Feed, lateral, and vertical response accelerations were measured using a 3-axis accelerometer when the steel block was excited by the shaker [23]. Ota et al. measured the vibration of the carriage moving at a constant speed in the feed direction. They also identified key vibrational components by frequency spectrum analysis of the detected response sounds and accelerations [24]. Through the impulse hammering test, Rahman et al. evaluated the natural frequencies and damping ratios of machine tool tables supported by linear rolling bearings. The natural frequency and damping ratio were identified based on the frequency response function [25].

In contrast, the author measured the acceleration of a carriage carrying a column which is imitated the long workpiece. The vibration measuring system for evaluating the dynamic characteristics of the linear rolling bearing is shown in Figure 14 [26]. The system comprises a rail, a carriage, a column made of S50C steel, an electrodynamic shaker, and a stinger made of S45C. The column was fixed on the carriage with bolts. The rail was fixed on the stone surface plate through the steel base. The shaker was supported with cloth belt to obtain a free boundary condition. The stinger has a circular cross-sectional shape. One of stinger ends was screwed to the free end of the column through the impedance head, and the other one was screwed to the shaker.

The column was used as a pseudo workpiece to satisfy the condition that the center of gravity is higher than the upper surface of the carriage. This condition is often observed in the general usage of linear rolling bearings, such as in the feed drive mechanism of a machine tool. The excitation force in the feed direction was applied to the free end of the column.

The sinusoidal excitation signal generated using the function generator was input to the shaker via a power amplifier. The excitation and acceleration signals detected at the free end of the column were amplified by a charge amplifier and analyzed using a digital spectrum analyzer. In addition, at the same time as the excitation, the eddy current displacement sensor was used to detect the response displacement of the carriage in the feed direction.

In the general case, forces with different amplitudes and frequencies act on the linear rolling guide. For example, in the case of a machine tool, cutting force acts on the rolling guide via the workpiece and table. Its frequency and amplitude vary widely depending on machining conditions such as depth of cut and spindle speed. Micro cutting exerts a force of several millinewtons, and heavy cutting exerts a force of hundreds of newtons. Therefore, the exciting force changed from 0.1 N to 100 N in half amplitude and up to 1000 Hz in frequency.

The lubricant (mineral oil) weighed by syringe was manually applied to the raceways (5 ml per raceway). When changing the lubricant oil, the all components of the linear rolling bearing were washed with kerosene and alcohol for degreasing.

4.2 Influence of excitation force in rotational direction

Figure 15 shows the FRF when excited in the feed direction. Two resonance peaks were observed in the FRF [26]. The vibration modes at these resonance frequencies were measured by experimental modal analysis. Figure 16 shows the mode shapes of the carriage and column at the two resonances [26]. At the lower resonance frequency, the carriage and column vibrate in the feed direction without any elastic deformation. It was caused by the nonlinear spring behavior of friction in the microscopic region discussed in the previous chapters. At the higher resonance frequency, the carriage and column vibrated in the pitch direction. This natural vibration is caused by the elastic deformation of the contact part between the raceways and the rollers. Since the center of gravity is higher than the upper surface of the carriage, not only the pitching motion but also the translational motion in the feed direction is occurred at the same time. In followings, these vibration modes are called “feed mode” and “pitching mode” in order from low frequency.

Figure 15 also shows the FRFs for different excitation force amplitude. As shown in figure, the dynamic characteristics of the linear rolling bearing change depending on the exciting force. In feed mode, the resonance frequency decreased as the excitation force amplitude increased. Eventually, the feeding mode became unobservable as the excitation force became large. The resonant peak due to the pitching mode has the same tendency as the feed mode for the excitation force amplitude. However, even if the exciting force became large, the resonance peak did not disappear.

Pitching mode is the natural vibration by the rigid body motion of the carriage due to elastic deformation at the contact point between the roller and the raceway.

In the previous study, Ota et al. used a single carriage model to analyze the resonant frequency in pitching mode [27]. In their case, the center of gravity was close to the center of the rail cross section, and the natural vibration characteristics were determined based on the stiffness of the contact parts. However, under actual usage conditions of linear rolling bearings, the center of gravity is higher than the top surface of the carriage. As a result, the natural vibration characteristics are affected by nonlinear friction. The carriage rotates around the center of gravity like a cradle and moves slightly in the feed direction.

In this state, the natural vibration characteristics of the pitching mode depend on the exciting force as shown in Figure 16, which is due to the influence of the nonlinear spring behavior.

As shown in the Figure 17, the resonance frequency of the pitching mode becomes high when the excitation force is small [26]. This indicates that frictional stiffness affects the pitching mode as “additional spring.” Considering the feed mode, the resonance peak disappeared because of an increase in the excitation force amplitude. It indicates that the influence of the nonlinear spring behavior of friction decreases as the response displacement amplitude increases. Thus, the influence of nonlinear friction on the dynamic characteristics of pitching mode becomes small. In addition, the damping ratio was constant in the small excitation force range up to about 10 N, but it decreased in the large excitation force range. According to our previous research, the damping ratio of the linear rolling guideway estimated by the impulse test with 2000 N in the excitation force was about 0.5% with a nonlubricated condition [28]. Thus, the damping ratio converged to about 0.5% when the excitation force increased further.

Figure 17 also shows the results when three types of oils with different kinematic viscosities are used. If the higher kinematic viscosity oil was used, the resonance frequency became higher. The frictional stiffness is high when oil with a high kinematic viscosity is used. These results indicate the difference in the resonance frequency of the pitching mode caused by frictional effect. The damping ratio tends to be higher when oil with higher kinematic viscosity is used. This means that the viscous damping due to the oil film was dominant in the pitching mode.

On the other hand, when the column is excited in the lateral direction, the rolling mode which the carriage and column vibrate around feed axis as shown in Figure 18 was occurred. However, the excitation force dependence seen in the pitching mode does not occur. This is because the rolling mode is not affected by the additional spring due to non-linear friction.